COLLECTED MATHEMATICAL PAPERS OF HENRY J. S. SMITH lonbon HENRY FROWDE OXFORD UNIVERSITY PRESS WAREHOUSE AMEN CORNER, E.C. leA N & or. MACMILLAN & CO., 66 FIFTH AVENUE THE COLLECTED MATHEMATICAL PAPERS OF HENRY JOHN STEPHEN SMITH M.A., F.R.S. LATE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD EDITED BY J. W. L. GLAISHER, Sc.D., F.R.S. FELLOW OF TRINITY COLLEGE, CAMBRIDGE WITH A MATHEMATICAL INTRODUCTION BY THE EDITOR, BIOGRAPHICAL SKETCHES AND A PORTRAIT IN TWO VOLUMES VOLUME II OXFORD AT THE CLARENDON PRESS I894 O)forb PRINTED AT THE CLARENDON PRESS BY HORACE HART, PRINTER TO THE UNIVERSITY CONTENTS OF VOLUME II PAGE XXIII. Memoire sur Quelques Problemes Cubiques et Biquadratiques.. 1 Annali di Matematica, Ser. II. Vol. III. pp. 112-165, 218-242. M6moire Couronn6 par l'Acad6mie Royale des Sciences de Berlin, avec une moiti8 du prix Steiner en Juillet 1868. XXIV. Arithmetical Notes.......... 67 Proceedings of the London Mathematical Society, Vol. IV. pp. 236-253. The three papers which form these Notes were read on Jan. 9 and Feb. 13, 1873. XXV. On the Integration of Discontinuous Functions..... 86 Proceedings of the London Mathematical Society, Vol. VI. pp. 140-153. Read June 10, 1875. XXVI. On the Higher Singularities of Plane Curves..... 101 Proceedings of the London Mathematical Society, Vol. VI. pp. 153-182. Read June 10, 1875. XXVII. Mathematical Notes.......... 132 Proceedings of the London Mathematical Society, Vol. VII. pp. 237, 238. Read Dec. 9, 1875. First printed in the Messenger of Mathematics, Vol. V. pp. 143, 144 (Jan. 1876). XXVIII. Note on Continued Fractions........ 135 Messenger of Mathematics, Ser. II. Vol. VI. pp. 1-14 (May, 1876). XXIX. Note on the Theory of the Pellian Equation, and of Binary Quadratic Forms of a Positive Determinant..... 148 Proceedings of the London Mathematical Society, Vol. VII. pp. 199-208. Read May 11, 1876. vi CONTENTS PAGE XXX. On the Value of a Certain Arithmetical Determinant.... 161 Proceedings of the London Mathematical Society, Vol. VII. pp. 208-212. Read May 11, 1876. XXXI. On the Present State and Prospects of Some Branches of Pure Mathematics........... 166 Proceedings of the London Mathematical Society, Vol. VIII. pp. 6-29. Read Nov. 9, 1876. XXXII. On the Conditions of Perpendicularity in a Parallelepipedal System. 191 Proceedings of the London Mathematical Society, Vol. VIII. pp. 83-103. Read Dec. 14, 1876. XXXIII. On the Conditions of Perpendicularity in a Parallelepipedal System. 213 Philosophical Magazine, Ser. V. Vol. IV. pp. 18-25. Read before the Crystallological Society, June 14, 1876. XXXIV. Sur les Integrales Elliptiques Completes...... 221 Atti della R. Accademia dei Lincei. Transunti, Ser. III. Vol. I. pp. 42-44. Read Jan. 7, 1877. XXXV. Memoire sur les Equations Modulaires...... 224 Atti della R. Accademia dei Lincei. Memorie della classe di Scienze fisiche, matematiche e naturali. Ser. III. Vol. I. pp. 136-149. Read Feb. 4, 1877. XXXVI. On the Singularities of the Modular Equations and Curves... 242 Proceedings of the London Mathematical Society, Vol. IX. pp. 242-272. Read Feb. 14 and April 11, 1878. XXXVII. Note on a Modular Equation of the Transformation of the Third Order 274 Proceedings of the London Mathematical Society, Vol. X. pp. 87-91. Read Feb. 13, 1879. XXXVIII. Note on the Formula for the Multiplication of Four Theta Functions. 279 Proceedings of the London Mathematical Society, Vol. X. pp. 91-100. Read Feb. 13, 1879. XXXIX. De Fractionibus Quibusdam Continuis....... 287 Collectanea Mathematica (in memoriam Dominici Chelini), Milan, 1881, pp. 117-143. The paper is dated 1879. XL. On some Discontinuous Series considered by Riemann.... 312 Messenger of Mathematics, Ser. II. Vol. XI. pp. 1-11 (May, 1881). XLI. Notes on the Theory of Elliptic Transformation..... 321 Messenger of Mathematics, Ser. II. Vol. XII. pp. 49-99 (August-November, 1882). OF VOLUME II Vii PAGE XLII. Notes on the Theory of Elliptic Transformation..... 368 Messenger of Mathematics, Ser. II. Vol. XIII. pp. 1-54 (May-August, 1883). XLIII. Memoir on the Theta and Omega Functions. 415 XLIV. Memoire sur la Representation des Nombres par des Sommes de Cinq Carres....... 623 From the Memoires presentes par divers savants i l'Acad6mie des Sciences de l'Institut National de France, Vol. XXIX. APPENDIX. I. Address to the Mathematical and Physical Section of the British Association at Bradford in 1873..... 681 II. Arithmetical Instruments....... 691 III. Geometrical Instruments and Models........ 698 IV. Introduction to the Mathematical Papers of William Kingdon Clifford. 711 XXIII. MEMOIRE SUR QUELQUES PROBLEMES CUBIQUES ET BIQUADRATIQUES. [Annali di Matematica, Ser. II. vol. iii. pp. 112-165, 218-242. Memoire Couronn6 par l'Acad6mie Royale des Sciences de Berlin, avec une moiti6 du prix Steiner en Juillet 1868.] ON admet qu'on peut resoudre par de simples intersections de lignes droites tout probleme de geometrie plane, qui n'a qu'une seule solution, et qui n'est pas transcendental; ou, pour nous exprimer avec plus de precision, tout probleme dont la solution unique peut s'obtenir par des intersections de courbes geometriques d'un ordre quelconque fini. On admet de meme, que tout probleme quadratique, qui n'est pas transcendental, peut se resoudre par des intersections de lignes droites, et de sections coniques; et, plus generalement, que tout probleme qui n'a que n solutions, et qui n'est pas transcendental, peut se resoudre par des intersections de droites, et de courbes de l'ordre n. Et en effet, la verit6 de ces theoremes parait decouler des premiers principes de l'Algebre. Mais quand il s'agit de problemes d'un degre superieur au second il se presente un choix de methodes, puisqu'on peut se servir, soit des intersections de droites par des courbes dont l'ordre n est egal au nombre de solutions du probleme, soit des intersections de courbes dont l'ordre est intermediaire entre les deux nombres 1 et n- 1. Ainsi, l'on peut faire dependre la solution de tout probleme cubique ou biquadratique, soit des intersections de courbes du troisieme ou du quatrieme ordre par des droites, soit des intersections mutuelles de courbes du second ordre, puisqu'on a la m6me evidence alg6brique de la generalite absolue des deux VOL II. B 2 MEMI/OIRE SUR QUELQUES PROBLEMES methodes. Or, c'est la derniere de ces deux m6thodes qui parait la plus simple, et qui A ete, a juste titre, preferee par les geometres. Ainsi, l'on a ramen6 la recherche des points d'intersection d'une droite par une courbe du troisieme ou du quatrieme ordre a la recherche plus simple des points d'intersection de deux coniques, tandis que personne, que nous sachions, n'a suivi la marche inverse, qui a la v6rit6 serait peu naturelle. II nous a done paru qu'il fallait avant tout apporter quelque perfectionnement theorique aux moyens dont on se sert pour trouver les intersections de deux coniques donnees. Pour cela, nous avons cru devoir imiter, autant que possible, les methodes dont on s'est servi si heureusement pour les problemes quadratiques. En effet, on salt que tout probleme, qui n'admet que deux solutions, et qui n'est pas transcendental, peut 6tre resolu par les intersections de lignes droites, et d'une seule circonference de cercle, tracee d'avance dans le plan. On doit ce beau r6sultat, base veritable de la partie operative de la science, aux travaux des illustres fondateurs de la geometrie moderne; il suffira de citer a cet egard le Traite'des Proprietes Projectives de M. Poncelet, et l'ouvrage elementaire de Steiner, ayant pour titre: Die geometrischen Konstructionen ausgefiihrt mittelst der geraden Linie und eines festen Kreises. On suppose ordinairement que le centre du cercle trace est donne, afin d'avoir en meme temps la ligne droite a l'infini, et les deux points imaginaires conjugues oh cette droite est coupee par une circonf6rence quelconque; elements dont la connaissance est indispensable pour la solution d'un grand nombre de probleimes dont les donnees en dependent, soit explicitement, soit implicitement. Nous ajouterons, qu'au lieu d'une circonf6rence de cercle, on peut prendre pour courbe auxiliaire une section conique quelconque completement d6crite, dont on doit connaitre, non seulement le centre (ou deux diametres, si la courbe est parabolique) mais aussi le foyer. En passant aux problemes d'un ordre plus eleve, nous supposerons qu'une seule section conique (qui d'ailleurs ne doit pas etre un cercle) soit completement donnee et decrite, et nous demontrerons qu'en se servant de cette courbe auxiliaire on resout tous les problemes cubiques et biquadratiques avec la regle et le compas seulement, en les ramenant, pour ainsi dire, dans les limites de la geometrie elementaire. Nous ferons aussi remarquer que pour chaque probleme il suffira de tracer une seule circonference de cercle, et qu'en supposant connus cinq points seulement de la section conique, on peut remettre la description de cette courbe jusqu'au moment ou l'on veut en determiner les points d'intersection par le cercle qu'on doit tracer. II est bien entendu que, lorsqu'on veut operer de cette maniere, on doit supposer que la ligne droite a l'infini, et les points cycliques CUBIQUES ET BIQUADRATIQUES. 3 sur cette droite soient connus; c'est a dire que doivent etre donndes (1) un parallelogramme quelconque, (2) deux angles droites, ou trois angles egaux quelconques, ces angles dans les deux cas pouvant avoir ou le m6me sommet, ou des sommets differents, mais etant assujettis a ne pas former un parallelogramme. Sans ces donnees, on ne pourrait determiner ni le centre du cercle, qu'on aura a tracer, ni son rayon. La solution de chaque probleme cubique sera purement lineaire, sauf le trace du cercle; mais pour les problemes biquadratiques il y aura en general une construction quadratique, qu'on ne pourra eviter, mais qu'on pourra effectuer, soit avant, soit apres le trace du cercle. Mais puisqu'au point de vue pratique on abrege beaucoup d'operations lineaires en se servant d'une section conique, nous n'insisterons point sur la linearite absolue des constructions, et, le plus ordinairement, nous ferons usage en chaque probleme de la conique auxiliaire des le commencement meme de la solution. La methode que nous venons d'esquisser n'a rien de bien nouveau, puisqu'on la trouve deja, comme resultat analytique, dans la Geometrie de Descartes * Mais nous ne connaissons pas un seul probleme qui ait ete resolu geometriquement par cette methode. Les auteurs qui on traite de cette matiere, parmi lesquels nous signalerons De la Hire, Maclaurin, et Joachimsthal t, nous paraissent ne pas avoir cherche a demontrer par des considerations de geometrie pure la proposition si remarquable de Descartes, ni N s'en servir pour la solution geometrique des problemes. Ainsi Maclaurin ne s'occupe-t-il guere que de la construction geometrique des racines des equations algebriques; De la Hire, qui s'est propose de trouver les normales a une section conique qui passent par un point donne, n'a rattach6 sa solution de ce probleme a aucune theorie geometrique; enfin, Joachimsthal, qui a donne une autre solution, d'une elegance admirable, de ce meme probleme biquadratique, s'est encore servi de formules analytiques, dont, comme il l'avoue lui-meme, sa construction geometrique n'est pas entierement affranchie. Il restait done a trouver les liens qui rattachent la proposition de Descartes aux theories modernes de la science, et a montrer le parti qu'on peut en tirer pour les constructions actuelles de la geometrie pure. Nous nous empressons d'ajouter que dans ces developpements nous n'aurons qu'a nous occuper de considerations tres elementaires. En effet, nous admettrons que la solution de tout probleme cubique ou biquadratique peut se reduire a la determination des points d'intersection de deux coniques, qui ne sont pas decrites, mais dont chacune est determinee par un nombre suffisant d'elements; et nous * Note I (p. 50). t Note II (p. 50). B2 4 MIEMOIRE SUR QUELQUES PROBLEMES rambnerons ce dernier probleme h celui de trouver les points d'intersection d'une circonference de cercle par la section conique decrite d'avance, au moyen de simples transformations homographiques ou correlatives, dont on pourra se servir en plusieurs manieres diff6rentes. Les problemes que nous aurons a resoudre, et surtout le problme cubique dont la solution est specialement demand6e par l'Academie, impliquent necessairement des considerations relatives aux elements imaginaires. Nous jugeons done a propos, avant de venir au sujet qui doit nous occuper principalement, d'entrer dans quelques details sur cette matiere. Nous sommes loin de penser que nous pourrions ajouter a cet egard quelque chose qui fit inconnue aux geometres; et nous n'avons aucune pretention de pouvoir eclaircir la vraie nature de ce phenomene singulier qu'on nomme l'imaginaire en geometrie. Mais nous avons remarque que les questions th6oriques qui concernent les imaginaires ont ete traitees avec beaucoup plus de detail que les questions pratiques. On a bien donn6 les moyens necessaires pour determiner les elements reels, qui dependent pour ainsi dire immediatement d'elements imaginaires donnes. On salt, par exemple, trouver les axes de symptose reels de deux couples de droites imaginaires conjuguees, etc.; mais il s'en faut beaucoup que de telles determinations suffisent aux besoins actuels de la science. Il parait resulter de considerations alg6briques tres eldmentaires que tout probleme qui peut se resoudre lorsq'on en suppose les donnees reelles, devra 6galement rester resoluble lorsqu'on substitue deux elements imaginaires conjugues a deux 6l6ments quelconques reels qui entrent symetriquement dans la question. Mais pour operer generalement cette extension de la solution d'un probleme aux donnees r6elles t la solution d'un probleme aux donnees imaginaires, il faut souvent, du moins dans l'6tat actuel de la science, isoler l'un de l'autre les deux elements d'un m~me couple d'elements imaginaires conjugues, afin de faire sur chacun d'eux une suite d'operations plus ou moins longue, ayant pour resultat des couples d'elements imaginaires conjugues, dont la combinaison donne enfin les elements reels qu'on cherche. Cette isolation des elements imaginaires n'a rien d'absolu; elle est relative h un systeme de deux elements imaginaires conjugues arbitrairement choisis. En effet, etant donnee une seule couple d'elements imaginaires conjugues, il parait tout aussi impossible de distinguer entre ces elements que de distinguer entre les racines de l'equation x2+1 =0. Mais, de m6me qu'en distinguant hypothetiquement entre les deux racines de cette equation (ce qu'on fait, par exemple, quand on les designe par +i et -i), Art. 1.] CUBIQUES ET BIQUTADRATIQUES. 5 on parvient a distinguer en meme temps entre les deux racines imaginaires de toute autre equation quadratique; de m6me, en creant une fois pour toutes une distinction fictive entre les deux elements d'une couple quelconque d'elements imaginaires conjugues on arrive a etablir en m6me temps une distinction pareille entre les deux elements de toute autre couple; puis, en admettant cette isolation relative, on peut op6rer sur les elements imaginaires tout aussi bien que sur les elements reels; quoiqu'il faut avouer que, dans le cas des premiers, les operations auxquelles on se trouve conduit sont presque toujours d'une longueur rebutante. L'illustre et regrettable auteur de la Ge'omtrie de Situation a savamment effectue cette isolation hypothetique des elements imaginaires, en rattachant chaque element d'une m6me couple $ l'un des deux sens opposes qu'on peut observer en toute formation geom6trique qui contient des elements imaginaires. Mais cette maniere de considerer la chose, quoique tres utile pour les developpements theoriques, nous semble se pr6ter aux constructions avec moins de simplicite que celle que nous avons du pref6rer pour notre but actuel. D'apres ce qui vient d'etre dit, nous diviserons ce memoire en trois parties. Dans la premiere nous traiterons des constructions dont les donndes sont imaginaires; dans la seconde nous demontrerons la th6oreme de Descartes; dans la troisieme nous 6tudierons divers problemes cubiques ou biquadratiques, et notamment celui qui a ete signale par l'Academie. PREMIERE PARTIE. 1. Dans cette partie de notre travail nous ferons usage du mot grec dyade pour exprimer l'ensemble de deux elements conjugues imaginaires. Par chaque dyade de points imaginaires il passe une seule droite reelle; elle sera pour nous l'axe de la dyade. Dans l'espace, il existe, comme on sait, des dyades de droites imaginaires qui ne se coupent pas, et qui ne passent par aucun point reel. Mais les deux rayons d'une dyade de droites imaginaires, qui se trouvent dans un plan reel, se coupent en un point qui est toujours reel, et que nous nommerons le centre de la dyade de lignes droites. Pour plus de simplicite, nous considererons chaque dyade comme formee par les elements doubles d'une involution reelle, dont les segments, ou les angles, empietent les uns sur les autres. Cette definition d'une dyade aura toute la generalite necessaire, puisqu'on sait que toute determination quadratique se reduit, en derniere analyse, a la recherche des elements doubles de deux systemes homographiques, elements qui sont en m6me temps les elements doubles d'une involution, qu'on deduit lineairement des deux systemes homographiques, en prenant dans chacun des deux systemes, 6 MIMOIRE STUR QUIELQUES PROBLIMES Pt. T. 'Fele'ment conjugue d'un m6me Blement P, dont le conjugu6 harmonique par rapport a ces deux elements sera aussi le conjugue dans l'involution cherchee. Nous dirons qu'une dyade est donnee, quand on a deux couples d' elements reels de linvolution dont la dyade represente les 1e6ments doubles. Maintenant, si l'on prend une premiere dyade a a2 sur l'axe A, et une seconde dyade b, b2 sur l'axe B, different de A, qu'il coupe au point (A1, B1), les deux couples de droites imaginaires a1 b1, a2 b, et a b2, a2 b1, seront evidemment des dyades de droites. Soient P et Q les centres de ces dyades; nous les appellerons les centres d'homologie des dyades a, a2, b, b2. On trouve lineairement la droite PQ, en joignant les deux points A2, B2 qui sont conjugues de (A1, B1) dans les deux involutions qui d6terminent les dyades a1 a2, b, b2. Mais la construction des points P, Q eux-m6mes est essentiellement quadratique. Soient aa2, p132 des couples de points appartenant aux deux involutions respectivement; les points P, Q seront les points doubles d'une involution, dont A2B2 est une premiere couple, et dont on trouve deux autres couples en prenant les intersections de PQ par les deux couples de droites a, 13, a2,2 et a1 2, a2 1e. Cependant, on. voit facilement que les segments de cette involution ne peuvent pas empieter les uns sur les autres; done les points PQ seront toujours reels, comme on peut voir a priori. L'un des points PQ etant donne, l'autre s'en deduit lineairement, puisque PQ, A2 B2 sont quatre points harmoniques. Pour avoir une image nette des deux dyades et de leurs deux centres d'homologie, on peut remarquer que si al a2 est la dyade cyclique a l'infini, b1 b2 est la dyade commune a un systeme de cercles, ayant B pour axe radical, et PQ pour points limites. Nous ne nous arreterons pas a la construction correlative des axes d'homologie, ou de sympt6se, de deux dyades de droites imaginaires, qui n'ont pas le m6me centre. En considerant deux dyades de points imaginaires, qui n'ont pas le m6me axe, comme homologiques par rapport a l'un de leurs centres d'homologie, on etablit une certaine correspondance entre les points des deux dyades, en telle sorte que si l'on echange entr'eux les points imaginaires de l'une, il faut en m6me temps echanger entr'eux les points imaginaires de l'autre. Puisqu'il y a deux centres d'homologie, cette correspondance peut s'etablir de deux manieres differentes, qui se rapportent a ces deux centres respectivement. Ainsi nous dirons que l'homologie des deux dyades est donnee par le centre P, ou par le centre Q, selon que l'on a choisi le premier ou le second de ces deux centres pour etablir la correspondance. Pour les dyades de droites imaginaires dont les centres sont differents, on a une definition correlative. Si l'on a deux dyades de m6me espece, ayant le meme centre, ou le m6me Art. 2.] CUBIQUES ET BIQUADRATIQUES. 7 axe, on doit etablir leur homologie d'une maniere indirecte, en prenant une troisieme dyade, dont le centre ou I'axe soit different, mais dont l'homologie avec chacune des deux premieres soit donnee. L'homologie d'une dyade de points, et d'une dyade de droites, est donnee immediatement, lorsque les droites de la seconde dyade passent par les points correspondants de la premiere. En tout autre cas on etablit indirectement l'homologie des deux dyades; soit en determinant le centre d'homologie de la premiere dyade, et de la dyade de points qui resulte de l'intersection de la seconde dyade par un axe quelconque reel, soit en determinant l'axe d'homologie de la seconde dyade et d'une dyade quelconque de droites imaginaires passant par la premiere dyade *. 2. Quand on ne considere que deux dyades, inddpendantes l'une de l'autre, on peut choisir arbitrairement entre les deux manieres de determiner leur homologie. Mais en considerant un plus grand nombre de dyades, on voit qu'on peut disposer comme on veut de l'homologie de la seconde avec la premiere, de la troisieme avec la premiere, et ainsi de suite, mais qu'alors l'homologie de deux dyades quelconques de cette serie sera completement determin6e. Nous aurons done a resoudre le probleme tres g6neral, mais tres elementaire, que voici. Etant donnee l'homologie des dyades a1 a2, b, b2, et aussi l'homologie des dyades a, a2, c1 c2, trouver l'homologie des dyades b, b2, c1 c2. Ce probleme est toujours resoluble lin6airement. I1 y a plusieurs cas a considerer, mais pour abreger nous n'en considerons que la moitid, d'ohi l'on pourra deduire la solution des autres en s'appuyant sur le principe de dualit6. I. Soient A, B, C les axes des trois dyades de points a1, b b2, c1 c2; nous supposerons en premier lieu que ces droites soient toutes differentes entr'elles. Soient Q et R les centres d'homologie de al a2, c. c2 et de a, a2, b, b2 respectivement. I1 s'agit de trouver P le centre d'homologie de b, b2, c1 c2. On se rappelle que si P' est le centre d'homologie oppose a P, la droite PP' peut se construire lineairement t. Premiere solution.-Du point Q projetons les points de l'axe A sur la droite C, et du point R projetons les m6mes points sur la droite B. Soient a, a2 deux points conjugus de l'involution determinant la dyade a, a2; (1 e2 et 7y 72 les projections de ces points sur B et C respectivement. Les droites /3 71, 2 72', etc., envelopperont une section conique, tangente aux deux droites B et CI; de plus, les couples de tangentes, telles que /,y1, 0272, formeront un systeme de tangentes * Note III (p. 51). t Note IV (p. 54). 8 MfMOIRE SUR QUELQUTES PROBLEMES [Pt. I. en involution de cette conique; done les points d'intersection des deux tangentes de chaque couple seront en ligne droite. Le pole de cette droite, par rapport a la conique, est le point P cherche. Pour le determiner, il suffira de trouver le point de contact sur chacune des deux tangentes /31, y 2 72: la corde joignant ces points ira couper PP' au point P. Si les trois axes A, B, C se coupent au meme point 0, les deux droites B, C, qui sont d6ja homographiques par rapport aux points /02..., y72..., deviendront homologiques par rapport a ces memes points, puisque le point 0 se correspondra a lui-meme sur chacune des deux droites. Dans ce cas le centre d'homologie des deux droites est pr6cisement le point cherche P, qui se trouvera a l'intersection des droites PP', QR. I1 y aurait aussi une simplification, si les trois droites QB, B, C concouraient en un meme point. Seconde solution.-On determinera, comme nous avons fait dans la premiere solution, deux divisions homographiques sur les droites B et C. L'axe * de ces deux divisions coupera PP' an point P'. Ce point trouve, la position du point P s'en deduira lineairement. Cette solution se simplifie (1) si les divisions homographiques sur les deux droites B et C deviennent homologiques, (2) si l'axe de ces deux divisions se confond avec PP'. En ce dernier cas, on echange entr'eux, sur l'un des deux droites B et C, les deux points conjugues de chaque couple de l'involution qui determine la dyade que l'on considere sur cette droite. Alors les deux droites deviennent homologiques, et leur centre d'homologie est le point P qu'on cherche. Si l'axe A se confond avec lun des axes B et C, on pourrait encore trouver l'homologie des dyades b, b2, c c2, en faisant de le6geres modifications dans les solutions precedentes sans rien changer a leur principe. Mais on peut aussi operer de la maniere suivante. Soit z z2 la dyade auxiliare qui etablit l'homologie de a1 a2, b6 b2, dont on suppose les axes coincidents. L'homologie des dyades a, a2, za z2 sera donnee, et aussi celle des dyades a, C1 c2; on en deduira l'homologie des dyades z1 2, c1 c2. Mais l'homologie des dyades i z2, b, b2 est connue; done on pourra trouver l'homologie des dyades b, b2, c1 c2. Cette solution suppose seulement, que l'axe de la dyade z1 z2 ne se confonde pas avee C. Si les trois axes A, B, C coincident, soit y Y2 la dyade auxiliaire qui etablit l'homologie de a, a2, c c2; on trouvera par le cas precedent l'homologie * L'axe des deux divisions homographiques a, a2..., i, /12... est la droite lieu des points d'intersection des couples de droites telles que a, f3, a, i3. Art. 3.] CUBIQUES ET BIQUADRATIQUES. 9 de b, b2, y, Y2 Mais alors y, Y2 sera une dyade auxiliaire qui etablira l'homologie de b b2, 1 C2. II. Ce qui precede suffit pour tous les cas oh l'on ne considere que des dyades de la m6me espece: nous allons maintenant supposer que les trois dyades a1 Ca2, b b2, c1 c soient d'especes diff6rentes. Nous ne consid6rerons que les deux cas suivants, dans lesquels nous supposerons toujours que l'homologie de ac a2c, b1 b2, et de ac1 2, c1 c2, soit donn6e, et que l'on cherche l'homologie de bl b2, c1 c2. (i.) Soient a a2, b, b2 des dyades de points, c1 c2 une dyade de droites: soit aussi Y1 Y2 la dyade de points auxiliaires qui sert h etablir l'homologie de a, a2, C1 c2. Pour determiner l'homologie de bl b2, c1 c2, on n'aura qu'a trouver l'homologie de Y1 Y2, b, b2, ce qu'on pourra faire, puisqu'on connaitra l'homologie de c, a,, bl b2, et aussi celle de a, a2, yi y.2 (ii.) Soit a, a2 une dyade de droites, b1 b2, cx c2 des dyades de points. Designons par Y1 y2, zI z2 les dyades de points qui etablissent l'homologie de aLa2 avec c1 c2 et b, b respectivement. Dans la s6rie de dyades b b2, zx z2, yi Y2, c1, on connaltra l'homologie de chacune avec celle qui la precede, donc on pourra trouver l'homologie de la premiere avec la derniere. On peut donner un enonce plus general du probleme de cet article. ' Soient oa1a2, b, b,..., x1 x2 des dyades en noinbre quelconque, de mnme espece ou d'especes difterentes; etant donnee l'homologie de chacune d'elles avec celle qui la suit immediatement, on peut trouver l'homologie de la premiere avec la derniere.' 3. Maintenant, prenons arbitrairement une dyade fixe, qui doit nous servir comme terme de comparaison pour toutes les autres dyades que nous aurons a considerer. Nous dirons qu'un element imaginaire est donn6, quand la dyade a laquelle cet element appartient est donnee, l'homologie de cette dyade avec la dyade fixe etant aussi donnee. On voit que pour connaitre un element imaginaire, il faut connaitre (1) l'axe, ou le centre, de la dyade a laquelle Fl'lement appartient, (2) deux couples d'ele'ments reels de l'involution qui determine cette dyade, (3) l'homologie de cette m6me dyade avec la dyade fixe. Il faut convenir que de cette maniere on n'isole pas l'un des deux elements imaginaires d'une m6me dyade, et que, par consequent, l'expression 'element imaginaire donne,' n'est pas rigoureuse. Cependant on voit que si l'on echange un element imaginaire donne avec son conjugue, il faut en meme temps echanger tout autre element donne avec son conjugue. Et c'est la tout ce qu'il faut pour qu'on puisse operer avec des donnees imaginaires, de la m6me maniere qu'avec des donnees reelles. En effet, nous pourrons maintenant r6soudre les deux problemes correlatifs que voici. VOL. II. C 10 MIEMOIRE SUR QUELQUES PROBLaMES [Pt. I. Trouver le point d'intersection de deux droites donnees, dont l'une au moins est supposee imaginaire. Trouver la droite qui passe par deux points donnes, dont l'un au moins est suppose imaginaire. I1 suffira de considerer le second de ces deux problemes. Soient done a1, bk, deux points imaginaires donnes, dont les axes ne se confondent pas. Puisque les points a1, b, sont donnes, les dyades ac a2, b, b2, auxquelles ils appartiennent, sont aussi donnees, de m6me que l'homologie de chacune d'elles avec la dyade fixe. On peut donc trouver lineairement l'homologie de ces deux dyades. Soit P leur centre d'homologie; P sera le centre de la dyade a laquelle appartient la droite cherchee (a1, b1). Les involutions qui correspondent aux dyades a, a2, bib2 determinent la m~me involution au point P; cette involution determinera la dyade Pa b1, Pa2b2. Enfin l'homologie de cette dyade de droites avec chacune des dyades a1 b1, aO2 b est donnee immediatement; on peut done trouver son homologie avec la dyade fixe; ce qui acheve la determination de la droite (a1, b.). 4. Il resulte de la solution de ces deux problemes, que lorsqu'on peut resoudre lineairement un probleme quelconque, dont les donnees ne contiennent que des points et des droites reelles, on pourra encore resoudre lineairement ce m6me probleme, apres qu'on aura substitue, en tout ou en partie, des droites et des points imaginaires donnes, aux donnees reelles du probleme. On n'aura qu'a suivre de pas en pas la solution du probleme aux donnees reelles pour en conclure la solution du probleme correspondant aux donnees imaginaires. Seulement, puisque les determinations des droites joignant des points donnes, et des points d'intersection des droites donnees, qui sont des Postulata pour les cas reels, exigent des constructions detournees pour les cas imaginaires, on conyoit que la methode, quoique parfaitement generale, doit conduire a des operations assez longues. Supposons qu'on ait resolu lineairement un probleme dont les donnees soient toutes reelles; si, parmi ces donnees, il y en a deux qui entrent symetriquement dans la question, on pourra substituer une dyade donnee a ces elements symetriques, sans que le probleme cesse d'etre resoluble lineairement. En effet, apres cette substitution il n'y aura qu'une seule dyade independante; en la prenant pour la dyade fixe, on trouvera lineairement l'homologie de toute dyade derivee qu'on aura a considerer, et l'on n'aura qu'a operer sur des elements imaginaires donnes. Mais il en serait autrement, si l'on voulait substituer a la fois plusieurs dyades a la place d'un nombre 6gal de couples de donnees reelles. Toutes ces dyades seraient independantes, et pour qu'on put appliquer la methode prece Art. 5.] CUBIQUES ET BIQUADRATIQUES. 11 dente, il faudrait qu'on en connAt les centres, ou les axes d'homologie. Dans les applications, on fera bien de determiner actuellement ces centres ou ces axes par la construction que nous avons rappelee ci-dessus, Mais, puisque cette construction est quadratique, il importe de faire voir qu'on pourrait s'en passer a la rigueur. Pour cela, il faut seulement qu'au lieu d'introduire toutes les dyades a la fois, on les introduise l'une apres l'autre. Toutes les fois que l'on substituera une seule dyade a deux donnees symetriques reelles, on aura un nouveau probleme, dont on pourra trouver la solution par ce qui precede. Mais on peut considerer ce nouveau probleme comme n'impliquant que des donnees reelles, puisque la dyade que nous y avons fait entrer, peut 6tre remplacee par deux couples de l'involution qui la determine. On pourra done substituer une seconde dyade a une seconde couple de donnees reelles, et ainsi de suite, sans qu'on ait besoin d'aucune construction quadratique. 5. Nous allons maintenant indiquer quelques problemes dont on puisse trouver les solutions par les principes precedents. Tout element, qui n'est pas dit expressement etre reel, pourra 6tre imaginaire. (i.) ARtant donne's trois points d'une meme droite, ou trois droites passant par un meme point, trouver le conjugue' harmonique d'un de ces ele ments, par rapport aux deux autres. Soient a1, b1, c, des points donnes d'une m6me droite w1, et proposons nous de trouver le point cd, conjugue harmonique de c, par rapport a a, b1. Soient A, B, C les axes dyades a, a2, b b2, c1 c2; a,, y, les trois sommets du triangle ABC. La droite A sera l'axe d'homologie des dyades (aal, a a2) et W1 2; pareillement, B sera l'axe d'homologie des dyades (/3b, /3 b) et wI W2; on pourra done determiner l'axe d'homologie X des dyades (aal, aa), (3 b, 3 b2); soit x1x2 la dyade de points determinee sur X par ces deux dyades, et D l'axe d'homologie des dyades (7x1, yx2) et Dwi; D sera l'axe de la dyade cherchee d1d2, dont l'homologie avec 'I w2 sera evidemment connue. (ii.) tlant donnes, dans deux series homographiques, trois elemnents de l'une, et les trois elezments correspondants de I'autre, trouver l'elemient de l'une des deux series qui correspond dc un ele'ment quelconque de l'autre. On resoudra lineairement ce probleme general, en imitant, de la maniere que nous avons expliquee, les solutions qu'on a donnees du probleme dans le cas particulier ou les donnees sont reelles. Le principe de ces solutions (auxquelles on peut donner des formes tres variees) consiste, comme on sait, a trouver une troisieme serie qui soit homologique avec chacune des deux series donnees. On aura en m6me temps la solution du probleme important, C 2 12 MEMOIRE SUR QUELQUES PROBLEMES [pt. I. Sltant donnes cinq ele'ments de msnme espece qui determinent une conique, trouver autant d'ele'ments de cette courbe qu'on voudra, qui est au fond le m6me que le precedent. I1 y a deux cas tres particuliers qui meritent quelque attention. (1.) ]Jtant donnees deux dyades de droites ac a,, b, b2 ayant le mnme centre P, et deux autres dyades a, a2, /1 /2, ayant le mnsme centre I, et veriiant l'Iquation * n. [a, a2, /31 2 = P. [a1 a2, b1 b2 trouver la droite d'un des deux faisceaux qui correspond Z une droite donnee de l'autre faisceau. Pour que ce probleme n'admette qu'une seule solution il faut qu'on connaisse 1'axe d'homologie A des dyades ac a2, a, a2, et l'axe d'homologie B des dyades bl b2, 1 i32. I1 est vrai que lun des deux axes etant donne, on pourrait trouver l'autre lineairement, mais, pour abreger, nous supprimerons la demonstration de cette assertion, et nous supposerons que les axes A et B soient tous les deux donnes f. Soient Plp2, q1 q2 les dyades qu'on aura sur les droites A et B. En faisant passer une conique par le points P11p2, qlq2, PH on ramenera le probleme a celui de determiner autant de points qu'on voudra d'une conique dont on connait deux dyades et deux points reels. Mais, comme on sait, le theoreme de Carnot conduit a une solution lineaire de ce dernier probleme, meme en supposant qu'un seul point reel de la conique soit donne. (2.) Par un point P d'une conique dont on connalt cinq points reels i passe une droite imnaginaire donnee; trouver le second point d'intersection de cette droite par la conique. Soit (10) P un point reel. On sait trouver l'axe de symptose reel de la conique et de la dyade a laquelle appartient la droite imaginaire: le point d'intersection de cet axe par la droite imaginaire donnee sera le point cherche. Soit (20) P un point imaginaire. L'axe de la dyade a laquelle ce point appartient sera l'un des axes de symptose de la conique et de la dyade a laquelle appartient la droite imaginaire donnee. L'autre axe se trouvera lineairement et fera connaitre la solution du probleme. (iii.) Etant donne le trace complet d'une conique reelle, trouver les points d'intersection d'une droite w1 par une conique C, reelle ou imaginaire, dont on donnait cinq points. Soit C reelle. On trouvera par une construction quadratique connue, pour * Voir la note Art. 3, troisibme partie (p. 36). tt Note V (p. 54). Art. 5.] CUBIQUES ET BIQUADRATIQUES. 13 laquelle on se servira de la conique tracee, les deux axes de sympt6se de C et de la dyade de droites o w,; les points d'intersection de ces deux axes par w, seront les points cherches. Soit C imaginaire. On ramenera la question a la recherche des points doubles de deux divisions homographiques sur une m6me droite. On projetera ces deux divisions sur la conique tracee; et l'on cherchera l'axe des deux divisions projetees. Cet axe sera en general imaginaire; on determinera les points ou il est coupe par la conique tracee; ces points feront connaitre la solution du probleme. (iv.) Jltant donnes cinq points imaginaires, trouver des coniques reelles appartenant aufaisceau qui contient les coniques imaginaires determineies par ces cinq points, et par leurs cinq points conjugues. Soit P un point reel, et proposons-nous de determiner la conique du faisceau qui passe par le point P. I1 importe d'observer que si l'homologie des cinq dyades etait inconnue (hypothese que nous excluons, puisque nous supposons que les cinq points sont donnes) il y aurait seize solutions correspondant a chaque point P, puisque les cinq dyades determinent trente deux coniques dont chacune passe par l'un de deux points de chaque dyade. Cette remarque montre clairement l'importance qu'on doit attacher a l'homologie des dyades. Pour trouver la conique passant par P, on prendra un point quelconque reel q, et l'on en determinera la polaire relativement a l'une des coniques imaginaires. Cette determination sera toujours possible puisqu'elle n'exige que des operations lineaires que nous avons deja enseigne a faire. Soit Q le centre de la dyade a laquelle appartient la polaire de q; toute conique du faisceau divisera harmoniquement le segment reel qQ. En prenant quatre points qj, q2, q q, et leurs quatre points reciproques Q1, Q2, Q3, Q, on aura la solution du probleme, puisqu'on sait construire la conique qui passe par un point reel donne, et coupe harmoniquement quatre segments reels. On resoudrait de la meme maniere le probleme suivant. ]~tant donne's deux systemes polaires reels, dont les coniques peuvent etre imaginaires, trouver la conique qui passe par un point reel donne, et qui appartient au faisceau ditermine par les deux coniques. Nous ferons remarquer a cette occasion qu'il y a deux especes diff6rentes de coniques imaginaires qu'il importe de distinguer entr'elles. Une conique imaginaire de la premiere espece n'a aucun point reel; elle est coupee par toute transversale reelle en deux points imaginaires conjugues; elle a un systeme polaire 14 MEMOIRE SUR QUELQUES PROBLIMES [Pt. I. reel. Une conique imaginaire de la seconde espece a une base formee, soit de quatre points reels, soit de deux points reels et d'une dyade, soit de deux dyades. Elle coupe toute transversale reelle (autre que celles qui joignent deux des points de la base) en deux points qui ne sont ni tous les deux reels, ni des points imaginaires conjugu6s. La polaire de tout p6le reel qui n'est pas un sommet du triangle harmonique de la base, est imaginaire. (v.) Jitant donnees deux dyades a a2, i/33, ayant le mnme axe, et deux autres dyades al a2, b1 b2, ayant des axes A et B diffrents l'un de l'autre; trouver la conique Z qui passe par les deux dyades a1 a2, ba b2, et qui satisfait a& 'equation [O1, a2, bl, b2]= [a, a2, 1, 321. Pour que ce probleme soit lineaire il faut que l'homologie des dyades a a2, a, ca, et aussi celle des dyades i, /2, b b2, soient donnees. On prendra le centre d'homologie de a1 a2, a, 2; de ce centre on projetera 1 /32 sur A; et l'on determinera le centre d'homologie de b, b2, et de la projection de 1i32 sur A. Ce dernier centre sera un point de la conique cherchee 2, qui sera des lors completement determin6e. Si l'on supposait que les dyades a a2, /312, n'ont pas le m~me axe, mais qu'elles appartiennent a une conique reelle r, l'expression [a1 a2, /1S2] etant relative a cette courbe, on pourrait encore determiner la conique Z par la construction precedente. On n'aurait qu'a projeter les dyades aa2, /31i sur une droite L, en prenant pour centre de projection un point quelconque de r. On voit que les deux coniques r et Z seront homographiques par rapport aux points al a2, 1 /32, et a a2, b1 b2. Soit ~ un point donne reel de r; on trouvera de la maniere suivante le point correspondant x de la conique I. Soient a/ a, /3 i3, r' les projections de a, a2, /3132, sur la droite L: du centre d'homologie des dyades a ac,, a2 on projetera /3fi3, ' sur A: soient /3, /', F" les projections de ces points. On trouvera le centre d'homologie p des dyades /3i'T/, b, b2; ce centre appartiendra a la conique 1, et le point x cherche sera le second point d'intersection de la droite pr" par 2. Si, au lieu d'un point reel r, une dyade 12, appartenant h r, etait donnee, on trouverait x,1x, la dyade correspondante de 2, par une construction toute semblable. On aurait sur la droite A une dyade "', dont lhomologie avec 2 serait connue; puis on determinerait l'axe de symptose de Z et de la dyade des droites p0' p '; ce qui suffirait pour faire connaitre la dyade x1 xz, et l'homologie de cette dyade avec la dyade donnee 4;2. On peut enoncer le probleme que nous venons de resoudre de la maniere suivante. Art. 5.] CUBIQUES ET BIQUADRATIQUES. 15 ttant donnees deux dyades d'une premiere figure correspondant 4 deux dyades d'une seconde figure, transformer la premiere figure homographiquement en la seconde. (vi.) Eitant donne'es deux dyades a, a,, b, b2, ayant des axes differents A et B, et quatre points 1, 2, 3, 4 formant la boase d'un faisceau de coniques reelles, trouver la conique qui passe par les deux dyades a,1a2, b, b2 et qui satisfait c' I'equation (1, 2, 3, 4) [a1, a2, bl, b2]= [a, a2, bl, b2] Soit r une conique quelconque passant par les deux dyades ac1a2, b1 b2. Qu'on prenne un point reel y de cette conique et que de ce point on projette sur la conique les involutions qui sont determinees par le faisceau (1, 2, 3, 4) sur les droites A, B. Soient p et q les poles des deux involutions qu'on aura maintenant sur la conique; les droites des deux faisceaux (p) et (q) correspondront anharmoniquement aux coniques du faisceau (1, 2, 3, 4): donc ces deux faisceaux de droites seront homographiques; soit C la conique, lieu des points d'intersection des rayons correspondants. Qu'on determine les axes de symptose des dyades (pa1, pa2), (qb1, qb2) avec la conique C; soient a, f 1302 les points d'intersection de ces axes par C; les points p et q seront les centres d'homologie des dyades a, a 1, a2a et b, b2, $i1 12 respectivement. Dans le faisceau (p) les droites imaginaires pa1, pa2 correspondront aux coniques (1, 2, 3, 4, a1), (1, 2, 3, 4, a2); de meme dans le faisceau (q) les droites imaginaires qb,, qb2 correspondront aux coniques (1, 2, 3, 4, b)), (1, 2, 3, 4, b). Done le rapport anharmonique des quatre points al, a2, 1, 02 appartenant a la conique C, sera egal au rapport anharmonique (1, 2, 3, 4). [al, a, bl, b2]. On determinera la conique qui passe par les deux dyades a1, a2, bl, b2, et qui satisfait a l'equation la1 a & bl, b2] =[al, 2a2, 102]; cette conique sera celle qu'on cherche. Cette solution est purement lineaire, puisqu'on ne trace pas les coniques dont il y est question. On pourra l'abreger en operant de la maniere suivante. Soit le point d'intersection des droites A et B;, ri les points oi ces m6mes droites coupent pour la seconde fois la conique (1, 2, 3, 4, w). Soit y un point quelconque de la droite t; en prenant pour r la conique (a1, a2, b,, b2, 7) les faisceaux (p) et (q) deviendront homologiques, et les dyades ax a2, /3, seront connues, des que lon aura determine l'axe d'homologie de ces deux faisceaux. 16 MEMOIRE SUR QUELQUES PROBLIMES [pt. I. Si l'on se permettait une construction quadratique, on pourrait encore raccourcir la solution. On trouverait un centre d'homologie des deux involutions determinees par le faisceau (1, 2, 3, 4) sur les axes A et B; en prenant ce centre pour le point y, les deux points p et q viendraient se confondre en un seul point, qui appartiendrait a la conique cherchee. A la verite, cette methode ne serait pas applicable, si les centres d'homologie des deux involutions etaient imaginaires. Nous ajouterons quelques corollaires de ce probleme, qui nous seront utiles plus tard. (1.) -]tant donnees quatre dyades ac a2, bl b2, 1 c2, di d2, et un point reel p, trouver le point w oppose 'a deux de ces dyades cl c2, d2d, par rapport A la courbe cubique (a,, a2, bl, b2, C1, c, d1, d2, p). On determinera la conique Z qui passe par les points a,, a2, b1, b,, et qui satisfait a l'equation (C1, C2, dl, d). [a, a2, b1, b2] = [a,, a2, b, 2]. Ensuite on trouvera le point p' de cette conique pour lequel on a (ci, c2, d, d,) [a1, a2, b, b2,p]=[a, a2, bl, b2, p]. Pour cela, on remarquera que dans la solution generale precedente (oi l'on peut remplacer les points 1, 2, 3, 4 par les points c1, c2, di, d2) on trouve immediatement le point?r de la conique auxiliaire C, qui correspond a la conique (c1, C2, d1, dp, p). Mais ce point trouve, on en deduira le point p' par une construction qui a ete deja indiquee (v). Enfin, le point ohi la droite pp' rencontre pour la seconde fois la conique (a1, a2, bl, b2, p') sera le point a cherche. Ainsi on pourra trouver lineairement autant de points qu'on voudra d'une cubique dont on ne connaitra que quatre dyades de points imaginaires, et un seul point reel. Ce cas nous parait avoir ete omis par les auteurs qui ont traite des constructions des courbes cubiques. Si, au lieu d'un point reel p, une dyade pi 2 de points imaginaires etait donnee, on pourrait encore trouver la dyade w1 w2, dont les elements sont les points opposes au systeme des quatre points cl, c2, dl, d2, par rapport aux deux courbes cubiques imaginaires (a, a2, bl, b2, c1, cI, dC, d2, p) et (a1, a2, bi, b2, ci, c2, d, d2, P2). On commencera par trouver la dyade -1 7r2, qui appartient a la conique auxiliaire * Note VI (p. 55). Art. 1.] CUBIQUES ET BIQUADRATIQUES. 17 C, et dont les elements correspondent aux coniques imaginaires (c1, c2, di, d2,,p) et (cx, c2, di, d2, p2). Le centre d'homologie de cette dyade et de plps sera un point connu. Puis on determinera la dyade p'p2, qui appartient a la conique 1, et qui verifie l'equation (c1, C2, d, c. d [ d, a2, b,, b2, p, p2]=[O [, a2, b,, b2, pI, p].Soit P le centre d'homologie des dyades pp', 2P1p2. Un des axes de sympt6se de la dyade de droites (Ppl, Pp2), et de la conique T, sera l'axe de la dyade pIp'2; l'autre pourra se determiner lineairement; les points d'intersection des droites Pp1, Pp2 par ce second axe de symptose seront les points wi w2 cherches. (2.) lEtant donnees quatre dyades de points imaginaires a, a2, b1 b2, c c2, d d2, trouver le neuvieme point 0, appartenant A toute courbe cubique qui passe par les quatre dyades. C'est ce qui se fera par une construction connue. Soient x, y deux points reels; soient, r les points opposes au systeme c1c, dc dd par rapport aux deux courbes (a1, a2, b1, ba, 01, C0, dC, d2, x) et (a1, a2, b, b2, c, C2, d1, d2, y) respectivement, 9', n' les points opposes au systeme aaa, b1b2 par rapport a ces m6mes courbes; les droites n', r' se couperont au point 0. SECONDE PARTIE. 1. Soit Z une section conique, qui ne soit ni evanescente ni un cercle, et que nous supposerons completement tracee; nous reviendrons plus tard sur le cas oh l'on n'aurait qu'une partie de la courbe. Toutes les autres coniques dont nous aurons a nous occuper ne seront point tracees (a moins qu'elles ne soient des cercles, ou des couples de lignes droites); elles seront seulement definies par un nombre suffisant de conditions. Voici le probleme que nous aurons a resoudre. ' Etant donnees deux coniques S1, S2, trouver leurs quatre points d'intersection, ou, ce qui revient au meme, leur triangle harmonique commun, avec la regle et le comoas seulement, mais en se servant de la conique tracee S.' Les coniques S1, S2 peuvent 6tre imaginaires de premiere espece, ou imaginaires de seconde espece et conjuguees l'une a l'autre. Mais puisque, dans ces deux cas, on peut construire autant de points qu'on voudra sur les coniques reelles du faisceau (Si, S2), nous pourrons supposer, sans perte de generalite, que Si, S2 soient elles-m6mes reelles. I1 y a trois reseaux de coniques qu'on est conduit naturellement a considerer relativement au faisceau (S,, S2), ou plutot relativement a son triangle harmonique ajy7; ce sont (1) le reseau des coniques circonscrites au triangle a/3^, (2) le VOL. II. D 18 MIOMOIRE SUR QUELQUES PROBLIMES [Pt. II. reseau des coniques inscrites a ce triangle, et (3) le reseau harmonique, c'est a dire le reseau des coniques dont ce meme triangle est un triangle harmonique. Une conique du premier reseau est determinee par deux points; une conique du second reseau par deux, tangentes; une conique du troisieme reseau, soit par deux points, soit par deux tangentes. Le second reseau contient quatre cercles, dont la construction est deja un problme biquadratique. Mais chacun des autres reseaux ne contient qu'un seul cercle; ce sont le cercle circonscrit, et le cercle polaire, du triangle af3y. On peut se servir de l'un ou de l'autre de ces deux cercles pour resoudre notre probleme, et lon est ainsi conduit a deux methodes de solution diffrentes. D'apres la premiere methode on transforme homographiquement une conique quelconque du premier reseau dans la conique 2, ou bien on transforme correlativement une conique du second reseau en cette m6me conique. D'apres la seconde methode, c'est sur une conique du troisieme reseau que l'on opere, et la transformation peut 6tre de l'une ou de l'autre espbce. Dans tous les cas on cherche le cercle unique du reseau transforme: les intersections de ce cercle avec la conique 2 font connaitre la solution du probleme. 2. Avant de donner les details necessaires sur ces deux methodes, nous allons resoudre quelques problemes preliminaires, qui se rapportent aux coniques des trois reseaux: surtout, nous ferons voir comment on peut construire, dans tous les cas possibles, les cercles du premier et du troisieme reseau. Les coniques du r6seau circonscrit sont precisement les coniques r6ciproques de droites, qui ont ete considerees par MM. Poncelet et Chasles. Ainsi chaque conique K de ce reseau est le lieu des points qui sont reciproques des points d'une certaine droite k relativement au faisceau (S1, S2); K est aussi le lieu des poles de k, relativement aux coniques du faisceau (Si, S2); enfin, l'ensemble des lignes K, k est la courbe Jacobienne du rdseau determine par S1, S2 et la droite k, prise deux fois. Pour construire la conique du reseau circonscrit qui passe par deux points p1p2, on construit les points -P P2 r6ciproques de P1P2; la conique reciproque de la droite P1 P2 est la conique cherchee. De m6me, on pourra trouver la conique du reseau circonscrit qui soit tangente en un point donn6 p v une droite L. On prendra la conique reciproque de L; sa tangente, au point reciproque de p, sera rdciproque de la conique cherchee. Pour les coniques du second reseau on a des constructions correlatives; mais la construction de la conique a- appartenant au r6seau harmonique, et passant par deux points plp3, est un peu moins facile. Soient x1 x2, yi y2 deux couples de points qui divisent harmoniquement le segment P1 p2; construisons les points rciproques X1 X2, Y1 Y2, et menons les droites Xi X, Y1 Y, se coupant au point q. Soient encore abc les Art. 2.] CUBIQUES ET BIQUADRATIQUES. 19 points o(i la droite P1P2 rencontre les c6tes du triangle a,3y; et designons par a'b'c' les points conjugues harmoniques de abc par rapport t P1p2. En considerant l'involution dont les points doubles sont Pip2, on voit que le quatribme point commun aux coniques (a, j3, y, x1, X2), (a, /,,y y, y2), (c'est-a-dire le point Q reciproque de q) est le point d'intersection des trois droites a a',,, c'; mais ces trois droites sont 6videmment les polaires de a, b, c, relativement a cr; done le point Q reciproque de q est le pole de la droite 1 p2, relativement a cette conique. Soit (a, f, y, x, y) une conique du reseau circonscrit qui passe par deux points quelconques xy de la droite pI p2; cette conique est aussi une conique du reseau circonscrit appartenant au faisceau (S, a-); par cons6quent les points r6ciproques des points de (a, B, y, x, y), consideres relativement au faisceau (S1, o-), seront en ligne droite. Pour avoir cette droite, on determinera les points XY, reciproques de xy, qu'on sait trouver, puisqu'on connait les polaires de x, y relativement aux coniques - et S1. Soit 0 un point quelconque de (a, 3, 7, x, y); les polaires de 0, relativement h a- et S, se croisent sur la droite XY; on connait aussi le point oil la polaire de 0 relativement a a- coupe la droite Pi P2; on pourra done construire cette polaire, qui determinera completement la conique a. On ne doit pas prendre pour xy les points doubles de l'involution determin6e par (Si, ar) sur la droite pI p2; ce qui ferait coincider la droite X Y avec p1P2; de m6me, on ne doit pas prendre pour 0 le point reciproque du point d'intersection de XY, P P2. De ce que nous avons dit on tire la construction suivante pour les deux cercles. Soient x1 x2, yY Y2 deux couples de points rectangulaires a l'infini; X X2, Y, Y2 leurs points reciproques relativement au faisceau (Si, S2). Le centre Q du cercle polaire est le point reciproque de l'intersection q de X1X2, Y YY. Soient rlr2 les deux autres points diagonaux du quadrilatere X X2!, Y Y2. Le cercle circonscrit est reciproque de la droite r1 r2; de plus les points RR2, reciproques de r1 r2, sont les extremites opposees d'un meme diambtre de ce cercle. En effet, la droite r, r2 coupe la conique C, reciproque de la droite a l'infini, en deux points imaginaires Qi Q2, qui sont reciproques des deux points wi w2 de la dyade cyclique. Car les points d'une conique reciproque d'une droite correspondent anharmoniquement aux points de cette droite; de sorte qu'aux points doubles de l'involution x1 X2, yi y, sur la droite a l'infini correspondent les points doubles de l'involution X1 X2, Y, Y sur la conique C. De plus, les deux points r1 r2 sont harmoniquement conjugues aux points QN 2,; done, sur le cercle reciproque de la droite r1 r2, les points RI R2 sont harmoniquement conjugues 'a 1 2; ce qui veut dire que RI R est un diametre du cercle. On connait done le cercle circonscrit; D 2 20 MEMOIRE SUR QUELQUES PROBLIMES [Pt. II. quant au cercle polaire on achevera sa determination, soit en observant qu'il coupe orthogonalement le cercle directeur du cercle circonscrit, soit en trouvant par la methode que nous avons indiquee la polaire d'un point quelconque. Dans les applications, on pourra toujours remplacer la conique non tracee S, par la conique tracee Z. On prendra pour x, y dans la construction precedente les points x, x2 dont on s'est deja servi, et dont on aura construit les polaires relativement a Z. On abaissera de Q des perpendiculaires sur ces polaires; la droite qui joindra leurs pieds sera r6ciproque de l'hyperbole 6quilatere '(a, /3, y, x1, xi par rapport au faisceau (E, -). Soit 0 un point quelconque de l'hyperbole, O' le point d'intersection de la droite reciproque de cette courbe par la polaire de 0 relativement 'a; la perpendiculaire abaissee de 0' sur Q sera la polaire de 0 relativement au cercle polaire. On arriverait aussi ~ des constructions assez simples, en prenant pour xy, soit les deux points h l'infini sur 2, soit les deux points de la dyade cyclique. Il est inutile d'ajouter qu'on ne peut determiner lineairement aucun point du cercle polaire, ce cercle pouvant 6tre imaginaire. On sait que le cercle circonscrit a un triangle harmonique de la conique S coupe orthogonalement le cercle directeur de S; et que le cercle polaire d'un triangle circonscrit a S coupe aussi orthogonalement ce mdme cercle directeur. On pourrait donc trouver le cercle circonscrit en construisant les cercles directeurs de trois coniques quelconque du reseau harmonique, et le cercle polaire en construisant les cercles directeurs de trois coniques quelconques du reseau inscrit. On aura aussi les resultats particuliers que voici, dont la plupart etaient connus: 'Le lieu des centres des hyperboles equilateres du reseau circonscrit, est le "cercle des neuf points " du triangle harmonique fondamental. 'Le lieu des centres des hyperboles equilatbres du reseau harmonique est le cercle circonscrit. Ces hyperboles se coupent aux quatre centres des cercles inscrits au triangle harmonique. 'Le lieu des foyers des paraboles du reseau inscrit est le cercle circonscrit. 'Les droites directrices des paraboles du reseau inscrit se coupent au centre du cercle polaire. 'Les droites directrices des paraboles du reseau harmonique se coupent au centre du cercle circonscrit.' 3. Ce qui precede suffit pour notre but actuel; cependant nous croyons devoir ajouter quelques observations elementaires qui pourraient dtre utiles en d'autres occasions. On peut considerer les sommets du triangle harmonique commun h deux coniques comme representant d'une certaine maniere les trois Art. 3.] CUBIQUES ET BIQUADRATIQUES. 21 racines d'une equation cubique. Ainsi au probleme analytique 'Trouver les fonctions symetriques des racines d'une equation cubique' correspondra le probleme geometrique 'Trouver les lieux et les enveloppes qui dependent symetriquement des trois elements du triangle harmonique commun a deux coniques non tracees.' D'apres ce qui a etI dit, on aura des moyens assez simples pour trouver les points, les droites et les coniques qui dependent symetriquement des elements du triangle, consider6 par rapport au systeme de deux points ou de deux droites quelconques. Mais nous ne nous sommes pas occupe des lieux et des enveloppes qui se rapportent au triangle consider6 relativement a un seul point ou a une seule droite. Parmi tous ceux qu'on pourrait imaginer ce sont surtout les derivees polaires du triangle (considere comme ligne soit du troisieme ordre, soit de la troisieme classe) qui offrent quelque int6ert. Pour abreger, nous ne parlerons que du centre des distances moyennes du triangle, et de la droite a l'infini consideree relativement au triangle, puisqu'on tirera facilement de ce que nous allons dire tout ce qui est relatif a un point ou a une droite quelconque. En revenant donc sur la notation de l'article 2, nous nous proposerons de trouver (1) le centre des distances moyennes, (2) l'ellipse minimac circonscrite, (3) l'ellipse maxima inscrite. Supposons que le point q' soit harmoniquement conjugue a q par rapport aux points X1 X2; si Q' est le point reciproque de q' relativement au faisceau (S1, S2), il est evident que QQ' sera un diametre de l'hyperbole equilatere (a, 3, 'y, X, X2). Soit 0 le point milieu de RIR2, O' le point milieu de OQ; 0 et 0' seront les centres du cercle circonscrit, et du cercle des neuf points. Ce dernier cercle sera completement determine, puisque le point milieu de QQ' sera un point de sa circonf6rence. Soit r le point harmoniquement conjugue a Q par rapport k 00': r sera le centre des distances moyennes du triangle; on sait, en effet, que le centre des distances moyennes et le centre du cercle polaire sont les deux centres de similitude du cercle circonscrit et du cercle des neuf points, et que les deux centres de similitude divisent harmoniquement le segment 00'. On voit en m6me temps, qu'en generalisant convenablement cette construction, on pourra trouver avec la regle seule le p6le d'une droite quelconque, et la polaire d'un point quelconque par rapport au triangle a3,y. Les deux ellipses maxima et minima sont les coniques polaires du centre des distances moyennes et de la droite a l'infini, par rapport au triangle. Mais elles sont aussi les coniques du premier et du second reseau pour lesquelles la droite a l'infini est la polaire du centre des distances moyennes. Il suffira donc de resoudre ce problUme ' Trouver la conique du premier reseau pour laquelle un point donne rest le p6le d'une droite donnee A.' 22 ME]MOIRE SUR QUELQUES PROBLUMES' [Pt. II. Soit x1 un point quelconque de A; on trouvera la conique du troisieme reseau qui est tangente aux droites xr et A, et l'on menera du point r la seconde tangente k cette conique, qui coupera A au point x2. Les deux points x1 x2 seront conjugues par rapport a la conique qu'on cherche. Soient yi Y2 une seconde paire de points conjugu6s sur la meme droite; prenons X1X2 YY les points r6ciproques de x1 x2 y, y2; la diagonale du quadrilatere X X2 Y Y2, qui ne passe pas par le point d'intersection de X1X2, Y Y2 sera la droite reciproque de la conique cherchee. On peut aussi operer de la maniere suivante. Soit x un point quelconque de A; toutes les coniques du r6seau circonscrit, qui coupent harmoniquement le segment x, passent par le m6me quatrieme point, dont nous allons trouver le point reciproque. Soient r", X les points reciproques de r, x; les coniques dont il s'agit auront pour droites reciproques des droites conjuguees i la droite F'X par rapport a la conique r6ciproque de Fx, puisque quatre points harmoniques en ligne droite ont pour rdciproques quatre points harmoniques sur la conique reciproque de la droite. Soient a1 a2, b, b2 deux couples de points harmoniquement conjugu6s aux points r, x; A1 A2, B1 B2 les points r6ciproques correspondants; le point E oi les cordes A A2, B B2 vont concourir sera le point reciproque du quatrieme point d'intersection. En prenant un second point y sur la droite A, on trouvera un autre point n tel que t; la droite tq sera reciproque de la conique cherchde. On trouverait la conique du second r6seau, qui aurait un p6le et une polaire donn6s, par la construction correlative. Si c'6tait la conique du troisieme reseau qu'on voulait trouver, on prendrait la conique reciproque de la droite A, puis la polaire de r' par rapport a cette conique, et enfin la conique reciproque de la polaire. Cette derniere conique coupe la droite A aux m6mes points ot elle est coup6e par la conique harmonique qu'on cherche. On pourra done trouver avec la regle seule, autant de poles et de polaires relativement a cette courbe que l'on voudra; puisque pour cela il n'est pas necessaire de connaitre les deux points oh la courbe est coupde par la droite A, mais seulement l'involution qui determine ces deux points *. Nous ajouterons encore un exemple de l'application des principes precedents. Soit propose de determiner le cercle qui passe par les pieds des perpendiculaires abaiss$es d'un point donn6 P sur les cotes du triangle a/3y. On d6terminera la conique du r6seau inscrit qui a pour foyer le point P. Le cercle decrit sur laxe focal de cette conique comme diametre sera le cercle cherch6. * Note VII (p. 55). Art. 4.] CUBIQUES ET BIQUADRATIQUES. 23 Quoiqu'on puisse representer les racines d'une dquation cubique par les trois 6elments du triangle harmonique commun a deux sections coniques, il arrive ordinairement que la solution d'un probleme cubique ne conduit pas a la recherche d'un tel triangle, mais a la recherche de trois des points d'intersection de deux sections coniques, dont un des points d'intersection est donn6 d'avance; ou l la recherche des trois points communs a un reseau de coniques circonscrites au m6me triangle. Or, le second de ces deux cas se roduit au premier, puisqu'en considerant les coniques d'un faisceau dterin par deux coniques du reseau, on voit qu'elles correspondent anharmoniquement aux points old elles coupent une troisime conique e du rseau qui n'appartient pas au faisceau que l'on considre. On pourra done determiner lineairement le quatribme point commun h une conique quelconque du faisceau et t la troisieme conique. De plus, Btant donne un des points d'intersection P de deux coniques, le triangle a/3y est homologique au triangle abc, dont les sommets sont les trois autres points d'intersection; il est aussi inscrit N ce m6me triangle. Le centre d'homologie est le point P; quant h l'axe d'homologie, c'est l'axe de symptose d'une conique circonscrite h Pa 3y et d'une conique du r6seau harmonique qui est tangente a la premibre conique au point P. L'homologie des deux triangles pourra done se determiner lineairement; et lon aura ainsi une methode generale, dont on pourra se servir pour d6duire les lieux et les enveloppes nrelatives a Fun de ces triangles des lieux et des enveloppes relatives h l'autre. 4. Premiere methode.-On determinera cinq points a, b, c, d, e d'une conique quelconque a- du reseau circonscrit appartenant au systeme des deux coniques (S1, S2). On choisira a volonte trois points a'b'c' de la conique S, qu'on fera correspondre homographiquement h abc; et l'on en d6terminera un quatrieme d' de maniere que le rapport anharmonique des quatre points a'b'c'd' de la conique Z soit egal a celui des points abcd de a-. Puis on transformera homographiquement les coniques (S1, S2) de manibre qu'aux points abed de la figure donn6e correspondent les points a'b'c'd' de la figure nouvelle. La conique ac se transforme en 2, laquelle sera ainsi une conique du r6seau circonscrit appartenant aux transformees de S1 et S2. Le cercle appartenant h ce reseau coupera Z en quatre points, dont un qui est r6el sera connu d'avance, et pourra se construire lin6airement, sans tracer ni le cercle, ni la conique 2. Les trois autres points (dont un sera toujours reel) seront les points de la nouvelle figure qui correspondent aux sommets a/Sy du triangle harmonique commun h S1 et S2. On pourra done construire ce triangle; ce qui permettra de trouver par une construction quadratique les points d'intersection de S1 et S2. 24 MIMOIRE SUR QUELQUES PROBLIMES [Pt. II. On voit que cette construction est purement lineaire, jusqu'au moment ot l'on veut determiner les intersections du cercle et de Xo Mais on peut l'abreger de beaucoup, en se servant des le commencement du trace de la conique E. On prendra deux points P112 de cette conique, et l'on choisira pour Ca la conique (a, B, 7, pi, p2). Puis on considerera la droite pip 2 comme axe d'homologie de a et 2; en se servant du trace de I on pourra trouver les deux centres d'homologie des deux courbes, appartenant a cet axe. On voit qu'on fera bien de prendre pour p1P2 deux points imaginaires conjugues appartenant: 2; si l'on prenait deux points reels, il pourrait arriver que les points d'intersection de 2 et ca fussent tous reels, sans qu'il y eAt aucune tangente commune; mais alors les deux centres d'homologie appartenant a l'axe 11ip2 pouraient devenir imaginaires. En supposant done que pi P2 soient imaginaires, on fera avec deux couples de l'involution qui determine ces points les m6mes operations que nous avons faites avec les deux couples de points rectangulaires x1 x2, y, Y2 de l'article 2; et, de m6me que nous y avons trouve les deux extremites d'un meme diametre du cercle circonscrit, on trouvera ici les deux extremites reelles a a2 d'une corde de a passant par le pole de la droite pip2 relativement a a-. On joindra le point d'intersection de a, a2, pip2, au p6le de p1ip2 relativement 'a 2, par une droite qui coupera I en deux points b1 b2, qui seront toujours reels, puisque le p6le de P1P2 est interieur a 2; les intersections de a, b., a2 b2, et de acb2, a2 b, seront les deux centres d'homologie de a et 2, appartenant a l'axe d'homologie (PIp2). Avec l'un de ces centres d'homologie, on determinera la droite X et la dyade 0102, homologues de la droite a l'infini et de la dyade cyclique; puis on trouvera le segment reel intercepte par la conique (a, f3, o, 01, 02) sur une droite passant par le pole de X relativement a cette conique: enfin on determinera le quatrieme point d'intersection de (a,,3, y, 1, 02) avec (a, 3, y, p, p2) ou -. En passant 2 la figure homologique, la premiere de ces deux coniques deviendra un cercle, dont on connaltra un diametre et une des intersections avec E. On n'aura donc qu'a tracer le cercle pour avoir les trois points homologues des points cherches. Si Z est une ellipse on prendra pour P1P2 les deux points imaginaires ohl cette courbe est coupee par la droite k l'infini. Dans ce cas, on determinera un diametre du cercle circonscrit au triangle af3y, et un diametre de a-, qui sera l'ellipse circonscrite au meme triangle et homothetique avec Z. On trouvera les deux centres de similitude des deux ellipses, en determinant les points extremes d'un diametre de Z parallele au diametre de a-; ce qui exigerait une construction quadratique, si 2 n'etait point tracee. Alors, pour avoir la solution du probleme, on n'aura qu'a tracer le cercle correspondant au cercle circonscrit a a/3y. Nous Art. 5.1 CUBIQIJES ET BIQUADRATIQUES. 25 croyons que cette solution est peut-6tre la plus simple qu'on puisse trouver; cependant elle ne laisse pas d'6tre fort longue. Pour l'abreger autant que possible, on prendra les deux points a l'infini sur les axes principaux de 2 pour l'un des couples de points dont on se servira pour trouver un diametre, soit du cercle circonscrit, soit de l'ellipse a; le second couple ne pourra pas 6tre le m6me pour les deux courbes, mais on fera en sorte que les deux couples aient un point commun. Ainsi il faudra chercher les points r6ciproques, par rapport au faisceau donne, de dix points diff6rents, afin d'avoir un diametre de chacune des deux courbes, et leur quatrieme point d'intersection. Cette recherche sera un peu penible, puisque la determination de chaque point reciproque exige la construction de la polaire d'un meme point par rapport a chacune des deux coniques non tracees S1 et S. On pourrait eviter toute construction quadratique, sans cesser de se servir d'une conique homologue de 2, en prenant pour a- la conique du reseau circonscrit qui est tangente a Z en un point donne, puisqu'en ce cas, un des centres d'homologie des deux coniques 6tant connu d'avance, on peut trouver l'autre lineairement. C'est ce qu'on serait conduit naturellement a faire si la conique 2 etait une parabole. On prendrait pour o- la parabole homoth6tique du reseau circonscrit; on en trouverait trois points, dont un serait son quatrieme point d'intersection par le cercle circonscrit. En menant des tangentes a 2 paralleles aux tangentes de a- en deux de ces trois points, et en joignant les points de contact des tangentes homologues, on aurait le centre de similitude. En ce cas on n'aurait B trouver que [neuf] points reciproques, mais, par compensation, la determination du centre de similitude serait un peu plus compliquee. 5. Cette premiere methode va nous fournir une demonstration bien simple de ce beau theoreme de Descartes, qu'il n'y a pas un arc de section conique si petit qu'il ne suffise pas pour resoudre geometriquement tout problkme cubique ou biquadratique. Prenons deux points a, b sur une conique quelconque c du reseau circonscrit; faisons correspondre a ces deux points deux points a'b' de la partie tracee de 2. Soit a le sommet reel, ou l'un des sommets reels, du triangle a$3y. La conique a est divisee par la corde ab en deux parties; en determinant, avec une approximation m6me tres grossiere, la position du point a (ce qui se fera dans tous les cas au moyen de quelques points qu'on prendra sur une conique quelconque du reseau circonscrit, autre que a) on distinguera l'arc a a b, qui passe par a, de l'autre partie de la courbe. Selon que l'on prend pour c un point de l'arc a ab, ou de l'arc oppos6, on prendra pour c' un point de l'are trace a'b', ou un point de la conique Z qui ne soit pas compris entre ces deux points. D'aprbs VOL. II. E 26 MEMOIRE SUR QUELQUES PROBLtMES [Pt. II. cela, les points de 'arc trac6 a'b' correspondront sur la conique Z aux points de Fare a ab de la conique a-; done le point correspondant de a tombera entre les points a', b' de l'arc trace, et y sera determine par le cercle circonscrit, appartenant a la figure transform6e, qui viendra couper l'arc a'b' en ce point. Mais on connait un second point d'intersection de ce cercle et de 2; c'est celui qu'on peut construire lineairement; done on pourra trouver par une construction lineaire la secante commune, dont les intersections avec le cercle feront connaitre les deux autres sommets du triangle a /7. 6. Seconde methode.-Lorsqu'on se sert du cercle polaire pour trouver les intersections des coniques donnees S1 et S2, on peut operer d'un grand nombre de manieres differentes. Nous n'indiquerons ici que les constructions que nous croyons les plus simples, mais nous n'osons pas affirmer qu'on ne pourrait en trouver d'autres, dependant du m6me principe general, qui auraient quelque avantage sur celles que nous allons exposer. (1.) Supposons que a/37, le triangle harmonique commun de S, et S2, n'ait qu'un sommet reel. En ce cas, deux coniques quelconques reelles du r6seau harmonique se coupent en deux points reels, et en deux points imaginaires conjugues; de plus, ce r6seau ne contient aucune conique imaginaire de la premiere espece. D'apres cela, on peut operer avec une conique quelconque de ce reseau de la meme maniere qu'avec une conique circonscrite. Soit a la conique qu'on choisit; on la transformera soit homographiquement, soit correlativement, en 2, et apres avoir determine le cercle appartenant au reseau transforme, on trouvera les deux intersections reelles du cercle et de 1. On aura ainsi le triangle harmonique commun a ces deux coniques, triangle qui est le correspondant dans la nouvelle figure du triangle a 3y dans l'ancienne. On pourra aussi employer une transformation homologique, au lieu de la transformation homographique generale; seulement on remarquera que la determination des centres d'homologie de cT et Z est loin d'6tre aussi facile dans le cas actuel que dans le cas d'une conique circonscrite. En effet, il convient de prendre pour a une conique du reseau harmonique ayant avec 2 une secante ideale commune. Or, on ne peut determiner lineairement aucun point d'une telle conique, quoiqu'on sache trouver son systeme polaire. On aura done besoin d'une construction quadratique pour trouver les points extr6mes d'une corde de a- passant par le pole de l'axe d'homologie relativement a cette conique; points dont on fait usage pour trouver directement les deux centres d'homologie. II est vrai qu'on pourrait operer avec une conique a- tangente a i en un point donne, mais la construction d'une telle conique serait un peu penible. Art. 7.] CUBIQUES ET BIQUADRATIQUES. 27 La m6thode que nous venons d'indiquer s'applique aussi au cas que nous avons exclu, mais elle conduit quelquefois a un resultat inutile. En effet, quand le triangle a 7y est reel, les coniques du reseau harmonique qu'on determine par deux points imaginaires conjugues peuvent 6tre imaginaires; ainsi, le cercle du reseau transforme peut 6tre imaginaire, ou, tout en restant reel, il peut ne recontrer Z en aucun point reel. D'ailleurs, quand m6me cette methode reussirait, elle conduirait a des operations assez prolixes. 7. (2.) Supposons que les points d'intersection de S1, S2 ne soient pas tous imaginaires; nous allons voir qu'on pourra determiner ces points directement, sans chercher prealablement le triangle harmonique. Soit (1~) Z une ellipse, A1 A2, B1 B2 ses deux axes principaux. Qu'on mene une transversale L qui coupe en deux points imaginaires une conique quelconque reelle S du faisceau (S,, S2). Ce faisceau determine une involution sur la transversale L, dont les points doubles,u A2 sont necessairement reels. Des points I1, t2 menons des tangentes a S; soient a a2, bI b2 les points de contact des tangentes issues de IA, et,2 respectivement; ces points, qui seront tous r6els, diviseront la conique S harmoniquement. Qu'on transforme la figure homographiquement de maniere qu'aux deux couples ca a2, b, b2 correspondent les deux couples A1A2, B1B2; ce qui peut se faire de huit manieres diff6rentes. La conique S deviendra S, puisqu'elle doit devenir une conique divisee harmoniquement par les deux couples de points A1 A2, B1 B2. De plus les quatre points d'intersection de S1 et S2 se transformeront en quatre points de l'ellipse Y, situes sur une m6me circonference de cercle. Car les points I1 i2, qui sont des points conjugues par rapport a toutes les coniques du faisceau (S1, S2), se transformeront en les points situes a l'infini sur les axes principaux de I; donc toutes les coniques du faisceau transforme auront leurs axes principaux paralleles a ceux de 1, et par cons6quent se couperont sur une circonf6rence de cercle f2. Pour avoir ce cercle, on cherchera son centre par la construction ci-dessus; puis on prendra un couple de points reciproques P1 P2 par rapport au faisceau transforme; le cercle Q coupera orthogonalement le cercle dont P P2 est un diametre. On peut aussi determiner f2 en prenant trois couples de points reciproques par rapport au faisceau transforme: ce qui donnera trois cercles que l coupera orthogonalement. Soit (20) Z une hyperbole ayant A1A2 pour axe rdel, et passant par les points x1 4b2 a l'infini. On coupera le faisceau (S1 S2) par une transversale sur laquelle on ait une involution aux points doubles reels I 2. Soit S une conique du faisceau qui coupe la transversale en des points reels Pi 02. L'un des points AxA2 sera exterieur a s; soit x1, ce point, et designons par a a2 les points de E 2 28 ME]MOIRE SUR QUELQUES PROBLIMIES [Pt. II. contact de tangentes a S issues de,l; la conique S sera divisee harmoniquement par les couples de points Q 02, a1 a2. En transformant la figure de maniere que aC, a2 deviennent A1, A2, et que 'j, qj2 deviennent 4 12, S deviendra 2, et aux points A,2 correspondront respectivement les points a l'infini sur l'axe imaginaire et sur l'axe reel de 2. Donc les points d'intersection du faisceau transforme appartiendront a une circonference de cercle, qu'on pourra determiner comme precedemment. Soit enfin (30) 2 une parabole, ayant A pour sommet et tangente au point M la ligne droite a l'infini. On coupera toujours le faisceau (S1S2) par une transversale sur laquelle on ait des points doubles reels f/ U2. Soit S la conique du faisceau qui touche la transversale au point!,; de A2 menons la seconde tangente a S; soit a son point de contact. Choisissons a volonte deux points x et X sur les coniques S et I respectivement; soient y et Y deux autres points sur ces memes coniques, tels que le rapport anharmonique des points I4, a, x, y de S soit egal a celui des points M, A, X, Y de 2. En faisant correspondre homographiquement les points M A X Y aux points,u axy, on oura la m6me construction que dans les deux cas precedents. Nous remarquerons qu'apres avoir mene la transversale on est plus restreint, quant au choix de la conique qui doit 6tre transformee en 2, quand cette courbe est une parabole, que quand elle est une conique centrale. Mais, dans le premier cas, la transformation de la conique en Z peut etre opere d'une infinite de manieres diff6rentes. Cette methode offre de grands avantages. On n'opere que sur les coniques du faisceau donne, sans avoir besoin d'aucune autre conique du reseau harmonique; ce qui facilite beaucoup la construction. De plus, on arrive directement a la solution du probleme biquadratique des intersections, et on ne la fait pas dependre du probleme cubique du triangle harmonique. Quand Z est une parabole on peut prendre une droite tangente a l'une des courbes S, et S,, pour la transversale qui coupe le faisceau; en operant ainsi, on n'aura qu'une seule construction quadratique a effectuer; c'est celle qui est inevitable (a ce qu'il parait) quand on veut trouver le rayon du cercle polaire au moyen duquel on resout le probleme. 8. (3.) Quand aucun des points d'intersection de S, et S2 n'est reel, la d6termination du triangle harmonique precede n6cessairement la determination des points d'intersection. En ce cas on peut faire usage de l'un ou de l'autre des procedes suivants. (i.) Prenons une droite tangente a S, en un point quelconque x,; ce point Art. 9.] CUBIQUJES ET BIQUTADRATIQUES. 29 sera un des points doubles de l'involution determinee par le faisceau donnd sur la droite; soit x, le second point double, qui pourra 6tre trouve lineairement; en designant par S2 la conique du faisceau qui passe par x,, on aura deux coniques du faisceau tangentes k la m6me droite reelle. Puisque ces coniques n'ont aucun point reel commun, et qu'elles ont une tangente reelle commune, elles en ont quatre. Soient acr et o2 les coniques reciproques polaires de S1 et S2 par rapport a; a- et -2 se couperont en quatre points reels qu'on determinera par la methode precedente, dont l'application sera tres facile, puisqu'un des quatre points d'intersection etant connu d'avance, il suffira de trouver le centre du cercle polaire pour determiner completement ce cercle. Les polaires des points d'intersection de a-, et 0-2 seront les tangentes communes de S1 et S2, et feront connaitre le triangle harmonique du faisceau. Au lieu de prendre les coniques reciproques polaires de S, et S2, il sera plus facile, en beaucoup de cas, de faire usage de la transformation correlative suivante. En supposant, pour abreger, que I soit une ellipse, soit L un point interieur a SX, soient,1, M2 les rayons doubles reels du faisceau en involution determine au point L par les coniques S1, S2 considerees comme formant un faisceau tangentiel. Soient enfin a1, b1 et a2, b2 les deux couples de tangentes a S. aux points oi6 cette conique coupe les droites x,, 2-. En faisant correspondre correlativement les deux couples de sommets de I aux deux couples de droites a1 bl, a2 b2, on transformera S, en Z, et les quatre tangentes communes de S1 et S2 deviendront quatre points de 2, situes sur une m6me circonf6rence de cercle, dont un sera connu d'avance. On deduira la demonstration de celle de l'article 7. Cette methode pourra servir a trouver directement les tangentes communes de deux coniques donnees, en supposant toutefois qu'il y en a de reelles. (ii.) Deux coniques quelconques du reseau harmonique, qui se coupent en un point reel, se coupent aussi, dans le cas qui nous occupe, en trois autres points reels. On pourra done considerer, au lieu de S, et S2, deux des coniques du reseau harmonique qui passent par un m6me point, et l'on appliquera la methode de larticle 7. Ici encore il y aura l'avantage qu'on connaitra d'avance un des points communs "a et au cercle polaire du reseau transforme. 9. (4.) Les cas du probleme general, ou il s'agit de trouver les intersections d'une conique tracee Z avec une conique non tracee S, m6rite quelque attention. On pourra en ce cas se servir des principes de l'article 7, et l'on transformera le faisceau (S, 2) en un autre faisceau qui contienne a la fois la conique Z et un cercle, soit en transformant S en 2, soit en transformant 2 en elle-m6me. Quand le faisceau (S, 2) determine, sur la ligne droite a l'infini, une involution ayant 30 MEMMOIRE SUR QUELQUES PROBLEMES [pt. III. des points doubles reels (ce qui arrivera toujours si 2 est une ellipse ou une parabole), on pourra op6rer cette derniere transformation de maniere que la ligne droite a l'infini se corresponde a elle-meme. Pour cela, on n'aura qu'a prendre la ligne droite a l'infini pour la transversale L de l'article 7, et la conique 2 elle m6me pour la conique S. Lorsque Z est une conique centrale, en transformera les quatre points d'intersection de S et 2 en quatre points d'une m6me circonference, en transformant Z en elle-meme, de maniere que les points extremes d'un certain couple de diametres conjugues deviennent les sommets de la courbe. Lorsque I est une parabole, on aura une transformation encore plus simple. [Soit A le sommet de 1, a le point de contact de la tangente a I menee parallelement a la polaire par rapport a S du point a l'infini sur l'axe de 2; on n'aura qu'a transformer I homologiquement en elle-m6me, en prenant pour centre d'homologie le point a l'infini sur Aa.] TROISIPME PARTIE. 1. Probleme.-Abaisser d'un point donne P des normales sur une conique completement decrite. Pour abreger le discours nous supposerons que la conique soit centrale. On sait que ce probleme a et6 resolu par Apollonius de Perge, qui a demontre que les pieds des normales se trouvent sur une hyperbole equilatere, passant par P et par le centre de la conique, et ayant ses asymptotes paralleles aux axes principaux de cette courbe. L'hyperbole est le lieu des points d'intersection des diametres de la conique par des perpendiculaires abaissees de P sur les diametres conjugu6s. En employant les diverses methodes prec6dentes, on parviendrait a resoudre ce probleme d'un grand nombre de manieres diffirentes. Mais c'est surtout la belle solution de Joachimsthal qui peut interesser les geometres; c'est pourquoi nous la reproduirons ici avec une demonstration fondee sur les principes precedents. Soit Z la conique trac6e; r l'hyperbole 6quilatere d'Apollonius: S, C les centres respectifs de ces deux courbes. Soit encore A1 A2 l'axe reel, ou l'un des deux axes reels, de I; a le point a l'infini sur cet axe,,3 le point a l'infini sur l'axe conjugue. Qu'on d6signe par 7 un point quelconque de r, et qu'on abaisse du sommet A1 une perpendiculaire sur la droite Py. Cette perpendiculaire, qui sera en mome temps parallele au diametre de Z conjugue a So, coupera i en un second point que nous d6signerons par a-. Il est evident qu'on pourra considerer les deux courbes r et 2 comme homographiques par rapport aux deux series de Art. 1.] CUBIQUES ET BIQUADRATIQUES. 31 points 7 et o. Qu'on transforme F en 2, et que, dans la seconde figure, Z' soit la courbe correspondant a Z dans la premibre. Nous allons voir que les points d'intersection de 2 et 2' appartiendront a une m6me circonf6rence. Aux points a et,B de r correspondront evidemment les points A1 et A2 de 2; d'oh l'on conclut qu'au point C de la premiere figure correspond le point 3 de la seconde. Soit L la droite de la premiere figure qui correspond a la droite a l'infini de la seconde: il faut que cette droite passe par C, puisque le point correspondant de C est a l'infini. En se rappelant la propriete caracteristique des points de r, on verra que, si S est une ellipse, la droite L est l'axe d'homologie de la dyade asymptotique de S et de la dyade des perpendiculaires abaissees de P sur la dyade asymptotique; et que les points imaginaires, oi L est coupee par l'une ou l'autre de ces dyades, appartiennent a r. Soit D le point a l'infini de la droite L; les deux points CD seront evidemment des points conjugues par rapport ] la conique r; done, au point D de la premiere figure il correspondra le point a de la seconde. Mais les points CD sont aussi des points conjugues par rapport a E, puisqu'ils sont harmoniquement conjugues aux points oui la droite L rencontre la dyade asymptotique de 2; d'ouN il s'ensuit que L est parallele a la polaire de C par rapport a 2. Par consequent, les points a, f3 seront des points reciproques par rapport aux deux courbes 2 et 2'; done il passe une circonference de cercle par les points d'intersection de 2 et 2'. La demonstration serait tout aussi simple si l'on supposait que Z fit une hyperbole. Soit Q2 le cercle appartenant au faisceau (2, Z'); pour le determiner completement, il faudra trouver trois couples de points reciproques par rapport a ce faisceau. Pour cela, soit s le point de Z qui correspond au point S de r, et qui se trouve sur la perpendiculaire abaissee de A1 sur PS; les droites sA1, sA2 de la seconde figure correspondront aux droites Sa, S/3 de la premiere, c'est-a-dire, aux deux axes de 2. Qu'on prenne les points reciproques de a et /3 par rapport au systeme des deux courbes 2 et r; les points de la seconde figure qui correspondront a ces deux points reciproques seront evidemment les points d'intersection de la tangente A1/ par sA2, et de la tangente A2f par sA,; points qu'on designera par a, et a2. Donc les deux points A a,, et les deux points Aa, seront reciproques par rapport au systeme (2, 2'). Le cercle 9 coupera orthogonalement les cercles (Aa,) et (A2 a2) ddcrits sur A1 c1 et A2a2 comme diametres; mais il est evident que ces cercles coupent orthogonalement le cercle (A1A2); de plus la droite qui joint leurs centres est precisement la tangente a 2 au point s. On aura donc le beau resultat ddmontr6 par Joachimsthal que les cercles (A A2) et (Q) ont pour axe radical la tangente a Z au point s. 32 MiMOIRE SUR QtELQUES PROBLhMES [Pt. III. Pour achever la determination du cercle Q, il reste a trouver un troisibme couple de points reciproques par rapport a Z et Z'. C'est ce qu'on peut faire d'une infinite de manieres diff6rentes. Soit, par exemple, F le pied de la perpendiculaire abaissee du point P sur la polaire de ce point relativement a 2; P, F seront des points reciproques par rapport au systeme (1, r), puisque PF est la tangente a r au point P. On trouvera le point p de 2 qui correspond au point P de r en menant par le point A une parallele a la polaire de P relativement a 2. Pour trouver le pointf de la seconde figure qui correspond au point F de la premiere, on menera dans la conique I la corde Alf' parallele a la polaire de F; l'intersection de la droite sf' par la tangente a au point p sera le point f cherch6. Le cercle QL coupera orthogonalement le cercle (pf), et se trouvera par cela completement determine. On pourrait aussi se servir de la methode suivante. Qu'on fasse varier le point P sur la droite fixe SP, et qu'on prenne sur cette droite SP' = - SP. Les points s et p resteront fixes; les cercles Q2 passeront toujours par les points d'intersection de la tangente au point s et du cercle (A1 A2). Aux deux points P et P' correspondra le m6me cercle; de plus les cercles l2 varieront anharmoniquement avec les segments de l'involution P1 P1, P2 P', etc. On en tirera la construction suivante. Soit 7r le centre du cercle Q qui passe par le point p; et qu'on designe par p l'intersection de SP par la polaire de P. Le centre du cercle (2 cherch6 sera le point d'intersection de la ligne des centres par une droite men6e de P parallelement a p7r 2. On salt qu'un grand nombre de problemes cubiques et biquadratiques conduisent a l'etude des correspondances determinees par les equations (A yl + 2 B 2 + C1 Y2)+x2 (A2+2B2 y +C2y)=O,.. (1) x (A, y + 2B1 Y Y2 + C1 y2) + 2 x2 (A2 y + 2 B2y Yy2 + C2 y2) + x (A y + 2B3 yl y+ C3 Y2),... (2) xI (A, y- + 3B1 y y2 +- 3 C1 Y Y22 + D1 y3) +x2(A2y'+3B2y y2+3C2y y2+D2y3)=O,... (3) dans lesquelles on peut supposer que les quotients 1, Y1 sont des rapports anX2 Y2 harmoniques qui determinent la position des points variables x et y, soit sur une droite, soit sur une conique. Une thdorie complete de ces correspondances * Note VIII (p. 55). Art. 2.] CUBIQUES ET BIQUADRATIQUES. 33 ddpasserait de beaucoup les limites de ce memoire. I1 nous suffira de placer ici quelques observations qui sont d'une grande importance dans cette theorie. (1.) Soient a, b, c trois points d'une conique determinds par l'6quation cubique F= Ax~ + 3Bx2 x + 3 Cx1 X2 + Dx = O. Soient aussi ABC le triangle circonscrit, L l'axe d'homologie des deux triangles ABC, abc; ac', b', c' les points d'intersection des droites Aa, Bb, Cc par la eonique. Alors les trois points a'b'c' sont les points determines par le covariant cubique de F; et les deux points d'intersection de L par la conique sont determines par le covariant quadratique. Supposons qu'on donne sur une droite trois points P, Q, R, et trois autres points p, q, r harmoniquement deriv6s des premiers par rapport a un systeme F de trois points inconnus. Proposons-nous de determiner (1) le point s harmoniquement derive d'un point donne S par rapport au systeme F, (2) les deux points covariants de ce systeme, (3) le systeme de deux points S1 S2 harmoniquement derives d'un point donne s, (4) les trois points inconnus eux-m6mes. De ces problemes le premier n'est que lin6aire, le second et le troisieme sont quadratiques, le quatrieme est cubique. On projetera les points PQR, pqr sur une conique, en prenant pour centre de projection un point de la conique; nous designerons les points projetes par les memes lettres. Soient P'Q'R' les poles des droites Pp, Qq, Rr relativement a cette conique. Le point X, pole de la droite L, satisfait h l'equation.[P', P, Q', Q, R', R]=[P, p, Q, q, R, r]; done ce point pourra 6tre determine lineairement, puisqu'on pourra trouver deux sections coniques dont il sera le quatrieme poifit d'intersection, les trois autres etant connus. Le point s sera determine lineairement par l'6quation X. [P, Q, R, S]= [p, q, r, s]; de m6me, en supposant que s soit donne, la droite X S, d6termin6e par cette equation anharmonique coupera la conique aux deux points S, S2. Enfin, en prenant un point quelconque o de la conique, et en faisant correspondre anharmoniquement les faisceaux X. [P, Q, R,...] et w. [p, q, r,...], on aura une section conique S qui coupera. au point connu w, et aux trois points inconnus. On pourra se passer, comme on voit, du trace de la conique Z, si l'on ne veut determiner que les points s et X, et la droite L. Dans la solution du probleme cubique on remarquera que le point & peut VOL. II. F 34 ME'MOIRE SUR QUELQUES PROBLUMES [Pt. III. 6tre pris k volont6 sur la conique 2; on pourra m6me determiner d'avance la position de ce point de sorte qu'on puisse faire passer une circonf~rence de cercle par les quatre points w, a, 3, y. Pour cela, on observera que les points w correspondent anharmoniquement aux coniques du faisceau (a, 3, y, X). On prendra trois positions du point w, et l'on determinera les points rectangulaires a l'infini appartenant aux axes principaux des coniques correspondantes (a, 13, y, X, w); ou, plut6t, les trois points p harmoniquement conjugues h un point fixe par rapport a, ces trois systbmes de points rectangulaires. II est 6vident que les points p correspondront anharmoniquement aux points w; done, en ddsignant par a- le conjugue harmonique du point fixe par rapport aux points a l'infini appartenant aux axes principaux de 2, le point w sera ddtermin6 lin6airement par 1'6quation [Pl, P2, P3, ]=[ 1, [ 2, 5 35 2 ]. (2.) Dans l'Iquation (2), qu'on consid6rera relativement h une conique 2, on fera varier le point y, ou ce qui revient au m6me, le rapport y,: y2. Les cercles qui joignent les deux points d6termines par les valeurs correspondantes de x,: x, envelopperont une section conique X. Pareillement, on aura une section conique Y, enveloppe des droites joignant les points y correspondant a une m6me position de x. Soit 0 un point quelconque de Z; soient 1 n2 les deux points qui correspondent h 0, consider6 comme appartenant h la serie des x; 42 les deux points correspondant au m6me point 0, considdr6 comme appartenant a la s6rie des y. Qu'on prenne la corde 0o 02, conjugu6e harmonique de a-0 par rapport aux deux droites 1 72, 6 2 se coupant au point a. Cette corde sera la polaire de 0 par rapport a une troisieme section conique, que nous designerons par 0. Pour les deux coniques X et Y, les droites enveloppantes, ou, si l'on veut, les points de contact sureces droites, correspondront anharmoniquement aux points de la conique 2. I1 y a trois problRmes biquadratiques qui se presentent naturellement, quand on considbre la correspondance doublement quadratique (2). Trouver les quatre points (x) pour lesquels les deux points (y) correspondants deviennent coincidents. Trouver les quctre points dont chacun represente deux points (y), qui sont devenus coincidents. Trouver les points oA le point (x) coincide avec l'un des points (y) correspondants. Qu'on prenne sur la conique z les quatre points de contact des tangentes communes a Z et X; chacun de ces points representera deux points y devenus Art. 2.] CUBIQUES ET BIQUADRATIQUES. 35 coincidents. On en deduira les quatres points x, auxquels correspondent ces quatre points doubles, en se servant de la relation anharmonique que nous avons indiquee. Enfin le point x coincide avec l'un des points y correspondants aux quatres points de rencontre de 2 et O. La thdorie de l'dquation (2) se simplifie, si elle est symetrique relativement aux deux series de points x et y. En ce cas, les deux coniques X et Y coincident l'une avec l'autre, et avec la conique, polaire reciproque de I par rapport a 6. Les points d'intersection des deux coniques X et '2 sont precis6ment les points x pour lesquels les points y correspondants deviennent coincidents; et les points de contact sur 2 des tangentes communes de 2 et X sont ces doubles points. En supposant toujours que l'equation (2) soit symetrique, prenons les points yY2 correspondant a un point quelconque x: soit x' l'un de ces deux points; l'un des deux points correspondant a x' sera x, l'autre sera un nouveau point x". Determinons successivement de la meme maniere les points x"', x",...; il peut arriver, comme on sait, qu'apres un nombre fini d'operations on retombe a la fin sur le point de depart x. Ces cas particuliers ont et6 beaucoup etudies par les g6ometres; mais c'est surtout le cas oi l'on aurait x"' = x, qui est important (ainsi que nous allons voir) pour la theorie des problemes du troisieme et du quatrieme ordre. Les problemes lineaires et quadratiques qui se rattachent a l'dquation (2) peuvent se resoudre en beaucoup de cas par les methodes connues. Par exemple, l'on voit qu'8tant donnes huit points x, et un point y correspondant a chacun de ces points, les deux points y correspondant a un point x quelconque doivent s'obtenir par une construction quadratique. Et, en effet, on peut operer cette construction, en se servant des propridt6s des courbes du quatrieme ordre, ayant deux points doubles. (3.) Considdrons la correspondance d6finie par l'6quation (3); et supposons que cette equation soit relative a une conique Z. A chaque point x correspondront trois points de la s6rie y: et l'on aura ainsi une s6rie de triangles inscrits a 2. Mais cette s6rie de triangles sera en m6me temps circonscrite a une seconde conique; elle sera de plus une serie de triangles harmoniques par rapport a une troisieme conique. Ainsi, cette sdrie de triangles pourra 6tre d6finie par une correspondance quadratique double, qui sera sym6trique, et dans laquelle le troisieme element d6rivd coincidera avec l'61ement d'ou l'on est parti. Il s'ensuit que la plupart des questions relatives a l'involution cubique definie par l'dquation (3) pourront 6tre r6duites aux recherches analogues relatives a cette espbce particulibre de correspondances quadratiques doubles *. * Note IX (p. 59). F 2 36 MMMOIRE SUR QUELQUES PROBLIMES [Pt. III. 3. Nous designerons par [a, b, c, d] le quotient ad:. bc qui est un des rapports anharmoniqueses quatre points a, b, c, d en ligne droite; de m6me, nous repr6 -sin aPc sin bPc senterons par P. [a, b, c, d] le rapport anharmonique sin Pd sin bPd des quatre sin aPd sin bPd droites Pa, Pb, Pc, Pd se coupant au m6me point P *. Cela pose, nous aurons les deux lemmes suivants. Lemme I. Soient Pp, Pp1, Pp2, Qq, Qq1, Qq, Rr, Rr, Rr, neuf droites donnees; le lieu d'un point x, qui satisfait a l'equation P. [x,, 1, 2] X Q.[x,, q, q2] x R. [x, r, r, r2]=,.. (1) a etant une constante, est une courbe cubique passant par les neuf points d'intersection des droites Pp1, Qq1, Rr1 avec les droites Pp2, Qq2, Rr2. Demonstration. (1~). Le lieu du point x qui satisfait a l'equation Q. [, q, ql, q2] X R. [x, r, r1, r2 =.... (2) JL etant une constante, est une conique (,u) passant par les quatre points d'intersection des droites Qq,, Rr1 avec les droites Qq2, Rr2. Soit Rp le rayon du faisceau (R) qui satisfait a l'equation R. [p, r, r, r2] =.......... (3) En divisant membre B membre l'equation (2) par l'6quation (3), on aura Q. [x, q, q, 2] x R. [x, p, r, r2] = 1, ou, ce qui revient au meme, R. [x, q,, q2] =. [x, p, r r1], equation qui demontre ce qui a ete avance. * En general, si a, b, c, d sont quatre 1eements quelconques, dont on peut definir le rapport anharmonique (par exemple, quatre points d'un meme conique, ou quatre courbes d'un meme faisceau) nous exprimerons ce rapport pas la formule [a, b, c, d]; et, toutes les fois qu'il sera necessaire de distinguer entr'eux les divers rapports anharmoniques du meme systeme de quatre elements, la formule [a, b, c, d] designera pour nous le rapport anharmonique analogue au rapport -- b de quatre points, ad bd en ligne droite. Nous nous servirons, avec quelques g6ometres, des parentheses pous exprimer des courbes passant par des points donnes; ainsi (a, b) sera la droite qui joint les points a, b; (a, b, c, d, e) sera la conique des cinq points a, b, c, d, e. Si P est la base d'un faisceau de courbes d'ordre quelconque, (P, a) sera la courbe de ce faisceau qui passe par le point a; et P. [a, b, c, d] sera le rapport anharmonique des quatre courbes (P, a), (P, b), (P, c), (P, d). Art. 3.] CUBIQUES ET BIQUADRATIQUES. 37 (20.) En donnant a,u des valeurs successives diff6rentes, les coniques (,u) correspondantes, lieux des points x, seront toutes circonscrites au m6me quadrilatere. Or, ces coniques correspondront anharmoniquement aux valeurs de,u. En effet, supposons que Qx soit une direction fixe; on sait que les coniques correspondront anharmoniquement aux points oh elles coupent cette droite. En d6signant par p, rT, T2 les points d'intersection de Qx par Rp, BRr, Rr2, la position du point x, appartenant a la conique ()u), sera determinee par l'equation [x, p, r2, r] = Q. [x,, q1, q2], dont le second membre est une constante C. En multipliant les deux membres de cette equation par les deux membres de l'equation = [p,,, r, r2], on aura [X, r,, = C d'oui il s'ensuit que le rapport anharmonique de quatre points x est egal au rapport anharmonique des valeurs correspondantes de u. (3.) La position de la droite Px, qui satisfait a 1'equation P. [x, p, p1, 2] P2 variera anharmoniquement avec les valeurs de u. Done le lieu des points d'intersection d'une conique (,u) par la droite correspondant a la meme valeur de u sera une courbe cubique; mais il est evident que ce sont precisement ces points d'intersection qui satisfont a l'6quation (1). On voit d'ailleurs que la courbe cubique passera par cinq des neuf points d'intersection des droites donnees; elle passera aussi par les quatre autres, puisque dans la demonstration on peut echanger entre eux les faisceaux (P), (Q), (R). Lemme II. Soient S1, S2, S3 trois courbes du m6me ordre; A1, A2, A3, B1, B, B3 les courbes des faisceaux (S2, S3), (S3, S), (S1, S) qui passent par les points a et 3 respectivement; on aura l'equation [S2, S3, A4, B]x[,S, S1, A,, B2] x [S, S, A3, B] = + 1... (3) Demnonstration. Puisqu'il y a toujours une courbe du faisceau (S2, S3) qui appartient en m6me temps au faisceau determine par deux courbes quelconques des faisceaux (3, S1), (S,, S2), il y a une courbe commune aux faisceaux (S2, 53), (A2, A3). Mais cette courbe commune ne peut 6tre autre que A,, puisque A, est une courbe du faisceau (S2, S3), et qu'elle passe par a, un des points d'intersection 38 3MEMOIRE SUR QUELQUES PROBLIMES [Pt. III. de A2, A3. Donc les trois courbes A1, A2, A3 appartiennent au m6me faisceau; le m6me raisonnement s'applique aux trois courbes B1, B2, B3. Soient a et 3 les points oti vont concourir les droites polaires d'un m6me point 0 par rapport a A1, A2, A3, et a B1, B2, B3 respectivement; soient aussi a-, o2, 0- les points d'intersection de (a, 3) par les polaires de 0 relativement h SI, S2, 53; on aura evidemment [S2, S3, A1, B1]=[C2, 03, a, a] [S3, S1, A2, B2]=[o3, C a, /3] [^S, X2, A3, B3]-[bl, [2r a2 I J], valeurs qui satisfont identiquement a 1'equation (3). Avant de terminer ces preliminaires nous rappellerons qu'6tant donnd trois points sur chacune de deux cubiques, et en outre six des neuf points d'intersection des deux courbes, on trouve aisement les trois autres points d'intersection par une construction cubique qu'on doit a M. Chasles. Soient 1, 2,..., 6 les six points d'intersection donnes, 7, 8, 9 les trois points cherches; on d6termine lineairement deux coniques telles que (5, 6, 7, 8, 9), (4, 6, 7, 8, 9), dont les quatre points d'intersection sont le point connu 6, et les trois points cherches. Pareillement, etant donne trois des points d'intersection d'une conique et d'une cubique, qu'on suppose determinees toutes les deux par un nombre suffisant de points, on trouvera les trois autres points d'intersection par une construction cubique facile. Soient, b, c les points dons,,, des points oesdonnes de la cubique et de la conique respectivement, to le point oppose au systeme a, b, c, d, relativement a la cubique. En designant par x un point quelconque de la cubique, les coniques (a, b, c, d, x) et les droites (w, x) se correspondront anharmoniquement. Soit t le quatrieme point d'intersection de la conique donnee par (a, b, c, d, x); les deux faisceaux (S, r), (w, x) seront homographiques, et la conique, lieu des points d'intersections des rayons correspondants, coupera la conique donnde au point connu S et, en outre, aux trois points cherches. 4. PROBLUME. Stant donne' treize des points d'intersection de deux courbes du quatrieme ordre, trouver les trois autres. Nous supposerons que les treize points soient tels qu'on peut faire passer actuellement par ces points une vraie courbe du quatribme ordre. Nous exclurons donc absolument les cas ou l'on aurait, soit cinq points en ligne droite, soit neuf points sur une m6me conique, soit treize points sur une m6me cubique. Mais, afin de simplifier la discussion g6n6rale, nous en exclurons aussi, pour le moment, les cas ot l'on aurait Art. 4.] CUBIQUES ET BIQUADRATIQUES. 39 (1) Neuf points formant la base d'un faisceau de cubiques. (2) Huit points sur une m6me conique. (3) Onze points sur une m6me cubique. (4) Quatre points en ligne droite. Pour tous ces cas la solution g6nerale se simplifie plus ou moins; nous les consid6rerons separement plus tard. Enfin, nous supposerons que les treize points soient tous reels, et tous differents; nous reviendrons ci-aprbs sur les cas oi l'on aurait des points imaginaires. Soient 1, 2, 3,..., 13 les treize points donnes, 14, 15, 16 les points qu'il s'agit de trouver. Prenons six points quelconques des treize points, par exemple les points 8, 9, 10, 11, 12, 13; nous allons montrer comment on peut determiner la cubique qui passe par ces six points, et par les trois points inconnus. Pour cela, nous prenons un quelconque des six points que nous avons choisis, par exemple le point 8; nous le joignons aux sept points 1,..., 7, et nous considerons le systbme des huit points 1,..., 8 comme formant la base P, d'un faisceau de courbes cubiques. Determinons le point p8, de sorte que les cinq droites (Ps, 9), (Ps, 10), (8s, 11), (8s, 12), (P8, 13) correspondent anharmoniquement aux courbes cubiques (P8, 9), (P8, 10), (Ps, 11), (P8, 12), (P8, 13). La determination du point P8 se fera lineairement par une construction sur laquelle nous reviendrons plus tard; nous dirons que ce point est biquadratiquement oppose aux points 1,..., 8 de la base P8. La courbe du quatrieme ordre, qu'on peut faire passer par les treize points et par le point P8, aura pour faisceaux gen6rateurs le faisceau de droites (p,), et le faisceau de courbes cubiques (P,); de plus, les droites (P8, 14), (P8, 15), (P8, 16) du premier faisceau correspondront aux courbes cubiques (Ps, 14), (Ps, 15), (Ps, 16) du second faisceau, puisque 14, 15, 16 sont des points de la courbe du quatrieme ordre. Substituons successivement au point 8 deux autres points du systeme de six points, par exemple les points 9 et 10; soient P9, P,, les bases cubiques qu'on aura ainsi, p,, p,, les points biquadratiquement oppos6s a ces bases; nous allons 40 MiMOIRE SUR QUELQUES PROBLIMES [Pt. III. voir que les trois points oppos6s p8, P9, po0, et les trois points d'intersection des trois couples de droites p9 (10), Pio (9); pio (8), P8 (10); P8 (9), 9 (8), sont des points de la cubique cherchee, qui sera des lors completement determin6e puisqu'on en connaitra douze points. Considerons la courbe cubique, lieu des points x qui satisfont a l'equation p8. [x, 11, 9, 10] xp9. [x, 11, 10, 8] xp10. [x, 18, 9]= +1. D'apres le lemme I, cette courbe passe par les points _p8, p9, Pio, 8, 9, 10, et par les trois points d'intersection des trois couples de droites (p9, 10), (pl0, 9); (Pi1, 8), (P8, 10); (P8, 9), (]9, 8); elle passe en outre par le point 11, puisque chacun des trois rapports anharmoniques devient egal a l'unit6 positive si l'on fait coincider x avec ce point. Mais les points 12, 13, 14, 15, 16 appartiennent aussi a la m6me courbe. En effet, soit un quelconque de ces points; d'apres la relation anharmonique qui subsiste entre les faisceaux (p8), (Ps); (pg), (P9); (plo), (Plo), on aura 8s *[, 11, 9, 10]=P8.[, 11, 9, 10], p,. [ll,, 10, 8]=P.[ 11, 10, s 8], p1o.[11, 8, 9]=Plo. [ll, 8, 9]. Mais le produit des seconds membres de ces equations est l'unitd positive; comme il resulte du lemme II, en y ecrivant 1 = (P, 10)=(PIo, 9), S2=(Plo, 8)=(P8, 10), S3=(P8, 9)=(P9, 8). Il resulte de ce qui precede que la cubique, qui passe par les neuf points 8,..., 16, passe aussi par les neuf points biquadratiquement opposes aux systemes de huit points, qu'on obtient en joignant successivement aux sept points 1,..., 7 chacun des huit points 8,..., 16; et par les trente-six points d'intersection des couples de droites (pa, P), (ps, a), en designant par a, f3 deux nombres inegaux de la s6rie 9,..., 16. De ces cinquante-quatre points, on en connaitra ving-sept, qui ne dependent pas des points inconnus 14, 15, 16. Pour avoir une autre cubique passant par ces trois points, nous remarquons que, par hypothbse, la cubique C7 des neuf points 8,..., 16, ne peut pas passer Art. 5.] CUBIQUES ET BIQUADRATIQUES. 41 par tous les sept points 1,..., 7. Soit done 7 un de ces points qui n'appartient pas a C0; on echangera entr'eux dans la construction precedente le point 7 et un point quelconque 8 des six points 8,..., 13; et l'on determinera ainsi la cubique C2 des neuf points 7, 9,..., 16. Les cinq points 9,..., 13 seront des points communs aux deux courbes CQ et C2; le point biquadratiquement oppose aux points 1,..., 8 sera un sixibme point commun; enfin, les points qu'il s'agit de trouver seront precisement les trois autres points communs. Puisque par hypothese les huit points 1,..., 8 n'appartiennent pas tous h une m6me conique, on pourra dire autant des huit points 9,..., 16. Soient 11, 12, 13 trois des points 9,..., 13 qui n'appartiennent pas a une mnme conique avec les points inconnus. Qu'on determine les deux coniques (12, 13,..., 16) et (11, 13,..., 16), se coupant au point connu 13; les trois autres points d'intersection de ces courbes seront finalement les points cherches. On remarquera que les deux courbes du quatrieme ordre dont nous nous sommes servis dans la demonstration precedente, ne sont pas deux courbes quelconques du faisceau determine par les treize points donnes. Chacune des deux courbes est assujettie a passer par le neuvieme point appartenant a la base cubique formee par un systeme de huit points choisis parmi les treize points donn6s *. 5. La solution precedente depend essentiellement de la determination des points biquadratiquement opposes aux divers systemes de huit points qu'on peut former avec les treize points donnds. Soit P = [1, 2, 3,..., 8] l'un quelconque de ces systemes; pour avoir le point oppose biquadratiquement a P, il nous faudra avant tout un systeme de cinq points, ou de cinq droites, qui correspondent anharmoniquement aux cubiques (P, 9), (P, 10), (P, 11), (P, 12), (P, 13). A cet effet, on pourrait se servir, comme on sait, soit des tangentes a ces courbes en un point quelconque de la base P, soit de leurs points d'intersection par une des droites qui joignent deux des points P, soit enfin de cinq droites, polaires d'un m6me point par rapport aux cinq courbes. Mais nous preferons la methode suivante, qui conduit a des operations moins penibles. On choisira parmi les points P un systeme Q de quatre points quelconques 1, 2, 3, 4, et l'on ddterminera pour chacune des cinq cubiques le point oppose au systeme Q. Pour cela on determinera la conique r qui passe par les quatre points 5, 6, 7, 8, et qui admet le rapport anharmonique Q. [5, 6, 7, 8]; puis, on prendra sur cette conique des points 9', 10', 11', 12', 13', tels qu'on ait [5, 6, 7, 8, 9', 10', 11', 12', 13']= Q.[5, 6, 7, 8, 9, 10, 11, 12, 13]; * Note X (p. 63). VOL. II. G 42 MEMOIRE SUR QUELQUES PROBLiMES [Pt. III. les intersections de F par les droites (9, 9'), (10, 10'), (11, 11'), (12, 12'), (13, 13') seront les points opposes a Q, appartenant respectivement aux cubiques (P, 9), (P, 10), (P 1, 1 (P ), (P, ) ( 13). Nous les designerons par o9, wc, co 1,,12 013, et nous nous en servirons pour un systeme de points correspondants anharmoniquement aux cinq cubiques. Ensuite, on determinera la conique Z qui passe par quatre points quelconques 9, 10, 11, 12 des cinq points 9, 10, 11, 12, 13, et qui admet le rapport anharmonique [9, 0, o 1, 12]. Soit 13" le point de cette conique qui satisfait a l'equation [9, 10, 11, 12, 13"]=[Wg, o10, W12, 1, 3]; le point d'intersection de Z par la droite (13, 13") est le point oppose biquadratiquement au systbme P. On voit que la construction revient au fond a la construction si connue du point oppose a un systeme de quatre points appartenant a une courbe cubique. La construction ne reussit pas si trois des points Q sont en ligne droite, mais elle ne devient que plus facile si trois des points 5, 6, 7, 8, ou bien trois des points 9, 10, 11, 12, 13 sont en ligne droite, puisqu'alors l'une ou l'autre des coniques r, Z est remplacee par un systbme de deux droites. On peut done toujours faire en sorte que le systeme Q ne contienne pas trois points en ligne droite. Cependant, s'il y avait quatre points en ligne droite parmi les points P, on ferait bien de les prendre pour le systeme Q, puisque en ce cas la determination du point biquadratiquement oppose au systeme P se r6duirait tout simplement a la determination du point oppose au systeme des quatre points 5, 6, 7, 8, relativement a la courbe cubique (5, 6,..., 13). Encore, s'il y avait quatre des cinq points 9, 10, 11, 12, 13 en ligne droite, la construction precedente ne serait plus applicable. En effet, dans ce cas il n'y a aucun point oppose biquadratiquement au systbme P, mnoins que la condition P. [9, 10, 11, 12]=[9, 10, 11, 12] ne soit satisfaite par les quatre points en ligne droite. En supposant que cette condition eut lieu, on determinerait sur la droite (9,..., 12) le point 13' qui satisfait a l'6quation P.[9, 10, 11, 12, 13]=[9, 10, 11, 12, 13']; et on trouverait que tout point de la droite (13, 13') aurait la propri6te caracteristique d'un point oppos6 biquadratiquement au systeme P. Art. 6.] CUBIQUES ET BIQUADRATIQUES. 43 La construction cesse encore d'etre applicable si les points P et trois des points 9,..., 13 appartiennent a une m6me cubique. Elle deviendrait indeterminee si l'un des cinq points 9,..., 13 etait le neuvieme point appartenant a la base cubique P. Ainsi en supposant que 13 appartint a cette base, tout point de la conique Z serait oppose biquadratiquement au systeme P. C'est ce qui arriverait aussi si l'on avait la relation P. [9,..., 13] =[9,..., 13], ce dernier symbole se rapportant a la conique qu'on peut mener par les cinq points. 6. Nous allons maintenant revenir sur les cas particuliers que nous avons exclus de la discussion generale (Art. 4). Dans tous ces cas, comme on a pu voir par ce qui precede, on pourra simplifier la determination des points opposes biquadratiquement a certains systemes de huit points, si toutefois ces points ne cessent pas d'exister. (1.) Supposons que les neuf points 1, 2,..., 9 forment la base P d'un faisceau de courbes cubiques. La conique 2, qui passe par les points 10,..., 13, et admet le rapport anharmonique P. [10, 11, 12, 13] passera aussi par les points 14, 15, 16. Car on pourra faire passer par les treize points et par un point quelconque cr de cette conique, une courbe S du quatrieme ordre qui aura pour faisceaux generateurs le faisceau de cubiques (P), et le faisceau de droites (c). Les courbes 2 et S se couperont en huit points, dont a, 10, 11, 12, 13 seront cinq; nous allons voir que les trois autres seront precisement les points inconnus 14, 15, 16. Soit, en effet, x un des trois points d'intersection de I et 8, autres que a-, 10, 11, 12, 13; en prenant x pour centre du faisceau generateur, on aura une courbe X du quatrieme ordre, qui passera par x et par les treize points, mais qui ne pourra pas coincider avec 8, puisque, en designant par a un point quelconque qui n'appartient pas a la conique 2, les deux rapports anharmoniques x.[10, 11, 12, a], -.[10, 11, 12, a] ne sauraient 6tre egaux. Done le point x est bien un des trois points 14, 15, 16, puisq'il appartient en m6me temps aux deux courbes S et X. On arriverait au m6me -resultat en s'appuyant sur la proposition generale de larticle 4, d'oi l'on conclurait qu'un point quelconque de Z appartient a la cubique determinee par les sept points 10,..., 16 et deux quelconques des points P; c'est-a-dire que cette cubique est composee de Z et de la droite qui joint les deux points. Pour completer la solution du problkme, on determinera par la methode generale une cubique qui passera par les trois points inconnus, par trois des points 10,..., 13, G 2 44 MEMOIRE SUR QUELQUES PROBLiMES [Pt. III. et par trois des points P. Trois des points d'intersection de cette cubique par 2 seront connus; done on trouvera les trois autres par la construction cubique que nous avons deja indiquee. Nous ferons remarquer que si les points 10,..., 13 appartiennent, deux a deux, a deux courbes cubiques du faisceau (P), la solution du probleme sera lineaire. En effet, soient (P, 10, 11), (P, 12, 13) les deux cubiques; on pourra considerer la droite (12, 13) et la cubique (P, 10, 11), prises ensemble, comme une courbe du quatrieme ordre passant par les treize points; pareillement la droite (10, 11) et la cubique (P, 12, 13) composeront une autre courbe du quatrieme ordre passant par les m6mes points; done le point d'intersection de (10, 11) et (12, 13) sera un des points cherches; et l'un des deux autres sera le troisieme point d'intersection de la cubique (P, 10, 11) par (10, 11), point qu'on sait d6terminer lineairement. La construction ne sera que quadratique, si trois des points qui n'appartiennent pas a P sont en ligne droite. Soient 10, 11, 12 trois points d'une m6me droite X; soit aussi a le point de X qui satisfait a l'6quation [10, 11, 12, a]= P. [10, 11, 12, 13]; la conique Z sera remplac6e par l'ensemble des deux droites X, et (13, a). Qu'on determine par la m6thode generale ci-dessus une cubique qui passe par les trois cherches, par trois des points P, par deux des points 10, 11, 12, et enfin par 13; un des points cherches sera le troisieme point d'intersection de la cubique par X; celui-ci se trouvera lineairement; les deux autres seront les deux points d'intersection, autres que 13, de la droite (13, a) par la cubique; on les aura par une construction quadratique. Enfin, si tous les quatre points 10, 11, 12, 13 appartiennent a la meme droite, il n'y aura aucune vraie courbe du quatrieme ordre qui pourra passer par les treize points donnes, a moins que la condition P.[10, 11, 12, 13]=[10, 11, 12, 13] ne soit verifiee; done tous les fois que cette relation ne subsistera pas, ce cas sera un de ceux que nous devons rejeter. De plus, quand m6me la condition se trouverait r6alisee, on pourra faire abstraction du point 13, puisque toute courbe du quatrieme ordre qui passe par les points 1,..., 12, passera necessairement par ce point. II faudra done qu'un quatorzieme point soit donn6 pour que les seize points soient completement determines; mais ce point etant donne, les deux autres se trouveront par la construction quadratique pr6cddente. Nous avons deja remarqu6 qu'en designant par P le systeme 1,..., 8, tout Art. 6.] CUBIQUES ET BIQUADRATIQUES. 45 point appartenant k la conique (9,..., 13) est oppos6 biquadratiquement h P, si l'on a la relation P.[9,, 10, 11, 12, 13]=[9, 10, 11, 12, 13]. Ce cas se reduit immediatement a celui que nous venons de traiter. En effet, le neuvibme point appartenant a la base cubique P, est 6videmment un des trois points cherches, puisqu'il est un point commun B deux courbes du quatrieme ordre, passant toutes les deux par les treize points, et ayant le m6me faisceau generateur de courbes cubiques (P), mais ayant des points diff6rents biquadratiquement oppos6s h ces faisceaux. Donc on trouvera lineairement un des trois points 14, 15, 16; on obtiendra les autres par une construction quadratique, puisqu'on connaitra quatre des points d'intersection de la conique (9,..., 13) par une cubique qu'on fera passer par les deux points inconnus, par quatre des points 9,..., 13, et par trois des points de la base cubique. (2.) Supposons que les huit points 1,..., 8 appartiennent tous a une meme conique (P). La conique c = (9, 10, 11, 12, 13) sera une premiere conique passant par les points inconnus 14, 15, 16. Soit C une cubique determinee par les trois points inconnus, par trois des points 9,..., 12, et par trois des points P. On pourra determiner cette cubique par la methode generale, et puisqu'on connaltra trois des points d'intersection de C par cr, on pourra trouver une seconde conique a-', qui coupera a- en quatre points, dont un sera connu d'avance, tandis que les autres seront les points cherches. (3.) Supposons que onze des treize points appartiennent a une m6me cubique (P); soient 1, 2,..., 11 ces onze points, que nous designerons par P. On sait que toute courbe du quatrieme ordre, qui passe par onze points d'une cubique, rencontre la cubique en un douzieme point fixe. Ce point fixe sera un des trois points cherches; nous le designerons par 14; il pourra se determiner lin6airement; de plus, cette determination d6pendra uniquement des onze points P, et nullement des points 12, 13. On pourra done substituer a ces deux points deux autres points quelconques choisis a volonte, pourvu qu'ils ne soient pas situes sur la cubique (P). Nous prendrons actuellement, au lieu de 12 et 13, deux points 12' et 13' qui forment avec sept des onze points les neuf points basiques Q d'un faisceau de courbes cubiques, auquel la cubique (P) n'appartient pas. Soient 1, 2,..., 7 les sept points; on prendra pour 12' un point quelconque qui n'appartient pas a (P); 13' sera le neuvieme point appartenant a la base cubique 1,..., 7, 12', mais il ne sera pas necessaire de le trouver, quisqu'on n'en fait aucun usage dans la construction. On determinera la conique qui passe par les quatre points 8, 9, 10, 11, et qui admet le rapport anharmonique Q.[8, 9, 10, 11]: cette co 46 MEMOIRE SUR QUELQUES PROBLaMES [Pt. III. nique passera par le point 14 cherche, puisqu'elle doit passer (Art. 6, 1) par les trois points qui completent la base biquadratique 1,..., 11, 12', 13'. En echangeant entre eux un des quatre points et un des sept points, on aura une seconde conique, qui coupera la premibre en trois points connus d'avance; le quatribme point d'intersection sera le point 14. Pour trouver les points 15 et 16, on considerera un systeme de huit points Q, compose des deux points 12 et 13, et de six points quelconques des points P. Soient 1,..., 6 ces six points, et designons par w le point biquadratiquement oppose a Q. La cubique (P) et la droite (12, 13) composent une courbe du quatribme ordre passant par les treize points donnds. Donc les points 15, 16 sont les deux points d'intersection (autres que 12, 13) de la droite (12, 13) par la courbe du quatribme ordre qui a pour faisceaux generateurs le faisceau de cubiques (S2) et le faisceau de droites (w). Donc enfin les points 15, 16 sont les points doubles des deux divisions homographiques determinees par les deux faisceaux sur la droite (12, 13). Si douze des points donnes se trouvaient sur une m6me cubique, il faudrait qu'un quelconque de ces douze points fdt determine par les onze autres de la nanibre que nous avons indiquee; autrement on ne pourrait faire passer aucune vraie courbe du quatribme ordre par ces points. Mais en ce cas il faudrait aussi qu'un quatorzieme point fhit donne, afin de determiner completement le systeme des seize points; alors la construction des points 15 et 16 serait la m6me que cidessus. (4.) Lorsque quatre des treize points sont en ligne droite, on aura tout d'abord une cubique passant par les trois points inconnus, puisque ces points appartiendront evidemment a la cubique des neuf autres points. Mais on peut aussi operer de la manibre suivante, sans faire usage de cette cubique. Soient 10, 11, 12, 13 les quatre points en ligne droite; les points P^o, P11, p12, P13 pourront se determiner par la methode gendrale de l'article 5. Ces points, ainsi que les points d'intersection des droites (P0i, 11), (p11, 10), etc., appartiendront a la conique (8, 9, 14, 15, 16), qui se trouvera ainsi completement determinee. De meme on pourra determiner la conique (7, 9, 14, 15, 16), dont les intersections avec la premiere conique feront connaitre la solution du probleme. 7. I1 y a encore quelques cas particuliers du probleme qui ne sont pas d6pourvus d'int6ret, mais dont la discussion, d'ailleurs tres facile, depasserait les limites que nous nous sommes prescrites. Mais nous ne saurions nous dispenser de placer ici les observations suivantes qui serviront a eclaircir la solution gendrale. (1.) La determination de la cubique (8,..., 16) exige la connaissance de deux Art. 7.] CUBIQUES ET BIQUADRATIQUES. 47 seulement des points Ps, p9,..., P13. En effet, quand on aura trouv6 les deux points p, et p9 on aura neuf points de la cubique, puisque le point d'intersection des droites (Ps, 9), (ps, 8) appartient aussi a cette courbe. De plus, tant que le probleme reste cubique, il ne peut arriver que les neuf points forment la base d'un faisceau de courbes cubiques. Pour que le point d'intersection de (p8, 9), (P9, 8) fAt le neuvibme point appartenant a la base cubique 8,..., 13, P8, P, ii faudrait que les six points ps, 9,..., 13 appartinssent a une m6me conique. Or la conique (P8, 10,..., 13) n'est autre que la conique qui satisfait a la relation P. [10, 11, 12, 13] = [10, 11, 12, 13]; en outre, x etant le point de cette conique qui verifie l'equation P. [9, 10, 11, 12, 13]=[x, 10, 11, 12, 13], P8 sera le second point d'intersection de la droite (x, 9) avec la conique. Done, si le point 9 appartient lui-meme a la conique, P8 viendra se confondre avec 9, mais la position limite de la droite qui joindra ces deux points coincidents sera toujours la droite (x, 9); d'ohi il s'ensuit que les six points P8, 9,..., 13 (dont les deux premiers sont coincidents) ne peuvent pas 6tre censes appartenir a une meme conique, a mooins que le point x ne coincide avec 9. Mais si cela arrivait, l'6quation P8.[9, 10, 11, 12, 13]=[9, 10, 11, 12, 13 serait satisfaite; c'est-a-dire, le neuvieme point appartenant a la base Ps serait un des trois points cherches, et la determination des deux autres ne serait que quadratique. On conclura aussi, de ce qui vient d'etre dit, que si Ps, l'un des deux points opposes qu'on aura a determiner, venait a coincider avec 'un des points 9,..., 10, la cubique (8,..., 16) n'en serait pas moins completement determinee, puisqu'on aurait remplace deux points par une tangente et son point de contact. Nous ajouterons que si les deux points P8 et p9, tout en restant d6termines l'un et l'autre, venaient a se confondre en un seul point p, ce point serait un des trois points cherches; puisqu'en considerant successivement les deux bases Ps et Pg (qui auraient le m6me point oppose p, mais qui ne pourraient pas etre identiques, parceque nous avons suppose que la position du point oppose p n'est pas indeterminee), on aurait deux courbes du quatrieme ordre, passant par les treize points, et se coupant en outre au point p. Done on conclura generalement qu'afin d'avoir les deux cubiques, dont on a besoin pour determiner les trois points cherches, il suffira de trouver trois points 48 MEMOIRE SUR QUELQUES PROBLiMES [Pt. II. opposes biquadratiquement a trois systbmes de huit points, choisis convenablement parmi les treize points donnes. (2.) Puisque les deux courbes cubiques (8,..., 16) et (7, 9,..., 16) se coupent au point p8, il est evident que p8 est le neuvibme point qui appartient a la base cubique 9,..., 16. Nous aurons done le theoreme que voici:' Si l'on partage les seize points d'une base biquadratique en deux systbmes de huit points, le point biquadratiquement oppose a l'un de ces systbmes appartient en meme temps a la base cubique determinee par l'autre systbme. La courbe du faisceau qui passe par le neuvieme point appartenant N l'un des deux systames, passe aussi par le neuvieme point appartenant a l'autre systame.' 8. Nous allons maintenant supposer que quelques uns des treize points deviennent imaginaires. I1 suffira de consid6rer les deux cas (1) oi l'on n'aurait que trois points reels, (2) oi l'on n'aurait qu'un seul point reel. Pour abreger, nous supposerons que la position des treize points soit tout-8-fait generale, et nous ne nous occuperons pas encore des circonstances spdciales que nous avons considerees dans l'article 6. (1.) Supposons que 1, 2; 3, 4; 5, 6; 10, 11; 12, 13 soient des dyades de points imaginaires, mais 7, 8, 9 soient des points reels.. La determination des points a, b, c biquadratiquement opposes aux bases cubiques (1,..., 6, 8, 9), (1,..., 6, 9, 7), (1,..., 6, 7, 8), que nous designerons par A, B, C, se fera a peu pres comme si les treize points 6taient tous reels. En effet, d'apres ce que nous avons dit dans la premiere partie de ce memoire, on saura determiner (1) la conique F de l'article 5; (2) le point reel w)9 et les dyades woj, W 12 os13 appartenant a cette conique et correspondant anharmoniquement a la cubique reelle (C, 9) et aux cubiques imaginaires conjuguees (C, 10), (C, 11) et (C, 12), (C, 13); (3) la conique 2 qui passe par les points 10,..., 13, et qui satisfait a l'equation,,, ]=[o, 3=[10, 11, 12 13][w ]=C.[ 1, 12, 13]; (4) le point 9' de cette conique qui verifie la relation [9', 10, 11, 12, 13]=[9o, o, 1, 1, w13] =C. [9, 10, 11, 12, 13]; (5) enfin, le point c cherche, oui la droite (9, 9') rencontre pour la seconde fois la conique Z. Les points a, b, c une fois trouves, les cubiques (8, 9, 10,.., 16), (9, 7, 10,..., 16), (7, 8, 10,..., 16), dont deux suffisent pour notre but, seront completement determinees. En effet, on connaitra neuf points, dont trois sont reels, de chacune de ces courbes; et nous avons vu que ces neuf points ne peuvent pas appartenir a une m6me base cubique. (2.) On etendra la solution au cas ot l'on n'aurait qu'un seul point reel 7, au Art. 8.] CUBIQUES ET BIQUADRATIQUES. 49 moyen des principes generaux que nous avons etablis dans la premiere partie. La cubique (8,..., 16) sera reelle; la determination de cette courbe se reduira a celle des points b, c qui seront des points imaginaires conjugues. Pour les trouver il faudra substituer dans la solution precedente la dyade 8, 9 aux deux points reels que nous avons designes par les m6mes nombres. Quoiqu'en cette solution il soit question de points imaginaires, elle ne consiste actuellement que d'une certaine suite d'operations lineaires, portant sur des points et des droites reelles. Done, en substituant aux points reels 8, 9 la dyade de points imaginaires, on parviendra a determiner lineairement les deux points imaginaires c, b biquadratiquement opposes aux systemes 1,..., 7, 8 et 1,..., 7, 9; il est d'ailleurs evident que ces deux points appartiendront a la m6me dyade, dont on connaltra l'homologie avec la dyade 8, 9. Le centre d'homologie des deux dyades appartiendra lui-m6me a la cubique cherchee; il sera le seul point reel qu'on en connaltra. Pour eviter la consideration de courbes cubiques imaginaires, on pourra substituer successivement a la dyade 8, 9 les dyades 1, 2 et 3, 4; on aura ainsi trois courbes cubiques reelles se coupant en quatre points connus. On determinera, par la methode de M. Chasles, les trois coniques dont chacune passe par les cinq points d'intersection inconnus de deux de ces trois courbes. Ces coniques se couperont aux trois points cherches, qu'on pourra des lors determiner par une construction cubique. VOL. II. H APPENDICE. NOTE I (p. 3). Descartes, Geonme'rie, livre troisieme (UIuvres de Descartes, ed. Cousin, vol v. p. 409). 'Or, quand on est assure que le probleme propos6 est solide, soit que l'equation par laquelle on le cherche monte au carre de carr6, soit qu'elle ne monte que jusques au cube, on peut toujours en trouver la racine par l'une des trois sections coniques, laquelle que ce soit, ou meme par quelque partie de l'une d'elles, tant petite qu'elle puisse etre, en ne se servant au reste que de lignes droites et de cercles. Mais je me contenterai de donner une rbgle g6nerale pour les trouver toutes par le moyen d'une parabole, a cause qu'elle est en quelque fagon la plus simple.' NOTE II (p. 3). De la Hire, La construction des equations analytiques, opuscule qui fait partie (pp. 297-452) des Nouveaux 6lements des sections coniques, Paris, 1679. Le probleme des normales d'une section conique sert a l'auteur pour exemple de la methode alggbrique gendrale. Maclaurin, Treatise of Algebra, London, 1748, p. 352. Joachimsthal, Journal de Crelle, vol. xxvi, p. 172, vol. xlviii, p. 377. Voici la critique que fait Joachimsthal de la solution qui lui est propre, et de celle qu'il attribue a De la Hire. 'On sait que chaque probleme qui depend d'une equation du quatrieme degre, se resout par la regle et le compas, en supposant une seule section conique completement ddcrite. C'est sur ce principe que repose la solution du probleme en question, due a De la Hire. Tandis que le geometre grec qui a trait6 le premier cette question, Apollonius de Perge, se sert, outre la conique donnee, d'une hyperbole equilatere, De la Hire ne fait usage que d'une circonference de cercle, dont les intersections avec la courbe ont, a un facteur pres, les memes abscisses que les pieds des normales. Mais cette m4thode offre plusieurs inconvenients qu'il est bon de signaler. En premier lieu, telle que De la Hire l'a representde, sa Note III.] APPENDICE. 51 mnthode n'est qu'un rksultat de calcul, et ne se rattache a aucune proposition de geometrie; ensuite le choix de linconnue comporte ndcessairement une ambiguite, et, en dernier lieu, les formules qui determinent la position et le rayon de la circonfdrence sont trop compliquees pour se preter aisement aux constructions graphiques. On adressera peut-etre, et a juste titre, ce dernier reproche 6galement a la nouvelle solution qu'on va lire; et neanmoins je n'hesite pas a la publier; car, quoiqu'elle ne soit pas encore une solution d6finitive, les propositions si simples sur lesquelles elle repose, pourront servir de point de depart pour arriver a une solution purement geometrique du probleme dont il s'agit.' La solution, dont parle ici Joachimsthal, est due non pas a De la Hire, mais a M. E. Catalan. I est vrai que M. Catalan ne parle que d'une reproduction avec quelques simplifications de la solution ancienne (Nouvelles Annales de Matlkematiques, per MM. Terquem et Gerono, vol. vii, p. 332); mais il est pervenu a une solution qui est entierement differente de celle de De la Hire, et qui nous a paru plus d61gante, et surtout plus naturelle. Nous n'avons pas cherche l'interprdtation geometrique des formules analytiques un peu compliquees dont s'est servi De la Hire; mais nous sommes parvenus a traiter par les mdthodes de la gdom6trie pure la solution de M. Catalan, aussi bien que celle de Joachimsthal. (Voir la note VIII.) NOTE III (p. 7). Soient PIp2, q q2 deux couples de l'involution qui d6termine une dyade donn6e AX A2; nous dirons que cette dyade est repr6sent6e par [P1 2, Tq2]. Soit encore [xx 2, Y Y2] une representation d'une seconde dyade 2 it2; et supposons que l'dquation anharmonique [P2, 12, q, q2] = [1, T2, Y1, 2] soit satisfaite. Cette dquation entraine n6cessairement l'une ou l'autre des deux 6quations suivantes [PI p2 Y qS q2, 5 Al, 2]=[01 X, YI ' 2/2' I l2] [P1i 2 q1, P q2 2, l]=[x1, 2' Y1 Y2i d/', 1 F.2]; mais il importe d'observer que ces 6quations ne sauraient avoir lieu toutes les deux ensemble. Selon que la premiere ou la seconde equation est satisfaite, nous dirons que les elements imaginaires A 1A2, PFj2, ou bien les dlements imaginaires A2A2, 1 M2, sont repr&sentes homographiquement par les systemes [p1 P2, q1 q2, [x1 2, Y1 Y22]. D'apres la d6finition que nous avons donnee de l'homologie des dyades, il est 6vident que lorsqu'on connait l'homologie de deux dyades, on connait en meme temps deux representations homographiquea des elements homologues de ces dyades. Et, re6ciproquement, il rnsulte de la solution du probleme fondamental de l'article II, que l'on peut trouver lindairement le centre ou l'axe d'homologie de deux dyades de meme espece, dont les axes ou les centres ne sont pas coincidents, lorsqu'on connait deux representations homographiques des elements imaginaires qu'on veut regarder comme homologues dans ces deux dyades. De plus, on verra ci-apres qu'en designant par a a2, 162, 1 C2, trois dyades quelconques, on peut determiner lineairement deux reprdsentations homographiques de bl 62, c1c2, lorsqu'on connalt deux representations homographiques de al a2, 6162, et deux representations homographiques de al a2, c2. D'apres cela, on peut se servir des representations homographiques pour definir l'homologie des dyades, en disant que l'homologie de deux dyades est donnee, lorsqu'on connalt une H 2 52 APPENDICE. [Note III. reprdsentation homographique de ces dyades. Cette maniere de definir l'homologie des dyades a le double avantage de ne pas exiger l'emploi de dyades auxiliaires, ni dans le cas de deux dyades dont les axes ou les centres sont coincidents, ni dans le cas de deux dyades d'espece differente, et de se preter facilement a la theorie des dyades de droites qui ne se rencontrent pas dans l'espace. Ce que nous venons de dire se rattache immediatement a la belle theorie des imaginaires, qui a forme l'objet principal des savantes recherches de l'auteur de la Geometrie de Siluation. Nous croyons faire plaisir aux lecteurs de l'ouvrage de cet excellent geometre, en ajoutant les observations suivantes, qui feront voir combien nous nous sommes peu eloignes de la route qu'il a tracee. (1.) ]tant donne deux representations homographiques [)p1 A2 q 2], 1 [2,1 y] des elements imaginaires A1 A, 21 2, et une autre representation quelconque [r1 r2, s8 s] de la premiere de ces deux dyades; on pourra determiner lineairement les eldments u1lz2, v v2, qui satisfont a l'equation anharmonique [x1 2, YI Y2 u12), V 1 2] = [P1p2, q q2, r, 81 82], et on aura ainsi une nouvelle representation K[u u2, v1 v2] des elements P1 2, qui sera homographique a la representation [r, r2, s 2] des elements 1 A2. Soient donec aas, I b2 C, e2 trois dyades quelconques donnees; [A], [B] deux representations homographiques des elements a a2, b1 b2; [Al], [C1] deux representations homographiques des elements a a2, c1 C2. On determinera une representation [C] des 6elments c. c2 qui soit homographique a la representation [A] des elements aa; et on aura ainsi deux representations homographiques [B], [C], des dldments 61b2, c1 c2 (2.) itant donne une reprdsentation [PeP2, q2] d'une dyade A1A2, et un troisieme couple quelconque r, r2 de l'involution qui d6termine cette dyade, il existe toujours un quatrieme couple de cette meme involution, qui satisfait a l'equation anharmonique [Pl, P,, q2, A, A2]=[r1, 2, S1, S2, A1, A2]. On trouve lineairement ce quatrieme couple en prenant pour Ye s2 les elements conjugu6s a P2_1 dans l'involution determinee par qr2, q2r. On peut done trouver une infinitd de representations des eldments imaginaires hA Ah, homographiques a une representation donnee. (3.) Il convient d'attribuer un signe algebrique d6termine a chaque representation d'une dyade; ainsi nous dirons que la representation [P P, q1 q2] de la dyade Al A2 est positive ou negative, selon que le sens du mouvement continu indique par p q1P2 est positif ou ne'gatif. I1 s'ensuit de la que deux representations homographiques d'une meme dyade ont le meme signe, ou des signes contraires, selon que les elements imaginaires se correspondent directement ou reciproquement dans les deux representations. En effet, en designant par [P1 2P, q q2], ['1 r2, s1 82] deux representations homographiques de la dyade X A 2, on a, dans le premier de ces deux cas, l' quation anharmonique [P 2 P2q, 1 q2, A2]=[rl, r2, S1, SA, XA2] d'oui il s'ensuit que le' sens indiqu6 par p1 q1p2 est le meme que le sens indique par rs 81 r2, * Staudt, Beitrige zur Geometrie der Lage (Nfirnberg, 1856). Note III.] APPENDICE. 53 puisque deux lehments correspondants, qui parcourent deux divisions homographiques, dont les dlements doubles sont imaginaires, se mouvent toujours dans le meme sens. Done, en ce premier cas, les deux representations sont de meme signe; tandis que dans le second cas, on aura l'equation [PI, P2, q1, q2, 12, x2]=) I, 2, A 2, A, A 1], ou, ce qui est la meme chose, l'equation [PlPI, 12, 2 A 1\2=81) 82), s l, r2, AT, X2A] Cette equation implique que le sens indique par pI q1tp2 coincide avec le sens indique par s rT S,; d'oui l'on conclura que les deux sens indiques par p1 q, P2 et r1 s r2 sont opposes, c'est a dire, que les deux representations [PI12, q q2 ], [i, r2, S1 S2] sont de signes contraires. (4.) Etant donne deux systemes geometriques, dont chacun consiste d'une sdrie d'1lements simplement infinie, et dont l'un depend lineairement de l'autre, a chacun des deux sens qu'on peut attribuer a la succession des 6lements dans le premier, il correspondra un sens determine dans l'autre. C'est ainsi qu'en rattachant chacun des deux 6elments d'une meme dyade n dedadeux ens oppos s que l'on peut concevoir dans le systeme goometrique auuel cette dyade ap partieent, de ieemaniere a etablir une distinction fictive entre les deux elements de la dyade, on 6tablit en meme temps une distinction correspondante entre les deux elements d'une dyade quelconque lindairement ddrivee de la premiere; puisque les elements homologues des deux dyades peuvent tre censes appartenir aux sens correspondants dans les deux systemes dont elles font partie. Comme xemple e tres particulier de cette remarque generale, considerons deux dyades de points X2, 1m cl2 ayant le meme axe rM. Soient [p12, ql 21, [X1 c2, Y1 Y2] deux representations homographiques des elements imaginaireses A. 2, eo1 i2e Puisque ces deux reprdsentations peuvent etre de meme signe ou de signes contraires, et puisque la correspondance des elements imaginaires est differente dans les deux cas, on voit tres clairement dans ce cas particulier, qu'en considerant les deux sens oppose's dans lesquels un point peut dcrire une droite, on arrive a distinguer a priori, non pas entre les deux elements d'une mnme dyade (ce qui serait absurde), mais entre les deux manieres d'etablir l'homologie de deux dyades. D'ailleurs on peut observer que, dans ce meme cas particulier, il y a une autre distinction tres marquee entre les deux manieres d'etablir F'homologie des dyades donnees. En effet, si les deux representations homographiques donnees des elements A1 A2 5 /, 2 sont de meme signe, les points doubles de l'involution (A1 k, Ak 2) seront imaginaires, et les points doubles de l'involution (A12, A2 J) seront reels, tandis que, dans le cas contraire, les points doubles de la premiere involution seront reels, et ceux de la seconde seront imaginaires. Pour demontrer cette assertion, prenons un point quelconque sur la circonference d'une conique, et de ce point, comme centre de projection, projetons les points de la droite L1 sur la courbe. Pour plus de simplicite, nous designerons les projections de ces points par les memes lettres que les points eux-memes. Les dyades A1 A2,,t 12 considerees sur la conique, donnent lieu a deux centres d'homologie, dont l'un est interieur, et l'autre exterieur a la courbe. Soit 12 le centre interieur, et menons les cordes z 12$, & 2, $2& iXi, Y2Qn2; [$ 2' ^ 2] sera une representation de la dyade A1 A2, homographique a la representation [P1 P2, q1 q2] de cetfte meme dyade, et aussi a la representation [1 x2, &y Y&2] de la dyade,k 112; de plus, le sens de I n1 $2 est le meme que le sens de x X 2, puisque Q2 est un point interieur. Done la representation 54 APPENDICE. [Note IV. [6e 2, -2 1 2] comporte le meme signe que la representation [Pi p2, q1 q2], ou le signe contraire, selon que les deux representations [xx2, y Y12, [ 1 PI2, 1 q2] sont de meme signe, ou de signes contraires. Par consequent [12, 2 12] et [x1x2, Y1Y2] sont deux representations homographiques des elements A1 A2, PIIL 2 dans le premier cas, et des elements A2A, 1 [ /2 dans le second. Mais cela revient a dire que Q2 est le point d'intersection des cordes imaginaires A1 Li, A292 dans le premier cas, et des cordes A1 2, A2 y dans le second. Et de la, en revenant des points de la conique a ceux de la droite, on conclut immediatement la verite de la proposition qu'il fallait ddmontrer. NOTE IV (p. 7). Les deux solutions que nous avons donnees de ce probleme ne different pas essentiellement, puisque la droite, lieu des points d'intersection des couples de tangentes que l'on considere dans la premiere solution, est en meme temps l'axe des deux divisions homographiques que l'on obtient en 6changeant entre eux sur l'une des droites B, C, les deux points de chaque couple de l'involution qui ddtermine la dyade sur cette droite. La solution suivante, peu diffcrente d'ailleurs des autres, nous parait aussi simple qu'on peut le de'sirer. En se servant des centres d'homologie donnds on obtient deux representations homographiques [13 /,2, I31 3 2], [yI Y2, /iy /2] des dlements imaginaires 6, b2, cc1 c; puis on ddtermine les axes des quatre systemes homographiques que voici, [1, P2, PD, '] = -I[/, y/, y2, Y] [/3l, 5 2' li 2]=[yl 7Y25, yl I2] [01, 2, 1', /32]=[1Y2, Y1, Y2, Yl2] [31,, 2, p']=l [-/2 Y2,,Y1' 2, Y:]. Les deux premieres de ces droites se croisent au point P; les deux dernieres au point P'. Si le premier ou le second systeme devient homologique, le centre d'homologie est le point P'; pareillement, si le troisieme ou le quatrieme systeme devient homologique, le centre d'homologie est le point P. On peut encore remarquer que si les dyades a, a2, b b2, c c2 appartiennent a une meme section conique, les six systemes de trois points, P'QR, PQ'', PQ'R, P'QR', PQR', P'Q'R, seront chacun en ligne droite. NOTE V (p. 12). Supposons seulement que l'axe d'homologie A de al a2, a, a2 soit donne. Faisons passer une conique reelle Z par PlI, et par la dyade de points ddterminde par a, a2, ou a, a2, sur l'axe A. Soient k k2, K2K les dyades de points determinees sur 2 par les dyades donnees b1 b2, 1 f2. L'axe A passera par l'intersection des droites imaginaires conjuguees k~ K2, k2 hl; ce qui suffit pour faire voir qu'on peut distinguer lindairement entre les deux centres d'homologie de k~ k2, K1 K2, et pourtant entre les deux axes d'homologie de 1b 62, /1 f.2 La construction est entierement lineaire, puisque pour determiner les dyades k k2, K1 K2 il n'est pas necessaire de tracer la conique 1. Note VIII.] APPENDICE. 55 NOTE VI (p. 16). La determination de la dyade or r2 ne prdsente aucune difficultd thdorique. Soit [x1 a2, Y1y2] une representation donnee de la dyade P1ip2; a une conique qui passe par cette dyade. D'un point quelconque r6el de c projetons sur cette conique l'involution determinee par les coniques du faisceau [cl, c2, dC, d2] sur l'axe de p1 P2. Soit x le pole de l'involution qu'on aura ainsi sur la conique -. Aux quatre rayons 7. [x,, x2, Y1, 2] il correspondra anharmoniquement quatre coniques du faisceau. Soient $, 2), 77, 27 les points de la conique auxiliaire C, qui correspondent anharmoniquement a ces quatre coniques; [VI $2, 7715 2] sera une representation de la dyade 7I 7r2, et cette representation sera homographique a la representation donnee [xl x2, Y1 2] de la dyade PI P2. NOTE VII (p. 22). Toute conique du r6seau circonscrit qui passe par r rencontre la droite A en deux points harmoniquement conjuguds par rapport a la conique cherchde. De meme, les deux tangentes mendes du point r a une conique du rdseau inscrit, qui est tangente h la droite A, sont deux droites conjuguees par rapport a cette meme conique du reseau harmonique. On peut done trouver tres simplement l'involution que determine cette conique, soit au point r, soit sur la droite A; et, d'apres ce que nous avons dit dans l'article prdc6dent, cela suffit pour ddterminer le systeme polaire de la courbe. Les expressions 'ellipse minima circonscrite,' 'ellipse maxima inscrite,' dont nous nous sommes servis dans cet article, sont relatives au cas d'un triangle harmonique r6el. Lorsque ce triangle est imaginaire, les coniques polaires du centre des distances moyennes, et de la droite a l'infini, sont toutes les deux des hyperboles. NOTE VIII (p. 32). Soient X, Yles projections orthogonales de P sur les axes A1A2, B1B2 respectivement; ST la tangente a F au point S. Menons la corde sW' perpendiculaire 'a A A. D'apres la definition de r, ST est le diametre de 2 conjugud aux cordes perpendiculaires a SP. Done ST est parallele a A2 s; de plus, puisque l'hyperbole r est 6quilatere, les droites bissectrices de l'angle CST sont paralleles aux asymptotes de la courbe; done SC est parallele a 2 s'. On voit en meme temps que SC est le diametre de Z conjugu6 aux cordes perpendiculaires a XY, puisque XY et SP font des angles egaux avec les axes de 2. Mais la droite XY passe elle-meme par C, puisqu'elle est une des diagonales du quadrilatere SPaP, inscrit a r. Done C est le point d'intersection de XY avec le diametre de 2 conjugue aux cordes perpendiculaires a XY, ou, si l'on veut, avec le diametre parallele A 2s'; et la polaire de C par rapport a 2 est la perpendiculaire abaissde du pole de XY sur cette droite elle-meme. Ces determinations nous seront utiles plus tard. Puisque nous tenons a faire voir que la solution de Joachimsthal ne conduit pas a des operations impraticables a cause de leur longueur, nous indiquerons ici la maniere de les effectuer, qui nous parait la plus simple. On supposera connus les axes et l'un des foyers H de la conique centrale 2; on prendra pour A1. l'axe focal. En se servant d'un equerre, on 56 APPENDICE. [Note VIII. menera la corde A1s perpendiculaire a PS, la corde A2 p perpendiculaire a A1 s, la corde ss' perpendiculaire a A A2,; enfin, la normale et la tangente a E au point s (on sait que cela peut se faire tres simplement avec l'6querre). Soit le point d'intersection de l'axe focal avec la normale; o-, r les projections orthogonales de S, HI sur la tangente, Y la projection orthogonale de P sur l'axe conjugue, Y1 le point de ce meme axe pour lequel l'angle YHY1 est droit; enfin, AP1 p etant menee perpendiculaire a PY1, soit y le point d'intersection des cordes P12', sB, et 0 la projection orthogonale de y sur ss. Le centre o du cercle de Joachimsthal sera le point d'intersection des droites O6, So-; le rayon de ce cercle sera w-q. On decrira le cercle, et on joindra les points d'intersection des deux courbes au point Al par des cordes A1 x: les normales issues du point P seront perpendiculaires a ces cordes, et leurs pieds se trouveront sur les dialmetres de Z paralleles aux cordes A2 x. D'apres ce que nous avons dit, on verifiera facilement les ddtails de cette construction, dans laquelle on n'aura besoin du compas qu'au moment ou l'on veut tracer le cercle Q2. En effet, puisque D se transforme en a, et C en /3, les axes principaux de la conique transformee Y' sont les droites qui correspondent au dianmetre de Z qui passe par C, et a la polaire de C. L'un de ces deux axes est ss'; c'est celui qui correspond a SC; l'autre est yO, puisque y correspond au point Yi, qui se trouve sur la polaire de C. Done 0 est le centre de 2'; et, puisque la perpendiculaire abaissee de 0 sur sa polaire par rapport a Z doit passer par o, et aussi par /, la construction du point co se trouve justifiee. On remarquera qu'en general, etant donne un point quelconque Z, pour en trouver le point correspondant dans la figure transformee, on menera les cordes A PA, A 2s1, dont la premiere est perpendiculaire a PZ, et la seconde est parallele a SZ; les cordes ss, pp se croiseront au point z. Et, en effet, c'est ainsi que nous avons determine les points f et y, correspondant aux points F et F1, dans les constructions precedentes. La solution analytique du probleme qui nous occupe, donnee il y a presque deux cents ans par De la Hire, a te l'objet de recherches int'ressantes par M. E. Catalan (Nouvelles Anntles8 de I[al/e'rnaciqgtes par MM. Terquem et Gerono, Vol. vii. p. 332 et 396, annee 1848). Nous allons voir que la solution de Mi. Catalan, aussi bien que celle de Joachimsthal, se deduit naturellement de la methode dont nous nous sommes servis. Soient n 2, n2, no, n4 les pieds des normales abaissees de P sur E; a ces quatre points M. Catalan substitue quatre autres nl, '2,1 n1"3, W/4, qui sont les points d'intersection de E par une circonference de cercle, et dont les abscisses, mesurees du centre do 2 sur l'un des deux axes, sont proportionnelles aux abscisses de nlIn2,n 3, n4*. Soient toujours AA /2, B1BS les axes de 2; a, 3 les points a l'infini sur ces axes respectivement; en supposant que A1 A2 soit l'axe des abscisses, on aura l'equation anharmonique /. [a, S, nW, n3,,4] = /3. [a, 8, nI, n2, '3, '4]...... (A) Qu'on transforme la figure de maniere qu'aux points n de la figure donnee correspondent les points n' de la nouvelle figure. Puisque les points /3, n, n2, i3, n4 appartiennent a la * Dans la solution plus compliquee de De la Hire ce sont les ordonnees des points n' qu'on fait proportionnelles aux abscisses des points n. De plus, au lieu du centre de la conique, on prend pour origine un point tel que les sommes des ordonnees des points n', et des abscisses des points n, s'evanouissent separ6ment. On voit qu'il en doit resulter une construction geometrique entibrement differente de celle que nous allons deduire de l'analyse de M. Catalan. D'ailleurs, nous avons reconnu que la solution de De la Hire ne s'applique pas au cas oil il y aurait quatre normales reelles: mais, malgr6 cet inconvenient, cette solution nous paralt meriter une etude plus approfondie. Note VIII.] APPENDICE. 57 conique F, on conclut de l'dquation (A), que / appartiendra a r', la transformee de r. Or, F' ne peut etre qu'une parabole, dont l'axe est paralllel a B1 B. Car le faisceau (r', E), dont les points n' forment la base, contient par hypothese un cercle; done a/3 sont les points doubles de l'involution que ce faisceau determine sur la ligne droite a l'infini, et la conique du faisceau, qui passe par f, y touche cette droite. Cela posd, il s'ensuit de l'dquation (A) que le point 3 de F' correspond au point a de F; done a se transforme en /3, et l'asymptote Ca devient la droite a l'infini a/3. Soient a, b les projections de C sur les axes A, A2, B1 B2; a se transformant en /3, b se transforme en a, puisque a, /3a sont des points reciproques par rapport aux faisceaux (2, r), (2, r'), respectivement. De plus, d'apres l'dquation (A), le point S doit se transformer en un point S' situe sur 3 8; done A1 ou Sa, se transforme en S',O, ou B1 B2. Soit 21 la conique du faisceau (2, r) qui se transforme en 2; il faut que pour' cette conique les droites aS, aC soient des droites conjugudes. Cette condition ddtermine 2l sans ambiguite, puisque, des deux coniques du faisceau (1, r) qui y satisfont, l'une est F, qui ne se transforme pas en 2. En ddsignant par X1 Yl les points d'intersection des axes de I avec la polaire de C, soient c, x les points places symetriquement a C, X1, par rapport a l'axe B1, B2, y le point place symdtriquement a Y1 par rapport a l'axe A1 A2. La conique du faisceau (5, r) pour laquelle C est le pole de xy ne peut etre autre que 21. Car la polaire de e par rapport a r est l'axe A A2, puisque Se est tangent a r au point S; et la polaire de c par rapport a Z doit passer par x, a cause de la situation syme'trique des points CX1, ex; done c, x sont des points reciproques par rapport au faisceau (2, r), et la conique de ce faisceau par rapport a laquelle C est pole de xy, aura Cc, ou Ca, pour polaire de x; c'est a dire que aS, aC seront des droites conjugudes par rapport a cette conique. I1 resulte de la que abx sera un triangle harmonique par rapport a 21, et que pour satisfaire a toutes les conditions du probleme, il suffira de transformer 21 en 2, de manieire que le triangle abx devienne le triangle /3aS. Soit (1~) Z une ellipse; les involutions determindes par le faisceau (2, r) sur les axes de 2 ont evidemment des points doubles imaginaires; done toute conique du faisceau rencontre ces droites en deux points rdels. Soient A1 \2 les points d'intersection de 21 avec A1 2; x sera le point milieu du segment A1 A2; mais S, qui est le centre de l'involution, se trouvera aussi sur ce meme segment; done les droites b5, 6x seront situdes dans le meme angle formd par les droites b A, bA2, tangentes 21 en A1, A2. Mais l'axe bS rencontre 2, en deux points reels; done aussi bx rencontre ~2 en deux points reels vz v2. D'ailleurs) les points X1 A2, P1 v2 sont quatre points harmoniques de E2; done en transformant homographiquement vl, v2 en A1, A22 et Al, A2 en B,, B2 (ce qui peut se faire par une quelconque de quatre transformations diffdrentes), on transformera 2, en 2, a 6x en /3a, r en une parabole ayant son axe parallele a B1 B2, et, enfin, les quatre points n en quatre points n' situes sur la circonfdrence d'un cercle et satisfaisant a l'equation (A). Quelque soit l'axe qu'on a choisi pour A1 A2, ce sera toujours la meme conique 21 qui se transformera en E; puisque la definition que nous avons trouvd pour 21 est symdtrique par rapport aux deux axes. Et l'on peut ajouter que dans les quatre transformations, relatives a un meme axe, ce sera toujours la meme conique qui se transformera en un cerele, et que les quatre cercles resultants, ainsi que les quatre paraboles rF', seront places symetriquement par rapport aux axes principaux de E. Si (2~) 2 est une hyperbole, on voit d'abord qu'il faudra prendre pour A A2 l'axe qui ne rencontre pas la courbe. Car si les sommets A, 22 e'taient reels, les points A1 A2 seraient reels, comme dans le cas de lFellipse, et il faudrait transformer deux points reels A1 A2 en deux points imaginaires B1 B2. On doit donc supposer que les sommets VOL. II. I 58, APPENDICE. [Note VIII. Al A2 sont imaginaires; en ce cas, on aura a transformer les points A1 X2 en deux points rdels B1B2, mais, pour que les points A1 A2 restent eux-memes rdels, il faudra que SX2 soit plus grand que -8A2. Qu'on mene des perpendiculaires aux asymptotes de Z par les points ou ces droites rencontrent la tangente a l'un des sommets B1, B2; et qu'ensuite par les points d'intersection de ces perpendiculaires avec l'axe conjugu6 on mene des paralleles a l'axe focal. La condition ci-dessus revient a dire que le point P doit etre compris entre les deux paralleles qu'on a tracees. D'ailleurs, cette limitation de la mdthode de M. Catalan rdsulte clairement des formules analytiques dont il l'a fait drpendre. Lorsque la condition de possibilite est satisfaite, les angles Al bA2, Sbx empietent l'un sur l'autre; mais l'axe bS rencontre la conique 21 en deux points reels; done bx ne la rencontre pas, et ba la coupe en deux points rdels P1 2. En faisant correspondre A A2 aB B2, et Il T2 aux deux points a l'infini sur les asymptotes de S, on aura quatre transformations diffirentes, dont chacune pourra servir pour la solution du probleme. Lorsqu'on veut faire usage de cette methode, on commencera par la determination de C, et de la polaire de ce point par rapport a E; on aura ainsi les points a, b, c, x. Pour avoir les points A1A2, on projettera du point B1 sur l'axe AlA2 les deux extrdmit6s du diambtre de I conjugud a la corde B1 x. De meme, si I est une ellipse, on trouvera P1 k2, les deux points d'intersection de B, B2 avec El, en projetant du point A1 sur B1 B2 les extremit6s du diametre parallele a A1 b. En faisant correspondre X,1 a B, A2 a B, on prendra sur B1 B2 les points a', S', qui satisfont a l'equation anharmonique [a, A1, A2, a, S]=[3, B, B2, a', S'], et l'on menera par S' une parallele a /1 A2. Cette parallele correspond a B_ B2; elle rencontre. en deux points reels On '2 correspondant a c1 * On fixera a volontd la correspondance de O'1 P2, 01 02 et on prendra le point /' qui satisfait a l'dquation [8,D 025 8,]= [S",I OD V,], et qui, par consequent, correspond a /. Le point C', correspondant a C, est 4videmment le point a l'infini sur la droite a'/3'; donc 11, qui correspond a D, et qui est le point r6ciproque de C' par rapport au faisceau (1, F'), se trouve a l'intersection de 3'/3 avec le diametre de I conjugu4 a a'/3'. Du point 1)' abaissons sur a'd3' une perpendiculaire; soit w le point d'intersection de cette perpendiculaire avec une droite Sw, qui fait avec l'un des axes principaux le meme angle que la perpendiculaire, mais de l'autre cote de cet axe. Le cercle, dont o est le centre, et qui coupe orthogonalement le cercle dont a',8' est le diametre, appartient au faisceau transforme. C'est ce qu'on vdrifie en observant que cx, a13, CGD sont des couples de points r6ciproques par rapport au faisceau (1, r), et qu'en dsignant par c' le point a l'infini harmoniquement conjugu6 a C' par rapport a a/3, les points correspondants c'S, a'13', C'D)', sont dgalement des couples de points reciproques par rapport au faisceau (2, rF'). Pour avoir le rapport des abscisses des points n' aux abscisses des points correspondants n, projetons P sur les deux axes. En ddsignant, comme nous avons fait plus haut, ces projections par X, Y, prenons Y' le point correspondant a Y; le rapport cherch6 sera celui de S'Y' a SX. Avant de terminer cette longue note, nous indiquerons une troisibme m4thode, qui ne s'applique pas a l'hyperbole, mais qui conduit a une solution assez simple pour le cas de l'ellipse. Soient a1 a2, b1 b2 les diambtres conjugues egaux de l'ellipse E. Qu'on transforme Note IX.] APPENDICE. 59 la courbe en elle-meme, de maniere que, la ligne a l'infini restant la m~me, les axes A1 A2, B B2 deviennent les diametres 4gaux aa2, b1 2. C'est ce qu'on peut faire par quatre transformations diff4rentes, en supposant, pour abrdger, qu'on 4change entre eux les points a l'infini de la courbe. Mais quelle soit la transformation qu'on choisit, le faisceau, qui correspond au faisceau (:, r), contiendra un cercle, puisque les points doubles de l'involution que ce faisceau d6terminera sur la ligne droite a l'infini, seront les points rectangulaires a/3. Soient done aa25 b6b2 les points qui correspondent a A1 A2, B1 B2 respectivement; et soit X un point donne de la courbe. On trouvera le point correspondant x en menant la corde Xx parallele a A1 a1; on aura ainsi le diametre Sx correspondant au diametre SX; de plus, la droite qui joint deux points correspondants de ces diametres sera parallele a Xx; done on pourra trouver tres facilement, dans l'une des deux figures, le point correspondant a un point donnd de l'autre. Soient toujours a, b les projections orthogonales de C sur les axes A A2, B1 B2 respectivement, a', b' les points correspondants dans la nouvelle figure. Qu'on abaisse de a', b' des perpendiculaires sur b, b2, a12 respectivement; le point de concours de ces droites sera le centre o du cercle 12 qu'on cherche. Avec le rayon Sal, dc4rivons un cercle concentrique a 2; soit dl d2 le diametre de ce cercle qui fait un angle droit avec Sw; les points d, d2 appartiendront a la circonf6rence de Q2. On tire cette derniere conclusion d'un thdoreme qui n'est qu'un cas particulier d'un autre plus gdndral, mais qui vaut la peine d'8tre enonce: 'Toute hyperbole, ayant ses asymptotes paralleles a Sal, Sb1, et passant par S, d6termine par ses intersections avec I une circonference de cercle, qui coupe orthogonalement le cercle imaginaire, dont S est le centre, et - Sa2 le carr6 du rayon.' On peut encore remarquer que la somme des angles excentriques de deux points correspondants x et X est 6gale a un multiple impair de o 7r. Cela verifie que les points correspondants aux pieds des normales appartiennent a une meme circonfdrence. NOTE IX (p. 35). La theorie des correspondances (1), (2), (3), a &t6 dtudihe par M. Battaglini, dans un excellent Memoire ('Sulleforme binarie dei primi quattro gradi, apparlenenti ad unaforma ternaria quadratica,' Giornale di Matematiche, vol. v. p. 39), qui, malheureusement, nous 6tait encore inconnu, lorsque nous ecrivions l'article precddent. Cependant, nous placerons ici quelques remarques additionnelles, qui ne sont pas sans interet pour les constructions graphiques. La thiorie geomdtrique de la correspondaace (2) peut etre presentee de la maniere suivante. Soient (A) et (B) deux systemes corrdlatifs dans le plan d'une conique donnee 2; et soient X et Y les coniques qui correspondent dans les systemes A et B a la conique 2, considdr6e comme appartenant aux systemes B et A4 respectivement. Soit 0 la conique des poles des deux systemes correlatifs, c'est a dire, la conique lieu des points qui se trouvent sur leurs droites corrdlatives; de meme soit 0' la conique des polaires, ou la conique enveloppe des droites qui passent par leurs points correlatifs. En considdrant un point quelconque y de M comme appartenant au second systeme, la droite corr6lative rencontrera I en deux points x, qui seront lies au point y, par une equation de la forme (2). D'apres cela, on aura les theoremes suivants qui donnent immediatement la solution des problemes biquadratiques dependant de la correspondance (2). 1 2 60 APPENDICE. [Note IX. (1.) Les points d'intersection de X et 2 sont les quatre points x pour lesquels il y a coincidence des points y correspondants: et les quatre points de contact avec I des tangentes communes a Y et I sont les points y, qui sont devenus coincidents. La correlation des deux systemes (A) et (B) fera connaitre l'un des deux systemes de quatre points lorsqu'on aura trouve l'autre. Pareillement, les points d'intersection de Y et 2, et les points de contact avec I des tangentes communes a X et 2, sont respectivement les points y dont les correspondants coincident, et les points coincidents eux-memes. (2.) Les quatre points d'intersection de 0 et: sont des points de coincidence d'un point x avec lun des points correspondants y, et les tangentes men4es a ~' de lun quelconque 0 de ces quatre points (tangentes dont l'une est aussi tangente a X, et l'autre a Y), rencontrent la conique I en deux points, qui sont les points correspondants a 0, autres que 0 lui-meme. On remarquera qu'une seule construction biquadratique suffit pour trouver, soit les points 1 X2, soit les points y2y2, qui deviennent coincidents, soit enfin les points qui correspondent a ces points coincidents dans chacun des deux systemes; puisque, ayant trouv6 les points d'intersection de deux coniques, on n'a besoin que d'une construction quadratique pour trouver leurs tangentes communes. IMais, pour trouver les points x, qui coincident avec un de leurs points correspondants y, il faut une construction biquadratique nouvelle. Le probleme lineaire ' ]tant donn6 huit points x, et un point y correspondant a chacun de ces points, trouver la droite Y1Y2 correspondante a un point quelconque x' peut s'enoncer plus g4neralement de la maniere suivante, 'I tant donne huit points dans l'une de deux figures correlatives, et huit autres points situds respectivement sur les droites correlatives des premiers points, determiner la correlation des deux figures.' Or, c'est de ce probleme que dUpend (ainsi que l'ont fait voir MM. Seydewitz et Schroter) la construction de la surface du second ordre qui passe par neuf points donnes. On en trouvera la solution compllte dans le M4moire de M. Schr6ter (Journal de Crelle-Borchardt, vol. lxii. p. 215). Si l'equation (2) est symetrique, les coniques X et Y coincident, de meme que les coniques 0 et 0', et les deux systemes (A) et (B) deviennent polaires reciproques par rapport a 0 et 0'. En ce cas particulier la solution du probleme lin4aire est tout-a-fait 6l1mentaire. En passant maintenant a la correspondance cubique, definie par l'Fquation (3), supposons que yy'y" soient les trois points correspondants a un meme point x, YY' Y" le triangle des tangentes a I en ces trois points. L'axe d'homologie des triangles yy'y" YY' Y" enveloppera une section conique C1; pareillement, le lieu du centre d'homologie de ces deux triangles sera une seconde conique C,, polaire reciproque de CL par rapport a E. Soit, de plus, o- la conique inscrite aux triangles yy'y", 2 la conique par rapport a laquelle ces memes triangles sont harmoniques, de sorte que 12 est une des coniques r4ciproquantes de Z et 0. Les trois coniques ~2, C2, 2 ont les memes points d'intersection; ces points sont en meme temps les points de contact avec I des tangentes communes aux trois courbes o-, C1, 2; de plus, la tangente a Q2 en un quelconque de ces points rencontre 2, pour la seconde fois, en un des points d'intersection de I avec o-, et y est tangente a cette derniere conique. Les points de coincidence de deux des points y, qui correspondent a un meme point x, sont dvidemment les points de contact avec I des tangentes communes a I et -; d'ou l'on voit que, pour trouver ces points, il suffit de connaitre l'une quelconque des quatre coniques auxiliaires C, C2, o-, 12. Les axes d'homologie des triangles yy'y" et YY'Y", considcrds Note IX.] APPENDICE. 61 comme des tangentes a C1, et les centres d'homologie de ces memes triangles, consid&rds comme des points de C2, correspondent anharmoniquement soit aux triangles yy^'', soit aux points x; cette observation servira pour ddterminer les points x qui correspondent aux quatre triangles dvanescents que nous venons de trouver. La ddtermination des points x, qui coincident avec un des points correspondants y, se fait un peu differemment. Soient P, Q deux points quelconques de E, P' un point qui n'appartient pas a cette conique. Les coniques du faisceau (P, P', y', y'") correspondront anharmoniquement aux points x; par consequent, le lieu gdomdtrique des intersections des droites Qx et des coniques correspondantes (P, P', yy', y") sera une courbe cubique, qui passera par les points P et Q, et qui, en outre, rencontrera 2 en quatre points, qui sont ceux qu'on cherche. On les ddterminera en se servant de la construction biquadratique indiquee par M. Chasles. Comme verification des rdsultats prdcddents, nous ajouterons quelques unes des principales formules analytiques qui se rattachent a la thdorie des correspondances (2) et (3). Soient p, q, r les coordonndes homogenes d'un point quelconque du plan que l'on considere; on prendra l'e'quation pr -- = 0 pour l'6quation de la conique 2, et on reprdsentera par 0,2, 01 02, 022 les coordonnees d'un point quelconque 0 de cette conique. Les deux systemes correlatifs, dont ddpend la correspondance (2), seront ddfinis par l'dquation P1 ( 3p2+ J 2B, 2 3+ C. r2) + P2 (2 P2 + 2B2 q + 2r2)+ rl (43 p2 + 2B3 q2 + C3 r2) = o; et en mettant dans cette equation les coordonnres du point 0, soit pour p q r,, soit pour p2 q2 r2, on aura l'dquation, soit de la droite 7i rl, soit de la droite C1 62. Donc l'dquation de la conique X, enveloppe de l 52, sera (AP +2A2q + A3r) (C,p+2C2q+ C3r) = (Bp+2B2q+B3r)2; et, pareillement, l'Uquation de Y, enveloppe de, 12, sera (A1p + C,2B q + C A, ) (A3p + 2 3q + C r) = (Ap + 2 B2 + C, r)2. Enfin, on aura pour la conique des poles l'equation 0 = A p2 4B2 q2 + C r2 +2(C2 + B) r + (A3 +C) pr 2 (B1 + A42) = 0, et pour la conique des polaires l'equation 0' = 4 A 2 = 0 en ddsignant par A le ddterminant A1 A2 A3 B1 B2 B3 C1 C2 C3 et par D la fonction lindaire p[A2C, —1 C-A 3B1 +lAB3 + [ 3-3A B2-B C2+ B2 C] + r B3 C1-B1 C3-A3 + C 2 C3]. Passons a la correspondance (3). Soit toujours - = pr —q2 = 0 l'equation de la conique 2, et designons par P, Q, R, S, U les ddterminants du systeme A1, B1, C2, 1 a2, B,, C2, -D, 62 APPENDICE, [Note IX. pris dans leur ordre naturel, et par al, bl, ce l a12, b2, C12 a2 b 12 12 les neuf quantit4s A1 C1-B, AD1-1-B1 C1 B1 1 —C2, AlC2-PfB,-BA2+-42C., I +421 A ---BlC2-82 C1 B-1 D 2PC1C2+.B SID AC2-B, A22-1Bc2, B2 -C2. Soient, de plus, 12, 1A2, A2, J2A 22 les coordonnees de deux des trois points y qui correspondent a un meme point x. En 6liminant x1 et x2, et divisant par 3(A1i, 2 —A2 lx), on aura l'Fquation suivante: 1 1 + Q (A1 P2 + A2 11) A1 a1+ + R R (A1 2 2+ 1) + (38+ ~R) A1 A2 i I2 + T(A1 P2 + A2 I) A2 2 +-UA =o0, qui est celle d'une correspondance quadratique double. Cette correspondance est dvidemment symrtrique; elle est aussi triangulaire, puisque, la fonction K= PU+RS-QT etant identiquement zdro, la conique enveloppe de la droite qui joint les deux points A correspondants a un point / donnd, c'est a dire la conique =4( p + q+ Ur) (Pp + Q+ Rr)- {Qp+ (38+~R) + Tr}2 = 0, ne differe pas de la conique enveloppe de la droite qui joint le point g a l'un ou l'autre des deux points correspondants A; en effet, on trouve pour l'dquation de cette dernibre conique ( —4K2 =0. Cela pose, la conique ra est la conique inscrite a tous les triangles du systeme; et Q = Pp2+(3S+JR)g2+Ur2+2Tqr+Rpr+2Qpg = 0 est l'Uquation de la conique par rapport a laquelle ces memes triangles sont des triangles harmoniques. L'dquation X1 I Y( $ bZl y2 + y)+ x, x /n 012, 2 z1 {(al y + b/1 Y12^ + 6cy) + 1i ^2 (a12y1 + 612 Y1'2 + 1a2 y 2) + x2 (aYx + 6b2y Y22 + e2yg) = 0 exprime la correspondance doublement quadratique, mais non symdtrique, qui a lieu entre un point donna x et les deux points covariants du triangle correspondant. Done l'Nquation de la conique C1, enveloppe de la corde qui joint ces deux points, sera C1 = 4(alp+bq+er)(cap+&bg+ c2r)-(a12p + 2 +c2r)2 = 0; et on aura l'expression suivante pour l'quation de la conique C2, polaire rdciproque de C par rapport a 2, Note X.] APPENDICE. 63 fr, a12 2 r a12, a1 r, a1, a 2 -2q, bi2, b2 x -2q, b12, bl + -2q, b1, 2 = 0., C12, C2 PA C12, C1., c, C2 Chaque terme de cette derniere dquation est divisible par le dgterminant al, 61, c1 D = a12, b12, C12 a2, b2, C2 en supprimant ce facteur constant, on a l'expression plus simple: C2 = 3+ (R-3S) = 0. Les fonctions D et R-3S sont trbs connues; l'4vanescence de la premibre implique que les axes d'homologie des triangles yyy", YY'Y" passent tous par un meme point, et que le systeme contient un triangle dont les trois sommets se confondent en un seul; l'4vanescence de la seconde exprime que deux triangles quelconques du systbme sont harmoniquement conjugue's l'un a l'autre, et que, par consequent, le systeme donne coincide avec le systeme harmonique correspondant. (Voir le Mdmoire de M. Battaglini, pp. 44-49.) NOTE X (p. 41). Si les sept points 1,..., 7, appartiennent tous a une meme conique, la determination de la cubique (9,..., 16) ne rdussit pas. En effet, dans ce cas, le point biquadratiquement oppose au systeme (1,..., 7, a) est le point a lui-meme; ou, plus exactement, il n'y a aucun point biquadratiquement oppose a ce systeme, puisqu'il n'y a aucune courbe biquadratique passant par les treize points, et ayant (1,..., 7, a) pour base de courbes cubiques gdndratrices. II faudra done, dans cette hypothese particuliere, 4viter de faire usage de la cubique (9,..., 16), ce qui sera toujours possible. Si les dix points 1,..., 10 appartiennent a une meme cubique, la determination de la cubique (9,..., 16) devient illusoire, et doit etre remplac6e par une autre, puisque, dans ce cas, les trois systemes de points (P8, 9, 10), (p,, 10, 8), (Plo, 8, 9), sont respectivement en ligne droite; d'ou il rdsulte que l'dquation anharmonique.Ps [I, 11, 9, 10] X p9. [Ea, 11, 10, 8] x.Po. [x, 11, 8, 9] = + 1 devient identique quel que soit le point xa, et ne peut servir a ddfinir aucun lieu gdomdtrique. La meme chose arriverait si les droites p, pl, I1oP7 P8 sP passaient par les points 8, 9, 10 respectivement. Pour dviter cet inconvenient, on prendra arbitrairement les points 8, 9, et on ddterminera les points p,, p9 avant de choisir le point 10. La droite P8, P9 peut bien passer par un des points 10,..., 13, mais elle ne peut pas passer par deux de ces points a, /3, puisqu'il s'ensuivrait de cette supposition que les neuf points 1,..., 7, a, 3 forment la base d'un faisceau de courbes cubiques, ou bien que les onze points 1,..., 9, a, j appartiennent a une meme cubique. I1 y aura done au moins trois des points 10,..., 13, qui ne se trouvent pas sur la droite; de plus, de pl dces trois points il y aura au moins deux qui ne peuvent pas appartenir a la cubique (1,..., 9); on prendra a volont6 l'un ou l'autre pour le point 10. I1 correspond une analyse tres simple a la demonstration g6omdtrique du thdoreme 64 APPENDICE. [Note X. de cet article. Soient (a, b), (a, b, c, d, e),..., les fonctions alg6briques qui, 6gal6es a z6ro, donnent les 6quations de la droite (a, 6), de la conique (a, b, c, d, e),..., et ainsi de suite. En designant par X une constante ind6termin6e, les courbes cubiques du faisceau (1,..., 8, 9) = xA(1,..., 8, 10); correspondent anharmoniquement aux droites (ps, 9) = X (p, 10); et puisque les fonctions (P8, 9), (P8, 9),..., qui entrent dans ces equations, peuvent etre multipliees par des constantes quelconques, on peut faire en sorte que la droite (p,, 11) corresponde a la cubique (P8, 11). Cela pose, (1,..., 7, 8, 9) (P8 10) = (1,..., 7, 8, 10) (P8, 9)...... (a) sera l'6quation d'une courbe biquadratique qui passe par le point p. et par les treize points donn6s. Pareillement, les courbes biquadratiques (1,..., 7, 9, lO)(p, 8) = (1,..., 7, 9, 8)(po, 10)..... (b) (1,..., 71, 10, 8) (o, 9) =(1,.., 7, 10, 9) (po, 8)..... () passeront par les memes treize points, et par les points p9, po,, respectivement. En multipliant ces trois equations, membre a membre, et divisant par le produit (1,..., 7, 8, 9) (1,..., 7, 9, 10) (1,.., 7, 10, 8) on aura l'6quation cubique (p8, 10) (9, 8) (p10, 9) = (P8, 9) ( pO, 10) (P10, 8)..... (d) I1 resulte du choix que nous avons fait du point 10, que les deux membres de cette equation ne peuvent pas etre identiques, a moins que le triangle p8,p9,Plo ne coincide avec le triangle 8, 9, 10. Mais, en supposant toujours que les sept points 1,..., 7 n'appartiennent pas tous a une meme conique, on remarquera que, si cette coincidence a lieu, les courbes biquadratiques (a), (b), (c) doivent avoir des points doubles aux points 8, 9, 10 respectivement; c'est a dire, que les trois points cherchds coincident avec ces memes points, et que toutes les courbes biquadratiques qui passent par les treize points ont en ces trois points des tangentes communes, dont on trouve facilement la direction, en se servant des courbes (a), (b), (c). En revenant done au cas general, Hlequation (d) sera l'Fquation d'une courbe cubique, qui passe 4videmment par les six points 8, 9, 10, p), p9? Plo et par les trois points d'intersection de (P8, 10), (P2o, 8), (8s, 9); (Plo, 9), (P9, 10). Mais cette courbe passe aussi par les points 11,..., 16. Soit 6 un de ces points; la chose est evidente, si 6 n'appartient a aucune des courbes cubiques (1,..., 7, 8, 9), (1,..., 7, 9, 10), (1,..., 7, 10, 8); ensuite, si 6 appartient a une seulement de ces courbes, par exemple a la premiere, 6 sera le point d'intersection de (p9, 8) et (P8, 9), et ne cessera point d'appartenir a (d); enfin, si i appartient a la fois a deux des mmes courbes cubiques (ce qui par hypothese n peut pas arriver, a moins que 6 ne soit un des trois points inconnus), e sera le neuvieme point appartenant a une base cubique donnde, et pourra etre ddtermine lindairement, sans qu'il soit ndcessaire de chercher la cubique (C). Note X.] APPENDICE6 65 I1 est assez remarquable que la solution que nous avons donn4e du probleme de cet article, s'applique aussi & cet autre probleme plus g6dnral: ' tant donna 4n -3 des 4n points d'intersection d'une courbe d'ordre n avec une courbe biquadratique, trouver les trois autres points.' On suppose n > 4. Prenons 4n-8 des 4n-3 points donnas; ajoutons-y (n- 1) ) -(4n- 8) points choisis arbitrairement, et consid4rons l'ensemble des (- ) ( + 2) - 1 points comme 1.2 determinant la base P4_-s d'un faisceau de courbes d'ordre n -1. En considdrant cette base par rapport aux cinq points 4n-7, 4n-6, 4n- 5, 4 n-4, 4n —3, on aura un point oppose P4,n-S qui sera le centre d'un faisceau de droites correspondant anharmoniquement aux courbes du faiscean P4,n-. Le lieu des intersections des lignes correspondantes des deux faisceaux sera une courbe de l'ordre n, qui passera par les 4n-3 points donnds, et, par consequent, par les trois points cherches. En permutant cycliquement les points 4n-8, 4n-7, 4n-6 (sans changer autrement la base des courbes de l'ordre n-1), on aura deux autres courbes de l'ordre n passant, comme la premiere, par les 4n points. De la en suivant, soit notre demonstration gEomdtrique, soit l'analyse pr4dcdente, on conclura que la courbe cubique des neuf points 4n-8,..., 4n passe aussi par les points oppos4s 4 n-?3 P4n-7 etc..., et par les points d'intersection tels que celui des droites (P4n-s, 4n-7) et (P4-7, 4n.-8). On voit done que tout se rdduit a la ddtermination des points opposes, dUtermination qui sera facile, lorsqu'on connaitra les rapports anharmoniques des faisceaux P4n-8.[4n-7, 4n-6, 4n-5, 4n-4, 4n-3]. Or, il n'est pas douteux, qu'4tant donn6 des points en nombre suffisant pour determiner une courbe geomdtrique d'ordre n, on ne puisse trouver lindairement, soit la tangente a cette courbe en un point donn6, soit la droite polaire d'un point donne par rapport a la courbe; mais, puisqu'il paratt qu'on n'a pas encore cherche la solution de ce probleme gdneral de g4omdtrie lindaire, nous ferons voir qu'on peut s'en passer ici, en se servant d'une mdthode particuliere qui se pr4sente naturellement. Consid6rons les points 1,..., 8, comme ddterminant la base A d'un faisceau de courbes cubiques. Soit (a) la conique qui satisfait a l1'quation (9, 10, 11, 12)=. [9, 10, 11, 12]. En prenant successivement diffirents points a de cette conique pour des points opposds a la base A, on aura un faisceau de courbes biquadratiques, correspondant anharmoniquement aux points a. Soit B la base du faisceau biquadratique; il est evident, qu'4tant donn6 un point quelconque x, on pourra trouver lindairement le point a correspondant a la courbe (B, x). DWterminons la conique (b) qui satisfait a l'quation (13, 14, 15, 16)= B. [13, 14, 15, 16]. Prenons un point quelconque b de (b); nous le considerons comme un point oppose a la base biquadratique B; et nous aurons de la sorte un faisceau de courbes du cinquieme ordre qui correspondront anharmoniquement aux points 6. Soit C la base de ce nouveau faisceau; pour trouver lin6airement le point b qui correspond a une courbe quelconque (C, x) du faisceau, on d4termine sur la conique (b) le point x' qui satisfait a l'4quation anharmonique [13, 14, 15, 16, a'] = B. [13, 14, 15, 16, ']; VOL. II. K 66 APPENDICE, [Note X. le point b est le second point d'intersection de la conique par la droite xx. Apres avoir dktermin4 une troisieme conique (c), qui satisfait a l'6quation (17, 18, 19, 20) = C. [17, 18, 19, 20], on sera conduit a consid6rer une base D de courbes du sixieme ordre, et, en continuant de la sorte, on arrivera enfin a une base M de courbes de l'ordre n - 1; cette base comprendra tous les points 1, 2, 3,..., 4n-8, puisque chaque base de la serie ascendante comprend evidemment tous les points de la base prec6dente; de plus, les courbes du faisceau 3 correspondront anharmoniquement aux points d'une certaine conique (1) passant par les points 4n-11, 4n-10, 4n-9, 4n-8; de sorte qu'6tant donne un point quelconque a, on pourra trouver lineairement le point I de cette conique qui correspond a la courbe (3, x). Enfin, on determinera la conique (m) qui satisfait a l' quation (4n-7, 4n-6, 4n-5, 4ni-4)= M. (4n-7, 4n-6, 4n-5, 4n-4), et le point m'4-3 de cette derniere conique qui correspond a la courbe (M, 4n- 3); le second point d'intersection de la conique (m) avec la droite (4n-3, m'-A3) sera le point P4,-,, qu'il s'agissait de trouver. On peut dire que cette construction est composee de transformations homographiques successives. On commence par determiner la conique r qui satisfait a l'Nquation [5, 6, 7, 8] = (12, 3, 4). [5, 6, 7, 8]. Soit v' [ = 9,..., 4n-3] le point de cette conique qui corresponde anharmoniquement a la conique (1, 2, 3, 4, v): le second point d'intersection de la droite (v, z/) avec F est le point o, de r qui correspond anharmoniquement a la cubique (A, v). On transforme homographiquement la figure de maniere que les points o9, (010o, l (12 deviennent les points 9, 10, 11, 12; la transformee de r sera (a); soit a' [v = 13,..., 4n-3] le point de cette derniere conique qui correspond au point o, de la conique r; le second point d'intersection de la droite (v, a',) avec (a) sera le point a,, qui correspond anharmoniquement a la courbe biquadratique (B, v). On transforme encore la figure de maniere que les points a,3a a14, a15, al. deviennent les points 13, 14, 15, 16; on ddtermine la conique (6) correspondant a (a), et le point ' [r = 17,..., 4n —3], correspondant a a^; on mene la droite (v, b'), qui determine le point bv, correspondant anharmoniquement a la courbe (C, v); et en continuant cette s6rie uniforme d'op6rations lineaires on parvient enfin a determiner le point 4n-3 de la conique (m), qui n'est autre que le point oppos6 4Nn-8* XXIV. ARITHMETICAL NOTES. [Proceedings of the London Mathematical Society, vol. iv. pp. 236-253. The three papers which form these Notes were read on January 9 and February 13, 1873.] I.-On the Arithmetical Invariants of a Rectangular Matrix, of which the Constituents are Integral Numbers. 1. LET |aij 11 represent a rectangular matrix of the type n x (n + m), and let V, be the greatest common divisor of the _ _- x 11( +m) is. II (n - s) Is. II (n + m - s) minor determinants of order s which appertain to it; so that, in particular, Vn is the greatest common divisor of the (m + n) determinants of the matrix; the Un. mrn numbers Vn, Vnl,..., V1 are the arithmetical invariants of the matrix. We suppose that the matrix is asyzygetic; i.e. that the (m + n) determinants are II n. In not all equal to zero. We then have the theorem (Memoir, p. 388 sqq. *): Theorem (a). 'The quotient -V is the greatest common divisor of the quotients obtained by dividing each minor determinant of order s by the greatest common divisor of its own first minors.' Let p be a prime dividing Vi, and let Ii be the exponent of the highest power of p which divides Vi; any minor of order i which is divisible by pli + * The references in this and the two following notes are to a Memoir 'On Systems of Linear Indeterminate Equations and Congruences' (Phil. Trans. vol. cli. pp. 293-326.) [This Memoir is No. XII of vol. i, pp. 367-406. The references in the text are to the pages of the Memoir as printed in vol. i.] K 2 68 ARITHMETICAL NOTES. [I. may be said to be divisible in excess by p6. Using this abbreviated mode of expression, we may enunciate the two following corollaries: Corollary (b). 'Any minor determinant, which is not divisible in excess by p, contains at least one first minor which is not divisible in excess by p.' Corollary (c). 'Any minor determinant, which is not divisible in excess by pI, contains at least one first minor which is not divisible in excess by pa.' Or, which is the same thing, 'If all the first minors of a given minor are divisible in excess by pl, the given minor is itself divisible in excess by pa.' Of these corollaries, the first is a particular case of the second; and the second is only a re-statement, in other words, of the theorem (a). 2. It is the object of this note to establish a theorem, which may be regarded as reciprocal to the theorem (a). Theorem (A). 'The fraction V_- is the greatest common divisor of the Vs fractions obtained by dividing each minor of order s- 1 by the greatest common divisor of its first majors; or, which is the same thing, the integral number V.is the least common denominator of these fractions.' - Any square matrix which contains a given square matrix is here called, for brevity, a major of that matrix; if the given square matrix be of order s- 1, its first majors are the square matrices of order s which contain it. The theorem (A) admits of two corollaries, corresponding to the corollaries (b) and (c): Corollary (B). 'Any minor determinant, which is not divisible in excess by p, is contained in at least one first major which is not divisible in excess by p.' Corollary (C). 'Any minor determinant, which is not divisible in excess by p', is contained in at least one first major which is not divisible in excess by po.' Or, which is the same thing, 'If all the first majors of a given minor are divisible in excess by p", the given number is itself divisible in excess by p6. 3. To prove the theorem (A), we consider, in the first place, a square matrix 1 aj 11 of the type n x n, so that Vn = 2 + a,, a22... - a,; and we represent the minor determinant d by Aij, so that the reciprocal matrix is 1 Aij. The determidaij n-I nant of this matrix is V, and the greatest common divisor of its minor detern i-1 minants of order i is V x V,_i. n Art. 4.] ARITHMETICAL INVARIANTS OF A MATRIX. 69 Let M= 1 (s-1).I(n-s+)' N=n-s+1; let k., [> = 1, 2, 3,..., M2] represent any one of the M2 minors of order s - 1 which appertain to the matrix lIai3ll; let c,, [v = 1, 2, 3,..., N2] represent any one of the first majors of k,J; let d, be the greatest common divisor of these N' numbers; lastly, let K, and Ky,, represent the minors which, in the reciprocal matrix 11 Aif 1l, are reciprocal to the minors ks, and k,, in the matrix || a Is; so that, for example, if k,A= -+ all a22.a* as-l, -1 s,s s+l *A,* n, n, and if k,,^= =t1- a22... as, s,, K, =2+ l+l A n, n Thus the N2 first minors of K- are precisely the determinants K,; we also have n-, n-s-1 the equations K,= V x k, K,,= V x k,, ^; so that, applying the theorem w n (a) to the matrix IAUtl, we find that the greatest common divisor of the M2 integral numbers, ~n-s ~ —s-1 k [V x]' [ V xd,]-Vxxd, is the quotient n-s n-s+l [ x ][ v x v xV =Vx V that is to say, the greatest common divisor of the M' fractions k is the fraction di in accordance with the theorem (A). The corollaries (B) and (C) may be immediately verified by observing that, if either of them were supposed untrue for any given minor k,,, the corresponding fraction kz would acquire a denominator which could not be a sub-multiple of V. 4. To extend the demonstration to the case of an oblong matrix of the type n x (m + n), we retain the preceding notations, so far as they are applicable; and we put M = - (n + m N = n n+m - s + I; so that the values of H (s - 1). H (n + m - s+ 1) /i and v are now [1, 2,..., M x M1] and [1, 2,..., N x N1] respectively. Let Q be a common multiple of the Mx Ml numbers d,, and let us complete the given rectangular matrix into a square one by adding rows of constituents each of which is divisible by Q. In the resulting square matrix, any given minor k,. (appertaining to the given rectangular matrix) will have N 2 first majors; but it 70 ARITTHMETICAL NOTES. [I. is evident that the greatest common divisor of these N/ first majors will be the same as that of the N x N first majors which appertain to the given rectangular matrix. Hence the MxMl fractions obtained by dividing each minor of dl order s- 1 in the given matrix by the greatest common divisor of its first majors, occurs among the M' fractions similarly derived from the completed square matrix; also the numbers V8 and V,,i are evidently the same for both matrices. Applying, therefore, the theorem (A) to the square matrix, we see that -1 is a common divisor of the Mx M fractions,; that it is the greatest common divisor of these fractions, may be proved by considering a minor determinant of order s in the given matrix, which is not divisible in excess by p; this minor determinant contains [by the corollary (b)] a first minor which is itself not divisible in excess by p; if k, be the first minor, the denominator of the corresponding fraction Z- necessarily contains the prime p raised to the power I - I_-. It will be observed that, whether we consider a rectangular or a square matrix, the corollary (C) is an immediate consequence from the theorem (A); but the theorem (A) does not follow conversely from the corollary (C). For the absence of factors prime to V8 from the least common denominator of the fractions - is asserted in the theorem, but is not asserted in the corollary. 5. Every common divisor of the first minors of a given minor is evidently a common divisor of the first majors of that minor; and, if the matrix be square, a consideration of the reciprocal matrix shows that, conversely, every common divisor, prime to the determinant of the matrix, of the first majors of any given minor, divides the first minors of that minor. We thus obtain the self-reciprocal theorem: Theorem (dD). 'In any square matrix, the greatest common divisor of the first minors of any given minor is identical with the greatest common divisor of its first majors, so far as factors prime to the determinant of the matrix are concerned.' This theorem is not universally true in the case of oblong matrices, at least if we suppose that, in its enunciation, the words 'determinant of the matrix' are replaced by the words 'greatest common divisor of the determinants of the matrix.' If, for example, II a is a square matrix of order n - 1, and IlblI is a Art. 5.] SYSTEMS OF LINEAR CONGRUENCES. 71 matrix of the type n x (n + m), of which the determinants are relatively prime, the symbol 0 b may serve to represent a matrix of the type n x (2 n + -1), a in which the first n- 1 constituents of the uppermost row are zeros. In this matrix the greatest common divisor of the first majors of |a] is evidently la itself; and this greatest common divisor is prime to the greatest common divisor of the determinants of the matrix, because these determinants are relatively prime; but unless fl a l is an unit matrix, its determinant cannot be equal to the greatest common divisor of its own first minors. 6. It may be added, that the properties to which this note refers admit of being stated in a generalized form. Thus, we may replace the enunciations of the theorems (a), (A), and (dD) by the following: Theorem (a'). 'The quotient -s is the greatest common divisor of the vs-i quotients obtained by dividing each minor determinant of order s by the greatest common divisor of its own first minors of order s - i.' Theorem (A'). 'The fraction -i is the greatest common divisor of the fractions obtained by dividing each minor of order s - i by the greatest common divisor of its own majors of order s.' Theorem (dD'). 'In any square matrix, the greatest common divisor of the minors of order s - i appertaining to a given minor of order s is identical with the greatest common divisor of the majors of order s+i appertaining to the given minor, so far as factors prime to the determinant of the matrix are concerned.' A very slight modification of the proof of the theorem (a), (Memoir, pp. 397 -399,) supplies a proof of the theorem (a'), and from it the theorems (A') and (dD') may be inferred by means of the methods employed in this note to establish the theorems (A) and (dD). II. —On Systems of Linear Congruences. 1. Let Ai, x+ Ai,2 2 +... + Ai x, - Ai +1 (mod M), (1) i= 1, 2, 3,....n, represent a system of n linear congruences; let Dn, D-l....... D, V,, Vn-,,......, V1, 72 ARITHMETICAL NOTES. [II. respectively denote the arithmetical invariants of the augmented and unaugmented matrices of the system; i.e. of the matrices AII | i=, 2...,n,}.. (2) 11Aall j=1, 2,...,, } ' ) and lAl1 2; 'I,..... (3) j = 1,2.,...,; let d and S be the greatest common divisors of M with D and of M with JDs-/a I Vs respectively; lastly, let d = d, x d2 x... x d,, 3 = i x 2 x... x -,. We then have the two theorems (Memoir, p. 399): 'The necessary and sufficient condition for the resolubility of the system (1) is d=. 'When this condition is satisfied, the number of its incongruous solutions is d =S' There are similar theorems (Memoir, p. 402) relating to defective and redundant systems of congruences. Some observations which may serve in certain cases to facilitate the applications of the theory are contained in the present note. 2. Consider separately the powers of the different prime numbers dividing the modulus M. Let p be one of these primes; and let,u, a,, a, be the exponents of the highest powers of p which divide M, D,, V, respectively. Then, because D8+1. D 8+1 Vsv1, a Vn D_ fD D ' I V + DS, and., are all integral (Memoir, D VD ' V V Ds_~ pp. 396-398), we have the inequalities a,-a. -1 - a - n-2 2a_ a..., _- al o;...... (4) a.>.. (6) as.- _ a s(-a-l..a......... (7) Let a, - a,_ be the first term in the series (4) which is less than M; so that, if a, - an-l < u, we have - = n, and in every other case a. + - a x. f> a a-...... (8) We may then replace the two theorems of the Memoir by the two following: Art. 3.] SYSTEMS OF LINEAR CONGRUENCES. 73 'The necessary and sufficient condition for the resolubility of the congruences (1), considered with regard to the modulus pi, is C a=a,.... (9) 'When this condition is satisfied, the number of incongruous solutions is paor + (I-<). For the condition of resolubility is (Memoir, p. 400) that the greatest common divisor of the numbers pa, plan+ +, pan2+2a,... p....... (10) should be the same as the greatest common divisor of the numbers pan pa_; +1L pa_-2+2l,..., pn......(11) The greatest common divisor of the numbers (10) is paa, + (-6); for it follows from the inequalities (8) and (4) that, in the series (10), the exponent a, + (n - a), is less than any exponent which follows it, and not greater than any which precedes it. Again, if a6 = a, pa +(n- ) is also the greatest common divisor of the numbers (11); for it is equal to one of them, and it certainly divides all of them, because it divides all the numbers (10), each of which, by virtue of the inequalities (6), divides one of the numbers (11). But if a, > a, the greatest common divisor of the numbers (11) is a power of p having an exponent higher than a. + (n- o),u. For the inequality a, > a, combined with the inequality (7), implies the successive inequalities a + 1 > a +1, a,+2> a +2,..., an > a,; so that every exponent in the series (11), which precedes a, + (n - cr) Mu, surpasses the corresponding exponent in (10), and therefore certainly surpasses a + (n - a-). And again, the exponents following a + (n - a-), in the series (11) also surpass a, + (n - a) tm, for these exponents are not less than the corresponding exponents in (10), each of which is greater than a, + (n - ),. Thus the condition for the equality of the two greatest common divisors, i.e. for the resolubility of the congruences (1), is a = a6. And, by transforming the proposed system (Memoir, p. 401) into an equivalent system of the type V 1 b c ( mod,.......od s (12) n-s it is immediately ascertained that, when this condition is satisfied, the number of incongruous solutions is equal to pa + (n - 6) i.e. to the common value of the two greatest common divisors. 3. The criterion of resolubility just established differs in form only from that VOL. II. L 74 ARITHMETICAL NOTES. [Il. given in the Memoir; and the coincidence of the two may be shown by observing that the equation a, =a,, combined with the inequalities (7), implies the equations ao_- = a,-, a,_2= a,-2,..., a, = 1. As here stated, the criterion of resolubility, and the expression for the number of solutions, depend only on the numbers a, V,, DZ, and do not involve explicitly the consideration of the two complete series of invariants. When, surpasses a,-a,_1, the condition of resolubility is a. = an, and the number of solutions is pan. In this case we only need to determine the invariants V, and Dn, or rather the exponents of the highest powers of p dividing those numbers. But when < a,-c, i, it is necessary to calculate successively the invariants D, D_., Dn_2,... in order to ascertain the index a for which the inequalities (8) are satisfied. It will be observed, however, that since every minor of order s, which is not divisible in excess by p, contains a minor of order s- 1 which is itself not divisible in excess by p, it is always possible to restrict the examination of the minors of order s - 1 (which is required in order to obtain the value of D_-1) to those which are contained in a single minor of order s. We may also notice that if, in the series of differences an -an) an-l a —a,..., a al-a,, ar - a is the first which is equal to zero, pas+l-as is the highest power of p for which the proposed congruences are resoluble. 4. In the case of a redundant or defective system, as well as in the case of a system such as (1), which is neither defective nor redundant, it will be found that the equation of condition d= S may be replaced by the equation a,= a,. And in all these cases alike, whenever the system admits of any solution at all, the number of incongruous solutions is par + (f"-), n denoting the number of the indeterminates. 5. In the case of a redundant system, the condition d = I, or ac = a,, although necessary is not sufficient (Memoir, p. 404). The complete condition of resolubility in this case is that the greatest common divisor of the numbers pan+, pan+A, 2pp-+2,+1.., ("),..... (13) should be the same as the greatest common divisor of the numbers p2a,+, pan.+2,... p(n+l)+...... (14) And the identity of these two greatest common divisors implies not only (as in Art. 2) the equation a, = a,, but also the inequality < an+ - an; (15) Art. 5.] SYSTEMS OF LINEAR CONGRUENCES, 75 since, if this condition be not satisfied, a,,1 will be less than any of the exponents following it in the series (13), and therefore less than any exponent in the series (14). The two conditions expressed by the equation aC= a, and the inequality (15), are sufficient as well as necessary. But it is remarkable that the exponent Au not only satisfies the inequality (15), but also the inequality < a a....... (16) _ n+1-,........ (16) which is, in general, closer than (15), because a, > a,. To prove this, consider, in the given redundant system, any set of n + 1 congruences such that the determinant of their augmented matrix is not divisible in excess by p; and multiply these n +1 congruences, taken in order, by the determinants of their unaugmented matrix. The sum of the products is the determinant of the augmented matrix; we infer that this determinant is divisible by pan + I/, i.e. that c,,+1 an + tt. Since (as has been said) the two conditions of resolubility, taken together, are sufficient, they must involve the inequality (16). To deduce it from them, we may employ the following lemma: 'Let 1lQ1 W = 3 2 n =Let, 2,..., n+m, represent any square or oblong matrix of integral numbers, and let j[qj\ be a partial matrix consisting of any s of the n horizontal rows of I IQ I\; also let Q1, Qni, n; qs, q_- *... be the arithmetical invariants of II QI and q II respectively; then, if r is any number not surpassing s, Q is divisible by -s Qn-r 2T8-r To establish this lemma, let us suppose that Ilql consists of the last s rows of II QI, and let us replace IQ II by a compound matrix of the form IIQ'II x IIQ"II, in which 1 Q"II is a prime matrix of the same type as 1 Q 1, and I1 Q'2 is a square matrix of the form l), kl, 2, kl, 3... 1, n o, h2, k2,3,... 2, n 0, 0, h3,..., k3, n 0 O, o,...,h Then the greatest common divisor of any horizontal row of minors in I Q 11 is the same as the greatest common divisor of the corresponding row of minors in 11 Q'I L 2 76 ARITHMETICAL NOTES. [II. (Memoir, p. 389); it is sufficient, therefore, to verify the lemma in the case of this last matrix. In it we have h x h, x... x h,, = Q hn-_+i x hAn+2x... Xh,= qs; and if we multiply h x h2 x... x hn_ = Qn by any minor of order s - r contained qs in the last s rows of 11 Q'I, we obtain a product which is either a minor of I| Q'll of the order n - r, or else is equal to zero this product is therefore in every case divisible by Q_ r; i.e. Qx q_ r is divisible by Qn r, or is divisible by qS qs Qn-r qs-r To apply this lemma to the case which we are considering here, we change n into n + 1, and we put s = n, r = n - a-. Attending only to the powers of p, we thus find (+n -a+ 1 a, +1 - a, But a, = a,, and a, - a, > t; therefore a,, + - a, > i. 6. When the determinants of the augmented or unaugmented matrix of a proposed system of congruences vanish, we may add multiples of the modulus to the constituents of these matrices, so as to obtain determinants which do not vanish. We proceed, however, to show that this preliminary operation is unnecessary, and that the preceding theory is immediately applicable, notwithstanding the evanescence of the determinants. If, in any matrix of which the arithmetical invariants are V,, V_,,..., V the minor determinants of order s +1 are all equal to zero, we must attribute the same value to the invariant Vs+,; and vice versd, if V8+, = 0, the minor determinants of order s+1 must themselves be all equal to zero. Let V +1 = O, Vs > O, so that Vs, = 0 (r = 1, 2,..., n-s), V_ > 0 (r =, 1, 2,..., s-1). The quotients Vs V, V-2' Vo are determinate positive integers, and the quotient Vl is Vsro, VV zero; but the quotients 2,...,, assume the indeterminate form. To Vs+1 ^n-1 0 these quotients we attribute the value zero: this may be regarded as an arbitrary definition of the value of these indeterminate numbers; it is, however, suggested by the observation that if the constituents of the matrix, instead of being integral numbers, were rational and integral functions of an indeterminate quantity, every factor of Vi would also be a factor of Vi +, and would be contained in Vi + oftener than in Vi; i.e. V-+1 would vanish with Vi. vi Art. 7.] SYSTEMS OF LINEAR CONGRUENCES. 77 7. Let us first suppose that the proposed system of congruences, Ai,,+ Ai,2 X2+...+Ai,,+mX n+m-A,+m+ (mod * ), * (17) i= 1, 2, 3,..., n, m > 0, is not redundant; and let the invariants V,, Vn,1... of its unaugmented matrix vanish as far as V + 1 inclusively, but let V, not vanish. The integers an) anl,..., as+l, an an-l* *a*, a+1a -- an,-1... as+l-as, 09n, (J^n-l~ '''5 (8+2? aa' — l? **l..s +2'-' [S + 1 may then be regarded as greater than any assigned number; and if, in the inequalities (8), which determine the index a, we adopt this interpretation of these symbols, the criterion of resolubility, and the expression for the number of solutions, will continue applicable to the system (17) without any modification whatever. For the criterion of resolubility this is evident, because the demonstration of Art. 2 subsists unchanged. To prove the same thing for the formula which expresses the number of solutions, we employ the following lemma: 'The matrix |A, 12 I A, j= 1, 2,..., n+m, (in which Vs I vanishes, but V, does not vanish,) satisfies an equation of the form x = | ~|. (A) 11!311 xKilx|Hl= o, o.. (A) in which ||/ 3| is an unit matrix of the type n x n, 11y71 an unit matrix of the type (n + m) x (n + m), qio j a matrix of the type n x (n + m), which has all its con0, 0 stituents equal to zero, except those composing the partial matrix |lqijll; this partial matrix being of the type s x s, and having for its arithmetical invariants Vs, Vsl,..., Vo/' Consider the system Ai 1 xl+Ai,2 X2+ -. +Ain+m, +m 0,.... (18) i i i i i 2, 3,... n, which is equivalent to s independent equations; and take for |[7|| any unit 78 ARITHMETICAL NOTES. [II. matrix of the type (n + m) x (n + m), in which the last n + m - s columns are the n + - s rows of a fundamental set of solutions of the system (18). The compound matrix [|A[[ x H7[ will be of the form \lcU, O|i, where l|cjll is a partial matrix of the type n x s, and 110 11 is a partial matrix, composed entirely of zeros, of the type n x (n + m - s); so that I cij, 0 II is of the same type as iiA i[. Again, consider the system Cl,j X + C2, j X2 +.. * + Cn, j Xn =, o o o... (19) j=1, 2, 3,...,, of which the equations are independent; and take for ]|3] any unit matrix of the type n x n, in which the n - s lowest rows are the n - s rows of a fundamental set of solutions of the system (19). The compound matrix |Iil3 x |IA|l x i\\71 is then of the form 0: 0 0, 0 If therefore, in the system (17), we transform the indeterminates by the substitution (X) X2.., ) Xn +n)=||7|| I. I Y, (Ym* Yn n), and at the same time replace the congruences themselves by others linearly derived from them by multiplication with the constituents of the rows of |] i|, we obtain an equivalent system of the form qi, j Y + q2, j Y2+... + qs, j y+ s hj (mod pl),..... (20) j=l, 2,..., s, in which the indeterminates s+1,..., Yn+m do not appear explicitly. By the theorem of Art. 2, the number of solutions of (20) is par+(8-6)+; and to each of these solutions there correspond p(+m-S) solutions of (17), because each of the indeterminates y+l,..., yn+m may have any one of py values. Thus the number of solutions of (17) is p-a+(n+m-)L. 8. Next, let the proposed system of congruences be redundant; and, as before, let V8+, vanish, but not Vs. In this case also the demonstration relating to the criteria of resolubility subsists unchanged; to determine the number of solutions, we select from the proposed system a partial system of s equations such that the determinants of its unaugmented matrix are not divisible in excess by p. Every solution of this partial system satisfies the remaining congruences of the proposed system; i.e. the number of solutions of the proposed system is pa +(~ - ), if n be the number of indeterminates. 9. It is worth while to observe that, in the equation (A), we may attribute Art. 10.] SYSTEMS OF LINEAR CONGRUENCES. 79 to the partial matrix llqijll the simple form V- Vs V8_, V1 For if lull and VS-, VS_2 Vo lvll are reducing units of ll|q|l, so that Vl x x ll-2 V' we may replace l\uil and v II, which are both of the type s x s, by unit matrices of the types n x n and (n + m) x (n + m) respectively; and may then compound these substituted units with the unit matrices /3P11 and |y71. The unit matrices thus substituted for 1 u I and II v contain II u and vIll as partial matrices; their forms are sufficiently indicated by the symbols I, vjO; where I and J are 0, J matrices of the types n x (n - s) and ( + m - s) x (n + m) respectively, and are subject to no restriction other than that implied by the equations v,0 | Tf I-| = 1 I s - =n1 0, ' =1. Another transformation of a matrix, in which the first n- s invariants vanish, is sometimes useful; viz., we may write Vs8 VS_2 Vo where I11w| is an unit matrix of the type n x n, and \\Q\I is a matrix of the same type as \IA 11, of which the s lowest rows form a prime matrix, while the others are arbitrary. The proof of this transformation presents no difficulty, and may be omitted here. 10. When the absolute terms in a linear system of congruences are all congruous to zero, the criteria of resolubility are always satisfied, as they ought to be, because the system is satisfied by attributing to the indeterminates values congruous to zero, and therefore certainly admits of one solution at least. In this case, however, it is usually of importance to determine, not the whole number of solutions, but the number of solutions prime to the modulus (i.e. the number of solutions in which one at least of the indeterminates is prime to the modulus). Reducing the proposed congruences to an equivalent system of the form (12), which can always be done by the transformations indicated in the Memoir (p. 401), and in Art. 7 of the present note, we find that if N is the whole number of solutions, N *..pn- is the number of solutions in which every indeterminate has a value divisible by p, so that N (1 - l a) is the number of solutions prime to 80 ARITHMETICAL NOTES, [II. the modulus. And, in general, if ki is the number of terms in the series (5) which satisfy the inequality a +1 - aS ~ - - i, so that, in particular, /o = n - a, the number of solutions in which every indeterminate has a value divisible by pI will be found to be N -pko+ki+ +.. k-i 11. The transformation of a given square matrix indicated by the equation V_ vn-_1 vn-2 V2 V1 IIAill =llal x vI v vn2 v_,...n V v-' Vol.. 111) forms the basis of many of the demonstrations in this and the preceding note. It has been already pointed out (Memoir, p. 392) that this equation always admits of an infinite number of solutions: this observation may be verified as follows. If IIuijll is any unit matrix of which the constituents are defined by the equations t,j V+I,, j <i, Vi._n+l. Vnj+l u 1= +, j., n-i Vn-j the matrix liviil, of which the constituents are defined by the equations Vi.-Xj V= iX., j V i,!11 — [ x |I~1, I1-x |1|,B will also be an unit matrix; and, if ff al, a |3| be two given unit matrices satisfying the equation (27), that equation will also be satisfied by the unit matrices comprehended in the formulae and by them only III.-On an Arithmetical Demonstration of a Theorem in the Integral Calculus. 1. Let ail Y + a2 y2 +... + an yn = i (A i=1, 2, 3,..., n represent a system of linear equations, in which the numbers aoj and bi are integral, the determinant V = al being different from zero and positive. If, considering the numbers bi with regard to the modulus V, we attribute to them in Art. 2.] THEOREM IN THE INTEGRAL CALCULUS. 81 succession the VI systems of values of which they are susceptible with regard to that modulus, we have the arithmetical theorem (Memoir, p. 405): (i.) 'The system (A) is resoluble in integral numbers y for precisely VW-1 of the Vn systems b.' 2. Let Z represent a complex, or multiplicity, of order n, of which the indeterminates y, y2,..., y, are independent. Any system of values assigned to the indeterminates is called a point of Z; and if the values are integral, the point is an integer point. Let [Xil), Xi2, Xi3, *,? xin] z = 1, 2, 3,..., n, represent n given integer points of E; we shall suppose these n points to be asyzygetic, so that the determinant V = x is different from zero, and may be assumed to be positive. We then obtain the theorem: (ii.) 'The number of integer points [X1, X2,..., X.] which satisfy the inequalities dV dV dV dl + dx 2 * dx, = zi= 1, 2, 3,..., n, is precisely V.' For, by the theorem (i.), the equations dV dv dV X, d + X2 d +... +X d B dx1 1d2 rdx,,,..... (B) i=1,2, 3,..., n of which the determinant is V"'1, and in which each of the numbers Bi is supposed to receive in succession every value from 1 to V"1- inclusive, are resoluble in integral numbers X for V(-1)2 of these Vn(n-l) systems of values. But the solution of the equation (B) is given by the system VXj = B, xj + B2x2j +... + B, nj =1, 2, 3,..., n. If therefore the equations (B) are resoluble in integral numbers for any particular system of values of the numbers Bi, considered with regard to the modulus V, they are resoluble for the vn("-2) systems of values congruous to that particular system for the modulus V, but incongruous to one another for the modulus Vn-l. Hence, if k be the number of systems of values, taken with regard to the modulus V, for which the equations (B) are resoluble in integral VOL. II. M 82 ARITHMETICAL NOTES. [III. numbers, we have kx vn("-2) = V(e-1)2, or k = V; a result which establishes the theorem (ii.). 3. Another proof of the same theorem may be obtained as follows; The number of systems of values for which the equations (B) are resoluble in integral numbers is equal to the number of solutions of the congruences B, xlj + B2 x2j +.. B xj - 0, (mod V), j= 1, 2, 3,..., n, in which B1, B2,..., B, are the indeterminates. And the number of solutions of this system of congruences is V. (Memoir, p. 399, or Arts. 1-3 of the preceding note.) 4. Let [yi, Yi2,..., Yin] z= 1, 2, 3,..., n, represent any n asyzygetic points of I; and let Xj be a positive common measure of the n quantities yj, yj,.., y, j, so that we have n2 equations of the form yi = xij x Xj, in which the numbers xi are integral, and \xi\l = V is different from zero and positive. j==n Let D= yjl = V x I Xj; the number of points of 2, which are of the type j=l [X XI, X2 X2, X, XnX] X, X2,... denoting integral numbers, and which satisfy the inequalities dD dD dD 0 < (X,) d + (X2 2) d...+(Xn Xn) dD. (C) is, by the theorem (ii.), equal to D -I X = V; that is to say, the product I x, taken as many times as there are points (XI X, X2X,..., X, X) satisfying the conditions (C), is equal to D. Let X,, X2,..., X\ be infinitesimal, and let U represent the integral f...ffdY dY2... d Y,, extended to all values of Y1, Y2,..., Yn which satisfy the inequalities dD dD dD O < Y d + Y2 d+....+ Yn d~-D =1, 2, 3,..., n we have, by the definition of a definite integral, U= D. 5. This well-known result (which may be described as a generalization of the expression for the volume of a parallelepiped in space referred to three rectangular axes) may of course be otherwise obtained by employing the formula Art. 6.] THEOREM IN THE INTEGRAL CALCULUS. 83 for the transformation of a multiple integral; viz., if we put U dD Y d+ + dD the limits of uz are 0 and D, and the functional determinant is D"-1; hence rD D rD... ddu du.... du= D-'x U, or D= U. But as we have obtained the value of the definite integral U by a direct process, we may employ the result to demonstrate the formula for the transformation of a multiple integral; and this it may be worth while to do, because some of the demonstrations which have been given of that formula are not free from obscurity. 6. For convenience of expression, we may regard the indeterminates X1, X2, X3,..., X. as rectangular coordinates of a homogeneous and isotropic space of n dimensions. On this supposition we may define the volume unit of Z to be that portion of Z which is comprised between the limits oXi<l, i=l,2,3,...,; so that dX1 dX2... dXn represents an element of 2 having the form (in n dimensions) of a rectangular parallelepiped. In any multiple integral fVd 2, of order n, we must regard the integration as extending over a certain 'integral space,' or portion of X; this integral space is to be resolved into infinitesimal elements, which may be of any form whatever, but which must exhaust the whole of it; each element is to be multiplied by the function V of its coordinates; and the definite integral is the sum obtained by adding the results. When the indeterminates are transformed by the equations X =..,i (X 3) D) i=1, 2, 3,..., n we have to consider a new space a-, in which the new indeterminates x,, x2,..., x may be regarded as rectangular coordinates; to each element of the first space corresponds an element of the second space, and vice versa; the equations (D), whether by their own nature or by restrictions imposed on their interpretation, being supposed to establish a 'one-to-one' correspondence between the two integral spaces. The problem of the transformation of a multiple integral then reduces M 2 84 ARITHMETICAL NOTES. [It. itself to the determination of the ratio of corresponding elementary spaces d I and d o. By the nature of infinitesimal magnitudes, this ratio is independent of the particular form assigned to either element (the form of one of course determines the form of the other). Let us take for the element d a- the parallelepiped (in n dimensions) of which one vertex is at the point [x,, x2,..., xn], and of which the adjacent n vertices are x + S XI, x2 + i x2,..., xn + Si x, i=1 2, 3,..., n, i=1, 2,...,, j, 2, 3,..., n,, determinantities; a conclusion which nfinitesimal and asyzygetic so that, by the theorem ofranformation. Art. If we deomposee of the parallelepipe and (which into elementary parallelepipeds ofn) is Isijxl. If o TT /dX dX dX f UdS=U x |j dXi| x d' in the form d~sru~x,~lx,~dx=s~/dx, =In ts ea dXl dX... dX, =and d, d... d n, t cg ments; but the substution of elements which do not corresponding to the element -= is evidently Buwhich d ie. the ratio of corresponding spaces is the fue e i ntegrational determinant; a conclusion which establishes the theor elem of transformation. 7. If we decompose the spaces i and on into elementary parallelepipeds of the type dX dX... dX, dx d... dxobtaining we may write the equation fUdS=fUx d xdain the form f...ff UdX dX2...dX,=f...ffUx dx. xdxdx2...dx,. 8. In this equation dX dX.. dwhen a definite order is established amnot correspong the ing eleterminates (X) and (x), the substitutio of corresponding elements which do not correspond for elements which do correspond is admissible, because the value of a definite integral is not altered by substituting one mode of decomposition into elements for another. It may be added that the symbol f Ud S, which involves no hypothesis as to the form of the element d 2, supplies the most general and abstract expression of the definite integral; while the symbol f.. ff U dX, dX2... dX,, which indicates a particular mode of decomposition into elements, suggests at the same time a particular method of obtaining its value by successive integration. 8. It will be seen that, when a definite order is established among the indeterminates (X) and (x), the ratio of corresponding elements in the two spaces is Art. 8.] THEOREM IN THE INTEGRAL CALCULUS. 85 determined in sign as well as in magnitude; but that if two indeterminates in either set be interchanged, the sign of this ratio is also changed. The Jacobian loci di = 0 and X3 = o (which for our present purpose we take together) may be regarded, in general, as dividing the space / into two regions A and B, in the first of which d is positive, and in the second negative. Similarly, a dx.d xj dXi -1 is divided by the corresponding loci -dX = = 0, into two regions a and b respectively corresponding to A and B, in such a manner that corresponding elements have the same sign in the regions A, a, and opposite signs in the regions B, b. But in using the formula for the transformation of a multiple integral, it is in general convenient to give to the functional determinant its absolute value, thus considering the elements of both spaces as positive throughout. And when the space over which the integration extends is traversed or bounded by the Jacobian loci, it is always necessary to examine the circumstances which present themselves at these loci. xxv. ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Proceedings of the London Mathematical Society, vol. vi. pp. 140-153. Read June 10, 1875.] 1. RIEMANN, in his Memoir 'Ueber die Darstellbarkeit einer Function durch eine Trigonometrische Reihe' (Abhandlungen der k. Gesellschaft der Wissenschaften zu Gottingen, vol. xiii., p. 87), has given an important theorem which serves to determine whether a function f(x) which is discontinuous, but not infinite, between the finite limits a and b, does or does not admit of integration between those limits, the variable x, as well as the limits a and b, being supposed real. Some further discussion of this theorem would seem to be desirable, partly because, in one particular at least, Riemann's demonstration is wanting in formal accuracy, and partly because the theorem itself appears to have been misunderstood, and to have been made the basis of erroneous inferences. 2. Let d be any given positive quantity, and let the interval b-a be divided into any segments whatever, = x- a, 2 =x2-x, -., =.. b -x_, subject only to the condition that none of these segments surpasses d. We may term d the norm of the division; it is evident that there is an infinite number of different divisions having a given norm; and that a division appertaining to any given norm, appertains also to every greater norm. Let ce, 62,... en be positive proper fractions; if, when the norm d is diminished indefinitely, the sum S= l (a + e1 + 2f)+ (x, + 62 ) +... + f (a- I + n) converges to a definite limit, whatever be the mode of division, and whatever be rb the fractions 62, 62,..., e, that limit is represented by the symbol I f(x) dx, and a Art. 3.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 87 the function f(x) is said to admit of integration between the limits a and b. We shall call the values of f(x) corresponding to the points of any segment the ordinates of that segment; by the ordinate difference of a segment we shall understand the difference between the greatest and least ordinates of the segment. For any given division, r2,..., S., the greatest value of S is obtained by taking the maximum ordinate of each segment, and the least value of S by taking the minimum ordinate of each segment; if Di is the ordinate difference of the segment Si, the difference 0 between these two values of S is 0 = SA DI + J2 D2+... +.n D. But, for a given norm d, the greatest value of S, and the least value of S, will in general result, not from one and the same division, but from two different divisions, each of them having the given norm. Hence the difference O between the greatest and the least values that S can acquire for a given norm, is, in general, greater than the greatest of the differences 0. To satisfy ourselves, in any given case, that S converges to a definite limit, when d is diminished without limit, we must be sure that 0 diminishes without limit; and it is not enough to show (as the form of Riemann's proof would seem to imply) that 0 diminishes without limit, even if this should be shown for every division having the norm d. 3. Let A (d) be the greatest value of S appertaining to a given norm d, and let B (d) be the least value of S appertaining to the same norm. If d, and dg are any two norms, of which d, is greater than d2, it is evident that A (d1) ~ A (d2), B (dj) < B (d2), because every division appertaining to the norm d2 also appertains to the norm d1. And it may be proved (although, for brevity, we omit the demonstration here) that, given any norm dj, we can always assign a norm d2, less than d1, which shall satisfy the inequalities A (d,) > A (d2), B (d,) < B (d2); except only when the function is such that the maximum (or minimum) ordinate is the same, throughout the whole interval, for all segments however small. In this excepted case, which is one by no means inconceivable, the value of A (d), [or of B (d),] is independent of d, and is simply h (b - a), where h is the maximum (or minimum) ordinate common to all segments of the interval b - a. In all other cases, it is possible to assign a series of norms, decreasing without limit, and such that the corresponding maximum values of S form a decreasing series, while the corresponding minimum values of S form an increasing series. Besides the maximum and minimum values of S corresponding to a given norm, we have also to consider the maximum and minimum values of S corresponding to a given division. Let P (d) be the maximum value of S appertaining to a given division of norm d, and let Q (d') be the minimum value of S 88 ON TEE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 4. appertaining to a different division, having the same norm or a different norm. It is important to observe that we shall always have P(d)> Q (d'), the sign of equality being inadmissible, except when the function is such as to be represented geometrically by a single segment, or a system of segments, parallel to the axis of x. Leaving out of consideration the excepted case, we may enunciate the theorem-' The least value of S that can be obtained by taking, in any division whatever, the greatest ordinate of each segment, is greater than the greatest value that can be obtained by taking, in any division whatever, the least ordinate of each segment.' To prove this theorem, let the two divisions, which give the values P(d) and Q(cd), be simultaneously applied to the interval b-a. To obtain P (d), each segment in the resulting division will have to be multiplied by its greatest ordinate, or by a still greater ordinate in some adjacent segment; whereas to obtain Q (d') each segment will have to be multiplied by its least ordinate, or by a still less ordinate. It follows that we have, in general, P (d) > Q (d'). If, however, we regard the interval b - a as composed of segments 1, 1,..., each of which has for its extremities points which are also extremities of segments in each of the two given divisions, we shall find that the inequality P (d)> Q (d') must be replaced by the equality P (d) = Q (d'), if it should so happen that the maximum ordinate of each segment I is the same as its minimum ordinate; i.e., if the function f(x) is represented geometrically by a series of segments parallel to the axis of x, and respectively equal to the segments ll, 12,.... 4. Again, let B'(d) be the least value of S corresponding to the division which gives A (cd); and let A'(d) be the greatest value of S corresponding to the division which gives B (d); it is evident from what has been said that we shall have the inequalities A (d) > A'(d) > B'(d) > B (d). Now A (d) -B (d) = [A (c) - B' (d) + [A'(d) -B (c)]- [A' (d)- B' (d)]; and A' (d) B' (d); therefore A (d) - B (d) < [A (d) - B' (d)] + [A'(d) - B (d)]. Hence, to prove the evanescence of A (c) - B (d) or (, it suffices to prove the evanescence of A (d) -B'(d), and of A'(d)-B (d), which are, in fact the two values of 0 corresponding to the two divisions which give the absolutely greatest and least values of S for the norm d. 5. The theorem of Riemann may be enunciated as follows:'Let oa be any given quantity, however small; if, in every division of norm d, Art. 7.1 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 89 the sum of the segments, of which the ordinate differences surpass a-, diminishes without limit, as d diminishes without limit, the function admits of integration; and, vice versd, if the function admits of integration, the sum of these segments diminishes without limit with d.' The following (with a slight modification suggested by the preceding considerations) is Riemann's demonstration of the first part of the theorem: Let s, be the sum of the segments which, in the division corresponding to A (d) and B'(dc), have ordinate differences surpassing a-; and let Q be the greatest ordinate difference in any division appertaining to the norm d; Q is necessarily finite, because all the ordinates are finite. The contribution of the segments s, to the difference A (d) - B' (d) cannot surpass s, x Q, and the contribution of the remaining segments cannot surpass o- x (b - a - s); i.e., A (d) - B' (d) s1 x Q + a (b - a - s). Similarly, if s2 is the sum of the segments which, in the division corresponding to A' (d) and B (d), have ordinate differences surpassing a-, A'(d) - B(c) x Q+(b - a - s). Adding these two inequalities, we find A (d) - B (d) (s + s2) (Q - a) + 2a (b - a). But a may be taken as small as we please, and, by hypothesis, however small a may be, d can always be taken so small as to render s, and s2 as small as we please; i.e., the difference A (d) - B (d) = 6 diminishes without limit with d, and f(x) admits of integration between the limits a and b. 6. Riemann's demonstration of the second part of the theorem requires no modification. For, if S converges to a definite limit, 0 must be comminuent with d, and, a fortiori, each of the quantities 0 must be comminuent with d. But, evidently, in any given division in which s is the sum of the segments having ordinate differences which surpass ac, as 0. Hence, however small the given quantity a may be, we can always, by taking d small enough, make - less than O' any assigned quantity; i.e., if S converges to a definite limit, s must diminish without limit at the same time with d. 7. It will be observed that, in order to establish the convergence of S to a definite limit, it is sufficient to know that the sum of the segments, having ordinate differences surpassing a-, is comminuent with d in each of two specified divisions [viz., in the division which gives A (d) the maximum value of S, and in VOL. II. N 90 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 8. that which gives B (d) the minimum value of S]. Hence, if these two sums are comminuent with d, the corresponding sum in any other division of norm d is also comminuent with d. 8. Let us suppose that the function f(x) has any number of discontinuities between a and b; and let there be, (a) points at which there are discontinuities surpassing v-. (We say that a discontinuity surpassing cr exists at a given point, when any segment, however small, being taken which includes that point, the ordinate difference of the segment surpasses a.) If + (Ca) has a finite and assignable value for every value of o-, however small, the condition of integrability is certainly satisfied, even if + (rr) increase without limit, when a- diminishes without limit. For, in any division of norm d, the sum of the segments having ordinate differences which surpass cr-, cannot surpass 2 dx J (a-); and, however small Ca may be, d can be taken so small that 2d x + (c) shall be less than any quantity that can be assigned. As an example, we may take the function considered by Riemann, viz., (x) (2x) (3x) (x)= + 4 + 9 + where, by (x) we are to understand the (positive or negative) excess of x above the whole number nearest to x; or, if x lies half-way between two whole numbers, the arithmetical mean between the two differences. and -, i.e., zero. In this function, if x = 2n, where m and 2n are relatively prime, we have + 0) =f( - 12 -2 2 no) - 16n2 Thus the number of discontinuities in any given interval is infinitely great. But the number of discontinuities which in any given interval surpass a given quantity a, is always finite. For example, the number of discontinuities between O and 1 which surpass a, is equal to the number of irreducible proper fractions, having even denominators 2n, which verify the inequality 2> a; or, if P (m) be the number of numbers not surpassing n and prime to m, and if h be the greatest integer not surpassing 2, the number of discontinuities in question is 2 2 Art. 11.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 91 which is evidently finite for any given value of a-, although it increases without limit when a- diminishes without limit. 9. Next, let us suppose that f(x) in the interval b - a has an infinite number of discontinuities surpassing a given quantity ao. The points at which these discontinuities occur may either 'completely fill' one or more finite portions of the interval b - a, or there may be no finite portion of that interval which is 'completely filled' by them. A system of points is said to 'fill completely' a given interval when, any segment of the interval being taken, however small, one point at least of the system lies on that segment. Thus the rational points on any line, i.e., the points of which the abscissae are rational, completely fill any segment whatever upon the line. We may observe that the assertion, that any given segment of an interval contains at least one point of a given system, is equivalent to the assertion that any given segment contains an infinite number [i.e., a number greater than any that can be assigned] of the points of the system. For we may divide the given segment into as many parts as we please, and each of them must contain at least one point of the system. 10. When the points at which there occur discontinuities surpassing a- completely fill any finite portion of the interval b - a, the function f(x) is certainly incapable of integration. For, if I be the total length of the segments which are completely filled, we have evidently 0 > a-i for any division of any norm d; i.e., it is impossible that 0 should diminish without limit with d. But points may exist in an infinite number within a finite interval, without completely filling any portion of that interval. Whenever this happens, it must be possible in any given segment of the interval, however small, to take a finite part such that it shall contain no point of the system; otherwise, the segment in question would be completely filled. We give a few examples of such systems of points, the limits of the interval being in each case 0 and 1. We shall say, for brevity, that points are in close order on any segment when they completely fill it, and in loose order when they do not completely fill it, or any part of it however small. 1 11. (i.) Let the system of points be defined by the equation x = -, a being any positive integer. It will be seen, (1) that these points are infinite in number; (2) that they are indefinitely condensed in the vicinity of the origin; (3) that they are in loose order over the whole interval, no segment, even in the immediate vicinity of the origin, being completely filled. For if cl be any given quantity, however small, we can always find a finite integral number such N 2 92 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 12. that - < d, and then the finite spaces (, -) (, &c.... all men\ l,+ n m+2 m+1 lie on the segment (0, cl), and are all free from points of the system, if we leave their initial and terminal points out of account. 1 1 12. (ii.) Let the system of points be defined by the equation x = + - a, a2 where a, and a2 are any positive integers. Here, it is evident that the points are indefinitely condensed in the vicinity of each of the points of the system (i). But it can also be shown that they are in loose order over the whole interval from 0 to 1. Let x=L1, x=L2, (L1<L2,) be two consecutive points of the system (i); let, be any positive quantity whatever, and consider the segment ~L~ + L~ (I +-~, L2). If x -+ lies on this segment, we must have L1, 1 l +1 a a2 a~ -< L,, because no point of the system (i) lies on the interval (L1, L2); and also 1 1J+1;+ L q- L. -+ - > L whence a < -, a L -. These inequalities show aC a2 /A +1 L -L 2L -L - that, if, from the beginning of any free segment in the system (i), we cut off as small a part as we please (which we may do by taking M great enough), the remaining portion of that segment will contain only a finite number of points belonging to the system (ii). And this suffices to prove that the points of the system are in loose order; for if d be any segment, however small, situated anywhere in the interval from 0 to 1, we can certainly find on this segment a part free from points of the system (i), and, by what has just been proved, parts of that part will be free from points of the system (ii). 13. (iii.) Let a system of points P, +, be defined by the equation 1 1 x= ---t- +-+... + — 01 2 as +1 where a1, a2,..., ca + are positive integers. Assuming (what has just been proved for s = 2) that the system P8 is in loose order over the whole interval from 0 to 1, we shall prove the same thing for the system P,+1. Let x= L, x= L2 be any two consecutive points of the system P8; and consider as before the interval 1 1 1 (t + 12, L2). If the point P. +i, or x =- + +. + -, lies on this interv +1 2 xl aS a +I val, we must have, besides the inequality a, a. s +1 - 1 Art. 14.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 93 the s + 1 inequalities included in the formula 1 1 1 1 1 — + +... - <L+ — a a2 as +1 ai because no point of the system P8 can be between L1 and L2. These inequalities give a, i=l, 2, 3,..., +l, L2 - LI whence we may infer, precisely as in the case in which s = 2, that the points Ps+ are in loose order over the whole of interval from 0 to 1. 14. Letf(x) be a function, which coincides with a given continuous function p (x) for all values of x between 0 and 1, except at the points P +,; and let the difference between f(x) and k (x) at those points not exceed the finite quantity cr. It may be shown that f(x) is integrable between the limits 0 and 1, and that 1 1 Xf (x) dx = / (x) dx. For, take any small interval from 0 to A; the points P1 which lie outside it, between J to 1, are finite in number and at finite distances from one another. Let there be A1 of them; from each of them measure a space Al to the right; the number of points P2, lying outside of the measured spaces S + A, 1, is necessarily finite; and these points are at finite distances from one another. Let their number be A2, and measure a distance S3 to the right of each of them. Proceeding in this way, we shall obtain measured spaces amounting in all to +Al 11+ A + A+... + A I-= H. Let e be any given quantity however small; and in the preceding construction let 3s 2 +2) 2 ( 2)A ** < + < 3( 3 (s + 2) ' 3 (s + 2) U 3$ (s+ 2)A2 +1 3($ 2) As + we shall thus have H< ~e. Let d be the least of the spaces S, S1,,., +; it may be shown that, in any division of norm d, the sum of the segments containing points PS + cannot exceed 3 H. For all the points P +1 lie on the measured spaces; and supposing (which is the most unfavourable case) that one of those spaces begins and ends with a point P+ 1 we can at most triple it, by imagining a segment equal to cl placed on each side of it. Thus, in every division of norm d, the sum of the segments containing the points of discontinuity is less 94 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. LArt. 15. r1 than e; whence we infer, by Riemann's theorem, thato f S(a) dx has the same value as (x) dx. 15. (iv.) Let m be any given integral number greater than 2. Divide the interval from 0 to 1 into m equal parts; and exempt the last segment from any subsequent division. Divide each of the remaining m -1 segments into m equal parts; and exempt the last segment of each from any subsequent division. If this operation be continued ad infinitum, we shall obtain an infinite number of points of division P upon the line from 0 to 1. These points are in loose order: for if d be any segment however small, situated anywhere in the interval from 0 1 to 1, we may take an index k which satisfies the inequality -< dd; and then a a+1 m determine a segment of the type (-I ) lying entirely on the segment d. But this segment is either itself an exempted segment or its mth part is so. It will be seen that, after k operations, the sum of the exempted segments amounts to 1 - (1 - -); so that, as k increases without limit, the points of division P occur upon segments which occupy only an infinitesimal portion of the interval from 0 to 1. And it may be inferred that a function, having any finite discontinuities at the points P, would be integrable. For, if d be any given small 1 1 quantity, let the index k be determined by the inequalities -u> d > -; the L *1. Ml+1 number N of excepted segments which surpass -k is 1 + (m- 1)+ (m- 1)2 +... + (m- 1)k-2; and the sum of the remaining segments is It is evident that in any division of norm d, the sum of the segments containing points P cannot exceed 1 \k1 (1- 1 - + 2.Nd. But, as d decreases, and k increases, without limit, (1- ) and 2Nd, which is less than,-, both decrease without limit; i.e., in any division of norm d, the Art. 17.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 95 sum of the segments containing points of discontinuity diminishes without limit with d; and the function is integrable. 16. (v.) Let us now, as in the last example, divide the interval from 0 to 1 into m equal parts, exempting the last segment from any further division; let us divide each of the remaining m - 1 segments by in2, exempting the last segment of each segment; let us again divide each of the remaining ( - 1) (m2 - 1) segments by M3, exempting the last segment of each segment; and so on continually. After k - 1 operations we shall have N= 1 + (m- 1) + (m- 1) (m2- 1) +... + (m-1) (m2- 1)... (-2 _- 1) exempted segments, of which the sum will be 1- 1- 1) (1-1)...(1 -_1) This sum, when k is increased without limit, approximates to the finite limit 1 - E (); where E (-) is the Eulerian product l (i - -7) and is cer\m/ \m 1 tainly different from zero. The points of division Q exist in loose order over the whole interval. For, if d be any small segment of that interval, and if 1 ca a+1 a.k(_ <1 d, a segment of the type (mI, 'a -loi)) can be found lying entirely on the segment d, and this segment is either itself exempted, or its 1 \th,:,A ( ) part is exempted. But a function having finite discontinuities at the points Q would be incapable of integration. For, if d be any norm, and 1 J < k(k-l) < d, in the division + _, =0, 1, 2, 3,... (which is a division of norm d), the sum of the segments containing points of discontinuity is ) (1-)...(1 M1 m+ k(k-i) which approximates to the finite limit E (1 when d is diminished, and - is increased without limit. 17. The result obtained in the last example deserves attention, because it is opposed to a theory of discontinuous functions, which has received the sanction of an eminent geometer, Dr. Hermann Hankel, whose recent death at an early 96 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 17. age is a great loss to mathematical science. In an interesting memoir (' Untersuchungen ueber die unendlich oft oscillirenden und unstetigen Functionen,' Tiibingen, 1870), Dr. Hankel has laid down the distinction, here adopted from him, between a system of points which completely fill a segment, and a system of points which do not completely fill any segment, but lie in loose order. [The term employed by Dr. Hankel is 'zerstreut'; the use of the equivalent English words 'dispersed' or 'scattered' has been avoided in the present note, because they might seem to exclude the sort of condensation in the vicinity of a finite or infinite number of points, which, as we have seen in the examples (i.), (ii.), (iii.), may present itself in the case of systems of points in loose order.] Dr. Hankel then asserts (see p. 26) that, when a system of points is in loose order on a line, the line may be so divided as to make the sum of the segments containing the points less than any assignable line. The proof of this assertion is, in effect, as follows:-Divide the line into segments, of which each contains a point of the system, and imagine each segment to be diminished to its nth part, yet so as still to have upon it the point of the system which it contained before. The sum of the segments can thus be made less than the nth part of the whole line; i.e., less than any line that can be assigned, because we may suppose n as great as we please. It must be conceded that this demonstration is rigorous, if the number of points in the system is finite; but the construction indicated ceases to convey any clear image to the mind, as soon as the number of points becomes infinite. If we are allowed to divide the line from 0 to 1, in example (iii.), in such a manner as to include every point (Ps+) in a segment of its own, these segments, in the vicinity of the points P,, will have to be less than any line that can be assigned; and, if such a mode of division is admissible, it is difficult to see why it should not also be considered admissible so to divide the line as to include every rational point in a segment of its own: in which case Dr. Hankel's proposition would extend to systems of points in close order, as well as to systems in loose order. But whether we do or do not admit the truth of Dr. Hankel's proposition, the use which he makes of it (p. 31) to establish the applicability of Riemann's criterion to a certain class of functions would seem to be erroneous. To prove that Riemann's condition of integrability is satisfied for a given discontinuous function, we have to show that, given any finite quantity d, however small, the sum of the segments, which, in any division whatever of norm d, contain the points of discontinuity, is evanescent with d. And it is evident that this cannot be shown, if we confine ourselves to considering modes of division in which some of the segments Art. 19.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 97 are from the very beginning assumed to be less than any quantity that can be assigned. While, therefore, we nay safely admit the theorem that no function can be integrable which has discontinuities, surpassing a given quantity a-, at an infinite number of points forming a system in close order; the converse assertion that, when the system of points of discontinuity is in loose order, the function is integrable, would seem to be established by no satisfactory demonstration, and to be negatived by the result obtained in example (v.). 18. Another proposition, contained in the same memoir (p. 28), appears open to a similar objection. It may be admitted that a function f(x) having discontinuities, which surpass a given quantity a- however small, only at points which form a system in loose order, is necessarily continuous over finite portions of any interval however small. But it would seem to be untrue that such a function is necessarily continuous in the vicinity of any one of its points of discontinuity. If, for example, f(-+-) =1, and f(x)=0, for every other value of x, it is evident that, however small the given quantity e may be, the difference oscillates an infinite number of times between the values 0 and 1, as J decreases from e to 0; i.e., the function f(x) is discontinuous in the vicinity of the point - to the right. a, 19. We add a few remarks which may serve still further to illustrate the meaning and use of Riemann's theorem. (i.) The problem, 'Given a system of points upon an interval (a, b), to find, among all divisions of norm d, that in which the segments containing the points have the maximum sum,' is perfectly determinate. We may say that a point of the system is isolated, when it is separated from the next preceding and next following point by a distance > 2d. Similarly a group of points may be said to be isolated, when the distance between any two consecutive points of the group is less than 2d, but the distance between the extreme points of the group, and those which immediately precede and follow it, is greater than 2d. It is evident that, for any given value of d, the given system of points resolves itself into a finite number of isolated groups. The first and last point of each group determine a segment; on either side of each of these segments, and on either side of each isolated point, we may place a segment VOL. II. O 98 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 20. equal to d. The sum of the segments thus obtained is the maximum sum required. It will be observed that in this solution each point of the system is regarded as double; i.e., as capable of affecting two segments at once, one on each side of it. If the discontinuity of a function at any point can be removed by changing the value of the function at that point only, for example, if f(x- 0) =a, f(x) = a +-, f(x+ 0) = a, the point must be regarded as single (its contribution to the difference 6 of Art. 2 would be only a x d). But if the values of the function preceding and following the point of discontinuity are different (i.e., if f(x - 0) = a, f(x + 0) = a + o-), the point of discontinuity produces a double effect, its contribution to the difference 6 being 2 a- x d. Similarly, in the case of functions which, like cos (-) in the vicinity of the origin, admit of an infinite number of maxima and minima within a finite interval, the contribution to 6 of each point at which there is a maximum, or minimum, is two-fold. For the practical application of Riemann's criterion, the distinction between points producing a one-fold effect and those producing a two-fold effect is immaterial. 20. (ii.) When a function, which is discontinuous but never infinite, does not admit of integration between the limits a and b, the symbol f(x) dx becomes indeterminate. But the maximum and minimum values attributable to that symbol are perfectly determinate; and if it should become advisable to attribute a definite value to the symbol, we- might select for that purpose the arithmetical mean between these two extreme values. If, for continually decreasing values of d, we calculate the corresponding maximum values of the sum S of Art. 2, these values will, as shall now be shown, converge to a determinate limit A. And similarly the successive minimum values of S will converge to a determinate limit B, different from A in the case under consideration. The difference A - B is, of course, the limit of the successive differences 6. From the two sets of inequalities A (d) > A (d,)> A (d) >.. B (d) <B (d) < B(d3)<., combined with the inequality A (dc) > B (d.), which holds for any value of n however great, we infer that each of the two series, A (d,) -A (), A (d)- A (d,), A (d3)-A (d,),., B (d2)-B(d), B(d)- B (d), B (d) - B(d),., Art. 21.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 99 consists of positive terms, and that, however many terms of either series we add together, we can never surpass A (d) - A A(d&) in the first, and B (d,) - B (d,) in the second; i.e., in neither of them can we ever surpass A (d,) - B (d,). But if a series of positive terms be such that the sum of any number of its terms, however great, can never surpass a given finite quantity, the sum of the first n terms of the series converges to a finite and determinate limit, when n is increased without limit (see Riemann, Vorlesungen, pp. 39, 40). The sums A (d) - A (dn), B (d) - B (d6), therefore converge to finite and determinate limits; or, which is the same thing, the two series of terms A (di), A (d), A (d),.. B (di), B (d), B (d),..., converge to finite and determinate limits. If, for example, the function f(x) have the value a-, at every point of the system considered in Art. 16, example (v.), and the value O2 < a- at every other point; we shall find B = 2, A= 2+ (Cl- 2) xE 21. (iii.) Riemann's criterion of integrability is applicable to the case of any multiple integral extended over a finite space. For example, in the case of a triple integral, we must imagine the whole space of the integration divided into small spaces such that any one of them could be comprehended within a sphere of a diameter d; and any such division into spaces is a division of norm d. The criterion of integrability, then, is that, in any division whatever of norm d, the sum of the spaces in which the ordinate-differences surpass a given quantity a-, must diminish without limit with d. The ordinate-difference of any space is, of course, the difference between the greatest and least values of the function within the space. Considering, for simplicity, the case of two dimensions only, we observe that the space of integration may not only contain points of discontinuity finite or infinite in number, but may be intersected by curves of discontinuity. The function may have values differing by a finite quantity on either side of such a curve; or its values at points along the curve may be discontinuous, or both of these kinds of discontinuity may be combined at the same curve. If L (a), the total length of the curves at which the discontinuities surpass a, be finite, the function can be integrated over the given space; since, if we draw curves parallel to the curves of discontinuity and at a distance d from them on either side, the 0 2 100 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 21. area of the channel-like spaces thus obtained will be 2 dL (a), and will surpass the greatest sum of spaces, including the curves in any division of norm d. But the function may be integrable even if the total length of the curves of discontinuity is infinite; because an infinite number of contiguous curves may be enclosed in one and the same channel. And, provided that the curves can all be included in channels of which the length is L, and of which the breadth ~ is comminuent with d, the condition that L x d should be comminueni with d, will suffice to ensure the integrability of the function. XXVI. ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Proceedings of the London Mathematical Society, vol. vi. pp. 153-182. Read June 10, 1875.] THE ordinary singularities of a plane curve are its double points and double tangents, its stationary points and stationary tangents; or, as they have been also called, its nodes and links, its cusps and inflexions. The fundamental theorem, that any of the so-called higher singularities of a plane curve may be regarded as equivalent to a certain number of ordinary singularities of each of these four kinds, has been enunciated by Professor Cayley, who has also given a method for determining in every case the four. indices,r, K, i, proper to any given singularity. Several enquiries, which appear to possess some interest, are suggested by this theorem. Among them we may mention the two following(1) It is important to prove that the indices of singularity, as defined by Professor Cayley, satisfy the equations of Pliicker; and that the 'genus' or 'deficiency' of the plane curve is correctly given, by these indices. (2). It is also of interest to examine whether any given singularity can be actually formed by the coalescence of the ordinary singularities to, which it is regarded as equivalent; in other words, whether a singularity of which the indices are S, a, K, i, and which is therefore to be regarded as equivalent to S double points, r double tangents, K cusps, and i inflexions, possesses a penultimate form, in which all these singularities exist, distinct from one another, but infinitely close together. The present paper relates chiefly to the first of these enquiries; the second is reserved for a future communication. 1. Consider a plane curve C of order m and class n, defined by an equation 102 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 2. F (p, q) = 0 between the parameters of two pencils, of which the corresponding rays intersect on C, and which are represented by equations of the form p (QP) + (QR) = 0, q (PQ) + (PR)= 0; P, Q, R denoting the three vertices of a triangle, (PQ)= 0, (QR) = 0, (PR) = 0 the equations of its sides. It is convenient to suppose that Q and P, the centres of the two pencils, have no speciality of position with regard to C; or, more precisely, that neither Q nor P lies on the curve, nor on any singular line appertaining to the curve. Under the general name of singular lines we include (1) lines joining two singular points, (2) singular tangents, (3) tangents at singular points, (4) tangents passing through a singular point; we shall also suppose that PQ is not a tangent to C, and does not pass through any singular point. Thus to every finite value of p there will correspond in finite values of q, and vice versd; and, in particular, to any singular point on the curve there will correspond a finite pair of values ofp and q. To an infinite value of q there will correspond m infinite values of p, and vice versd; these answer to the m intersections of PQ with the curve, no two of which, by hypothesis, are coincident. We may, if we please, project the line PQ to an infinite distance, and regard p and q as Cartesian coordinates; we prefer, however, for our present purpose, to consider them as parametric ratios; i.e., as purely numerical quantities (real or complex). 2. Let f(q) be the discriminant of F(p, q)= 0, considered as an equation of the order m in p; we may suppose the coefficient of pmn, which is certainly different from zero, to be unity. The first polar of P with regard to C is dF dF,d= 0, andf (q) is the resultant of the elimination ofp from F and, so that the roots off(q) = 0 are the parameters of the lines drawn from P to the points of intersection of C with the first polar of P. Attending to the suppositions which have been made as to the situation of P and Q relatively to the curve C, we infer (a) that f(q) has no infinite roots, and is therefore of the full order In (mn - 1) in q; (p3) that f(q) has n, and only n, non-multiple roots q'; (7y) that for each of these n roots q' the equation F (p, q') = 0 acquires two equal roots p', its remaining roots being all different from p', and from one another; ({) that q' is not a multiple root of the equation F(p', q) = 0. The n sets (p', q') give the n points of contact of tangents from P; the remaining factor of f(q), viz., fi (q) =f(q) I (q - q'), consists exclusively of multiple factors, and appertains to the singular points of the curve. The index of its order, i.e., n (n - 1) - n, we may term the total discriminantal index of the singular points of the curve. Let q0 be a root of f, (q) = 0 of multiplicity v; the equation F (p, qo) = 0 has but Art. 3.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 103 one multiple root; let this be p,, and let its multiplicity be,u; then (po, qo) is a singular point 0 on the curve, of which the order (i.e., the least number of points in which it is cut by any straight line passing through it) is,u, and of which v may be termed the discriminantal index. It is evident that the number of singular points is equal to the number of unequal roots of fi (q) = 0, and that the total discriminantal index is equal to the sum of the discriminantal indices of the separate singular points. We shall presently (Art. 8) see that the discriminantal index of a singular point can in general be further subdivided into parts, appertaining respectively to the different branches of the curve which pass through the point, taken singly, and in pairs.. 3. It is a well-known theorem of Cauchy, that so long as the analytical modulus of q- q0 is less than the least of the modules of any of the quantities 9l - qO, where q, is any root off(q) = 0 other than q0, the mr roots of the equation F (9p, q) = 0 are developable in convergent series of the form —. o= A +A, (q- q) + Al (q- q)al +Aq- ( q - a2 "+.. (A) the exponents al, a2,... being rational and positive numbers, which satisfy the inequalities 1 < a < a2 <.... Of the equations (A), m-, give the values of p corresponding to the m -, points not in the vicinity of O, in which C is cut by the line (q). The series in the right-hand members of these m - u equations we shall designate by A1, A2,..., A_,: we observe that in them the quantities A are all different from one another and from zero; because (q0), not being a singular line, intersects C in m - 1, points, which are different from one another and from 0; also, in these equations, the exponents ac, a2, a3,... are integral. In the remaining Ju equations, which give the developments appertaining to the branches of C that pass through 0, the quantities A are all equal to zero: these equations divide themselves into groups of conjugate equations, the equations of any one group being of the type 01 p -op = Bo (q- qo) + B ll (q- qo) A+.*, where the numerators.I are positive, integral, and increasing; A is less than P,, and is the least common denominator of the fractional exponents; c is any root of wA = 1: so that, if we use one and the same value of the radical in all the A equations of the group, they will differ from one another only by containing different values of w; each of the tt equations defines a branch of the curve passing through O., If A = 1, the branch is linear or of order 1; if A > 1, the A conjugate equations are regarded by Professor Cayley as defining A partial branches 104 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 4. forming a single superlinear branch of order A; in every case the sum of the orders of the branches is equal to the order of the point, i.e.,:A =,. The coefficients Bo are all different from zero, and the indices -L are all greater than unity, because neither (po) nor (qo) is one of the tangents at 0; but these coefficients and indices are not necessarily different in two developments belonging to two different linear or superlinear branches-indeed any two such developments may coincide for any finite number of terms; and to ascertain the true nature of any singular point it is indispensable to continue the developments until they all become different from one another. The series in the right-hand members of the, equations we denote by B1, B2, B3,..., B,. 4. The series A and B of the preceding article are absolutely convergent within the assigned limits; i.e., any one of these series would continue to be convergent within these limits if its terms were replaced by their analytical modules. For the multiplication of two absolutely convergent series we have the theorem:'If the product of two given absolutely convergent series, proceeding by ascending powers of a variable, be arranged in a series proceeding in the same manner, this series is absolutely convergent for all values of the variable for which the given series are absolutely convergent, and its sum is equal to the product of the sums of the given series.' (Cauchy, 'Analyse Algebrique,' cap. vi.) Multiplying together the m series p- o- A, and p -p0 - B, we obtain, by virtue of this theorem, the equation F(p, q) = n (p -o - A) x n (p -~o - B). This equation is an identity; i.e., if the multiplication be actually effected on the right-hand side, all powers of q - q0 above the mth will disappear, and the terms that remain will be precisely the terms of F(po +p -o, oP 0 + q - qo) or F (p, q). But an arithmetical equality between the two sides of the equation subsists only so long as the analytical modulus of q - q0 does not surpass the limit assigned in Cauchy's theorem (Art. 3). Subject to the same limitation, f(q) is equal to the product of the squares of the differences of every two of the series A and B. 5. The number of the intersections at any point 0 of two branches of the same curve, or of different curves, which pass through the point, and which are there represented by equations of the form p(l) -_ = B1) ( - qo) + (2) -2 = B2) (q - q) +... Art. 6.] ON THE HIGH-ER SINGULARITIES OF PLANE CURVES. 105 is defined by Professor Cayley to be the number which expresses the order of evanescence of (l) -p2), i.e., the integral or fractional exponent X for which o(1) -,(2) (_- -i) X has a finite limit, when q -o is diminished without limit. We may (q - q0) justify this definition by proving that, whenever two curves C, and C2 have a multiple intersection at any point, its multiplicity is correctly obtained by adding together the numbers (as thus defined) of the intersections of each branch of 0C by each branch of C2. If we suppose (as we may do) that the points P and Q have no speciality of position with regard to the curves Ci and C2 considered as one curve, the resultant D (q) of the equations 0C (p, q) = 0 and C2 (p, q) = 0 is of the order m1 x m2; and if t,x branches of C, and /2 branches of C2 pass through 0, we shall have, for Cm, ml-,u equations A(1), and, equations B(); and similarly, for C2, m2 - 2 equations A(2), and u, equations B(2). Denoting by IT (B(1 - B()) the product of the l x Au2 differences obtained by subtracting them in succession, each series B(2) from each series B1), and by X the number of intersections, as above defined, of any one branch of C7 by any one branch of C2, we see that the limit of II (B(1) - B(2)) (q q0)- is finite. But () (q) = II (B(1) - B(2)), the sign of multiplication now extending to all the m1 x m, differences obtained by considering the mi series A(') and B(1), and the m2 series A2 and B,; and of these m1 x m, differences, none, except the AuL x Au2 differences already considered, are evanescent with q - qo (for the hypothesis that P and Q have no speciality of position with regard to the system of the two curves C, and C2 implies that none of the constants A(I) can be equal to any of the constants A(2)). Hence 2 X is the multiplicity of the factor q - qo in d) (q); i.e., since (po, q0) is the only intersection of C( and C2 which lies on (q0), 2 X is the multiplicity of that intersection. If we regard the equation F(p, q) = 0 as determining a correspondence of points on a line, the coincidences of corresponding points (except indeed the coincidences p = q = oo) answer in number and multiplicity to the intersections of C by the straight line p = q. We are thus led to a theorem given by M. Zeuthen ('Bulletin des Sciences Mathematiques,' Vol. V., p. 186). 6. As it is only the hypothesis that the points P and Q have no speciality of position with regard to C which gives us a right to assert that every one of the developments B contains a term linear in q - q, and no term in which the exponent of q - qo is less than unity, it is worth while to see how far the results of the preceding article can be depended on when this hypothesis is dispensed with. It will be found that if (po) is one of the tangents at 0, i.e., if one, or more, of the coefficients BP is zero, the discriminantal index of the point is still VOL. II. P 106 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 7. equal to the order of evanescence of II (Bi - Bj)2. But this conclusion would no longer hold, if (qo) were one of the tangents at O. In this case the developments appertaining to the branches to which (q0) is a tangent would contain powers of q-q, inferior to unity; and the order of evanescence of II (B -Bj)2 would exceed the discriminantal index of 0 by r, if n - r is the number of tangents other than (qo) which can be drawn to the curve from P. But the multiplicity of the intersection of two different curves, at a point which is singular for one or both of them is correctly obtained by the process of Art. 5, even when the developments contain positive powers of (q - q0) inferior to unity. Thus, in the 1 curve p -Po = (q - qo)a, a being an integer, the order of evanescence of 1 [B - Bj]2 is a- 1, whereas the point is not a singular point at all, and has consequently a discriminantal index equal to zero: its tangent (qo) however is a singular tangent, and counts as a-1 tangents drawn from P. On the other hand, if we 1 1 consider the two curves (p -po) = (q- qo)a, p -0o = (q - q)b, in which a and b are both integers and b < a, the order of evanescence of II (B - Bj)2 is b; and this is the multiplicity of the intersection at (po, q0). 7. We can now prove that the discriminantal index of the singular point 0 is equal to twice the number of the intersections of C by itself at that point; and, again, that this discriminantal index is equal to the number of the intersections at the same point of C by its first polar with regard to any point not having a special position. For (1), considering the gi equations B, we see that twice the number of intersections of C by itself at the point (pc, q0), is the order of evanescence of II (B - Bj)2, the sign of multiplication extending to all the -i (IA-1) differences; or, observing that f(q) - I (B -Bj)2 is a product of nm (m -1)- m (Iju, -1) squared differences, none of which vanish with q- q0, twice the number of intersections of C by itself at the point (po, qo) is equal to the order of evanescence of f(q) with q - q0, i.e., to the discriminantal index v. dF And (2), since the polar of P is - = 0, and since the resultant of F= 0 and dF M d- = 0 is f(q), we infer (Art. 5) that v is the number of intersections at (po, q0) of C by its polar with regard to P. 8. Considering a superlinear branch, of which the component branches are defined by the A equations, p -op = Bo (q - 0o) + B1 w (q - q0)A+*,... (A) let A, be the greatest common divisor of A and 3/3; and, if r, is the first of the Art. 8.] ON THE HIGHEER SINGULARITIES OF PLANE CURVES. 107 numbers /3 which is not divisible by A,, let A2 be the greatest common divisor of A, and y,71; if, again, 72 is the first of the numbers 3 which is not divisible by A2, let A3 be the greatest common divisor of A2 and 7,, and so on continually. Since the numbers / have no common divisor with A, we shall at last arrive in the series A1, 2, A,... at a term equal to unity, when the series will terminate: and twice the number of the intersections of the superlinear branch by itself will be expressed by the formula 2N= 7 ( -,) +, (A - A2) + 72 (A2- 3) +..., in which 7 is written for i,. For if w denote any given root of the equation WA = 1, of its remaining roots x there are A-l1 which verify the equations xY = wC, XY = W71,..., xai- = cyi-i; because Ai is the greatest common divisor of A, Y) 71?, *., 7i-1: similarly there are A, - 1 roots other than w1 which verify the same equations, and in addition the equation x7y = ao. Thus, of the A (A -1) differences obtained by subtracting each of the series (A) in turn from every other, there are A (Ai- Ai+) which are of the order; i.e., 2N= 27, (Ai - A +). The value of N depends, therefore, not on every exponent in the series (A), but only on certain critical exponents i in the denominators of which, when reduced to their lowest terms, a new factor appears for the first time. The number 2N, which is the 'discriminantal index' of the superlinear branch is, not itself necessarily even, but the difference 2N- (A- 1) is always even, since we have 2N-(A- 1)= (y- 1)( - Al,)+ (7- 1)(,l- A2) +..., and in this expression, if A is uneven, so also are A\, A2,...; if A is even, let Ai be the first of the numbers A, A2,... which is uneven; then y-1, is uneven, and so are all the subsequent numbers Ai+, A+,.... In either case, therefore, every term in the expression of 2N- (A- 1) is even. Again, if two superlinear branches of the orders A and A' have the same tangent, let (q - q) be the lowest power of q-qo which has not the same coefficient B in the two sets of series (A) and (A'): it may, of course, in one of these sets have a zero coefficient. Then the terms of lower exponent are common to the two sets; and if the exponents be reduced to their least common denominator, these initial terms will be of the form a, a2 aj Bo (q - o) + B, Oa (q - q0)d + B a2 (q - q0)d +... + Oat (q -qO)d, P2 108 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 9. where d is a common divisor of A and A', 0 is any root of 0a= 1, and - is the exponent next inferior to h. The number of intersections of the two superlinear branches is then a A' A A ' N'=h a + [(d-)+ (- + [ (- ) + (...], the numbers a-, i 2,..., d,, d2,... (of which in particular a-= ai) being detera mined from the series of exponents d, in the same way that the numbers 7y, 7,..., A, A2,... were determined from the series of exponents. For if 0 represent a given root of the equation Od = 1, the d roots of the equation w = 1, A which satisfy the equation cd= 0, will give the same initial terms; and we may thus divide the equations (A) into A groups, each containing d equations; the equations of the same group differing from one another by containing different values of 0, but the different groups not differing from one another, so far as the initial terms are concerned. Similarly we may divide the equations (A') into d groups. Considering only one group of each set, we find (by the same reasoning as before) for the order of the product of the d x d differences obtained from them, the expression hd + a (d - d,) + -, (, - d) +..., the additional term hd appearing because we have now to take into account the d differences in which all the initial terms vanish: the result, multiplied by A A' d x d gives the value of N'. Lastly, when two superlinear branches have not the same tangent, the number of their intersections is evidently N" = AA'. By means of these formulae the discriminantal indices of the branches at any singular point, taken by themselves or in pairs, may always be obtained as soon as the developments appertaining to the branches have been found. The sum of these separate discriminantal indices is of course the discriminantal index of the point, or v = 2 N+ 2 EN' + 2 EN". 9. Every singular point of a plane curve is regarded by Professor Cayley as being equivalent in a certain manner to 3 common nodes and K common cusps; and, correlatively, every singular tangent as equivalent to r double tangents and L inflexional tangents. For any superlinear branch of order A passing through a Art. 10.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 109 singular point, the cuspidal index K is by definition -1; thus, for a linear branch K = 0. The cuspidal index of a singular point is the sum of the cuspidal indices of the several superlinear branches passing through it; so that, for any singular point, K = Z (A - 1) = - X, if is the order (Art. 2) of the point, and X the number of distinct linear, or superlinear, branches passing through it. The nodal index ~ for a singular point, and for its branches, taken singly or in pairs, is defined, not directly, but by equating 2 +3K to the discriminantal index; thus, for any superlinear branch of order A, we have 2 = (7- 3) (A - A) + (71- 3) (Ax- A2) +..., which is always even (Art. 8), and positive, except when 7= 3, A = 2, in which case = 0, and the superlinear branch is a common cusp. For r and t we have correlative definitions. 10. Adopting these definitions, we have now to prove that the numbers 2S, IK, 2r, z2 (the summations extending to all the singularities of the curve) satisfy the equations of Plucker, and further that the deficiency of the curve is correctly given by the formula:= ~ (m -1) (m- 2) - - K. It is sufficient to establish the four equations, (i.) n=m(m- l)-22S-32K, (ii.) m=n (n - ) - 2 r - 3, (iii.) H== 2 (m-1) (m- 2)- 28- 2, (iv.):= 2 n - 1) n - 2) -z 2-r,, -, because the three equations (i.), (ii.), and (iii.) = (iv.) are equivalent to the six equations of Plucker. But the equation (i.) has been already proved; for we have found (Art. 2) that n = rn (m -1) - 2v; and by definition 2v = = (2 A + 3K). The equation (ii.) is the correlative of (i.) and needs no separate proof. In the equations (iii.) and (iv.) it is important to take a definition of H which does not involve any special supposition as to the nature of the singularities appertaining to the curve. The simplest, though not the most direct, course is to adopt the method of Riemann, and to define 2H+ 1 as the index of multiplicity of connexion of the m-leaved spirally connected surface [Q], which is such that if the complex values of q be represented upon it in the usual manner, p may be regarded as a one-valued function of q. In any such surface the index of multiplicity of connexion 2H+ 1, the number of leaves m, and the number of spires (spiral points, windzngs-puznte) N are connected by the equation N= 2H+ 2m - 2. This equation Riemann himself demonstrates by comparing the values of certain 110 ON THE HIGHER- SINGULARITIES OF PLANE CURVES. [Art. 1 1. contour-integrals (' Theorie der Abelschen Functionen,' Art. 7). But he observes that it is entirely independent of considerations of magnitude, and that it belongs properly to the geometry of situation. The demonstration of it from this point of view, which has been given by M. Neumann (' Vorlesungen,' p. 309, ~ 99), is also independent of any supposition as to the special nature of the singularities of the curve C; and is therefore available for our present purpose. But we may observe that the algebraical demonstration of the same equation, which is given by MM. Clebsch and Gordan (in their 'Theorie der Abelschen Functionen,' p. 54, ~ 16), would here be inadmissible, because in that demonstration it is expressly supposed that the singular points of C are only common nodes and cusps. (See the note at p. 11, loc. sit.) It is not difficult to find the number of spires N on the surface [Q]. There is a one-fold spire for every tangent from P to C; for, if (po, qo) be the point of contact of any such tangent, we have for values of q in the vicinity of q, two conjugate developments of the type (p -o) = B ( - qO) +B2 (q - q0) +..., in which B1 is different from zero; all the other developments (Art. 3) being of the type (A), because the point P has no speciality of position. Again, there is a (a - 1)-fold spire for any singular branch which is superlinear and of order A; this is apparent from the form of the A developments appertaining to the branch (see Riemann, loc. cit., Art. 6; M. Puiseux, 'Liouville,' 1st series, Vol. XV. pp. 384-404). We have therefore N= n + (A - 1) = n + K, and Riemann's equation becomes n+ K = 2+ 2 (m- 1); or, since n + IK = M (- 1)- 2 Z- 2 ZK, 7= 1 (m - 1) (m -2) -2l -2K, which is the equation (iii.) Again, it is an immediate consequence of Riemann's definition of the number H (see his 'Abelsche Functionen,' Art. 11) that this number remains unchanged by any unicursal transformation of the equation F(p, q) =0. But (as has been already observed by MM. Clebsch and Gordan) any tangential equation of the curve C may be regarded as an unicursal transformation of the equation F (p, q) = 0, because the points and tangents of a curve correspond to one another one to one. The equation (iii.), therefore, involves the equation (iv.); a result which, as we have seen, implies that the six equations of Plicker are satisfied by the numbers 2Z, ZK, Er, ZI. 11. The indices 7 and i appertaining to any superlinear branch at a singular Art. 11.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 111 point, and the number of tangents common to two osculating superlinear branches, may be ascertained directly from the point-equations B, without actually forming the corresponding line-equations. To prove this, we shall establish a relation which subsists between certain terms in the two sets of equations. If Q and R are given constants, p = Qq + R is the equation of a straight line in the system of parametric point-coordinates which we have been employing. In passing to line-coordinates, we may take Q and R as the coordinates of this straight line; and we may regard Q and R as the parameters of two ranges of points, lying on the lines PQ, PR, respectively, and represented by equations of the form Q(P) +(Q) =, R (P) + (R)=; the line p = Q0 q + Ro or (Q0, Zo) being the line joining the points determined in the two ranges by the values Q0, Ro of the parameters. If to the hypothesis of Art. 1 we add the supposition that PR is not a tangent to C, and does not pass through any singular point of C, the line-equation of C, which we may represent by D (Q, R) = 0, will have the same sort of freedom from speciality which has been already attributed to the point-equation F(p, q)= O. The parameters of the tangent to C at the point (p, q) are IdF dF qQ. Q=_ (-dq) *-(d)' RP=p-qQ. Let (p, q) be a point lying on the branch B, of which the point-equation is 1 p -po = Bo ( - q0)+ B1 l (q- qo)A +...; and suppose (p, q) different from (po, q0), but sufficiently near to it (Art. 3) to ensure the convergence of the m series A and B. Writing F(p, q) = Mx (p-po- B), where M is a product of factors, none of which can vanish at the point (p, q), because no singular point other than (po, qo) exists within the range of values attributed to q, we find dB &=(dp)' n P-q^dq> Putting Bo= Qo, po - q B = RO, so that (Qo, Ro) is the tangent at (Po, qo) to B, we obtain the equations Q- =o B, (q - q0) + 2 ( o)^- + R- o= - qo(Q-Qo) + (1- )o q q + -) ( - qo) +..., 112 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 12. which determine the parameters Q, R of the tangent at any point of B. If we further write 1 B, W(q qO)^=, QQo x y 1-^, R-R+q(Q, -Qo)=Z, these equations become y~=i-A = 1- [I1 + - 2 2-I $+ " 3- 1 +...],... (a) Z=p l "1 + i p2 +.,.~ ~ ~ (3) where we have written o- for i B, and p, for (1 - ) Bi. It will be observed that w has disappeared from these equations, which therefore appertain equally to all the A branches composing the superlinear branch B. To obtain the tangential equation of that branch, i.e., the expansion of R - Ro in a series proceeding by powers of Q- Q0, we have three operations to perform. First, we have to raise each side of the equation (a) to the power - --; we thus obtain an expansion of the form 0Y=Yt(l +A +B~4b+...),....... (y) 0 denoting any root of the equation 0i-^ = 1. Secondly, we have to revert the series (Y), so as to obtain the series =OY{1 +A'(oY)a'+ B'(0 +... }...... () Lastly, we have to substitute, in the equation (P3), for t its value given by the series (f); the final result being of the form z= H(e )hl+H +,. (......... (Z) 1 or, if i=f B lh2 R- = -o R (Q - Qo) + H1,h ol (Q Qo)1- + QH2 (Ah2 o N2 (Q- Qo)-A +.... (H) 12. Certain of the terms of H, and indeed precisely those critical terms upon which the determination of 7 and l depends, can be assigned a priori by the help of the following considerations. (i.) If a, b, c,..., 1,... are positive and integral numbers, arranged in order of magnitude, of which I is such that it cannot be formed by addition of any multiples of the numbers which precede it, the coefficient of x1 in the expansion of [,+ (x)], where a is any real exponent, and + (x) = +Axa + Bx +Cc + +... + +..., is LTL; and, in particular, if all the numbers preceding I are multiples of any Art. 12.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 113 number a, of which I is not itself a multiple, a supposition which implies that I cannot be formed by addition of multiples of a, b, c,..., I is the least exponent in the development of [+ (x)]*, which is not divisible by a. (ii.) If the series y = x~ (x) be reverted so as to obtain the equation x= Y+ (y)=y( + yal+B yb+...), the exponents a1, b, c1,... are all formed by addition of multiples of a, b, c,.... For, if this is not so, let h,1 be the least exponent in Al (y), which cannot be formed by adding multiples of a, b, c,...; on substituting x (x) for y in y+, (y), a substitution which ought to have x for its result, we find that the coefficient of hl + 1 is H1; i.e. H1 = 0, or the exponent h, does not occur in 41 (y). Again, if the exponent I in + (x) cannot be formed by adding multiples of the exponents which precede it, the coefficient L1 of yl in 41 (y) is - L; for, on making the same substitution as before, the coefficient of x + is found to be L1 + L; i.e. L = - L. And, in particular, if I is the lowest exponent in + (x) which is not divisible by a, I is also the lowest exponent in xJi (y) which is not divisible by a. Let 3i = 7, be one of the critical exponents y, y7,... considered in Art. 8; then all the differences i2- 1 -/31, 3- i,... up to ii-1 -f1, are divisible by A,_i; but /i - 1i is not divisible by A_-,. Therefore, by (i.), the coefficient of ai-i in the development of - is i - i; by (ii.) the coefficient of (0 Y)-al in the expansion of f0 is - A, and -i_-3i is the least exponent in that expansion which is not divisible by A-1; finally, on substituting in the equation (3), we see that the term H(0Y)$i in the development of Z can arise only from the terms pi,,e and pi tai in (I); its coefficient H is therefore - P _ + Pi = Bi; Jai 'Ys and the coefficient of 0i (Q - Qo)1-A or 075 (Q - Qo0)-^, in the expansion of R - R, is 3i Bi. Nor can any exponent preceding 7Y have a numerator which is not divisible by As-_. Observing that the greatest common divisor of A3 - A, and i31, is the same as that of A and 0,1 we infer from this result that the numbers \,, a, A3,...; y, 71, 72,... are the same for the series H as for the series B; and since the numbers 7, 7T1 72, * *. have no common divisor with a, neither have they with 13 - A = 7 - A; i.e., 7- A is the least common denominator of the exponents of H. Hence we have, writing - A A,, VOL. II. 114 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 13. I = A,-1, y = A+ A,= I+K+ 2, 27+3t==7 (A,-A)+ 7 (A- A2) + 72(A2-A3)+...; or, subtracting the discriminantal index 2 8 + 3, (2 7 + 3) -(2+ 3K) = A,2 - A2, =1( - K) ( + K- 1), an equation which establishes a relation between the four indices of the superlinear branch. 13. If we consider any term whatever in the series B, for example the term (i) - q we shall in general find a corresponding term () in the series (i) = B, E ~(q- q)^a, we shall in general find a corresponding term (I) in the series H, containing Q- Q0 raised to the power _i -: (I) may be considered as the sum of two parts, I1 and 12, of which the first, I1, arises from the term (i) itself, the other, 12, from the terms preceding (i); (I) being in no way affected by the terms following (i). If /i is one of the critical exponents, we have just seen that hi 2 =0, 1 = iA Bi (Q - Qo)I1- A. If fi is not one of the critical exponents, the first of these equations ceases to subsist, but the second remains true, and its proof requires only a slight modification of the reasoning in Art. 12. Now let two series B, appertaining to two different superlinear branches, which have a common tangent, coincide as far as the term (i), but exclusively of it; the two corresponding series H will coincide as far as the term (I), but exclusively of it; we suppose i > 0. That all terms preceding (I) will coincide in the two developments H is evident, for these terms arise solely from the terms preceding (I), which are identical in the two developments B. And the terms (I) themselves 13s are different: for the difference of the two terms (i) is (Bi - B$) wa (q - q0)A, where one of the two B, B$ may be zero, but the difference Bi-BB is by hypothesis not zero; and the difference of the two terms (1) is hi i>i (B - B6) 0i x (Q - QO)- = I- I;, for these two terms have the same part 1z. Let D be the number of points, T the number of tangents common to the two branches B at the point (po, q0); T is given by the formula z= ~, a' + - a) (7 a')( -T=f ad +( (- ad) {)+2(d )+. which is derived from the expression for N' in Art. 8, by writing 7 - A for, Art. 14.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 115 -A' for', and - - d for d. Observing that whence 7'-A' A 7-A A A d -K + = K+ 1 we find T-D= d- ( —2d) =(-K) (I K + 2)=(' -,K') (t +K+ 2) =(t + 1) (+ 1) - ( + 1) (K' + 1) = A, - AA'. We have supposed in the demonstration that i > 1, or that the two developments of p-p0 - B ( - q0) coincide for at least one term. But, for the validity of the formulae, it is only necessary that the first exponent should be the same in the two developments; and indeed the last two expressions for T- D hold universally for any two superlinear branches having a common tangent. 14. The species of a superlinear singularity may be regarded as defined by the series of numbers A and A,; A1, A2,., y., 71,..., so that two superlinear singularities, for which these indices have the same values, may be considered as belonging to the same species. A rougher classification, however, which is sometimes useful, may be obtained in the following way. Leaving out of sight the case in which two superlinear. singularities present themselves as conjugate imaginaries, and attending only to the case of a real superlinearity, we may distinguish four varieties differing from one another in the appearance which they present to the eye. (See a Memoir by M. Stolz, 'Mathematische Annalen,' Vol. VIII., p. 440.) (i.) A uneven, A, uneven; no apparent cusp or inflexion. (ii.) A even, A, uneven; an apparent cusp, no apparent inflexion. (iii.) A uneven, A, even; an apparent inflexion, no apparent cusp. (iv.) A even, A, even; an apparent cusp, and an apparent inflexion. The form of (ii.) is that of the common or keratoid cusp; (iv.) has the form of the cusp of the second species, or rhamphoid cusp. There is an apparent infiexion at the rhamphoid cusp, because, if a person describing the curve continuously passes through the cusp, the concavity of the curve is to his right after he has passed through the cusp, if it was to his left hand before, and vice versd. We may further observe that, in case (iv.), A and A,, being both even, have a common Q 2 116 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 15. measure; thus A2 > 1, and the superlinearity is composite. The cases (ii.) and (iii.) are correlative; the cases (i.) and (iv.) are their own correlatives. 15. The curvature of a curve at two points infinitely near to a given superlinear point, and at equal distances from it on either side, is always the same; and is infinite, finite, or zero, according as A > A,, A = A,, or A < A,. Thus, in each of the cases (i.) and (iv.), there are three sub-varieties of form; and two in each of the cases (iii.) and (ii.). The following are the simplest examples of each of these sub-varieties: for the sake of completeness, the cases in which either of the two numbers A or A, is unity, are included. (i.) A and A, uneven, A>A,: y=x3; y =x5 Y-x2' A=A,: y=X2; y y=x2+. < A,: y=x4; y=xa. (ii.) A even, A, uneven, A > A,: y=x; y=xi. A<A, y= 2. (iii.) A uneven, A, even, A > A,: y = x. <A; y = x3; y = x (iv.) A and A, even, 3 7 A > A,: y =X2 +x. =,: y = x2 +. A<~A,: y=x3+~x. It should be noticed that in the equation y = x + x4, the only independent radical is xi, and that x~ is to be interpreted as (A4)2. Thus, supposing x positive, and understanding by Vx3 and C/x7 the real and positive values of the radicals, we have for the four partial branches the equations y = /X3 + 4/x, y= / -3 _ X7 y= -_ J3_-i X7, y= - /X3 + i/x7, of which the first two appertain to a real rhamphoid cusp. If we were to change the sign of /x3, we should pass from the equation U= (y2 + x3)2 _ X3 (x2 + 2y)2 = 0, which is the rationalised equivalent of y = x2 + X4, to the equation V= (y2 + x3)2 _ X3 (x2 + 2y)2 = 0, Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 117 which is the rationalised equivalent of y = - x + x4. It is, of course, quite possible that two developments, such as y= + +X +..., may both belong to the same curve (as indeed they do both belong to the curve UV+ x15 <p (x, y) = 0), but such a curve would have two distinct superlinear branches touching one another at the point x= 0, y = 0. 16. Let O be any point whatever on a curve line; let the arc OP = a-, P being a point on the curve infinitely near to 0; let M be the orthogonal projection of P on the tangent at 0; and let the tangents at 0 and P intersect at T, making the infinitesimal angle o. Then it will be found that A,_ logW OT LTPO _ logMP A log - TP L TOP log OMA The fraction - which admits of these various geometrical interpretations may perhaps be called the logarithmic curvature of the curve at the point 0. At any ordinary point it is unity; and in a geometrical curve it is always rational, but in a transcendental curve it may have any value rational or irrational. Since A or K +1 is the number of points in which the superlinear branch is cut by any line passing through 0, other than its tangent at the point 0, we infer that, correlatively, l + 1 or A, is the number of tangents drawn to the superlinear branch from any point on the tangent at 0, other than O itself. Thus, if d be the discriminantal index of 0, or the number of points in which the curve is cut at 0 by the polar of any arbitrary point, d + A, is the number of points in which the curve is cut at 0 by the polar of any point on the tangent at 0, other than 0 itself; there is, of course, a correlative definition of d + A. Lastly, since A + A, is the number of points common at 0 to the tangent and the curve, it is also, correlatively, the number of tangents drawn from 0 to touch the curve at that point. Thus the polar of the point 0 intersects the curve at 0 in d + A + A, points, and the tangent at 0 counts as d + A + A, tangents common to the curve, and to the tangential polar of OT with regard to the curve. For the numbers A, A2,..., 71, 72 *... no simple geometrical definition has as yet presented itself. 17. The proof of Plicker's formulae, which is indicated in Art. 10, may appear very indirect. Some further observations on these formulae, and on the various modes of demonstrating them, may not be out of place. (1.) If we write D = X (2 3 + 3K), T= z (27 + 3,), 1=, K= K:, 3 (I- K) = Q, Plucker's formulae become n = ( - 1)- D, m = n (n - l)- T; 118 ON TIlE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17. =nm (m-2)-D, -Q=n (n -2)-T; giving T-D=n2 _- 2, =3 (n-m), Q4-2Q2 (T+D)-4Q(T-D)+(T-D )2= o. It is thus apparent that Plicker's equations do not contain either K or I separately, but only the difference I- K. (2.) The discriminantal index d = 2 + 3K of any given point is defined geometrically as the number of intersections of the polar of an arbitrary point with the curve at the given point. But the definition which we have given in Art. 9 of the cuspidal index K is an analytical one, and does not readily admit of interpretation in coordinate geometry. The Hessian does not serve to define either i or K, for in all the cases that have as yet been rigorously investigated, it has been found that the number of intersections of the Hessian with the curve at a point of discriminantal index d is 3 d + - K, so that, even if the number of these intersections at any singular point should be determined by a general method, we should only obtain a definition of the difference I - K. Again, if several superlinear branches have a common tangent OT at the point 0, it will be seen that the geometrical definitions of Art. 16 only give the numbers 2 (t + 1) and 2(K + 1); viz., if d is the total discriminantal index of all the branches intersecting at 0, the first polar of any point on OT (other than 0) intersects the curve at 0 in d + Z (i + 1) points; the polar of 0 intersects the curve at 0 in d + 2 (I + K + 2) points; and there are correlative definitions of the numbers + (K + 1), and d + 2 (I + K+ 2). By combining these definitions, we obtain a geometrical definition of the difference 2 (I - K), the summation extending to all the branches which touch one another at O. But here it is to be observed (1) that to deduce the values of It and 2K from those of I (i + 1) and I (K + 1), we should require to determine the number X of distinct superlinear branches which touch OT at 0; and (2) that, even if Ii and 2K were known, it would still remain to determine the decomposition of these sums, and to assign the partial indices appertaining to each of the X branches; whereas no determination of the number X, or of the indices t and K of each separate superlinear branch, has as yet been obtained by considering the intersections of the given curve with any concomitant or system of concomitants. (3.) The difficulty, which thus presents itself in obtaining a definition of the indices i and K, ceases to exist when we leave the domain of coordinate geometry, and consider either the analytical expansions, or the geometrical representations Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 119 (depending on principles foreign to coordinate geometry) which correspond to those expansions. If several superlinear branches touch one another at a given point, the analytical expansions separate them, and assign the cuspidal and inflexional indices proper to each of them. If we apply to the equation F(p, q) = 0 the geometrical methods of double algebra, the cuspidal indices appear in the cycles of values of p, which present themselves at the points answering to the discriminantal values of q. (See the memoir of M. Puiseux, 'Liouville,' Vol. XV., p. 384.) If, instead of the simple plane of double algebra, we use the multiple plane of Riemann, the cuspidal indices are represented by the spires which connect the leaves of the multiple plane. But it is important to remember that, in employing the methods of double algebra, and & fortiori in employing the surfaces of Riemann, we are entirely abandoning the methods of coordinate and projective geometry. The present question is perhaps not directly affected by the fundamental distinction between the 'infinite' of double algebra, which is a point, and the infinite of projective geometry, which is a straight line. But the duality, characteristic of projective geometry, is lost in double algebra; so that, when the complex values of p and q which satisfy the equation F (p, q) =0 are regarded as developed on a plane, or on one of Riemann's surfaces, we do indeed obtain a direct representation of the cuspidal index K, but no corresponding representation (unless we first transform the equation into its reciprocal) of the correlative index c. Indeed, it may be asserted that, whereas the character of any given superlinearity mainly depends on a series of indices A = K + 1, a, = 1 +1, A, A2,..., 71 72,...*, the modes of geometrical representation, to which we are here referring, offer a sensible image of the first of these indices only. If we employ a simple plane, any one of the A values of p, which come to coincide with one another at the discriminantal point, must describe A elementary contours around that point before it acquires again its original value. If for simplicity we suppose that A2 = 1, the A values of p, which form the cycle, will divide themselves into A, sub-cycles, each containing - values; and any value, belonging to one of these sub-cycles, will acquire approximately its original value, after describing A elementary contours around the discriminantal point, the order of the error being 7 if the order of the infinitesimal radius be taken as unity. And upon this approximate return to the original value depends the only indication which the method affords of the existence of sub-cycles, and of the values of the 120 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17. numbers A1 and y7. If we employ the multiple plane of Riemann, we may perhaps represent the relations of the A expansions to one another by taking a A1 -leaved plane, repeated o times, and having a spire of order - 1, so arranged that after - revolutions we return to the same Al-leaved plane upon which we Al were when we set out, but not to the same leaf of that plane. And we can give to this image a certain amount of clearness by supposing that the Al leaves of any Al-leaved plane are infinitely nearer to one another than are any two of the - repetitions of the A,-leaved plane. (4.) The demonstrations of Plucker's formulae, which are usually given, apply only to the case in which the singularities are simple; the cases of multiple points, or multiple tangents, or of branches having contact with one another of any order, being made to depend, by the method of limits, on the simple cases of double points, or double tangents (see Dr. Salmon's 'Higher Plane Curves,' p. 53). But these demonstrations do not admit of immediate extension to the case of the higher singularities properly so called, because it has not as yet been established, in any general manner, that a higher singularity may be regarded as the limit of an equivalent number of lower singularities situated infinitely near to one another. It would seem that Plicker himself was well aware of the incompleteness (in this respect) of the demonstration of his equations; for he supplements that demonstration by separately considering the case of a common cusp of the second species. Assuming the equation n = m (m - 1) - D, and its reciprocal, (about the rigorous proof of which there is no doubt,) we have only to establish one other equation of the system. Two different methods are given by Plicker (' Theorie der Algebraischen Curven,' Part ii., Arts. 77-81): (i.) He establishes directly the theorem that, at a cusp of the second species, the curve d2F (dFd2 d F d2F d dF\2 dp2 dq dp dq dp dq dp (which may be used for our present purpose instead of the Hessian) intersects the given curve in 3d + - K = 15 points. We have already stated that, in all the cases which have been examined hitherto, the number of intersections of the Hessian with the curve at any point has been found to be 3d + - cK; but no general demonstration of this theorem has as yet been given. The only method at present known for determining the number of intersections of two curves at a point which is singular on each of them, consists in obtaining the developments Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 121 of the various branches of the two curves at the point, and in comparing these developments with one another. The discussion in Art. 18 of the development of the polar curve in the vicinity of a superlinear branch, may serve to show that the corresponding enquiry in the case of the Hessian is one of considerable intricacy. (ii.) The other method employed by Plucker depends on a determination of the number of double tangents lost by a curve of the fourth order in consequence of the presence of a cusp of the second species. In the absence of any demonstration that a higher singularity can be regarded as the limit of simple singularities existing infinitely near to one another, it is difficult to see how this mode of proof can be rendered universally applicable. (5.) We have seen (Art. 10) that the theorem of the invariance of the number 1 (m - 1) (m - 2) - L - /c in any unicursal transformation of the curve suffices to establish the equation ( 1) (m - 2) - 2 (n - 1) (n -2)... (A) and thus to complete the proof of the formulae of Plucker. Among the demonstrations of this theorem which have been given in recent times that of MM. Bertini and Zeuthen (' Giornale di Mathematica,' Vol. VII., p. 105; 'Mathematische Annalen,' Vol. III., p. 150; Dr. Salmon's 'Higher Plane Curves,' p. 314) is remarkable for its simplicity; and appears, as we shall now attempt to show, to admit of extension to the case in which the curves have any singularities whatever. We begin by assuming that when a curve is subjected to an unicursal, or one-to-one transformation, the continuity of its branches is invariably preserved, even when the position of these branches with regard to one another has undergone great distortion. For example, if a curve have two branches intersecting at the point 0, these two branches will certainly be represented by two corresponding branches in the transformed curve; but these two branches may have no point of intersection, and the point 0 may be represented by two different points one on each of the two branches. Again, two branches which osculate one another with any degree of approximation may be transformed into branches having no contact and no point in common. But a superlinear branch behaves as one branch, and always is transformed into one branch and one only. Consider, for example, a real branch which is superlinear at 0, and suppose for simplicity that no other branch passes through 0; whatever be the nature of the superlinearity, we have one continuous branch passing through 0, and if a point describe this branch, the track of the image point in the transformed figure cannot be anything but one continuous branch. Let C1, C2 be two curves of the orders mn, m2, and of the classes n1, n2 lying VOL. II. R 122 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17. in the same plane and corresponding to one another unicursally; and let P1, P2 be points upon them corresponding unicursally. Taking two arbitrary points S1, S2, we consider, with M. Zeuthen, the locus r of the intersection of the rays S, P1, S2 P2; and we propose to determine the number of tangents that can be drawn to r from each of the two points S1 and S2. We may suppose that SX S2 cuts each of the two curves in points which do not have singular points of the other curve for their corresponding points; then it is evident that r will have rnm ordinary branches passing through S~, and ma ordinary branches passing through S2. We may further suppose that the n, tangents drawn from S, to C, are none of them singular tangents, and that to the points of contact of these n, tangents there answer on 0C points having no singularity: each of these tangents will then be a tangent of F, but not a singular tangent of that curve. Beside the 2 in2 + n. tangents, which we have now drawn from S, to r, there may be others, coinciding in direction with the rays running from S, to the singular points of Cl. Let X, be a superlinear point on CQ, having the cuspidal index K,; and to X, let X2 answer on C2, the cuspidal index of X2 being K2, where K2 > 0. We may suppose at first that only one branch passes through X1 and only one through X2. The ray S1 X1 meets C, at X1 in precisely K1 + 1 coincident points, because S~ XX is not a tangent at X1; similarly S2X2 is not a tangent at X2, but meets C2 in precisely K2+ 1 points at X2, since we may attribute to S2 the requisite generality of position with regard to C2. Thus, if Q is the intersection of S X1, S2 X2, the locus r is intersected at Q K1 +1 times by S, X1, and K + 1 times by S2X2. The points of the curve r answer, one to one, to the points of C, or C2; thus at Q there is but one branch answering to the one branch at X1, or to the one branch at X2. If K1= 2, the cuspidal index of this branch is K = K2, while its inflexional index remains unknown. If K1> K2, its cuspidal index is K2, its inflexional index is K1 - K2- 1; similarly, if K2 > K,, these indices are K1 and K2 - K1- 1; i.e., in the first case, Si X1 counts K, - K2 times as a tangent to r at Q, and S2 X2 is not a tangent at all; in the second case, S2 X2 counts K2 - K times as a tangent, and Sl X1 is not a tangent at all. When K1 = K, neither S1 X1 nor S2 X2 are tangents. The preceding reasoning will not be affected, if we now introduce the supposition that several linear or superlinear branches intersect or osculate at X1, and that branches corresponding to some or all of them pass through X2. Several branches will now pass through Q, but each of them may be considered separately, and the number of times that it is touched by S, Q or S, Q may be ascertained as above. Equating the results appertaining to the points S, and S2, we now obtain Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 123 2 m + n, + z' (K - K2) = 2 m + n2 + 2' (K2 - K); where Z' extends only to those differences which are positive. Written in the form n, + K1 - 2mv = n + K2 - 2m2, this equation coincides with the formula 2 ( I l(ml -2) - (+1 C+ K) =(M2 -1) (nm2-2) - (J2 + K2)n which it was required to prove. The assumption, which we have explicitly made, that a linear or superlinear branch is always transformed by a one-to-one transformation into one branch, and one only, is indispensable in the preceding proof; as upon it depends the determination of the number of times that r is touched by S1 X1 or S2 X2. In the case of a real branch transformed by a real transformation, the assumption may be regarded as evident; in the general case, we should have to consider, instead of two plane curves, the two corresponding surfaces of Riemann. For our immediate purpose, however, we do not need to establish the assumption as universally true in all cases; because the only one-to-one transformation (beside that of C1 or C2 into r) which is here employed is the transformation by polar reciprocation; and the investigation of Art. 11 affords a direct proof that in this transformation any one linear or superlinear branch is always transformed into one branch (linear or superlinear). (6.) Abandoning for a time the hypotheses of Art. 1, let us suppose that P is a singular point on the curve C, Q retaining its generality of position. And first let P be a point through which only one superlinear branch passes, having the indices K = - 1, L =, - 1; let us also suppose that no singular tangent of C (other than the tangent at P) passes through P. The order of p in the equation F (_p, q) = 0 is now m - A, instead of m; and the number of tangents that can be drawn from P to the curve C (other than the coincident tangents at P itself) is n-A -a, (see Art. 16), instead of n. To all the singular points of C, other than P, there will appertain developments of precisely the same form as in the case in which P has no speciality of position. Let q0 be the value of q corresponding to the tangent at P; the parameters of the point P are p = Ac, q = qg. We cannot therefore, in examining the superlinear branch at P, develope p in a series pro1 ceeding by powers of q - q0; but we may so develope -, or any linear function c+dp d of p, such as r b, which assumes a finite value po = - when p= c. The exa+bp ~ _p ' ponents in any such development will have a, instead of A, for their least comR 2 124 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17. mon denominator, because the tangent to C at P meets the curve (Art. 16) in A + A, points, so that, if q = q0, A, of the m - A values of c dp become equal to a+ bp Po. Setting out from the given equation F (p, q)= 0, let us form the developments appertaining to all the singular discriminantal values of q; and in each group of conjugate developments let us consider the greatest common divisor 0 of its exponents. The sum 2 (0-1) will be equal to 2Kc + A - A,, instead of Y-K,; and the three numbers, by which we have now replaced m, n, and 2K, will satisfy the equation (n - -,) + ( + A, - A) - 2 (m - ) = n + K - 2 m. The cases in which (a) more than one branch passes through P, (3) one or more singular tangents pass through P, (7) Q as well as P has some speciality of position with regard to C, may all be treated by the same method. In any of these cases, let E (p) be the highest exponent of p in the equation F (p, q) = 0; and let w (p) = 2 (0 - 1), the sign of summation now extending to all the discriminantal values of q, so that (0 - 1) contains an unit for every ordinary tangent that can be drawn from P to touch the curve elsewhere. If any of the discriminantal values of q, or any of the corresponding equal values of p, are infinite, we are to employ linear functions of p and q themselves, in forming the developments from which we are to infer the numbers 0. We shall thus obtain the equation (p)- 2E (p) = (q) - 2E(q) = n+ K-2m,.... (B) from which, as Clebsch has shown, the general theorem of the invariance of the deficiency may be immediately deduced. (See a Memoir by M. Nother, 'Mathematische Annalen,' Vol. VIII., p. 497.) In the memoir to which we have just referred, M. Nother offers a demonstration of the equation (B). But this demonstration is perhaps not wholly free from obscurity. (See the words, p. 499, loc. cit., 'Dieses findet... ergiebt,' with the accompanying reference to the Gottingen ' Nachrichten.') A similar remark applies to a second demonstration, in the same memoir, of the invariance of the deficiency. [See p. 501, ' Man hat aber dann.. das Glied 2i, (i - 1).'] M. Nother has returned to the same subject, in a recent memoir of great interest (' Mathematische Annalen,' Vol. IX., p. 166), in which he considers the resolution of a higher singularity by successive applications of a simple quadratic transformation, and infers (though by a method which can hardly be accepted as rigorous) that any higher singularity may be regarded as the limit of a certain number of lower singularities situated infinitely near to one another. We may Art. 18.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 125 observe (a) that the use of a quadratic transformation for the resolution of complicated singularities is due to Cramer (' Analyse des Lignes Courbes'); (P3) that to establish the complete system of the formulae of Plicker, M. N6ther selects the same three equations, which we have been led to employ in the present paper [viz., the equations (i.), (ii.), and (iii.) = (iv.), of Art. 10]. 18. The expansions of Arts. 3 and 4 enable us to examine the relation of a curve at a singular point to its polar curves. Putting for brevity p-p=o, q- q0 = F(p, q)= F1 (,, ), we have F (, ) = (, - A) x I (n - B), dF d F 'dp-d q From the expression of F1 (?i, r) as a product of m factor-series, we infer that if, on writing K1 t for n in F1 (y, (), we obtain a result of which the order of evanescence with t is higher than ax, r = K], +... is the beginning of one at least of the expansions B. Again, let us substitute for n in F1 (?, a) an expression of the form KI+Ka+K a+...+K Kr= K- + KK a,, + KK 2,a +... + KI, a., in which 1 a< a3 < <... < a,. If the order of evanescence of F1 (K, ) with ~ experiences an abrupt diminution when either ay or Ky (the exponent and coefficient of the last term of K) is affected by any small variation, the terms K are the initial terms of one at least of the expansions B. This observation (which admits of some useful applications) enables us to deduce the developments appertaining dF to the polar curve d-, in the vicinity of the point (Po, q0), from the developments appertaining to C. Let k of the developments B coincide with one another and with K, as far as the term Ky fa. inclusively, so that for any one of these k developments we have B = K+ L 4If,, l >aB =,.. a... (K) Li = Xi + X, 41,+.., the terms Xi\ li not being all identical. Put V=-K, n-I (V-L, 1i) =(V); dF, dH, (1)+M~,(V), then F1 (), ) = Mx (V), and d1 = d (d- ) + '(), r, tl M being a product of m - k factors, viz., of the m - ju factors r - A, and of those m - k factors n - B which do not coincide with r - K as far as the term K, a_ inclusively. Suppose, at first, that 11, 12,..., 4 are all unequal, and arranged in order of magnitude; it is easily ascertained that the first terms in the expansions 126 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 18. of the roots of 0' (V)= 0 are k-1 k-2 k-3 =_ X1, V2=JlX2'2, V3= \3_X,..., V2-i=X-_-1. Substitute for ji in - an expression of the form j = K+ H^, where h > a, and H is independent of r. If H-h is not the same as any one of the quantities V1, d M V2,..., vk-_, the order of evanescence of p (V) -surpasses that of M+'(V); di7 for the order of evanescence of M cannot surpass that of by a number greater than a,, whereas the order of p (V), on the supposition that none of the equations HEh = V is satisfied, surpasses the order of p' (V), at least by one of the numbers 11, 12,..., lk. If we now suppose H and h to vary continuously, the order of evanescence of (' (V) is abruptly increased when H1h comes to coincide with any one of the roots V0, Vi,..., Vkl; and, since the order of evanescence of M remains unchanged, that of d is also increased abruptly. Hence k- 1 of the dF developments appertaining to -are of the type k - 1 kc - 2,=K+J X1i+...,,=K+k- X222+..., =K+ Xk-llk-l+.... Again, suppose that s of the indices I are equal; let, for example, the s lowest indices be equal; then s roots of the equation +'(V) = 0 are of the form Hi +..., where 1= 1 = 12 =... = ls; and if + (0) = (0 - x) (0 - x)... (o - Xs), the s coefficients Hi are the roots of the equation (k- s) + (0) + 0+' () =... (0) If the s equal indices 11... 12 are followed by another set of s' indices equal to one another and to l', l' being > 1, put (0-Xs +1) (0-Xs +2)... (0- x +') = 1 (0); then the equation P'(V) = 0 has s' roots of the form H/1z' +..., the coefficients Hi[ being the roots of the equation ( - s - s') +1 (0)+ co; (0)=o,...... (o') and so on continually. Lastly, considering any group of equal indices I, for Art. 18.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 127 example the group 1i,,, 1s+, 27... l+,, let C of the corresponding coefficients X be supposed equal (in which case a- of the developments K coincide with one another for one term at least after Ky av); the corresponding equation (0') will have a- - 1 roots (and no more) equal to one another and to the equal coefficients X; so that -1 of the developments appertaining to the polar will coincide, as far as the term next after Ky %a, with the a developments appertaining to C. To carry on these a- developments until their complete separation from one another, we must repeat the preceding process as often as may be necessary, using in the first instance KI+ X1 instead of K, and confining our attention to the a- developments, appertaining to C, in which K+ X41 are the initial terms. As the roots of the equations + (0) = 0,,1 (0) = 0,... are all different from zero, so also are the roots of the equations (0), (0'),..., except when the highest index I is one of a group of equal indices. In this case, if f (0) = II (0 - X), the sign of multiplication extending only to those coefficients Xi which occur in terms having the greatest exponent 1, the last of the equations (0) is of the form +' (0) = 0, and r of its roots may be equal to zero. When this happens, in the r polar developments corresponding to the zero roots, the terms K are not followed by a term of the form HZl, but by a term of higher exponent. To determine this term in each of the r developments, we must use, in forming + (0), not simply the quantities Xi, but as many terms of the series Xi + XA $1 +... as may be necessary. The zero roots of +' (0) = 0 are then replaced by roots of the form H4a, a being positive, and the initial terms of the r polar developments are given by the formula K + Hi + a. We shall employ the preceding method to examine the nature of the polar branches in the vicinity of a superlinear branch. We suppose the superlinear branch to be of the type [An, A1, 2,..., A As+l=l; 71, 72', ", Ys]; and we consider only the case in which this superlinear branch (A) is not touched by any other branch. The polar has A -1 branches (A') touching the superlinear branch. Their developments coincide with one another, and with those of (A), as far as the term [xA] exclusively. But at this term -1 of them cease to osculate any branch of (A); they do not contain the term [xA], which is replaced in each of them by a term of higher exponent, yet so that the aggregate of the — 1 exponents cannot exceed — 1. The remaining (A-1) branches Al Al1Al 128 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 19. divide themselves into groups of, - 1 each. The A, -1 branches of each group are identical with one another, and with A2 of the branches (A), as far as 71 the term [IA] exclusively. At this term - 1 branches out of each group (2 cease to osculate any branch of (a), and the remaining (2 -1) divide themA1 selves, in the same way as before, into - groups of A - 1 each; the branches of 2 each group being identical with one another, and with a2 of the branches of (A), 72 as far as the term (xa) exclusively. In this way we obtain the following theorem in which i is to have every value from 0 to s, both inclusively. A ra 'The polar curve of an arbitrary point has branches which form Ai&~~~~~~~~Ai +1 Ai -1 superlinear branches of the type [ A A All Ai-,, A 1i-1Y,,7, ' i These superlinear branches coincide with one another, and with the branches of 7i 7i (a) as far as the term [WxA] exclusively; instead of the term [xA] each of them contains a term of higher exponent; the -1 superlinear branches may, but Ai+i do not necessarily, group themselves into higher superlinear branches.' 19. The development appertaining to a superlinear branch can always be obtained from the equation of the curve by successive applications of the ' analytical triangle.' The process has been described by M. Puiseux in his important memoir ' Recherches sur les fonctions algebriques.' (' Liouville,' Vol. XV., p. 384; see also a paper by M. de la Gournerie, ibid., 2nd series, Vol. XIV., p. 425, Vol. XV., p. 1.) We propose to conclude the present paper by showing how the numbers 7, 7,..., A, \,... present themselves in the course of the operation. Putting, as in Art. 18, n for p-po, 5 for q - q, we first of all write the equation F1 (,, () = 0 in the form u, + u + +..., where 6u, is a homogeneous function of ~ and, of the order,c, which is that of the singular point. If (?n - Bo O)a is a multiple factor of u. the line? - Bo t is touched by branches (linear or superlinear) of which the aggregate order is a. Put - Bo0 = v; the resulting equation between v and t will give precisely a values of v in which the order of v surpasses that of 6. Form, by the analytical triangle, the equations (of the Art. 19.] ON THE HIGHER SINGULARITIES OF PLANE CURVES.e 129 aggregate order a in v) which give the initial terms of the expansions of these a values. These equations are of the type where X and v are relatively prime, X > v, and a,1 v a; they are always obtained linearly, except when there are s of them in which the numbers al, X, v are all the same; in which case the analytical triangle determines an equation, of order 8, having constant coefficients, of which the roots are the s quantities K. There are four cases to be considered: (i.) ac = 1, v = 1; (ii.) a = 1, v > 1; (iii.) a, > 1, v = 1; (iv.) Ca > 1, v > 1. (i.) To the equation v - KH = 0, X > 1, answers a linear branch which, considered by itself, has no point-singularity (if X is > 2, it is an inflexion). (ii.) To the equation v^- Kx4 0 answers a superlinear branch of which the character is defined by the equations A = v, A = 1,? = X; its develop1 ment proceeds by integral powers of (v, and the successive terms are obtained linearly by the analytical triangle. (iii.) To the equation (v - Kx)a =0 answer a, branches, which may be all linear, but which also may group themselves in whole or in part into superlinear branches; if A', A",... are the orders of these separate linear or superlinear branches, we have XA = a1. (iv.) To the equation (vv - K x)a, = 0 answer a, v branches, which may belong to ca distinct superlinear branches of the type (Ae= v, A1 = 1, =y =X); these superlinear branches may however themselves be grouped, wholly or in part, into branches of higher superlinearity; if A, A', A",... are the orders of the distinct superlinear branches, these numbers are all divisible by v, and Z - = a1; we have also for every one of them - =, X, A, having to be determined subsequently for each of them separately. With the cases (i.) and (ii.) we have nothing further to do; the case (iii.) may be regarded as included under (iv.); we therefore continue the process 1 A 1 in this last case only. Put v- K v = v, ~v representing any one determinate value of the radical; and form by the analytical triangle equations of the type Al l 1 X1 (~ - 1 2 V)a = 0, of which the aggregate order in uv is a, and which give the initial terms of those oi values of vj, of which the order surpasses that of wv; we have of course - > -, or - > X; X\ and vl are relatively prime, but we observe that X\ is not '1a / v /1 VOL. II. s 130 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 19. necessarily prime to v. We consider the same four cases as before. (i.) To the equation v - K -v = 0, or more properly to the v equations comprehended in it, answers a superlinear branch of the type (A = v, =, = v). (ii.) To the v equations vv -K K v =0 there also answers a single superlinear branch for which A= vV1, A = v, IA2; = 1;? = XV, 71 = \; i.e., a superlinear branch of the type (-=, 1-=, A2= 1; 7?=-X, 7 =Xi 2) In this case, as well as in (i.), the discussion of the superlinearity is complete. A (iii.) To the v equations (v, - K V)a2 = 0 there may answer a2 superlinear branches of the type (f =v, =1; 7=XA); or these may group themselves in any manner into higher superlinear branches for each of which = v, y = X A; the A, numbers al (which have to be determined for each branch separately), satisfying Al the condition tA=1a2. (iv.) To the equations (v' -K V)a2 answer a certain number of superlinear branches, for each of which A Al -, I 1,; y/=XA1, Ayi-Xla2; while A2 and the subsequent numbers of the series have still to be determined, and may be different for each of them; we have however the equation -a 2 - - =a2. The process, which we need not follow further, may be considered to terminate for any particular development, when that development is separated from every other, and can be continued linearly. This will happen when, in the series a a a2..., we arrive at a term equal to unity. And we shall eventually arrive at such a term; for, though the second of two consecutive indices a may be as A great as the first (the equation (v- K4v) may, for example, at the next step in the process, lead to only one equation; and this may be of the type Al ('1- K1 Yv)al = 0, so that we should have a2 = aC), yet it is impossible for two branches to osculate one another indefinitely, because the discriminantal index is necessarily finite. Art. 19.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 131 If a, be the first of the indices a which is equal to unity, we have A A1 A... a=v,.. A =l; A1 -, 2 ' ' A ss8 + - and the development appertaining to the superlinear branch is of the type (A =VlV2...Iv, Al= =v2...Vs,...., As=v8, Ai8+=l, 7=XA1, '7=\lA2,......, s-l=X8-1As,, z=X). XXVII. MATHEMATICAL NOTES. [Proceedings of the London Mathematical Society, vol. vii. pp. 237-238. Read December 9, 1875. First printed in the Messenger of Mathematics, vol. v. pp. 143-144 (January 1876)]. (i.) ON a Problem of Eisenstein's. If p is an uneven prime, the function 4 1- = Z can always be expressed in the form Y2-(-)(- ')(1X2, where X and Y are rational and integral functions of x having integral coefficients. This is a theorem of Gauss. Eisenstein's problem (' Crelle's Journal,' vol. xxvii., p. 8') is 'To determine the cases in which the equation Z= 2 - ( - ) (-)pX2 admits of a multiplicity of solutions, and to ascertain the law connecting the various solutions, when there is more than one.' The solution of this problem is as follows: If [T, U] is any solution whatever in integral numbers of the equation T2-(-)(-l)-pU2= 4, and [X, Y] is any one given solution of Gauss' equation, then all the solutions of Gauss' equation are comprised in the formula [ TX+ ( - )(^P-1)2 UV }, (UX+TY)]. Thus, if p = 4n + 3, the equation admits of but one solution (the four solutions [~ X, ~ Y ] being regarded as but one) except in the case p = 3, when it admits of three; if p = 4n+1, the equation admits of an infinite number of solutions. That the functions [T (TX+ pUY), 2 (UX+ TY)] are all of them solutions of Gauss's equation, is evident; the proof that this formula comprises all the solutions of the equation is less elementary, because it depends on the irreducibility of the function Z. There exists a general theory of the representation of rational and integral functions of x by quadratic forms; such representation being, of course, only possible when the given function of x is capable of resolution into two factors by the adjunction of a quadratic surd. MATHEMATICAL NOTES. 133 (ii.) On the Joint Invariants of Two Conics or Two Quadrics. Let P and Q be two conies, and let 123 be any triangle self-conjugate with regard to P. Let also P1, P2, P3 be the rectangles of the points 1, 2, 3 with regard to the conic P, these rectangles being taken upon transversals measured in any fixed direction; and let Q1, Q2, Q3 have similar meanings with regard to the conic Q, the direction of the transversals being also fixed. Then the expression - + P + has the P,. P2 P3 same value for all self-conjugate triangles of P, and is, in fact, that invariant of P, Q which is linear with regard to Q and quadratic with regard to P, and the evanescence of which expresses that Q harmonically circumscribes P. The corresponding theorem in the geometry of the straight line is ' If Q1 Q2, P1 P2 are two pairs of fixed points on a line, and if A1 A2 is any pair of harmonic conjugates of P1P2, the value of the expression Al Q AlQ2 + A2Q1A2Q2 is independent Ae Po. A1 P + A P-. A2 P2 of the particular pair A A2 considered.' From this theorem the result given above for two conics follows immediately; from it the corresponding property for two quadrics may be inferred, viz. + 2 + + p = constant; and so on for P1 P,2 P3 P4 quadratic functions containing any number of indeterminates. (iii.) On the Equation P x D = constant, of the Geodesic Lines of an Ellipsoid. From this equation (in which P is the perpendicular from the centre upon the tangent plane at any point of the geodesic, and D is the semi-diameter parallel to the tangent line of the geodesic), it is convenient to be able to infer directly the principal properties of the geodesic line, without having first to transform the equation into M. Liouville's form g2 cos2 i + V2 sin2i = a2. In Dr. Salmon's 'Geometry of Three Dimensions,' the theorem of the constancy of the sum or difference of the geodesic radii vectores, drawn from any point of a line of curvature to two umbilics, is thus demonstrated. And it is worth while to add (though it is very improbable that the point has not been noticed before), that a proof of the theorem, that two geodesic tangents of a line of curvature, which intersect at right angles, intersect on a sphero-conic, may similarly be obtained without transforming the equation. Let Q be the point where the two geodesic tangents intersect at right angles, 0 the centre of the ellipsoid; let c = OQ, and let a, b be the semi-axes of the central section parallel to the tangent plane at Q. The two geodesics make angles of 45~ with the lines of curvature at Q; hence, for either of these geodesic lines, D= a2 b2 Let Q' be a second point a+ b~ 134 MATHEMATICAL NOTES. where two geodesic tangents to the same line of curvature intersect at right 2 P2 ab2 2P'2 a'2 b/2 angles; then +b2, because P x D has the same value for all a2 + b2 a 2 + b/2 geodesic lines touching the same line of curvature. But P2 a2 b2 = P2 a2 b'2 because parallelepipeds circumscribing an ellipsoid with their faces parallel to conjugate diametral planes are equal. Hence a2 + b2 = a'2 + b'2. But also a2 + b2 + c2 C 2 + bc2 -+ c'2; therefore c = c' and Q and Q' lie on the same sphero-conic. XXVIII. NOTE ON CONTINUED FRACTIONS.* [Messenger of Mathematics, Ser. ii. vol. vi. pp. 1-14 (May 1876)]. 1. LET 2- = + -, p and q being two numbers relatively prime, q A2"+ I3 +.. ~8 1 I1 of which p is the greater. Writing, for convenience, P = - and = - we divide P a line 01 of unit length (measured from left to right) into p equal parts at the points 1 P, 2P, 3P,..., (p - 1) P; and also into q equal parts at the points 1 Q, 2 Q, 3 Q,..., (q- 1) Q. We do not reckon 0 either as a point P or as a point Q, but we reckon 1 both as a point P and as a point Q, so that we have in all p points P, and q points Q, of which none are coincident, excepting the two extreme points, which coincide at 1. 2. It is the purpose of this note to show that the arrangement of the points P and Q upon the line 01, or, which is the same thing, the arrangement in order of magnitude of the proper fractions x and Y, may be inferred from the development of P in a continued fraction; and that, vice versd, the development of q q q may be inferred from an inspection of the arrangement of the points. An example will serve to explain the nature of the relation which we have to establish. 3. Let p = 39, q = 17, so that we have the development 3 = 2 + - 3 2+ 2 the arrangement of the points P and Q is indicated in the following scheme, in * The substance of this note was commnnicated to the Mathematical Section of the British Association, at the Bristol meeting in 1875. 136 NOTE ON CONTINUED FRACTIONS, [Art, 3s which transverse lines are placed at the close of each of the sequences* to be presently defined. P, 2P, Q 3P, 4P, 2 Q 5P, 6P, 3 QI7P, | 8P, 9P, 4 Q, 10P, 11P, 5Q)12P, 13P, 6 Q 14P l15P, 16P, 7Q ll, 17P, 18P, 8Q 19P, 20P, 9Q, I21P, 22P, 10Q123P I 24P, 25P, 11Q 26P, 27P, 12Q 28P, 29P, 13QI30P,||I 31P, 32P, 14 Q |I 33P, 34P, 15 Q 35P, 36P, 16 Q 37P, 38P, 17Q139P 111. In this scheme, because x, = 2, we have two points P before we come to a point Q; the sequence PPQ, which consists of 1A, points P followed by a point Q, we term a sequence of order 1; this sequence is repeated three times, because e2 = 3, and is then followed by a single point P (which is a sequence of order zero); a sequence, such as PPQ \ PPQ I PPQ j P, consisting of x2 sequences of order 1, followed by a sequence of order zero, we term a sequence of order 2; it contains 8I, p2 + 1 points P, a2 points Q; this sequence of order 3 is, in the scheme before us, repeated twice, because I3- = 2, and is then followed by a sequence of order 1; the sequence thus obtained, consisting of a3 sequences of order 2, followed by a sequence of order 1, we term a sequence of order 3. This sequence, containing I I2 I3 + 1 + 1+3 = 16 points P, and I2 I3 + 1=7 points Q, is in the * These sequences have been already noticed by M. Christoffel, in an interesting paper entitled 'Observatio Arithmetica,' (Annali di Mathematica, 2nd series, vol. vi., p. 148), with which I unfortunately did not become acquainted until my own investigation was completed. M. Christoffel considers the least positive remainders of the series of numbers q, 2q, 3q,... for the modulus p, and designates any remainder by the symbol c or d, according as it is less or greater than the remainder immediately following. It is easily seen that the sequences of the symbols c and d coincide with the sequences of the points P and Q. For if the remainder of sq is greater than the remainder of (s+ 1) q, we shall have, for some integral value of h, the inequalities (h —1)p < sq < hp < (s+ 1)q < (h+ l)p, s h s+l whence - < - < - 2 q p or the point hQ lies between the points sP and (s + 1) P. And, again, if the remainder of sq is less than the remainder of (s + 1) q, we have the inequalities (h-1)p < s < hs, (h-1)p < (s+l)q <hp, which give immediately h-1 s h s+l — <, -> -- q p q p proving that no point Q can lie between the points sP and (s + 1) P. Art. 5.] ANOTE ON CONTINUED FRACTIONS. 137 scheme repeated twice, because u, = 2, and is followed by a sequence of order 2. We thus obtain a sequence of order 4, consisting of M4 sequences of order 3, followed by a sequence of order 2, and containing iM2l3A + tI 2+ [l i4 + l34+ 1 = 39 points P, and P2 M3 g4 + 2+ + Yq = 17 points Q. This sequence, in the instance which we are considering, exhausts the whole system of points. We observe that all sequences begin with P, and that sequences of an uneven order end with PQ, sequences of an even order with QP. 4. In general, when the continued fraction is given, and it is required to obtain the arrangement of the points P and Q, we denote a sequence of order i by Si, and we then find successively S = P "Q, S2 = S1 P, S3= S3' S",..., the final sequence (which exhausts the whole series of points) being S, = S' SS_2. Vice versd, when the arrangement of the points is given, and it is required to infer from it the development in a continued fraction, we count the points P till we come to the first point Q; if there are 1uA of them,, is the first quotient, and S, = Pli Q. If we can repeat this sequence /2 times, without departing from the given arrangement, the second quotient is D2, and the sequence of order 2 is S2= S12P. This sequence we now repeat as often as we can do so without departing from the given arrangement, observing, however, that the last repetition of S2 is to be followed by a sequence S,. If, subject to this condition, we can repeat S2 M3 times, the third quotient is /3, and the sequence of order 3 is S3' S1. The subsequent quotients and sequences are to be determined in the same manner; and, if P- is the convergent,it + -, pi and qi are respectively the qi 2 +... 1i numbers of points P and points Q in the sequence Si. 5. Since y, << A~ + 1, or, P < Q< ( + 1) P, it is evident that the arq rangement of the first,1 + 1 points of the series is represented correctly by the sequence S1 = PL1 Q. We therefore proceed to show that the arrangement of the first ju2 + 1 points P, and the first /2 points Q is correctly represented by the sequence S = S2= P. Since 1 p 1 1 + - > - > I + IA2 q ^12+1 we have (1 k + 1 P > kQ, for all values of kc < 2, but (, k + 1) P < hQ, if k > 2. VOL. II T 138 NOTE ON CONTINUED FRlACTIONS. [Art. 6. If we write down the sequence Pi Q, 1 + times over, so as to obtain the series 1P, 2P,... 1P, Q, (1 + )P, (2 + i)P,*.. 2 P. 2, Q (1 + 2ll) P, (2 + 2,) P,... 3t1 2, 3 Q, (1 + 2 - 1] M) P, (2+ [p,2-l 1)... * P, 1 P Q, (l+ M2 M)P, (2+ M2 I1) P,... (1 + 2)P, (1 + /2) Q, (1+ [t2 + 1] Ml) P,........... the inequalities k,cu P < kQ < (kcL + 1) P, which hold as long as k <,2, show that all these points, with the exception of the last of them (1 + /2) Q, succeed one another in the proper order. But the last is in error, for, putting k= 1 +u2, (1 + M1 + 1 1u2) P < (1 + /2) Q, and consequently (1 + /M) Q does not follow immediately after (1 + /12) 1, P. We conclude, therefore, that we can repeat the sequence Pti Q /2 times, but that we cannot repeat it 1 + 2 times. And, since two points Q cannot come together, the series (P'i Q)f2 is necessarily followed by a point P, so that the sequence S2 = S12 P correctly represents, as far as it goes, the arrangement of the points. 6. We have thus shown that the relation between the continued fraction and the sequences S1, S2, S,... holds as far as S2. Assuming, therefore, that it holds as far as Si, where i >2, we have to prove that it holds as far as Si+l. The proof depends on an elementary theorem relating to continued fractions, which was first established by Lagrange. 'If Pi-, P1 are consecutive convergents to the same rational or irrational qi - qi quantity 0, i-1 - qi-_ 0 is less in absolute magnitude than any quantity of the form y - xO, where x and y are positive integers, of which x is less than qi.' Supposing, for brevity, that i is uneven, we infer from this principle that the least segment in the sequence Si is its last segment qi Q -pi P, and that the next least segment in Si is the last segment of Sii, viz. pi-1P - qi-i Q. We have to add that pi- P - qi_- Q is also less than the segment P (1 +p) - qi Q which immediately follows Si. For if qi+1 > +- Mi w+l, a condition which is certainly satisfied when i > 1, we have i~-p ~-+ > pi+1, qi - li-I qi+l Art. 6.] NOTE ON CONTINUED FRACTIONS. 139 i.e. >, because i + 1 is even. Let us write down the sequence Si 1+ k times over, and let yQ - xP be any segment of Si contained between two consecutive points P and Q, of which Q is to the right of P; the corresponding segment in (1 + k) S will be (kqi + y) Q - (kp, + x) P = k (q Q -2p, P) + (yQ - P); i.e. Q will be still further to the right of P, and the distance between P and Q be increased. Next, let xP-yQ be a segment of Si, contained between two consecutive points P and Q, of which P lies to the right of Q; or, again, let xP- yQ represent the segment (1 +_p) P - qi Q, which immediately follows Si. The corresponding segment in (k + 1) Si, or immediately following (k + 1) Si, will be (kpi + x) P - (kqi + y) Q = xP - yQ - k (qi Q -pi P); so that, if k be not too great, the two new points P and Q will lie in the same relative position with regard to one another as the two points originally considered, the distance between them being diminished; but, for values of k which surpass a certain limit, the point Q will be shifted to the right of P, and the segment QP will be replaced by a segment PQ. As long as this interchange of places between two consecutive points Q and P does not occur, so long the successive repetitions of S,, will represent with accuracy the arrangement of the points P and Q. Now the least of the segments xP - yQ is pi-l P - qi- Q, and i- P-qi-Q-i + (qi Q-Pi P) is still positive; therefore we may repeat Si 1 +i + times, but we cannot repeat it 2 + pi +1 times, for p-l P-qi-1 Q- (1 + S +1) (qi QQ-pi P) is negative. The sequence S' S+i_, will therefore truly represent, as far as it goes, the arrangement of the points P and Q; but the sequence SI+ i+1 Si_ would fail to do so. We should in fact come to an error in the last two points of Sil, which, according to the law of that sequence, we should have to write down as QP, whereas the true arrangement of these points is PQ. This suffices to establish the general theorem of Art, 4; but it is of interest to add, that the error which we have just shown must occur in the last two points of the sequence Sil+i' Si_i, is the only error that can occur in that sequence. And this is certain; for, in the first place, we have seen that there is no error in Si+"'; and, in the second place, if xP - yQ be any segment of Si_ of the same positive sign as pi1 P - qi-1 Q, xP - yQ - (1 + xi + ) (qi Q -Pi P) is necessarily positive; for, by T 2 140 NOTE ON CONTINUED FRACTIONS. [Art. 10. the theorem of Lagrange, xP - yQ > qi-2 Q -pi-2 P; and xP-YQ +p P- q, > (-Pi-2) P - (q- i-2) Q>p P - qi- Q by the same theorem; whence xP - yQ - (1 + i +l) (qi Q — Pi P) is positive, because pi.P - qi + - Q i, (qi Q -p P) is positive. 7. It will be noticed that the sequence S can only be repeated M2 times, whereas any subsequent sequence Si can be repeated 1l+ ui + times. The exception in the case of SI is apparent rather than real, and arises from the fact, that the period So consists of only one term. If we were to attempt to repeat the sequence S, 2+ u2 times, the sequence SO, which commences the last repetition of S1, ought, according to the general theory, to be in error; viz. its last point must be interchanged with the preceding point; and, as S0 contains but one point, this interchange vitiates the sequence S, immediately preceding. 8. Any finite continued fraction may be written either with an even or with an uneven number of quotients, because the last quotient may be made either equal to unity or greater than unity. If the number of quotients be even, the two extreme points P and Q, which coincide with 1, must be written in the order QP; if the number of quotients be uneven these points must be written in the order PQ. 9. If we omit these two last coincident points, the remaining p- I points P and q - 1 points Q evidently form a symmetric series, being similarly distributed on either side of the middle point of the line. And, similarly, if we remove from any sequence whatever its two final points, we obtain a symmetrical series, because the sequence Si corresponds to the division of a line into Pi equal parts and also into qi equal parts. 10. If we wish, from the arrangement of the points P and Q, to infer the arrangement corresponding to the fraction mi+,+ — +..., obtained from the i M+2 fraction p by omitting its first i quotients, we have only to replace the sequences Si and Si-_ by single points. Thus, in the example of Art. 3, if we put S = A, S, = B, we find A, 2A, 3A, B 4A, 5A, 6A, 2B 7AI 8A, 9A, 10A, 3B\ 11A, 12A, 13A, 4B[ 14A 1115A, 16A, 17A, 5B\ll, Art. 12.] NOTE ON CONTINUED FRACTIONS. 141 corresponding to z = 3 + 2. And, again, if we wish to obtain the arrange1 1 ment corresponding to the fraction g + - -, where i +j < s, we first mi+l+ "** 'i+j replace Si, Si_- by single points, and then consider in the resulting arrangement the sequence of order j. Thus the arrangement A, 2A, 3A, B 4A, 5A, 6A, 2B17A I corresponds to the fraction 3 + -. Addition to the preceding note. 11. The theorem of Lagrange, on which the demonstration in the preceding note depends, will be found in the second paragraph of his 'Additions to Euler's Algebra.' But as this theorem is no longer included in elementary treatises, we shall here place Lagrange's demonstration of it. If /i is the complete quotient of order i in the development of 0, we have 0 = iPi1+Pi-2 or _i= Pi-2-qi-2 i qi- +qi-2 i- - qi-20 But (i is positive and greater than unity; hence, Pi-2- i-2 0, and pi-l - qi_ 0 are of opposite signs, and pi_ - qi_ 0 is less in absolute magnitude than Pi-2- qi-20. Again, since pi qi -Pi -1 qi = (- l)i, we can always find, whatever the given integral numbers x and y may be, two integral numbers X and u satisfying the equations x = X q_ - + M y=XP_ + u Pi, whence we obtain y- X (pi_-qi- 0) + (pi- qi 0). As pi-_-qiO0 and pi-qiO are of opposite signs, if y-xO is less than pi_- -q-,_ 0, X and u must be of the same sign; that is to say, x and y are either respectively equal to qi and pi, or else they are respectively greater than qi and pi. 12. In the same place Lagrange has also established the converse theorem, that if b - a0 is a minimum difference, i.e. if b - a 0 is less in absolute magnitude than any difference y - x 0, in which x is less than a, - is a convergent to 0. a Writing pi for b, and q_1 for a, we first determine the positive numbers pi-2 and qi_2, respectively less than pi-1 and qi_, which satisfy the equation Pi-l qi-2~ -i- 2 qil =, denoting an unit of the same sign as pi - - qi- 0. If 142 NOTE ON CONTINUED FRACTIONS. [Art. 14. we write pi- -qi- 0 = ui_l, Pi-2- 2q-2 0 = -U1, we find, on eliminating 0, E -1 _i -2 In this equation ui_2 is greater, by hypothesis, than i-1, because qi-2 < qil _i_- ni-i~a fortiori il is greater than Uil. But ui-_ is of the same sign as e; thereqi-2 qi-1 fore, ui_j and ui-2 must have contrary signs. Consequently the quotient Ui-2.)i-12-qi-2 0 is positive, and greater than unity; and if "-1 be developed in a continued fraction having Pi-2 for its last convergent (which is always possible), we obtain - -2 Q Pi-2 +iPi-1 y +J 1 1 1 Ei-2 + Hi!i-l2 + 3 +...** e-l + i i.e. Pi-2 and Pi-l = - are successive convergents to 0. qi-2 qi —1 a 13. Combining the two theorems of Lagrange, we see that if we have ascertained, by observation, that p2i-i-qi- 0 is less than any difference y-xO in which x is less than qi-, we can at once infer that p —qi - 0 is also less than any difference y-xO in which qi- < x < qi. 14. The two theorems of Lagrange serve to define the successive minima of the expression y- x0. The theory of the successive minima of the expression? - 6 is perhaps less complete. Thus we have the elementary theorem, that the difference Pi' - 0 is less than any difference ~ - 0 in which x does not surpass xi-l x Q_ I, and is also less than any difference of the same sign with itself, in which x does not surpass qi; but there may be differences of a contrary sign to P - 0, ii-i in which x does not surpass qj, and which are less in absolute magnitude than 0. And again, if — 0 be a minimum difference (i.e. if — 0 be less in qii ~ a a absolute magnitude than any difference - 0, in which x is less than a), we canb7 x not in general infer that - is a convergent to 0. We shall attempt, in what follows, to define accurately the successive minima of the expression Y - 0, and thus x Art. 15.] NOTE ON CONTINUED FRACTIONS. 143 to give a greater amount of precision to this part of the theory of continued fractions. 15. We still consider a rational or irrational quantity 0, of which the development is 0 ciPi-l+Pi-2_ 1 1 (Pi q-l+ 9 -2 1 2+... i-1+ (i and, adopting the designation of Lagrange, we term the fractions = kpi- +Pi-2 Q7 kqi-l -i_2' where 0 > k < li, intermediate fractions. These fractions are evidently intermediate between Pi-2 and Pi; hence 0 lies between any one of them and P~-'b qi-2 qi2 qi-1 If -- 0 is a minimum difference, we can, by reasoning as in Art. 12, arrive at an equation of the form 1 1 1 1 1 t2+ /3+ *-^j-l+ X+ + b 1 1 where - = + a,u2+... / j_i + and we can prove that in this equation -, is positive. But we cannot prove that, is greater than unity; i.e. instead of the equation X= j, we have the inequalities 0 < X <; and thus from the hypothesis that - -0 is a minimum b~ a ba difference, we cannot infer that - is a convergent to 0, but only that - is either a a a convergent or an intermediate fraction. But not every intermediate fraction can give a minimum difference; for in order that k -0 should be a minimum differQk ence, Pk - must (at any rate) be less than Pi-l 0, because qi_-< Q. The Vjc 9i-1 absolute value of Pi-1 is 1 qi-1 qi-1 (~iqi-l + qi-2) and the absolute value of P 0 is Qk (kqi-1 + qi-2) (ij qi-1 + qi-2)' whe if k0 is a minimum difference, we ust have whence, if - is a minimum difference, we must have Qk 144 NOTE ON CONTINUED FRACTIONS. [Art. 16. 0p < 2k + qi-2 1 q..... (A) or + — -+... <2k + ial + I i_ -2+ * i_2 Pk, And this necessary condition is also sufficient. For, since 0 lies between k and P'-1, and since (if the condition (A) be satisfied) 0 also lies nearer to Pk than ( -i j- PI1 toi-, any fraction which is nearer to 0 than must lie between -and -i1, and must therefore have a denominator greater than Qk, because 4i~Pk -1 -Qk qi-1i i-1 Qk We are thus led to divide the fractions, intermediate between 2i-2 and Pi qi-2 qi into two sets, according as they do or do not satisfy the condition (A). We may call those fractions which do not satisfy that condition the inferior, and those which do satisfy it the superior intermediate fractions. We then have the theorem: 'The complete series of successive minima of the expression Y - 0 is obtained x by taking in succession for z the convergents, and the superior intermediate fractions in their natural order.' 16. If k> |ui, the condition (A) is satisfied; if =k = i, the condition is satisfied if ui-1 <,ui+; if k =- i, yui_ = l l, the condition is satisfied if _i2>Mi+2, and so on continually. If the continued fraction be finite, symmetrical, and of an uneven number of quotients,,i =2k being the middle quotient, we have a singular case in which the errors of Pi- and are qi-1 r exactly equal; we may in this case regard - as an inferior fraction. It will be seen that, as nearly as possible, one-half of the fractions intermediate between Pi-2 and P- are superior. Thus, if i = 2h * 1 is uneven, there are h inferior and qi-2 qi h superior intermediate fractions; if,. = 2 h is even, there are certainly h - 1 inferior and h-1 superior intermediate fractions; but whether - is inferior or Qh Art. 18.] NOTE ON CONTINUED FRACTIONS. 145 superior, can only be decided (as we have just seen) by comparing the quotients which precede, with those which follow it. 17. The difference P'- 0 is of the sign (- 1)i-; the differences - 0, Yi-1 Q which, in forming the complete series of minima of Y -0, we have to intercalate x between - - 0, and P - 0 are of the same sign as the latter of these differqi-1 qi ences, i.e. they are of the sign (-1)i. Thus, after every convergent there is a change of sign in the series of minimum differences, and the minimum differences formed with convergents are distinguished by this criterion from the minimum differences formed with superior intermediate fractions. 18. Again, if b 0 be any minimum difference, and if qi- < a < qi, the a only differences Y - 0, which are less than - - 0, and which have denominators x Gl x less than qi, are the minimum differences which lie between - - 0 and Ah 0. a It is sufficient to prove this for the case in which a = qi, b =pi-. Let P-1 be the last of the inferior fractions, intermediate between pi-l and P -; then 0, which lies between - and P~i —, is nearer to 'the latter than to the former of those fractions. If then Y be nearer to 0 than pi-l is, Y must itself lie x — i x between P"i- and PX. But, if x<q,, cannot lie between Pi- and 2-i; (qi Q- I Q-1 qi I hence, Y must lie between - and P-. But the only fractions between these X qi QXAlimits, which have denominators less than qi, and lie nearer to 0 than P1i-, are Qi - 1 the superior fractions intermediate between pi-2 and pi. For all such fractions G'P~-l Tji2 Sqi-2 i are of the type Pi —l+ r-2, the relatively prime numbers a and r satisfying the inequalities o X- < -< i,.......... (1) o-:+Ti-~ <i, whence < <i,...... (2) ~ q — <2-+ *.... e..... (3) r!i VOL. II. IT 146 NOTE ON CONTINUED FRACTIONS. [Art. 20. Now if,i =2h+1, we have X-l=h, and the inequalities h<-, and <2 h +1 (from which equality is excluded), show that unity is the only Ph admissible value for r. Again, if i = 2h and h is an inferior fraction, we have Qh h < -, < 2 h, and unity is the only admissible value for 7. In both these cases, therefore, the only fractions having denominators less than qi, which lie between P-^- and P, are the superior intermediate fractions. If, however, = 2 h, and QA-1 qi Ph. is a superior fraction, the inequalities (1) and (2) are satisfied by the values -O =2h - 1, = 2, so that, besides the superior intermediate fractions, the fraction (2h - 1)p -1 + 2P-2 lies between the limits Q-_ and,. But this fraction is (2h - 1) qi- + 2 qi-2 Qh-1 qi more remote from 0 than Pi-l is, because the equation C = 2h- is inconsistent qi-1 T 2 with the inequality (3). 19. The inferior intermediate fractions kP, < X-1, do not give minimum differences, because qi- < Qk, and Pi-1 0 is less in absolute magnitude than p ~i'-1 ^._y-1 Pk - 0. But, with the single exception of Pi-l1 -0, all other differences - 0, in Qk qi-1 x which x is less than Qk, are greater than - 0. For, if Y lie between and Qk x Qk Pi -, x must be greater than Qk; if P- lie between Y and i-1, the difference Y- 0 is certainly greater than - 0, because 0 lies between and Pi-'; x k k ~-l lastly, if -l lies between Y and V, we find (taking the case in which i is x uneven)ri> 1 1 > Y->Y zi-I1 1 P- Pk P x x qi-1 xqi-1 Qk qi- qi- Qk 20. The theorem of Lagrange admits of an important geometrical interpretation. If with a pair of rectangular axes in a plane we construct a system of unit points (i.e. a system of points of which the coordinates are integral numbers), and draw the line y = Ox, we learn from that theorem that if (x, y) be an unit point lying nearer to that line than any other unit point having a less abscissa Art. 20.] NOTE ON CONTINUED FRACTIONS. 147 (or, which comes to the same thing, lying at a less distance from the origin), Y is a convergent to 0; and, vice versd, if Y is a convergent, (x, y) is one of the x x 'nearest points.' Thus the 'nearest points' lie alternately on opposite sides of the line, and the double area of the triangle, formed by the origin and any two consecutive 'nearest points,' is unity. In particular, if 0=, p and q being relatively prime integers, the coorq dinates of the two 'nearest points' above and below the finite line joining the origin to the unit point (q, p) satisfy respectively the equations px - qy = 1, and px - qy = - 1. We thus obtain a simple geometrical method of finding the least solution in integral numbers of either of those indeterminate equations. UI 2 XXIX. NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT.* [Proceedings of the London Mathematical Society, vol. vii. pp. 199-208. Read May 11, 1876.] 1. LET 0o = o + 1 1 be any continued fraction, of which 0, 0,,... are 1 1+ A2+ 0 1 - ar the complete quotients; po, P1 P.. the successive convergents; so that q0 qi oPO =! o = 1 =... P Ai=i-Pil+ - i2, g+i =i qi-l+ i-2) 0 = _ipi-1+ P-2 3 _ O-2 qi- 2 i qi-l + qi —2 ' i-l -OqiAlso, let pti - Oqil (- 1)i ~i-, so that = e-2; 1-1 we have 01 -, and hence ei-=0 0- C0 =... * The following summary of the contents of this Note may be of use to the reader:Art. 1.-The relation, in a continued fraction, being the quantities e and 0. Art. 2.-The theorem that T+ UV/D is equal to the product of the complete quotients in the development of /D. Art. 3.-The same theorem for the period of complete quotients in the development of any quadratic surd. Art. 4.-Theorems as to the number of different periods of complete quotients; viz., equations (1)-(4). Art. 5.-Theorems as to the number of non-equivalent classes of quadratic forms; viz., equations (5) and (6). Art. 6.-Equations arising from a comparison of the formulee (5) and (6) with those of Dirichlet. Arts. 7-13.-Discussion of the nature of the periods in the more important special cases. Art. 14.-On the symmetry of any periodic series. Art. 15.-On the arithmetical conditions under which the various special cases present themselves. (It would be difficult to say that anything in the Addition (Arts. 7-15) is new: the discussion there attempted has never been given completely (see Art. 7); but this may have been because no one has thought it worth giving.) Art. 3.] NOTE ON THE THEORY OF THE PELLIAN EQUATION, ETC. 149 This expression, which supplies a measure of the rate of decrease of the difference eii, admits of an interesting application to the theory of the Pellian equation, and of binary quadratic forms of a positive determinant. 2. Let D be any positive integer, not a perfect square. In the development of VD in a continued fraction, let VD + Q /D +Q2 VD+ Q 1 2 PPi be the period of complete quotients; so that, if a is the integral number next inferior to A/D, V/D + Q, _ D + V a D+ Q, a Pi `-D-& ' Pi Let Tand U be the least integral numbers satisfying the equation T2 - DU2 ( -)i and let 1- 1 1 1 and let VD = a + — /j-I+ /u+..-. l+ 2a+..' we have T=pi_, U=qi-l, (- 1) ei_ = T- UVD; whence, by the preceding theorem, - V/D + Q T+~ U/D = nI:1 -~ Example.-The continued fraction equivalent to V/13 is 1 1 1 1 11 v13=3++1+ 1 + 1+ 6+...' and the period of complete quotients is 1/13 +3 v/13+1 -/13+2 1/13+1, ~-4 — — 3X, — g, — XX4 /13 + 3o giving (/13 + 3)2 (/13 +1)2 (V13 + 2) x -4 = 18 - 5 /13, and 182-25x13= -1. 3. Again, let Q = a + 2 bu + cu2 = 0 represent any properly primitive equation of determinant D (i. e. any quadratic equation whatever, in which a, b, c are integral numbers satisfying the equation b2 - ac = D, and a, 2 b, c have no common divisor). If ^/D+Q1, AD+Q2 VD+ Qi P P.. Pi 150 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 3. is the period of complete quotients obtained by the development of either root of Q, we shall have, as before, T+ UVD = D1 V- For the equations of the period are of the type ao + 2/ o U-a 1 = 0, - a, - 231i it + a2 U = 0, a2+ 2 /2 - a2 u- = 0, (-1)t-], _,+(-l ),- 2,8,__1 u-+( 1)ia o Ll_=, (- 1)'[a + 23uo a- a ] = 0, so that, if Po, Il, P2?..., -, is the period of integral quotients, we have 1 1 1 o =MO + J --- ---- - p 3+... i- + Uo ro + 8/D where zu0 = - a1 1 _ '0 1 Q2 i- 1 =uiu, ^u.i-1,U = D QD+Qs s=l p Ps But, by a known theorem (see the 'Report on the Theory of Numbers,' in the 'Report of the British Association for 1861,' Art. 96 (i.), p. 315*), we have T= 2 (p + q-2), U= Pi-2 _ Pi — qi-2 qi-.. (Q) a2 a, a0, 2,30 al whence Pi_ -- qi-1 O =Pi- - qi- -+ — = T- U/D, a1 (- 7)' ~:, /iVD + Q. or T+ UD= (- (-^) = Hns=, pD - T- UVD ' P3S If the given equation (Q) is improperly primitive (i.e., if the numbers a, 2b, c have 2 for their greatest common divisor), we have to replace the Vol. i.p. 195. Art. 4.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 151 numbers T and U in the equations (Q), by I T1 and I U1, where T1 and U1 are the least numbers which satisfy the equation T - D U2 = (- 1)i 4; and we find I (T1 + U1VD) - = H VDl p+ Q WI -S=1 PS 4. Every primitive quadratic equation of determinant D, of which one root is positive and greater than unity, and the other negative and less in absolute magnitude than unity, occurs in one, and only in one, of the periods of equations of determinant D (see the Report cited, p. 309, Art. 93*). Hence, every expression of the form D p, in which, P and Q being positive, Q is less than 1VD, P is a divisor of D - Q2 intermediate between VD - Q and I/D + Q, and D- Q2 the three numbers P, Q, D - are relatively prime, is the root of an equation contained in a period of equations of determinant D; and, for any given determinant, the number of periods of complete quotients is equal to the number of periods of quadratic equations, if we regard two quadratic equations such as a + 2bO+ c 02 = 0, - a- 2 b 0- c02= 0, which differ only in sign, as identical with one another. e D-Q2 Let P'= -, and let k be the number of periods of properly primitive complete quotients of determinant D; we have evidently (T+ UD)n.= e + D (1) or1 D VD+ Q r = log (T+ UVD) log p, the sign of multiplication IH, and the sign of summation A, extending to all positive numbers Q which do not surpass V/D, and to all divisors P of D- Q2, which are intermediate between /VD + Q and VD - Q, and are such that the three numbers P, 2 Q, P' admit of no common divisor other than unity. Let + (Q) be the number of such divisors of D- Q2; observing that, if V/ Q is a complete quotient in a period, Q == /P Q is also a complete quotient in the J /.D - Q * Vol. i. p. 186. 152 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 5. same or in a different period, we may write (T+ U2vD)k = n [ + Q;... (2) where it will be noticed that, if P = P', D = P2 + Q2, the two identical complete VD/ + QI~ Q quotients ID- and %PQ are each of them equal to [I D +Q JL VD L) - Q^ L -~v'D- QJ When D =1, mod 4, let k, be the number of periods of improperly primitive complete quotients of determinant D, we find as before ( T + I ET, D)7 D+Q (- iT.P+.-.VGa:v,. =....p. (3) 1 IJD+Q or kir2: lob - or t~ = log 2 (T + U, VD) g P the symbols II and 2 extending to all positive uneven numbers Q which are less than VD, and to all divisors P of D - Q2 which are intermediate between V/D + Q and /D - Q, and are such that the three numbers P, 2 Q, P' have 2 for their greatest common divisor. If +4 (Q) be the number of such divisors of D- Q2, we may also write.2 U, D)7=... =.D,I (4) It will be observed that, if Q is even, we have always +, (Q)= 0. 5. Let h and h, respectively denote the numbers of properly and improperly primitive classes of quadratic forms of determinant D, and let [7, v], [vr, v1] be the least numbers which satisfy the equations 72 - Dv2 =+ ~1, 1 - DU2 = + 4; so that, when the equations x2- Dy2 =, x Dy2 = _ 4..... (T) are resoluble [they are either both resoluble or both irresoluble], T+ VD = (T + U /D)2, 2 7 + v, D = (2 T, + I U1 VD)2; and when the equations (T) are irresoluble, 7 +v1D = T+ UIVD, rl+ uvlD= T + U1 D. We can now establish the equations _+7/D + Q ). (5) (r ++ 1 D)7"= II[..... Q(6) For this purpose it is only necessary to show that, when the equations (T) are not resoluble, we have h = 2 k, h = 2k;....... (a) Art. 6.] QUADRATIC FORMS OF A POSITIVE DETERMINANT. 153 but when these equations are resoluble, we have, instead, h =, h =......(b) To the single period of complete quotients VD/D +3o ~VD + 31 VD +132.,,........ (7) a1 a2 a3 there correspond two periods of reduced quadratic forms, viz., (ao,0 ) - 1)(-, ala2), (a2, 2, 3)..... (8) and (-ao, d, a), (al, {,- a2), (- a2,, a),..... (9) For the complete quotients (7) are the positive roots of the equations (Q); they are also the positive roots of the s sae equations with their signs changed; and the period (9) is related to the period (- ) exactly as the period (8) is related to the period (Q); viz., the coefficients of the forms are the same as the coefficients of the equations, except that in the periods of equations the middle coefficients are alternately positive and negative, whereas in the periods of forms these coefficients are all positive. The two periods of forms are, in general, but not always, distinct; and we shall now prove (what is indeed well known) that these two periods are, or are not, identical, according as the equations (T) are, or are not, resoluble. We may observe that the form (a, - b, c) is termed the opposite of the form (a, b, c), and (- a, - b, - c) the negative of (a, b, c): thus (- a, b, - c) is the negative of the opposite of (a, b, c). (i.) If the equations (T) are resoluble, any form (a, b, c) of determinant D is properly equivalent to the negative of its opposite; viz., (a, b, c) is transformed into (- a, b, - c) by bU-T, -cU -aU, bU+T Hence the reduced forms (ao, o3, -a,), (-a0, 0o, a1) are properly equivalent; either of them is therefore contained in the period of the other; i.e. the two periods are identical. (ii.) If the two periods (8) and (9) are identical, the form (- ao, o, a,) must occur in the period of (a,, /0, - a); and because its first coefficient is negative, it must occupy an even place in that period. Hence the period of complete quotients (7) consists of an uneven number of terms; and we infer from the formulae (Q) of Art. 3 that the equations (T) are resoluble.. 6. If D=- 1, mod 8, we have h = hA; if D 5, we have h = 3hi when 7- and VOL. II. X 154 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 7. v1 are even, but h = h, when r1 and v, are uneven, in which case (71 4TvVD)3 =T+vVD. We thus find, in both cases alike, ii /D+ Q ~(Q) = D + Q (Y& 6 (Q [VD Q] =H[VD+QQl ' where o = 1, or = 3, according as D 1, or _5, mod 8. Again, since, by the formule of Lejeune Dirichlet, we have (see cited, Art. 101 ) hl 1 hlog[T+ vD]=2VDI ( v, \~~~n n.. (12) the Report where the sign of summation extends to all numbers prime to 2D, and is the generalised symbol of quadratic reciprocity, we obtain 2 ( 1 -= + (Q)log D, v/ 'DE(D 6 v/D- Q' or I= + _Q + +5 + ] or n _ [ 3D 5...].... Similarly, from the formulae (see ibid.) h log [ I+ 1 v^D]== 2vD (-) 01 2 a' flf I (1) (13) where oa has the same meaning as before, we infer () 1 2By (Q) [ + 3 + + ] (14) n n -D 3D2 5D33 It is probable that a direct demonstration of the equations (12), (13), (14), of which any two involve the third, would offer considerable difficulties. Addition to the preceding Note. 7. As the preceding determination (equations 5 and 6) of the number of non-equivalent classes for a positive determinant depends on the equations (a) and (b), which assign the relation between the number of periods of complete quotients and the number of periods of reduced forms, it is worth while, for the sake of distinctness, to describe fully the characteristic appearances presented by these periods in certain special cases which are of some importance. Every form, or class of forms, is, of course, properly equivalent to itself, and improperly equivalent to its opposite. But a form, or class of forms, may be(i.) Properly equivalent to its opposite, and improperly equivalent to itself (in this case the class is ambiguous); * Vol. i. p. 217. Art. 8.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 155 (ii.) Properly equivalent to its negative, and improperly equivalent to the negative of its opposite; (iii.) Properly equivalent to the negative of its opposite, and improperly equivalent to its negative. Since, if any two of these specialities coexist, they necessarily involve the third, there are four cases to be considered, viz., the cases (i.), (ii.), (iii.), in which the specialities (i.), (ii.), (iii.) exist singly, and the case (iv.) in which they all exist simultaneously. We shall briefly refer to each of these cases in succession. We may observe, however, that the case (i.), which is that of an ambiguous class, has been fully considered by Gauss ('Disq. Arith.', Art. 187, Obs. 6, 7, 8); of the rest, the case (ii.) has, perhaps, attracted less attention than might have been expected. 8. If the period of reduced forms equivalent to (a, b, c) is (a0, o, -a1), (-al,, a2),..., (-a-2k -, 2k-i, ao),.. (8) the periods of reduced forms equivalent to (a, - b, c), ( - a, - b, - c), ( - a, b, - c) are respectively (aO, P2k-1, - a2k-1) (- ak-, P2k-2, a2k-2),..., (- al, o, ao),.* (10) (-ao, P/2k-i, ak-1) (a2k-1, P2k-2, -a2k-2), *.., (a, P0, - a), * * (11) (-a, P0, a1), (a,, -,..., (ak-1, 12k-i, -ao). (9) As in Art. 3, we designate the period (7) of complete quotients, or, which is the same thing, the period formed by the positive roots of the equation (Q), by U0, U1,...* *, 2k —1 The negative roots of the same equations we represent by 1 1 1 Vo VI "' _l so that, if Mo,,u,.., 12k-1 is the period of integral quotients, we have, by a wellknown theorem,= s =Is = I+1, the symbol Ix denoting the greatest integral number not surpassing x. The four periods of forms (8), (10), (11), (9) we represent for brevity by the symbols 00, 1 2,,**.., _. * * _2k-1, - _2k-2, _2k-.3, ***,......** _2k-, 2~_,2k-2, 2-3_ ***, *o,........ ** (oP, O, ~ 21 *. 2,..., *... X 2 156 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 10, Two such forms as (a, b, c), (c, b, a) are said to be associated; thus, 0, and As, or again (p and As, are associated forms. The periods q) and k, and again the periods 4 and,, are themselves termed associated periods. The period of complete quotients corresponding to the periods 1 and 1P is (Art. 5) Uto, U1,..., U2k —; and similarly the period of complete quotients corresponding to the periods ' and,k is ~andtv P is2l, V2k 2k_2,..., Vl, V0 9. Case (i.)-If (a, b, c) is properly equivalent to (a, - b, c), the periods {D and I must coincide; i.e. we must have, for some value of o., o= +2a,,, the suffix 2 r + 1 being uneven, because the extreme coefficients of +2a+1 must have the same signs as the extreme coefficients of o0. Hence q, = +2a, (2 = +2,-,...; and finally q, = +, +, a + 1= +,, or there occur in the period two consecutive forms of which each is the associate of the other. As we may begin the period with any form we please, we may suppose that $o, and (P, are these two consecutive forms, so that =3o, a, = a0, 2/3 O0, mod ac. It will be seen that, if o = +2 + 1, we have not only = +a + 1, (a + i= a,, but also 0(P+k = +a+l-k-+a+k+1D 0Pa+k+1 = ++k. Thus a sequence of two associated forms occurs twice in the period; and, assuming (as we have done) that - = 0, the period of forms is of the type 1 5 01; (Pl 1 2 P3 *.. * nk-1; (k) nAk; +k-l5 +k-2 5 *** 3 +2? where ( + 1 =,+2k- for every value of s. The period of integral quotients is of the type Xo,,1, Fa,.., ik-~; Xk, 1k-1, Mk-2,., 1, where Xo - A 2k,al ak +1 The period of complete quotients is of the type V^ 1; U1 3,* *, r-l; U^k5..A.; Ukk 1k 2-1k, k-2 I V 29 where %.+ 1= V2k- for every value of s. The periods of the coefficients a and a are respectively of the types a,; a2,..., ak; ak +1; a,..., a2; P1, P3 P12, ** *, k-l; PkI Pk; Pok-In ** * P2* The two ambiguous forms are (pb and +k. 10. Case (ii.)-If (a, b, c) is properly equivalent to (-a, -b, -c), the periods b and 4 must coincide. Hence we must have (for some even value of the suffix 2 O) 00 =2a; whence P=A*, and also + k=+ ak =ff++k. The Art. 12.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 157 equation P<, = =, is equivalent to the equation a, = ac +; we thus see that the period contains two forms, in each of which the extreme coefficients are equal in absolute magnitude; so that (supposing, as before, that - = 0), we have D= + a2, D= 3 + a. The period of reduced forms is of the type +0 =0o; 1, 02 *,..., f-; k = k; k-1, +k-2, *... X 15 where 2 = -2k-; the period of complete quotients is of the type Uo =V; U1, U2..., 6 k-; Uk=Vk; Vk-1-, Vk —2 **. V, where z = 2k-. Lastly, the periods of integral quotients and of the coefficients a and / are of the types MO 1l) * ** en C k-1; A - 4l M -.1? *.*. * l * O 1; ao; I,./., /3 k -; / 3k ak —, k-2,, l 3. o0; On 2, *..., -; Ok-I; 07-,; k-1,..-2, 0. 11. Case (iii.)-If (a, b, c) is properly equivalent to (-a, b, - c), the periods ) and P coincide, and we must have 00o= (2+,,I, o= =2y+] where 2o-+I is less than 2k. From these equations we infer P0 2 (2 + 1), or 2 r+l=1k, since if (P = m,, m is a multiple of 2k. The period of forms is therefore of the type (0 **..., _k-i, 0 ( 0 *..., _k-1, k being an uneven number; the periods of complete quotients, of integral quotients, and of the coefficients a and /3, consist each of a period of k terms, twice repeated. The period of equations (Q) in like manner consists of a period of k equations, twice repeated; but each equation appears in the second half of the period with its sign changed. 12. Case (iv.)-If (a, b, c) is properly equivalent to any two of the forms (a, - b, c), (- a, - b, - c), (- a, b, c), and therefore to all three of them, the nature of the periods is most readily ascertained by considering the series of integral quotients. Since the conditions characteristic of the cases (i.) and (iii.) must be united, the semi-period XO, /1, 2,...**, _-1 must be term for term identical with the semi-period Xkn Mbk-1, a2n *.. M1, k being an uneven number 2 i + 1. Hence 0 = Xk, and the period is of the type - Ml? P2>..., f) Miu,...Pi M, 2 MJI 158 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 13. twice repeated; which combines the characters of the periods of integral quotients in the cases (i.), (ii.), (iii.). The period of forms is of the type +1, 01; 02, 3,..., (,i+l, '), %_i-,, *, +~2; +1 (1; (,2 ) **... pi + 1, -1, *. * * 2; where Pb = '2i +2-s = <Ps-2i-1 = 'l4i+3-s; the period of complete quotients is of the type VI t1 l9 tt2) 83 **5 Sti+1; 1 it Vi+ + **.. V2; twice repeated, where u8 = v2i +2-; and the periods of the coefficients a and 3 are respectively a; a2, a3,.., ai+, ai+, ai,..., a2; P1, P1; 2,.23.., f? Pi+1 Pi, **.. P2; each of them twice repeated. 13. If therefore we develope the two roots of a given primitive quadratic equation, we obtain, in the general case, two distinct associated periods of complete quotients and four distinct periods of reduced forms. In the special cases (i.) and (ii.), we have but one period of complete quotients and two periods of reduced forms; in case (i.) the two associated periods of each pair combine; in case (ii.) each period of reduced forms becomes identical with the negative of the opposite of its associated period. In case (iii.) we have two distinct periods of complete quotients; but only two distinct periods of reduced forms; the period of any form being identical with the period of the opposite of its negative, and consisting of an uneven number of forms followed by the opposites of the negatives of the same forms; the period of complete quotients contains only half as many terms as the period of reduced forms. Lastly, in case (iv.) we have but one period of complete quotients, and but one period of reduced forms, the four periods, which in the general case are distinct, being all identical with one another. We may observe that, if the equation va- 2- b+c = O......... (p) can be satisfied by three numbers X, Ix, v, which also satisfy the condition - A,..,........... (q) the form (a, b, c) is transformed into (a, - b, c) by the substitution ', and consequently has a period of the type (i.). If, instead of the condition (q), the condition 2- -....., (r) Art. 14.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 1591 is satisfied by the three numbers (X, g,, v), (a, b, c) is transformed into (-,- b, - c) by 1' IV and the period of (a, b, c) is of the type (ii.). Lastly, if the equation (p) can be satisfied by two different sets of numbers, of which one set satisfies the condition (q) and the other the condition (r), the period of (a, b, c) is of the type (iv.). 14. The periods which we have to consider in the cases (i), (ii.) and (iv.) afford examples of each of the three kinds of symmetry which can exist in a periodic series. Let... Co, cl, c2,... c,_,... be a period of n terms repeated indefinitely in both directions; it will be found that the series thus formed may be symmetrical (i.e. may be the same whether we follow it forwards or backwards) in three, and only in three, different ways. (i.) Let n be even; and let the series continued from Co forwards coincide with the series continued from c,,, + backwards, so that Co = 21 +1, c0 = c2k,...: the period then is c c c c c c period then is COI, C..., Ck; Ck,...* Co, C2k+2,... Ca+l1 or, if n = 2v, C__+k+l, C-V_,k+2,... o, Co C,..., Ck, Ck,..., Co, C__,.., Cy.-Vk+;.. (A) where there are two centres of symmetry, one falling between the two terms c,,, the other falling between the two terms c_+ k +. Of this type is the period of the coefficients f3 in case (i.); and the periods of the integral quotients and of the coefficients a in case (ii.). (ii.) Let n still be even, but let the series continued forwards from Co coincide with the series continued backwards from c2,; the period is Co, C1...', Ck —l Okj Ck —l *-,' CO; C2k+l2.1. * Cn-l, or, if n = 2 v, C_y+k; C_+tkil, *.., C, Cl,... Cl; Ck; Ck_l,..., CO,..., C_- Y k+l, ~. (B) where there are two centres of symmetry falling on the terms c_v k and c, respectively. The symmetry of the integral quotients, and of the coefficients a in case (i.), and of the coefficients /3 in case (ii.) is of this type. (iii.) Let n = 2 v +1 be uneven; and let the series continued forward from c, coincide with the series continued backward from C2,k+. The period is of the type cO, C1,... ck, Ck,..., c0, C2k+2,..., C-l, or Cv+ck; C —v+k+l, C —v+k+2,..., CO, I1,... Ck, Ck..., C0o C-0-l... C —y+k+l, where again there are two centres of symmetry, one falling on the term c_V+^, the other between the two terms c,. If we had supposed C, = c2, we should have obtained a period of the same form. It is evident that, if this period be doubled, 160 NOTE ON THE THEORY OF THE PELLIAN EQUATION, ETC. [Art. 150 it combines the symmetries of the periods (A) and (B). Of this type are the periods of integral quotients and of the coefficients a and 3 in case (iv.) In case (iii.) there is no symmetry; but an unsymmetrical uneven period is twice repeated. 15. Every determinant has ambiguous classes; and every ambiguous class has a period of the type (i.). But developments of the types (ii.), (iii), (iv.) can only present themselves in the case of determinants of the form P or 2P, where P is a product of uneven prime numbers of the form 4n + 1. For in case (ii.) D must, as we have seen, be the sum of two square numbers prime to one another, and in case (iii.) the equation T'-DU2 = -1 must be resoluble, whence again D is the sum of two squares prime to one another. If u is the number of different primes dividing P, the number of ways in which P can be decomposed into the sum of an even and uneven square prime to one another is 2~-1. Let D = P, and let D = A'+ B' be one of these decompositions, A being uneven and B even; the forms ( - A, B, A), (A, B, - A), ( - B, A,B), (B, A,- B) are all reduced, and the first two are properly, the last two improperly, primitive. We thus have 2/-1 properly primitive periods of reduced forms of the type (ii.); and as many improperly primitive periods of the same type; i.e. since there are 2'-1 properly and as many improperly primitive ambiguous classes, there are as many classes having periods of the type (ii.) as there are classes having periods of the type (i.). If D = 2P, we have a similar result; viz., there are 2"-1 equations of the form D = 2P = A2 + B' in which A and B are both uneven. We thus obtain 2' properly primitive periods of reduced forms of the type (ii.); i.e. as many as there are of type (i.) There are, of course, no improperly primitive classes of a determinant of the form 2 P. When the equations (T) are not resoluble, but the determinant is of either of the forms P or 2P, the developments of the type (i.) and those of the type (ii.) are entirely distinct from one another. On the other hand, when the equations (T) are resoluble, the developments of the types (i.) and (ii.) coincide, giving rise to developments of the type (iv.), and all the remaining developments are of the type (iii.). It is known that, when D is an uneven power of an uneven prime of the form 4n + 1, the equations (T) are always resoluble. But when D has any other value of either of the forms P or 2P, there is no known criterion for deciding whether these equations are or are not resoluble. XXX. ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. [Proceedings of the London Mathematical Society, vol. vii. pp. 208-212. Read May 11, 1876.] LET (m, n) denote the greatest common divisor of the integral numbers im and n; and let + (m) be the number of numbers not surpassing m and prime to mn; the symmetrical determinant A,=z~(1, 1)(2, 2)... (m, mz) is equal to (1) x + (2)x... x, (m). This theorem may be established as follows. Let p1, P2,... be all the different primes dividing rn, and consider the columns (P) of which the indices are mn m in i n m, - 99-, 999, -.~7. Pi 2 P lP2 P lP2P3 Take these columns with the signs of the corresponding terms in the product ^m(l)=m (l-1i l —1)...; and, attending to these signs, replace the terms of the last column of A, by the sum of the corresponding terms in the columns (P). The value of A. is not changed: the term (m, m) is evidently replaced by + (m); and we shall now show that every other term (m, k) in the last column is replaced by zero; i.e. that A, == + (m) x A,_, which is the theorem to be proved. First, let k be prime to mn; then (m, )= 1, (I, k )= 1, &c. PI3 VOL. II. y 162 ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. and (in, k) has to be replaced by a sum of units, of which as many are negative as positive; i.e. by zero. Secondly, let k be a divisor of m, other than unity or m itself; and let us separate the primes p into two classes, q and r, in such a manner that k does not divide any quotient of the form -, but does divide every quotient of the form m-. There may or may not be any primes q, but there must be at least one prime r, or we should have k= m: we further observe that (mi k) = ( ) =(rl ) =... =k; ( -,k)= fi-, (in k)= k q q q qq Thus, if we were to attend only to those columns of which the indices are i, -,...,,..., we should have to replace (in, k), or k, by k H(I -- ), just q q1!2 ' T as before we replaced m by + (m). But we have to attend to the complete series of columns (P); and thus we have to replace (in, k), not by kHT(- -) taken once, but by k(l - -) taken as often as there are terms in the product I(1- -), and taken each time with the sign proper to the corresponding term of that product; i. e. (m, k) is replaced by zero. Lastly, let k= h 5, being the greatest common divisor of k and m; so that (m, k) = (m, S) = '. If d is any divisor of m, we have the elementary theorem ( N, =5 (, h~). For, if ($, ) = =t, we have also (in, d5) = ds'; and hence 8', which is a common divisor of mn and dcl', divides dS', which is the greatest common divisor of those two numbers. But h is prime to therefore, a in.,in J nm. 5' fortiori, h is prime to d; z. e. ( y-, h ) =1, for d ' is prime to, as well as to h, or, which is the same thing, (, h) = = (, 5 ). It appears from this d ' ha>= s~=(, 3. It appearsfromthis that in the columns (P) the terms which lie in the row of which the index is hU, are precisely the same as the terms which lie in the row of which the index is S; and hence (m, h3) is replaced by terms of which the sum is zero, because (m, 5) is replaced by terms of which the sum is zero. ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. 163 The following remarks are suggested by the preceding theorem, or by its demonstration: (1) If we denote by la the greatest integer not surpassing the positive quantity a, the theorem may be expressed by the equation AM I.(A) 1.-2.3... m.... (A) 1. 2. 3... m ( 29) the sign of multiplication extending to all primes not surpassing m. (2) Instead of the greatest common divisors themselves, we may consider their powers of exponent s; writing,I (m) = W8 II (1 — ),andfollowing the same course of demonstration, we obtain the theorem a, +(, (1)s (2, 2)s... (, m)8 = 8 (1) x (2) x (3) x... x (m),...(B) A m, s or =IT J) ~ (1. 2.3... m) -) from which we infer, as a particular result, A, 2 =1.2.3... mx n(l +P). Am,1 P (3) The equation (B) is an identity with respect to the exponent s, which may have any value whatever: the case in which s = - 1 is especially interesting. Let [m, n] be the least common multiple of m and n, so that [m, n]= (in, n) we find VM = ~ [1, 1] [2, 2]... [m, m] =1.2.3... m. 1 (1 -p), in, Im whence U= +Hi.p 1 Am and, in general, if V, = 2 ~ [1, 1]S... [mn, m]s, Vm, _ ((n.p ')s: 7ln, s the sign being that of (- ) z-. (4) If, for the greatest common divisor ~ of n and n, we substitute any function whatever p (S) of S, and denote by d (in) the function (m) ( +..., 2 Y 2 164 ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. we arrive at the identity +q(1, 1) (2, 2)... (m, m) = (1) x Q (2) x I (3) x... x (m). Two particular cases are worth attention. (a) Let s be an integral number, and let S(m)=1s- +2S+3s+...+ (m- + 1)S+m, so that, when s > 1, s(m)=S m+1 + s 2m- s(sl-)(3s-2)- +, qS~(m) s+- + +/ +B1 O H S1.2 31.2.3.4 B1, B3,... being the fractions of Bernoulli, and the last term being s-3 s-2 (-1)2 B2_ m2, or (- 1)2 BSm, according as s is uneven or even. Let also +,s (m) be the sum of the powers s of the numbers prime to m and not surpassing m; we shall have z + S s (1, 1) ~> (2, 2)... Os (m, m)= +8 (1) x As (2) x + (3) X... x As (m, m). The forms of the functions,s(m) are deducible from the expression for (min) (see a paper by Mr. Thacker in ' Crelle,' Vol. XL., p. 89); we thus find l(m) =m2l (1-), ( I) = m (1-) +- (1-), the general rule being that, in order to obtain s(m(), we are to substitute Mkl (1 -pn-) for mnk in the expression for (n)(m). (i3) Let o-r(m) be the sum of the powers s of the divisors of m; it will be found that i (m)- z (^)+ - (^.. For, if m =p'lp2..., and if we put P = ( +p+...+pas), P'= (1 +s +... +p(a-l)s), we have s-(m) =P1XP2x..., s P() P2 xP... Ts(A = P1 x P2 x P3..., whence ( S()+S( ) - =(P1-P)(P2-P )... =. We thus obtain the equation z + s(1, 1) %,(2, 2) cr,(3, 3)...% (m, m) = (1.2. 3... m), ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. 165 in which s may be any quantity whatever: the cases in which s- =0, s= -1, s = + 1, are equally remarkable. (5) Returning to the equation A, = + (1, 1) (2, 2)... (m, m) = (1)x +(2)x... x y (rm), we may observe that it is by no means necessary that the numbers 1, 2, 3,..., m should be the natural series of numbers. We may, in fact, take any different numbers il, A2,.., Am that we please, subject only to the condition that, if x be any one of these numbers, every divisor of ax must also appear among them, a condition which implies that unity is always one of the numbers S. Subject to this condition, we have always z + (1, 1) (I2, 2)... (xm, m) = I () X + (2) * *+(m), or, more generally (see 4 supra), z + ( (Al, E1) ) (92, )... M (=, n) = A (, Al) ) (2 ~ 2), **)... (iUn, 1 m). The most obvious cases are-(a) when we reject the multiples of given primes; e.g., when the numbers 1 are the uneven numbers in their natural order; (3) when we consider only numbers composed with given primes, e.g. when the numbers are all the divisors of one of them iu; (7) when we consider only linear numbers, i.e. numbers not divisible by any square. In all these cases the results are immediately obtained by the methods which we have already used, and which it is unnecessary to exemplify further. (6) Lastly, the symbols ux need not represent integral numbers at all, but may be any quantities which admit of resolution into factors in a definite manner. If, for example, ai = xi-l or xj- according as i < j, orj < i, we have + X11, 22,..., am = (XTn-1 _ X-2) (m-2 m- 3)... (X 1) = (x- 1)m-1x-(m1)(M-2) XXXI. ON THE PRESENT STATE AND PROSPECTS OF SOME BRANCHES OF PURE MATHEMATICS. [Proceedings of the London Mathematical Society, vol. viii. pp. 6-29. Read November 9, 1876.j I HAVE been led to believe that the Society may not be unwilling to allow a certain latitude in the scope of the remarks which they permit their Presidents to address to them upon retiring from the Chair. Relying upon this belief, I propose, on the present occasion, to invite your attention to some considerations relating to the present state of Mathematical Science, with especial reference to its cultivation in this country, and to our own position as representing a great number of those who are interested in its advancement. The subject is so extensive that I am sure you will excuse me if I endeavour to limit it in every way I can. I propose, therefore, to exclude from what I have to say all that relates to Applied Mathematics, and to ask you to confine your attention to questions of Pure Mathematics only. I am well aware how much by this exclusion I restrict the field before me; but the restriction is forced upon me, not only by the limit of time, but by the far narrower limits of my own knowledge. And I cannot help adding that I shall regard it as a fortunate circumstance, if the attention of my successor, when he in his turn is looking round him for a subject for his own Presidential Address, should be attracted by a domain, upon which I must myself decline to enter, but of which he, better perhaps than anyone among us, is fitted to take a clear and comprehensive view. The restriction which I have mentioned is far from being the only one which I must impose upom myself. I can only presume to offer fragmentary remarks upon great subjects, in the hope that even such casual and hasty notices may not be without their use, if they serve to remind us of the vastness of our science, and yet of its unity; of its unceasing development, rapid at the present ON SOME BRANCHES OF PURE MATHEMATICS. 167 time, and promising to be no less rapid in the immediate future; of its marvellous power of assimilating to itself the accessions which each year brings to our knowledge of external nature, while yet it derives strength and vitality from roots which strike far back into the past, so that the organic continuity of its gigantic growth has been preserved throughout. In every science there is a time and place for general contemplations, as well as for minute investigations. And it is a rule of sound philosophy that neither should be neglected in its proper season. 'Itaque alternandse sunt istse contemplationes,' says Lord Bacon, 'et vicissim sumendse, ut intellectus reddatur simul penetrans et capax.' * Perhaps it is the besetting sin of mathematicians to concentrate the mental vision upon as narrow and definite a field as possible. And there is much to be said in excuse for our indulgence of this tendency. If we are to find anything worth finding in the mines of mathematical research, we must dig deep; and if we want to dig deep, we must, if we are not gifted with Herculean force, confine our efforts to a narrow superficial area. But the tendency is not without its peril. The illustrious mathematician under whose auspices this Society was founded, felt it right in his opening address to warn us against the danger. 'Our subject,' said Augustus De Morgan, on the 15th of January, 1865, witha characteristic irony of expression, 'Our subject is really rather a wide one. But there are mathematical publications in which it is contracted; and it is often treated as a narrow subject.' He cautioned us against falling into 'a line which may be useful, but which is still confined and partial'; and, while exhorting us to do our part in the additions to the more rapidly developing 'branches of the science,' he bid us at the same time take care 'not to let any one particular branch overgrow us.' It would not have been Augustus de Morgan if he had not added some pointed criticisms upon examinations in general, and on Cambridge examinations in particular, and if he had not cautioned us against any excessive admiration for that part of mathematical ingenuity which devotes itself to the narrowest of all the narrow fields ever chosen by a mathematician, the invention and solution of 'ten-minute conundrums.' It is now nearly twelve years since these warnings were given to the infant Society by its first President; and perhaps the time may have arrived when we might put to ourselves the question whether its subsequent history has shown that we have profited by the lessons of that eminent and large-minded teacher. It would ill become me to attempt to answer such a question. I * Novum Organon, Lib i., Aph. 168 ON THE PRESENT STATE AND PROSPECTS OF would only venture to express, and that with great diffidence, the double opinion-that, on the one hand, the mathematical world will wholly acquit the Society of having devoted its energies to little or trivial subjects; but that, on the other hand, while it would be universally conceded that the volumes of our Proceedings contain memorable additions to mathematical knowledge, it might be alleged by an 'advocatus diaboli' (if such a character should be assumed by some severe critic) that we, in this respect resembling the other mathematicians of our country, have shown, and still continue to show, a certain partiality in favour of one or two great branches of the science, to the comparative neglect and possible disparagement of others. Perhaps it would be well to begin our reply by denying the charge; but, having done so, if we should be advised to urge a second and somewhat contradictory plea, we might with great plausibility rejoin that ours is not a blameable partiality, but a wellgrounded preference. So great (we might contend) have been the triumphs achieved in recent times by that combination of the newer algebra with the direct contemplation of space which constitutes the modern geometry-so large has been the portion of these triumphs which is due to the genius of a few great English mathematicians-so vast and so inviting has been the field thus thrown open to research,-that we do well to spend our time and our labour upon a country which has, we might say, been 'prospected' for us, and in which we know beforehand that we cannot fail to obtain results which will repay our trouble, rather than adventure ourselves into regions where, soon after the first step, we should have no beaten tracks to guide us to the lucky spots, and where the daily earnings of the searcher for mathematical treasure are (at the best) but small, and do not always make a great show even after long years of work. Such regions, however, there are in the domain of pure mathematics, and it cannot be for the interest of science that they should be altogether neglected by the rising generation of English mathematicians. I propose therefore, in the first instance, to direct your attention to some few of these by us comparatively neglected regions; and foremost among them I must name the Theory of Numbers. Of all branches of mathematical enquiry this is the most remote from practical applications; and yet, more perhaps than any other, it has kindled an extraordinary enthusiasm in the minds of some of the greatest mathematicians. We have the examples of Fermat, Euler, Lagrange, Legendre, and Gauss, of Cauchy, Jacob, Lejeune Dirichlet, and Eisenstein, without mentioning the names of others who have passed away, and of a few who are still living. But, somehow, the practical genius of the English mathe SOME BRANCHES OF PURE MATHEMATICS. 169 matician has in general given a different direction to his pursuits; and it would sometimes seem as if we in England measured the importance of the subject by what we find of it in our text-books of Algebra, or as if we regarded its enquiries as problems of mere curiosity, without a wider scope, and without direct bearing on other branches of mathematics. I might endeavour to remove this impression-if indeed it exists in the minds of any of those who hear me-by enumerating instances in which the advancement of Algebra and of the Integral Calculus appears to depend on the progress of the arithmetic of whole, numbers. But, instead of wearying you with the details which would be necessary to make such an enumeration intelligible, I would rather ask you to listen to what is recorded of the most eminent master of this branch of science. 'Gauss,' we are told by his biographer, 'held Mathematics to be the Queen of the Sciences, and Arithmetic to be the Queen of Mathematics.' 'She sometimes condescends '-so spoke the great Astronomer and Physicist-'to render services to Astronomy and the other natural sciences, but under all circumstances the first place is her due.' * In a more serious mood he wrote, 'The higher arithmetic presents us with an inexhaustible storehouse of interesting truths-of truths, too, which are not isolated, but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions, which have the stamp of simplicity upon them, but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process, while the simpler methods of proof long remain hidden from us.'t Or, again, let the young mathematician, who feels an instinctive liking for arithmetical enquiry, be encouraged by the observation which has been put on record by Jacobi in his brief notice of the life of Gopel, that many of those who have a natural turn for mathematical speculation find themselves in the first instance attracted by the Theory of Numbers. There are three great departments of arithmetic, not, it must be admitted, wholly separable from one another, which seem to me at the present time to offer a very inviting field to the researches of the mathematician. Of these * 'Gauss. Zum Gedachtniss.' Von W. Sartorius v. Waltershausen. Leipzig, 1856. t Preface to Eisenstein's 'Mathematische Abhandlungen.' Berlin, 1849. + Notiz fiber A. Gopel, 'Crelle's Journal,' vol. xxxv. p. 313. VOL II. Z 170 ON THE PRESENT STATE AND PROSPECTS OF I will name first the arithmetical theory of homogeneous forms, or quantics, as we in England have now learned to call them. It is worthy of remembrance that some of the most fruitful conceptions of modern algebra had their origin in arithmetic, and not in geometry or even in the theory of equations. The characteristic properties of an invariant, and of a contravariant, appear with distinctness for the first time in the ' Disquisitiones Arithmeticse'; " and in that treatise also the attention of mathematicians was for the first time directed to the study of quantics of any order and of any number of indeterminates.t Again, Eisenstein in the course of his researches on the arithmetical theory of binary cubic forms was led to the discovery of the first covariant ever considered in analysisthe Hessian of the cubic form.4 But the progress of modern algebra and of modern geometry has far outstripped the progress of arithmetic, and one great problem which arithmeticians have before them at the present time is to endeavour to turn to account for their own science the great results which have been obtained in the sister sciences. How difficult this problem may prove is perhaps best attested by the little progress that has been made towards its complete solution. One or two instances may serve to illustrate the actual position of the enquiry. The algebraical problem of the automorphics of a quadratic form, containing any number of indeterminates, may be regarded as completely solved by a formula due to M. Hermite and Professor Cayley. Q The arithmetical formula ' Disq. Arith.,' Arts. 157, 267, 268, where binary and ternary quadratic forms are considered. t 'Sed manifesto hoc argumentum' [the theory of binary quadratic forms] 'tanquam sectionem maxime particularem disquisitionis generalissimae de functionibus algebraicis rationalibus integris homogeneis plurium indeterminatarum et plurium dimensionum considerare... possumus.' ' Sufficiat hunc campum vastissimum geometrarum attentioni commendavisse, in quo materiem ingentem vires suas exercendi, arithmeticamque sublimiorem egregiis incrementis augendi invenient.' ('Disq. Arith.,' Art. 266.) + See ' Crelle's Journal,' vol. xxvii. p. 89. It is remarkable that the cubic covariant does not explicitly appear in this paper (December, 1843) or in the note (p. 105) which immediately follows it. This omission is supplied in a subsequent note, dated March 3, 1844 (ibid. p. 319). The earliest papers of Boole, who approached the study of linear transformations from a geometrical point of view, belong to the years 1841 and 1843 (Cambridge Mathematical Journal, vol. ii. p. 64 and vol. iii. p. 1 and p. 106). In these papers (of which perhaps only the first should be cited here) covariants do not appear, but the first general theorem of invariance ever enunciated, the theorem of the invariance of the discriminant of any quartic, is distinctly stated and proved. ~ M. Hermite, in ' Crelle's Journal,' vol. xlvii. p. 309, appears to have considered forms of three indeterminates only; his solution was subsequently generalised by Professor Cayley (ibid. vol. i. p. 288). See also a later memoir by Professor Cayley 'On the Automorphic Linear Transformation of a Bipartite Quadric Function,' in the 'Philosophical Transactions' for 1858. SOME BRANCHES OF PURE MATHEMATICS. 171 which gives the automorphics of a binary quadratic form has long formed a part of the elements of the Theory of Numbers; and the corresponding investigation for an indefinite ternary quadratic form may be now regarded as completed by the memoirs in which M. Paul Bachmann has followed up the earlier researches of M. Hermite.* Again, the problem of the equivalence of two positive or definite ternary quadratic forms was completely solved by Seeber; and the problem of the arithmetical automorphics of such forms, by Eisenstein. t The corresponding but far more difficult problem of equivalence for indefinit ternary forms has received its first solution only in very recent times from M. Eduard Selling; and perhaps it is not too much to hope that these profound researches may receive some further development from their distinguished author, and may be brought into closer relation with other parts of arithmetical and algebraical theory. So far, then, as binary and ternary quadratic forms are concerned, we have not much reason to complain of the slowness of the advances made by arithmetic. But if we pass to quadratic forms of four or more indeterminates, we shall find that the limits within which our arithmetical knowledge is confined are indeed restricted. The fundamental theorem of M. Hermite, that the number of non-equivalent classes of quadratic forms having integral coefficients and a given discriminant is finite, and the recent researches of M. Zolotareff and Korkine on the minima of positive quadratic forms, mark the extremest limit to which enquiry has been pressed in this direction. * The solution of the problem is made to depend on the solution in integral numbers of the indeterminate equation p2 + F (ql, 2, q,) = 1, where F is the contravariant of the given ternary form. No general method, however, of obtaining the solutions of this equation has as yet been given. (See M. Hermite in the memoir already cited, 'Crelle,' vol. xlvii. p. 307 sqq.; M. Bachmann, 'Borchardt,' vol. Ixxi. p. 296, and vol. Ixxvi. p. 331, together with the note by M. Hermite completing his former solution, ibid. vol. lxxviii. p. 325.) t See L. Seeber, 'Untersuchungen ueber die Eigenschaften der positiven ternaren quadratischen Formen,' Freiburg, 1831; and, in connection with this work, the review of it by Gauss (in the Gottingen ' Gelehrte Anzeige' for 1831; or in 'Crelle,' vol. xx. p. 312; or in the collected edition of Gauss' Works, vol. ii. p. 188), and the subsequent and simpler investigations of Dirichlet (' Crelle,' vol. xl. p. 209), and M. Hermite (ibid. p. 173). The theory of the automorphics of positive ternary quadratic forms is given by Eisenstein in the Appendix to his ' Table of Reduced Positive Ternary Quadratic Forms.' (' Crelle,' vol. xli. p. 227.) He observes (see the note at p. 230) that Seeber, without actually solving the problem, had come extremely near to its solution. 4 'Borchardt's Journal,' vol. Ixxvii. p. 143. ~ See the letters of M. Hermite to Jacobi (' Crelle,' vol. xl. p. 261, sqq.), and the papers of M. Zolotareff and Korkine (' Clebsch,' vol. v. p. 581, and vol. vi. p. 366). I may perhaps also be allowed to refer to my own papers ' On the Orders and Genera of Quadratic Forms containing more than three Indeterminates,' in the Proceedings of the Royal Society, vol. xiii. p. 199, and vol. xv. p. 387. Z 2 172 ON THE PRESENT STATE AND PROSPECTS OF As a second and much simpler instance of the difficulties which remain for arithmetic after the work of algebra is done, let us consider the system of two binary quadratic forms. The first question that we naturally ask, is, what is the arithmetical meaning of the evanescence of their joint invariant? I gave myself an answer to this question some years ago in the following theorem, which for brevity I express in the proper technical language. 'If the joint invariant of two properly primitive forms vanishes, the determinant of either of them is represented primitively by the duplicate of the other.' * This theorem is very far from exhausting the subject to which it refers. But it may serve as a fair illustration of the class of enquiries which I wish to propose to the attention of the Society as likely to be not unfruitful. The geometrical interpretation of the invariantive character to which the theorem relates is (as we all know) that the two pairs of elements, represented by the two quantics, are harmonically conjugate; and I think it especially deserving of notice that the same invariantive character has an important meaning both in arithmetic and in geometry, but that neither of the two interpretations seems in the least likely to suggest the other. If we pass on to the case of two ternary quadratic forms, the geometrical signification of the evanescence of either of their joint invariants is now embodied in well-known elementary theorems; but I do not think that any answer has been given to the corresponding arithmetical question, nor indeed do I know that anyone has occupied himself with it. I would, however, venture to hazard a conjecture that the arithmetical interpretation of these invariantive conditions may have an important bearing on the researches of M. Selling, to which I have already referred. I do not wish to weary the Society with too many particular examples; but I will venture to point to one more instance from which it would appear that modern geometrical and analytical conceptions may help us a little, if only a little, on our way in the trackless wilds of arithmetic. Let us take a question which has some relation to the familiar notions of 'unicursality' and 'one-to-one correspondence.' It is an old theorem, that if the homogeneous indeterminate equation of the second degree containing three variables admits of one solution, it admits of an infinite number; and there is a Memoir of Cauchy t showing how * Report on the Theory of Numbers in the Reports of the British Association for 1863, p. 783, Art. 123 [vol. i. p. 284]. t 'Exercices de Aiath6matiques,' vol. i. p. 233. Cauchy considers the ternary cubic as well as the ternary quadratic equation in this memoir. SOME BRANCHES OF PURE MATHEMATICS. 173 from one given solution all the solutions are to be derived. But here two things deserve our notice-(1) that no geometry (so far as I am aware) helps us in any way to decide whether the given equation does or does not admit of solution. The criterion turns on the definition (first given by Eisenstein) of the generic characters of ternary quadratic forms-a definition which itself depends on a simple arithmetical inference from the algebra of such forms.* But (2), in strong contrast to what I have just stated, when once we have a single solution, the rest is a matter of intuitive geometry. Our equation represents a conic; the conic is real, because we have a single rational point on it; every rational line drawn through this point meets the conic again in a rational point; and thus the unicursality of the conic enables us to deduce from any one solution all the solutions which exist, and, what is more, to obtain them all in a natural sequence. To pass to a question of a somewhat higher order of difficulty, there is no known criterion (so far as I am aware) for deciding whether a ternary homogeneous cubic equation does or does not admit of solution in integral numbers. Such a criterion would be of great interest, and ought not, one would suppose, to lie beyond the present scope of analysis. But here again geometry shows us at once, that if we have one solution, we have, in general, an infinite number. For the tangential of a rational point on a rational cubic curve is itself a rational point, and the line joining two given rational points meets the cubic in a third rational point. (There are, of course, cases of exception to this mode of derivation of one integral solution from another, but I need not advert to them here.) Advancing a little further, we have not to look very far into the connexion which modern algebra has established between ternary cubic and binary quadratic forms in order to satisfy ourselves that the Diophantine problem of rendering a biquadratic expression a perfect square (a problem which has been the subject of numerous researches ever since the time of Euler) is the same as the problem of finding rational points upon a cubic curve of which the equation is rational; and that, in particular, the tangential method to which I have just referred enables us in general to deduce an infinite number of solutions of the Diophantine problem from any given solution. A second department of Arithmetic which, as it seems to me, has in quite recent times received less attention than it deserves, is the Theory of Congru* See Eisenstein, in ' Crelle,' vol. xxxv. p. 117; and a note of my own in the Proceedings of the Royal Society for 1864, vol. xiii. p. 110 [No. XIII. vol. i. p. 410]. t Special cases of the equation here considered have attracted much attention. It will be sufficient to mention Fermat's theorem of the impossibility of the equation x3 + y -- Z = 0. 174 ON THE PRESENT STATE AND PROSPECTS OF ences. Some time before the notation of congruences had been introduced into the Theory of Numbers by Gauss, Lagrange, to whom the conception of a congruence (apart from any special notation) was perfectly familiar, had established the elementary theorem:' f(x) 'If an expression of the form () where f(x) and 5 (x) are rational functions of x having integral coefficients, acquires an integral value for any given integral value of x, the value of p (x) must be a divisor of the resultant of f(x) and (p (x).' This theorem naturally suggests another which was subsequently given by Cauchy: t 'If two congruences which have the same modulus admit of a common solution, the modulus is a divisor of their resultant.' These propositions suggest the possibility of transferring to the Theory of Numbers some at least of the results which have been obtained by modern researches in the theory of algebraical elimination. For example, we are led to consider the problem: 'Given two congruences, to find the number of their common roots when we take in succession for modulus each divisor of their resultant.' In all such inquiries we shall find that the considerations which suffice for the solution of the algebraical problem enter as indispensable elements into the arithmetical investigation, but that this investigation compels us to take notice of other elements also, with which, in algebra, we were not concerned. Thus, in the solution of the problem which I have mentioned, and which I hope at some future day to bring more fully under the notice of the Society, we should have not only to consider in a general manner the system of the divisors of the resultant itself, but we should also have to distinguish, in that system of numbers, those which are at the same time common divisors of certain systems of minors in the dialytic matrix of which the resultant is the determinant. If I may be allowed to regard the subject of complex numbers as belonging to the theory of congruences, I must also be allowed to modify to a certain * The 'Disquisitiones Arithmeticae' were published in 1801. The Memoir of Lagrange, entitled 'Nouvelle methode pour resoudre les Problemes ind6termines,' appeared in the Transactions of the Academy of Berlin for 1768. See also paragraph 4 (p. 528) of his Additions to the French translation of the Algebra of Euler (Lyons, 1774, and often reprinted since); the reference here is to the edition of the 'an iii de l'ere republicaine.' t 'Exercices de Mathematiques,' vol. i. p. 164. SOME BRANCHES OF PURE MATHEMATICS. 175 extent what I have said as to the indifference with which that theory has been looked upon in very recent times; for there is no reason to complain that complex numbers have received insufficient attention, at least from the mathematicians of the Continent. Without referring here to the results obtained by Lejeune Dirichlet, or to the splendid series of researches upon the complex numbers formed with roots of unity which we owe to M. Kummer, we may notice that the general theory has attracted the attention of M. Kronecker, whose investigations relating to it have unfortunately not as yet been published, and are only known from the application which he has made of them to the equations which present themselves in the Theory of Elliptic Functions.' In a Supplement, added to the second edition of Lejeune Dirichlet's Lectures on the Theory of Numbers, M. Dedekind has given the outlines of a complete and very original theory of complex numbers, in which he has to a certain extent deviated from the course pursued by M. Kummer, and has avoided the introduction (at least in a formal manner) of ideal numbers.t I may remind my hearers that the inapplicability, in general, of Euclid's theory of the greatest common divisor to complex numbers formed with the roots of equations having integral coefficients renders it impossible to define the prime factors of such numbers in the same way in which we can define the prime factors of common integers, or of complex numbers of the form a + bya/ - 1; and that, in the effort to overcome the difficulty thus arising, M. Kummer was led to introduce into arithmetic an entirely new and very important conception-that of ideal numbers. I shall ask leave to mention a very recent and very interesting application of this conception which has been made by M. Zolotareff to the solution of a problem of the Integral Calculus, which was first attempted by Abel, and has since attracted the attention of MM. Tchebychef and Weierstrass. We owe to Abel the remarkable theorem that the differential expression d = (x + X) dx V/ (X4 + ax3 + bx2 + cx + d) can or cannot be integrated by logarithms, for some value of the parameter X, according as the radical can or cannot be developed in a periodic continued * See the 'Monatsberichte' of the Academy of Berlin for June 26, 1862, p. 370. M. Dedekind also refers to the investigations of M. Kronecker at p. viii. of his Preface to the second edition of Dirichlet's Lectures, presently to be noticed. t M. Dedekind has also commenced the publication of a resume of his theory in the ' Bulletin des Sciences Mathematiques et Astronomiques' for December, 1876. 176 ON THE PRESENT STATE AND PROSPECTS OF fraction. But Abel gave no criterion for deciding whether the development is or is not periodic-a question which obviously cannot be decided by mere trial. The problem was first solved by M. Tchebychef for the case in which the coefficients a, b, c, d are rational; and the complete solution has at last been obtained by M. Zolotareff, with the help of a new theory of complex ideal numbers.' The last part of arithmetical theory to which I would wish to direct the attention of some of the younger mathematicians of this country is the determination of the mean or asymptotic values of arithmetical functions. This is a field of enquiry which presents great difficulties of its own; and it is certainly one, in which the investigator will not find himself incommoded by the number of his fellow-workers. 'Vix ullus reperietur geometra,' wrote Euler in the last century, 'qui non, ordinem numerorum primorum investigando, haud parum temporis inutiliter consumpserit.' But I do not think that, as a rule, the * The Memoir of Abel, 'Sur l'intgration de la formule diff6rentielle 1p?, R et p etant des fonctions entibres,' will be found in 'Crelle,' vol. i. p. 185, or' (CEuvres Completes,' vol. i. p. 33. In this memoir Abel demonstrates the general theorem that, if R be a rational and integral function of x of any order, such that VR can be expressed by a symmetrical and periodic continued fraction of the form 1 1 1 1 1 V/R = r+, -1_......__ ---.. + + (1) 2M+ 2 1A2 + 2i+ 2Mu+ 2r+ it is always possible to find a rational and integral function p of x such that pdx log(Y R)+ c,........... (2) y being a rational function of x; and that, conversely, if the equation (2) can be satisfied, the development of VR is of the form (1). Abel further shows that y is always the convergent [r, 2y, 21,..., 2 1, 2 /]; and he gives the application of his theorem to the case in which R is a biquadratic function. The memoir of M. Tchebychef was first published in the 'Bulletin de l'Acad6mie de St. Petersbourg' for 1860 (tom. iii. pp. 1-12), and has been reprinted in the 'Melanges Mathematiques et Astronomiques de St. Petersbourg' (tom. iii. pp. 182-192), and also in 'Liouville's Journal' (Second Series, 1864), vol. ix. p. 225. M. Tchebychef has given his method without demonstration; this was first supplied by M. Zolotareff in a paper, 'Sur la m6thode d'int6gration de M. Tchebychef' ('Mathematische Annalen,' vol. v. p. 560). The complete solution of the problem is contained in a work by M. Zolotareff (' Theorie des Nombres entiers complexes,' St. Petersbourg,) which I have not yet seen. The result obtained by M. Weierstrass (however remarkable in itself), that the integrability by logarithms of the differential du depends on the possibility of expressing a certain constant in the form aK+i/3K', where a and 13 are rational numbers, and K and K' are certain complete elliptic integrals, does not supply an available criterion. (See the 'Fortschritte der Mathematik,' vol. vi. p. 118.) + Dec. 1, 1760, ' Novi Commentarii Petropolitani,' vol. ix. p. 99; or, 'Commentationes Arithmeticze Collectee' ('Petropoli, 1849), vol. i. p. 356. SOME BRANCHES OF PURE MATHEMATICS. 177 mathematicians of the present day have any reason to reproach themselves on this score, or stand in any need of the apology which Euler proceeds to deliver. Nevertheless, much has been done in this direction since the days of Euler; enough, certainly, to give abundant encouragement to further inquiry. The first asymptotic results that were obtained are due to Gauss, and are given without demonstration in the 'Disquisitiones Arithmetice '; they relate to the average number of classes of binary quadratic forms of a positive or negative determinant. The general principles on which such inquiries depend were laid down by Lejeune Dirichlet, forty-eight years later, in a memoir entitled 'Ueber die Bestimmung der mittleren Werthe in der Zahlentheorie,' and inserted in the Transactions of the Berlin Academy for 1849. The subject has recently been resumed by M. Mertens of Cracow, who, in an interesting memoir ('Borchardt's Journal,' vol. lxxvii. p. 289), has determined the asymptotic values of several numerical functions. In particular, he has demonstrated the expression given by Gauss for the mean value of the number of quadratic forms of a negative determinant; this mean value being, in the vicinity of n, when n is a considerable number, the quotient obtained by dividing 2. Vn by the sum of the cubes of the reciprocals of the natural numbers. As to our knowledge of the series of the prime numbers themselves, the advance since the time of Euler has been great, if we think of the difficulty of the problem; but very small if we compare what has been done with what still remains to do. We may mention, in the first place, the undemonstrated, and indeed conjectural, theorems of Gauss and Legendre as to the asymptotic value of the number of primes inferior to a given limit x (the former theorem assigns for this value e integrim i () = the tegr lgartlatter an expression of the form - ).ft But these theorems are only approximately log x -a * See Arts. 302, 304, and the Additamenta to Art. 306, X; also various passages in the posthumous fragments, 'De nexu inter multitudinem classium,' &c. (Gauss, 'Werke,' vol. ii. pp. 269-303.) t See Legendre, 'Theorie des Nombres' (Paris, 1830), vol. ii. p. 65: Gauss to Encke (Dec.'24, 1849), in Gauss's Collected Works, vol. ii. p. 444. Legendre assigns to a the conjectural value 1 08366. Gauss compares his own formula and that of Legendre with the results of actual enumerations of the prime numbers, and finds that, as far as the end of the third million, the comparison is in favour of the formula of Legendre, but that the error of that formula shows a tendency to increase more rapidly than the error of his own. He also observes that the mean value of the constant a tend.s to decrease, and that its true limit may possibly be 1, or may differ from 1 by a quantity of the order -. Encke log x VOL. II. A a 178 NOTE ON THE PRESENT STATE AND PROSPECTS OF consistent with one another, and are perhaps still less approximately true. The memoir of Bernhard Riemann, 'Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse,' contains (so far as I am aware) the only investigation of the asymptotic frequency of the primes which can be regarded as rigorous." He shows that, if F(x) be the number of primes inferior to x, there exists an analytical expression for the series F (x) + F(x12) + F(x3)+ which consists (1) of a term which does not increase without limit with x; (2) of a non-periodic term Li (x); (3) of an infinite series of periodic terms of the type Li (-ai) -Li (x +ai), the constants a being the roots (infinite in number) of a certain transcendental equation. It follows that the non-periodic part of the expression for F (x) is of the type Li (x) - I Li (xl) - I Li (Zx) - 15- Li (x) + I Li (x) - Li () +...; and thus the equation of Gauss, F(x) = Li(x), can, roughly speaking, be correct only as far as quantities of the order x. No less important than the investigation of Riemann, but approaching the problem of the asymptotic law of the series of primes from a different side, is the celebrated memoir, 'Sur les Nombres Premiers,' by M. Tchebychef,t in which he has established the existence of limits within which the sum of the logarithms of the primes P inferior to a given number x must be comprised. The limits assigned by M. Tchebychef are not very close, not even close enough to determine llogP the asymptotic value of the quotient, to which the value 1 has been x conjecturally assigned. But the theorem of M. Tchebychef may be, perhaps, said to mark the furthest point to which our knowledge of the series of prime numbers has yet been carried; and while it is truly remarkable that, in a matter of so much difficulty, a process so apparently simple as that which he has employed 1 had communicated to Gauss a formula of his own, lg x 10 2 log, which, as Gauss observes, may, for log x very great values of x, be regarded as coinciding with - (See the letter cited, and also log x- Ilog 10 the note of M. Schering, ibid. p. 521.) * See the 'Monatsberichte' of the Academy of Berlin for November, 1859; or Riemann's 'Mathematische Werke,' p. 137. In the ' Annali di Matematica,' tom. iii. p. 52 (1860), M. Genocchi has given a very interesting account of the method of Riemann, and has arrived at a result differing in one respect (see p. 58) from that of Riemann. t 'Liouville,' First Series, vol. xvii. p. 366. The memoir was presented to the Academy of St. Petersburg in 1850. SOME BRANCHES OF PURE MATHEMATICS. 179 should be capable of leading to a result of so much interest and importance, it is somewhat disappointing to find that this method, even in the hands of its eminent inventor, should seem incapable of being pursued further, and unlikely to furnish any nearer approximation to the truth. The method of M. Tchebychef, profound and inimitable as it is, is in point of fact of a very elementary character, and in this respect contrasts strongly with that of Riemann, which depends throughout on very abstruse theorems of the Integral Calculus. I do not know that the great achievements of such men as Tchdbychef and Riemann can fairly be cited to encourage less highly gifted investigators; but at least they may serve to show two things-first, that nature has placed no insuperable barrier against the further advance of mathematical science in this direction; and, secondly, that the boundaries of our present knowledge lie so close at hand that the inquirer has no very long journey to take before he finds himself in the unknown land. It is this peculiarity, perhaps, which gives such perpetual freshness to the higher arithmetic. It is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths. I do not know that a more striking example of this could be found than that which is furnished by the theorem of M. Tchebychef. To understand his demonstration requires only such algebra and arithmetic as are at the command of many a schoolboy; and the method itself might have been invented by a schoolboy, if there were again a schoolboy with such an early maturity of genius as characterised Pascal, Gauss, or Evariste Galois.* * In addition to the memoirs to which reference has been made in the text, we may also mention the following:-(1) M. Tchebychef, 'Sur la totalit6 des nombres premiers inferieurs a une limite donnee' (' Liouville,' 1st Series, vol. xvii. p. 341). In this paper (which was presented to the Academy of St. Petersburg in 1848) M. Tch6bychef proves (among other things) that, if the expression - log x has a limit at all, the value of that limit must be- 1. This result shows that in the Fr (x) approximate formula of Legendre, F (x) =, we ought to take a = + 1. (2) A paper by the late Judge Hargreave, in the 'Philosophical Magazine' for 1849 (vol. xxxv. p. 36), in which it is shown (but not by any very rigorous demonstration) that the average interval between two consecutive primes in the vicinity of any very great number x is log x; a result at which Gauss had arrived while still a boy, as may be inferred from his letter to Encke, quoted above. (3) A paper in the 'Mathematische Annalen,' vol. ii. p. 636, in which the author, M. Meissel, by the aid of a method suggested by those employed by Legendre ('Theorie des Nombres,' vol. ii. p. 86), obtains a formula which greatly facilitates the determination of the number of prime numbers contained between given limits; he has thus found that the number of primes in the first ten millions is 664579: and in a later note ('MIathematische Annalen,' vol. iii. p. 523) that the number of primes in the first hundred millions is 5,761,460. (4) A Aa 2 180 NOTE ON THE PRESENT STATE AND PROSPECTS OF I pass on to speak of some other branches of analysis which appear to me at the present moment to promise much in the immediate future. I will first refer to one or two points to which the transition from the arithmetic of whole numbers is easy and natural. We owe to Jacobi the first suggestion of a method of approximation which forms a natural extension of the theory of continued fractions, but which still remains in an incomplete condition. In the memoir 'De functionibus duorum variabilium quadrupliciter periodicis,' ('Crelle,' vol. xiii. p. 55,) Jacobi demonstrated the theorem that an uniform function of a single variable can at most be doubly periodic; and that, if it be doubly periodic, the ratio of the two periods is necessarily imaginary. He effected this by proving that, if aa'a", bb'b" are independent irrational quantities, it is always possible to find integral numbers mm' m" such that the value of each of the two expressions ma + m' a +m"a", mb + m'b' + i" b", shall be less than any quantity that can be assigned. This idea of Jacobi was subsequently further developed by M. Hermite, who showed its connexion with the theory of the reduction of quadratic forms (see his letters to Jacobi in' Crelle's Journal,' vol. xl. p. 261 sqq.). The same conception lies at the basis of Lejeune-Dirichlet's researches on complex units, and led him to his celebrated generalisation of the theory of the Pellian Equation.* Since the death of Jacobi, a memoir of his (apparently left incomplete) has been published in 'Borchardt's Journal' (vol. Ixix. p. 29), in which he examines the relations between the successive sets of integral numbers x0, xl, x2, which render an expression such as Xo + X1 m1 + X2 02 (where w, and w2 are irrational quantities) approximately equal to zero. He note by Mr. J. W. L. Glaisher in the Report of the British Association for 1872 (' Transactions of the Sections,' p. 19), in which the results of some enumerations of the primes are given, and are compared with Mr. Hargreave's theorem as to their average frequency. (5) A paper by M. Mertens ('Borchardt's Journal,' vol. lxxviii. p. 46), in which he determines the asymptotic values of the functions 2 p, II (1 — ). These functions had been already considered by Legendre ('Th6orie des Nombres,' vol. ii. p. 67), and by M. Tchebychef in the memoir already cited in this note: but M. Mertens obtains a more precise result by more rigorous reasoning. (6) A preliminary note on an enumeration of primes, by Mr. Glaisher, in the Proceedings of the Cambridge Philosophical Society, Dec. 4, 1876. * See the ' Monatsberichte' of the Academy of Berlin for 1842, p. 95, and for 1846, p. 105. SOME BRANCHES OF PURE MATHEMATICS. 181 applies the theory to the examples w = 2, = 3, 5; W=2, =3 =3 5 and finds that in each case the development is periodic; but he appears not to have obtained any demonstration of the general theorem that the corresponding development in the case of the root of any cubic equation having integral coefficients is always periodic." These unfinished researches of Jacobi, to which M. Borchardt has called the special attention of mathematicians (in the preface to the 68th volume of his Journal) have been resumed by M. Bachmann, t and still more recently, though from a somewhat different point of view, by M. Firstenau. The latter of these mathematicians defines a continued fraction of the second order to be a continued fraction in which each element is itself a continued fraction; and, availing himself of this definition, he has succeeded in showing that we can always approximate to the real root of an equation of the order n, having integral coefficients, by means of a periodic continued fraction of the order n- 1. It is evident that the discovery of such a mode of approximation to the root of an equation may lead to theoretical considerations of great interest, though it is hardly likely that the method itself will be found practically useful. Closely allied to the investigation of new methods of approximation is the problem of determining the arithmetical or transcendental character of irrational quantities. It was first shown by M. Liouville ~ that irrational quantities exist which cannot be the roots of any equation whatever having integral coefficients; a proposition which certainly required the proof which it has received from him, although it might easily seem incredible cd priori that such irrational quantities should not exist. We may, perhaps, be allowed to designate by the terms arithmetical and transcendental the two classes of irrational quantities between which the theorem of M. Liouville has taught us to distinguish; and it becomes a problem of great interest to decide to which of these two classes we are to assign the irrational numbers, such as e and 7r, which have acquired a fundamental importance in analysis. To Lambert, the eminent Berlin mathematician of the last century, the first great step in this direction is due. He showed that neither?r nor 7r2 is rational; with regard to e he was even more successful, for he was able * This theorem had been already obtained by M. Hermite. See the Letters already cited,' Crelle,' vol. xl. pp. 286-289. t ' Borchardt's Journal,' vol. lxxv. p. 25. + E. Fiirstenau, 'Ueber Kettenbriicke hiherer Ordnung,' Wiesbaden. I regret to say that I only know this work from the notice in the 'Jahrbuch fiber die Fortschritte der Mathematik' for 1874. ~ 'Comptes Rendus,' vol. xviii. (1844), p. 883, and p. 910; reproduced with additions in ' Liouville's Journal' (1st series), vol. xvi. p. 133. 182 NOTE ON THE PRESENT STATE AND PROSPECTS OF to prove that no power of e, of which the exponent is rational, can itself be rational. * There (with one slight exception) the question remained for more than a century; and it was reserved for M. Hermite, in the year 1873, to complete by a singularly profound and beautiful analysis, the exponential theorem of Lambert, and to prove that the base of the Napierian logarithms is a transcendental irrational. t But, in the letter to M. Borchardt already cited, M. Hermite declines to enter on a similar research with regard to the number wr. 'Je ne me hasarderai point,' he says, 'a la recherche d'une demonstration de la transcendance du nombre 7r. Que d'autres tentent l'entreprise; nul ne sera plus heureux que moi de leur succes; mais croyez m'en, mon cher ami, il ne laissera pas que de leur en coAter quelques efforts.' It is a little mortifying to the pride which mathematicians naturally feel in the advance of their science to find that no progress should have been made for one hundred years and more toward answer* ' Memoire sur quelques propri6tes remarquables des quantites transcendantes circulaires et logarithmiques,' in the 'M6moires de l'Academie des Sciences de Berlin' for 1761, p. 265; the demonstration depends on the continued fraction ex-e- x x 2 x2 x2 ex+ e- 1 + 3 + 5+ 7 +.. A different method of proving the incommensurability of e (depending on the exponential series) has found its way into many elementary treatises; it would seem that this simple method cannot be applied to prove the more general proposition that e" is incommensurable; but it has been successfully employed by M. Liouville ('Liouville's Journal,' 1st series, vol. v. p. 192) to show that neither e nor e2 can be the root of a rational quadratic equation. This result forms the only extension which the theorem of Lambert had received up to the date of the memoir of M. Hermite, 'Sur la fonction exponentielle,' to which we shall presently refer. The only elementary work in which (so far as I know) the incomm mensurability of e' is demonstrated, is Mr. Todhunter's 'Algebra,' ed. 5, p. 530. The theorems as to the incommensurability of 7r and 2 are excluded from English text-books; the only exception that occurs to me being Sir David Brewster's English edition (Edinburgh, 1824) of the Geometry of Legendre, where Legendre's reproduction of the demonstration of Lambert is given in Note iv. p. 239. The exclusion of these theorems is a matter of regret; for they constitute the only 'short method with the circle-squarers'; and perhaps the extraordinary prevalence within the United Kingdom of the form of delusion known as circle-squaring may partly arise from the appearance of an 'ipsi dixerunt' on the part of the mathematicians, which is certainly suggested by the omission in elementary works of any rigorous demonstration of the irrationality of ir. M. Hermite has given a demonstration of the irrationality of X and 7i2, which is very beautiful and entirely different from that of Lambert (Letter to M. Borchardt in 'Borchardt's Journal,' vol. lxxvi. p. 342); with this single exception, the theory of the quadrature of the circle rests to-day where Lambert left it in the year 1761. * See the Memoir 'Sur la fonction exponentielle,' already cited in the preceding note, 'Comptes Rendus,' vol. lxxvii. pp. 18 etc.; and also published separately by Gauthier-Villars, 1874. SOME BRANCHES OF PURE MATHEMATICS. 183 ing the last question which still remains to be answered with regard to the quadrature and rectification of the circle. But mathematical discovery is like electricity; it follows the lines of least resistance; and an adherence to the rule which this analogy suggests is certainly conducive to the comfort of the individual mathematician, and is probably also, in the long run, conducive to the progress of mathematics themselves. It has often happened in mathematical history that a difficulty, which had for ages resisted all direct attempts to overcome it, has yielded at last to the gradual advance of science; just as in the operations of strategy a strong position, which cannot be carried by a front attack, may nevertheless be turned and taken in the rear by an enemy who has possessed himself of the country round it.* I may, perhaps, mention yet one more class of questions lying on the border land of arithmetic and algebraic analysis; I mean the questions which relate to the transcendental or algebraic character of developments in the form of infinite series, infinite products, or infinite continued fractions. The theorems of Eisenstein and M. Heine, of which a simple and beautiful demonstration has lately been laid before us by our colleague M. Hermite, are amply sufficient to awaken the expectation of great future discoveries in this almost unexplored field of enquiry. t I have detained you so long over arithmetical and quasi-arithmetical subjects that I can only venture to glance hastily at some topics on which I could have wished to have dwelt much longer. I am afraid that I have only given you an additional instance of that one-sidedness against which, as I have reminded you, we were cautioned by our first President. In the hope of convincing you that I have not wholly forgotten the claims of other parts of our science, I will now hazard the assertion, that (after all) the advancement of the Integral Calculus is at once the most arduous and the most important task to which a mathematician can address himself. In the applications of mathematics to physics * I find that I am here closely following (haud passibus equis) some observations of the late Dr. Hermann Hankel in his inaugural address, 'Die Entwickelung der Mathematik in der letzten Jahrhunderten,' Tuibingen, 1869. This discourse, by one who was at once a learned scholar and an original investigator, contains much which deserves the attention of all who are interested in the progress of mathematical science, or who wish to see a higher spirit infused into the mathematical teaching given in the schools and universities of this country. t Eisenstein, 'Monatsberichte' of the Berlin Academy for 1852, p. 441; M. Heine in 'Crelle's Journal,' vol. xlv. p. 285, and vol. xlviii. p. 267. The note of Eisenstein is reproduced in the first of these memoirs. 184 NOTE ON THE PRESENT STATE AND PROSPECTS OF the Integral Calculus is confessedly of ever increasing importance; and it is especially interesting to observe that some of the most recent developments which it has received have had their origin in considerations of pure analysis, and yet have come just in time to furnish us with the most appropriate instruments for dealing with the problems which at the present moment are the most prominent in physical enquiries. But I must not dwell on the prospects of great future extension which are thus opened up for the various branches of mathematical physics I can only advert (and that very hastily) to some of those parts of the integral calculus which, even from the point of view of the pure mathematician, seem to promise an abundant and immediate harvest. Let me first mention the theory of ordinary differential equations. This is a subject which ought to have a special interest for ourselves, as one of the latest advances that have been made in it-the introduction of symbolical methods-is due in great measure to English mathematicians, and above all others to George Boole. Nor have Englishmen been behindhand in the cultivation of another branch of the subject (intimately connected with the use of symbolical methods) -the representation of the solutions of differential equations by means of definite integrals. But, simultaneously with these investigations, a line of research has been pursued on the Continent to which we in England have not paid equal attention. I refer to the endeavours which have been made to determine the nature of the function defined by a differential equation, from the differential equation itself, and not from any analytical expression of the function, obtained by first 'solving' the differential equation. The generality and importance of such an enquiry (whatever be its difficulty) cannot be overrated; for, just as we long since learned to regard integration in a finite form (or, more properly, integration by means of algebraic or exponential and logarithmic functions) as only a very small part of the problem which the Integral Calculus has to solve with regard to differential expressions containing a single variable, so also, when we come to differential equations, we are forced to remember that the variety and complexity of the functional relations expressed by them may altogether transcend any other means of expression at our disposal. Perhaps we may regard as the fundamental theorem in the whole subject the proposition of Cauchy, that every differential equation admits (in the vicinity of any non-singular point) of an integral which is synectic within a certain circle of convergence, and which is consequently (within that circle) developable by the series of Taylor. Various applications of this theorem (together with a demonstration somewhat simpler than that given by Cauchy) will be found in the classical treatise of MM. Briot SOME BRANCHES OF PURE MATHEMATICS. 185 and Bouquet.* Closely allied to the point of view indicated by the theorem of Cauchy is that adopted by Riemann, who regards a function of a single variable as defined by the position and nature of its singularities, and who has applied this conception to the linear differential equation of the second order which is satisfied by the hyper-geometrical series. In the memoir, 'Beitrage zur Theorie der durch die Gauss'sche Reihe F (a,/3, 7, x) darstellbaren Functionem,' t Riemann sets out with the conception of a function which possesses three discriminantal points (I venture to propose this word as the most natural English rendering of 'Verzweigungs-punkte'), and which is further characterised by the property that any three of the values which it admits at any point are connected by a linear and homogeneous equation with constant coefficients. Such a function Riemann shows, by reasoning of great beauty and originality, necessarily satisfies the linear differential equation of the hypergeometric series; and thus the nature and mode of existence of the functions defined by that equation are put before us with a precision and clearness which could not, perhaps, have been attained by any application of the ordinary methods of analysis to the discussion or integration of the equation. The collected works of Riemann include another, but unfortunately unfinished memoir (' Zwei allgemeine Lehrsatze ueber lineare Differential-gleichungen mit algebraischen Coefficienten'), relating to the case in which the number of independent functional values is any whatever instead of only three. And the fertility of the conceptions of Cauchy and of Riemann is further attested by the researches to which they have given rise, and are still giving rise, in Germanyresearches among which I must especially mention those of L. Fuchs, whose papers on linear differential equations, in the 66th and subsequent volumes of 'Borchardt's Journal,' must form, it seems to me, the basis of all future inquiries on this part of the subject. There is one celebrated problem connected with differential equations which, after all that has been written and said about it, remains a problem still; I mean the problem of Singular Solutions. If it were not for the papers of M. Darboux ('Bulletin des Sciences Mathematiques et Astronomiques,' tom. iv. p. 158), and * 'Theorie des Fonctions Elliptiques,' ed. 2, Paris, 1875, p. 325. See also a memoir by the same authors in the 'Journal de l'Ecole Polytechnique,' cahier 36, p. 137. For an enumeration of Cauchy's Memoirs on Differential Equations, see his Life by M. Valson, Paris, 1868, vol. ii. cap. 9, pp. 104-117, t 'Transactions of the Academy of Gottingen' for 1875, vol. vii., or in Riemann's Collected Works (Leipzig, 1876). VOL. II B b 186 NOTE ON THE PRESENT STATE AND PROSPECTS OF of Professor Cayley (' Messenger of Mathematics,' vol. ii. p. 6, and vol. vi. p. 23), I do not know where I should advise a student to turn to acquire any distinctness of insight into this important question. These papers have at any rate rendered one great service; they clearly show that there is a difficulty, and a difficulty not yet surmounted. The point of the difficulty I presume to be, that whereas a singular solution, from the point of view of the integrated equation, ought to be a phenomenon of universal, or at least of general, occurrence, it is, on the other hand, a very special and exceptional phenomenon from the point of view of the differential equation. The explanation suggested by M. Darboux is (to say the least) deserving of very careful consideration. He says, at p. 167 of the memoir just cited, ' Since differential equations are formed by the elimination of constants from an equation in finite terms and its derived equations, writers have supposed (and it would seem erroneously) that, when we are given a differential equation of the first order (for example), it always possesses an integral of the first order expressible in the form f(x, y, c) = 0, where f is a function having in the whole extent of the plane the properties generally recognised in analytical functions. This function f might be more or less difficult to find, but it was conceived of in every case as existing. Now this is just the disputable point, and we think that recent researches on the theory of functions ought to lead us to adopt a different view.' It is evident that, if the observations of M. Darboux are well founded, an important series of questions arises as to the nature of the integral equation answering to a given differential equation; and, further, that some of the elementary considerations with which it is usual to introduce the subject of differential equations must be abandoned as untenable. The rules that can be given in aid of mathematical discovery are, I suppose, very few, and I have already ventured to call your attention to one of them-the rule that bids us follow up any opening that may present itself, rather than try to force a way against obstacles which may prove insurmountable. In the case before us, I think we come upon an illustration of another rule, which is of less general application, but nevertheless often useful. The rule is, that an apparent contradiction (as distinct from a mere misunderstanding) is always to be regarded as an indication of some undiscovered truth. Yet it is remarkable what a tendency there is in the minds of men to ignore or soften down such apparent contradictions, instead of looking for the reality which lies at the bottom of them. The equation J - = log x must have been familiar to mathematicians for a century at least before they set themselves seriously to examine the apparent contra SOME BRANCHES OF PURE MATHEMATICS. 187 diction presented by the equation of a single-valued to a multiple-valued expression. And yet what a flood of light was thrown on the whole theory of functions by the researches to which Cauchy and others were led when the endeavour was at last made to account for this and similar apparently inexplicable phenomena. If the clue offered by such familiar instances as the equations rx dx dx j -=log x, I -= Sin- x, Ji x ' Jo./1-X2 had been seized and followed up, it is difficult to believe that the main outlines of the Theory of Elliptic Functions would not have been discovered much sooner than they actually were. If we look back on the history of the past, the discovery of the 'principle of double periodicity,' and, with it, of the essential characteristic of an elliptic function, cannot but appear to us as one of the most extraordinary efforts of mathematical genius. The integration of the differential equation of elliptic integrals by Euler-an integration obtained by a sort of divination, which has deservedly remained celebrated in the history of science; the systematisation of the calculus of elliptic integrals by Legendre; the simultaneous discovery by Jacobi and Abel of the double periodicity latent in the equation of Euler,-these were the successive steps-and each one a gigantic step-by which those great mathematicians arrived at the theory of elliptic functions in the form in which we now possess it. But if we compare the actual history of the discovery with the outlines of the theory, as we find them, for example, in the work of MM. Briot and Bouquet, it is impossible not to be struck with the contrast. Each step in the theory, as exhibited in that work, appears to follow from those that precede it in such a natural and necessary order that we are inclined to wonder why those who discovered the great results themselves should have failed to find the easiest path of access to them. If I had had the honour of addressing the Mathematical Society ten years ago, I think I should have had to complain of the neglect in England of the study of elliptic functions. But I cannot do so now. The University of Cambridge has given this subject a place in its Mathematical Tripos; the University of London in its examination for the Doctorate of Science. The British Association has supplied the funds requisite to defray the cost of printing Tables of the Theta function-Tables of which the mathematicians of this country may justly be proud, and which will form an enduring memorial of the great ability and indefatigable industry of our colleague, Mr. Glaisher. We further owe to Professor Cayley an introductory treatise on elliptic functions, the first which has appeared B b 2 188 NOTE ON THE PRESENT STATE AND PROSPECTS OF in our language. I consider that the service which he has thus rendered to students is an important one, and one for which we ought to be very grateful. I am convinced that nothing so hinders the progress of mathematical science in England as the want of advanced treatises on mathematical subjects. We yield the palm to no European nation for the number and excellence of our text-books of the second grade-I mean, such text-books as are intended to guide the studies of the undergraduate within the courses prescribed by our University examinations in honours. But we want works adapted to the requirements of the student when his examinations are over-works which will carry him to the frontiers of knowledge in various directions, which will direct him to the problems which he ought to select as the objects of his own researches, and which will free his mind from the narrow views he is too apt to contract while 'getting up' subjects with a view to passing an examination, or, a little later in his life, while preparing others for examination. Can we doubt that much of the preference for geometrical and algebraical speculation which we notice among our younger mathematicians is due to the admirable works of Dr. Salmon; and can we also doubt that, if other parts of mathematical science had been equally fortunate in finding an expositor, we should observe a wider interest in, and a juster appreciation of, the progress which has been achieved? There are, of course, other treatises besides those of Professor Cayley and Dr. Salmon to which I might refer; there is, for example, the work of Boole on Differential Equations; and there are the great historical treatises of Mr. Todhunter, so suggestive of research and so full of its spirit; we have also a recent work by the same author on the functions of Laplace, Lame, and Bessel. But the field is not nearly covered, though, indeed, my enumeration is not complete; and, even without leaving the domain of the Integral Calculus, I might point out that there are at least three treatises which we greatly need-one on Definite Integrals, one on the Theory of Functions in the sense in which that phrase is understood by the school of Cauchy and of Riemann, and one (though he should be a bold man who would undertake the task) on the Hyperelliptic and Abelian Integrals. I fear that our colleague, Professor Clifford, would hardly listen to us if we were to appeal to him to undertake this task; but at least we may express the hope that he may be able to continue the profound researches which he has commenced on this great branch of analysis. I feel I must now bring these somewhat desultory remarks to a conclusion; though, if your time and patience were unlimited, there are many things I could wish to say. Among other matters I should have adverted to the great efforts SOME BRANCHES OF PURE MATHEMATICS. 189 which have been made in very recent times, in Germany, in Russia, and in Norway, to advance the theory of partial differential equations; and I should have noted with pleasure that our own Society has received important communications on this subject from Professor Tanner. And again, leaving the field of the Integral Calculus, I think I should have hazarded some references to the Theory of Substitutions, and its applications to the Theory of Equations; and though I should have been relapsing into a region dangerously near to the Theory of Numbers, I should have exhorted the younger mathematicians of our time not to turn away from a subject which, if forbidding at the first aspect, contains so much promise of future development, and lies so near the very centre and fountain of much that is important in Algebra. Lastly, I should have endeavoured to make my peace with Geometry, which all this time I have been treating with such marked neglect, and would have invited your attention, though it were but for a few moments, to some of those questions of geometry which the natural advance of science has brought to the forefront at the present time. Even here, I might have found one example more of a study which we in England too much neglect; and I might perhaps have reminded you of the great hopes which Gauss entertained of the Geometry of Situation, containing, according to him, vast and as yet quite uncultivated regions over which our present analytical methods can pretend to no dominion. I certainly should not have forgotten to congratulate the Society on the part which its members, under the guidance and inspiration of Professor Sylvester, have taken in the development of the great geometrical theory of link-work movements. 'Verum haec ipse equidem spatiis exclusus iniquis Praetereo, atque aliis post commemoranda relinquo.' I will not sit down without again offering my excuses for the fragmentary and disconnected nature of the reflexions I have laid before you this evening. But, indeed, over so wide a field I could only take a wandering course. My object has been to impress upon those who have been indulgent enough to listen to me that the vast increase which this century has witnessed in the extent of the ground already covered by mathematical science has been accompanied by a proportionate increase in the number and variety of the objects of interest to which the mathematician may turn his attention, and by an even more than proportionate increase in the opportunities of discovering new truths which have been brought within his reach. Our border on every side is the unknown; and the further our boundary line is extended, the more multitudinous become the points at which we may hope to penetrate beyond it. In these days when so 190 NOTE ON SOME BRANCHES OF PURE MATHEMATICS. much is said of original research and of the advancement of scientific knowledge, I feel that it is the business of our Society to see that, so far as our own country is concerned, mathematical science should still be in the vanguard of progress. I should not wish to use words which may seem to reach too far, but I often find the conviction forced upon me that the increase of mathematical knowledge is a necessary condition for the advancement of science, and, if so, a no less necessary condition for the improvement of mankind. I could not augur well for the enduring intellectual strength of any nation of men, whose education was not based on a solid foundation of mathematical learning, and whose scientific conceptions, or, in other words, whose notions of the world and of the things in it, were not braced and girt together with a strong framework of mathematical reasoning. It is something for men to learn what proof is, and what it is not; and I do not know where this lesson can be better learned than in the schools of a science which has never had to take one footstep backward, which has never asserted without proof, or retracted a proved assertion; a science which, while ever advancing with human civilization, is as unchangeable in its principles as human reason; the same at all times and in all places; so that the work done at Alexandria or Syracuse two thousand years ago (whatever may have been added to it since) is as perfect in its kind, and as direct and unerring in its appeal to our intelligence, as if it had been done yesterday at Berlin or Gottingen by one of our own contemporaries. Perhaps also it might not be impossible to show, and even from instances within our own time, that a decline in the mathematical productiveness of a people implies a decline in intellectual force along the whole line; and it might not be absurd to contend that on this ground the maintenance of a high standard of mathematical attainment among the scientific men of a country is an object of almost national concern. But I need not ask your assent to such wide assertions; I shall be more than satisfied if anything that may have fallen from me may induce any one of us to think more highly than he has hitherto done of the first and greatest of the sciences, and more hopefully of the part which he himself may bear in its advancement. XXXII. ON THE CONDITIONS OF PERPENDICULARITY IN A PARALLELEPIPEDAL SYSTEM. [Proceedings of the London Mathematical Society, vol. viii. pp. 83-103. Read December 14, 1876.] 1. THE conception of a parallelepipedal system (i.e., of a space divided by three systems of equidistant parallel planes into similar and equal parallelepipeds) which forms the basis of the usually received theory of crystallography," has also led to important researches in a domain belonging partly to arithmetic and partly to geometry.t It is the object of the present paper to discuss, more completely than perhaps has been done as yet, the conditions for the perpendicularity of lines and planes in a parallelepipedal system. The arithmetical principles required in the discussion are very simple, and are to be found, for the most part, in articles 1-7 of a memoir on 'Linear Systems of Indeterminate Equations and Congruences' in the 'Philosophical Transactions' for 1861t. The geometrical considerations involved, although well known, are less elementary, and relate to the theory of * See Dr. Leonhard Sohncke 'Die unbegrenzten regelmissigen Punktsysteme als Grundlage einer Theorie der Krystallstruktur,' Karlsruhe, 1876, for a different and more general hypothesis as to the ultimate structure of a crystal. See also a memoir by Dr. Sohncke, 'Die regelmassigen ebenen Punktsysteme von unbegrenzter Ausdehnung' in 'Borchardt's Journal,' vol. lxxvii. p. 47, and a memoir by M. Camille Jordan 'Sur les groupes de mouvements' in the 'Annali di Matematica' (Brioschi e Cremona), vol. ii. p. 167. t See (1) Gauss's Review of Seeber's 'Untersuchungen ueber die Eigenschaften der positiven ternaren quadratischen Formen' in the G6ttingen ' Gelehrte Anzeige' for 1831, or in ' Crelle's Journal,' vol. xx. p. 312; (2) Lejeune Dirichlet,' Ueber die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen,' 'Crelle's Journal,' vol. xl. p. 209; (3) the 'Etudes Crystallographiques' of Auguste Bravais, Paris, 1866; (4) A Letter of Eisenstein to M. Charles Hermite (' Journal de Mathematiques,' vol. xvii. p. 473). The researches to which this letter refers are purely arithmetical, but they have an important bearing on a class of questions which are naturally suggested by the present enquiry, and of which the following may be taken as an example, 'In what cases can a parallelepipedal system, possessing a spherical symmetry, be regarded as containing a system in which the parallelepipeds are cubes?' + No. XII. vol. 1. p. 367. 192 ON THE CONDITIONS OF PERPENDICULARITY [Art. 2. contravariant systems of cones. For the definition and some of the principal properties of such systems, we may refer to a paper 'On some Geometrical Constructions' in the 'Proceedings of the London Mathematical Society,' vol. ii. p. 85*; the most important case, that of two-fold systems, forms an essential part of the theory of cubic cones or curves; the considerations relating to that theory which we shall have occasion to employ, will be found in the 'Introduzione alla Teorica delle Curve Piane' of Professor Cremona, or in the 'Treatise on the Higher Plane Curves' of Dr. Salmon. The results of the present enquiry (which has been undertaken at the request of Professor N. S. Maskelyne, and owes much to his suggestions) are submitted to the Mathematical Society with great diffidence; because, while they do not seem likely to admit of any direct application to the measurement of crystals, there is also some uncertainty as to their exact relation to other parts of crystallographic theory. 2. It is, perhaps, hardly necessary to explain that by a 'line of the system' we understand a line joining any two points of a given parallelepipedal system; by a 'plane of the system,' a plane containing any three points of the system; the 'points of the system' being the points of intersection of the three sets of equidistant parallel planes by which the system is defined. For our present purpose it will be sufficient to consider exclusively origin lines and planes; i.e., lines and planes passing through a fixed point of the system taken as origin. Whenever a line of the system is perpendicular to a plane of the system, the system has a certain 'symmetry of aspect' with regard to that plane. Let Q2 be the plane, and let 0 be any point of the system lying in it. The planes and lines of the system which pass through 0 are symmetrically distributed with regard to Q; but the points of the system are not (in general) symmetrically distributed with regard to Q; thus, if OP is any line of the system, and OQ is the reflexion of OP in the plane 2, OQ is a line of the system, but the points of the system which lie on OQ are not, in general, the reflexions of the points which lie on OP. Hence, while the points of the system are not themselves symmetrically distributed with regard to Q2, the directions in which they would be viewed by an eye situated at 0, are symmetrically distributed; and this we may express by saying that the system has a 'symmetry of aspect' with regard to Q. It will be seen-(1) that ti is, or is not, a plane of absolute symmetry (i.e., a plane of symmetry with regard to the points of the system), * No. XIX. vol. i. p. 524. Art. 2.] IN A PARALLELEPIPEDAL SYSTEM. 193 according as the point of the system, that lies the nearest to 0 upon the normal at 0, does, or does not, lie upon the nearest plane of the system parallel to Q; (2) that two parallelepipedal systems may have the same aspect without coinciding. Suppose, for example, that the given parallelepipedal system has a symmetry of aspect with regard to three rectangular planes intersecting in 0 (this is the case of an 'ellipsoidal symmetry' to which we shall presently refer). Let OA, OB, OC be the three lines of intersection of these planes, A, B, C being the points of the system nearest to 0 on these lines respectively. The four points 0, A, B, C determine a new parallelepipedal system (OABC) composed of rectangular parallelepideds. This system is contained in the given system, because every point of (OABC) is a point of the given system; but the two systems are not equivalent, because every point of the given system is not a point of (OABC). In general, a certain number of points othe given system lie in each rectangular parallelepiped of (OABC), and are similarly distributed in each of them, though not symmetrically in any one of them. But the 'aspect' of the two systems from the point 0 (or from any point common to both of them) is the same; and the question whether the given system has an 'ellipsoidal symmetry of aspect' is ame as the question whethter it contains a rectangular parallelepipedal system. Similar considerations apply to every case of symmetry of aspect; and indeed, whenever one of two parallelepipedal systems contains the other, the two systems have the same aspect at any one of their common points." As we shall have no occasion to consider planes of absolute symmetry, we shall, henceforward, for the sake of brevity, use the word symmetry in the sense of 'symmetry of aspect.' Thus any line, and any plane of the system, which are at right angles to one another, are an axis and a plane of symmetry. The cases of symmetry, as thus defined, which can present themselves in a parallelepipedal system, are four in number. There is (1) the case of simple symmetry, where there is only one axis and one plane of symmetry; and there are three cases of triple symmetry, which may be characterized as (2) the ellipsoidal, (3) the spheroidal, and (4) the spherical. In an ellipsoidal system there are three rectangular planes of the system which are planes of symmetry; in a * The relation between a contained and a containing parallelepipedal system is the same as the relation between a contained and a containing ternary quadratic form (see Gauss, Ioc. cit.). The problem, 'To find the conditions that a given parallelepipedal system should have an ellipsoidal symmetry of aspect,' may be stated in a purely arithmetical form as follows:-' To find the conditions that a definite ternary quadratic form (of which the coefficients may be rational or irrational) should contain a form of the type Ax2 + ByC2.+ C VOL. II. C C 194 ON THE CONDITIONS OF PERPENDICULARITY [Art. 3. spheroidal system there is one equatorial plane of symmetry, but every plane of the system at right angles to this plane is also a plane of symmetry; in a system having spherical symmetry, every plane of the system is a plane of symmetry, and every line of the system an axis of symmetry. Two simple symmetries cannot coexist without forming a triple symmetry, which is ellipsoidal, if the axis of one of the symmetries lies in the plane of the other, but is spheroidal in every other case; three simple symmetries form an ellipsoidal symmetry, if the three axes are at right angles to one another; a spheroidal symmetry, if one of the axes is at right angles to the plane of the other two, which are not at right angles to one another; a spherical symmetry in every other case. These assertions are, in part, simple consequences of the well-known theorem that if a plane contains two pair of lines of the system at right angles to another, every line of the system in that plane is at right angles to a line of the system; in part, however, their demonstration requires a fuller discussion of the conditions of symmetry, to which we now proceed. 3. General Conditions of Perpendicularity. We begin by recording, in the form which is most suitable for our present purpose, the conditions for the perpendicularity of lines and planes in space referred to oblique axes. The coordinates of any point being x, y, z, and X, Y, Z being the angles between the positive directions of the axes, the formula F (x, y, z) = 22 y 2 + 2yz cos X + 2xzcos Y+ 2xycosZ, which denotes the square of the distance of the point x, y, z from the origin, supplies a convenient mode of expressing the condition for the perpendicularity of any two lines x y z lm n' x y z 1-7 — / x ^ 1. = x_. I ' - n viz., this condition is dld,dF,dF -' r +mn + n * (1) dl F drn dn dF' dF' dF' or I -- + n, =0 dl' dmrnf? dan where F=F(l,m, n), F = F(', n', n'), Art. 3.] IN A PARALLELEPIPEDAL SYSTEM. 195 Similarly the condition for the perpendicularity of the line x y z m __n and the plane Xx + i y + vz = O, dF dF dF dl dm dn 2 is (2= )=..... () X /U V To obtain, in a form similar to (1), the condition for the perpendicularity of two planes, we introduce the contravariant of F, i. e., the function q ( r 0) = 2sin2 X+ 2sin2 X n2+ Y sin2Z + 2 nr(cos YcosZ- cos X) + 2 A (cos Z cos X- cos Y)+ 2, (cos X cos Y-cos Z), or, as it may be written, if X1, Y1, Z1 are the angles between the normals to the coordinate planes, (a,, ) = "2 sin2 X+ 72 sin2 Y + 2 in2 Z+ 2sin Ysin Z cos X1 + 2 A sin Z sin X cos Y1 + 2 sin X sin Y cos Z,. The condition for the perpendicularity of the two planes xx+f y+vz= 0, 'x + py + yvz = O, may now be expressed in either of the two equivalent forms,dc,dc, d X "TT + A a — + v -r O dcX d (3) Xd d'V + / = 1( d7_ +d Et + v, = 0; * lX1 and the condition (2) for the perpendicularity of a plane and a line may be expressed in a second, and equally convenient form, dx du dv _.= =.- _._... _=.v (4) 1 m n The formulae (1), (2), (3), (4) are well-known, and are only given here for convenience of reference; their demonstration by the ordinary methods of analytical geometry presents no difficulty. We may, however, call attention to the following points: (i.) The equation F (x, y, z) = 0 represents a sphere of evanescent radius having its centre at the origin; or, which is the same thing, a certain imaginary cone having its vertex at the origin, and asymptotic to any sphere concentric with the origin. Such an imaginary sphere-cone possesses the characteristic properties, C 2 196 ON THE CONDITIONS OF PERPENDICULARITY [Art. 4. that (a) any two lines, (,3) any line and plane, (7) any two planes, which are harmonically conjugate with regard to it, are atright angles to one another. The conditions (1) and (2) are the analytical expression of the properties (a) and (P). (ii.) Again, the tangential equation of the imaginary sphere-cone, i. e., the condition that the plane (x + ny + Yz = 0 should be one of its tangent planes, is 6(, o;)=0; and the conditions of perpendicularity (3) and (4) are the analytical expressions, by means of the tangential equation JD, of the properties (7) and (3). (iii.) As the formula F(x, y, z) expresses the square of any rectilineal segment in terms of its oblique projections upon the three coordinate axes; so the formula I< (~,,, ') expresses the square of any plane area in terms of its oblique projections t sin X,, sin Y, ' sin Z, upon the three coordinate planes. The imaginary sphere-cone possesses, it will be noticed, the two paradoxical properties, that the square of any segment lying on one of its lines, and the square of any area lying on one of its tangent planes, are each of them equal to zero. 4. Conditions of Perpendicularity in a Parallelepipedal System. Adopting the notation of the classical treatise of Professor Miller, we designate by a, b, c the parameters of a parallelepipedal system; we thus have, for the square of the distance between any two points of the system, the expression f(x, y, z)= F (ax, by, cz) = a 2x2 + b2y2 + c2z2 + 2bcyz cos X+ 2 cazx cos Y + 2 abxy cos Z, where xyz now denote integral numbers. Again, if d (.,,, ~) = (bca, can, abe) = b2c2S2 sin2 X+ c2a22 sin2 Y+ a2b2'2 sin2 Z + 2 a2bc, sin Ysin Z cos X+ + 2 b2ca sin Z sin X cos Y, + 2c2ab a sin X sin Y cos Z1, the form A, which is the contravariant off, characterises (in the manner indicated by Gauss) a new parallelepipedal system (the polar system of Bravais) in which every line is perpendicular to a plane of the given system, and in which the parameter corresponding to any line is the elementary parallelelogram in the plane to which the line is perpendicular. Writing, for brevity, f= Ax2 + By2 + Cz2 + 2 A'yz + 2B'zx + 2 Cxy, = =A12 +B q2 +C2 +2A[ + 2B ~ "+2 C02, Art. 4.] IN A PARALLELEPIPEDAL SYSTEM. 197 (so that A = 2,...,A = be cos X,...A = b2c2 sin2X,..., A= a2be sinYsin Zcos X,...,) and denoting the determinant A, C', B' C', B, A' =a2b2c2(1-cos2X-cos2Y-cos2Z+2cosXcosYcosZ) B', A', C by A, we have the well known, relations A = BC-A'2, B1= CA-B'2, C=AB- C'2, A = B'C'-AA', B = C'A'-BB' C1 = A'B'- CC', A1, C1, B1 CB, B,, Al = A, B[, A', C1 AA = BC, - A2, AB = CAI - B12, A C = A1B, - C2, AA'= BCA - AlA,, AB' = CA - BB, A C' AB; - C1C; which serve to show that, if we suppose A = 1, or the volume of the elementary parallelepiped of the system f to be unity (a supposition which is admissible, because, if A be not unity, we may alter the parameters in the ratio of V/A: 1), the relation between the two parallelepipedal systems is reciprocal; i.e., the system f is derived from the system / in the same way in which the system q has been derived from the systemf. The five quantities upon which the nature of the parallelepipedal system ultimately depends are unquestionably the angles X, Y, Z, and the ratios of the parameters a, b, c. But the combinations of these quantities which enter into the conditions of perpendicularity and symmetry are precisely the six covariant coefficients A, B C, A', B', C', and the six contravariant coefficients A1,B1, 'II, A", B1, C, X y z au - bv cw ' Thus, if ucw..(5) X y z au7 bvy cwl are any two lines of the system, the condition for their perpendicularity is df df df du + V1 d W + d 198 ON THE CONDITIONS OF PERPENDICULARITY [Art. 4. dfi dfi dfi or =0,+If, + z+ =o, dul dvi dw, i. e., the condition that the lines (5) should be perpendicular is the same as the condition that the lines x_y x y z X =X _ -_, = -..... (6) U V W U1 V W1 should be harmonically conjugate with regard to the cone f= 0. Again, the condition that the two planes of the system hx ky 1z +- -{ =0O, a b c '.....(7) h^x k, y 11z r -+ -+ — = 0, a b c should be perpendicular to one another is the same as the condition that the planes hx+ ky + lz= 0,..... (8) hPx+kly+lz=0, 3. hl x + kI y + I1 z = 0, should be harmonically conjugate with regard to the same cone f= 0, of which the tangential equation is p = O; viz., this condition is dh dP dq hi + k, + 11 T = o, ddho, dk1 dql or~ hda + kd +ld1 = 0. dh, dk, dl, And similarly the condition that one of the lines (5) should be perpendicular to one of the planes (7), is the same as the condition that the corresponding line (6) and plane (8) should be a polar line and polar plane with regard to the cone f=O or 5 =0. We shall, in what follows, adopt the mode of expressing the conditions of perpendicularity which has just been explained. By rational lines, planes, or cones, we shall understand lines, planes, or cones, of which the equations have integral numbers for their coefficients. We shall term the lines and planes (6) and (8), the rational lines and planes corresponding to the lines and planes of the system (5) and (7). Except in the case of a spherical symmetry, the cone f= 0, or q = 0, is not itself a rational cone: and thus (in other cases) the problem, 'to determine all the lines and planes which are perpendicular to one another in a given parallelepipedal system,' is the same as the problem, 'to find all the rational lines and planes which are harmonically conjugate to one another with regard to a given irrational cone.' Art. 5.] IN A PARALLELEPIPEDAL SYSTEM. 199 5. Linear Relations connecting the Coefficients. Let the parallelepipedal system contain a pair of perpendicular lines (5); the condition of perpendicularity gives immediately the relation A uu +Bvv+ v ww+ + A' (vw + wv1) + B' (wu, + uwu) + C' (uv + ztv) =. (9) Unless therefore the six covariant coefficients are connected by a linear homogeneous equation having integral coefficients, no two lines of the system can be perpendicular to one another; and, correlatively, unless the six contravariant coefficients are connected by a similar relation, no two planes of the system can be perpendicular to one another. But the existence of such a relation connecting the six coefficients (or the six contravariant coefficients), though a necessary condition, is not a sufficient condition for the existence in the system of a pair of perpendicular lines (or planes). We proceed, therefore, to examine more closely the circumstances which present themselves, when the coefficients are connected by one, two, three, four, or five relations. By a relation connecting the coefficients we understand a linear homogeneous equation of the type pA + qB+ r C+2pA'+ 2qy'B'+2r'C= 0, (10) in which p, q, r, p', q', r' are integral numbers; in connection with such a relation we shall have to consider the quadratic form + =p2+qn2~+r2 + 2pr' + 2q' + 2r'i, and its reciprocal form = (p2 - qr)x2 + (q2 - rp) y2 + (r'2 -pq) z2 + 2 (p' - q'r') yz + 2 (qq' - r')zx + 2 (rr' - pq') xy; these we shall term the quadratic form and the reciprocal quadratic form appertaining to the given relation. For brevity, we shall attend only to the cases in which given relations exist between the six covariant coefficients A, B, C,...; the cases, in which given relations exist between the six contravariant coefficients, are simply the correlatives of these; conditions for the perpendicularity of two planes answering to conditions for the perpendicularity of two lines, and every condition of symmetry admitting of a two-fold expression according as we employ the covariant, or the contravariant coefficients. It is remarkable that, in every case, the conditions of perpendicularity and symmetry depend solely on the coefiients soof thoe linear relations connecting the crystallographic coefficients; so that two parallelepipedal systems in which the crystallographic coefficients have different ratios, but satisfy the same linear 200 ON THE CONDITIONS OF PERPENDICULARITY [Art. 6. relations, would resemble one another exactly in respect of symmetry and perpendicularity. 6. Case of one Linear Relation between the Coefficients. Here we have the theorem: 'The system contains a single pair of perpendicular lines, or contains no such pair whatever, according as the reciprocal form appertaining to the given relation is, or is not, a perfect square.' For (i.) if the system contain a pair of perpendicular lines, the coefficients are connected by a relation of the type (9); and the reciprocal form appertaining to this relation is the perfect square X, y, z 2 -x U V, W 4X vZ, W Conversely, (ii.) if the coefficients are connected by a relation of the type (10), which satisfies the condition = 1, the system contains a single pair of perpendicular lines. Let I = (ax+ 3y+ -z)2, and consider the matrix P, r-, q'+/3 r +7,, 'p -a q'-3, p'+a, r in which it is easily proved that every first minor is equal to zero, and in which we may suppose that the nine elements are freed by division from any common divisor which they may possess. If in this matrix we represent by u, v, w the greatest common divisors of the elements in the first, second, and third rows respectively, we find immediately jp = unU, q = vvl, r= ww1, 22p/= vw + wvl, 2 q'=wu + uwl, 2r' = uv + vul; whence it follows that the equation (10) assumes the form of the equation (9) and that the two lines x y z au bv cw' x y z au buy cw1' are perpendicular to one another. Art. 7.] IN A PARALLELEPIPEDAL SYSTEM '201 For the condition that k must be a perfect square, we may, if we please, substitute the condition that the form + appertaining to the given relation should resolve itself into two linear factors having integral coefficients. If these factors are U4+Vn +W^ 1,U, t + V1 B + W1 ( the lines (5) are two perpendicular lines. It is important to observe that the reciprocal form k cannot be identically equal to zero. If it were so, the form 4+ would itself be a perfect square, and the cone f=0, which is certainly imaginary, would contain a real line. Again, when we say that k is to be a perfect square, we understand that Ik itself is to be the square of a linear function of x, y, z (and not merely a multiple of such a square by a number which is not itself a square). Thus the condition k = EI may be replaced by the two conditions, (1) that the discriminant of + is to be zero, (2) that the greatest common divisor of the first minors of this discriminant is to be a perfect square; and of these the second may, if we please, be replaced by the condition that, of the three principal minors, one, which is not zero, is to be a perfect square. (These three minors cannot all be zero, and if one of them is a perfect square different from zero, the other two are either zero or perfect squares.) 7. Since the formf is positive and definite, the coefficients A, B, C, A', B', C' are subject to certain inequalities. But, subject to these, they may be any quantities whatever, rational or irrational. Hence no relation of the form (10) can in general subsist between them; or, which is the same thing, a parallelepipedal system cannot, in the most general case, contain a single right angle. The problem, 'Given any number of irrational quantities to determine the rational linear relations (if any) which subsist between them,' presents great difficulties, and, in the present state of indeterminate analysis, is perhaps insoluble. We observe, however, that if m + n quantities Yi, Y2,..., Y,+n are connected by m independent equations of the form gi,1 Yl+g=,2 Y2 + + g,m n Y+n =, where i= 1, 2, 3,...,m, and the numbers g are integral, we can always (and indeed, unless n = 1, in an infinite number of ways) assign n quantities Z1, Z2, Z, such that the m + n quantities Y can be expressed in terms of these by means of a system of equations of the type VOL.. D. d 202 ON TEE CONDITIONS OF PERPENDICULARITY [Art. 8. Y.=hjZ,+h,2,Z2+...+ hjn Z, where the numbers h are integral, and j = 1, 2, 3,..., m +.* It follows from this theorem that, if the coefficients are connected by one, and only one, relation, there is an equation of the type f(xy,Z) ='Wll+tz212+...+W5 25 where 2:, 2,..., 5 are quadratic forms in x, y, z having integral coefficients, and 1, C, 2..., C5 are irrational quantities between which no linear rational homogeneous equation can exist. In general, if i +j = 6, and if there are i relations between the coefficients, we have an equation of the type f=02lEl+... +ojj. The rational cones of which the equations are 2,=0, = 2 =0,..., j=0, and again the rational cones of which the tangential equations are +1 (b, 7, Y) = ( ~ 92(4?75 ) = ~)...*) i(4; n,.)=.., determine two systems of contravariant cones; every cone of the system (Z) circumscribing harmonically every cone of the system (*), and, vice versd, every cone of the system (+) being harmonically inscribed in every cone of the system (Z).t We have seen that the existence of a pair of perpendicular lines in a parallelepipedal system implies the existence of a relation between the coefficients; but that not every such relation implies the existence of a pair of perpendicular lines. When the reciprocal form k appertaining to a given relation is not a perfect square, we can only assert that the cone f harmonically circumscribes the cone -. But this statement seems to have no crystallographic meaning. 8. Case of two linear relations between the coefficients. Let '1, 42;,, l,2 be the forms appertaining to the two given linear relations; and let us at first suppose that = n, k2= 1; so that the system contains two pairs of perpendicular lines. Three particular cases require especial notice. * See the memoir to which we have already referred, 'Phil. Trans.,' 1861, p. 299, sqq. [vol.i.p. 375]. t 'Proceedings of the London Mathematical Society,' vol. ii. pp. 90, 91 [vol. i. p. 528]. Art. 8.1 IN A PARELLELEPIPEDAL SYSTEM. 203 (1) If the squares 1i and k2 are identical, or differ only by a numerical factor, so that, retaining the notation of Art. 6, we have the equations, al _ 1 -_ T 2a2 72 the planes of the two right angles coincide; and every line of the system that lies in this plane has a line of the system at right angles to it in the same plane. (2) The plane of one of the right angles may contain one of the rays of the other right angle. (3) The two right angles may have one ray in common; their common ray is then at right angles to the plane of the other two rays, and this plane becomes a plane of simple symmetry. The condition that this should happen is that 4i and 42 should have a linear factor in common; viz., if these two forms have a common linear factor, this factor is of necessity rational, the two reciprocal forms are perfect squares, and the common factor is al, 31, /l ~ a2, A2) 72 Still retaining the hypothesis that 4, and 'k2 are perfect squares, but excluding the particular cases to which we have just referred, we observe that, besides the two pairs of perpendicular lines which we obtain immediately from the two given relations, the system contains one, and only one, other pair of perpendicular lines. This is a consequence of the elementary theorem that if AOB, COD are two right angles in different planes, the intersection of the planes AOD, BOC is at right angles to the intersection of the planes A C, BOD; a theorem which is itself a particular case of a general property of two pairs of conjugate lines with regard to any cone (or conjugate points with regard to any conic), first given by Otto Hesse. We now quit the hypothesis that '1 and "2 are squares. When only one linear relation subsists between the coefficients, we are certain that, if the reciprocal form appertaining to it is not a perfect square, the parallelepipedal system cannot contain a single pair of perpendicular lines. But, when we have two such relations, we have still to inquire whether, by combining them linearly with one another, we cannot obtain a new relation which shall satisfy the condition ~ = t. Let 0, 01, 02 be the three roots of the discriminantal cubic of k1 + 0f2. If Dd2 204 ON THE CONDITIONS OF PERPENDICULARITY [Art. 8. these roots are irrational, the system contains not a single pair of perpendicular lines. If one of them, for example 0, is rational, we have still to examine whether the factors of, 1+0+2 are rational; if they are, we have a pair of perpendicular lines. If all the three roots are rational, we have to examine the factors of the three forms *1 + 0 2, 'I, + 012,, 1 + 02* 2; according as these factors are or are not rational (if the factors of two of the forms are rational, the factors of the third are also rational), we obtain one or three pairs of perpendicular lines, or no pair at all of such lines. If two of the roots of the discriminating cubic are equal, the cones +\ and 2. touch one another; if the three roots are all equal, these cones osculate (the cases of double contact and of super-osculation cannot present themselves, as they would imply the existence of a real line on the cone f= 0). When two of the roots are equal, we have either the special case (2) considered above, or we have one, and only one, pair of perpendicular lines; when the three roots are all equal, we have a single pair of perpendicular lines. Lastly, the coefficients of the discriminating cubic may all vanish. If this happens, either (a) 'k and '2 differ (if at all) only by a numerical factor, or (3) * and *2 have a common linear factor. The case (a) is the particular case (1) to which we have already referred. For each of the equations = 0, 2G =0 represents a pair of lines in the plane V^'k = 0 or V2, = 0; these two pairs of lines may themselves be irrational, but, even if they are, every rational line lying in their plane has a rational conjugate in the involution determined by them, and the pairs of this involution are conjugate pairs with regard to the cone f= 0. The case (3), when *1 and *2 have a common linear factor, is the case of simple symmetry (the particular case (3) already mentioned). We may therefore enunciate the theorem: 'The conditions that a parallelepipedal system should possess a simple symmetry are '(a) That the coefficients should be connected by two linear relations; '(b) That the two quadratic forms appertaining to these relations should have a linear factor in common.' For the condition (b) we may substitute the following:'The four invariants of *1b and *2 must vanish; the first minors of the discriminants of *1 and *2 not being proportional to one another.' This is only an explicit statement of the conditions which are necessary and sufficient to ensure the presence of a common factor in +1 and *2. Art. 9.] IN A PARALLELEPIPEDAL SYSTEM. 205 9. Case of three linear relations between the coefficients. If the parallelepipedal system includes three pairs of lines at right angles to one another (these pairs being asyzygetic, i.e., such that the existence of one of them is not a necessary consequence, in the manner already explained, of the existence of the other two), the six coefficients are connected by three independent linear relations of the type (10), and the reciprocal forms appertaining to these relations are perfect squares. Conversely, when these conditions are satisfied, the system contains not merely three pairs of lines at right angles to one another, but (as we shall presently see) an infinite number of such pairs; indeed, this will happen even if only one of the three reciprocal forms is a perfect square. And, even when none of these forms is a perfect square, the system may contain rectangular pairs of lines, because (as in the last case) it may be possible to combine linearly the given relations in such a manner as to obtain a new relation satisfying the condition -=D. Such a combination, as we now shall prove, is, or is not, possible, according as the intermediate equation C= d+1 d+l d+l =0 d ' d/' d72 d+2 d+f2 dxf2 d ' ds ' d' d+3 d~+3 dJrf3 d ' d ' d does, or does not, admit of resolution in integral numbers. In accordance with the theorem of Art. 7, we represent f by an equation of the type f= W1 + W2z2+W33; and since no linear relation can exist between w,, w2, w3, we infer that any pair of rational lines which are conjugate with regard to f are also conjugate with regard to 21, 22, and 23; the problem before us, therefore, is, 'To determine all the rational lines which are conjugate with regard to Z1, 2 ', 23.' With reference to this problem, we consider the system of contravariant cones + *2, 3,, and 2, 2,,3; the equations = =0 being equations in plane coordinates, and the equations Z =0 being equations in point coordinates. The Cayleyan cone of the system is the cone of the third class C= 0, of which the equation has been already given. This cone is the Jocobian of the cones 206 ON THE CONDITIONS OF PERPENDICULARITY [Art. 9, X1, k2, +2 3; the Jacobian of the cones E1, 2, 23 is the Hessian cone of the system; its equation is H= d1 d dI 1 =0. dx' dy' dz d;2 d12 d12 dx ' dy' dz dE23 d23 d13 dx ' dy' dz The Cayleyan C is a combinantive covariant of the three forms, 2, 3; the Hessian H is a combinantive contravariant of the same three forms; the developed expressions of C and H have been given by Dr. Salmon,* and it is unnecessary to repeat them here. All the pairs of lines harmonically conjugate with regard to the system of cones (2) lie on the cubic cone H= 0; and vice versd, every line lying on this cubic cone has a corresponding line lying on the same cone which is its conjugate with regard to all the cones of (Z); it only remains, therefore, to ascertain whether any of these pairs of conjugate lines are rational. The problem, 'To determine whether a homogeneous cubic equation, containing three indeterminates, and having integral coefficients, does, or does not, admit of solution in integral numbers,' is one of which, in its general form, no complete discussion has as yet been given. It is obvious, however, that, if the equation H= 0 admits of a single solution in integral numbers, it admits of an infinite number of such solutions. In general this is true for any homogeneous cubic equation containing three indeterminates, and having integral coefficients. For we may regard the cubic equation as defining a rational cubic cone; if there is a rational line appertaining to this cone, the successive tangentials of this rational line are themselves rational lines; and the plane meeting the cone in any two rational lines meets it in a third rational line. This mode of construction is subject to certain exceptions; if, for example, the given rational line is a line of inflexion, it is its own tangential, and gives rise to no second rational line. But in the case before us (when the cubic cone appears as the Hessian of a rational system of cones) the line conjugate to any given rational line of the Hessian cone is always rational, and distinct from the given line; and the plane joining the two lines meets the Hessian cone in a third line distinct from either of them, of which we may again take the conjugate line, and so on continually. The line * 'Conic Sections,' Ed. 5, pp. 346, 347. Art. 10.] IN A PAEIALLELEPIPEDAL SYSTEM. 207 conjugate to a given line of the Hessian cone is most readily found by taking the polar planes of the given line with regard to the cones (Z); these polar planes are all rational, and intersect in a rational line, which is the line required. We may also note-(1) That if AA', BB' are two pairs of rational conjugate lines, the lines (AB', A'B), (AB, A'B) are themselves a pair of rational conjugate lines; (2) that if C is any rational line of the Hessian, the planes CA and CA' meet the Hessian again in a pair of rational conjugate lines; (3) that A, A' have the same rational tangential D, which is the conjugate of the line in which the plane AA' meets the Hessian again. From what precedes, it appears that, if the equation H=0 admits of a single solution in integral numbers, the parallelepipedal system contains an infinite number of pairs of perpendicular lines all lying on the cubic cone H(xa, yb, zc) = 0. And the same thing is true (as has been already said) if the equation C=0 admits of a single solution in integral numbers; for the two equations C= 0 and H= 0 are simultaneously resoluble or irresoluble; or, which is the same thing, if the Cayleyan cone C(7 =0 has rational tangent planes, the Hessian cone H H=0 has rational lines, and vice versa; this, indeed, is evident, because the plane containing any two conjugate lines of the Hessian is a tangent plane of the Cayleyan; and the intersection of any two conjugate planes tangent to the Cayleyan is a line of the Hessian. 10. Three Relations between the Coefficients-Conditions of Symmetry. The system may have a simple symmetry, or an ellipsoidal symmetry, or none at all. But it cannot have a spheroidal or a spherical symmetry. If there is a simple symmetry, the cones (Z) must have in common a rational polar line and polar plane; but must not have in common a rational self-conjugate system of diametral planes. On the other hand, when there is an ellipsoidal symmetry, the cones (Z) must have such a system of diametral planes. (A) Conditions that the cones of the system (2) should have in common a single polar line and polcr plane (the line and plane are necessarily rational). We take for the axis of z the polar line, the polar plane for the plane of xy, and for the axes of x and y the lines in which this polar plane is intersected by the tangent planes drawn from the axis of z to touch any one of the cones (J). (Except in a special case, which we shall notice presently, these cones all touch the same pair of planes intersecting in the axis of z.) We may then take for the representatives of the system (Z) the three loci '208 ON THE CONDITIONS OF EPERPENDICULARITY [Art. 10, 2=0, y2=0, z2-2Xxy=0; and for the representatives of the system (4k) the three envelopes Y?=0, YF=0, X'2+2Sj=0; X being a constant different from zero. Thus the Hessian and the Cayleyan respectively become xyz and (24n-X 2), the former resolving itself into a product of three linear factors, the latter into a product of a linear by a quadratic factor. The axis of z is conjugate (with regard to the cones L) to every line in the plane xy; every line in either of the planes xz or yz is conjugate to a line in the other of those two planes; and the planes containing these pairs of conjugate lines envelope the Cayleyan cone 2-X2 = 0. The conditions that a ternary cubic should resolve itself into three linear factors is that the Hessian of the form should coincide with the form itself; and the condition that a ternary cubic should resolve itself into a linear and a quadratic factor is that the evectants of its two principal invariants should coincide. *. The conditions, therefore, for a simple symmetry are that the Hessian of H should coincide with H, and that the two evectants of C should coincide with one another; but that the Hessian of C should not coincide with C. The rational line corresponding to the axis of symmetry is the line, and the only line, common to all the quadratic cones represented by the first minors of the Hessian; the linear factor of C represents it tangentially. The rational plane corresponding to the plane of symmetry is the polar plane of this rational line with regard to the quadratic Cayleyan cone. One of the linear factors of the Hessian represents this polar plane, and is certainly rational; the other two factors (x and y) may be irrational, or even imaginary. When they are rational, there exist in the parallelepipedal system two planes, not at right angles to one another, intersecting in the axis of symmetry, and each containing an infinite number of lines of the system at right angles to one another. These two planes may coincide; if they do, the plane of coincidence is certainly rational. The special case at which we thus arrive has been excluded from the preceding discussion; for we have supposed that the cones 4+ all touch two distinct tangent planes intersecting in the common polar line of the cones 2. * Dr. Salmon's 'Higher Plane Curves,' pp. 190 and 202, sqq. Art. 10.] IN A PARALLELEPIPEDAL SYSTEM. 209 But the cones + may instead touch two coincident planes; i. e., they may all pass through the polar line, and touch a fixed plane along that line. If we take this plane for the plane of yz, and any other plane through the polar line for the plane of xz, we may represent the systems (2) and (4') by the loci X2 =, xy = O, y2 + X2 = 0, and the envelopes n==0, ^=O0 -X12+2=0. The Hessian and the Cayleyan are respectively x2z and (Y2 +X 2), the former being a product of a linear by a square factor; the latter being a product of three linear factors, of which two are certainly imaginary, because, if X were negative, the cones I would have two real lines in common, which would consequently appertain to the conef. The conditions for this special case of simple symmetry, which is characterised by the presence of a single plane passing through the axis of symmetry, and containing an infinite number of pairs of perpendicular lines, are that the Hessian of H should vanish identically, and that the Hessian of C should coincide with C. (B.) Conditions that the system (2) should have in common a self-conjugate system of diametral planes, of which one at least is rational. The cones (4+) must have three tangent planes in common (viz., the three self-conjugate planes of the system 2). Thus the Cayleyan cone resolves itself into three lines (the three lines of intersection of the diametral planes), and the Hessian resolves itself into those three diametral planes themselves. For this the necessary and sufficient conditions are that the Hessian of the Hessian should coincide with the Hessian, and the Hessian of the Cayleyan with the Cayleyan. The three linear factors of the Cayleyan (or the Hessian) may, however, be all irrational, or one of them only may be rational, or they may all three be rational; and accordingly there may be either no symmetry at all, or a simple symmetry, or an ellipsoidal symmetry. To distinguish between these three cases it is sufficient to examine the discriminantal cubic of any two of the cones 2, or of any two of the cones +; for it is not difficult to verify that the linear factors of the Cayleyan, or Hessian, are rational or irrational according as the roots of any one of these equations are rational or irrational. VOL. II. E e 210 ON THE CONDITIONS OF PERPENDICULARITY [Art. 12. 11. Three Relations between the Coefficients-Special Cases. Some special cases, in which there is no symmetry, have (nevertheless) a certain interest. Of these we may briefly mention two. (1) The cones + have one, and only one, tangent plane A in common. The Hessian in this case consists of A, and of a quadratic cone which is the reciprocal cone of A with regard to the cones of I. Thus, in the parallelepipedal system, there is a cone and plane such that every line lying in the plane has a line at right angles to it lying in the cone, and vice versd; the lines of intersection of the plane and cone are at right angles, and may be lines of the system. (2) The cones + have a polar line and polar plane in common. The result is as follows:-The parallelepipedal system has an infinite number of pairs of lines at right angles to one another, all lying in the same plane; and it may have a second set of such pairs lying on the surface of a quadratic cone, the plane of each such pair passing through the polar line of the first-named plane with regard to the cone. 12. Case of Four Linear Relations betweenthe Coefficients. In this case the formf may be expressed by an equation of the type f= Wl1l + 2 2:2, where the ratio of w1 and w2 is irrational, and where the cones 21, 22 cannot have any real line in common. Every rational line has a rational line conjugate to it with regard to each of the cones 21 =0, O2 = 0, and consequently with regard to the cone f= 0. Hence every line of the parallelepipedal system has a line at right angles to it; and this distribution of pairs of perpendicular lines may exist without the presence of any symmetry whatever. Conditions of Symmetry. The symmetry (if any) may be simple, or ellipsoidal, or spheroidal, but cannot be spherical. There is a simple symmetry when the discriminantal cubic of the system (2, 12) has one rational root, and an ellipsoidal symmetry when this cubic has three rational and unequal roots. When two of the roots are equal, the cones I; and 12 have an imaginary double contact, and the axis and plane of the double contact are the axis and equatorial plane of a spheroidal symmetry. Art. 13.] IN A PARALLELEPIPEDAL SYSTEM. 211 If we represent by S the harmonic covariant of X; and 12, the equation dS dS dS dx' dy ' dz d21e d 1 di =0 dx ' dy ' dz d12 d 2 d 2 dx' dy' dz represents the three diametral planes which are self-conjugate with regard to the cones Xi and X2; and its left-hand member J vanishes identically when 2, and 12 have double contact. The Jacobian J is a combinantive covariant of E] and 22; its coefficients therefore involve only the determinants (of the second order) which can be formed with the coefficients of E2 and 12. But these determinants* are proportional to the reciprocal determinants (of the fourth order) formed with the coefficients of +1, +2, 3,) +4 i.e., the coefficients of the Jacobian can be expressed in terms of these reciprocal determinants. To avoid introducing a new notation, we abstain from giving the developed expression of J, which indeed is not required here. It will be observed, however, that, in the case which we are now considering, the conditions for the various kinds of symmetry, and the determination of the planes of symmetry, depend on a single ternary cubic form (resoluble into three linear factors) which is a combinantive contravariant of the four forms J. (See the concluding remark in Art. 5.) 13. Case of Five Linear Relations between the Coefficients. In this case the ratios of the coefficients are themselves evidently rational; and the parallelepipedal system has a spherical symmetry. For, the coefficients off being rational (after division, if necessary, by a common irrational factor), every rational line has a rational polar plane; i.e., every line of the parallelepipedal system has a plane of the system at right angles to it. And, conversely, if a parallelepipedal system has a spherical symmetry, the ratios of the coefficients are rational; for, if we consider any three lines of the system at right angles to one another, we obtain three independent relations (see Art. 5) between the coefficients; and if we consider any line of the system * See 'Philosophical Transactions,' 1861, p. 301 [vol. i. p. 376]. Ee 2 212 ON THE CONDITIONS OF PERPENDICULARITY, ETC. [Art. 13. (not lying in the same plane with two of the first three), and the plane perpendicular to it, we obtain two more relations between the coefficients, independent of one another and of the former three; i.e., the ratios of the coefficients are rational.* * The question of the rationality or irrationality of the ratios of the crystallographic coefficients had already attracted the attention of Gauss, who, as appears from his Life (' Gauss. Zum Gedachtniss;' by W. Sartorius von Waltershausen; Leipzig, 1856), had in the year 1831 devoted himself with great ardour to the study of crystallography. See the concluding paragraphs of the Review of Seeber's 'Untersuchungen' already cited. XXXIII. ON THE CONDITIONS OF PERPENDICULARITY IN A PARALLELEPIPEDAL SYSTEM. [Philosophical Magazine, Ser. v., vol. iv. pp. 18-25. Read before the Crystallological Society, June 14, 1876. 1. THE conception of a parallelepipedal system (i.e. of a space divided by three systems of equidistant parallel planes into similar and equal parallelepipeds) may be regarded as forming the basis of the usually received theory of crystallography. It is the object of the present note to state some of the conditions for the perpendicularity of lines and planes in such a system. The results of this enquiry (which has been undertaken at the request of Professor N. S. Maskelyne, and owes much to his suggestions) are submitted to the Crystallological Society with great diffidence, because they do not seem likely to admit of any direct application to the practical work of the crystallographer. Such interest as they possess belongs to a domain which borders on the one hand on pure arithmetic, and on the other hand on pure geometry. 2. It is perhaps hardly necessary to explain that by a 'line of the system' we understand a line joining any two points of the given parallelepipedal system, by ' a plane of the system' a plane containing three points of the system, the points of the system being the points of intersection of the three sets of equidistant parallel planes by which the system is defined. It will be sufficient to consider origin-lines and planes, i.e. lines and planes passing through a fixed point of the system taken as origin. 3. Whenever a line of the system is perpendicular to a plane of the system, the system has a certain 'symmetry of aspect' with regard to that plane. Let Q be the plane, and let 0 be any point of the system lying in it. The planes and lines of the system which pass through 0 are symmetrically distributed with regard to Q; but the points of the system are not (in general) symmetrically 214 ON THE CONDITIONS OF PERPENDICULARITY [Art. 5. distributed with regard to Q: thus, if OP is any line of the system not lying in the plane 9Q and if OQ is the reflection of Q with regard to the plane Q, OQ is a line of the system as well as OP, but the points of the system which lie on OQ are not (in general) the reflections of the points of the system which lie on OP. Hence, while the points of the system are not themselves symmetrically distributed with regard to uQ, the directions in which they would be viewed by an eye situated at 0 are symmetrically distributed; and this is what we intend to express by saying that the system has a 'symmetry of aspect' with regard to the plane Q. As we shall have no occasion in what follows to consider planes of absolute symmetry, we shall for the sake of brevity use the word symmetry in the sense of 'symmetry of aspect.' Thus any line and any plane of the system which are at right angles to one another are an axis and a plane of symmetry. 4. The cases of symmetry, as thus defined, which can present themselves in a parallelepipedal system are four in number. There is (1) the case of simple symmetry, when there is only one axis and one plane of symmetry; and there are three cases of triple symmetry, which may be characterised as (2) the ellipsoidal, (3) the spheroidal, and (4) the spherical. In an ellipsoidal system there are three mutually rectangular planes, which are planes of symmetry; in a spheroidal system there is one equatorial plane of symmetry, but every plane of the system at right angles to this plane is also a plane of symmetry; in a system having spherical symmetry every plane of the system is a plane of symmetry, and every line of the system an axis of symmetry. Two simple symmetries cannot coexist without forming a triple symmetry, which is ellipsoidal if the axis of one of the symmetries lies in the plane of the other, but is spheroidal in every other case: three simple symmetries form an ellipsoidal symmetry if the three axes are at right angles to one another, a spheroidal symmetry if one of the axes is at right angles to the plane of the other two which are not at right angles to one another, a spherical symmetry in every other case. 5. Adopting the notation of the classical treatise of Professor W. H. Miller, we designate by a, b, c the parameters appertaining to the three lines of the system taken for the coordinate axes; we also denote by X, Y, Z the angles between the coordinate axes, and by X1, Y, Z1 the angles between the normals to the coordinates planes. We thus have for the square of the distance between any two points of the system the expression f (x, y, z) = a2x2 + b2y2 + c22 + 2bcyz cos X+ 2cazx cos Y+ 2abxy cos Z, Art. 6.] IN A PARALLELEPIPEDAL SYSTEM. 215 where x, y, z denote any positive or negative integral numbers; and this ternary quadratic form may be regarded as characterising the given parallelepipedal system. Again, if p (I,, =) =b2 C22 2sn2 X+ C2a 2 2sin2 Y+a2 b2 (2 sin2 Z + 2a2 be r sin Ysin Z cos X, + 2 b2 ca (t sin Z sin X cos Y, + 2 c2ab, sin X sin YcosZ,, the form <, which is the contravariant of f characterises (in the same way in whichf characterises the given system) a new parallelepipedal system (the polar system of Auguste Bravais) in which every line is perpendicular to a plane of the given system, and in which the parameter corresponding to any line is the elementary parallelogram of the given system lying in the plane to which the line is perpendicular. 6. We write for brevity f= Ax2 + By2 + Cz2+ 2A'yz +2 B'zx + 2 C'xy, = A 2 +B1 2+C(2+2Al' +2B ( +2Clt (so that A =a2,..., A'= be cosX,..., A = b2 c2 sin X,..., A a2 bc sin Ysin Zcos X,...); and we observe that although the five quantities upon which the nature of the parallelepipedal system ultimately depends are the ratios of the parameters a, b, c, and the three angles X, Y, Z, yet the combinations of these quantities which it is most convenient to consider in discussing the conditions of perpendicularity are precisely the six coefficients A, B, C, A', B', C', and the six contravariant coefficients Ai, B1, C1, A1, Bi, C'. Thus the condition that the lines of the system x y z au vb cw (i) x y z au, by, cw, should be perpendicular to one another is df df df U1 ~ + V1 + W-=, du dv f dw or - df+ df + df =; du, dv, dwl 216 ON THE CONDITIONS OF PERPENDICULARITY [Art. 7. the condition that the planes of the system hx + ly kz = a b C ___ b c (ii) hx l'y k, z | a- + b + = should be perpendicular to one another is dc k d~~ d 0 hi d + kd + 11 d= O; A Ady dI or h do + kdo+ dl 0; dhl dk dI and the conditions that the first of the lines (i) should be perpendicular to the first of the planes (ii) may be written in one or other of the equivalent forms (df) (df) ( df) du dv dw h k I (d) (d ) (d ) dh d dl u v w 7. Let us now suppose that the given parallelepipedal system contains a pair of perpendicular lines (i); the condition of perpendicularity gives immediately Auu, + Bvv, + Cww + A' (vw, + wv1) + B' (wu, + w) + C' (uVl + v u) = 0. Unless, therefore, the six covariant coefficients are connected by a linear homogeneous relation having integral coefficients, no ttwo lines of the system can be perpendicular to one another; and correlatively, unless the six contravariant coefficients are connected by a similar relation, no two planes of the system can be perpendicular to one another. But the existence of such a relation connecting the six covariant coefficients (or the six contravariant coefficients), though a necessary condition, is not a sufficient condition for the existence of a pair of perpendicular lines or planes. We proceed, therefore, very briefly to describe the principal cases which present themselves when the coefficients are connected by one, two, three, four, or five linear relations. By a linear relation connecting the coefficients we understand a linear homogeneous equation of the type pA + qB +rC+ 2p'A'+ 2q'B'+ 2r' C'= O, where p, q, r, p', q', r' are integral numbers which we may suppose free from Art. 9.] IN A PARALLELEPIPEDAL SYSTEM. 217 any common divisor. In connexion with such a relation we shall have to consider the quadratic form '1 =2p2 + qj2+ r 2 + 2p',+ 22q' + 2r/ and its contravariant or reciprocal form I = (p'2 - qr) x2 + (q'2 - rp) y2 + (r'2 -pq) 2 + 2 (p' - q'r') yz + 2 (qq' - r'p') zx + 2 (rr' - 'q') xy. These we shall term the quadratic form and the reciprocal quadratic form appertaining to the given relation. For brevity we shall attend only to the cases in which given relations exist between the six covariant. coefficients A, B, C, A', B', C', the cases in which given relations exist between the six contravariant conditions being simply the correlatives of these. It is remarkable that in every case the conditions of perpendicularity and symmetry depend solely on the coefficients of the linear relations connecting the crystallographic coefficients; so that two parallelepipedal systems, in which the crystallographic coefficients have different ratios but satisfy the same linear relations, would resemble one another exactly in respect of symmetry and perpendicularity. 8. Case of one linear relation between the coefficients. Here we have the theorem, 'The system contains a single pair of perpendicular lines, or contains no such pair whatever, according as the reciprocal form appertaining to the given relation is or is not a perfect square.' For the condition that the reciprocal form k should be a perfect square, we may if we please substitute the condition that the quadratic form + appertaining to the given relation should resolve itself into two rational factors. Or, again, we may replace this condition by the two conditions, (1) that the discriminant of + is to be zero, (2) that the greatest common divisor of the first minors of this discriminant is to be a perfect square. 9. Case of two linear relations between the coefficients. We represent the quadratic forms and the reciprocal quadratic forms appertaining to these relations by +1, *I, x2), *2, and by 0, 0', 0" the roots of the discriminantal cubic of!1 + 0,2. If these roots are irrational, the system contains not a single pair of perpendicular lines. If one of them, for example 0, is rational, we still have to examine whether the factors of *1 + 0*2 are rational; if they are, we have a pair of perpendicular lines. If all the three roots 0, 0', 0" are rational, we have to examine the factors of each of the three forms V! 1+ i02, *1 + 0' 2, N\1 + 0" *2; according as these factors are or are not rational VOL. II. F f 218 ON THE CONDITIONS OF PERPENDICULARITY [Art. 10. (if the factors of two of them are rational the factors of the third are so too), we obtain one or three pairs of perpendicular lines, or no pair at all of such lines. When two of the roots 0, 0', 0" are equal, we have either one, and only one, pair of perpendicular lines; or we may have two pairs, the plane of one of the right angles containing one of the rays of the other right angle. When the three roots are all equal we have a single pair of perpendicular lines. Lastly, the coefficients of the discriminating cubic may all vanish. If this happens, either (a) '1 and '2 differ, if at all, by a numerical factor, and every line of the system that lies in a certain plane has a line of the system at right angles to it in the same plane; or (/) *i and +2 have a common linear factor, and the system possesses a simple symmetry. We may thus enunciate the theorem: 'The conditions that a parallelepipedal system should possess a simple symmetry are (a) that the coefficients should be connected by two linear relations, (b) that the two quadratic forms appertaining to these relations should have a linear factor in common.' 10. Case of three linear relations between the coefficients. We represent by +1, +2, +3 the quadratic forms appertaining to the given relations, and we obtain the following theorem: 'The system contains no right angle, or an infinite number, according as the indeterminate cubic equation da d41 dl, do ' dl' dy C =d+2 d+2 d 2 d 0 dc'' d4' d d*3 d+3 d+3 do' d ' d' does or does not admit of solution in integral numbers.' By virtue of the three given relations the characteristic expression f(x, y, z) of Art. 5 assumes the form f(x,, Z) = wIf1+ 2f2+3 + 3, the ratios of the quantities co, 2, w,3 being irrational, but the coefficients of the quadratic forms fl, f, f3 being integral numbers. If H(x, y, z) denote the Jacobian of these three forms, we have the theorem: 'When the indeterminate equation C=0 admits of solution, the infinite number of right angles which the system contains all lie on the cubic cone Art. 11.] IN A PARALLELEPIPEDAL SYSTEM. 219 H (xa, yb, zc) = 0; viz. an infinite number of lines of the system lie on this cone, and every line of the system which lies on it has a line at right angles to it, also lying on the cone.' The system may have a simple symmetry or an ellipsoidal symmetry, or none at all; but it cannot have a spheroidal or a spherical symmetry. The conditions for a simple symmetry are that the ternary cubic form C((, n,,) should resolve itself into a rational linear factor and a rational quadratic factor, and that the ternary cubic form H (x, y, z) should resolve itself into three linear factors. These conditions admit of being further developed (see Dr. Salmon's 'Higher Plane Curves,' pp. 190 and 202 seqq.); it is sufficient for our purpose to observe that the coefficients of the Jacobian H(x, y, z), no less than those of C(7, (,, ), depend solely on the coefficients of the forms +4, *2, *3, i.e. on the integral numbers entering into the given linear relations. The conditions for an ellipsoidal symmetry are that C (J, Y, ') should resolve itself into three rational linear factors, and that H(x, y, z) should resolve itself into three factors. Two special cases of the general theory (which, however, are not cases of symmetry) deserve attention. (1) There may exist in the parallelepipepal system a quadratic cone and a plane, such that every line of the system lying in the plane has a line of the system at right angles to it lying in the cone. (2) Or, again, the parallelepipedal system may have an infinite number of pairs of perpendicular lines all lying in the same plane; and it may also have at the same time a second set of such pairs lying on the surface of a quadratic cone, the plane of each pair of this second set passing through the polar line of the first-named pair with regard to the cone. 11. Case offour linear relations between the coefficients. Here every line, without exception, of the parallelepipedal system has a line at right angles to it; and this distribution of pairs of perpendicular lines may exist without the presence of any symmetry whatever. The symmetry (if any) may be simple, or ellipsoidal, or spheroidal, but cannot be spherical. The characteristic form f(x, y, z) may be expressed by an equation of the type f= lfl + W2f2, the ratio of cw and w, being irrational, but the coefficients of the quadratic forms f, and f2 being integral numbers. There is a simple symmetry when the F f 2 220 ON PERPENDICULARITY IN A PARALLELEPIPEDAL SYSTEM. [Art. 12. discriminantal cubic off, + Of2 has one rational root, and an ellipsoidal symmetry when it has three rational and unequal roots, a spheroidal symmetry when it has two equal roots. (It cannot have its three roots equal, because the cone f(x, y, z)= 0 is imaginary.) We suppress the further discussion of these conditions, only observing that they may be so expressed as to show that they depend only on the coefficients of the four given relations, and not on the six coefficients A, B, C, A',B', C' themselves. 12. Case of five linear relations between the coeffcients. In this case the ratios of the coefficients are themselves evidently rational, and the parallelepipedal system has a spherical symmetry. It is also true, conversely, that when there is a spherical symmetry the ratios of the coefficients are rational. We may mention that the question of the rationality or irrationality of the ratios of the crystallographic coefficients had attracted the attention of Gauss, who, as appears from the memoir of his life (' Gauss. Zum Gedachtniss,' von W. Sartorius v. Waltershausen: Leipzig, 1856), had in 1831 devoted himself with great ardour to the study of crystallography.* * Some of the demonstrations, which have been omitted in the present note, will be found in a paper inserted in the 'Proceedings of the London Mathematical Society,' vol. vii. p. 83 [No. XXXII. yol. ii. p. 191.] XXXIV. SUR LES INTEGRALES ELLIPTIQUES COMPLETES. [Atti della R. Accademia dei Lincei. Transunti Ser. iii. vol. i. pp. 42-44. Read January 7, 1877.] EN suivant la notation usuelle, posons O = x + iy, la quantite y etant positive, K= 2- (7 _ e'i0- )2, K' = - K, [ 14 + e27ni 7 co 8 k2 =1 - k'= 8 (w) = 16e TII1 1 +(2m - Ii;w' designons aussi par 1) (k2) et ' (k2) les integrales rectilignes [ Zdy --- et - Jf //(1 - 2) (1 - /() - ~) ( - la valeur initiale de chacun des deux radicaux etant + 1. Lorsque k2 est reel, positif et plus grand que l'unite, la definition de la premiere integrale presente une ambiguite; celle de la seconde se trouve pareillement en defaut, lorsque k2 est reel et negatif. Nous conviendrons done, dans ces deux cas, de prendre pour 4P (k2). celle des deux valeurs admissibles dans laquelle le coefficient de i est positif; et pour P (k2) celle des deux valeurs de l'integrale correspondante dans laquelle le coefficient de i est negatif. On sait que, si la partie reelle de s'evanouit, c'est a dire, si k2 et k'2 sont reels, positifs et moindres que l'unite, on a les deux equations K= (k2), K' = (k2)....... (1) Mais il a ete de1nontre par M. Hermite, que ces equations ne peuvent pas avoir 222 SUR LES INTEGRALES ELLIPTIQUES COMPLETES. lieu pour toutes les valeurs de w. II est vrai qu'en doit avoir, dans tous les cas, dx dx ~~-K= -K '.. (2) K o A(1 -X2)(1 - ) X2A) (1 - k'x)5 mais ces integrales ne sauraient 6tre, en general, rectilignes; et la determination du chemin de l'integration a paru offrir quelque difficulte. On y parvient de la maniere suivante. Soient a, /3, 7, S des nombres entiers, positifs ou negatifs, qui satisfont a l'equation aS-,-y=1, et aux congruences a- S=l, mod 4; /3 70, mod 2; soit aussi Q = X+ Y, Y etant positif, et les quantites reelles X et Y etant assujetties a verifier les inegalites -1<X<1, _X<X2+Y2>X.... (3) On conclut facilement de la theorie de la reduction les formes binaires a determinant negatif, qu'on peut toujours satisfaire, et cela d'une maniere unique, a l'quation y+ ~ a + /3Q Les nombres a, 13, y, e, et la quantite complexe Q2 etant ainsi determines, la theorie des transformations lineaires donne aussitot K(,) = aK (Q) + i K' (Q), iK' )= -K(,Q)+ 'J' (Q); de plus l'on verifie sans peine que les equations (1) subsistent, tant que w satisfait aux inegalites (3). En effet, tant qu'on a -t < < 1, - x< 2 +y2 < X, les fonctions k2 et k'2 ne peuvent pas atteindre aucune des valeurs pour lesquelles les integrales rectilignes cessent d'6tre completement determinees. Et, quant aux cas limites qui se presentent lorsqu'on a x =1, ou bien x2 + y2 = x, les conventions, que nous avons adoptees pour lever les ambiguites dans les definitions des integrales F et I, ont ete choisies de maniere a faire accorder dans * Voyez la Note sur la theorie des fonctions elliptiques ajout6e au Calcul differentzel de Lacroix, torn. ii. pp. 420-425, Paris, 1862. SUR LES INTEGRALES ELLIPTIQUES COMPLETES. 223 ces memes cas limites, les valeurs des integrales 1 et V avec celles des fonctions K et K'. On a done K () )= ( (Q)), K' (Q) = (~(p)8 ou bien, en observant que q8 (Q) = 8 2 () = k2, K () = ~(k), K' (Q) = (k); d'ou l'on tire finalement K o) -= a (k2) + i3t (k2), iK' (w) = y ( (k2) + iS, (k2). Ces equations remplacent les equations (1), et font connaitre le chemin que l'integration doit suivre dans les equations (2). Nous ferons remarquer, en terminant, que la transformation de la quantite complexe w, qui donne la solution du probleme, est la m6me que Jacobi avait employee pour la reduction des fonctions 0, sauf la diff6rence qui provient des congruences auxquelles nous avons assujetti les nombres a, 13, 7, S. XXXV. MEMOIRE SUR LES EQUATIONS MODULAIRES. [Atti della R. Accademia dei Lincei. Memorie della classe di Scienze fisiche, matematiche e naturali. Ser. iii. vol. i. pp. 136-149. Read February 4, 1877.] 1. ON connait les beaux resultats auxquels sont parvenus MM. Kronecker et Hermite, en etudiant les rapports qui existent entre les equations modulaires et les formes quadratiques binaires ~ determinant negatif. Mais les points de rapprochement, qu'on a trouves jusqu'ici entre la theorie des equations modulaires et celle des formes quadratiques t determinant positif, ont ete peu nombreux; et, a cet egard, nous ne saurions citer que le Memoire si remarquable de M. Kronecker, 'sur la solution de l'equation de Pell par le moyen des fonctions elliptiques.' Cependant, nous avons ete conduits p reconnaitre qu'il existe entre ces deux theories des liens tres intimes. C'est ce que nous nous proposons de faire voir dans ce Memoire, en demontrant que, si l'on designe par N un nombre entier quelconque, et par (k2, 2) = (2, k2) = 0 une des equations symetriques, qui definissent les transformations modulaires du Nieme ordre, la courbe representee par l'equation cartesienne 'D('+1X+iY, + X- iY)=O aura la propriete singuliere de presenter une veritable image geometrique du systeme complet des formes quadratiques reduites appartetantes au determinant positif N. C'est en suivant la route tracee par les illustres geometres que nous venons de nommer, que nous avons ete conduits h ce resultat, qui nous a paru offrir une interessante application de l'arithmetique h la geometrie, aussi bien qu' la th6orie des fonctions elliptiques. 2. Soit = x + iy une quantite complexe, la valeur de y etant positive. P osons ow =? + Posons o) a + a ); a, 3, 7, J etant des nombres entiers qui satisfont h l'equation a~ —3y=l, Art. 3.] MEMOIRE SUR LES EQUATIONS MODULAIRES. 225 et aux congruences a,',m 1,0,mod2; a-_ 1, mod4. En representant, comme on fait ordinairement, les quantites complexes par les points d'un espace de deux dimensions, nous exprimerons la relation qui subsiste entre w et Q., en disant que les deux points correspondants sont equivalents. Cette definition de l'equivalence est plus restreinte, et par consequent, moins naturelle que celle qu'on emploie ordinairement en arithmetique; et c'est uniquement pour abreger le discours que nous l'admettons ici. Dans le plan xy, dont toutefois nous ne considerons que la partie situee au dessus de l'axe des abscisses, tragons les deux droites P = x- 1 = 0, P- = x +1 = 0, et les deux demicercles Q =x2+y2_x=0 Q -1 =x2y2+X=0. Soit 2 l'espace compris entre les deux droites, mais exterieur aux deux cercles; P et Q etant censes appartenir t cet espace, mais P-l et Q-1 en etant exclus. Cela pose, on aura les propositions suivantes, qu'on deduit sans peine de la theorie de la reduction des formes quadratiques binaires k determinant negatif. ' tant donne un point quelconque o, il existe toujours un point reduit (c'est a dire, un point appartenant a l'espace reduit 2), qui est equivalent a co; et il n'en existe qu'un seul.' 'La substitution r6duisante est aussi unique.' Pour abreger, nous conviendrons de nommer normales les substitutions telles a, 3 que, i 3. Soit N = b - ac; a, b, c 6tant des nombres entiers. L'equation a+2 bx + c (2 + y2) =0 appartient v un cercle reel; nous representerons ce cercle par [a, b, c], et la forme quadratique correspondante par (a, b, c). Nous conviendrons d'appeler cercle rationnel tout cercle tel que [a, b, c]; mais nous ne considererons toujours que les demi-circonferences situees au-dessus de l'axe des x. Lorsque N est un carre, on peut avoir c = O; dans ce cas le cercle rationnel devient une droite. Soit (A, B, C) une forme equivalente a (a, b, c) par la substitution normale ' |; les cercles correspondants [A, B, C] et [a, b, c] seront aussi equivalents par la m6me substitution. En effet, l'equation w = + etablit, entre les points w et Q des deux cercles [a, b, c], [A, B, C, cette espece de correspondance geometrique qui a ete appelee affinite circulaire par Moebius, et qui, t la verite, ne differe point essentiellement de la relation si connue de l'inversion. I1 est bon VOL. II. G g 226 MEMOIRE SUR LES EQUATIONS MODULAIRES. [Art. 4. de remarquer que la transformation par affinite circulaire du cercle [a, b, c] dans le cercle [A, B, C], est en m6me temps une transformation homographique. Ainsi, lorsque N est non-carre, les substitutions automorphiques de la forme quadratique (a, b, c) sont representees geometriquernent par des transformations homographiques du cercle [a, b, c] dans lui-m6me. Et de m6me que les substitutions automorphiques normales sont les puissances, positives ou negatives, d'une seule d'entr'elles; de meme les transformations homographiques, que nous avons a considerer par rapport au cercle [a, b, c], proviennent de la repetition, dans les deux sens, d'une seule transformation fondamentale. En supposant que [a, b, c] soit un cercle primitif, et en designant par t, u les moindres nombres positifs qui satisfont a l'equation t2 - Ni2 = 1, u etant pair, t impair, on trouve que l'ellipse, regulatrice de cette transformation fondamentale, est representee par l'equation a + 2 bx + cx2 + ct2y2 = 0. L'excentricite de cette ellipse est donnee par l'equation /1 +e =t+u/N; d'oti l'on voit que les transformations homographiques, qui correspondent aux substitutions automorphiques normales, sont semblables pour tous les cercles primitifs du meme determinant. 4. Soit a= 2 ou = 1, selon que le plus petit nombre u, qui satisfait k l'equation t2-Ni2 = 1, est pair ou impair. I1 est facile de verifier que la restriction, que nous avons du apporter h la definition de l'equivalence, entraine la repartition des formes de chaque classe proprement primitive en 3 classes subalternes, qui satisfont, en nombre 6gal, aux conditions exprimees par les congruences (A) a -c 1, mod 2, (B) aO0, c1=, mod 2, (C) a-l, c=0, mod 2. Pareillement, lorsque N=1, mod 8, chaque classe de formes improprement primitives se partage en six classes subalternes, dont il y a toujours deux qui satisfont h chacun des systemes (A') a c 0, mod 4, (B') a-O, c=2, mod 4, (C') a22, c0O, mod 4. Enfin, lorsque N 5, mod 8, chaque classe improprement primitive contient 2 -' classes subalternes, a-' etant = 1, ou = 3, selon que l'equation t2 - Nu2 = 4 est resoluble ou irresoluble en nombres impairs; de plus, chacune des congruences (A"), b 1, mod 4, (B"), b — 1, mod 4, est satisfaite par -' de ces classes subalternes. Art. 5.] MEMOIRE SUR LES EQUATIONS MODULAIRES. 227 Soient h, h' les nombres des classes proprement et improprement primitives, qui appartiennent au determinant N; soient H, H' les nombres correspondants des classes subalternes de cercles. En observant que les deux fermes (a, b, c), (- a, - b, - c) ne correspondent qu'a un seul cercle [a, b, c], et que ces deux formes appartiennent toujours a des classes subalternes differentes, bien que, dans certains cas, elles peuvent 6tre comprises dans la m6me classe, on parvient a Btablir les deux equations H=- 3 h, H'= 'h'. 5. Maintenant, aux points du cercle primitif [a, b, c] substituons les points reduits correspondants. Ce cercle se changera en un assemblage d'arcs circulaires reduits, dont la totalit6 formera une ligne L, qui en apparence sera brisee, mais dont on mettra en evidence la continuit6, en repliant sur lui-m6me l'espace X, de maniere a former une surface fermee tricuspide, les droites P, P-1 etant reunies ensemble, et aussi les cercles Q, Q-1. Il importe surtout de savoir quels sont les cercles equivalents a [a, b, c] qui traversent 1; et quelle est la loi qui, dans la ligne composee L, gouverne la succession des arcs reduits. Voici la solution de ce probleme pour le cas d'un determinant non-carre. Soit 0 l'une ou l'autre des racines de l'equation a + 2 b 0+ c 02 =. Developpons 0 en fraction continue, en prenant toujours pour quotient integral le nombre pair, positif ou negatif, qui est le plus rapproche du quotient complet correspondant. Soit ~Soit ^=2l 1 1 1 01- 2 — 8: +-2 e2 U2+... + 2 62s -2s + 01 la periode de la fraction continue; les nombres entiers,x2, **,... M2 e'tant positifs; E1, e2,... * 62 designant des unites positives ou n6gatives; et le quotient complet 01 etant de rang impair dans le developpement de 0. Posons aussi 1 1 1 01 = 2 + 1, + 0 7 = 2 E., 2S = 0 2 e2 I2s+; et considerons les Zu quantites irrationnelles l 01- 62el) 01-4ei el, *. * 2 (-2(l-1) el, 03, 03-23, 03-4e3,...,3-2(3-1) 63, 1 1 1 02s 02' - 22 e2s -462s) 02s-2( 2s - 1)2s Gg 2 228 MEMOIRE SUR LES EQUATIONS MODULAIRES. [Art. 7. Chacune de ces + + +... quantites est la racine d'une equation quadratique equivalente a a + 2 b 0+ c 02 = 0; par consequent, les cercles correspondants sont equivalents t [a, b, c]. Or, tous ces cercles traversent l'espace Z, et la ligne L est composee des parties de leur circonf6rences qui se trouvent dans l'interieur de cet espace, ces parties etant prises dans l'ordre indique par le developpement. 6. Les formes quadratiques correspondantes aux cercles dont nous venons de parler, different des formes reduites de Gauss, (10) parceque nous nous servons ici d'une fraction continue avec des quotients pairs; (20) parceque nous admettons parmi les formes reduites, non seulement les formes reduites principales qui correspondent aux quotients complets 01, -a,..., mais aussi les formes intermediaires, correspondantes aux racines 01 —2~i, *e..-, 0 -,_ *.... 01 - 2 ej (30) parceque nous prenons pour les racines des equations qui correspondent aux quotients de rang pair, les quantites 1 1 1 02 ~4 02s au lieu des quantites - 02, - 4,..., - 02s. I1 est presque inutile d'ajouter que, pour avoir la suite des arcs r6duits, on pourrait se servir, au lieu du developpement en fraction continue, de l'algorithme des formes contigues de Gauss, en y apportant une legere modification. 7. Lorsque le determinant est carre, les arcs reduits, equivalents a un cercle donne, forment toujours une suite continue; mais cette suite, au lieu d'6tre periodique, commence avec un arc passant par un des points singuliers de la surface tricuspide, et se termine de la m6me maniere. Designons les points (0, oo), (0, 0) par p et q; et l'un ou l'autre des deux points equivalents (1, 0), (-1, 0) par r. La suite des cercles reduits, equivalents a un cercle proprement primitif donne, aura pour ses points extremes rr, qq, pp, selon que le cercle donne satisfait aux congruences (A), (B), (C) de l'article 4. Pareillement, dans l'ordre improprement primitif, les points extremes de la suite des cercles reduits, equivalents t un cercle donne, seront pq, qr, rp, selon que Fl'quation de ce cercle satisfait aux congruences (A'), (B'), (C') de l'article cite. Pour determiner complitement la ligne L qui correspond a un cercle donne [a, b, c], dont le determinant est un nombre carre, il suffira de connaltre les equations des deux cercles extremes de L, et d'en deduire la substitution normale unique qui transforme l'un d'eux dans l'autre. Soit, en effet, y+ + =a + Art. 8.] MEMOIRE SUIR LES EQUATIONS MODULAIRES. 229' cette substitution; on en deduira le developpement fini 1 1 1 = 2eIA~ + 2=2 e2 e + + -...+ 22 E2sf2 + 1 ohi il faut remarquer qu'on peut avoir a, = 0, 12 = 0. Ce developpement remplacera la fraction continue periodique de l'art. 6, et fera connaitre tous les cercles de L dans leur ordre naturel de succession. Tout se reduit done a trouver les equations des deux cercles extr6mes de L. Pour cela, soit (a, b, c) (x, y)2 = m (px +p'y) (qx + q'y), m etant le plus grand diviseur commun de a, 2b, c. En designant toujours par a, ' une substitution normale, et en regardant comme inconnus les nombres entiers a, f, 7,, 7, ', X, X', 1'equation indeterminee p,.p Xa, _ r 4, 2' x 7^ " X, x, admet une solution unique, dans laquelle les valeurs absolues de, n' ne surpassent pas l'unite, et celles de X, X' ne surpassent pas + (pq'-p'q) = 2- /N. Le cercle [msX, ^m (r/X'+,^A), mr'X'] est l'un des deux cercles cherches; l'autre peut s'obtenir en echangeant entr'eux dans la solution precedente, les deux facteurs de (a, b, c). Mais, un des deux cercles extremes etant trouv6, il vaut mieux partir de l'6quation nouvelle X, X' | ' |,? 1 | est a, puisque ainsi on est conduit immediatement h la substitution a, qui transforme le premier cercle dans le second. On tire aussi de cette 6quation la conclusion, tres importante pour notre but actuel, que si l'on connait la substitution a,, et les deux points extremes de L, on a tout ce qu'il faut pour pouvoir determiner les equations des deux cercles extr6mes et, par consequent, les equations de tous les cercles reduits, pris dans leur ordre naturel. 8. Le nombre N etant quelconque, les arcs reduits equivalents B un cercle donne sont de six especes differentes, qu'on peut distinguer entr'elles par les symboles (PP-1), (Q Q- 1), (P Q), (P Q- ), (P- Q), (P- Q- 1), qui indiquent les differentes parties du contour de S, sur lesquelles se trouvent les points d'entree et de sortie de l'arc que l'on considere. Lorsque N est carre, les cercles 230 MEMOIRE SUR LES EQUATIONS MODULAIRES. [Art. 8. extremes restent exclus de cette classification; on pourrait, au besoin, les exprimer par les symboles (p Q), (p Q-1), (q P), (q P-1), (r P-1), (r Q-1), (r 'P), (r 'Q). I est aussi convenable de distinguer entre deux symboles tels que (P, Q) et (Q, P), pour pouvoir indiquer le sens dans lequel l'arc reduit est cense d'etre parcouru. Cela pose, la table suivante fera connaitre l'espece de l'arc reduit qui, dans une fraction continue quelconque, correspond 2 un quotient donne. r impair; s = 1, 2,..., /ir-1. r pair; = 1, 2,..., -1. er-i er r Or - 2 s e +1 +1 (Q-1 P) (P-P) +1 -1 (Q- 1) (P-1 P) -1 +1 (Q.P) (P-1P) -1 -1 (Q P-1) (P P-1) Er- Er r,- 2 s Er +1 +1 (P-l Q) (Q-1 Q) +1 -1 (P-Q-1) (QQ-1) -1 +1 (P Q) (Q-' Q) -1 -1 (P Q-1) (Q Q-1) Reciproquement, etant donne le trace des arcs reduits, equivalents a un cercle quelconque, la table servira pour retrouver la fraction continue, et, par consequent, les equations des cercles reduits. Mais on peut obtenir le m6me resultat sans faire usage de la table. Posons I P 1, 0 I 1, 2. IT)!l = 2, 1 12, 8 n;/=:a 3 et observons que toute substitution normale,' peut se mettre, et cela d'une maniere unique, sous la forme JP JEl 1X Ij Q E22... Q 2s8J2s; les exposants e1P1, E622,..., E 2s 2s, (dont le premier et le dernier peuvent s'evanouir) etant les m6mes nombres qui se presentent dans le developpement -y+Q 1 1 1 1 o = = 2el~ - a+= =2 + 262,2 +...+ 2'2, 2 s + Q D'un autre cote, un arc reduit qui se termine en P, P-l, Q, Q-1 est toujours suivi par un arc reduit qui commence en P-l, P, Q -1, Q; et les substitutions qui correspondent a ces quatre cas sont respectivement IP|I PI-1, P I Q, Q1-1. I1 suit de l1 que, pour avoir les exposants 1 I, 62 f,..., il suffira de compter les arcs reduits, en faisant attention a leurs points d'entr6e et de sortie. Ainsi, Art. 9.] M1~MOIRE SUR LES EQUATIONS MODULAIRES. 231 par exemple, dans le cas d'un determinant non-carre, supposons qu'on commence la periode avec un arc reduit (0) qui prend son origine en Q ou Q-1 et qui se termine en P'i; tu sera le nombre des arcs reduits, y compris (0) lui m6me, qu'on aura a parcourir avant de venir a un arc qui se termine en Q ou Q-1; supposons encore que le premier arc reduit, qui n'aboutit pas en PE1, se termine en QE2; A2 sera le nombre des arcs reduits qui se terminent en Qe2, avant qu'on arrive a un arc qui aboutit en P ou P-l; et ainsi de suite. On remarquera que les arcs des deux premieres especes correspondent aux quotients intermediaires 01-2e,"...,._2; tandis que ceux des quatre dernieres correspondent aux quotientes complets 01,.... On peut ajouter que tout cercle qui coupe le cercle 2+ y2= 1 est un cercle reduit d'un de ces quatre especes, et reciproquement. 9. Pour faire l'application de ce qui precede aux fonctions modulaires, soit toujours = x + iy, et posons, avec M. Hermite, k2= - 8 (w), k'2 = +8 (w); faisons aussi s8 (w)= + X+iY, +8 (c )= X-i Y, X et Y etant des quantites reelles. A chaque point c du plan xy ou, si l'on veut, k chaque point seulement de l'espace reduit 2, faisons correspondre le point X +i Y du plan illimite XY; c'est pour mettre en evidence la symetrie des figures, que nous avons choisi pour origine O des axes rectangulaires OX, 0 Y le point q8 (c) = 8 () = }, qui a pour correspondant le point co=i. On sait que, dans une telle transformation, les parties infinitesimales correspondantes des deux figures sont, en general, semblables. Dans le cas actuel, il n'y a exception que pour les trois points singuliers p, q, r, auxquels correspondent respectivement les points (- 2, 0), (-, 0), et l'infini du plan XY. De plus, si l'on ne considere que l'espace reduit, la correspondance sera parfaitement determinee; de sorte qu'I chaque point reel de 2 il correspondra toujours un seul point reel de XY, et reciproquement. Maintenant il est facile de voir que, si l'on suppose r6elles les quantites a, b, c, d, x, y, l'6quation c +d(x + iy) -x+zy= a+b (x+ iy) entraine les deux suivantes a = d, c+2dx + b(x2+y2) =0. De cette seule observation on tire immediatement le theoreme que voici: 232 MEMOIRE SUR LES }QUATIONS MODULAIRES. [Art. 11. 'Tous les cercles rationnels du plan xy sont representes, dans le plan XY, par des courbes algebriques.' 'Les equations des ces courbes se deduisent des equations modulaires en posant k2= +X+i Y, X2= +X-iY 10. Considerons d'abord quelques cas particuliers, et designons par A~, A2 les points (-, 0), ( —, 0). Le cercle x2 + y2 = X, et ]es droites x= 0, x = 1 sont representes dans le plan XY par les trois parties de l'axe des X, depuis +oo jusqu'a A1, depuis A1 jusqu'I A2, et depuis A2 jusqu' - co. L'axe des Y depuis - coo jusqu'a + cc, correspond au cercle x2 + y2 = 1, qui est le seul dont la demicirconf6rence soit entierement comprise dans l'espace I. Enfin, les deux cercles equivalents x2+y2 = 2x, et les deux droites 6quivalentes 2x = +1, sont representes respectivement par les cercles (X+ 1)2 + Y2 1, (X_- )2 + Y2 = 1. On voit que les cercles rationnels de determinant + 1 sont representes par des droites et des cercles dans le plan XY; ce sont les seuls cercles rationnels qui jouissent de cette propriete. Ces cercles divisent l'espace Z en douze parties distinctes; la consideration de cette division, et de la division correspondante du plan XY, est tres importante pour la theorie des transformations du premier ordre. On peut aussi remarquer que deux points, symetriques par rapport n une droite de determinant +1, ou inverses par rapport n un cercle de determinant +1, sont remplaces dans le plan XY par deux points qui ont la meme proprietd par rapport a la ligne correspondante. 11. Ceci suffit pour les cercles de determinant+1, tant proprement qu'improprement primitifs. La table suivante, dans laquelle nous avons pose R2=X2+ Y, = (XI- )2 + 2, R2=(X+)2+ Y2, donne les resultats correspondants pour les determinants 2, 3, 4, 5. n= 2 Fraction continue Cercles reduits Courbe modulaire | [2, 2] (x - 12 _ 1)2 + y=2 2 2 (x+ 1)2 +y2= 2 (x-2)2+y2= 2 [4,-2] X +y2 = 2 1= 16 R (x+ 2)2 +y= 2 2 (+ 1)2+ 2y2 = 1 [-2,4] 2 (x-1)2+ 2y2 = 1 R4 = 16R2 2X+2+2y2 = 1 Art. 11.] MEMOIRE SUR LES EQUATIONS MODULAIRES. 233 n = 3 (x-2)2 + y2 = 3 x2 +- y2 = 3 [4,-4] (3+2 16 (R12 —) X [(R2+-)2- 1] +X4 = 0 3x22+ 3y2= 1 (x-_1)2+ y2=3 (f+ 1)Y+ y2= 3 1121 21 - - X 212z3R2 nR (2R2_R —2- 1) —(R2 — 1)4 0 [2, 2,-2, +l+2 21+(2)-+-l)-(2-l)==0 3 (x+ 1)2 + 3y2 = 1 1 3 (x-1)2+3y 2 = 1 2(x2+y2)-2x-1 = 0 2 (X2+y2)+ 6x+ 3 = 0 [2, -2, 2, 2] ( = 0 27R1R2(2 R —l) —(R1-1)4 = 2 (X2 + y2)- 6x + 3 1 1- 0 2(x2+y2)+2x-l = 0 n= 4 (-+ 1)2 + y2 = 4 (1 + 27 RP R2)2 - 214 R2 R2 R4 (=+ 1)2+y2 = 4 = 213 (Y2_X2)R4+ 28. 3. 7.R4 = 1 3 (X+y2)+ x-1 = O — 214X2y2+ 27.32 (y2_-2) 343 = 0 2+^ 3 (x2+y2) —2-x-1 =0 (x- 2)2 + y2 = 4 o =4+2 x2y2 = 4 (x+2)2+y2 = 4 (R + 27 R2 )2- 210 R [( + )2 + y2]2 = o 3x2+3y2+4x = 0 CO=-2+- 1 3x2+3y2-8x+4=0 +-2+Q 3x2+3y2+8x+4 = 0 3a2 +3 y2 —4x = 0 1 4x+1= -0 _4+ 4X24+4y2= 1 2 -4x-1 = 0 ( 2R + 27 R2)2-210R [(X_~)2+ y2] = 0 4x —3 =0 =-2 1 X 1 42+4y2+8x+3 = 0 2+ -2+- 1 y2 2+ 4x24+4y2-_8x+3 = 0 4x+3 = 0 VOL. II. H h 234 MEIMOIRE SUR LES AQUATIONS MODULAIRES. [Art. 12. n-5 (x- 2)"+ = 5 |(x-2)'2 + y2 = 5 [(1+ 4 R2)2-_ 16X2] [4, 4] (x+2)2+ y2 =5 x [4R2(9+4 R2)2_-9X2(4R2+45)] 44(x+2)2 + 2= +X6=O (x-3)2+ y2 = 5 (x-1)2 + 2 = 5 (R-1)6 + 29 R2R (x+1)2+ y = 5 x[(l+R2) (25 R-7 (1-R2)2+3. 28Y2) [6, —2, 2,2 —22 [6, -2, 2, -2] (x+3)2+ 2= 5 +(2X+ 1) (25R2+78 (1 -R2)2-29Y2)] 5(x —1)2+5y2= 1 = 0 5(y+1)2+5y2= 1 4(X2+y2)+ 6x+ 1=0 4 (ax2+y2) _ 10 +5=0 (R12-1)6+ 29 RR2 4( 24+ y2)+ 10 +5=0 x [(1 + R) (25 - 7 (1 -R1)2+ 3. 28y2) [-2, 2, -2, 6] [-2, -2, 6 4(x2+y2) — 6+1=0 -(2X-1) (25RN2+78 (1 —R)2 29Y2)] 4(x2+y2)+ 2x-1=0 =0 4(x2+y2)- 2x-1=0 2(x2+y2)-2x-2 = 0 [(1-4R)2+16Y2] 2 (x2y2)++6x+2 = 0 2 (22 + y2) +6z+2- 0 x [9 Y2 (45-4R2)-4R (9- 4R2)2] [2,-2,-2,2 ] 2( 2+ y2)+2x-2 = - y 0 22(x2+y2)-6x+- 2 = 0 12. Maintenant, pour 6claircir le theoreme general de l'article 9, il faut rappeler quelques resultats relatifs aux equations modulaires. (i) Soit N un nombre impair; designons par F (k2, X2, N)= 0 l'quation modulaire normale pour les transformations du Niyre ordre, dans laquelle, lorsque N admet des diviseurs carres, nous supposerons qu'on ait supprime les facteurs correspondants aux transformations d'un ordre inf6rieur. En posant k2 = 18 (W), on sait que les racines de cette equation sont comprises dans la formule X2 = 8(7w +2k), dans laquelle 7, ' sont des diviseurs conjugues de N; k designant un terme quelconque d'un systeme de residus pour le module y', et les nombres y, y', k etant assujettis i ne pas avoir un diviseur commun. Cette equation est symetrique par rapport a k2 et X2; en outre, elle ne change pas lorsqu'on substitue k2 et X2 deux fonctions semblables, prises parmi les six fonctions anharmoniques que voici Art. 12.] M1MOIRE SUJR LES ]IQUATIONS MODULAIRES. 235 1 1 k2 k2 - 1 k2, 1 -k2, k2' 1-k2' k2-1' k2 X2 1-X2 1 X2 X2- I 1X 1-X21 X2- X2 En substituant des fonctions dissemblables, on obtient le systeme complet des equations modulaires du Ni^me ordre. Nous ecrirons ces equations, comme il suit: (i) F (, 1 - x2) =, (ii) F (k2, 2) = 0, (iii) F(k, ) =, (iv) (k2, X2) =0, (v) F(k, x2 )= ~; (i.) F (k2, = ~ elles correspondent, comme on salt, aux six formes differentes O,P 1 11 1,0 1, 0 1, 1 0, 1 1,0 0,1 ' 1,1 ' 0,1 ' 1,0 10,1 que peut avoir une substitution de determinant impair par rapport au module 2. Les equations (v) et (vi) s'echangent entr'elles lorsqu'on permute k2 et X2; les quatre premieres sont symetriques par rapport a k2 et X2, Done en 6crivant k2= -+X+iY, X2 X - i Y, dans les equations (i), (ii), (iii), (iv) on aura les equations reelles de quatre courbes geometriques, que nous appellerons desormais la premiere, la seconde, la troisieme et la quatrieme courbe modulaire; nous observons toutefois que lorsque N 3, mod 4, la quatrieme courbe se reduit aux deux points conjugues (+, 0). Des equations (v) et (vi) on deduit les equations de deux courbes imaginaires conjuguees, dont nous ne nous occuperons pas dans ce Memoire. (ii) Soit N = 2'. Dans ce cas, on a l'equation modulaire normalef(k2, X2, 2) = 0, dont les racines sont donnees par la formule k2 = 8 (w), X2 = 8 ( 2+ ), en designant par h un terme quelconque d'un systeme de residus pour le module 29. Cette equation n'est pas symetrique par rapport a k2, X2; mais elle jouit des deux proprietes de ne pas se changer (1i) lorsqu'on 6crit 2 pour k2, (2C) lorsqu'on X2 ecrit X2 1 pour X2. I1 suit de la que les trente-six substitutions anharmoniques ne donnent que neuf equations differentes: H h 2 236 MEMOIRE SUR LES ]QUATIONS MODULAIRES. [Art. 12. (i) f( xi2 = (iv) f (k, 1-2)=0, (vi) f(-, )- )=0, (viii) f(k2, 2) = 0, (iii) f ( -k2, X2)=0, (v) f(1-k2, 1-2) = 0, (Vii) f (-1 - 2) = )0 (iX) f( X2) I2) = ~0 qui correspondent respectivement aux neuf formes diff6rentes 1, 1 0,1 0,0 1,0 1 0, 0 0, 0 1, 0 1,1 0, 1 1,1 ' 0,0 ' 1,0 0,0' 0,1 1 1,0 ' 0,0 ' 0,1 ' que peut avoir une substitution de determinant pair (la forme 0 etant exclue) 0, 0 par rapport au module 2. En effet, les racines de ces equations sont donn6es par les formules: (i) '=( 12h+l+2)' (iii) X2 (-l+2h)co (ii) X2 (8 (-2/) 2h+~ ' (iv) X\ = / 2h+w) (Vi) 2 -08( -1+2ho) (vi) - 1+(2h +2h) )' (viii) X2 = 8 ( 2hho )o 2/-+2h+co (V) = ^ (l+ 2 h)' (vii) X2'= 8 (- 1 -(20 ) (ix) X2=~(l1+-(2h+l) ). Les equations (i), (ii), (iii) sont symetriques par rapport a k2 et X2; en y ecrivant k2= +X+i I y, X2 - +X- i Y, comme ci-dessus, on obtient les equations de la premiere, de la seconde, et de la troisieme courbe modulaire. Les equations (iv) et (v), (vi) et (vii), (viii) et (ix) s'echangent entr'elles, lorsqu'on echange k2 et X2; par consequent, elles ne fournissent que des courbes imaginaires conjuguees. (iii) Soit enfin N= 2n, n etant impair. Dans ce cas encore il y a neuf equations modulaires; on les obtient successivement, en eliminant z de l'equation F (z, X2, n)= 0, et de chacune des neuf equations modulaires de l'ordre 2, dans lesquelles on remplace X2 par z. En effet, en designant par N~ et N2 deux nombres premiers entr'eux, et par fJ (k2, X2, N1) =0, f2 (k2, X2, N) = 0 deux equations modulaires appartenantes aux ordres N1 et N2 respectivement, le resultat Art. 13.] M-MOIRE SUR LES ~QUATIONS MODULAIRES. 237 de 1'elimination de z des deux equations f1 (k2, z, N) = 0, et f (z, X2, N2) = 0, est toujours une des equations modulaires de lordre N1 x VN. Si N, N2 avaient un diviseur commun, cette proposition serait encore vraie; mais le resultat serait encombr6 de facteurs etrangers, qu'on peut assigner A priori, et qui appartiennent a des transformations d'ordre inferieur. Dans le cas actuel, on verifie facilement que les eliminations indiquees fournissent le systeme complet des equations modulaires pour les transformations de l'ordre 2 x n. Les trois premieres de ces equations sont les seules symetriques, les autres etant conjuguees par couples. On en deduit (comme dans le cas precedent) les equations de trois courbes modulaires reelles, et de trois couples de courbes imaginaires conjugu6es. Le theoreme suivant resulte de cette discussion: 'En designant par N un nombre quelconque positif, les cercles proprement primitifs de determinant N, qui appartiennent aux classes subalternes (A), (B), (C), sont representes respectivement dans le plan (X Y) par la premiere, la seconde, et la troisieme courbe modulaire.' 'Lorsque N- 1, mod 4, les cercles improprement primitifs sont representes par la quatrieme courbe.' 13. Soit, en premier lieu, Nun nombre non-carre. Les courbes modulaires ne peuvent rencontrer la ligne droit a l'infini que dans les deux points cycliques imaginaires; de plus, aucune de ces courbes ne peut avoir une branche reelle passant par A1 oU A2: toutefois le point A1 peut appartenir, comme point conjugue, a la seconde courbe, le point A2 a la troisieme, et tous les deux k la quatrieme. Chacune des trois premieres courbes est composee de - H=-1 crh parties fermees, entierement distinctes entr'elles, et dont la forme generale est celle d'une spirale lemniscatique entrelacee, qui s'entortille alternativement autour des deux points A1, A2. Chaque spirale peut etre consideree a volont6 comme representant, soit une classe subalterne de cercles proprement primitifs, soit un cercle unique choisi arbitrairement dans cette classe subalterne. De m6me, lorsque N- 1, mod 4, la quatrieme courbe modulaire est composee de H' = -' h' spirales, qui representent respectivement les H' classes subalternes des cercles improprement primitifs. Soit, en second lieu, N= n2 un nombre carre. Designons par A le nombre des solutions de la congruence a2 +10, mod 2n, de sorte qu'on ait A = 0, si n est pair, ou divisible par un nombre premier de la forme 4 m + 3. Comme dans le cas precedent, chaque courbe modulaire est composee d'un certain nombre de spirales distinctes; mais ici chaque spirale doit passer soit par le point A1, soit par le point A2; ou bien elle doit avoir un point a l'infini; de plus une spirale peut 6tre simple, ou multiple; c'est h dire qu'elle peut satisfaire a ces conditions 238 MEMOIRE SUR LES PQUATIONS MODULAIRES. [Art. 14. soit une fois, soit plusieurs fois. Dans chacune des trois premieres courbes il y a A spirales simples, et (H-3 A)= (h - A) spirales doubles; chaque spirale ayant soit un point, soit deux points, a l'infini dans la premiere courbe, et passant soit une fois, soit deux fois, par le point A, dans la seconde courbe, et par le point A2 dans la troisieme courbe. Enfin, dans la quatrieme courbe, qui n'existe que lorsque n est impair, il y a A spirales triples, 1 ('- 3 A)= - (h'- A) spirales sextuples; chaque spirale ayant, soit un point, soit deux points a l'infini, et passant, en outre, soit une fois, soit deux fois, par chacun des deux points A1, A2. Une spirale simple represente une seule classe subalterne de cercles; les deux arcs reduits, qui appartiennent aux cercles extr6mes, etant representes par les parties de la spirale les plus voisines, de part et d'autre, soit du point a l'infini, soit de celui des points A1, A2 qui appartient a la spirale. Une spirale double represente, au contraire, deux classes subalternes, comprises dans la m6me classe, mais non equivalentes entr'elles; les deux parties de la spirale, qui correspondent a ces deux classes subalternes, etant separees l'un de l'autre par le point double A1 ou A2, ou bien par les deux points a l'infini. On remarquera que lorsqu'une spirale double a deux points h l'infini, ces deux points sont toujours distincts, ayant chacun une asymptote a distance finie; et de m6me, lorsqu'une spirale double passe deux fois par le meme point A, les points extr6mes de chacune des deux parties dans lesquelles elle est divisee, bien que reunis au point A, appartiennent a deux branches qui s'y croisent a un angle fini. Les spirales triples et sextuples de la quatrieme courbe modulaire donnent lieu a des observations tout-a-fait semblables, que nous pouvons passer sous silence. Quelque soit le nombre N, toutes les courbes modulaires sont symetriques par rapport a l'axe des X; la premiere et la quatrieme sont aussi symetriques par rapport a l'axe des Y; la seconde et la troisieme sont symetriques entr'elles par rapport a ce m6me axe; de plus, ces deux courbes sont les inverses de la premiere par rapport aux cercles R= 1, R- = 1; enfin, lorsque N- 1, mod 4, la premiere courbe se change dans la quatrieme par la substitution X = i Y, Y= - iX. Ajoutons que dans chaque courbe modulaire, quelques unes des spirales sont ellesmemes sym6triques par rapport B l'axe des X; pareillement, dans la premiere et dans la quatrieme courbe, il y a certaines spirales qui sont symetriques par rapport a l'axe des Y, ou bien qui ont le point 0 pour centre. Mais pour abreger, nous supprimons la discussion de ces particularites, qui dependent de l'ambiguit6 des classes et de la resolubilite de Fl'quation t2 - Nu2 = - 1 en nombres entiers. 14. Imaginons que dans le plan XY on ait opere une coupure, suivant l'axe des X, depuis A2 jusqu' - oo, et depuis A1 jusqu' + oo, en y comprenant les Art. 14.] MEMOIRE SUR LES EQUATIONS MODULAIRES. 239 deux points A1, A2 eux-m6mes. Chaque spirale modulaire sera divisee dans un certaine nombre de parties, que nous nommerons les spires de la spirale, et que nous considererons separemment. I1 est evident que, si l'on remplace le cercle rationnel, qui est represent6 par la spirale, par l'assemblage des arcs reduits qui lui sont equivalents, chacun de ces arcs reduits aura pour image geometrique dans le plan XY une certaine spire de la spirale. Done, en negligeant, pour abreger, les spires qui ont un point a l'infini, et celles qui passent par un des points Al, A2, on pourra en distinguer six especes differentes, dont voici la description generale. Une spire de premiere espece (P-l P) prend son origine dans un point du segment (-oo A2) et se dirige vers la partie inf6rieure du plan; ensuite elle traverse le segment AO0 pour passer dans la partie superieure du plan, et aboutit a un point de (- oo A2), different en general du point d'origine. Les spires de seconde espece (Q-1Q) sont symetriques, par rapport a l'axe des Y, aux spires de premiere espece. Une spire de troisieme espece (PQ) commence dans un point de (- oo A,) et aboutit a un point de A1o, en restant toujours au-dessus de l'axe des X. Une spire de quatrieme espece (PQ-1) prend son origine dans un point de (- oo A2) et se dirige vers la partie superieure du plan, mais ensuite elle traverse A2A1, pour continuer son chemin au-dessous de l'axe des X, et se termine a un point de (A1). Enfin les spires de la cinquieme et de la sixieme espece, (P-1Q) et (P-lQ-1), sont respectivement symetriques par rapport a l'axe des X, aux spires de la quatrieme et de la troisieme espece. On voit que les differentes especes se distinguent entr'elles par des caracteres qui se rapportent a la geometrie de situation. Et il resulte de ce qui a ete dit dans les articles 5, 6, 7, que, si la courbe modulaire etait une fois decrite, on n'aurait qu'a suivre des yeux le trace d'une spirale quelconque pour avoir, en premier lieu, la fraction continue qui correspond a cette spirale, et pour retrouver ensuite le systeme complet des formes quadratiques reduites, representees respectivement par les diff6rentes spires dont la spirale se compose. Voici le principal resultat auquel nous voulions parvenir dans ce Memoire. Nous nous proposons, dans une autre occasion, d'exposer quelques applications, que nous avons faites des principes precedents, et qui nous semblent propres a montrer le parti qu'on peut espe'rer d'en tirer en diverses questions d'arithmetique et de la theorie des fonctions elliptiques. 240 MllMOIRE SUR LES ]EQUATIONS MODULAIRES. [The following abstract of the preceding paper was published in the Transunti, Ser. iii. vol. i. pp. 68-69.] I1 Socio L. Cremona presenta una Memoria: Sur les equations modulaires del signor Henry J. Stephen Smith professore all'Universita di Oxford. L'auteur se propose dans ce Memoire d'etudier la liaison qui existe entre les equations modulaires, et les formes quadratiques binaires B determinant positif. Cette liaison, qui peut-etre n'avait pas ete remarquee jusqu'ici, est h la verite purement analytique; mais on parvient a la mettre en evidence, en regardant d'une part les equations modulaires comme definissant des courbes geometriques, et en se servant, de l'autre part, d'une nouvelle representation geometrique des formes quadratiques, dont voici le principe. A chaque forme (a, b, c) du determinant positifD = b2 - ac, on fait correspondre un cercle a + 2 bx + c (x2 + y2) = 0, trace dans un plan (xy) dont toutefois on ne considere que la partie situee au dessus de l'axe des x. On appelle espace reduit la partie de ce plan comprise entre les deux droites x = +1, mais exterieure aux cercles X2 + y2 =; et l'on regarde comme arc reduit tout arc de cercle qui se trouve en dedans de l'espace reduite. Cela pose, la periode des formes reduites appartenantes a une classe donnee est repr6sentee par les arcs reduits correspondants, dont l'assemblage forme une ligne en apparence brisee, mais qui peut etre envisag6e comme continue. En suivant la notation usuelle des fonctions elliptiques, qu'on pose w = x + iy = iK K' k2= 08(W)= + X + i Y, et qu'on fasse correspondre aux points x + iy de l'espace r6duit les points X+ i Y d'un nouveau plan illimite (XY), par le moyen de l'equation + X + i Y= 8s(x + iy), dans laquelle on suppose reelles les quantites x, y, X, Y. Les lignes brisees correspondantes aux differentes classes de formes quadratiques de determinant D se trouveront representees dans le plan (XY) par autant de spirales formees distinctes, dont l'ensemble formera une courbe geometrique complete. L'6quation cartesienne de cette courbe sera simplement F(2+X+iY, +X-i Y)=0, en d6signant par F(k2, DX2) - une des equations modulaires pour les transformations elliptiques de l'ordre D. On tire de 1l ce resultat remarquable que si, sans penser aux formes quadratiques de determinant D, on trace la courbe modulaire, on aura sous les yeux une image exacte du systeme complet de ces formes; de sorte que, par un simple denombrement des spirales, et des differentes spires dont chaque spirale se compose, on pourra deter M-MOIRE SUR LES EQUATIONS MODULAIRES. 241 miner (1) le nombre des classes non 6quivalentes; (2) le d6veloppement en fraction continue, propre a chaque classe, developpement qui donne, comme on sait, le systeme complet de formes reduites, representantes de la classe. Le cas d'un determinant carre est compris (sauf quelques particularites) dans la th6orie g6ndrale. Il est bon de remarquer que les notions arithmetiques de classe, d''quivalence, et de forme reduite, doivent subir une legere modification pour les adapter a la theorie des fonctions modulaires. On donne dans le Memoire toutes les explications necessaires a cet egard. Au lieu des equations modulaires entre k2 et X2, on aurait pu introduire les equations plus simples de Jacobi entre y/k et W/X; mais la theorie arithmetique en deviendrait un peu plus compliquee. Les m6mes principes peuvent s'appliquer avantageusement a d'autres questions de la theorie des fonctions elliptiques, parmi lesquelles on peut signaler le probleme pose par Jacobi dans la note ajoutee k la page 75 des 'Fundamenta.' L'auteur espere de soumettre prochainement a I'Acadedmie la solution qu'il pense avoir trouve de ce problRme difficile. VOL. II, I I XXXVI. ON THE SINGULARITIES OF THE MODULAR EQUATIONS AND CURVES. [Proceedings of the London Mathematical Society, vol. ix. pp. 242-272. Read February 14 and April 11, 1878.] 1. IT is proposed, in this paper, to examine the characteristic singularities of the modular equations and curves. The method employed is applicable to all the modular equations hitherto considered by geometers;* but, for brevity, the discussion is confined to the equations containing the squares of the modules, and to the case in which the order of the transformation is uneven. * The modular equations considered by Jacobi in the 'Fundamenta Nova' are (2) the equation between u = q (co), and v = < (Q2), (see Art. 4 of this paper, equations (3) and (4),) and (5) the equation between u8 and v8, of which the characteristics are discussed here. M. Kronecker, in his researches on the modules which admit of complex multiplication, would seem to have also employed (3) the equation between u2 and v2, and (4) the equation between u4 and v4. (See the account of these researches in the Reports of the British Association for 1865, pp. 332 and 358 [vol. i. pp. 301 and 336]; see also Professor Cayley, ' Phil. Trans.,' vol. clxiv. p. 450.) M. Joubert (' Comptes Rendus,' vol. xlviii. pp. 290 -294) was the first to consider (6) the equation between u8 (1 - u8) and v8 (1 - v8). Dr. Felix Miller, in his Inaugural Dissertation (Berlin, 1867), drew attention to the equation (7) between (1-U + V16)3 (1 8 +16)2 Tr(.^ ()and T ( Q)(-lV () = (1-8 + u15) v 16(1 - V8)2 and the discussion of this equation has recently been resumed by Professor Klein ('Mathematische Annalen,' vol. xiv. p. 112, May 1878). These geometers have expressed T(w) and T7() rationally in terms of a third indeterminate, in the cases N = 2, 4; 3, 5, 7, 13; the deficiency of the equation (7) being zero in these six cases; but neither of them has given any example of the equation in its explicit form. Writing x for T(co) and y for T(Q2), I find, when N= 3, x (x + 27. 3. 53)3+y (y + 27. 3. 53)3 - 2163y3 + 211. 32. 31 X2y2 (x + y) - 22. 33. 9907xy (2 + y2) + 2. 34. 13. 193. 6367xay2 + 28. 35. 53. 4471xy (x + y) - 25. 56. 22973xy = 0. The equations (2)... (7) are symmetrical with regard to the two indeterminates, and, the number N Art. 2.] SINGULARITIES OF MODULAR EQUATIONS AND CURVES. 243 2. We represent by q, the square of the modulus of a given elliptic function; by p, the square of the transformed modulus, the transformation being primary,* and of the uneven order N; by F(pq,).......... (1) the modular equation subsisting between p and q; in connection with this equation, we consider the modular curve C, of which the trilinear equation F(a,,.......... (2) is obtained by writing 2p =, q =- We denote by P, Q, R the vertices of 7 7 the triangle afly; and by S the point a = / = 7, or p = q = 1; the three sections (PS, QR), (QS, lRP), (RS, PQ) we represent by a, b, c. We r p and q as the parameters of two pencils of lines a-py, -qy, of which the centres are Q and P, and between the rays of which a correspondence is established by the equation (1); we observe that, to the values 0, 1, oo of either para- meter, there answer in the two pencils respectively the rays a QR, QS, QP; PR, PS, PQ; and that, by a known property of the modular equation, the pairs of rays (QR, PB), (QS, PS), PQ) are corresponding rays in the two pencils, either ray of any of inter*egard (QP, these being uneven, they are of the order A +B (Art. 3) in the indeterminates separately, and of the order 2A in the indeterminates jointly. I have recently found that the Eulerian functions X (o) and X(Q), defined by the equation ir X (O) = 4/2. e24 x n(l:_ (_ I) e si), satisfy an equation (1) having the same properties; for the cube X (co) this had already been shown by M. Kcenigsberger (' Borchardt's Journal,' vol. lxxii. p. 182 sqq). Writing uz= X (co), v1 = X (2), we have, in the cases N= 5, 7, 11, A 2 Uo + 2 u 1 - 23 zu + 6 = 0, u}2' + 8. 2 1 V l I — 44 ~ 9 v9 + 44. 2' 77 7- 44. 23U5 V+1 +22 13 3 -2.23 21- + V12 = 0, - - 1667 + 7 V4 - 2Fu1 + v = 0. The function X (to) is a twenty-fourth root of u8 (1 - us8); the formulae relating to its linear transformation have been given by M. Hermite ('Comptes Rendus,' 1858, vol. xlvi. p. 721). In respect of simplicity of form, the equations (1), (2)... (7) appear to arrange themselves in this numerical order; but, in respect of simplicity of algebraical theory, the order is reversed, as the deficiency decreases from (1) to (7). * A transformation of the uneven order N is primary when it satisfies the congruence |, j l,O ~, mod 2. For the theory of the elliptic multiplier it is convenient to fix the signs of c, d 0, 1 a, b, c, d by the additional condition a = 1, mod 4; but for our present purpose this restriction is unnecessary. Ii 2 244 ON THE SINGULARITIES OF THE [Art. 4. pairs being the only ray answering to the other ray of the same pair; the modular curve C is the locus of the intersections of corresponding rays in the two pencils.* We denote by m and n the order and class of C; by H its deficiency; by K and I its cuspidal and inflexional indices; by D and T its discriminantal order and class; by F(p)=E(q) the highest, by E' (p) = E' (q) the lowest, exponents of p and q which present themselves in the equation (1). ARTS. 3-8. N not divisible by a square. 3. We confine ourselves, in the first instance, to the case in which N is not divisible by any square; and we represent, in this case, by A the sum of the divisors of N which surpass VN; by B the sum of the divisors of N which are less than vN; by v the number of divisors of either sort, so that 2v is the whole number of divisors of N. We then have the formulae (i.) m= 2A, (ii.) n=3A -B, (iii.), H= I(A + B) - 3v+ 1, (iv.) K=2A+2B-6v, (v.) I-K= 3(A -B), (vi.) D = 4A25A + B, (vii.) T- D = (A - B)(5A-B). To these we may add the equations (viii) E(p)=E (q)=A+B, E'p) = E' (q)= 2 B. Of these formula (i.) and (viii.) are well-known; of the remainder, it will suffice to attend to (iii.) and (iv), because, when the values of m, H, and K are given, the values of n, I, D, and T are known from the equations of Plucker. 4. The Deficiency.-The demonstration of the formulae (iii.) and (iv.) depends on the simultaneous expression of the modular parameters p and q as * In a paper which I hope shortly to lay before the Society, I have discussed, with some fulness of detail, the relation of the algebraical singularities of the parametric equation F(p, q, 1)=0 to the characteristic singularities of the curves of which the equations are included in the formula F(~B,, 1)=0, A, B, C, D being the equations of straight lines. The discussion comprises an examination of the effect of any quadric transformation on the singularities of a curve. Art. 4.] MODULAR EQUATIONS AND CURVES. 245 transcendental functions of the quotient of the periods of the given elliptic function. As we have already developed these considerations elsewhere,* we * In an Introduction (now in the press) to Mr. J. W. L. Glaisher's Tables of the Theta Functions [No. XLIII. of the present volume]. The method indicated in this article has been already employed by Professor Klein, in the paper to which reference has already been made (' Math. Ann.,' vol. xiv. p. 111; see especially ~~ 6-8, 7-13), and by Professor Dedekind, in a letter addressed to M. Borchardt (June 1877, see 'Borchardt's Journal,' vol. lxxxiii. p. 265, especially ~~ 1-4 and 7). In the year 1873, I submitted to the Academy of Sciences in Paris a Memoire 'Sur les Equations Modulaires' (' Comptes Rendus,' August 1873, vol. lxxvii. p. 472). In this Memoire (which was ultimately presented, without alteration, to the Accademia dei Lincei, and was printed in their 'Atti,' vol. i. series 3, p. 136, February 1877) [vol. ii. pp. 224, etc.], I had employed the same method (see Arts. ii, ix, and x of the Memoire) to establish the relation which exists between the modular equations of order N and the binary quadratic forms of the positive determinant N. The Memoire was devoted to that theory alone, as I attached more importance to it than to any other result relating to the modular equations at which I had then arrived. But I had already in the year 1873 obtained:-(i) a proof of the existence of the modular equations, simpler perhaps than that of M. Dedekind, and based solely on those elementary properties of the function b(co), which were deduced from the theorem of Fourier by Cauchy and Poisson, without employing any elliptic formulae; (ii) a determination in the simpler cases of the Pluckerian characteristics of the modular curves; (iii) a solution of one part at least of the problem relating to complex modules, proposed byJacobi in Art. 32 of the 'Fundamenta Nova.' I communicated to Professor Cayley, in 1873, the formulae for the deficiency of the equations (2)... (5) when N is an uneven prime (see his Memoir on the Transformation of Elliptic Functions, presented to the Royal Society in that year); the formulae for the cuspidal index I obtained by transforming into normal developments the parametric developments which give the deficiency (see Art. 5 of this paper): thus, the order of the curves being known, all their Plutickerian characteristics were determined. The case when N is a product of uneven primes presents no greater difficulty than the case when N is a prime; and I had (in fact) obtained the formula for this more general case as early as 1873. The case when N is divisible by a square, and still more the case when N is itself a square, appeared to involve some difficulty; and these I left untouched till the spring of the present year, when I found that the introduction of the arithmetical function f' (see Art. 9 of this paper) caused the supposed difficulty to disappear. To the more exact determination of the indices characteristic of each special singularity of the modular curves, I was guided by the methods employed in a former paper on the Higher Singularities of Plane Curves. A complete system of formulae, analogous to that given in the present paper for the modular equation (5), I have already obtained for the equations (2), (3), and (4); with the equation (7), and with the eight equations between corresponding powers of X (co) and X (&), I have not advanced equally far, but I have not found that they offer any peculiar difficulty. In the M6moire 'Sur les Equations Modulaires,' I have confined myself (as in the present paper) to the equation (5) between the squares of the modules. At the time when the Memoire was written, I was well acquainted with the characteristic property of the function T (w); viz. that it is unchanged by any linear transformation of the elliptic functions; and I even thought of employing it in the Memoire instead of the function 48 (co). I had conjectured (erroneously however) that the modular curves T7 derived from the equation (7) would represent ordinary periodic continued fractions with positive integral quotients, in the same way in which the modular curves C derived from the equation (5) represent periodic continued fractions with even quotients. But I was deterred from employing 246 ON THE SINGULARITIES OF THE [Art. 4. shall in this place assume the results of the discussion as known, and shall confine ourselves to their application to the formulae (iii.) and (iv.). Denoting by x and y real quantities, of which y is essentially positive, and by w the complex variable x+iy, which is thus subject to the restriction that the coefficient of i in its imaginary part is positive, we define the function p (c) by the equation ',s=oo 1 + e2siir W () = /2. e [nI::-1 +e(r s;...* (3) and we consider the A + B quantities Q, of which the values are given by the equation g'w +2k (4 -^, ~ ~ ~ ~ ~ ~ ~ ~ ~ (4) g, g' being any two conjugate divisors of N, and k being any term of a complete system of residues for the modulus g. We then have the fundamental theorem, 'If q = 8 (w), the A + B corresponding values of p are included in the formula or, which is the same thing, F(p, q, 1) = [p - 0 (Q)], the sign of multiplication extending to the A + B values of.' It results from the discussion to which we have referred, that, if we regard q as an independent complex variable, and represent its values in the usual the equation (7) by the consideration that there was not a single calculated example of it; indeed, at that time I was not acquainted with the researches of Dr. Felix Muller, and did not know that the equation had attracted any attention. I have since found that the curves T do not precisely represent the reduced forms of Gauss, but instead a system of forms determined by a different regulative principle. I am disposed to think (notwithstanding the considerations mentioned in the note on Art. 1), that there is some advantage in continuing to regard the equation between the squares of the modules, as the principal modular equation, rather than either the equation (1) or (7). At least, as far as concerns the arithmetical theory to which the Memoire relates, and which I have since extended to the equations (1), (2), and (7), (the theory of the equations (3), (4), and (6) hardly requiring a separate discussion), the modular curves (5) present phenomena in some respects simpler than those presented by the curves (7), and in all respects simpler than those presented by the curves (1)... (4). Both in the Memoire and in this paper, I have given especial attention to the case in which N is a square, because the solution of the problem of Jacobi for the transformations of order N depends on a consideration of the spaces into which modular curves of order N2 divide a plane. A note published in the 'Transunti' of the Accademia dei Lincei (vol. i. p. 42, 7 Jan. 1877) [vol. ii. pp. 221 etc.], contains what is in fact a solution of Jacobi's problem for the case N= 1; to this particular case of the general problem, the attention of geometers had been called by M. Hermite in the note appended to the second volume of M. Serret's edition (1862) of the Differential Calculus of Lacroix, pp. 421-425. Art. 4.] MODULAR EQUATIONS AND CURVES. 247 manner by the points of a plane, p, considered as a function q, has no spiral points other than the three q = 0, q = 1, q = oo (it will be remembered that in the plane of double algebra the infinitely distant is a point). Thus, if we cause q to describe any closed contour, not including one of the three points 0, 1, ao, the values of p will undergo no interchange; but each root of the equation F (p, q, 1) =0 will return, when the contour is closed, to the same value which it had at the beginning; although the contour may include points (other than 0, 1, C ) at which two of the values of p become equal. But the case is different if we cause q to describe a closed contour round one of the three exceptional points. At each of these points all the values of p become equal to one another, and to the value of q indicated by the point. As the general result is the same for each of the three points, it will suffice to attend to one of them only; for example, to the point q = 0. If, then, we cause q to describe a closed contour round 0, the g values of p, or of the expression 08 (- +2 which contain the divisor g in the denominator, change into one another cyclically, and thus the g = A + B roots of the equation arrange themselves in 2 v cycles, corresponding to the different divisors of N; or, which is the same thing, the developments obtained by expanding the different values ofp in series proceeding by ascending powers of q are singular; being, in fact, of the type p =Xqg +............. (5) Similarly, at the points +1 and so, we have singular developments of the types p- 1 = (- 1)+...,......... (6) ().............. (7) As these are all the singular developments that can exist, we infer that, if W (p) represent the number of the spiral points of p, each point being reckoned with its proper multiplicity, W(p) = 3z (g- 1) = 3(A +B)- 6v. Substituting, in the equation of Riemann, 2H= V(p)- 2E (p)+2,........ (8) the value of E (p) (equation viii., Art. 3), and the value just obtained for W(p), we find H I (A + B) - 3 + 1, which is the equation (iii.). 248 ON THE SINGULARITIES OF THE [Art. 5. 5. The Cuspidal Index.-To determine the cuspidal index of C, we first consider the developments (5) which appertain to the point R. Since N is not a square, we cannot have g = g'; if g' > g, we have the normal development* "=X () +...;.......(9) if g > g', we have the normal development ( )+........ (10) The branches corresponding to the developments (9) and (10) have, for their cuspidal indices, g- 1 and g' -1 respectively. Hence each of the lines QR, PR is touched at the point R by a set of branches of which the aggregate cuspidal index is B - v In the same manner, it will be found that at the point S each of the lines QS, PS is touched by a set of branches having the same cuspidal indices as the branches which touch QR or PR at R. Lastly, from the developments (7), we deduce normal developments of the types 9' Z= +~.., ->g;........ m g' = )+..., g>.... (1) and from these we infer that the line PQ is touched at each of the points P and Q by a set of branches of which the aggregate cuspidal index is A - B - v. Since C can have no other cuspidal branches, we find K= 2(B - v) + 2 (B - v) + 2 (A - B -) =2 (A +B) - 6v, which is the formula (iv.). * If A, B, C represent straight lines forming a triangle, a development of the type A B mL (B 2 A X(c) + 2(C) +... in which al, a,... are positive and increasing, and a, is greater than unity, is termed a normal development; A is, of course, the tangent to the branch, B is any line passing through the point to which the development refers. Art. 6.] MODULAR EQUATIONS AND CURVES. 249 6. The ciscririnant of F(p, q, 1).-The values of m, E(p), E(q), E' (p), E' (q) are inferred from the equation (1), by a method, due (as it would seem) to M. Kronecker, of which examples are given in the Report on the Theory of Numbers ('Reports of the British Association' for 1865, p. 349 sqq.).* This method is also applicable to the discriminant of the equation (1);t i.e. to the expression V (q) = n[p (Q) - (Q2)....... (13) where the sign of multiplication extends to every pair of values of Q. Let (A+B)2=A,+B2......... (14) where A2 comprises all the terms in (A + B)2 which are greater than N, and B2 all the terms which are less than N; as for the terms which are equal to N (of which the sum is evidently 2vN), we divide them equally between A2 and B2; thus, if g1, g2,...g, are all the divisors of N (unity and N included), we have I — t pS=t A2 = N+ L.= -g grg > N B2= -N+.=I =ggs gg <N; < (15) A2 + B2 =: 2s:, g = (A + B)2. Applying the method of M. Kronecker, we find that the highest power of q in V (q) is 2 A - A - B, and that the lowest power of q is 2 B- A -B. By a known property of the modular equation, F(1-p, 1 -q, 1)=F(p, q, 1); hence V (q) must be divisible by 1 - q as often as by q; we have, therefore, V ()=(q_-q) 2)2B2-A-B X X(q);...(16) where X (q) is a rational and integral function of q, not divisible by q or I - q, and of the order 2 A - 4 B2+ (A + B). By another property of the modular equation we have the identity (pq)A+BxF(1, q, I)=F(pq, 1). p q * [Vol. i. p. 325.] t It has been applied by M. Koenigsberger ('Vorlesungen fiber die Theorie der Elliptischen Funktionen,' vol. ii. p. 154) to the discriminant of the modular equations between tu and v, in the case in which N is not divisible by any square; the result had already been given by M. Hermite in his M6moire sur la Theorie des Equations Modulaires (' Comptes Rendus,' vol. xlviii. p. 1079). The discriminant of the modular equation between X3 (co) and X3 (2) has been similarly treated by M. Krause (' Math. Annalen,' vol. xii. pp. 1-3). VOL II. k 250 ON THE SINGULARITIES OF THE [Art. 7. If therefore we write p -p' for p, and q -q' for q, the dialytic discriminant of the bipartite binary quantic (pq)A + B x F(,, 1) is symmetric with regard to q and q'; i.e. it is of the form [qq' (q'- q)]- 2A- BX (, ),....( (17) where X (q, q') is symmetric and of the order 2A2 -4B2 + A + B. It will be observed that the order of the discriminant (17) is 3 (2B2-A -B) +2A-4B2+ A + B =2 (A + B2) - 2 (A +B) =2(A+B)(A+B-1), as it ought to be; and that the equation V (q) = 0 is to be regarded as having 2 B2- A - B infinite roots, and as having lost the same number of dimensions. We may add that X (q) is a perfect square. For, if q-qo, be any factor of X (q), all the developments of p, proceeding by powers of q - q,, have integral exponents; the exponent of q - q, in x (q) is the sum of the' discriminantal indices of these developments taken in pairs; and this sum is always even. 7. The discriminantal indices of P, Q, R, S.-We next examine the discriminantal indices in the curve C of the points P, Q, R, S. Representing these indices by D (P), D (Q), D (R), D (S), we have D(P)=2A2-A2-A=D(Q), D (R) = 2 B- 2 A =D (S).j ** * To establish these formulae, it will suffice to consider the points P and R. At P we have v superlinear branches of the aggregate order A -B touching PQ; we may symbolize the branch of which (11) is the normal development by (g', g), where gg'=N, and g'>>VN>g. The discriminantal index of the branch (g', g) taken by itself is g' (g' -g - 1); the joint discriminantal index of the two branches (g', g) and (g[, g,) is 2g' (g' - g), if g' > g', and consequently g < gl. Hence we have D (P) = g' (g'g- 1) + 2 g' (g - g), (19) the summations extending to all values of g' and g' which satisfy the inequalities g'>V/N, gi<g, g >^N.... (20) Attending to these inequalities, we find Zg+'2 + 2 Yg'g1 =AA2, g' = A, 2g'g + 2 Zg'gl = A2- A2; Ait. 8.] MODULAR EQUATIONS AND CURVES. 251 and substituting these values in (19), we obtain the value of D (P), given by the formula (18). Again, at the point R, we first consider the branches touching PR, the normal developments of which are of the type (10). For the discriminantal index of these branches, taken singly and in pairs, we have the expression - Zg (g' - 1)+ 22g2g', the summations extending to all divisors g and g, which satisfy the inequalities g>V/N, g< g, g >N........ (21) The discriminantal index of the branches touching QR is evidently the same as that of the branches touching PR; and as the aggregate order of each of the two sets of branches is B, they intersect one another in B2 points, and the part of D (R) which arises from their crossing one another at R is 2 B2. We have, therefore, D (R) = 2g (g'- 1)+ 4 Z:gg'+2B2; but, attending to the inequalities (21), we also have -g=A, 2g'g+2 gg'= B2-B2, whence, in accordance with (18), D (B)2B2-2A. 8. The intersections with C of the polar curves of P and Q.-Since the branches which touch PR at R are of the aggregate class A - B, the line PR, considered as a tangent drawn from P, counts A - B times as a tangent at R. Similarly PS counts A- B times as a tangent at S. Again, since the branches which touch PQ at P and at Q are of the aggregate order A - B, and of the aggregate class B, PQ, considered as a tangent drawn from P, counts A- B + B = A times as a tangent at P, and B times as a tangent at Q. Thus the three lines PB, PS, PQ count as (A -B)+(A-B) + (A + B) = 3A -B tangents from P; i.e. no other tangents can be drawn to C from P. Again, the polar curve of P intersects C at P, Q, R, S, in D(P)+ A =2A2- A, D(Q)+B =2A2-A2+ B-A, D(R)+A-B=2B2-B-A, D (S) +A-B=2B2-B-A, points respectively; or, in all, in 4A2 + 4 B2 - 2 A - 3 A - B points. The whole number of intersections of C by any one of its first polars is 2 A (2 A - 1); hence K k 2 252 ON THE SINGULARITIES OF THE [Art. 9. the polar of P intersects C in 2A (2A-1)-(4A2+4B2-2A2-3A-B)=2A2-4B2+A+B points, other than P, Q, R, S. These 2 A2 - 4B + A + B intersections correspond in the discriminant of F(p, q, 1) to the factors of X (q), of which the aggregate order is the same. As the intersections other than those at P, Q, R, S correspond to the factor X (q), so also the intersections at R correspond to the factor q2B2-A-B, and the intersections at S to the factor (1 - q)B2-A-B. But it is proper to observe that the remaining intersections at P and Q (of which the aggregate number is 4 A2- 2A2+ B- A) surpass the order of the remaining factor q'2B2-A-B of the complete dialytic discriminant; the difference 4A2-2A2+B-A-(2B2-A-B)=2A (2A-1)-2(A+B) (A +B-1) being, as it ought to be, equal to the difference between the number of intersections of C by any one of its polars, and the order of the dialytic discriminant. We reserve, for a future communication to the Society, a complete discussion of the relations which subsist between the exponents of the factors of the dialytic discriminant of any parametric equation, and the corresponding intersections of the locus curve by the polars of the centres of the generating pencils. The points, other than P, Q, R, S, in which C is intersected by the polar of P, are all ordinary double (or it may be multiple) points, free from superlinearity, and having tangents which do not pass through P. For C has no superlinear branches beside those at P, Q, R, S, and the only tangents which pass through P are PQ, PR, PS. The same thing is also evident from what has been said in Art. 6 of the exponents of the factors of X (q). ARTS. 9-11. N divisible by c square. 9. Definition of certain Arithmetical Functions.-We now pass to the general case in which N is any uneven number whatever. Let N = a la23..., a,, a2,... being different uneven primes; let g, g', as before, be two conjugate divisors of N, and let a be the greatest common divisor of g and g'. We resolve g into the product of two factors 71 and 72, of which 71 contains only those prime divisors of g which do not occur in r and g'; and 72 contains only those prime divisors of g which do occur in, and g'. Representing by f(z) the number of numbers prime to any given number z and not surpassing it, we write f' (g) = lf (2), Art. 9.] MODULAR EQUATIONS AND CURVES. 253 and we observe that we have the equations f () _ f' (g) f'(g') g g each of these quotients being equal to I(l-l), if e denote any prime divisor of?. We still retain the symbols v, A, B, A2, B,; but with extended significations, which we proceed to explain. We define v by the equation 2v= f(),........ (22) the sign of summation extending to every divisor g of N. We observe that, in general, each term f () occurs twice in If(,), because r is the same for each of any two conjugate divisors; but that, if N= 02 is a perfect square, the term f(0) =f' (0) occurs only once in If(?). Again, we define A and B by the equations = f'(g), g > N, (23) the summations extending to all divisors g of N which satisfy the inequalities (23) respectively; when N = 02 is a perfect square, we divide the termf' (0) =f (0) equally between A and B. We thus have in every case A +B= f'(g), the summation extending to every divisor g of N. Lastly, we define A2 and B2 by the equations A2= f'(g1)f'(g2), gg2 N, ) B,-= Zf'(g1)f'(g2), gg 2<N; in which gt and g2 are any two divisors of N (the same or different) which satisfy the inequalities specified; so that, if g, and g2 are different, the termf'(g1)f'(g2) occurs twice in A,, or in B2, as the case may be. If g1g2=N, we divide the double term 2f' (g)f' (g2), corresponding to these two divisors, equally between A2 and B2; if, in particular, N= 02 is a perfect square, the single term [f(0)]2 is to be divided equally between A2 and B2. It is evident that we have, in every case, A2 + B= (A + B)2. The sums 2 = -2f(q), and A + B= lf'(g), may be conveniently expressed in terms of the prime divisors of N. Observing (1) that the terms of the product I [1 + a +a2 +...aa] 254 ON THE SINGULARITIES OF THE [Art. 10. represent, after development, all the divisors of N, and (2) that f'(g) =f' (h) xf' (h2), if g be a product of two relatively prime factors h, and h2, we find f' (g) =I [f' (1) +f'(a) +... f'(a")] = n[ +(a- 1)+ a(- 1)+... + aa-2 (a -l)+aa] = n [a- a+ aQ]; whence A+B=Nx (1 + )........ (25) Again, if we write f"(g) for f(?), and give to hl, h2 the same signification as before, f"(g) satisfies the equation f" (g) =f" (h) xf" (h2) whence we infer that f (/) = [f"(i) +f"(a) +... +f"(aa)]. First, let a = 2 t + 1; then f"(1) +f"(a) +... f"(Ca2y+1) = 2 [1 + (a -1) + a(a-l) +... + al (a - 1)]= 2at. Secondly, let a = 2,; then f" (1) +f" (a) +... +f" (a 2 ) = 2 [ + (a- 1)+a(a- 1)+...+a-2 (a- 1)]+a'-l (a-l) = am + ca/- 1. If therefore N = H b2 + 1 x ITc2', where b,..., c,..., are different prime numbers, we have Zf(q)=2P=H2bxnHc(l-)...... (26) It will be observed that the definitions which we have now given of the symbols v, A, B, A2, B2, coincide, in the case in which N is not divisible by any square, with the definitions of Arts. 3 and 6. 10. Case when N is not a square.-Excluding, for the present, the case in which N is a perfect square, we have to show that, in all other cases, the formulae of Arts. 3-8 hold without further modification. For brevity, we shall establish only a few of the assertions included in this general statement, as the method to be pursued with regard to all of them is the same. (i.) If d (N) is the sum of the divisors of N, and eq, e2,... are the primes of which the squares divide N, the order of the irreducible modular equation of order N is N Art. 10.] MODULAR EQUATIONS AND CURVES. 255 (See M. Joubert, 'Comptes Rendus,' vol. i. p. 1041; Report on the Theory of Numbers, loc. cit., p. 332*). But this expression has for its value NI (1 + ); i.e. the order of the modular equation in p or in q is A + B (equation 25). (ii.) If we write q= -8 (w), the A +B corresponding values of p are given by the equation ( + 2k) in which k is any one of the ~()g =f'(g) residues of g which are prime to a, the greatest common divisor of g and g'. If, as in Art. 4, we cause q to describe a closed contour round O,f'(g) values of p, which answer to any given divisor g, arrange themselves in f(q) cycles each containing g roots; and thus the developments (5), which appertain to the simultaneous values p =0, q =0, assume, in the general case which we are now considering, the form g'. g p=x(q)V '+..........; (27) the least common denominator of the exponents being 2. It will be observed that there are f(C) developments, in which g and g' have the same values; the coefficients X having different values in these f(Ql) developments. Similarly, there are f (rn) developments of each of the types o' g ( -l) = (- ~),7 +........... (28) - X............. (29) (28) P q Hence Wp) =3f(/ ) [ 9-1 = 3 f'(g) - 3 f(rl) = 3 (A +B) -6 v; and, consequently, as in Art. 4, I=- (A + B) -3v+ 1. (iii.) From the developments (27), (28), (29), we can deduce the normal developments of the six sets of branches which touch PR and QR at R, PS and QS at S, PQ at P and Q. Each set comprises v branches; if h2 is the greatest square dividing N, h of these are linear in each of the first four sets; all * [Vol. i. p. 302.] 256 ON THE SINGULARITIES OF THE [Art. 10. the rest are superlinear. It will suffice here to determine the cuspidal and discriminantal indices of the branches touching PQ at P. The normal developments of these branches are of the type g' —g a7() +........ (30) g' >g. Hence their aggregate cuspidal index is If(?)() - -1), < N; or, which is the same thing, g>VN, g<V N, g<VN, f'(g) - zf'(g) - Zf(7 ) = A - B - v. To obtain the discriminantal index, we first consider a single group of f(,) branches corresponding to given values of g, g'. The discriminantal index of one of these branches, taken by itself, is the joint discriminantal index of two different branches of the group is 2g (S -); so that the aggregate discriminantal index of the group (g', g) is g'0 911 gg, _ f(q.f) x - ( - g +() f(r)-1 - l]x ( -) =f' (') f'(g') -f'(g) - 1]. We next consider the two groups (g', g) and (g', g1) consisting respectively of f(q) and f(71) branches. If g' > g', or, which is the same thing, if g'g1 > N, the joint discriminantal index of the two groups is 2W(f(1) x g (g - ) = 2f'(g') [f (g1) -f'(g)]. Thus the aggregate discriminantal index of the branches touching PQ at P is given by the equation D(P) = f' (g') [f'(g') -f'(g) - 1]+2 If' (g') [f' (g) -f'(gl), the summations extending to all values of g' and g> which satisfy the inequalities g'> N, g<g', gl>vN. Art. 11.] MODULAR EQUATIONS AND CURVES. 257 But, as in Art. 7, [f' (g') ]2+ 2 Ef' (g')f'(g;)= A2, f (g') A, f'(g')f'(g) + 2 1Zf'(g')f(gl) = A2 - A2; whence, as before, D (P) = 2 A2 - A2 - A 11. Case when N is a square.-The case in which N= 02 is a perfect square requires separate consideration, because the modular curve of order 02 meets the line PQ inf(0) points distinct from one another and from P and Q; and again, at each of the points R and S, it has f (0) linear branches, of which the tangents are different from one another, and from the lines RP, RQ, SP, SQ. Thus some of the characteristics of the singularities at PQRS are changed; and with them some of the characteristic indices of the curve. We write O' for f(O)=f'(0). It will be found, on referring to Art. 10, (i.) and (ii.), that E (p)= E (q)= A + B, m=2A, H=2 (A+B)-3v+1, as in the case when N is not a square. Again, the cuspidal index of each of the four sets of branches which touch PR and QR at R, PS and QS at S, is, as before, B- v; but the cuspidal index of the branches at P and Q is A - B - v + I 0' instead of A - B - v. For this index is g>VN g < N g< VN f'(g') - If'(g) - If(.), (see Art. 10, iii.); and f' (g) = A -1 0', g<VN zf' (g) = B- 0', g<VN zf (.) v - -'2. To find the discriminantal indices of PQBS, we denote by A, B, A2, B2 the numbers obtained by omitting in ABA2B2 the terms depending on 0; we thus have A =A-A 0' B=B- 20' A2 = A2-20'A + 1, B2 = B2-2 0'B + 0. Using these expressions, we find, as in Art. 10, (iii.), D (P) =D (Q) = -A2-A2-A; VOL. II. L 1 258 ON THE SINGULARITIES OF THE [Art. 13. or, substituting for A and A2 their values, D(P)=D(Q) = 2A2-A-A+1'. To determine D (R), we have:-(i.) For the discriminantal index of the set of branches touching either PR or QR, B2 - A - B2; (ii.) for the joint discriminantal index of these two sets of branches, 2B2; (iii.) for the joint discriminantal index of the 0' linear branches, O' (' - 1); (iv.) for the joint discriminantal index of the linear branches taken with the branches touching either PR or QR, 2 0' B. Hence D (R) = D (S) = 2 [B- A - B2] +2B2 + ' (0' - 1) + 4 o'B = 2(B2- A), as in the case when N is not a square. There is no change in the expressions for the order of V (q), and for the exponents of the factors q and 1 - q in V (q) (see Art. 6); and these expressions agree with the values which we have obtained for D (R) = D (S), and for D (P)= D (Q). For PR, touching the curve at R, counts as A - B tangents drawn from P; and hence the order of q in the discriminant ought to be, what in fact it is, 2(B- A)+(A -B) = 2B2-A -B. And again, PQ, considered as drawn from P, counts as A - O' tangents at P, and as B - ' tangents at Q. Thus, the number of intersections of C by the polar of P, which lie on the line PQ, is 2(2A2 - A-A + 0') +(A — ') + (B - 0')= 4A -2A2+B-A; and this number is, as it ought to be, the excess of 2 A (2A - 1) above the order of V (q); i.e., the excess of the whole number of intersections above the intersections lying on PQ. ARTS. 12-14. Formulce applicable to all values of N. 12. If, in the formula relating to the case when N is a square, we omit the terms containing the symbol 0' defined by the equation 0' =f () =f' (') =f (VN) we obtain the corresponding formulae for the case when N is not a square. We shall henceforward denote by 0' a number which is equal to zero when N is not a square, and which is equal to f(VN) when N is a square; and we shall treat the two cases simultaneously, except when it is necessary to call attention to the difference between them. 13. The discriminantal class of the superlinear branches.-In the paper on the Higher Singularities of Plane Curves* (Arts. 12 and 13), it has been shown * Proceedings of the Society, vol. vi. p. 153 [vol. ii. p. 112]. Art. 14.] MODULAR EQUATIONS AND CURVES. 259 that, if d and t are the order and class of a superlinear branch, D and T its discriminantal order and class, we have the equation T- D t2 d. And again, that if there be a second superlinear branch of the order d' and class t' touching the first, and if we represent by T and D the joint discriminantal indices (of order and class) appertaining to the two branches, we have the equation T- D =2 (tt'- dd'). Combining these two results, we obtain the theorem'If any number of branches touch one another at the same point, the difference between the discriminantal order and class of the singularity is equal to the difference between the squares of its order and of its class.' Employing a notation explained in Art. 14, we apply this theorem to determine the discriminantal class of the branches (PPQ), (PQQ), (PRR), (QBR), (PSS), (QSS). We thus find T (PPQ) - D (PPQ) = T (PQQ) - D (PQQ) -(B-20')2- (A -B)2...... (31) T(PRB) - D (PRR) = T(QRR) - D (QRB) = T (PSS) - D (PSS) = T (QSS) - D (QSS) = (A - B)2 -(B- );......... (32) so that T(PPQ) = T(PQQ) = B2- B - A - O'B + 0' + '2 = B2- B-A = D (P1PR) = D (QBR) = D (PSS) = D (QSS), and T(PBR) = T(QBR) = T(PSS) = T(QSS) =2A2-A,-A+e0 =2A2- - A = D (PQQ) = D (PPQ). 14. Summary of the results.-For convenience of reference, we exhibit the preceding results in a tabular form. Characteristics and Singularities of the Modular Curve C. I. Explanation of the symbols: (1) The order of the transformation is.the uneven number N. (2) g and g' are conjugate divisors of N; h2 is the greatest square dividing N. L1 2 260 ON THE SINGULARITIES OF THE [Art. 14. (3) X is the greatest common divisor of g and g'. (4) f (r) is the number of numbers not surpassing n and prime to it. (5) f'(g) and f' (g') are defined by the equation f'(g) _ f()! _ f/(gi') g t! g (6) 2.=I f(I), A + B f'(g), A2+ B2= (A+B)2. In these equations the summations I extend to all divisors g of N; A comprehends all the terms f'(g) in which g> VN, and also, if N= 02, the term 0O'= f(0); A2 comprehends all the terms of If'(g) x Zf'(g2), in which g1 g2 > N, and one-half of every term in which g, g2 = N, g, and g2 denoting any two divisors of N, the same or different. The definitions of B and B2 follow from those of A and A,. (7) m, n, K, I, D, T, H denote respectively the order, the class, the cuspidal index, the inflexional index, the discriminantal order, the discriminantal class, and the deficiency of the curve. (8) The symbol (XXY) or (YXX) denotes a branch, or an aggregate of branches, touching the line XY at the point X. (9) The symbols O(XXY), C(XXY), K(XXY), I(XXY), D(XXY), T(XXY) denote the order, class, cuspidal index, inflexional index, discriminantal order, discriminantal class of the branches (XXY). The symbols 0 (X), K(X), D (X), C (X Y) I(X Y), T(X Y) are to be similarly interpreted with regard to the branches which pass through a given point X or touch a given line XY. Lastly, the symbols D (XX Y, XXZ) and T(XXY, XYY) denote respectively twice the number of points common to the branches (XXY), (XXZ), and twice the number of tangents common to the branches (XXY), (XYY). II. Characteristics of the Curve.* m=2A, n=3A-B- O', H= = (A+B)-3v+1, K=2(A+B)-6v+O', I=5A-B-6v-20', I-K=3A -3B-30', D=4A2- 5A+B+O', T= (3A -B-O')2-5A B+ 0' T- D = (3A - B- 0)2 - 4A2. * Several of the formule which follow may be more simply expressed by using the symbols A, B A2, B2 of Art. 11, and by writing v = v - 0/. Art. 14.] MODULAR EQUATIONS AND CURVES. 261 III. Characteristics of the Special Singularities. (i.) Characteristics of (PPQ) and (PQQ). O(PPQ)=A-B; C(PPQ)=B- O', K(PPQ)= A - B-v+- ', I(PPQ) = B- v, D(PPQ) = 2A2 - A,- A + ', T(PPQ) = B -B2 -A - 'B +0o' + '2. The number of distinct branches is v- 0' They are all superlinear; viz. corresponding to every divisor g of N, which is less than VN, there are in (PPQ) f(,) superlinear branches, each of the order -g, and of the class g; (PQQ) is of the same type as (PPQ). (ii.) Characteristics of (PRR), (QRR), (PSS), (QSS). All these singularities are of the same type. O(PRR) =B - '; C(PRR)= A-B, K (PRR)=B-v, I (PRR) = A - B - v + ', D (PRR) = B - B2 - A - O'B + ' + 0'2, T (PRR)= 2A2 - A - A + 20'. The number of distinct branches in (PRR) is v- 0'; of these, h- ' are linear (Art. 10, iii.); the characteristics of (PRR) and (PPQ) are reciprocal; viz. corresponding to any divisor g of N, which is less than VN, there are in (PRR) f(n) branches, each of the order g and of the class - (iii.) Characteristics of (PQ). C(PQ)=2C(PPQ)=2B-'; I(PQ) = 21 (PPQ) = 2B- 2v, T(PQ) = T(PPQ) + T (PQQ) + T(PPQ, PQQ) = 2 T(PPQ)+2 (B- 2 o')2 =2B- 2A - 40'B+ ' (0' + 1). (iv.) Characteristics of (R) and (S). These are the same for the two points. (1) 0 (R) = (PRR)+ 0 (QRR) + (0) -2B. (2) K(R) = K(PRR) + K(QRR) =2B-2 v. 262 ON THE SINGULARITIES OF THE [Art. 14. (3) D (R) = DP ( R) + D (QRR) + D (PRR, QRR) + D (o) + D (o, PRR) + D (O, QRR) =2[B2-B2 -A- O'B +10'+ 0/2]+2( 0 ( 1 o' + 'e ( - )+ 4 '(B - ') =2 (B2-A). The symbol (0) is used to represent the O' linear branches which pass through R, having tangents distinct from one another and from PR, QR. (v.) Tangents to the Curve from PQRS. (1) PQ, considered as drawn from P, counts as A - 0' tangents at P, and as B- -0' tangents at Q; PR counts as A - B tangents at R; thus PQ, PR, PS count as 3 A - B - ' = n tangents drawn from P. (2) The tangents to the branches (PRR), (QRR), and (0) count as 2 (A - 0') + 2 0' = 2 A + 0' tangents drawn from R. Thus, there are A - B - 2 0' other tangents which can be drawn to the curve from R.* (vi.) Intersections with the Curve of the sides of the quadrangle PQRS. (1) PQ meets the curve in A- I 0' points at P, and in as many at Q; and in 0' non-singular points distinct from either P or Q. (2) PR meets the curve in A - B points at P; at R it meets the branches (PRR) in A - 0' points; the branches (QRR) in B -1 O' points; the branches (0) in 0' points; in all in 2A points. The same statements hold, mutatis mutandis, for the lines QR, PS, QS. (3) RS meets the curve 2 (B - -1 0')+ 0' = 2 B points at R, and in as many at S; and also in 2 (A - 2B) other points.t IV. Residual singularities of the Curve. Designating by K1, Ix, D1, T1 the parts of the indices K, I, D, T which arise from the singularities connected with the points and lines of the quadrangle * If X,( 0)=- represent the equation of the multiplier, which is of the order A B in i, and of the order I (A - B) in q, the values of q appertaining to the points of contact of these tangents are determined by the equations X(V' q )=, X (-Vn, q 1)=0. WhenN='2, thefirst of these equations has 0' roots equal to zero, and 0' infinite roots; both these sets of roots are to be rejected. t At each of these points we have p=q. The equation F(p, p, 1)=0 is divisible by [p (p - 1)]2B; the remaining roots, which are 2A - 4B in number, give the intersections of the curve by RS at points other than R and S. These roots may be determined by the method (due to M. Kronecker) described in the Report on the Theory of Numbers, Arts. 131-133 [vol. i. p. 325]. Art. 15.] MODULAR EQUATIONS AND CURVES. 263 PQRS, we find, from the preceding formulae, K1=2 (A +B)-66 v+0', 1=4A -2B- 6v+20', D1=4A2 +4B2-2A2-6A+0', T, = 8A2+2B2-4A2-6A - 40'B+30'+ 0'2; and for the residual singularities we have K2=0, I=A+B-40', D2=2A2- 4B2+A+B, T = 4A2+4B2+A2- 5B+ 60'(B - A)+A+B-20'. ARTS. 15, 16.-Case when N is a square. The Linear Branches (0). 15. The developments appertaining to the 0' linear branches U, which intersect PQ at points other than P and Q, are 1 eiu ei(1 + eiu) ei(l1 + eiu)(21 + 11e /u) -- p = + 2^ + ---- +*...). (33) p q 2q 2 64q2 2/11 where u = 210 r, 21+ 1 being any term of a system of residues prime to 20. We hence obtain the normal developments 2 r + e (a - 2?) = 5 (1 - e2i)2 +... (34) so that the tangents of the 0' branches are the lines - 27+,eiu (a - 27)=0, which meet one another in the point a = = - 1; i.e. in the point 0 in which RS intersects ab. The developments appertaining to the 0' linear branches at R are p =iv q + ei (1-ei) +,.. (35) where v = -h r, h being any term of a system of residues prime to 0; so that the tangents are a- ev 3 = 0, none of the branches being inflected at R. Similarly the developments appertaining to the 0' linear branches at S are p-l= eiv(q- 1) + eiV(1 - ei) (q- 1)2+.... (36) and the tangents are a - 7 = ei (/3- 7), there being no inflexion. 264 ON THE SINGULARITIES OF THE [Art. 16. The two sets of tangents at R and S meet PQ in the same points in which it is intersected by the linear branches U; for, if 21+ 1+2h=(2k+1) 0, we have u + v = (2k + 1) 7r, whence e" = - ei. 16. The developments (33-36) may be obtained as follows, with the help of formulae established in the Report on the Theory of Numbers already cited. If = 1 + -, where C- is positive and increases without limit, q = ()= 1 - -8(i) increases without limit; and the limit of q- 16 e1 is unity. The corresponding values of p are comprised in the formula g( +) +2k) 9 where g and g' are conjugate divisors of N, and g, g', k have no common divisor. But this expression may be exhibited in a form from which the dimensions of p (Q), as compared with p (w), may be inferred; viz., we have (Report, loc. cit., p. 350) (9' (i ) ~ a +d2k 2 9 d where d' is the greatest common divisor of g' + 2k and g, d = and 21 + 1 is determined by a certain congruence for the modulus d. In order that the development of 1, in a series proceeding by powers of, should correspond to a 1 1 branch intersecting PQ elsewhere than at P or Q, - and - must be of the same dimensions. But ' P Lim.p e = 1, Lim. q s- eg= 1; hence d'= d, or N is necessarily a square, and d = d'= 0. Since 6 = c' is the 0 greatest common divisor of g'+2k and g, let g= A 0, ' =; then 0 divides 0 0 +2 k; i.e., - divides g, g', and 2k, which are relatively prime. Hence X = 0, g = 02= N, g' = 1. Now there are just 0' values of 2 k for which 0 is the greatest common divisor of 02 and 1 + 2k; viz., if 2u + 1 be any number less than 2 0 and prime to 0, the 0' values of 2k are included in the formula 2k = (2 A + 1) 0 -1; Art. 17.] MODULAR EQUATIONS AND CURVES. 265 and it will be found that the congruence determining 21+1 is (2x +1) (21+1) = -1, mod 0. Hence we have, for the 0' values of 2k which we are considering, (i+1+2k -s (c+ \8 (i+ Expanding the values of q 1 - (8 (a), ( r+) by means of the formula s (a) = 16 ei(1r - 8ei' + 44e2i -...),..... (37) which arises from the expansion of (3); and equating the coefficients of like powers of e-"7 in the series 1 A B C - + +... p q q q3 we obtain the developments (33). Similarly the developments (35), which appertain to the linear branches at R, may be obtained by substituting the expansions of p=8 (ic +T) and q = 8 (ir) in the assumed series p= Aq+Bq2+.... ARTS. 17-19. The Six Modular Curves. 17. If we represent by e (x) any one of the six anharmonic functions 1 1 x x-l1 x, -x, ' 1-X X-1' X (38) the modular equation (1) is unchanged by the simultaneous substitution of e (p) for p and e (q) for q. Hence, if e (x), E2 (x) denote any two, the same or different, of the functions (38), the thirty-six substitutions F[e1(p), 62(q), 1] -....... (39) give only six different equations. As representatives of these, we take the following: (i.) F (I-p, q, 1) =0, (ii.) F(p,, I)=, VOL. II. M m 266 ON THE SINGULARITIES OF THE [Art. 18. (iii.) F(-, _ -, l)=0, (iv.) F(p, q, 1)=0, (v.) F(1-p,, )=o0, (vi.) F(p, )=o. The equation F(p, q, 1)= 0 is symmetric with regard to p and q; and it will be found that the equations (i.), (ii.), (iii.) possess the same property; thus, for example, the equations F(p,, 1)=0, and F (q,, 1)=0, are the same, because F(x, y, 1)=F(y, x, l)=(xy)+BF(, 1). The fifth and sixth equations, on the other hand, are changed, each into the other, by the interchange of p and q. 18. Denoting by X and Y rectangular Cartesian coordinates, and writing in the equations (i.),..., (vi.), +.(40) q=- 2 +X+iY ) we obtain the equations of six curves, which, in the Memoire 'Sur les Equations Modulaires,' * we have called the first, second, third, fourth, fifth, and sixth modular curves. The equations of the first four of these curves are real, as appears from the symmetry of the equations (i.)-(iv.) with regard to p and q; the equations of the fifth and sixth curves are imaginary and conjugate to one another. The first and fourth curves are each of them symmetric with regard to both axes; the fourth curve is its own inverse (anallagmatic) with regard to each of the two real circles (X~ )2 2 = 1 and the first curve with regard to each of the two imaginary circles X2+(Y+ ~ i)2= -1. The second and third curves are symmetric with regard to the axis of X, and symmetric to one another with regard to the axis of Y; the fifth and sixth (imaginary) curves are symmetric with regard to the axis of Y, and symmetric * LVol. ii. p. 224.] Art. 19.] MODULAR EQUATIONS AND CURVES. 267 to one another with regard to the axis of X. The second and third curves are the inverses of the first, with regard to the circles (X-)2 + Y2 1, (X+y)2+ Y2:1, respectively; similarly the two imaginary curves are the inverses of the fourth curve with regard to the two imaginary circles X2+(Y+ i)2=-1. Lastly, the substitution X = i Y', Y= - iX' changes the first curve into the fourth, the second into the fifth, the third into the sixth, and vice versd. These assertions are the geometrical equivalents of the properties of the modular equation stated in Art. 17; it will suffice to verify one of them. The equation of the first modular curve is F( —X+iY, +X+iY, 1)=0; its inverse with regard to the circle (X + _)2 + 2 =1 is obtained by writing l —X'-iY" 1 {-X-iY= x - X 'iY"/ 1 so that +X+iy= - The equation of the inverse curve is therefore F -X'-Y1 1- _X, iY 1)=0; and this is identical with the equation F +X'-iY, 1-X+tY' 1)=0; i.e. with the equation of the second modular curve, because F( -- 1-, I)=0 is identical with F (x, -, 1) = 0. 19. The equations of the first and fourth modular curves are included in the general equation F(a, I, ) =0; M m 2 268 ON THE SINGULARITIES OF THE [Art. 19. viz. to obtain the first curve, we write a=- X+iY, 1=-i+X+iY, 7=1; and, to obtain the fourth curve, we write a=+X-it, 3-= +X+i y = =. Thus the theory of the singularities of these two curves is implicitly contained in the preceding discussion of the singularities of C. In both curves the points P, Q are the cyclic points, and (ab, RS) or 0 is the origin: in the first curve ab and RS are the axes of X and Y; a, b being the points (~ 1 0), and R, S the points (0, + i): c is the point at an infinite distance on the axis of Y; in the fourth curve R, S are the points (+, 0), and a, b the points (0, + ), c being the point at an infinite distance on the axis of X. Both equations (as has been already said) are real; and it follows, from the theory explained in the Memoire cited, that both of them represent real curves, except when NX 3, mod 4; in which case the fourth curve reduces itself to the pair of conjugate points (+, 0). When N is not a square, both curves are completely and parabolically cyclic, having at each cyclic point v branches, of the aggregate order A - B and class B, touching the line at an infinite distance. When N is a square, each of the two curves has 0' real infinite branches. The fourth curve has also 0' real branches passing through each of the points ( + I, 0); (these two points always belong to the curve, though, when N is not a square, only as isolated points:) the tangents to the 0' branches are parallel to the asymptotes of the curve. Similarly the first modular curve acquires 0' linear, but imaginary, branches at each of the points (0, ~- i-); the tangents to these branches being imaginary lines parallel to the real asymptotes. The equations of the asymptotes of the first and fourth curves are respectively Y Cos u - X sin u =,. (41) Y sin u + Xcos u = 0;j 2+1~-2 u denoting 1 r, as in Art. 15. And it may be inferred from the developments (33) and (33), given in that Article, that the rectangular hyperbola (Y cos ut- X sin -u) (Y sin u + X cos u) = 5 sint osculates at an infinite distance the branches asymptotic to the two lines (41). Art. 19.] MODULAR EQUATIONS AND CURVES. 269 2h Lastly, if v= - r, as in Art. 15, the tangents of the fourth curve at the points (+, 0) are r cos —v + (X+ ~) sin -v = 0; the tangents of the first curve at the imaginary points (0, + 2i) are (Y+ I i) sin I v- Y cos = O: and these tangents are parallel to the asymptotes of the curves to which they respectively appertain; because, if 21+1+ 2h=(2k+ 1)0, tan - = cot I v. 2 2 The points (+-, 0) and (0, iz) are foci, and indeed the only foci, of both curves: of these, the points (0, ~ li) lie on the first curve, and the two real points (~ -, 0) are its two foci (properly so called); the axis of Y being the only corresponding cyclic axis, or directrix. The points ( +~, 0) belong to the fourth curve (only as isolated points, when N is not a square), and this curve has, properly speaking, only the pair of imaginary foci (0, ~ -i). 19. The second and third modular curves may be regarded as derived from the equation (1), by the substitution a 7 2 - j:e, q=d, =1+_(X+iY), 7=1. X and Y being rectangular Cartesian coordinates, and the upper signs relating to the second curve, the lower to the third. Thus the theory of each of these curves is comprehended in that of the curve C', of which the trilinear equation is (/)B-a X F(af, 72, /37) = 0,(43) or (a7)B- X Ft(a/3, 2, a ) =0. The singularities of C' may be examined by the method already employed in the case of C. Attending, for brevity, only to the case in which N is not divisible by any square, we write p =, q in the parametric developments of Art. 4, and we deduce, as follows, the normal developments of the singular branches of C'. 270 ON THE SINGULARITIES OF THE [Art. 19. (i.) From (6) we obtain.9 a-T 7 - (7 or, multiplying by = 1 +' a- x (7 )...... (44) which is itself a normal development, if g' > g, and gives rise, by reversion, to such a development, if g' < g. Hence C' has a singularity at S, having the same characteristics as the corresponding singularity of C. (ii.) From (5) we infer g9 a=x(;3) +.... a 9 Here, when p and q are small, a must be small compared with 7, and y compared with /3; i.e. the coordinates of the point (p = 0, q = 0) are a = 0, y = 0; and the normal development is g +g' i(^) 9+.9......... (45) (iii.) Similarly from the development (7) we deduce g9' a= (-)+.... a \y7 Thus the coordinates of the point (p = oo, q = o) are /3=0, = 0; and we find, after reversion and multiplication by 7, the normal development g'+g -=Q) +.*.......... (46) which is of the same type as (45). Thus the curve has at P and Q singularities of one and the same self-reciprocal type, not resembling the singularities which C has at the same points. The point R does not lie on C'. Art. 20.] MODULAR EQUATIONS AND CURVES. 271 ART. 20. Characteristics and Singularities of the Modular Curve C'. I. Characteristics of the Curve. m=2A+2B, n=3A+B, H= (A+B)-3v+1, K=2A+4B-6v, I=5A+B-6v, I-K= 3 (A -B), D=4(A +B)2-5A- 3B, T=(3A+ B)2-5A-3B, T-D=(3A+B)2-4(A+B)2 = (A - B) (5 A + 3B). It will be noticed (1) that these formulae do not contain 0', although the case when N is a square is included in them; (2) that, when N is not a square, I- K has the same value for C' as for C. II. Characteristics of the Special Singularities. (i.) Characteristics of (PPR) and (QQR). 0 (PPR)= A + B= C (PPR), K(PPR) = A+B - 2 v=I(PPR), D (PPR)= A2 +3B2- 2A - 2B = T(PPR). The number of distinct branches at each of the points P and Q is 2v; viz., corresponding to every divisor g of N, there are in (PPR), f(rn) branches of the order J and of the class g; of the 2 v branches, h are linear, and, in particular, when N is a square, O' of these are also non-inflexional. (ii.) Characteristics of (PSS), (QSS), and (S). These are the same for C' as for C (see Art. 15, III., (ii.) and (iv.)). When N is a square the equations of the tangents to the linear branches are a-7= ei (y- 3) (see Art. 15, equation (36)). (iii.) Tangents to the Curve from PQRS. (1) PR, considered as a tangent drawn from P, counts as 2 A + 2B tangents at P; and PS counts as A - B tangents; hence PR and PS are the only tangents from P. 272 ON THE SINGULARITIES OF THE [Art. 20. (2) RP, considered as a tangent drawn from R, counts as A + B tangents at P; and so does RQ at Q; thus there are A - B other tangents which can be drawn to C' from R. (3) Besides the tangents at S, there are A -B -20' other tangents which can be drawn to the curve from S (see Art. 14, III., (v.) (2)). (iv.) Intersections with the Curve of the sides of the quadrangle PQRS. (1) PQ meets the curve in A + B points at P, and in A+ B points at Q. Thus it never meets the curve again, and touches it nowhere. (2) PR and QR each meet the curve in 2A+2B points, touching it at P and Q respectively, and meeting it nowhere else. (3) PS and QS meet the curve in A + B points at S, and in A + B points at P and Q respectively; thus they never meet the curve again. (4) RS meets the curve in 2B points at S, and in 2A other points.* III. Residual Singularities of the Curve. Employing the notation of Art. 14, IV., we have K=2A+4B-6v, 1=4 A -6v+ ', D1= 2A2+ 8B2-6A-4B, T=4A2+6B2-6A-4B+ '. I2=A+B-0', D2=2A2-4B2+A +B, T2=2A2-2B2+3A2-3B2+ A + B-'. The indices K2, D2 have the same values for C and C', because these indices refer to singularities which do not lie, in either figure, upon the fundamental triangle of the quadric transformation by which the curves are changed into one another. The equality of the indices 12, when N is not a square, implies the theorem: 'Each of the first three modular curves has as many non-singular inflexional tangents as it has osculating circles, which pass through the point (), 0); or again, through the point (- 2, 0).' * The points of contact of the A -B tangents, (iii.) (2), and of the A -B- 20' tangents, (iii.) (3), and the 2A points of intersection, (iv.) (4), can be determined by methods similar to those indicated in the Notes on Art. 14. Art. 20.] MODULAR EQUATIONS AND CURVES. 273 Of the formulae contained in the preceding enumeration, we shall demonstrate only one; viz., the expression for D (PPR) or D (P). We have, as in Art. 10, (iii.), D (P) = Z (,) g +) [g) 1] +2 Sf() f(,l) g(g ~') where g is any divisor of N, and g' is the conjugate divisor of g; g1 is any divisor less than g, so that g > g'; the summations X2 and 2 extend to every value of g1 and g respectively. Hence we find D (P) = Zf'(g) [f'(g) +f'(g')] - 2 [f' (g) +f'(g)] +2 2lf' (gl) [f'(g)+f' ()].. (47) But we have, evidently, zf'(g)f'(g) + 2.ZZf'(g)f'(g) = (A + B)2=A2+B2; [f '(g) +f'(g)] = 2A +2B; and, observing that g'g < N, we also find f' (g)f' (g') + ef '(g1)f' (g') B2 Introducing these values into the equation (47), we obtain D(P)= A+ 3B2- 2A - 2B in accordance with the formula II. (i.) supra. VOL. II. N n XXXVII. NOTE ON A MODULAR EQUATION OF THE TRANSFORMATION OF THE THIRD ORDER. [Proceedings of the London Mathematical Society, vol. x. pp. 87-91. Read February 13, 1879.] I HAVE given elsewhere* the modular equation for the transformation of the third order between =zf (,2) k (1 - )2+ ) and y =f (X2 X ( X)3 2=f^ji)-= kl(i- 2)2 X4 (I _ X2)2; viz., F (x, y) = x (x + 27. 3. 53)3 + y (y + 27. 3. 53)3 - 216 X3 y3 + 211. 32.31 x2y2 (x + y) -22. 33. 9907xy (x2 + y2) + 2.3. 13.193. 6367 x2y + 28. 35. 53. 4471 xy (x + y) -215. 56 22973 xy = 0. The following is the process by which the coefficients were determined: It follows from the general theory of modular equations that F (x, y) is symmetrical with respect to x and y; and that it is of the order 4 in x and y separately, and of the order 6 in x and y jointly. Hence F (x, y) is of the form,, 3 xy3 + A4, 2 2 (X2 + y2)+ 4, y (x3 + y3) + (X4 + y4) + A3,2 2y2 (x + y) + A3,1 x (X2 + y2) + A3,0 (3 + y3) + A2,2 X2y2 + A2,1 xy (x + y) + A2,o (X2 + y2) + A,,l xy + Al,o (x + y) + Ao,o. Several of the coefficients may be conveniently found by employing the method of Sohncke; i.e. by substituting for x and y their expressions as series * 'Proceedings of the London Mathematical Society,' vol. ix. p. 243, note [vol. ii. p. 242]. NOTE ON A MODULAR EQUATION. 275 proceeding by powers of q, and equating the coefficients of the powers of q to zero. We have, by a known formula, k2 (1-k2) =24q lm' (1 + q2M)-24; calling this quantity z, we have x =- Z we also write Z=24q3 F=J1 (1+ q6m-)-24 so that (- Z)3 Substituting these values, and equating to zero the coefficients of q-28, q-26, and qp24, we find successively A4,2=O, A4,1= O, A3,3= -216. To determine A3,,, we observe that q-22 presents itself only in the terms 216 x3y3 + A,2 x2y3 Its coefficient in x2y3 is 2-40; its coefficient in x3y3 is 32 31 x 2-45; viz. this is the coefficient of q2 in 1 [ ( l+q)144- 9 (1 )120 9 2 2 22 L2 (1+2q)4 220Q (1 + Hence A,2 x 2-4 + 2-45. 32. 31. A3,3 = 0, or A3,2 = 211. 32.31. Again, to determine A3,1 we consider the coefficient of q-20; this power of q presents itself only in the terms y3 [A3, X3 +A3,2 x2+A3, x]. It will be found that in the development of y3 there is no power of q intermediate between q-~1 and q-12; hence, the coefficient of q-2 in A3, x3 x+ A3,2 x2+A3,1 x, or the coefficient of q4 in ri, [2 T 32 +214 q(l +q) 196 (210 (l+ )72 27 ] + A3,2 [ q2 (1+q) - q(l+q )7+ - ++ 4 q4 is equal to zero. Substituting the values of A3,3, A3,, and reducing, we find A3, = - 22. 33. 9907. n 2 276 NOTE ON A MODULAR EQUATION OF THE All the coefficients may successively be determined by this method, but the work becomes very laborious for the later coefficients. They may be more easily obtained by the consideration that, if ~, r are any two corresponding values of k2 and X2, f (/) and f () are corresponding values of x and y. Availing ourselves of this principle, we find (i.) F(0, y) =y (y + 27.. 53)3; 3 2 33. 113.233 2 (ii.) F(,y) =(y2- x 133283y- *124 (iii.) F(y, y)= -2'6y (y- 353) (- 5)(y+2)2 From (1) we infer that F (x, y) is of the form y (y + 27. 3. 53)3 + x ( + 27. 3.5 3)3 21633 + 211. 32.31 x2y2 (x + y) 22. 33 9907 xy (X2 + y2) (A) + A2,2 xy2 + A2,1 xy (x + y) + A1,1 xy = 0. From (iii.) we find Al1 = 21,. 56, 22973, A2,1 =28. 3.53.4471, A2,2 = 211. 1262587 + 23. 33 9907 - 2 =2.34. 15974803 =2.34.13.193. 6367. This completes the determination of the coefficients; a verification is supplied by the equation (ii.); and it only remains to show how the equations (i.), (ii.), (iii.) are inferred from the modular equation between = k2 and 7 = X2; viz.: I (, r) = (52 + 6 5 + n2)2 _ 16n (4 5r - 3 5 - 3s + 4)2 = 0. (i.) Let p denote either root of the equation p2 - p + 1 = 0; if k2 = p, we have x =f(p)= 0; we also find (p, ) = (r - p) (3 - [128 - 253p] 2 _ [128 + 253p2] + 1). If (w, w2, 3o are the roots of the cubic factor, the four roots of F (0, y) are f(p)=, f(l), f(2), f(,(). But the cubic factor is one of the two conjugate factors of the expression (112 - + 1)3 + 27. 3. 53. 2 (-_ -1)2. Hence f (1) =f (W2) =f (3) = - 27. 3.53, and F(0, y)=y(y +27.. 53)3. TRANSFORMATION OF THE THIRD ORDER. 277 (ii.) Let k2 =, so thatf (k2) = 7; we find v (I-, y) = (-I 2)2- 482 ( - 12) + 16 If,, 2, 3,5 ~ 4are the roots of 1 (2, i), the four roots of F (, y) are f/(i), f("2), f(i3), f (4); i. e. if zI, 2 are the roots of Z2 481 z + I these four roots are zj2 (1 - z1)3, z2 (1 - 2)3, each taken twice. Hence F(2, y) =[y - z2- (1 - z)3]2 [y z-2 (1 - Z2)3]2 and by the ordinary methods we find 32 33 113 233 [y - (1- )3] [y (1 - ]2 =y2- x 133283 y - (iii.) Since the only roots of the equation f('?) =f(0) 1 1 _1 are are~, 1-,, (~, -1-',- ' F -1' the roots of the equation F (y, y) = 0 are all of the form f(), where 0 is a root of one of the six equations: + (0, 0) 2= 2(0- 1)2 (o- + )=0,............. (1) (0, 1-0) (16 0-16 + 1) (4 2-4 0-1)20,....... (2) 04(0, 1)=(0_1)2(02-60+1)2(02+140+1)=0,........ (3) (1-0)4 ~ (0, 1 ) = (02 - 0 + 1) X [(1 -0 +2)3 + 128 02 (1 0)2] =;.. (4) 4 (, 0 02 — I) ( + ) [(02 _- + l) + 1t2802 (0 1 -)2] =o,.... (5) (0- 1)4 (0, )=2 (02 -16+16) (2+4 4)2=...... (6) These equations are of order 8, the first and second having each two infinite roots. The 48 roots give, in all, five distinct values for f(0) according to the following scheme: A. f(0)= oo; 12 roots, viz.: 0= co, 2 roots in (1) and in (2); = 0, 2 roots in (1) and in (6); 0=1, 2 roots in (1) and in (3). 278 NOTE ON A MODULAR EQUATION. B. f(0)= 0; 6 roots, viz.: 02-0+1 = 0, in (1), (4), and (5). 33. 53 C. f(0)= 24; 6 roots, viz.: 1602-160+1 =0, in (2); 02+140+1 =0, in (3); 02-160+16=0, in (6). D. f(0) = - 128; 12 roots, viz.: (02 _ + 1)3 +128 02 (0 - 1)2 = 0, in (4) and (5). 53 E. f(0) =;2 12 roots, viz.: (402 -40- 1)2=0, in (2); (02-60+ 1)2=0, in (3); (02+40-4)2=0, in (6). 33. 53 53 Hence the roots of F(y, y)=0 are 0, 2, each once; o, -27, 2, each twice; i.e. FPyy)= - 216 ( 33 53 ( 53 2 )2 ) -2y y 24) (y- 2(y+27). The multiplicity of the roots of F (y, y)= 0 may be otherwise, and more simply, determined as follows. The form of F (x, y) (see the equation (A), supra) shows that F(y, y) has one root equal to zero and two roots equal to infinity); the multiplicities of the finite roots are determined by the equation [33 53 53 - 216 24 +2 x -2 x27J=2 3,2=212.32.31. XXXVIII, NOTE ON THE FORMULA FOR THE MULTIPLICATION OF FOUR THETA FUNCTIONS. [Proceedings of the London Mathematical Society, vol. x. pp. 91-100.* Read February 13, 1879.] THE normal formula for the multiplication of four Theta Functions is ('Proceedings of the London Mathematical Society,' vol. I., part viii., p. 4 t) 2 i,,, l ( ) ~,2, ( 8) N o31, 3, (%3) 04, Y, (x '= Y:l^ a f -e, a_ - (s - x j) + (- 1)',1, Aj +1 - iC) O (1) +( l) Rn a.j +1, _ (-) +(-):. In this formula we have O, (x) =,-+ (- 1)w2 4'q(m+"e26i(2+) = _- ( l)l'ei(2m+t").e ( + i(; (2) 2s = x+x++ x2+ x 4, 2(0 =/ 1 + 2+ j3 +/4......... (3) 2-'= 4A' + 1 +A +I4;, q=eirwc being a constant of which the analytical modulus is inferior to unity, and the indices P, (F' being integral numbers, which render the sums 2a-, 2 a' even. I. The Eleven Cases of the Formula. Since o0f+2, ^A () =- x=(-1)y', t () (4) o, a, +2 (x) = 0,/ (W), we need only attribute the values 1, 0 to the indices,, u'. We may also * [This paper is referred to on p. 75 of vol. x., and appears in the Contents of the volume under the title, 'On the formula for four Abelian functions answering to the formula for four Theta functions.'] t [Vol. i. p. 443.] 280 NOTE ON THE FORMULA FOR THE permute the four arguments x1, x2, x, 4 in any way we please; thus the matrix,tl, 2 3 4 may have eleven different values, which are enumerated in the M1, 2, i3, P4 following table: TABLE I. A. -0, ' 0, mod 2: Cases i.-iv. 1111 1111 0000 0000 1 1 O O O O I I O O O O I 1111 0000 ' 1111 0000 B. a( O '-0, / mod 2: Cases v., vi. 1111 0000 1100' 1100' C. cr 1, -=0, mod 2: Cases vii., viii. 1100 1100 1111 ' 0000 D. a —1, o-1, mod 2: Cases ix.-xi. 1 1 00 1100 11 00 1100 ' 10101 0011 The formulae appertaining to these eleven cases are all different from one another; i.e., none of them can be derived from any other by a permutation of the four arguments x; we can, however, pass from any one of them to any other by means of the formula which connects any two different Theta functions; viz., 7r [V tv + (+ ) Using a notation with a single suffix, and writing 00,1 (x)=0 (x), 01, 1 ()=1 (x), 01, 0(x))= (x), o0, (x) = 3 (x), we may exhibit the eleven formulae as follows. TABLE II. Formrulefor the Multiplication of Four Theta Functions. A. (i.) 25ax3xaxa = 1X al X a3 X ax +3a X 3 X a3X a2 +ao X o X rO X ao -%3 X x3 X 3 X a3. (ii.) 2~xaxSi x S~= 2,x x.5x S 2x,x 2+,x x x + a3 X a3 X a3 X a3 - ao X Qo X oQ X 3o. MULTIPLICATION OF FOUR THETA FUNCTIONS. 281 (iii.) 2 ao x o x,0x 0= oX - X a, X 3, X, + a3 X a,3 X a3 X 3 + ~ x l, X x - X, X a2 x X a2. (iv.) 2 a x rX 33 X 3 X 33-3 X x- X >( X >+0 X o Xo + 2 X a2 X 2. x 42- al X l X al x al. B. (v.) 25 xax3 x x2= 2> X a,>x ax+, X +x2X - X X3 X x o X ao + 3o x o 3 x a3 x 2. (Vi.) 2 ao X aQ X 3 X x = a3 X p3 X o X o + x o 0 X o X oa3 X Jx3 - r X >4 X 'a X oj + a~ x >l X X 2* C. (vii.) 2 1X o x x r0o X = o x oX a1+ax +3 x >4 x 2X +, X x X alx X xl - >. x X x>3 x 2. (viii.) 2a2 x >a x 3 x a = X3 x X x 2 + a x 2 2 o x a1 x l + x S x S3 3- l x 1 l X o x oaO. D. -(x.) >4x x3= >X4 oX- >x4 +x x >, x3. (X.) 2aj~1X~a2Xa1Xa3= 3XalXa2X a1xa0X33 X alXa (xi.) 2 a, x 52 X.0 x o = ao X ao X 3> X 2- + >3 X,1 X a, -, x, x, x and the arguments s- x - For brevity, the arguments xz1, x2, x, x4, and the arguments s- x, s- x2, s- X3, s- X4 are omitted in the left and right-hand members respectively. It will be observed that of the 256 sets of values which may be attributed to the constituents of the matrix A1 2 [3 P4 Vl F2 F2 V4 192 are excluded by the condition that 2a- and 2cr' are even; of the remaining 64, 4 are represented by the formulae (i.)-(iv.), which are symmetrical with respect to all the arguments; 24 by the formula (x.), which is entirely unsymmetrical; and 6 by each of the formulae (v.)-(ix.) and (xi.), which are symmetrical with respect to the arguments taken in pairs. VOL. II. 0 0 282 NOTE ON THE FORMULA FOR THE II. Application of the Formula to the Abelian Functions. The Abelian functions are defined by the equations ( ) 2 ( )7 \ Ali(x) = e() x ---- 9/")( o (0) A1o(x)= e-) x ' 1-2q+2q4-2q9+2q6........ () (8) Al,/x) -2)2 < 2i The formule of Table II. give rise to a corresponding system of formule for the multiplication of four Abelian functions. To obtain this second system, we have only to express the Theta functions as Abelian functions, and to attend to the equations,, 3(o) - k'2 - 2 (= ) =r + 2+2(O ) (o).. (9) = 2o'+2q - 2K ' +3 ' 'TABLE III. Formulce for the Multiplication offour Abelian Functions. A. (i.) 2Al xAlAl x AlxA1 = Al, x Al, x Al x Al + A2 A12 x A l2 x A 12 + Alo x Alo x Alo x Alo - k,Al3 xAl3 xAl3 x A1. (ii.) 2Al1 x A2 x A x A1 = Al12x Al, x A12 x A12+ k''2 Al x Al x Al, x Al, 1 k'2 + Al, x Al x Al x A 3- - A x Al x A10 x Alo. MULTIPLICATION OF FOUR THETA FUNCTIONS. 283 (iii.) 2Alo x Alo x Alo x Al, = (iv.) 2A13 x Al, x Al, x Al, = Alo x Al x Alo xAlo + AlAo x Al3 x Al, x Al k2 + k2All x AlI x All x Al - k A12 x A l x Ai2 x Al2. Al3 x Al3 x Al3 x Al + + k2Al xAl2 xAl2 xAl - k'2 Alo x Alo x Alo x Alo k2k'" Allx Al x Al x Alx. B. Al x Al x Al, x Al, + Al, xAl, xAl2 x AL2 (v.) 2Al1 x Al1 x A, x A2 = (vi.) 2Alo x Alo x A x Al3 = + Al3 x Al3 x Alo x Alo - Al3 x Al3 x Alo x Alo k2 A12 x Al A l, x AAl - C. - Alo x Alo x Al x Al. + Alo x Alo x Al3 x Al k2 All x All x Al2 x A12. (vii.) 2All x Al x Alo x Al, = 1 Alo x Al x All x AI, Al, x Al3 x Al, x Al, 1 1 k~~~/2 (viii.) 2 A,2 x Al, x Al, x Al, = + Al, x Al x Alo x Alo + Al3 x Al3 x Al x Al - + A2 x Al2 x Al3 x Al3 + D. 1 k'2 k'2 Al2 x Al x Al3 x Al3. Alo x Alo x Al, x Al Ali x All x Alo x Al. (ix.) 2 Al1 x Al1 x Al x Al, = (x.) 2Al x Al x Alo x A3 = (xi.) 2 A1 x Ala x Alo x Alo = Al3 xAl3 x AlAl x Al + Al2 x Al2 x Alo x Al, Al3 x Alo x Al x Al + Al2 x Al1 x Al3 x Alo Alo x Alo x Al2 x Al + Al1 x Al, x Al, x Al3 - Alo x Alo x Al2 x Al + Alx Al x Al3x Al3. + Alo x A1 x Ali x Al - AlI x A1 x Alo x Al3. - Al3x Al3 x Ali x Ali + A2 x Al2 x Alo x Al. III. Case when the Sum of the Four Arguments is zero. Putting s = xI + x + x3 + x = 0, and attending to the equation 0 - = (- 1),' 1....... (10 we find that in each of the formulae (i.)-(xi.), one of the terms on the right-hand side cancels one of the two equal terms on the left, and that the four formulae A., and the formulae of the three pairs B., C., D., (ix.) and (xi.), become respectively coincident; the formula D. (x.) remains sui generis. Introducing the elliptic functions Al, (x) Al, (x) Al, (x) sn x= cnX — = dn — 2 Al (x)' A (x)' Al (x) 002 284 NOTE ON THE FORMULA FOR THE into the five formulae thus obtained, we arrive at the following system: (i.) k2 k'2 H. snx- k2 II. cnx + I. dn x- k'2 = 0, (ii.) k2 sn x, sn x2 en x cn x4 - k2 en xl en x, sn x3 sn x4 -dn x, dn x2 + dn x3 dn x4 = 0, (iii.) k'2 sn x1 sn x2- k'2 sn x sn x4 + dn xl dn x2 en x3 en x4 - en xl en x dn x3 dn x4 = 0, (iv.) sn xl sn x2 dn x3 dn x4- dn x, dn x2 sn 3 xsn x4 + en X3 en 4 - en x1 cn x2 = 0, (v.) sn x en x. dn x4 + dn x ecn xc sn x4 + dn x2 sn x3 cn x4 + en xl sn x2 dn x3 = 0. The first of these, which alone is symmetrical with respect to the four arguments, is the formula given by Professor Cayley ('Proceedings,' p. 43). The formulae (ii.), (iii.), (iv.) are symmetrical with respect to the two pairs x x2, x3 x4, and with respect to the two arguments of each pair; thus each of these formulae represents a set of three. Lastly, the formula (v.) remains unchanged when any two of the arguments are interchanged, provided that the other two are interchanged at the same time; i.e. there are six formulae of this type. As a verification of these formula in a particular case, let x4 = 0; we find Ddl d2 - k2 Ccqc2 = k', D + k2 Css2 = dld2. DCc2- CdId2= k'2ss,....... (11) C+ Ds1s = CC2, Dcls2 -Sd = - s1c, (where we have written, for brevity, s, = sn x1, c, = en xl, di = dn x; s2 c d and SCOD having similar meanings with respect to the arguments x2 and x1 + x2 = - xI); and these equations are easily shown to be true by means of the formulae for the addition of elliptic functions. IV. Formula for the Multiplication of Four Multiple Theta Functions. We define a double Theta function by the equation 0(AU /A'; v I;v y) _ -f=oc a- /(_)mrz/ +nv x A+(2m + )2 xB (2m+/+)(2n+v) x C+(2n+V)2' X e(2m+p)ix+(2n+v)iy >m=_+ooEn=+ (_ l)mi+nv' e-4 (2m+l2n +v) X ei(2in+)x+iX(n+I (2n+)y where ( is the quadratic form (a, b, c); ia a, ib7r, i c being the hyperbolic MULTIPLICATION OF FOUR THETA FUNCTIONS. 285 logarithms of A, B, C; and where the condition of convergence is that the real part of iq must be a negative form of negative determinant. Considering four different Theta functions, and writing 2s =x+x2+x3+x4, 2t — 1+Y2+Y3+Y4,A 2f ' f ' i ' ' i f i / i / I 2 a==l+ 2+ 3+ 11 4 T -= Y+ V2+ 1'3 + 14 * (13) 2 =+f2+ 3+4, 27 = t2+v3+v4, we have the formula 411. (,x,; jl vjI; Xj, Yj) - '+aa'++- (14) x H. 0 (a- -, a, '- + a'; r- Ij + /3, 7 — V + x; s -j, t yj) the signs of multiplication II referring to the four values 1, 2, 3, 4 of the index j, and the sign of summation Z in the right-hand member referring to the symbols a, a', 3, /', each of which is to have the values 0, 1; so that the right-hand member contains sixteen terms. The formula may be demonstrated in the same manner as the corresponding formula for the single Theta functions (see the note already cited, 'Proceedings,' vol. I., part viii., Art. 2).* If, instead of a double Theta function, we consider a multiple Theta function containing X arguments x, y,..., and depending on X pairs of indices,uA, vi, pp,..., we have the equation of definition 0 q \ I. __n=+oX _-n-=+oo -- 'r=+oo ol,,r I; [)1: =oh1 X (15) h erm+ i' a q+rpa+... fr(2m+c 2n+n, 2r+p,) X i(2m+ia)x+i(2n+v)y+i(2r+p)z+... where p is a quadratic form, containing X indeterminates, and such that the real part of io is definite and negative; and we obtain a formula similar to (14); viz. taking four Theta functions, such that the sums of the homologous indices are all even, we have 2A.0 ( | j, M; | X | )= (-l)laa'+aal X IT. 0( | -j + a,- C- f + |; | S-aXj |, (16) where the symbols l, I - j + a, A' - + aI are placed by abbreviation for the X pairs Pji,; A Ij,; rj, and o - Jc+a, a -.+ (a; 'r - j +, '-j+ i;..., respectively; and the symbol j a-' a a'I represents the sum of the X terms (ac'a + aa') + (P' +/) +.... * [Vol. i. p. 444.] 286 FORMULA FOR THE MULTIPLICATION OF FOUR THETA FUNCTIONS. Each of the 2X indices aa', I3 3',.. is to receive successively the values 0, 1; and the sign of summation 2 extends to every combination of these values, so that the right-hand member consists of 22" products, each affected with its proper sign, of four Theta functions. The multiple Theta function (15) satisfies the equations 0(tk+ 2,.u';...;...)= I.; x )17 0(,,'+2; 2; =o,/;...;...); ' * ' 0(,; -...; -...)=(- 1) 0(,;...; x...);.. (18) 0 (A,,'; v, '/; p, p';... x, y, z,...) = eirr(lvp,..) x ei(+vyv+P+') xO 0(0, 0; 0, 0; 0, 0;...xl, y1, Z1,...), x1=X+ 7r/'+-7rd Y, Y+l=y+ 7r +7v dv* d+ z = z + l p'+^ 4 7ra d,....... j which correspond to the equations (4), (10), (5), relating to a Theta function of one argument. The equations (17) show that we need only attribute to the indices the values 0, 1; thus, the formula (16) may be regarded as comprehending 11P different equations, if we regard two equations as identical which may be deduced from one another by permutation of the four indices j; or, as compre11. 12.13...10+X bending 1-2. 3 -.- X different equations, if we also regard as identical two equations which may be deduced from one another by a simultaneous permutation of the X arguments, and corresponding pairs of indices, in each of the Theta functions; viz. counting the different equations on this principle, we have as many of them as there are X-combinations, with repetition, of the eleven matrices of Table 1. But we can always pass from any one of the equations (16) to any other by means of the formula (19), which may be employed to express any one of the 4x Theta functions as a product of any other by an exponential factor; although (for brevity) we have supposed that the indices of one of the Theta functions compared in that formula are all equal to zero. XXXIX. DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Collectanea Mathematica (in memoriam Dominici Chelini), Milan 1881, pp. 117-143. The paper is dated 1879.] 1. PROPOSITA aequatione P1P2 - R = 1 olim demonstravimus fore"' Pi = (q2, q2,..., qi, q, **., q2, q1), P2=(q2, q3,... q,, qi, *.., q3, q2), R= (ql, q,2...,q qi, ~ *...* q3, q2), = (q2, q3, *.., q, qi,..., q2, qi), * 'Journal fur die reine und angewandte Mathematik,' vol. 1. p. 91 [No. III. vol. i. p. 33]. Verum hoc theorema ante nos invenerat vir clarissimus Serret. Numeratorem fractionis continume 1 q 2 + - 1 3 +. - - - q3+.. per formulam (ql, q2, q3,..., q,) denotamus; cujus proprietates a forma determinantali ql, -1, 0,............. 0 1, q2,- 1, 0,............0 0, 0,1 q1,................0 0, 0, 0, O,..., 1, qn, - 1 0, 0, 0, 0,..,0, 1, q, pendere in commentatiuncula supra memorata observavimus. Ipsam autem fractionem continuam ita significamus ut quotientium series uncinis quadratis includatur. Itaque habetur aequatio i'~1, q~2, 9''.v, jn.I = [, q25, q3...I q,] =(, q2, -., q) Cceterum in fractione periodica utimur punctis superpositis ad distinguendos primos atque ultimos periodbrum quotientes. Sic exempli gratia erit [ac b, c,..., el, p, 2, qn] fractio continua constans e quotientibus heteroclitis a, b, c,..., et periodo q, qp2..*? infinities repetita. 288 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 2. atque hinc haberi partitiones numerorum P1 et P2 in summas duorum quadratorurn. Eadem fere methodo sequationes PP2- 2R = + 1, PP,- 3R2 = +1, tractari possunt ope lemmatis quod viro clarissimo J. J. Sylvester accepturn referimus; ' Sit q, q2..,, qi, q..., q2 q1 quotientium series ordine symmetrico disposita, e quibus alterni per numerum ix multiplicentur; determinans K= (q1,. q2, q3,..., 3 q, q2, x q1) erit formse A2 u B2, designantibus A et B numeros inter se primos.' Fit enim, si numerus i par est, (q1 i q, q3, *.., q) = (q1, ~q, q3,...,3 ); si vero idem numerus impar est, (iq1i, q2, A q3, *..., Aq) = XX(ql, x2,l ***, q). Itaque erit aut K= (ql Ai q2, q *..., qi)2 +. + (ql, q2.... q-_l)2, aut K= (q, q2, q3,...,I qi1)2 +. (qi, i q2,..., ql, qi)2. Aperte autem confitemur totam hanc disquisitionem angustis finibus contineri; quam, annis ab hinc amplius viginti a nobis inchoatam, nunc demum longo post intervallo retractatam absolutamque, ideo arithmeticorum hominum judicio committere ausi sumus, quod elementis arithmeticae incrementa etiam tenuissima afferre operse pretium putamus. Qusestiones autem in eodem genere latius patentes et nos olim attigimus, et fusius tractavit vir clarissimus M.A. Stern in egregia commentatione.* 2. Initium operis facimus ab sequatione P, P, = 3 _R2+ 1; P1P2=3R I1; in qua fiat R2 = R - qlP2 p3R= ----+ =P1- 6qlJl + 3qlP2, P2 'Report on the Theory of Numbers,' Art. 123 (Report of the British Association for the Advancement of Science for 1863) [vol. i. p. 283]. 'Journal fur die reine und angewandte Mathematik,' vol. liii. pp. 1-102. Prieter autem Gcepelii dissertationem (ibid. vol. xlv. pp. 1-13) maxime memorabilis est cornmentatio clarissimi viri C. Hermite (ibid. p. 191); quae tamen a theoria continuarum fractionum paullo longius recedit. ' Art. 2.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 289 et generaliter Rn = RA^^-1 yn-l Pwl, p _ 3R+1 n+l- Pn1 ubi quotientes integros q ita determinari volumus, ut fiant numeri residui R quam minimi. Ponimus autem in sequatione data numeros PI, P2 (atque adeo omnes numeros P) esse positivos, et non impariter pares; R1 autem numerum esse sive positivum sive negativum, quem positive sumptum per [R,] denotabimus. Praeterea statuimus, quod licet, [R,] esse minorem quam -P1. Erit ergo P1 > P2, P2< [R1]; unde patet quotientem q1 evanescere non posse, et fore [R,] < -P2, P3 < [R2]; ideoque [R] > [R2], P2 > P3; sive generaliter [RS]> PS+1, [R]j< ps, [R,] > [R < [3]... P1 >P2 >P3,.... Itaque veniemus aliquando ad eequationem Pi Pi+ - 3 R2 = 1 in qua Pi+ =l, qi=Ri, quseque adeo omnium postrema erit. Tur vero habebuntur sequationes Ri =(qj), Pi = (, 3 q), Ri-l=(qi_-, 3q,, qj), P_:= (qi, 3, 3, i, 3 ), Pi- 2= (qi-x2, 3qi 2, qi, 3qi, -i- ), -Pi-2= (qi-2, 3 qi-2, yi, 3 qi, q-, 3),2) Pi-2=(qi-, 3yi-~, qi, 3qi, qi-:, 3qi-.), R,=(ql, 3q2, q3,..., 3q3, q2), P,=(ql, 3q,..., 3q3, Q2, 3q, ); quse facile aliae ex aliis probantur, ope formularum RS-1 = qS-1 P8 + R,, p _3Rl+1 Ps -1= P PS Itaque ex data sequatione P1 P2=3R+1 elicimus reprsesentationes numerorum PI et P2 per quadratum et triplicem quadrati. VOL. II. P p 290 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 3. Exemplum. Sit aequatio data, 1999 x 436 = 3 x 5292 +1: erit 1999 = (1, 3, 1, -6, 1, 3, -2, 3, 1, 3) = (1, 3, 1,- 6)2 + 3(1, 5, 1, - 6, 1)2 =262+3 x 212. 436 =(3, 1,-6, 1, 3,-2, 3, 1) = (3, 1, - 6, 1)2 + 3(1, 3, - 2)2 =172+3x 72. 3. Statuamus in fractione continua Pi 31 = -[q, 3q2, q3,..., 3q3, 2, 38q] 3 RI hunc haberi quotientium completorum ordinem 0, 302, 03,..., 0i sive 3 0,, 31) sive i,..., (, 321, ita ut fiat P 1 3 -01 =q+ 02= =+ 303 * 1 0. = qi. + ', qij+ 3+ _ ~ 3 (PI3 5*** Quantitatum 0 et q ea est conditio ut semper evadat [0] >, [l]>~> Quod ut certa demonstratione confirmetur, adnotamus primum, si inter quotientes q duse signis diversis unitates juxta positse inveniantur, continuari signum secundae unitatis in proximo quotiente. Sit enim, exempli causa, q = 1, q2= - 1; numeri RI et - R2 erunt positivi, quia signa numerorum q,, R, semper inter se conveniunt. Quum autem sit P2< R1i, sequitur fore -R2 > P, P3 = 21+2 < - = R2 + P3 <; P2 ideoque etiam q3 < 0. Monemus tamen duos ultimos quotientes qi_1 et qi unitates signis diversis esse posse; neque ulla lege adstringi signum quotientis duas unitates proxime antecedentis. His positis, apparet quantitatem s, + satisfacere conditioni supra memoratse, si eidem ab ipsis (,, s_,... satisfiat. Quum enim sit 1 1 (s+ I= + ^ + 3 g+3(Ps Art. 4.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 291 manifestum est signa ipsorum p(+1, p,,... congruere cum signis quotientium s+1, qs,... Sit igitur, brevitatis causa, q +1 numerus positivus; erit certe +is ++ >-> + + 1-*1 + + - + -- 3 ^= -3 -3 2 3 2_2 quia quotiens tertius q8_- unitas positiva esse nequit. Itaque ad extremum habebitur ( +1 >[,-3,2], sive 1,_ >Et similiter si est [08 +1] >., [0s] >,... erit etiam [08] > 1+ + 1- > -3 08+2 quia post quotientes qs = 1, q8+ = -, necessario obtinet proximum locum quotiens negativus. Sit 10 numerus integer qui omnium proxime accedit ad valorem ipsius 0; nanciscimur ex antedictis aequationes sequentes ql = I1, q = 0 2,..., qi-l = IOi-. At vero est qg= 0-; quumque fieri possit ut sit [j] < -}, mequatio qi = I0i non semper valet. Valet tamen praeterquam in eo casu, in quo est qi-_ unitati sequalis, [0] < 2, et in quo insuper alternantur signa quantitatum Li-2, 4i-1, Oi; quo in casu erit [jil < t, I-[Qi] = 2, qi unitas unitati qiI contraria. Cujus rei exemplum praebet mequatio 52 x 7 = 3 x 112 +1, ubi fit [2-3, 1 3,-1 6]. Hie enim est 0 = 14, ideoque 103 = 2, quum sit tamen q3 =1. Quodsi in fractione continua ad postremum quotientem, qui est 3ql, accedat quaevis ejusdem signi quantitas, valores 0, 02,..., I0 ipsos quidem aliquatenus turbatum iri manifestum est; sed nihilo secius permansuras sequationes q = 10,, q2 =02,..., q_1=ITi_.2 atque determinationem quotientis qi modo traditam. Quod patet ex ipsa demonstrationis forma. 4. AEquatio P1P2 = 3 R - 1 ita facillime tractatur si scribatur PP2= - 3R + 1, P 2 *'92 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 4. ut P2 et R1 numeri negativi evadant. Quo facto, determinentur numeri P3, P4..., Pi+; R2, R,..., Ri; per aequationes hasce R2=-RI- qP22 -3R+l1 P3 = 2 (in quibus numeros R quam minimos fieri intelligendum est)... donee perveniatur ad aequationem in qua sit Pi+ = e, Ri = eqi, designante litera e unitatem (- 1)... Turn vero erit e P1 = (q, -3q,..., q), - 3q), P2 =(q, - 33,..., q3, - 32), ER = (q, - 3 q2,..., - 3q3, q2); unde apparet ipsos Pi et P2 fore formse y2 - 3 2, aut formae 3x2 - y2, prout i par est aut impar. Coeterum in sequatione penultima Ri = R,_1 - qi-, Pi, si est P, = + 2, oritur ambiguitas in determinatione quotientis qi_1; quam ita tollimus ut quotientes qi-,, qi (e quibus hic certe unitas erit) afficiantur signis contrariis. Facile autem demonstratur in serie quotientium q7, q,...., q, si duse eodem signo unitates juxta veniant, mutationem signi fieri in proximo quotiente. Hinc in fractione continua = [,.. - 3!, 2 ** -33 q,, - 3ql] 3R, si fiat deinceps p 1 ~- 3l= =01 ql+3' 0= q2+_-3 I- qz - i, 0 3 0 — q+-3- ' o=q+ + - 33 3'" 1 1 et sic porro, erit ut supra []> ][ >. Quotientes autem q,,... determinantur per aequationes qI = 1, q2= I 0,...; inter quas etiam extrema illa qi = IOi locum sibi vindicat. Nam, si qi = + 1, (qui solus est casus de quo dubitatio oriri potest) erit necessario P = + 2, atque discrepabunt inter se signa quotientium qi_, et qi; quo fit ut valor absolutus ipsius /i=q= + + -3qi + — unitate major evadat. +i-2 Quum autem numerus e P1 infinitis modis per formulam y2 - 3 x2 reprsesentari Art. 4.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 293 queat, ostendere convenit ear repraesentationem que continetur formula eP1 = (q1, -3 q2,..., q2,-3q1) per numeros minimos fieri. Et generaliter quidem, si est y2Ax =N, t2 Au2 = 1, inveniri possunt in una eademque repraesentationum familia totidem repraesentationes repraesentatione data magis simplices quot habentur numeri u fractione 2y absolute minores. Etenim cum omnes indeterminati x valores in hac formula tx - uy comprehendantur, si qua est representatio repraesentatione data magis simplex, habetur inaequalitas (tx - uy)2 < x2; quae, eliminato numero t, convenit cum inaequalitate [u]< [2]- * Jam vero, si i= 2n, habemus oequationem P,==y- 32 =(q, -32,..., qn-, -3 q2n) -3(ql, -3Y2,..., q2n-) 2, in qua erit [ ] > 1;n ideoque, [] / I [y]2 [x] < ' Sin autem i = 2 n + 1, erit P = 3Sx2 y2=3(q, -3q2,..., q2n), -2(q, - 3q2,..., 3,3)2, w PI PI [y = [n] > 1, [XA] < 2 Hinc in utroque casu habetur [2 p < 1; unde apparet numeri P1 reprsesentationem magis simplicem exhiberi non posse in ea saltem familia, in qua fit 1 V3 =, mod Pi. Exemplum. Sit mequatio proposita 20306x -5197= -3(-5931)2+1; erit e= - 1, 20306= -(1, 6, 1, -12, -1, 3, 4, -3, -2, -3) = 3 (1, 6, 1, -12, -1)2- (1, 6, 1, - 12)2 =3 x 972- 892; 5197 = (6, 1, - 12, - 1)2- 3(- 2, - 3, 4)2 = 852- 3 x 262. 294 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 6. 5. Sit X numerus primus formse 4n+3; atque in aequatione X, = +AB, tribuantur literse A singillatim valores 1, 2, 3,..., 2n + 1: quo facto, erit B unus aliquis ex iisdem numeris vel positive vel negative sumptus. Hinc patet in hac sequationum caterva binas inter se invicem respondere; quarum numerus cum impar sit, una super erit, quae necessario hanc formam induet X = +3A2+1. Hinc per exclusionem nanciscimur demonstrationem non inelegantem notissimi theorematis; primos formae 12n +7 esse etiam formae x2 +3y2; primos autem formae 12n +11 habere formam 3x2 - y2. Ac simili fere modo demonstrari posset utramque formam x2+3y2, x2 - 3y2, primis 12n+1 competere; neutram vero primis 12 n +5. Sed primos formae 4n +1 in praesens missos facimus. Nam repraesentationes primorum 4n+3 per formas x2+3y2, 3x2-y2 indagari possunt per evolutiones radicum quadratarum in fractiones continuas; quae proprietas ad primos formse 4n+1 nequaquam pertinere videtur. Quo autem facilius intelligantur rationes evolutionum, quibus in hac qusestione utimur, pauca, licet aliunde nota, premittenda sunt. 6. Sit N integer non quadratus; a integer ipso VN proxime minor; quantitas a+^/N in fractionem continuam conversa dabit quotientium periodum ad hoc exemplar 2a, l, 2,..,, f, 5, _,...,; ubi primus quotiens quasi singularis est; coeteri ordine symmetrico circa quotientem medium 3 hinc et hinc disponuntur. Cujus symmetrie eo causa est, quod equatio a 2-N-2Cax+x=O,........ (A) a qua periodus originem trahit, anceps est, cum tertius in ipsa coefficiens secundum metiatur. Quod si evolutionem a quotiente medio incipimus, habebimus hanc periodi descriptionem, /5 An5, gn._-,..., tl, 2Ca, u1I, I2, *5, n 5,n quse est plane ejusdem formae cum periodo data. Hinc patet inter sequationes periodicas, praeter primam illam, mediam quoque ancipitem esse. Cujus Mequationis cognitio magnum momentum habet ad explorandam totius periodi naturam; sunt autem certi casus in quibus etiam sine evolutione innotescit. Sit, exempli causa, N= Xi x 2, designantibus literis X, et 2 numeros primos formse 4n+3; sit etiam A1 > X2; b integer surdo A/1 proxime minor. His positis, erit 2 b quotiens medius in evolutione radicis,/AI x A2, et aequatio media hanc formam habebit 72 9 - o7, —,,, I -9 2_- /R\ A2 - A1 -- OA2 -+- A2;=.. *. k-D) Art. 7.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 295 Cum enim X, x X, summa duorum quadratorum esse nequeat, duae illae oequationes ancipites, quee in periodo inveniuntur, diversae inter se esse debent; at praeter duas aequationes (A) et (B) nulla alia existere potest, quae et anceps sit, et characterem aequationes periodicae prae se ferat. Unde videmus evolutiones a I tiones radicum /JX x X2 et / - una eademque opera exhiberi; idque fieri per hujusmodi aequationes a + I/i xa2 = 2[2, i,)... ), ) b + x\/ - b + /A-[2 b, An, i An1,.., i 1, a +,/X X ]; e quarum utravis concludimus fore Xi X (, 2,...,n) =(a, A1,,,x2..., n b), (C) ita ut fiat X1 X (1, 2,..., 2- )2 — X (L1,,2, an) b)2 = (- l)n. 7. Sit primo - X2 ipsius AX residuum quadraticum; erit (- 1) = + 1, atque ideo e numeris (,l, A2,...,,n) et (iu, M2,..., i,, b) hic impar, ille par erit. Hinc sequitur, sequationem 4Xl-x2 Xy2=1 resolubilem esse; et, designantibus a' et ' numeros ipsis 2 /X1 x X2 et 2 A/ proxime minores, aequationem X2x2- 2 b'2x+X2 b2 - 4X = 0 occupare medium locum in periodo wequationis X2 - 2a'x + Ct2 -4N= O; hoc est, evolutiones quantitatum a + 2/1 x X2, + 2 / b+2 eodem modo alteram X^X2. ab altera pendere, quo evolutiones quantitatum a + /X x X2, b +/, Evoluta autem quantitate a' + 2/\ x X sit 2 a', v., V2,..., j, 2b', Vj,..., V2, V quotientium periodus; habebimus wequationes sequentes antecedentibus (C) consimiles 2...(V V, ) = (', V, D 4X1 x (v1, 2,.., vj) = (a', vl, v2,., v, V'); habebimus etiam solutionem aequationis 4 x2 -;2y2 = 1, scilicet, 4 X1 (v1, V2,..., Fj)2-_ X2(V1, V2,..., yj, bl) = 1; 296 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 8. quse quum convenire debeat cum solutione superius inventa (utraque enim minimos numeros exhibet qui aequationi satisfacere possunt) colligimus aequationes sequentes (/1, 2,,.5. = 2 (vl, Y2,..., vj), (~, y,., ~, 66): (,,, /,..., vj),) (a, 2, 2,,, ), b)= (va, 12,..., v b), (a, 81 82..., Mn, b) = 2(a'L V, V2,..., lj b); e quibus patet quomodo, ex evolutione quantitatis \/A\ x A2, evolutiones quantitatum 2,X x A2 et 2/ - derivari possint per conversionem fractionis vulgaris =B [a, A1, 2, *.n,,n b], erit etiam 2A 2A =Ca', Vi, ^,..., j, b J]. Sit, secundo, X2 ipsius X, residuum quadraticum; erit (-1) =-1, (1U1, 25,,,n, b) par, (ux, F2,..., My) impar; atque habebitur solutio sequationis Xx x2- 42 y2 = -1. Unde derivari poterunt evolutiones quantitatum /x x X2, et / 5, quibus tamen in prsesens opus non est. 8. Jam vero sit A2 = 3, XA = X numerus primus formme 12n + 7; erit -3 ipsius X residuum quadraticum; atque habebitur ex antedictis aquatio (a, 1A,,2 *...,, b) (,1, Fl2..., xn ) = 3 X (1, 2k,...) *, n b)2+ 1; quse tamen, quia (,Ul, 2x2,... (n) numerus est inipariter par, non potest commode tractari per methodum supra (Art. 2) traditam. Utendum est igitur evolutione quantitatis 2 AV/ qume si statuitur esse 2b', r~, j_i,,,,,... v 2a, ~..., v j, suppeditabit vequationem (a', r, v2...,,j, b) (vl, v2,..., (V)= 3 (vl,.,., j, b')2+ 1, in qua (v,, v2,..., ) impar erit, (a', vl, 2,..., Y, b') pariter par. Itaque si in hac sequatione fiat Art. 8. DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 297 (a', v1, V2,..., v, b )=(ko, 3k, k2,..., 3k2, k, 3ko), (V1, V2,.", vj) =(3k1, k2,..., 3k2, k). (V, 12,..., vj, b')=( ko, 3kl, k2,..., 3k2, kl), evolutio quantitatis 2 -/ ad sequentem formam redigetur ko+2 A/ x =[2ko, 3k k2,..., 3k, kl, 6k0o kl, 3k21,.., k2, 3k]. Quod facile demonstratur, si in evolutione vulgari quantitatis 2 -/ 3 quotientes 2 a', 2 b', interpositis cifris ita discerpantur ut fiat /a/ 3 = [a, vj,. b',, b', ', b,.V, a ', 0, a]; et si prseterea intelligatur, quod in fractione infinita licitum est, quotientium seriem ita terminari ut una ex cifris interpositis locum extremum obtineat. Sit - A - 2B0 + C02 = 0 mequatio quadratica; numeri ABC integri; determinans B2 + A C positivus. Sit item q,, q2,..., q, quotientium periodus per quam sequatio ista in se ipsam transit; ita ut fiat = [ql, q2,..*. qn, 0]; habentur sequationes notissimse (q2, q3..., qn) (ql,, n)-(2,..., qq- 1) (ql, q2,..., qn-_l) A B C C in quibus si forte fiat aut (q1, q2,..., q_)= (q2, q3,..., q2), aut (q2, q3 *..., qn)= (ql, q2, '*, qn-1), determinans sequationis erit formse x2+Uy2. Quodsi in evolutione quantitatis ko +2 / periodum ab illo quotiente inchoamus qui est aut triplex, aut tertia pars quotientis proxime antecedentis, habetur series aut hujus formae 3ki, ki-1,..., 3k2, k, 60o, k, 3k2,..., k2, 3k1, 2k0, 3k, k2,. 3k_l, k, aut hujus ki, 3k_-l..., 3k2, k 6ko, k1, 3k2,..., k2, 3k1, 2ko, 3k, k2,..., k-_, 3ki; prout numerus i par est aut impar. Unde patet in aequatione periodica Pi+l+ 2 Qi +l +Pi_2x2 =0, e qua istam quotientium seriem originem trahere statuimus, fore vel 3 Pi + + Pi 2 = 0, vel Pi + + 3 Pi + = 0, ita ut habeatur partitio VOL. II. Q q 298 DE FRACTIONIBUS QUIBUSDAM CONTINUTIS. [Art. 9, numeri 12X, ac proinde ipsius X, in quadratum et triplicem quadrati. Quo autem in wequatione = A2+3 B2 valores numerorum A et B facilius investigentur, aperienda est via ad evolutionem quantitatis k0 +2 / sine operoso calculo inveniendam. Nimis enim molestum esset eam evolutionem a vulgari evolutione radicis 2 A/ per transformationem supra traditam elicere. 9. Et primum quidem observamus in evolutione quantitatis ko +2 /, alterni quotientes per ternarium dividantur, haberi periodum ejusdem plane formne cum periodis primorum 4n+ 1. Patet autem ex iis quae antea (Art. 3) disputavimus quotientes ko, kj,..., ki-_ deinceps determinari posse eadem prope calculi forma quse vulgo usitata est ad radices quadraticas evolvendas. Habemus enim schema hujusmodi, si incipimus ab aequatione Po+ 2Q x + P 2 = O, in qua est P = 3 k -4X, Q1= 2koPI + Qo, Q2= 3k P2 + Q1, Q3= k2 P3 + Q2, Q4= k3 P4+ Q3, Q = -3 ko, P =3, 1=I2 /3 Q - 12 x - Q - 2_3_ X P2- - kl = 1 pQ_- 12X -Q2-213X - 2 k~ = I - - 2-, 2 k P3 2Q-12x 2 3I P4 ------ - --- P' 3 P,4 Q- 12X I-Q4+2/3X P4 ' P in quo videmus unumquemque numerorum Q esse ternarii multiplum, sed contra alternos tantum e numeris P. Si ergo scribatur Q, = 3be, P28 = a23, P2s+I = 3a2,+, habetur calculi forma paullo simplicior, bi = 2 ko a + bo, b2= k a2+ bi, 3b - 4\ 2 a, -bl-2 AV x k,- = I 3 (2 3b6-4 -b+2 3 3= -a2 kI -, a2 a3 Art. 9.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 299 3b-4X -bS+(-l)s2 AV bs= kl+a,.+b=-, a - -,, k-I I+ a. es a+ 1 vel, si notationem Gaussianam sequi placet 3b -4X b, b0,, mod a; a- - b2-b1, moda2; a 3b -- 4 5 b.-i, mod a,-,; a -,= i — asb, ~ ~ ~ ~ ~ e ~ ~ * b ubi numeros bs, k1_.=,- e congruentiis b, -bb, mod a,, ita determinari as- - oportet ut bs quam proxime accedat ad valorem radicis (- l)s+1 2/ His in schematis determinationes numerorum k2 = 11, k2 =02,..., kjl = IOi ex antedictis (Art. 3) accuratas esse apparet; sed contra determinationem quotientis ki = IQi falsam fore, si - bi+l+(- l)i+ 2 3 a,+, hoc est, si in aequatione 4X = a+1+3b+l1 fiat [bi +] < i2[af + 1]. Quocirca calculus eo usque producendus est donec perveniatur ad sequationem in qua aut ai ~+a = 0, aut ai + 3 b + 2ai+ =0. Nam, si [j]> %, aequatio quae conditioni a+1+ ai+2=0 satisfacit quaeque ideo in toto schemate ultima est, calculi normam sequenti suo loco se offeret. Si vero est [qJ] < 23, ultima ista squatio quodammodo extra ordinem calculi erit, quia fallit determinatio k = I0. Sed hoc in casu est necessario k = +1, unde sequitur penultimam mequationem talem fore ut sit L ac + 3bi + 2 ai =; cujusmodi sequatio si quando se obtulerit, ipsa certe est proxima ante ultimam, atque, ut ad ultimam perveniatur, quotienti ki tribuendus est valor + 1; idque observandum est etiam si forte fiat [0] > -. Calculum autem numerorum a et b Qq2 300 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 9. ulterius extendere omnino opus non est; cum in horum periodis semissis prior hanc habeat formam al; a2, a3, * ~., ai-, Ca+ 1, - j+1 - 'i,..., - 02; b, b2, b*,..., bi; bi +; bi, bi.-1.. bl; altera autem semissis eosdem numeros signis mutatis eodem ordine repraesentet. Quee omnia animadvertere operse pretium est propter similitudinem primorum formse 4n+1. Exemplum I. Sit X = 199; erit 12 - =16, ao= -28, b= -16, a = 1. iEquationes autem P8 + 2Q x + P +x2 =0 brevitatis causa per notas sic scribimus (a,, b6, a,8+); quo pacto habebimus ex calculi ordine has wequationes in primo periodi quadrante (-28, -16, 1), (1, 16, -28), (13, 14, -16), (-16, - s18,-11), unde fit 4 x 199 = 112+ 3.152; 16+2 -199=[32, 3, 2, 6, -3, -9, 2, 6, 1, 66, Exemplum II. Sit X = 607; erit 12 -=28, a,=-76, b,= 3. (-28, -12, 13), (-11, 15, 11): 1, 6, 2, -9, -3, 6, 2, 3]. -28, ac = 1. (-76, -48, -59), (-20, 36, -73), (-28, -24, 25), (61, 39, 35), (80, - 46, 49), Hinc oritur periodus (-76, -28, 1), (1, 28, -76), (-59, 11, 35), (35, -24, -20), (-73, -37, -23), (-23, 32, -28), (25, -26, -16), (-16, -22, 61), (35, -31, 13), (13, 34, 80), (49, 3, -49),... unde fit 4 x 607 = 492 + 3.3; 28 +2 /6- [56, 3, -1, -3, -3, 3, -3, 6, -2, 9, 1, -6, 5, -3, +1, 3, -1 115, -2, 3, 3, -6, 2, -9, -1, -9, -1, -3, 1, 168, 1, -3, -1, -9, 1, -9, 2, -6, 3, 3,-2, 15, -1, 3, 1, -3, 5, -6, 1, 9, -2, 6, -3, 3, -3, -3, -1, 3]. Art. 10.] DE FIACTIONIBUS QUIBUSDAM CONTINUIS. 301 Observandum autem est hic in penultima aequatione (80, -46, 49), sive (a14, b14, a15), haberi 40 - 3 x 46 + 2 x 49 = 0, unde sequitur esse k,, = 1, quamvis sit 46+28 3 49 2' 10. Quum autem numerus 4X per formam x2+3y2 trifariam reprsesentari possit, illa quidem repraesentatio, quae numeros x et y pares habet, cum nostra convenire non potest., Ex duabus, quae reliquae sunt, una sola est in qua x _ ~ 1, mod 12; atque hec illa est quam prebet equatio 4 X= cc+ 3b,. Si enim i impar est, habemus 3(ko, 3k,..., 3k)2- 4X (3k, k2,.., 3ki)2 =P,+2 = 3a,+2 sive (oj 3k1,.., 3k )2 12 \ (k, 32,..., ki)2 = ai+2 Unde fit ai+2-1, mod 12. Ac similiter, si numerus i par est, erit ai+1l, mod 12. Caeterum, ut ex inventa vequatione 4 X = a + 1+ 3 b +,, ipsius X representatio habeatur, fiat ai +l bi +, 0, mod 4: quo pacto erit X= (ai+ - l+3bi)2+3 (ai+b+l), contra, si data fuerit aequatio A = A2 + 3B2, fiat A = 6/ + 2E1, B = 2v + 2, designantibus literis ~e, 62 unitates, quarum altera ipsa oequatione definitur, altera ita capiatur ut,u + v par evadat. Quibus rite animadversis, habebuntur sequationes 4X=(32B -'eA)2+3 (elA +eB)2, 3 E A- elA =(- 1)i l + =(- l)i+ ai+, 1, mod 12 [1 A + c2B] = ( - )il+ b +l; nam (- )i+1 b + 1 certo numerus positivus est, cum debeat esse b+' + (- +1)+ 2 A - Hinc nanciscimur theorema: 'Designante X = A2 +3B2 numerum primum formse 12n +7, resolubilia sunt hsec tria mequationum paria x2-3Xy2= 3e2B- eA; =2e1A; =- 32 B -elA; X- 3 y2=3e2B- lA; = 2e, A; = - 3e2B- e A; 302 DE FRACTIONIBIJS QUIBUSDAM CONTINUIST [Art. 10. cujus veritas facile colligitur ex oequivalentia formarum (- 4X, 0, 3), (- 12X, 0, 1), (Pi+1, Qi+l, Pi+2). Tres isti numeri [3 2B- e1A], [2e A], [3 e2B + A], qui repraesentant totidem valores indeterminati x in aequatione 4X = X2 + 3y2 eo etiam nomine nobis memorandi sunt, quod, evoluta radice ~/3x in fractionem continuam vulgarem, inter denorninatores quotientium. completorum uno tractu veniunt. Atque, si est A > 3 B, erit 2A medius, si vero A < 3B, erit idem vel primus vel postremus. Nam, si A > 3B, colligimus sequationes 3B-A-2 (3B+A)x+2Ax2 =0, 2A -2 (3B-A) x- (3B+A) x2= alteram alteram excipere in periodo aequationis -3 X+x2 =0; utramque enim sequationi - 3 X + x2 = 0 2equivalere per demonstrationem theorematis prsecedentis evincitur; utraque autem characterem sequationis periodicae habet propter inxequalitatem A > 3B. Similiter, si A < 3B, sequationes - 2A - 2 (3B- A)x + (3B + A) x2= 0 (3B+A)+4Ax-(3B-A) 2 = deinceps occurrunt in eodem periodo. Hine nanciscimur methodum non inelegantem solvendi aequationem X = A2 + 3B2. Namque in evolvenda radice / 3, inveniemus tres juxta denominatores, quarum medius sequatur extremorum summae, idemque exsuperat radicem S/3X; ex his unus erit 2 A, reliqui duo habebunt valores impares [3B + A]. Exempluzm. Sit X = 139 = 82 + 3. 5; 4x 139=162+3. 102=(-23)2+3. 32=72+3. 132. Hic est e A = 8, 32 B - eA = -23, - 3 B - eA = 7. Atque erit ex evolutione nostra, cum sit 12 A = 14, (32, -14, 1), (1, 14, 32), (32, -18, 13), (13, 8, -28), (-28, -20, -23), (-23, 3, 23),... Unde fit 4 x 139 = (- 23)2 3.32; 14 + 2, = [ 28, -3, -2,3, -1, -3,1, -6, -1, 84, -1, -6, 1, -3, -1, 3, -2, -3]. Contra ex evolutione vulgari aequationis -17-40x+x2=O, Art. 11.] DE FRACTIONIBTJS QITIBUSDAM CONTINUIS. 303 orientur hujusmodi periodus (-17, - 20, 1), (1, 20, - 17), (-17, -14, 13), (13, 12, -21), (-21, -9, 16), (16, 7, -23), (-23, -16, 7), (7, 19, -),, -13, 31), (31,18, -3), (3, -18, 3 1 ),... in qua habemus 20+,/417 =[40, 2, 2, 1, 1, 1, 5, 4, 1, 12, 1, 4, 5,, 1, 1, 2, 2]; videmus autem numeros 16, - 23, 7, inter coefficientes sequationum periodicarum comparere, idque accidere fere vergente ad finem primo periodi quadrante. 11. Numeri primi forme 12n +11 similes aliquatenus proprietates habent. Namque ex evolutione vulgari b*+ /3= [2:b, j,yj-._,...., /x, 2a, l,...j,..... (A) elicimus aequationem - (a, ^,..., ~j, b) x (Al, j) - 3 (,, j 2 + 1) Quse cum sequatione PP2 = -3 R + 1 ita comparetur ut fiat P =(a, M1,..., j, b) =e(o, -k3ko 3., *, - 3o), P2 = -(1,..., j) - =e ( -'3 k, 2,..., -.3k2, kl), RI-= -(-1, 23,..., lj, b) =e(ko, 3 k,..., -3k2, lk), - 3R1= (c, A 2,...,., ) = E.(- 3k, k2,..., k, -3ko). Quibus positis, facile perspicitur in evolutione (A) quotientium seriem b, Pj, lj-x,...,, 2 A1, a cum hac serie ko, -3k,, k2,..., k7, -3ko sine fraude commutari posse. Erit itaque Co + /\ =[2ko -3kl, k2..., -3 k -, 6k, -6Co k, -3k, k -3. k (B) Qua in evolutione altera ex vequationibus, quee medium locum in prima periodi semisse tenent, dabit aut hanc aequationem 3X= h- 3X=P+-,, aut hanc 3 = Q+1 - + ^^^l^ -^~i+l, 304 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 11. prout scilicet numerus i par est aut impar. Ponimus autem eodem fere quo supra modo _ - Qs supras =3b, P2= 2 - a2, P28+ 1 =3a2+ 1, ao=X- 3k0, b = -k =-I /, a=l; O 0 0 A et calculum periodi instituimus secundum hoc schema x - 2 bb -=2koa + bo; a2= -b; k, I I a, a2 - b2. + x - 3 ~0 b2 = ka2+ b,; a3 =;3 k=I; a2 a3 b=, _ +b _.; as +,=;k0 =I! Os a +1 vel ex notatione Gaussiana b,-bo, mod a,; a2 =-; b2 b,, moda2; a3 =; a2 -3 b,2 b8_ b8, mod a,; as+=-3 b; a8 numeris bs ita determinaris ut quam proxime ad valorem radicis (-1)8 accedant. Cui calculo finis faciendus est cum primum vequatio ai+ + al+ +2 =0 sese obtulerit; eo enim usque valet formula ks = 0ls, quae ulterius progredientem destituit. Periodi autem numerorum a, b, k, eandem plane descriptionem habent, quam supra exemplis illustravimus. Quum autem in aequatione X = 3 bM + - a + numerus a + par esse nequeat, evenire potest ut repraesentatio ipsius X, quam ejus aequationis ope nanciscimur, non sit omnium simplicissima. Est tamen vel ipsa omnium simplicissima, vel proxima post simplicissimam. Habemus enim (Art. 4) - +(- 1; Art. 11.] ADE FRACTIONIBUS QUIBUSDAM CONTINUIS. 305 hoc est, VA+c a+I + > [a, + ^3]1 ideoque etiam _ / [a + ] <, /3 XI [bi+ + lb +l] < 2 X. At in sequatione t2 - 32 1, valores numeri n sic procedunt, 1, 4, 15,..., unde concludimus, si fuerit [bi + 1< [a+i, unam fore eamque solam repraesentationem qume per numeros minores fiat; qume si est X =3x2-y2, erit y par, x impar. Itaque si A et B minimi numeri exstant, per quos aequationi X = 12 B2 - AI satisfieri possit, vequationes x2 3Xy2=eA, 3A, =, 4(eA+3B), 3x2-Xy2=eA, =3eA, =4(eA+3B), resolubiles erunt, signo unitatis e ita determinato ut fiat eA A 1, mod 3. Facile autem perspicitur oequationi - 4(3B-A) - 2(6B-A)x +Ax2= 0 competere characterem sequationis periodicae: cum sit [A] < 3[B], [B] >. Quare vulgaris quoque evolutio radicis ^/3 \ suppeditat solutionem simplicissimam aequationis X= 12B2 A2; veniemus enim in ea evolutione ad sequationem ps+2q+,x+p,+1x2 =0 in qua erit 2ps= q,+Ps+,, ideoque etiam, qs,-S+, mod, mod 6 =12 ('6 -P2 +. Exemplum. Sit = 167; erit I A = 7; atque habebimus ex evolutione nostra hanc periodi formam (-20, 7, 1), (1, 7, - 20), (-20, -13, -17), (-17, 4, 7), (7, -3, -20), (-20, 17, -35), (-35, -18, -23), (-23, 5, 4), (4, -7, -5), (5, 8, -5),... Erit itaque 167 = 3. 82 52; 7 +/ 167= [i4, 3, -1, 3, -1, -, 3, -1, 3, -9, -3, 3, -1 3, 3-1,,, -142, 1, 3, -1, 3, 1, 3, 3, -9, 3, 9, - 1, 3 -1, 3,- 1, -3] VOL. II. R r 306 DE FEtACTIONIBUS QTUIBIJSDAM CONTINUIS. [Art. 12. Evoluta autem radice V/501 secundum methodum vulgatam invenimus periodum sequationum hancce (-17, - 22, 1), (1, 22, -17), (-17, -12, 21), (21, 9, - 20), ( 20, - 11, 19), (19, 8, - 23), (- 23, -15, 12), (12, 21, -5), (-5, - 19, 28), (28, 9, - 15), (-15, - 21, 4), (4, 19, 35), (- 35, - 16, 7), (7, 19, - 20), (- 20, - 21, 3), (3, 21, -20)... e quibus nona suppeditat solutionem sequationis 167 = 3x 82-52, cum sit 2 x 28 = 19 - 5. Erit praeterea 22 + /501= [4i, 2, 1, 1, 1, 1, 3, 8, 1, 2, 10, 1, 5, 2, 14, 2, 5, 1, 10, 2, 1, 8, 3, 1, 1, 1, 1, 2]. 12. IEquationes PP2= 2 R + 1, PP2 - 2R2 +1, et repraesentationes inde orientes numerorum primorum per formas 2x2+y2, 2x2-y2, a Goepelio in commentatione inaugurali* longe pulcherrima (ex qua disquisitionem nostram delibatam esse libenter fatemur) tanta felicitate ope fractionum continuarum illustrate sunt, ut ear rem iterum attingere prope supervacaneum sit. Tamen ne quis analogiam determinantiurn 2 et 3 a nobis praetermissam desiderit, in vestigiis ejus viri paullisper immorari liceat. Et primum quidem adnotamus, ab oequatione PmP2=2R2+1, eodem fere quo antea modo, perveniri posse ad equationem P1 = (k, 2k2, k3,..., 2k3, k2, 2k1); eamque transformationem ita fieri posse ut quotientem unitati mequalem excipiat ejusdem signi quotiens. Ex quo apparet in fractione continua P 2 RI =[kl, 2k2,..., k2, 2k1]; haberi oequationes k, = I0, k2 = I2,... eamque legem etiam in extremo quotiente ki observari, nisi forte fiant signa quotientium k,_ et Oi contraria, ac praeterea [0,]< 2; quo in casu erit I0,= +2, ki= +1. Nec secus in fractione continua quSe pendet ab sequatione PP2 = - 2R + I, erit ~P1=(k1, -2k2, 3,..., -2k3, k2, -2kl), R =[kl, -2k2, k3,... k2, -2kl], 2R 2 * 'Journal fur die reine und angewandte Mathematik,' vol. xlv. pp. 1-13. Art. 12.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 307 k = I0, k2= - I2,... atque adeo ki = 10i, cum sit utique i > 1 (2 + /V2): que res item docet in neutra formarum aequivalentium x2 - 2y2, 2x2 - y2, dari repraesentationem numeri P1 repraesentatione nostra magis simplicem. Turn vero, designante X numerum primum formae 4n+ 3, vel duplum talis numeri, atque evoluta radice /X secundum prmecepta vulgata erit in media periodo hujusmodi,equatio anceps - 1 ( - b2) -2 bx +2 x2 = (ea enim sola est quae exsistere potest), ubi numerus b ita determinatur ut 2 (X- b2) integer atque positivus, idemque quam minimus fiat. Sit a numerus integer surdo./X proxime minor; erit aut a =b, auta = b +1. Atque si est a = b, habebimus hujusmodi evolutionem N/-[a, 1 2..n 2 +....... (A) unde nanciscimur aequationes (1,,..., 2*, Yn a) = (,,1,, * n) (a, 141, a2),, =I, X A)= ( =, I..2,... ); e quibus concludimus fore (a,, A,..., * n) -x (2, A 2,..., In)2= ( —l)n+1 2, (1,,..., * An) (,..., n-1) = 2 (I, 1 2, *..., n_1)2 + (- 1) Itaque, si fiat e= (-1)n, erit e= +1, vel e -1, prout X (vel -X) est formse 8n+3 aut formse 8n+7; atque per transformationem jam saepius in hac commentatione usitatam habebimus hujusmodi evolutionem a + / = [2a, k1, 2eJ2,..., k2, 2e k, ca, 2kl, k2,..., 2 k2, kl] cujus veritas facili confirmatur, si formulam ex evolutione vulgari oriundarn paullulum immutatam adhibemus, a + /X = [a, O, a1, la,, n, -l a O. ia,, yn-2) **... 22 I]. Si autem est a = b + 1, erit /A = [a,, a 2,.* *,2, (a - 1 + /)], quee sequatio, si sic scribatur,,/-=[ +, 0, -~, [a...,, -, 0, (a + 1 + )], abit in formam precedenti (A) similem, atque eodem prorsus modo tractari potest. Itaque habebitur evolutio ad hanc normam concinnata a + 1 +X= [2(a+ 1), k, 2 2,..., k2, 2, (a+ ), 2e k, k2,..., 2 ek, kJ] in qua observandum est quotientem kJ negativum fore. R r 2 308 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 12. Schema autem calculi, quo ad has evolutiones utimus, hoc ferme est. Sit a = l, 2ea =2- X, numero negativo /o ita determinato ut a0 integer atque idem quam minimus fiat. Tur sumpto ab aequatione 2 ao +2 /x +aCx2 = 0 initio caeteras hoc pacto derivamus l/3fE3o, mod 2a1; a= 2 ca /2-, r1 mod 2 a; a3 = 2 e x - 82 SE/3s-1,mod2a,; a,+l= 2=as ubi numeros impares 3, ad valorem radicis (- 1)8,/ quam proxime accedere, quotientes autem k, per aequationes k, =O - O- determinari intelligendum est. 2 ea, Finis autem calculo faciendus est simul ac pervenerimus ad aequationem in qua est a++ ai+ 2 = 0, quaeque adeo suppeditat discerptionem qumsitam. Quando autem e = +1, hoc est quando X, vel X, est numerus primus formae 8n+ 3, fieri potest ut regula generalis in ultimo quotiente ki falsa sit; quae exceptio tum locum habet cum in discerptione quaesita X-A2 + 2B2 numerus B major est quam 2 A. Quo in casu erit in penultima 3equatione a ~+ 2i + 3a,+ = 0; atque, ut habeatur ad extremum a + +ai+ 2=0, quotienti ki tribuendus erit valor + 1, etiamsi fiat IO = + 2. Exemplis brevitatis causa supersedemus; illud unum adjicimus, pro numeris primis formae 8n+3, eorumque duplis sequationes x2- Xy2A, == -2A, = -A+2B, esse resolubiles, si quidem ponatur X = 2A2 + B2, et numerus A eo signo afficiatur, ut fiat (- 1)(A-l)= 1, vel (- 1)(-l)+(A- ) 1, prout est ipse X numerus primus, vel duplum numeri primi: quae quidem conditiones ex ipsa cTquationum forma oriuntur. Et similiter pro numeris primis formae 8n+7, eorumque duplis, oequationes - y2=A, = 2A,= 3A+ 2B, resolvi poterunt, designantibus literis A et B minimos numeross qui aquationi X = B2 - 2A2 satisfaciunt, et determinato ut supra ipsius A signo. Art. 13.] IDE FRACTIONIBUS QUIBUSDAM CONTINUIS. 309 13. Coronidis loco observamus eandem fere fractionum continuarum transformationem utilitate non carere in theoria numerorum complexorum. Sit enim A = a + b, A = a - bi, juY = a2 + 2 = JV. u = N. u'; habebimus hujusmodi aequationem K= (m!, *.., ) x (, 3, gA, /1) = (/l, /4,.03..., ) x (,.,, 1U) -(y1n 82n *,, 2) X (,2n *,** 82n l) + (8n y 25 *-* n 2n-1) x (2~2n-l *.*. /A2) Pl); unde sequitur determinantem K summam esse duarum, quas vocant, normarum, hoc est, quattuor quadratorum. Ponatur ergo PP2 = 1 + RR; designantibus literis Pi, P2 numeros positives reales, R1, RI numeros complexos conjugatos; atque in hac sequatione fiat = RI- l P2, P3 2 f2 P2 et sic porro, numeris R. ita determinatis ut norma R, x Rs evadat quam minima. Hinc erit P1 > P1 > P3,... atque ad extremum habebitur P1= ( M, ', **/ 3', /2' 14); quoe formula, cum universi numeri primi formulam 1 + x2 + y2 metiantur, continet demonstrationem theorematis Fermatiani omnes omnino numeros esse summas quattuor quadratorum. Plane autem eodem modo evincitur omnes numeros habere hanc formam X2+2 + 3 (u2 + v2), atque prseterea hanc etiam x2 + 2 y2 + 3 u2 + 6 v2. Quarum enuntiationum altera intra veritatem cadit, cum omnes numeri in alterutra harum formularum 2 + + u2 2 + 3 + 3 v2 comprehendantur; id quod a viro clarissimo Lejeune Dirichlet* jampridem demonstratum est: altera ea ipsa est quam olim summus in hac disciplina magister C. G. J. Jacobit ex notissima formula elliptica originem trahere indicavit. Hoc autem loco satis erit demonstrare, proposita aequatione p2 + p + 1 = 0, omnem numerum realem esse summam normae simplicis et duplicis. Sint igitur A, B, P numeri reales, A-Bp numerus complexus integer continens radicem aequationis p2 + p+ 1 = 0; erit A2 +AB+B2= N. (A - Bp). * 'Journal fir die reine und angewandte Mathematik,' vol. xl. pp. 231-232. + Ibid. vol. xxi. 310 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 13. Quod si ponatur A-Bp=/ P+ a-bp, numerus complexus l ita determinari potest ut sit N. (~ - bp) < (P2- ). Fiat enim, quod utique licet, a -A, b- B, mod P, [a] < P, [b]: - P. Hie, si signa numerorum a et b contraria sunt, habebimus manifesto N. (a -bp) < P2< (P2 ). Si' vero hec signa conveniunt inter se, atque est pr-eterea uterque numerus P-1 a et b minor quam -, erit iterum N(a - bp) < I (P - I)2 < 2 (P2 - 1) Quod si alter eorum, velut a, eum limitem superaverit, faciendum erit a'= a+ P, [a] < P, ita ut habeatur, si quidem P > 5, a. ('- b6) < (P/6 - P+ 1)2 < - (P2 _- ); experiendo autem invenitur eadem lege teneri numeros quinario minores. Hine patet calculum quo sepius jam usi sumus accommodari pose ad aquationem PP 1+2(A AB+B2) in qua si fiat A2 Bp=A - Bp- P2, 1 2 (A2 A2B2 + B2) P2 et sic porro, ea lege ut normoe A2+AB+B2 semper quam minimme evadant, veniemus tandem ad oequationem P1= (m, 2' 1, 2 ff, 23 82) 2 A1)~ qume continet demonstrationem theorematis, cum omnis numeruns impar metiatur formulam 1 + 2 x2 + 6 y2, vel, si mavis, formulam 1+ 2 (x + -y2). Simile fere theorema oritur ab mequatione 2 + c- +2 =0. Universi enira numeri primi, preiter septenarium, metiuntur formulam 1 +2 + 7y2, hoc est, formulam 1 + X2 + xy + 2 y2: ac preterea, posito P,1P2 = + A + AB + 2 B, utique fieri potest P2 > A + A,1B, + ~21 > Pi.' I I I~~ - kJ Art. 13.] DE FRACTIONIBUS QIIBUSDAM CONTIMUIS, 311 Unde patet numerum quemcumque integrum summam esse duarum normarum; vel, quod idem est, in hac formula x+ 2 y+2 y+ + uv +2 V comprehendi. Omnes itaque numeri pariter pares in hac forma continentur 2 + y2 + 7 u2 + 7 2; impares autem, atque impariter pares aut in illa aut in hac certe,x22.y2+ 7 (u2 +2v2). XL. ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. [Messenger of Mathematics, Ser. ii. vol. xi. pp. 1-11 (May 1881)]. RIEMANN, in a fragment published after his death in his collected works*, has determined the value of certain q series in the limiting case in which the analytical modulus of q is unity. If q = p ei, the series considered by Riemann contain 0 only, and converge and diverge for an infinite number of values of 0 between any two limits however near to one another; they appear, indeed, to have attracted Riemann's attention by this peculiarity. Two of these series, which may be regarded as examples of the rest, are considered in the present note, and some details relating to them are supplied which are omitted in Riemann's brief record of his results. Let S(q) =4.......(i) n denoting any positive integral number from + 1 to oo, so that (' Fundamenta Nova,' p. 103) S () = log 4/ - Put q =p e0, multiply each side of (i) by = and take the rectilineal q P integral from 0 to p es along the vector ei; we have fi S(q) dq=4 z( ) log(1+pen),...... (ii) --- ----- -- -q n2 * ' Mathematische Werke,' p. 427. ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. 313 the logarithms on the right being so taken as to vanish with p; or, which is the same thing, Pe"OS dq = i( t + (Z x, -,(-1)~ tan (iii) o n2 n2 \l+p cosnO ' where A\ is the arithmetical square root of 1+ 2pn cos n 0 + p2n, log A. is a real logarithm, and tan-'(I pnina) is an angle included between I + p& cosn a the limits - I 7r and + 7r. So long as p < 1, the series S (q), and the integrated series (ii) or (iii) are absolutely convergent. When p= 1, the series (i) and (iii) respectively become 4(- 1) e2lg2-2~I 4(f n 1-e-0 = 2ni 1og'-Z(-)n '- (n )n 62nit n 1 + e nt 0on n eCt + e- 1nf& =-21og2+2itz tan (nO),........ (iv) n7 and 2 2 ( ) log { 4cos2 ( 0)} + 4i( ) [( )];... (v) where by [1 (nO)] we understand the absolutely least angle which has the same tangent as [2 (n 0)], except when [2 (n 0)] is an uneven multiple of I rT, in which case [2 (nO)] = 0. The imaginary part of (v) is always convergent; the convergence or divergence of (iv) and of the real part of (v) depends on the nature of the value assigned to 0. First, let - be incommensurable. In this case the series (iv) is always di7r vergent. For if b,, a are two consecutive convergents to - of which the latter b'b bT7r has an uneven numerator a, and if d is the complete quotient immediately succeeding b, we have i b0 a- ~bb whence, when b is very great, the absolute value of tan bO is approximately,-~~~~~~~~~~~~- 2 b' __ b. Thus the series (iv) contains an infinite number of terms of which the 7r VOL. II. s s 314 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. absolute value is finite, and it is consequently divergent. Riemann says that when - is incommensurable, the real part of the series (v) is also always diverg7r ent. But this appears to be an oversight; for the real part of (v) is in fact divergent or convergent according as the incommensurable quantity 0 does, or does not, satisfy certain conditions; and, in particular, it can be shown that, whenever 0 is the incommensurable root of an equation of a finite order having integral coefficients, the real part of (v) is convergent. For each successive value of n we form the equation 0 l X -- m = -2 (1 - 0), 7r where m is an integral number, e = 1, and 0 < v < 1; it is evident that m, e, and v can always be determined in one way and in one way only. We then have log {4 cos2 (n0) } = log {4 sin2 2 (v7) }, 7r VJT 2 the series (-) 4s ) n2 log p2 1 is absolutely convergent; i.e. the series z (-~2 log {4 cos (n0)}.*eec.e (vi) is convergent or divergent according as the series 2;(-1) - 1 g ee..... (vii) is convergent or divergent. Let - be the root of an irreducible equation 7r f(x)=Axz8+Bx8-l +... +Mx+-N=; of which the coefficients A, B,... are integral; it may be shown as follows by a method due to M. Liouville, that the series (vii) is convergent. 0 Let x1, x2,... be the real roots other than - of f(x) y~i, y2i, y +iz2, its iaginay oots; and let 1 be a positive quantity greater than the greatest its imaginary roots; and let ~ - 1 be a positive quantity greater than the greatest ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. 315 0 0 of the differences - x, - -y; we have 7r r7 sf(2 — E)=A > -v xn xrI(2n~E x)x{(22 +-e — y)2+2}. nsf \ A n ^ \n X XI nz -Y Z But nsf(2 +-) is an integral number, which cannot be zero, becausef(x) is irreducible; hence 1 v > A S> s-' or - log v < (s - 1) log n + c, where c is a finite constant; this shows that (vii) is log n convergent, because 2; n is convergent. The same reasoning would suffice to prove, that when - is a root of any equation such asf (x); the series z log {cos21(n0) +(~n) always converges if Z log is absolutely convergent. On the other hand, between any two limits however near, there is always an infinite number of values of 0, for which the series (vi), and indeed any series of the form 2 ( log cos2( )}... (viii) p(n) becomes divergent, however rapidly the positive function +(n) may increase with n. If, for example, in the development of - in a continued fraction, it 7r happens an infinite number of times that a convergent C, having an uneven numerator a, is immediately followed by a complete quotient which surpasses cb (b), where c is any finite positive constant, the series (viii) contains an infinite number of terms of which the absolute value is finite, and is therefore divergent. 0 a Secondly, let - = b-, a rational fraction in its lowest terms. If a is uneven, 7r 0 the series (iv) and (vi) contain terms which are infinite. But when a is even the two series are convergent, and Riemann has succeeded in assigning their sums in the following manner. S S 2 316 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. Writing 2a for a, and observing that b is uneven, and designating the sum of the series (iv) by - 2 log 2 + 2iS, we have o ^n=oa(- (l)n. 1 / \ -=2b t=0 (-1)s /S) \ S =? =l — n= tan ( - - -s =0 2bt+s tan T7r) *.s=2b SCa 1 XS-ldx =s=lZ tan (-7r X 1 -X2b' i7r But if =e, an( ) 2 = as v =b-1 (_)-172acs and J X2b 2 b 1 7 1-; whence S- 2- - = ( -- 1)a- i -; = 1 - ]s=2i (- 1) SyT(2aa-) But Zs=2b(_ 1)ys(2a-*T)= 2b or 0, according as the congruence r -2ac + b, mod 2b, is or is not satisfied. We have therefore f 2a dx S -cr=-l (-1) o 7+adx z_1-( 1) 1 + 2aCaX 1 + - 2a a ) ( = 2,^ b- 1) MY (- - 1 - -2a7a (1+i'7 2aOx i+7-2a aX But f1 sin 0 dx =[1 1 + 2xcos 0 x 2 and hence, finally, S= 2i7r '=j (b- 1) (- [ a - i7r Z=b ( — 1) [{-] r ai a the symbol La' - denoting the excess (positive or negative) ofo- babove the integral number lying absolutely nearest to r a. The method employed by Riemann to effect the preceding summation is due to Lejeune Dirichlet, and was employed by him in his memoir on the Arithmetical Progression (' Transactions of the Academy of Berlin for 1837,' sections 4 and 10), and in his ' Recherches sur diverses applications de l'analyse infinitesimale a la thdorie des nombres (' Crelle's Journal,' vols. xix. and xxi; see sections 1, 9 and 10 of the Memoir). 317 0 OE DisCoNlTILNUObUS SERIES C~ SIlED Aan, denoting yby S' the sum o the series (- Ilog { 4cos2 I(n 0)}, Vz%7 I n 2o1 we have, if 0= -b 7 fm (_) S= l 2 1 Z - - L1) log Cos2 S (bt + s) S' -2log2xZS + y = _b ~,rr21, = ^ b-) (_ 1) log (4 cos2L xT -t= b + og6b2 + ' But, by a fornula due,t-~+ (+ -5)t Ls 'It= - 0 (bt +t S)2 S~,to Euler, 1 ~.__-CC —1 + I2+s)2 (4+s ' (2b + S)2 (2 b -4 I 1. -[^^ ^ 1 7= 4 --- cosec2" s 2-b) 4:b2 -C(2 b 4b' 72 2,OS(3b ) 7T 7T2 in( b2 sin-2 IS( whence P nllrr, nulliii, a, zr2 s2 5i (b -~ -l) ( g - ) s b)X =s 6b2 " vS Ir /ST\ cos ) b,- I si ('7r\ /S TT\ cos `(-; sin~ ^) 2= _ r2 s=b-l- (-y)s log (C4 CoS2s 7r log 2 + - l == — -b2 b x When the series -l) lo g 14 cos2 (n0)} 1 fb dverent we must regr dthe equation 'IS -7? I Io n O Cy> p x -ec~ 14 >e'6 ).(~~n., ~1 4: ( -c o i 2 I I. n o1 ( (iX) =2L~w2*.i2 onvergent, the equation */When that series iscnv ashaving no assignable xneanin. 318 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. always has a meaning, but some explanation is required to show clearly what its meaning is. As to the signification of the left-hand member no question can arise. For just as by p (x) dx we understand the limit of J (p (x) dx when h is increased without limit, so by ~1 (-1)n p enio p J~O n l+pneni 0 p we are to understand the limit of inp2(_ l)n pneni~ X dp Jo n l+pineneio p when 1 - p is positive and decreases without limit. We thus have the equation 4fI (-) 1+pe xn p =4xlimitof ( ) log(l+peni) Jo n +pnene p n2 log (1 + n en when p = 1. But before we can pass from this equation to the equation (ix), we have to show that the limit of (-1) log (1 +pneni), when p=l, is ( )n Z - log (1 + e n). And although this is true, it cannot be assumed without proof; any more than we could assume that the limit of the series (sin x)+ (sin 3x) + (sin 5x) + ".. when x = O, is the same (which it certainly is not) as the value of that series when x = O. It would seem that Riemann in his fragmentary note intended to give the required proof; for he developes the expression log log+ ) n. n log (1 + qn) in a series proceeding by powers of q (just as Jacobi has dealt with similar series in the ' Fundamenta'), and finds Z (-I) log (1 + qn) = Z1 + 1)2 (n) q, F (n) denoting the sum of the uneven divisors of n. He also enunciates and ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. 319 proves the following theorem (which is really due to Abel, CEuvres, vol. i, p. 69*): ' Si series ao + a + C2 + *, eo quo scripsimus ordine summata summam habet convergentem, functio ipsius r hac serie a0 + C1 r + a2 r +... expressa, convergente r versus limitem 1, convergit versus valorem eundem.' But the fragment breaks off abruptly with the words 'ex hoc theoremate...... facile deducitur,' and it is not very easy to see how he proposed to complete the demonstration, because the series zl1(-1) 2(n)q is certainly not always convergent when q =ei. The following considerations may serve to replace Riemann's intended demonstration. Denoting ei0 by q0, we have, in fact, to show that lim Z (- x log 1 + = 0, when p converges to unity, or q = p e to gq = ei0. The imaginary part of this series is il( ) lim[tan-' )-Q ]].... (X) -n2 1im tan- (1 + pn cosn 0 2 the real part is limi( log[p+(1p sec2(n)];.. (xi) 22 it will suffice to consider the latter only. Let h be any very great number; and let A be so great that 4Ah is greater than the greatest value of sec2 2 (nO) for any value of n not surpassing h (we observe that sec2L (no) cannot be infinite, because, in the cases under consideration, - is not a rational fraction having an uneven 7r 1 numerator); also let p be so near to +1 that 1- p < for all values of n not surpassing h; then, for all such values of n, p + { _(1-pf) 2 sec2I (no) lies 1 1 between 1 - -, and 1 + -; and, consequently, nlo (-g ) log OPn +. (I _ } 2 - )x * [The reference is to Holmboe's edition, 1839. In Sylow and Lie's edition (1881) the theorem occurs in vol. i, p. 223.] 320 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. is evanescent, being intermediate between the two values log (1 + x 1=h, which are both evanescent because - is as small as we please. Again, the series r= ( -) log [p + {-(1 sec2 (no)].. (xiii) n=h+l ' [PI +' is evanescent, for pn + {1 (1 - pn)}2 see 21 (nO) is certainly greater than ~ and less than sec2 (n); whence this series is less in absolute magnitude than the sum of the two series log4 1 2 log sec2 } (nO) log4x h+lnS. =-7&1 x n ( 2 l=l A^+12 n=h+l n n and of these the first is evidently evanescent when h is very great; the second is also evanescent because, the series Z log se2 (n0) being convergent in the n2 cases here considered, the remainder after h terms vanishes when h is very great. Hence, finally, the series (xi), which is the sum of (xii) and (xiii), is evanescent. XLI. NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Messenger of Mathematics, Ser. IT. vol. xii. pp. 49-99 (August-November, 1882).] I. On the Formulce of Transformation, with especial reference to the case in which the modular equation has equal roots. 1. LET y= F(x), where F (x) is a rational fraction containing x~, but no higher power of x, be an integral of the equation dy2 1 dx2 (1 y2) (1 \2y2) - 2 (1 - X2) (1 - k22) so that, if yo = F(x), ^ ___ ^/ Cdy 1 f dx I d~r 1 dx. (2) yo {(1 -y2) (1 - X2)} {(1-x) (-k2y2)} ( the track of the integration and the initial sign of the radical being assumed arbitrarily on the right-hand side, and being determined on the left-hand side in accordance with these assumptions by the equation y=F(x). If (4K, 2iK'), (4A, 2 i A') are pairs of conjugate periods of the integrals* -r cdx a ndy { x2)(1 k2X2)} and / (1 _ y2) (- y2)} * Any value of one of these integrals extended over a closed track, at the end of which the radical has the same sign that it had at the beginning, is a period of the integral. The periods, and the pairs dx of conjugate periods, of the integral {(1 - ) (1 -- k2 2X) are, in fact, the periods, and the pairs of conjugate periods of the doubly periodic function x = sin am u defined by the equation r_ dx _ du _ {, {(1 -x ) (1 -k2x)}' + 1. VOL. II. T t 322 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. respectively, the equation (2) implies the equations 4K M 4aA + 2biA' 2liKE1 (3) M - 4cA - 2diA' in which a, b, c, d are integral numbers; because whenever x describes a closed contour, at the end of which the radical under the integral sign has the same value that it had at the beginning, y describes a like contour, and consequently any period of the integral on the right-hand side of the equation (2), must be a sum of multiples of the periods of the integral on the left-hand side. iK' The determinant ad-bc==m is different from zero, because the quotient K is not real; and is positive, if (as we may suppose to be the case) the coiK' ZiA' efficients of i in the complex quantities tK, -- have the same sign. Let us now assume that the fraction F(x) satisfies the conditions F(O) =, F(1) =, F(o) = o; in this case, we have the equation _Y _ dy I L dx J V{(l -_ y2) (l - MX2y2) V ( Jo V { 2(l ) - (4) and as particular cases of it, the equations 1 dy _1 r d 1 (1 2) (X2 2)} M J (1 - X2) (1 - k X2)) rdy 1 rc*_dx Jo { - y2) (1-X2y2)} Mi V{ (1 -x) (1 - k2x2) The equation (4) implies that F (x) is an uneven function of x, and also that 1 the limit of Y when x =0, is +; the condition F(oo) = oo shows that the numerator of F(x) is of higher dimensions than its denominator. Consequently n is an uneven number, and the equation y = F(x) assumes the form _ x 1 +a lX +2x4+...+a(l_)n-xny =p 1 + bX2 +b2 x4 +... + b(n1) Xn-. (6) the coefficients a and b being connected by the relation l+2b= and the initial values of the radicals in the equations (4) and (5) being + and the initial values of the radicals in the equations (4) and (5) being + 1. Art. 1.] NOTES ON THE TTHEORY OF ELLIPTIC TRANSFORMATION. 323 We shall henceforward suppose that the pairs of conjugate periods (4K, 2iK'), (4A, 2iA'), employed in the equations (3), are prim)ary.* On this hypothesis, since K is one of the values of the integral r1 _ dx,t(il - ) (i - 2)} it follows that is one of the values of the integral J {(1 y. 2) (1-X2y2).(7 similarly, iK' is one of the values of the integral 00 dx o t{(l-x) (1- 2X2)} ' iK' and is one of the values of the integral J (ly..... y(7) o, /i J (- Y2) (i - x2 y2) 7. But all the values of the first of the integrals (7) are comprised in the formula (4p + 1) A + 2qi A', and all the values of the second of these integrals are comprised in the formula 2rA + (2s + 1) iA', p, q, r, s being whole numbers; we have therefore = (4 +1)A + 2qiA', iK' = 2rA + (2s + )iA', or, which is the same thing, the numbers a, b, d in the equations (3) satisfy the congruences mod 4, mod 2 a 1, b -0, mod 4; d-1, mod2, so that m = ad - be is an uneven number. Let u be one of the values of the integral Xp rdy jMoX V{(l y2) (1 x2y2)}' corresponding to a given value of y; then all the values of that integral, cor* The definition of primary pairs of conjugate periods, and a statement of the properties relating to them which are employed in the text, will be found in Note II. Tt 2 324 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I, responding to the same value of y, are comprehended in the formulae u +f+(4qA- 2iA')...(8) 2MA - u - Mu(4q A - 2pi ') ' where p and q are indeterminate integers. Writing 2mrMA for 2MA in the second formula, which is admissible, because m is uneven, and replacing 4A and 2iA' by their values in terms of 4K and 2iK', we obtain the formulae u -- [(pc + qd) 4K- (pa + qb) 2iKt'] m/n1. j.'. ~........ (9) 2K- t - - [(pc + qd) 4K- (pa + qb) 2iK'] in the second of which we have written 2K for 2dK- biK', as we may do, because d is uneven and b even, and because we may omit multiples of the periods 4K and 2iK'; any such omission being in fact equivalent to a change in the indeterminates p and q. Thus all the values of the integral X dx.... (o) JoV{(l-x2) (l -k2X2)} (10) which correspond in the equation (4) to the given value of y, are comprised in one or other of the two formulae (9). But since the congruences a + qb, mod....... (11) pc+qd-s) are resoluble for m and only m pairs of values of r, s, incongruous to one another for the modulus m, the values comprised in the formulae (9) group themselves in 2m sets not reducible to one another by the addition of multiples of the periods 4K and 2iK'. As representatives of these 2m sets we take the m pairs of values 4sK- 2riK' u, +. ---m L.... }.(12) 2K- 4sK- 2riK' 2K- uz - --- and we observe that to values comprised in the same set, or in two sets of the same pair, there answers one and the same value of x. Hence in the equation (4), and consequently also in the equation (6), to a given value of y there answer m values of x. We conclude that rn = n. Writing a = a, b = 2 i, y = 2 c, d = -, we may enunciate the theorem: Art. 2.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 325 'If y = F(x) is a rational integral of the equation (1), satisfying the conditions F(O)=O, F(1)=1, F(co)=oo; F(x) is of the form (6), the exponent n of the highest power of x contained in it being uneven, and any two pairs of primary periods of the elliptic integrals are connected by the equations M=aA + 3A', M 'yA+SiA',j where a, /3, y, ~ are whole numbers satisfying the equation as - /3y = n, and where the matrix ' fis primary; viz. it satisfies the congruences a=1, mod4; a, / 1 ~, mod 2.' 7, 0, 1 2. The foregoing theorem and its demonstration will be found in substance in the 'Traite des Fonctions Elliptiques' of MM. Briot and Bouquet. Both have been reproduced here, with some slight modifications, in order to establish with precision the relation between the rational transformations y = F (x) of the elliptic integral (10), and the arithmetical transformations (3) of its pairs of primary periods. The following observations, though not necessary for the immediate purpose of this note, may be useful as further illustrating this relation. (a) Every rational transformation must satisfy the condition F (x)= 0, as otherwise the equation y=F (x) would indeed be a rational integral of the differential equation (1), but would not transform the integral (10) into an2KJ other of the same form having the same lower limit. Since M must be equal to a sum of even multiples of A and A', this condition implies that b is even in the equations (3); or, which is the same thing, that every rational transformation y = F (x) corresponds to an arithmetical transformation of the periods which may be expressed in the form K K = lA + i'. the coefficient 7 being an even number. 326 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. Again, when the condition F (0) =0 is satisfied, a in the equations (3), or a in the equations (13) is uneven. For we can cause x to travel from x to -x by a route such that the right-hand member of the equation (4) shall be in2Kt creased by M; y will travel at the same time from y to - y, and the left-hand member of the equation will be increased by 2A together with certain multiples 2KV of the periods 4A and 2iA'; 2 is therefore equal to an uneven multiple of 2A, together with a multiple of 2iA'. The condition F (0)= 0 being satisfied, F (x) is (as we have already seen) a fraction of which the denominator is an even, and the numerator an uneven function of x. The order of the numerator may be either higher or lower by an unit than the order of the denominator; in the former case we have n uneven, and F (oo) =Go, in the latter, n even, and F(oO) =0. Further, if iK' F (c,) = ca, - must be the sum of an even multiple of A and an uneven iK' multiple of iA'; if F(oo) = 0, - must be the sum of even multiples of both A and iiA'; hence in the former case, d in the equations (3), or J in the equations (13) is uneven; in the latter case, this number is even. Thus the matrix a is of one of the four types, mod 2, m uneven, 0, 1 ' 0, 1 1 1, 0 1, 1 0, 0 ' 0, 0 (c) The demonstration that m =n, given in Art. 1, applies only to the case when m is uneven, and b even; but in all the cases alike the second of the formulae (8) may be written in the form - u - M[4 (q - ')A - 2piA'] t so that the second of the formulae (9) becomes -U - [{pc+(q-)d} 4K- f{a+(q- ) b}2iK'J. In discussing this expression we have to consider the congruences C 7a + qb - b + rm pc+qd=- dd- m+s mod m Art. 3.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 327 which, like the congruences (11), are resoluble for m and only m pairs of values of r and s, and are moreover resoluble for the same pairs of values of r and s as those congruences, because the congruences pa + qb — b 7d2 i - X mod nm, oc+qd^ -d-_m, are resoluble. Hence the values of the integral (10) answering in the equation (4) to a given value of y are comprised in 2m sets, which, as before, may be represented by the formulae (12); we have therefore m = n in all the four cases. (d) The remaining condition, F(1)= 1, renders a 1, b-0, mod 4, in the equations (3), or a- 1, mod 4, / -0, mod 2, in the equations (13). Thus, finally, if F(x) satisfies the two conditions F(0) = 0, F (1) = 1, we must have either n uneven, F(o))= o a, S- 1=', mod 2 a, O,1 or neven, I(cf)=0, | a, e 1 O, mod 2. T, S 0,0 When a or a is uneven, the condition that it is to be- 1, mod 4, can always be satisfied by changing the sign of i. It is important to observe that, if the pairs of primary periods have been chosen once for all in the equations (3), the rational transformation y = F (x) can correspond to only one arithmetical transformation of the periods; viz. an equation of the form aA + 3iA'= a, A + 0 iA' implies the equations a=al, 3=/l, because the ratio A: iA' is always imaginary. 3. If in the equation (1) we regard k2 as a given quantity having any value except one of the three 0, 1, oo, and Ml, X2 as quantities to be determined, any rational integral y = F (x) of that equation which satisfies the conditions F(0)=, F(1)=1, F(o) = oo, is termed a primary transformation of the elliptic integral (10)*. It results from * The word yrimary was employed by Gauss in the theory of complex integers of the form a + bi, to distinguish one of the four associated uneven numbers iR (a + bi), [ = 0, 1, 2, 3, from the other three; and the same expression has since been employed in other complex theories. The essential character of primary numbers in these theories is that te e they reproduce themselves by multiplication; viz. the product of two primary numbers is primary. It is natural to extend the use of the term to the 328 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. the theorem of Art. 1 that every primary transformation of this elliptic integral implies an equation of the form i, a+/3A where |, | is a primary matrix. Conversely, it is known from the theory of 7, the transformation of the Theta functions, that every equation of the form (14) supplies one, and only one, primary transformation of the elliptic integral; and further, that if two matrices a, are equivalent by post-multiplication with the transformations corresponding to matrices not so equivalent are always different follows from the concluding observation of Art. 2. Thus the number of different primary transformations of the order n is the same as the number of non-equivalent matrices* of determinant n; i.e. it is - (n), if ( (n) is the sum of the divisors of n; and there is a correspondence one to one between the primary transformations and the non-equivalent primary matrices. These o- (n) transformations are, however, not all primitive;t viz, if a, A3, 7, S have a greatest common divisor z, the transformation corresponding to a, 7, S is not primitive, but is compounded of a multiplication of the argument by M, and of a primary and primitive transformation of the order = -, corresponding theory of matrices; and to characterise as primary, matrices of the type defined by the congruences? c, d 1O., mod 2, a = 1, mod 4, because this type reproduces itself in multiplication. Again, it is convenient to designate as primary those pairs of conjugate periods of an elliptic integral or function which are formed from the fundamental pair (see note II) with primary unit matrices. Lastly, the introduction of the phrase primary transformation is justified by the consideration that the primary periods of the elliptic integrals transformed into one another by a primary transformation are connected by a primary matrix; and that, as a consequence of this relation, the composition of two primary transformations gives a primary transformation. * For brevity, matrices equivalent or non-equivalent by primary post-multiplication are here simply termed equivalent or non-equivalent. t The word primitive was applied by Gauss to quadratic forms of which the coefficients are relatively prime; it is natural to extend the use of the term to matrices of which the elements have no common divisor other than unity; and consequently to the rational transformations appertaining to such matrices. Art. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 329 a 3 /x / to the matrix;in fact, in this case the transformation (6) arises from /h / the combination of two primary transformations =F (x), y = F2 (), of the orders t,2 and v respectively, corresponding to the equations r _ dz rX dx JOZ vj/(l -( X2) (-2)} k22)} rdy 1 r d JoV{((1y2) (l-X2y)} = MJo {(1 -2) (-22)} of which, however, the former represents a multiplication, and not a transformation properly so called, of the elliptic integral. The number of primitive transformations is of course the same as the number of non-equivalent primitive matrices a and is given by the formula c'(n) = I(1 +), where the sign of multiplication extends to every prime divisor p of n. The primitive transformations are alone to be regarded as proper transformations of the order n, and in what follows we may confine our attention to them. iK' iA' _ + Q2 4. Let W = = K Q= so that co= -; and let 4 (0) be the function AL a+Q i i2 defined by the equations () 1 e 2mirO olI=2n1 +e:-^=;....... (m ) ~(o)=~/2e8;~enlqn l =++o 2 -~t +......(15) this function is characterized by the two converse properties (i) that the equation C+)DO 1- A~BO'.:. (16) AB. where C, D is a primary unit matrix, involves the equation (01) = (0),=....... (7) and (ii) that the equation (17) involves the existence of an equation of the form (16). VOL. II. U U 330 3NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I, It is known, from the theory of the Theta functions, that 2 = ~8(w), XI= 8 (Q); and that the a' (n) modules p8(Q), which present themselves in the different primary transformations of order n, are the roots of an equation f(k2, X2)= 0, termed the modular equation, which is of the order o-' (n), and of which the coefficients are rational and integral functions of k2 = 8 (). In the general case, in which the roots of this equation are unequal, the a' (n) primary transformations of the elliptic integral and the transformed modules correspond to one another one to one. We shall now show that, when the modular equation has equal roots, the values of M2 answering to the equal roots are different; so that, although in this special case two or more different primary transformations answer to one and the same transformed modulus, yet in all cases alike the a-'(n) different primary transformations correspond one to one to a' (n) different pairs of values of X2 and M2. To establish this assertion, the truth of which might be inferred from the concluding observation of Art. 2, we consider the equation of Jacobi 1 nk2(1 - k2) d.\2 ~ \2(l- X2) d.k ** ( 8) from which it appears that if for the same value of k2 we have simultaneously X2 - 2 =, M', we must also have d.x _ d.x2 ^dk2 d.k2 *. *.(19) But this equation is inadmissible. In a memoir ' On the Singularities of the Modular Equations and Curves,'* it has been shown that the modular curve represented by the equation (X, Y)=O, k =X, X2= Y, has no superlinear branches except at the points (X=O, Y=O), (X=1, Y=1), (X=co, Y=o0), which may be left out of consideration; and that it has no tangents at a finite distance parallel to either of the axes of X or Y. Hence the only points which need to be examined are multiple points free from any superlinearity; and at these it has only to be shown that two branches cannot touch one another. Let (hQ\ = 0 )\ ='Y1 + Jl O1 % + /2 92 0 (Q2)2 a, O, = Q 2 * 'Proceedings of the London Mathematical Society,' vol. ix. p. 242 [vol. ii. p. 242]. Art. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 331 the matrices a1, 1 and a2, being non-equivalent primary matrices of de71,1 - 72, 2 terminant n. The equation X2 = X, or 08 (Q1) = #8 (Q2), implies, as we have seen, an equation of the form C+ D fl, 2 A + Bi2' where A, B is a primary unit matrix; if then C,D a, a2, 2 | X A, B? 8 ^_, x C, D we may replace the equation a72 + 2 2 by the equation y+ Q2 a + o2Ql where a, | is a primary matrix of determinant n, equivalent to a2 2, and 7, S al, 72, 2 therefore non-equivalent to a, i 7 We now have = X2= (); but to calculate d.X2 in the equation (19) we must employ the equation a7 + 31, and to calculate d. X2, the equation _y +, Observing that a + Vw() = 5i 2 (Q) [1 - Q8 (w)] K2 (), '(,) = - 2 - f (iQ) [1 - 08 (Q,)] K2 (Q), where K(w) is the one-valued function of eiert defined by the equation A/2K(~) = 1 + 2eifTw + 2e4'i7 w+ 2e9ii +.., and that these derived functions are neither zero nor infinite, we infer from the equation (19) the apparently but not really identical relation d Q1), d Q1, dw) = dwIU 2) UU 2 332 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. of which the two members have different significations. This equation gives al+ Ql _ + a+_3Q Vn /Vn whence also + Q?l+31Q1 = _?+391 V/n Vn and finally, al=a, /31=, y=1-, = =, contrary to the hypothesis that,1 and 2 0 are non-equivalent. 71, 72' an 2 With a little modification this demonstration may be employed to show that no two modular curves can touch one another, except at the points [0, 0], [1, 1], [oo, oo]. Thus the -'(n) primitive transformations of order n are not only distinct from one another, but they are also distinct from any of the derived transformations of order n. 5. The result at which we have arrived, viz. that when two roots of the modular equation are equal the squares of the corresponding multipliers are always different, might seem to contradict the theorem, due to M. Kcenigsberger, that M is a rational function of k2 and X2. But it is to be observed that the demonstrations of this theorem fail (as they ought to do) when the modular equation has equal roots. For example, M. Kcenigsberger has observed ('Theorie der Elliptische Functionen') that a rational expression for M may be obtained by combining the equation (18) written in the form 1 - nkP(l-nk) (d.ik) 1- ).......X (20) X2 (I2(1x/2) dK (d. 2) with the equation of the multiplier; and this is of course in general true, but the right-hand side of the equation (20) assumes the form at a multiple point. Again, Prof. Cayley has shown (' On the Transformation of Elliptic Functions,' Phil, Trans., vol. clxiv, p. 423) how to form an expression for -, rational in k2 and X2, by considering the symmetrical functions s 2r s= 1, 2,...o'(n); r = 0, 1, 2,...; Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 333 but the expression thus obtained must assume the form ~ when two roots of the 0 modular equation are equal, for its denominator is the discriminant of the modular equation, and the multiplier M is never infinite. We may observe that the derived functions of X2 and M2 with respect to k2 are always finite, and rational in M2, X2, and k2; this may be inferred from the equation of Jacobi (18); because in the modular curve there is no superlinearity dY except at the excepted points, and because is never zero and never infinite; hence, even at a multiple point (Xo, Ye), the development of each branch of the modular curve is of the form 1.2 Y=u X+ 1822+.... where u, contains M2 as well as Xo and Yo, but 2t, i,,... can contain no new irrationality, because there is no contact of different branches. It may also be noticed that M, which, according to the theorem of M. Kcenigsberger, is in generalto, rational in k2 and, is always rational in 2, X2 This may be seen by evaluating according to the usual rule the satexpression for M, obtained by the method of Professor Cayley, and substituting for the derived functions of X2 with regard to k2, the equivalent expressions, rational in M2, k2, X2, which we have shown to exist. To obtain the values of M2 at the multiple point (Xo, Yo), we have only to form the equation, giving the directions of the tangents at that point. Substituting in this equation from the formula (18), we obtain an equation of the form X (-2 n Xe, YO) = 0. The coefficients of this equation are rational functions of Xo, Y,, having rational numerical coefficients; and, as we shall presently see, X can always be resolved into linear factors, rational in XT, YO, but having coefficients involving an imaginary quadratic surd. The coordinates (X0, Y,) of the multiple point are themselves numerical quantities, which may be irrational, because in the most general case the multiple points of a modular curve present themselves in sets; the abscissas, or the ordinates (as the case may be) of the points of any one set, satisfying an irreducible equation. 6. In the general case in which the roots of the modular equation are unequal, the coefficients a,, c2,.. b1, b,,... of the primary transformation are rational in k2 and X2, because the primary transformations and the values of X2 correspond to one another one to one; and, when two or more roots are equal, these coeffi 334 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. cients are at any rate rational in k2, X2, ] 2, because there is a similar correspondence between the primary transformations and the pairs of values of X2 and M2. In fact the coefficients a and b, as well as the multiplier M, can always be expressed as rational functions of k2 and X2; but in the general case the numerical coefficients entering into these expressions are rational; in the special case of equal roots the general expressions, or some of them, assume an indeterminate form, and new expressions present themselves, which may be deduced by evaluation from the general expressions, and in which the coefficients are no longer rational, but involve the same imaginary quadratic surd which enters into the expression of M. It is worth while to show how these assertions are consistent with the results of the algebraical theory of elliptic transformation. It is shown in that theory (see the 'Fundamenta Nova,' or the Memoir of Professor Cay]ey already cited) (i) that any primary transformation y=F(x) satisfies an equation of the form 1-y 1- x (P- Qx 2 1_= 1i x YQ)... (21) where P=a+?yIx2+..., Q=fI3+x2+..., are functions of x2, which are respectively of the orders 4(n-l), I(n-5); or (n-3), (n -3), according as n 1, or n_ 3, mod 4; the whole number of coefficients in P and Q being - (n +1) in both cases alike; (ii) that the determination of the ratios a:3: y... depends on the I(n + 1) equations included in the formula (k)x-1 [P2 ( ) + k2w2 P(22) Q F )+ k Q2 (12)]) = V/ 1k[P2+ 2PQ+ Q2 (2] which expresses the condition of Jacobi that the equation (6) or (21) remains 1 1 Y unchanged when we write in it for x, and - for y. Putting U for/ kx Xy U k we may write the (n + 1) equations in the form =?Un- 1-2- 1...(-1)=, l,. (...... (23) where 4(, and b are homogenous quadratic functions of a, 3,?,.... The elimination of a, 13, y,... from the equations (23) gives rise to an equation between U and k2, of which the order in U, according to the general theory of elimination is I (n + 1) x 2 (n-l) = (n + 1)2 - 3), but which must be capable of reduction to the Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 335 modular equation between U and k2 by the exclusion of extraneous factors; of this reduction, however, no complete account has as yet been given.* If we assume that to each value of U satisfying that modular equation, there corresponds but one set of ratios a:: 7..., we can infer that these ratios, and consequently the values of M and of the coefficients a and b, are rational functions of U and k2 with rational numerical coefficients. And, in the general case the assumption is justified, not indeed by anything which has been proved concerning the algebraical nature of the system (23), but by the one to one correspondence established in Art. 3, which suffices to show (as we have already seen in Art. 4) that, when the roots of the modular equation between X2 and k2 are unequal, only one primary transformation can correspond to given values, of k2 and X2, and, therefore, to given values of k2 and U. Again, to assert that M and the coefficients a and b are rational in Uand k2 is to assert that they are also rational in k2 and X2, for, as we shall now show, U is itself rational in k2 and X2, at least when X2 is not one of a group of equal roots in the modular equation between k2 and X2. In fact, if, as usual, u = l/k, v = vX, v U it may be shown that - and - are rational in u8 and vs. For, if u' Vs be any term in the modular equation between u and v, we have, by a theorem due to Sohnke (see the Memoir 'AEquationes Modulares pro Transformatione Functionum Ellipticarum,' Crelle's Journal, vol. xvi. p. 97), ns + r = n +1 + 8y, where /u is an integer, so that Ur V = +1 ( X U8 Hence, dividing every term of the modular equation by un+l, we have a relation of the form V2 u v2 1 f(L, 2 \)=f42 n U where in generalfy and 2 cannot vanish simultaneously; viz. if j and f2 vanish for the values u=u u', v = v', these functions also vanish for u = in', v - v', and * Professor Cayley in the memoir cited has shown how this reduction takes place in the case when n = 5; and Mr. Ely has given the corresponding determination of the extraneous factors for n = 7. ('Proceedings of the London Mathematical Society,' vol. xiii. p. 153; see also Prof. Cayley. 'Phil. Trans.' vol. clxix. p. 419.) The result in the case n = 7 possesses considerable interest, because it enables us to conjecture in what way the reduction takes place for higher values of n: viz. by the exclusion of factors corresponding to the modular equations of lower orders. 336 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. the modular equation between X2 and k2 has a pair of equal roots corresponding to k2 = u8. Thus - can be expressed rationally in terms of -z and is; but if Un Un U2 = U2, v2 = v2, and zuvg is any term in the modular equation between Ui2 and v2, we have ns+r=- n +l +4, V2 /V22 V4 whence, as before, -v is rational in ( 2) = and u4=us. Lastly, if iXs is any term in the equation between u2 = k and v2 = X, we have ns +-r= n +1 -+2J, X vX4 X2 v 2 whence, finally, 7 = - is rational in and k2; i.e. - is rational in X2 and k2. v /c2 u^~n2a Un Example. Let n=3; the modular equations between (u, v), {(u2, v), (k, X) may respectively be written in the forms Xk2 1 v _ k3 U2 -W I+ 6 k2 X2 k3 k 6 F+4 * x2 Us 1~k+ X2 X4 X2 4 k4-+ 6c2 + 1 43 4X2k2_33X2 3k;2+4 and give, by the elimination of -2 and - the following expression for -3 1 X4 (16k4 -13 k2 +1)+ 34X2k2 ( - k2)- k2(k4- 13k2 +16) i3 k2 32(1 -X2 - k2) + 5 (X4 + k4)+ 38 h2k2 - 8 2k` (X2 + k2) Of course, this expression is only one of an infinite number of equivalent forms, reducible to one another by means of the modular equation between k2 and X2; we have, for example, W _ X4(16k4- 13k2+ ) 342k2(1 - k2),- 2(k4- 13k2+16) v 32 2k2(2 + k2 - 2 k2) - 5 (x4+ k4) - 38X2k2+8 (X2+k2) Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 337 To obtain similar expressions for -, and -, we have only to interchange k2 and X2 in these formulae, and to change the sign. The values of the ratios a: -:..., or of the coefficients a, b, have been expressed in terms of u and v by Jacobi (in the 'Fundamenta Nova') for n = 3, 5, and by Professor Cayley (in the memoir already cited, and in the addition to it, 'Phil. Trans.' vol. clxix., p. 419) for n =7. It will be found that all these expressions contain only f and u8, or i and v8, and are consequently rational in k2 and X2, in accordance with what has been said here. Passing to the case in which the modular equation has equal roots, let (k2, X2) be a point of multiplicity s on the modular curve; we have already seen that some at least of the expressions for the ratios a::y..., and for the coefficients M, a, b, must assume indeterminate forms, and that the s different primary transformations may be elicited from these indeterminate forms by d. X2 differentiating with regard to k2, and by substituting for. k2' and for the higher differential quotients of X2, their values corresponding to the s different branches of the curve. But instead of employing a process of evaluation, we may return to the equations (23), and, before proceeding to eliminate, we may attribute in them to k2 and U the values which these quantities have at the multiple point. The system (23) will then admit of s different solutions; and will furnish an equation of order s for the determination of the ratio of 3: a, 1 2/ or, which is the same thing, for the determination of M = 1 + 2a It must be.21M a remarked however that the modular equation between U and k2 may have unequal roots, even in the case in which the modular equation between X2 and k2 has equal roots; (because in this case it is possible that U may not be rational in k2, X2, and that as many as four different values of U may answer to one value of X2); if this should happen, the equations (23) would enable us to express the coefficients a, b rationally in U and k2; but these coefficients would not be rational in X2 and k2. Example. Let n= 5; and, for simplicity, let us consider, instead of the equation between k2 and X2, the equation between u and v, viz. U - V6 + 5 2v2 (2 - v2) + 4uv (1 - U44) = 0. VOL. II. X x 338 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. The general formulae for a primary transformation of the fifth order are 2/ 1 U2 /1 5 U10 l+'{i(g-l) + (I+ ) X2+M^ X4 a-,4VX/v M v2 i M V 2 v5 X2 + U5 4 + 4 +1 2 +M+ t6 _(1) 1 v -U5 M v - v4u If 0 is a primitive sixteenth root of unity, (0, 05) is a double point on the dv modular curve, and the equation determining the values of -u is 5 d2+60-4dudv - 5dv =, dv 30-4+ 41 whence 3 0 = + The expression for - becomes indeterminate avaluatwc- 5 ~ M ing, we find 1 M= -1+_2 04, and we obtain two primary transformations of the integral J 7 4, included in the formula f ( Y dy __,. X d=4) x (2- 1) + X4 Y=X 1+ (2 i - 1) x in which either sign may be attributed to i. Again, writing k2 = - 1, U= 1 in the equations (23), we have a2- y2-0=, 2 2 + 2 7 +4a -+ 2a 7=O, 2/-2a7=0; whence we infer =a, 2a2+ 2 a3 + 32=0, or - -1 +i, = 1. a a The formula (21) becomes, when n = 5, (a2 + 2a o) + ( 2+ 2 a + 2 /3) X2+y2X4 Y ~a2 + (/2+2a7+2 a)X2 + (+72+2,7)z4 and gives, on substituting for the ratios a: A: 7 their values, the same result as before. It will be observed that, in accordance with the foregoing theory, the system (23) admits of two solutions, depending on a quadratic equation in 3: a or Y; it is a peculiarity, with which we are not here concerned, that the Art. 7.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 339 coefficients M, a, b are rational in u and v, though irrational in k2 and X2; the circumstance that k2 = X2 is also immaterial. The values of 1 might of course be obtained immediately by combining the dv equation (18) of Jacobi with the equation which gives the values of c at the double point. In this way we should find 3+4i 1 M2= ^25 A' - l-2, the extraction of the square root introducing a new ambiguous sign. This ambiguity may be removed by means of the equation of the multiplier, viz. 1 10 35 60 55 26 - 253k2 (1 -k2) 6A- Ms+ lM4 M3 2+5=, of which the left-hand member, when k2 = - 1, resolves itself into the product [(M ) ~] X [80 x M+5] shewing that = - 1 + 2i, as we have already found. 7. The ratio of the two unequal multipliers, which correspond to two equal values of X2, is always a simple imaginary quadratic surd. For, as we have seen in Art. 4, the equality of two values of X2 implies an equation of the form ai + 1i Q _ y+ SQ wa+PiQ a+Or' if therefore M1 and M are the multipliers corresponding to these two transformations, we have K-aA + =vA1% M-71A + il> -a +ifi M=?YA+3tA' J and consequently, Ml a +q 32 y_ +S M a= + i1iQ 1 + 2.(24) But Q is a quadratic surd; viz. if 27A = a71- a17, 47B = a1 —a+/ 3y- /3,7, -. (25) XX 2 340 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I. Q is a root of the quadratic equation A+2BQ + C22 = 0. In the equations (25) it is convenient to suppose that 27 is the greatest common divisor of 2(a 7i-al7), i(a 1-a-+ 137,-11, '(O-1/31), which are whole numbers, because the matrices a and a, 1 T 6 7i, S1 are primary, and which cannot vanish simultaneously because those two matrices are not identical; again, A and C cannot vanish, and AC- B2 = A cannot vanish or be negative, because the imaginary part of Q must be different from zero; thus we may attribute to r the sign of ay, - aly or 3S - 3bi, and may suppose that A, C and /VA are positive; we then have without ambiguity -B+iV^/ C (26) Let 2 Cr + 1 denote the uneven number, (a a~ + al-/3 l- y); we find immediately n2=(2 + 1)2 + 47A,.... 2. (27) and, substituting in (24) for Q its value given by (26), we obtain for M the expression M1 2 a + 1 + 2........ (28) showing that the ratio of the two multipliers is a simple quadratic surd, having unity for its analytical modulus. We might also arrive at the formula (28) by compounding the transformations a, 3 Q I, - Xxw,, c= x, I =. (29) -71, ai 7, and equating the product of their multipliers to the multiplier of the resulting transformation. The multipliers of the transformations (29) are (nM,)-l and M respectively; the resulting transformation Q a a- a71A, Oa 1- IX X ay-ayl,, a1i-Ay, which is in fact a complex multiplication of the argument, has for its multiplier [2 a + 1 2 i A]-1. Art. 7.]1 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 341 The equation (27) shows that, for a given value of n, the ratio of the multipliers corresponding to any two equal values of X2 can only be one out of a finite number of imaginary quadratic surds; for that equation can be satisfied only by a finite number of values of ao, r, and A. We may add that, conversely, every set of numbers 2 a- + 1, 2r, A, which are all different from zero, and which satisfy the equation (27), may be employed in the formula (28), and will serve to express the ratio of at least one pair of multipliers corresponding to equal values of X2 in the modular equation of order n. The demonstration of this converse proposition would, however, require a more complete discussion of the multiple points of the modular curves than can be undertaken within the limits of this note. Such a discussion would also show, that if 1M, M M2,... are the multipliers appertaining to as many equal values of X2, we may have Ml 1Mi Ml M2 an equation such as but not an equation such as = * For our present purpose it suffices to observe, that the ratio of two multipliers answering to equal values of X2, cannot be a root of unity, because the imaginary fourth and sixth roots of unity, which alone of all the roots of unity are simple quadratic surds, cannot be represented by the formula (28). Let a(^,h2,7X2)=0........ (30) be the equation, rational in k2 and X2, which determines the values of M at the multiple point (k2, X2). We may suppose that this equation is irreducible, i.e. that Iu cannot be resolved into factors without the adjunction of some irrationality other than those contained in k2 and X2; if,A =0 is not irreducible, we must consider successively instead of,x, the irreducible factors of a. Let M= pM1, M and M1 being two of the roots of (30), and p being a quadratic surd of the form (28). Then the equation ir( k2, 2)_A0. v 2(31) will have some, but not all of its roots in common with the equation (30); viz. - is certainly common to the two equations; if is also common to the two, 1 I M 1 1 we must have M =pM2, M being a root of (30), different from and M, 2]/[1 because M=pM1=p2M2; if is common to (30) and (31), we must have 1M2=pM3,, being itself a root of (30). Since p is not a root of unity, this Jyl 342 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. process cannot terminate until we arrive at a root of (30) which is not also a root of (31). Therefore - is the root of an equation which is of lower dimensions than (30), but of which the coefficients contain an imaginary quadratic surd iVA. If, is any second root of this equation, we shall have M= p'I', p' being a quantity of the form (28), different from p, but containing the same surd iV/A. We can therefore again reduce the order of the equation of which 1 t is a root, without introducing any new irrationality, and we shall at last arrive at an expression of the type 1= (i t/A9,X2)........ (32) ' being a rational function, with rational coefficients, of i/VA, k2, X2. It is proper however to observe, that each time that a new ratio p is employed, the radical zi/A is introduced with an independent sign; so that the expression (32) (in which we regard k2 and X2 as given) admits not of two values only but of 2r, if r is the number of different ratios p employed in the reduction. In the example, considered in Art. 6, the quotient of the two values of M is (-3T4i), in accordance with the foregoing theory. And in general, if p is a prime number of the form 4 m + 1, so that we may write p=a-+b2+; b_0, mod2; a~b, mod4;. x dx the integral /( C 4 —) ns transformed into itself by two primary transformations, answering to the two equations 1+i -ab x(1 +). a+b, a v The corresponding multipliers are a + i, of which the ratio is - ~ 1 abi in accordance with the formula (28). II. On the primary periods of the elliptic functions, and on the complete rectilinear integrals. 1. Two periods of a doubly periodic function are said to be conjugate, when the vectors representing them are adjacent sides of an elementary parallelogram in the parallelogrammic system appertaining to the function. If (P, P') Art. 2.] NOTES ON T TE TIEORY OF ELLIPTIC TRANSFOR:MATION. 343 is a pair of conjugate periods of the function, all the conjugate pairs are included in the formula Q = p + 1P'. Q OP+p? *,.. ( ) when a, f3, y, S are integral numbers, and a - 3y = + 1. The theory of the reduction of binary quadratic forms of a negative determinant is applicable in the manner explained by Gauss, in his review of Seeber's 'Untersuchungen tiber die ternaren quadratischen Formen,' to the parallelogrammic system appertaining to any doubly periodic function.* Thus there is always a reduced parallelism, one side of its elementary parallelogram being the vector which represents the absolutely least period of the function, and the adjacent side representing the least period which is not a numerical multiple of the absolutely least period. The sides of a reduced parallelogram cannot exceed its diagonals, and the included angles is less than 120~ and not less than 60~. If P= r+is, P'= r'+ is' A=r2 + s2 B=rr' +ss', C=r' %2 t* ~ () the quadratic form (A, B, C), or AX2 +Bxy+Cy........ (3) represents the parallelism determined by the given pair of periods (P, P'); and if thiS form be reduced by the method of Gauss, the equivalent reduced form represents the reduced parallelism. The reduced parallelism as well as the reducing substitution is unique, except when (A, B, C) is equivalent to a multiple by any real quantity of 2 + y2, or 2 S + xy +2y2 2. It will be convenient to restate, in a form adapted to the purpose of this Note, some of the principal results included in the theory of the representation of parallelogrammic systems by binary quadratic forms of a negative determinant. This theory is well known; but, in order to apply it to the elliptic functions, we have to introduce two modifications into it; viz. (i) we have to regard the sides of the elementary parallelograms as vectors, (ii) we have to restrict the conception of equivalence, confining it to primary equivalence only; this restriction is introduced in Art. 3; in this article we adhere to the Gaussian definition of equivalence. * Gauss, 'Werke,' vol. ii. p. 188. See also Dirichlet in 'Crelle's Journal,' vol. xl. p. 209; and the 'Report of the Theory of Numbers,' Part v Art.. 120 ('Report of the British Association' for 1863) [vol. i. p. 263]. 344 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. (a) When we are given the quadratic form (3), we are given the absolute magnitudes VA and V C of the vectors P and P'; we are also given the quotient of these vectors, but with an ambiguity of sign; viz. if p' 2= -, A=AC-B2, we have B iC-20 VA C-2Ag7+Bf2 -0, 12= i A To remove this ambiguity, we adopt the convention that the direction of rotation from the first vector P to the second P' is to be positive*; in accordance with this convention the amplitude of Q is positive, and if we denote by V/A the positive square root of A we have, without ambiguity, B + i /A B _iVA * *. * * *. e. v (4) The vector defined by this equation may be termed the vector associated to the form (A, B, C). Thus, when the quadratic form is given, the parallelism represented by it in any plane in which the positive direction of rotation has been assigned is given in species, but not in position; for, if we draw one of the two vectors in any arbitrary direction from the zero point, and set the other at the proper inclination to it, we obtain a parallelism which is represented by (A, B, C), and which by turning it about in its own plane may be made to coincide with any other parallelism represented in that plane by (A, B, C). If (A, B, C) is transformed into (A', B', C') by an unit matrix a, the vectors of the new parallelism are Q=aP+ P', Q' = P+ P', and, if 2'=, we have B'+ id4/a C' - 2 B'2'A +A'/2=, = O '= + _= d-/ +****,... * (5) 6'-.......... * If ReiO= X + i Y is any complex quantity of which R is the analytical modulus and 0 the amplitude, we shall suppose throughout this note that - r < 0 < 7r; so that the amplitude always has the same sign as Y. The angle of rotation from one vector P to another P' is the amplitude of the quotient p'; thus the angle of rotation is always a positive or negative Euclidean angle, or in the limiting case a positive angle of 180~; and the direction of rotation from P to P' is positive or negative, according as the amplitude of is positive or negative. the amplitude of - is positive or negative. Art. 2.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 345 By the convention which we have adopted improper equivalence is excluded; for, the amplitudes of 2 and Q' being both positive, the equation (5) implies that aJ- 3y is positive. Two improperly equivalent forms, such, for example, as the opposite forms (A, B, C) and (A, -B, C) represent parallelogrammic systems, which are symmetric to one another, but which, except in the cases of ambiguity to be noticed presently, cannot be made to coincide with one another by turning either of them about in the plane in which they lie. In the general theory of periodic functions there does not appear to be any reason for attending to the distinction between the positive and negative directions of rotation, or, which comes to the same thing, to the distinction between proper and improper equivalence. But, as we shall presently see, this distinction becomes of importance in the theory of the elliptic functions, and especially in the connexion of that theory with the theory of the Theta functions. (b) When the form (A, B, C) is reduced, the vector = X+iY is said to be reduced, and vice versa. The conditions that the form (A, B, C) should be reduced are (' Disq. Arith.' Arts. 171, sqq.) (i) A C, [B] A, (ii) If A = C, BO,;..... (6) [B] = A, B>O, hence, when (A, B, C) is reduced, Q satisfies the inequalities (i) X2+ y2>[, _.<X< }, (i) +r,.i....(7) (ii) If X2+Y2=1, X>0, or, which is the same thing, the extremity of the vector Q falls within the 'reduced space,' i.e. the space which lies above the real axis, between the lines X =- 0, and outside the circle X2+ Y2 1; the bounding line X+ 0, and the bounding arc, from - (-1+ iV3) inclusively to i exclusively, are not considered to belong to the reduced space. When (A, B, C) is reduced, the imaginary part of Q has a greater absolute value than the imaginary part of any equivalent vector; for Y= A and V/A is invariant, while A in the reduced form has the least value possible. * Square brackets are here used to denote the absolute value of the quantity included in them; the same notation may be conveniently employed in the case of complex quantities, so that [Rei ] = R. VOL. II. y y 346 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. When the extremity of Q falls on one of the boundaries of the reduced space, this statement requires an unimportant modification, viz. in this case the two vectors X + i Y, of which one only is reduced, are equivalent. The value of Y in a reduced vector can never be less than V/3. (c) A parallelogrammic system is ambiguous, when it can be brought into coincidence with itself by rotation through an angle of 180~ round one of the lines of the system; if (A, B, C) represent the reduced parallelism of the system, this takes place in the following three cases: (1) B= 0; the reduced parallelogram is a rectangle; the extremity of &2 falls on the line Y= 0. (2) A= C; the reduced parallelogram is a rhombus; the extremity of Q falls on the boundary X2 + Y2 = 1. (3) A = B; the reduced parallelogram has a side and a diagonal perpendicular to one another; the extremity of E2 lies on the boundary X- 1 = 0. In case (1) the axes of symmetry are the sides of the reduced parallelogram; in case (2) they are its diagonals; in case (3) they are its least side and the line of the system perpendicular to that side. The axes of symmetry are conjugate lines of the system in case (1) only; in the other two cases the vectors lying in the axes of symmetry contain a parallelogram which is double of an elementary parallelogram. There are two pairs of axes of symmetry, when the system consists of squares, and three pairs when it consists of equilateral triangles; in all other cases of ambiguity there is one pair only. (d) A parallelogrammic system can always be brought into coincidence with itself by a rotation in its own plane through an angle of 180~; this rotation corresponds to the automorphic. of the quadratic form. When the system consists of squares, it can be brought into coincidence with itself by a rotation through an angle of 90~, and when it consists of equilateral triangles by a rotation through an angle of 600; to these rotations there correspond the two pairs of opposite automorphics appertaining to the system (A, B, C)=1, o,1), Q=i, and the three pairs appertaining to the system (A, B, C)=(2, 1, 2), Q = (1 +i3); the identical pair + 1 0 counting in each case as a pair. 0, 1 Art. 3.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 347 The parallelogrammic system of sin2 am u consists of squares when k2 =and of equilateral triangles when k2 is an imaginary cube root of - 1. 3. We now restrict the application of the term equivalence to primary equivalence, and henceforward when we say that two forms or two vectors are equivalent, we shall understand that they can be transformed into one another by a primary unit matrix; i.e. by a matrix 'a, satisfying the equation a - 3y = 1, and the congruences a- 1, mod 4; f = 7 _0, mod 2. When this limitation is not intended, we shall say that the two forms or vectors are absolutely equivalent. In the general theory of doubly periodic functions we have only to deal with absolute equivalence; but in the case of the elliptic and modular functions, the consideration of primary equivalence is indispensable. It will be noticed that the two opposite parallelisms (P, P') and (-P, -P'), though represented by the same quadratic form, are not, strictly speaking, equivalent; for 0 0, -1 is not a primary unit matrix. In the reduced parallelisms which we shall have occasion to consider in the case of the elliptic functions, the real part of the first vector is always positive, and the opposite parallelism need not be considered. The restriction in the definition of equivalence introduces certain modifications into the theory of reduction, which we shall now briefly indicate. A quadratic form (A, B, C) is said to be (primarily) reduced, when its coefficients satisfy the inequalities (i) [B] _ A, [B] _< a (ii) If[B]=A, or [B]=C, B>0. () The associated vector 1 is reduced when the quadratic form is reduced, and vice versd. The 'reduced space' lies above the real axis, between the lines X+ ~1 = 0 and outside the semicircles X2 + Y2 ~ X= 0; the line X+ 1 = 0, and the semicircle X2 + Y2 + X= 0 are regarded as not appertaining to the reduced space. The extremity of a reduced vector falls within the reduced space, so that the vector satisfies the inequalities -1<X~1; -X<X2+ Y2X........ (9) Theorem. 'Every quadratic form is equivalent to one, and only one, reduced form, and by one, and only one, reducing transformation.' (1) Let a be the least number represented by (A, B, C) with values of the indeterminates which satisfy the congruences (x, y) (1, 0), mod 2; and, among y 2 348 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note 1I. all the forms equivalent to (A, B, C), which have a for their first coefficient, let (a, b, c) be that which has the least third coefficient; if the two opposite forms (a, b, c) and (a, - b, c) are equivalent, we take for (a, b, c) that one of the two in which the second coefficient is positive; we then have a - 4 [b] + 4c a, 4 a - 4 [b] + cc, whence it follows that (a, b, c) is reduced; i.e. a reduced form always exists equivalent to a given form (A, B, C). (2) Again, to show that only one reduced form can be equivalent to a given form, it suffices to establish the following lemma, which is in substance due to Legendre, and which serves to show that two reduced forms cannot be equivalent without being identical. 'If (a, b, c) is a reduced form, and if the numbers primitively represented by (a, b, c), or by any equivalent form, are divided into three series according as the indeterminates satisfy the congruences (x, yA) (1, 0), (x, y)( ), (, Y)(o 1), the least numbers in these series are respectively a, a-2[b]+c, c.' Let f(x, y) = x - 2 [b] xy + cy2, and let us suppose that x and y are positive; the identities f(x- 2, y)=f(x, y)-4a (- 1) + 4 [b]y, f(x, y- 2)=f(x, y)- 4c(y- 1) + 4 [b]x, combined with the conditions [b] = a, [b] = c, give rise to the inequalities f(x- 2, y) f(x, y), if x > y, f(x, y - 2) < f(x, y), if y > x, which show immediately that f(l, 0), f(l, 1), f(0, 1), or, which is the same thing, a, a - 2 [b] + c, c are the least numbers in the three series respectively, We may add that in the second series the least number but one is a + 2 [b] + c. (3) Lastly, it is readily seen that a reduced form cannot be transformed into itself by any primary substitution other than -o 1,; hence, the reducing substitution is always unique., 1 To obtain a reduced form equivalent to a given form, we may employ, with Art. 3.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 349 a slight modification, the algorithm of Gauss.* Let (ac, bo, c) be the given form; if [bo] > ao, let b, + 2 1a0O = bl, [b] < ao, and let 1, u, transform (ao, bo, cn) 0, 1 into (ao, b, cl); if [b] > c, let b, + 22cl = b2, [b2] cl, and let 1 0 transform 2/A2, 1 (ao, b1, cl) into (ac, b, c,); if [bj > a,, the process can be recommenced, and as long as it continues, we shall have a, > c > > a, > c2 >.... But a quadratic form of negative determinant can represent only a finite number of quantities less than any given quantity. Hence, we must at last arrive at a form (a,, b2,, ca), in which [by]j a,, [b,] ] c,, and which, if not reduced itself, has for its opposite an equivalent and reduced form. The coefficient of i in a reduced vector = X + i Y is greater than in any equivalent vector; for the first coefficient of a reduced form is less than the first coefficient of any equivalent form; the enunciation requires an unimportant correction when the extremity of the reduced vertex lies on a boundary of the reduced space. There is no inferior limit to the value of Y, which cannot vanish, but may be any positive quantity however small. From the inequalities (8), which are satisfied by the coefficients of a reduced form (a, b, c), it appears that in the corresponding parallelism the triangle of which the vertices are) (1), (o, +1....... (10) is acute-angled; one of the acute angles becoming a right angle in the limiting cases b = O, b = a, b = c; in the symbol (0, ~ 1), and throughout the rest of this article, the upper or lower sign is to be taken according as b is positive or negative. The triangle (10), which we shall designate by UVW, and which may be termed a reduced triangle of the system, is in fact a triangle contained by two adjacent sides and the lesser diagonal of an absolutely reduced elementary parallelogram. Every elementary triangle of the system, which is not obtuse angled, is a reduced triangle; for if UV, U4W, are the least sides of such a triangle, the elementary parallelogram contained by UV, UW, is an absolutely reduced parallelogram, because its sides cannot exceed its diagonals; all the reduced triangles of the system are equal to one another in all respects. The three sides of a reduced triangle, taken positively and negatively, can be paired in twenty-four different ways; in twelve of these the direction of rotation from the first side to the second is positive; we thus obtain twelve parallelisms * ' Disq. Arith.,' Art. 171. See also the account of the theory of reduction in the 'Report on the Theory of Numbers,' Art. 92 (' Report of the British Association' for 1861) [vol. i. p. 182]. 350 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. absolutely but not primarily equivalent to one another, if, however, we confine our attention to one of each couple of opposite parallelisms, such as P, P') and ( - P, - PI), it will suffice to consider the six exhibited in the following scheme: (U, ~ UW), (VW, ~ VU), (WU, ~ WV) ( UW, - UV), ( VU, -VW), ( WV, - WU) If -11 -1, 0, —1 PI pl =, P 1 0 ' '= 1, 0 so that p=p3 = +2= 1, =1 0,1 =1 the three parallelisms in the upper row of the scheme (11) are transformed into one another in cyclic order by the matrices 1, pi, p'; the three in the lower row by 1,5 p, p2, and any two in the same column by +~1. The six parallelisms are all primarily reduced; they are in general different in species from one another; they are, however, of the same species in sets of two, if UVW is isosceles and right-angled, and in sets of three if UVW is equilateral. The system has a pair of axes of symmetry if UVW is either rectangular or isosceles; two pair, if it is both, and three pair if it is equilateral. But the quasi-symmetrical pairs of parallelisms are equivalent, only when UVW is right-angled, so that in this case only is the system ambiguous in the sense of primary equivalence. 4. Let (4L, 2L') be a pair of conjugate periods of the elliptic function sin am (u, k2) = X (u), which we here regard as defined by the equations x dx x=X(u), _=J (1 -1k'... (12) A0 v5 (I _ X2) (1 - k2 x2)' (12) the initial value of the radical being +1, and k2 being any complex quantity whatever, other than 0, 1, oa. It is important in the theory of the function X (u) to consider specially the pairs of conjugate periods, infinite in number, which satisfy the equation x(L)=1,....... Xa(13) and in addition the condition that the direction of rotation from L to L' is to be positive. Such pairs of conjugate periods of X (u) we shall term primary." * There is a slight difference between the definition of the primary periods considered in the text and the definition of the elliptic periods of MM. Briot and Bouquet; viz. the pair [4K, 2iK'] is an elliptic, as well as a primary, pair of periods of A (u); but [4 Q, 2 Q'] is an elliptic pair of periods, only when /3 and y are evenly even in the equations (15); these authors, however, in the first edition of their classical work (Paris, 1859) had given a slightly wider definition of elliptic periods, requiring only that /3 should be evenly even. Art. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 351 One pair of primary periods of X (u) can always be assigned. Let dx dx.1 {4) = /r(l _ Xt) (t - k2X2)} Kl X_2) (1 k2 2)} k the integrations being rectilinear. The initial sign of the radical in the former integral is taken to be positive; the sign of the radical in the second integral iK' is so taken as to render positive the amplitude of the quotient K* The more accurate determination of these integrals will occupy us in Art. 5; for the 1 present either sign may be attributed to - in the lower limit of the second integral. By reducing to elementary contours the different tracks along which x may travel from the lower to the upper limit in the integral (12), it may be shown that [4K, 2iK'] is a pair of conjugate periods of the integral, or, which is the same thing, of the function X(u). But we have iK' evidently X (K) = 1, and the amplitude of -- is by hypothesis positive; therefore [4K, 2iK'] is a pair of primary periods of X (u). Theorem I. 'If [4L, 2 L] is any given primary pair of periods of X (u), all the primary pairs [4 Q, 2 Q'] are exhibited by the formula Q'=fL+ 7L'j ' Q=aL+?L'} in which a, | is any primary unit matrix.' 7, It will be sufficient to demonstrate this theorem on the supposition that L=K, L'=iK'. (1) Let [4 Q, 2 Q'] be a primary pair of periods of X (u). If X (v) = X (u), we have, by the known properties of the function X (u), either v = u, or v = 2K-u, omitting multiples of the periods of X (u) in either case. Hence the equation X (Q) = 1 = X (K) implies an equation of the form Q = (4r + 1) K+ 2 siK'. Again, 2Q', which is a period of X (u), must be a sum of multiples of the conjugate periods 4K and 2 iK'; we have therefore Q'= 2r'K+s'iK'. 352 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. Here s' is uneven, because, [4 Q, 2 Q'] being a conjugate pair, we must have (4 r+ 1) s' - 4r's = 1; the upper sign is to be taken because the amplitudes of ' iK' and - are both positive. Writing Q K a=4r+l, 3=2r', 7=2s, S=s, the matrix a, is primary, and the equations Q = aK+yiK' Q'= IK+SiK' *(' 1* 5) are satisfied. (2) Conversely, if [Q, Q'] is a pair of quantities satisfying the equations (15), in which the matrix 5' I is primary, we find immediately (Q) = X (L) = 1; also the amplitude of is positive; i.e. [4Q, 2Q'] is a primary pair of periods ofX (u). If [4 Q, 2 Q'] is any primary pair of periods of X (u) = sin am u, the pairs [Q, Q']; [2Q, 2Q']; [2Q-2Q', 2Q+2Q']; [2Q, 4Q]; are pairs of conjugate periods of 2 (2 u), of X2 (U), of cos am u, and of A am u respectively, and may be termed primary pairs of periods of those functions. Every primary pair [Q, Q'], in addition to the equation (13), satisfies the equations X(2Q)=0, X(~Q') = (2Q~Q') = oo X(Q~Q') 1..... 0(16) of which the first two result from the known formulae (see Art. 7) X (2K) = 0, X (+ iK') = X (2Kr iK') = o, the last is an immediate consequence of the definitions of K and iK'. If the sign of k be determined as in Art. 5, the upper or lower sign has to be taken in the right-hand number of the last equation, accordingly as f is evenly or unevenly even in the equations (15). The following theorem, in the enunciation of which 0(Q) is the function defined by the equations, a(Q)=V2q8n: +- 1 l, q=ei,...... (17) may serve to show the importance of considering the primary periods of elliptic functions. Ait. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 353 Q! Theorem II. 'If n= -, [2Q, 2Q'] being a primary pair of periods of X2(u, k2), Q satisfies the equation 8(2)= -k2 and, conversely, if this equation be satisfied by a complex quantity Q of positive amplitude, a pair of primary periods [2 Q, 2 Q] of X2(u, k2) can be assigned, such that Q2 = Demonstration (1). If n= -, [2 Q, 2Q'] being a primary pair of periods of X2(u, k2), let irn h=1r2' 2 h h' Q h =,a Q'. =e~', A= 2 + q2],, =Qh = It is known from the theory of the Theta functions that the function X [u, 8 (Q)] is identical with the function ~(2 h' q) X (18) and has, therefore, [4h, 2h'], or [4, Q, 2 Q'] for a pair of primary periods. The two functions X [ju, 0(8 (Q)] and X [v, k2] consequently have the same zero points 2mQ +2 nQ', and the same infinite points 2mQ +(2n +1) Q'; that is, they differ only by a constant factor, which may be determined by observing that lim () =( 1, when u = 0. We thus obtain the equation X [A u, f)8 (Q)] = x X [u, k2]. But X [Q, k2] = 1, X [h, p (2)] = 1; we have therefore = 1, X [, k2] = X8 [, ()], and, finally, differentiating with regard to u, 0s (Q) = k2. (2) Conversely, if 08 (Q) = k2, Q being a complex quantity of positive amplitude, the function X2 (u, k2) is identical with the square of the function (18), and has [2h, 2h'] for a pair of primary periods, of which the quotient is Q.* The primary pairs of periods of X2 (2u) are represented by a parallelogrammic system, which we may reduce by a primary transformation. If [M, M'] is the reduced primary pair, the least primary pairs of sin am u, cos am u, A am u, are [4M, 2M'], [2 M -2 M', 2M+2[MM, [2M,4M'], * The properties of the function f (Q2) employed in Note I, equations (16) and (17), are immediate consequences of the theorems of this article. VOL. II. Z Z 354 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IL respectively. For sin am u and A am u this is evident, and it may be proved for cos am u 'by observing that M~ M' are the two least values included in the formula (2r +1) M+ (2s+1) M (see Art. 3). We shall presently show that the rectilineal integrals K, iK' are the reduced primary pair of periods of X2 (2u, k2). In order, however, to give a precise signification to this assertion, we have in the first place to remove the ambiguity which attaches to the definition of iK', and in certain cases to the definition of K. We shall give in the next article the requisite determinations relating to these, and to some other elementary integrals which present themselves in the theory. 5. By the first of the equations (14), the integral K is completely determined, except when k2 is real, positive, and greater than unity. In this case, if k is the positive square root of k2, the radical vanishes when we come to the point x =; after that point its value is a pure imaginary and its sign can only be fixed by a new convention. The convention which we adopt is that it has a negative amplitude - T-r. This is the same thing as to suppose that x, when it arrives close to the point k, describes an infinitesimal semicircle round that point in the negative direction, i.e. above the real axis. We, in fact, regard k2 as having an evanescent positive amplitude, so that is a point lying just below the real axis, and infinitely near to it. The effect of the convention, in the case to which it applies, is to render positive the amplitude of every element of K, and therefore the amplitude of K itself; the real part of K is in all cases positive. To complete in the second equation (14) the determination of the integral iK', we understand by k that square root of k2, of which the real part is positive, or, if k2 is real and negative, that square root of k2 of which the amplitude is the positive angle par. The sign of the radical is so taken that the real part of the expression i x V{(1 - 2) (1 - k22) is positive; this real part is nowhere equal to zero in the course of the integration except at the two limits; for if x = +0 (1- 0 being a real variable, Art. 5.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 355 of which the limits are 0 and 1, the radical {t (1 - x2) (1 - k2X2)} may be written in the form lI (l-)e-lf(l-2)(2-)}x Av(l+I k)(l+2k)}, where the first radical factor is real, and is to be taken with a positive sign, and where it is readily seen that the real part of the second radical factor never vanishes, because the quantity under the radical sign never becomes real and negative. We thus have p {r1 ____M__dO I- 0 1 "J 40 (1 - 2)(2-I) (02- 1 + l k)(1+ 2 0 k) the real part of the radical under the integral sign being positive; whence it follows, that the real part of every element of K', and therefore the real part of K' itself, is positive. The convention by which we have fixed the sign of the radical in the expression for iK' comes to saying that at every point P in the rectilinear track from to 1 the direction of rotation from the vector 1- to the vector k k V{(1 - X2) (1 -k2x2)} is to be negative. The rule thus indicated is equivalent to the following: 'Let a simple closed contour be drawn passing through 0 and P, and 1 1 including +1, but not including -1, +, -; and let x, setting out from 0, describe this contour in the positive direction round +1; the radical, setting out with the initial value + 1, will arrive at P with the sign which it is to have in the integral expression for iK'.' To verify the coincidence of the two rules, it is sufficient to show that they agree at any one point of the rectilinear track. Let x travel from 0 along the real axis until it arrives at a point Q close to +1, and let it then describe in the positive direction a circular arc of infinitesimal radius round +1 as centre, until it arrives at a point P on the rectilinear track. Let a be the amplitude of 1-,, a the amplitude of k' = V(l - k2), the real part of this radical being supposed to be positive; it will be seen that a and f are of opposite signs; viz. the amplitudes of k2 and 1 - have the same sign, and the amplitudes of k2 and k' have opposite signs. When x arrives at Q the amplitude of the radical Z Z 2 356 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. is approximately /3; when x arrives at P, this amplitude is /3 + 1 a, or 'r + /3 + a, according as the amplitude of k2 is positive or negative; i.e. according as a is positive and f negative, or a negative and 3 positive. Subtracting the amplitude of 1- - we find that the difference a- a lies between -7r and 0 and the difference 7r +/3- a between 7r and 27r, because la and /3 are each in absolute magnitude less than 1-7r. The two rules are therefore in agreement with one another. This conclusion holds even in the cases in which k2 is real, and the demonstration may be applied to them, if we regard k2 as a quantity of positive amplitude, evanescent if k2 is positive, and differing from -T by an infinitesimal if k2 is negative. We may observe that when the amplitude of k2 is positive the radical arrives at the point P with its proper sign, if x travels in a straight line from 0 to P; the reverse is the case when the amplitude of k2 is negative. 6. The preceding determinations give immediately i o(1-x2) (l- X2 ) =K -TiK...... (19) the integration being rectilinear, and the initial value of the radical being +1; the upper or lower sign is to be taken according as the amplitude of k2 is positive or negative.* Again, writing x = k-in the integral (19), we find Y rk dx r _ dx Jo v {(-x2) (k-k2x2)} ( y2) d2) y being a real variable, and the real part of the radical under the integral sign on the right-hand side being positive; this determination is obtained by comparing corresponding elements, near to 0 and near to cX in the two integrals respectively; two such elements must have the same amplitude, viz. that of * Instead of employing the second of the equations (14), we might define 2iK' as the elliptic integral {( 2) (kd extended in the negative direction over a simple closed contour jo V { (1 - X 2) (1 -k Ix ) 1 1 including - and 1, but not -- or - 1. We should thus arrive immediately at the equation (19); and to show that in the equation (22) infra the upper sign has always to be taken, we might employ the principle of continuity; viz., when Pk is real, positive, and less than unity, the integrals in the equation (22) are each of them real and positive, and as they do not become infinite or zero for any value of ka, other than 0, 1, oc, the sign cannot change when k2 is made to pass from any one value to any other. Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 357 k approximately. The sign thus attributed to the radical on the right-hand side is that which it would acquire if y were to travel from 0 along the real axis and were to describe a semicircle round +1 in the positive or negative direction, according as ik' or - ik' has a positive real part, i.e. according as the amplitude of k2 is positive or negative. Hence, combining the equation (19) with the first of the equations (14), we find 0K0=2K -dy iK'1 = 2K-f0~ Wt(1 - sy) (1 - k2y2) (20) p00 dy iK' = (-y)(1 - y) (1 2y2) according as the amplitude of k2 is positive or negative, the integrals being rectilinear and the variable y being supposed to describe an infinitesimal semicircle in the positive direction round +1. We can now assign the values of all the different complete rectilineal integrals; i.e. of all the rectilinear integrals included in the formulae dx (a, b)=fv {(1 - xc) (1 - k2x2) T dx (a, Y) = (as ~) =n { (l - 2) ( - k22)}' when a, b are any two of the points 0, ~1, ~+, and Y is a point at an infinite distance. The following is a list of the complete integrals; the twenty integrals (a, b) are reduced to six by means of the formula (a, b) = - (- a, - b) = - (b, a). The upper or lower sign is to be taken throughout according as the amplitude of k2 is positive or negative. In the symbol (a, Y~) we understand by Y,, Y2, Y3, Y points lying at an infinite distance in the angles respectively contained by the vectors drawn from a to the pairs of points [1 [1, - [ - ] [-1.]; the vector drawn from a to a is to be interpreted as the vector from 0 to a produced. If P is a point infinitely near to a on the track ab, or aY, the initial sign of the radical in the integral (a, b), or (a, Y), is the sign with which the radical arrives at P, when x travels in a straight line from 0 to P, and when the radical sets out with the value + 1. 358 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION, [Note II. A. (O, 1)=K, ( )=KW;iK' (1, )=+iK', (1,-1)=-2K, (1,- )=-2K+iK';,' - k= - 2.K+ 2iK'. B. ( 0, ) = 2KTiK', (, Y3)=-2K+iK', ( 1, Y) = TKiK', (, 3) =-3K+iK, (-1, Y1) = 3K iK', (-1, Y3)= - K + iK', (,Y1)= -3K, ( Y3 = - 3K+ 2iK', (-, Y,)= 3K2iKT', (-y )= -K, ( 0,Y2)= +iK' (, Y4) = FiK'; ( 1, Y2)=-K+iK'; (, Y4) =-KTiK'; (-1,Y2)= K+iK'; (-1,Y4)= K+iK'; 1 y= -K+2iK" ( Y4)= -K; (-, Y)= K, ( -kY4)- K- 2iK'. In verifying the formulae (B), it is useful to notice (i) that Y,, Y3 are opposite points in the formulae (a, Y1), (-a, Y3), and Y2, Y4 in the formulae (a, Ye), (-a, Y4); (ii) that if the tracks aYT, aY, are separated by the track ab, and by that track only, we have (a, Y,) + (a, Y) = 2 (a, b). The formulae of this article have been constructed with reference to the general case in which k is any complex quantity whatever; but they retain their validity when k is real; we have only in this case to regard k2 as a quantity of evanescent positive amplitude, and to interpret accordingly the tracks ab, aY, and the rule by which the sign of the radical has been determined in each integral. For example, if k2 be real and less than unity, we have (- ) f(-xi,-k drx -Y, ~2 = j{(l~xS) (l~kx2)} =3K- 2 iK', kk(1 - x2) (1 -k the track of the integration lying along the real axis, above the points -1 Art. 7.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 359 and, but below the point + 1; and the initial sign of the radical being such that it acquires the value + 1, when x arrives at 0. 7. We shall now establish the equation I ~ dy = y2) (1- k.2y2),...... (21) the integration being rectilinear, and the initial value of the radical being +1; if k2 is real and negative, so that k' =- V(l - k2) is positive and greater than unity, the radical vanishes in the course of the integration; in this case we again suppose that k2 is a complex quantity, of which the amplitude is positive, and differs from or by an infinitesimal; or, which comes to the same thing, we suppose that, after we have passed the point x = I, the amplitude of the radical, which has now become a pure imaginary, is positive. The effect of this convention, in the case to which it applies, is to render negative the amplitude of the integral r1 dy. Jo ^(1- y2)k(1k'2y2)} the real part of this integral is in all cases positive. The substitution 1 - cy2 X2 - k2 transforms any given value of the integral 1 r1 dx 1/ J1 {(1-x2) 2(1 -x2)} into a corresponding value of the integral r1 dx Jo ^ (1 _- y2) (I - k2y2) To ascertain what value of the second integral is equal to K', we distinguish three cases. First, let k2 not be real, so that k has an imaginary part different from zero, and a real part different from zero and positive. Let =pe 0, x=X+iY. As y pursues the rectilineal track from 0 to 1, X+ iY describes the finite arc of the hyperbola X - P2.COS 20 X2y + 2 - p2 s 2 0 Y P sin 2 0 360 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. 1 1 from k or (p cos 0, p sin 0), to +1, or (1, 0); it will be observed that the points k and +1 lie on the same branch of the hyperbola, because the real part of 1. is positive. But the curvilinear integral along the arc of the hyperbola is equal to the rectilinear integral along the chord ( +, +1), because neither of the two remaining discriminantal points -1 and - can lie in the segmental area included between the curve and the chord. Hence K'= {(1 y2) ( -k'2y2)} (22) and the upper sign has to be taken because the real part of K', as well as the real part of the integral on the right-hand side, is positive. Secondly, let k2 be real and negative; as y passes from 0 to k, X+iY 1 1 describes the axis of Y from to 0, and as y passes from to +1, X+iY describes the axis of X from 0 to + 1. But the integral along the broken line ( 0, + 1) is equal to the rectilinear integral from { to +1. Hence the equation (21) subsists in this case also, the sign being determined as before by the consideration of the real parts of the integrals. It will be noticed that the convention of this article by which we have fixed, between the points - and +1, the sign of the radical in the integral expression on the right-hand side of (21) agrees with the convention of Art. 5, according to which - ik is positive; viz. these conventions render simultaneously negative the amplitudes of the two integrals in the equation (21). The remaining case, when k2 is real and positive, presents no difficulty, as the integrals in the equation (21) are both real and positive. We may, henceforward, define K and K' as the rectilineal integrals K= /(l - 2 sin2 v) o2 l-' d (23 Jo -(k sn )Jo -(1 k'sin"v)' ( the initial values of the radicals being +1, and the same conventions as before being retained when k2 is real and greater than +1, and when k2 is real and negative. Art. 8.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 361 In connexion with these integrals we shall consider the equation KXriK=fo dv ).....(24) 2K + i1C' 7T AA_ d(24),o /(k2 -sin2 v) ' ' v sin v which is obtained from the equation (19) by writing x = - The initial value of the radical is k, and the upper or lower sign is taken according as the amplitude of k2 is positive or negative. When k2 is real, positive, and less than unity, sin v must be supposed to describe a semicircle in the positive direction round k; but we shall not have occasion in what follows to employ this determination. 8. Theorem I. 'The vectors K and K' make acute angles with the vector +1 on opposite sides of it; and the angle between K and K' is an acute angle.' In this enunciation we understand by an acute angle an angle lying between the limits 0~ inclusively and 90~ exclusively. When k2 is real, positive, and less than unity, the vectors K and K' are real and positive; when k2 has any other real value one of them is real and positive, and the real part of the other is different from zero and positive; in these cases the theorem needs no demonstration. When k2 is not real, the vectors 1 - k2 and k2 lie on opposite sides of the real axis; and if we produce the vector k2 indefinitely both ways, the vectors 1 - k2 and + 1 lie on the same side of this indefinite line; hence the vectors k and k' (of which the real parts are positive) lie on opposite sides of + 1, making acute angles with it and with one another; the same things 1 1 are consequently true for the vectors k and k,. But for every value of v the a piu e o th ve t r1 1 amplitude of the vector,/(1- k2 sin2) is less than the amplitude of k, and is of the same sign; hence, the amplitude of every element of K, and therefore of K itself, is less than the amplitude of, and is of the same sign. Similarly, the amplitude of K' is less than the amplitude of K, and is of the same sign; i.e. the amplitude of K and K' are of opposite signs, and each of these amplitudes, K'. as well as the amplitude of the quotient K, is less in absolute magnitude than 90~. Cor. The amplitude of the quotient lies between 0~ and 180~, exclusively of both limits; it is less or greater than 90~ according as the amplitude of k2 is positive or negative. VOL. II. 3 362 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. Thus the ratio of the two periods of the elliptic integral od dx V{ (1 - 2) (1- k2x2)} is always imaginary. It is of some importance in the theory of the function X (u) to establish this elementary proposition by the direct consideration of the integral expressions for the periods themselves. It is indeed readily inferred from the equation of definition of X (u), that X (u) is a one-valued function of u, having the character of a rational fraction throughout the whole plane; and this character is incompatible with the existence of a pair of periods having a real incommensurable ratio; but is not incompatible with the existence of two periods having a commensurable ratio, and in fact reducible to a single period. Thus, in order to show that for all values of k2, other than 0, 1, oo, X (u) is a doubly periodic function of u, it has to be shown that the ratio of the two periods is not commensurable. And this may perhaps be most simply done by showing, as has here been shown, that this ratio is always imaginary. Theorem II. 'The triangles formed by the vectors K, -K K -iK',....... (a) K, -K-iK', iK',........ (b) are both right-angled when k2 is real; when k2 is not real, the triangle (a) is acute-angled, and the triangle (b) obtuse-angled, or vice versd, according as the amplitude of k2 is positive or negative.' Demonstration (1). If k2 is real, positive, and less than unity, the angle between the vectors K and iK' is a right angle. If k2 is real, positive, and greater than unity, the integral (24) is real, and the angle between the vectors K- iK' is a right angle; if k2 is real and negative, the integral (24) is a pure imaginary, and the angle between K- iK' and K is a right angle. (2) Let the amplitude of k2 be positive, i.e. greater than 0~ and less than 180~; in this case the amplitudes of - and -k are negative, the former amplitude being absolutely less than the latter, and each being absolutely less than 90~. 1 1 But the vector K - iK' lies between the two vectors k and -i, because /(k2-sin2v) always lies between k and ik'. Since K lies between +1 and k, the angle from K-iK' to K is acute and positive; and since iK' lies between k and i, the angle from iK' to - K+ iK' is also acute and positive. Art. 8.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 363 (3) Let the amplitude of k2 be negative. The vector K+iK' may be 1 i shown precisely as in the former case to lie between the vectors - and k, each of which has a positive amplitude less than 90~, that of k being the greater. k'f bi the greater Since K lies between and +1, the angle from K to K+iK' is acute and positive, and since iK' lies between i and -, the angle from K+ iK' to iK is also acute and positive. That the angle from K to iK' in case (2), and from - iK' to K in case (3), is acute and positive has been shown in the demonstration of Theorem I. Cor. The angle opposite to K in the acute-angled triangle is always absolutely greater than the amplitude of k', and the angle opposite to iK' is absolutely greater than the amplitude of k. Theoren III. 'When k2 is real, the right-angled triangles (a) and (b) are reduced triangles of the parallelogrammic system [K, iK']; when k2 is not real, the acute-angled triangle is a reduced triangle of the system.' Theorem IV. 'The pairs of periods [2K, 2iK'], [-2K~2iK', F2K], [T2iK', ~2K-2iK'], [2iK', -2K], [T2K, 2KT2iK'], [~2K-2iKK', 2iK'], are respectively the least pairs of primary periods of the functions X2 (, k2), X2(ik'u,, 'k), 2(ku, -2) 2 (iu, k'2), 2 (kug - k'2) X2 (ku, k2) or, which is the same thing, they are respectively the least pairs of periods of X2 (u, k2) which satisfy the equations (Qn)= 1 k21, 8 k2() 1 k2 08()=1 -_2, 08 (Q)= k21', 8(Q)= a Q2 representing the quotient obtained by dividing the second period of any pair by the first; the six quotients kQ are all primarily reduced, and one of them is absolutely reduced.' The pairs of periods, and the linear transformations of X2 (u), which appear in the enunciation of the theorem IV, correspond to the elementary matrices 3A 2 364 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. of Art. 3, which serve to pair in every admissible manner the sides of the reduced trianglet It will be observed, that the parallelogrammir system of X" (u) is primarily ambiguous, only and always when k2 is real; and that the angle between K and iK' is a right angle, only and always when k2 is real, positive, and less than unity. 9. Theorem I. 'Let Q be any primarily reduced complex quantity of which the amplitude is positive; let also, as in Art. 4, q=e, h (Q) = 2 x [M+ q2 k2 = 0 (Q), h' (Q)= Q x h (Q) the rectilinear integrals K and K' defined by the equations (23) are then expressed in terms of Q by means of the equations K= h (Q), iK' = h'().' For 2h(Q) and 2h'(Q) are a pair of primary periods of X2(u, k2); so also are 2Kand 2iK; we have therefore (Art. 4) h'(Q) = K~ +iK', iK' or, if w= K -f+a9 then= the unit matrix 1a, being primary. 7, I By the theorem IV of Art. 8, c is primarily reduced; Q is so by hypothesis; and by the theorem of Art. 3 two primarily reduced quantities cannot be equivalent without being identical; we have therefore co = Q, a = a- = = =0; or finally, K= h(Q), i' = h'(Q). Theorem II. 'Let Q be any complex quantity whatever of which the amplitude is positive; and let a be the primary reducing substitution of the quadratic form to which Q is associated, so that the vector 8-y2Q Art. 9.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 365 is primarily reduced; then, the same notations being retained, h () = aK+ yiK, h(Q) = SK+ SiK'.' For, as in the last demonstration, both [2h(Q), 2h'(Q)] and [2K, 2iKj are primary pairs of periods of X2(u); we have therefore h (Q) AK+ CiK h'() = BK+ DiK' kK' -B+AQ....... K - D-CQ) the matrix A, B being a primary unit matrix. Thus Q is equivalent to each CD. iK of two primarily reduced quantities K and Q'; these two reduced quantities, and the two reducing substitutions, are therefore identical; i.e. the equations (26) coincide with the equations which it is required to prove. The theorems I and II of this article may be demonstrated in a different manner. When k2 is real, positive, and less than unity, so that the quotient Q = is a pure imaginary, and q = eit' is a real positive quantity less than unity, the equations K= h(Q), iK'= x h(Q),........ (27) as has been shown by Jacobi, certainly subsist. But these equations cannot become untrue so long as the extremity of the vector Q lies within the primarily reduced space, because within that space the rectilinear integrals K and K' are continuous and one-valued functions of k2, which is itself a continuous and onevalued function of Q. When k2 is real and negative, there is, as we have seen, an ambiguity in the determination of the rectilineal integral K', and its differential coefficients with regard to k2 become infinite; the same things are true for the rectilineal integral K, when k2 is positive and greater than unity. But k2 cannot become real and negative at any point of the reduced space unless Q falls on the boundary X =; and k2 cannot become positive and greater than unity, unless Q falls on the boundary X2 + Y2 - X = 0; because, as we have already seen, if k2 is real and negative, K is real, and K- iK' is a pure imaginary * See a note 'Sur les Integrales Elliptiques completes' printed in the 'Transunti della R. Accademia de Lincei,' vol. i. (3rd series) [vol. ii. p. 221]. 366 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II. of negative amplitude; and, if k2 is positive and greater than unity, iK' is a pure imaginary of positive amplitude, and K-iK' is real and positive. Thus the equations (27) must continue to subsist at every point in the interior of the reduced space; and they do not become untrue at the boundaries of that space; for the conventions by which we have removed the ambiguities in the determination of K and iK' are so arranged as to correspond with the convention by which the boundaries X 1 = 0, X2 + Y2 + X= are excluded from, and the boundaries X- 1 = O, X2 + 2 - X = 0 are included in, the reduced space; viz. the values assigned to the integrals (23) in the cases of ambiguity are such as to make the equations (27) continue true at the included boundaries, while becoming untrue at the excluded boundaries. A complete demonstration is thus obtained of the theorem I of this article; the theorem II, and the theorems of Art. 8, may be regarded as corollaries from it. This mode of demonstrating the propositions of Art. 8 has some advantage in respect of simplicity; but the method adopted in this note is perhaps more direct and natural, as it depends only on the elementary properties of the rectilinear integrals themselves. iK' The theorem that the quotient K is always a primarily reduced complex quantity is of some importance in the theory of the connexion of the elliptic functions with the Theta functions." If k2 is given, the elliptic function X(u, k2) is given; but if we wish to express X(u, k2) as a quotient of Theta functions by means of the formula I —yQ --— h --- = \(U, k2), 1 h(2)' eirn) we have, first of all, to express the element Q of the Theta functions in terms of k2. The problem, as is well known, and as we have already seen in Art. 4, is indeterminate; viz. if Q is any admissible value of the element, Q' =- - + aQ 3 -?y2' the matrix being primary, is also an admissible value; or, which is the same thing, we may take for Q the quotient obtained by dividing the second period * See a note by Professor Weierstrass in the first volume of Jacobi's Collected Works (Berlin, 1881), p. 545. Art. 9.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 367 by the first in any primary pair of periods of X2 (). Since [2K, 2iK'] is a primary pair, the quotient is an admissible value of; but it is also the simplest of all the admissible values, for it renders the analytical modulus of q = ei the least possible, and the Theta series the most rapidly convergent. XLII. NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.* [Messenger of Mathematics, Ser. II. vol. xiii, pp. 1-54 (May-August, 1883)] III.-On the Functions Q(w) and Q' (o). LET Q (o)), Q' () be two functions of w defined by the equations iQ( ) =-2 d-K 2K2 d *d (1) <) ( ) i which imply the equality 1 because K'(o) = K( - 1); and which may be replaced by either of the following pairs: dK Q=2k2k'2 'K Q= 2k2k'2 d2 (3) Q -.k2 ' d.k2. Q =k2K- J, Q'= K'- J'..... (4) i7rd. k2 of which (3) was obtained by writing in (1) for dw its value - kk,2K2, and dK dK' (4) by substituting in (3) the values of d and d.k taken from the equations (i) and (v) of Art. 12. t [* These Notes contain the fragments, relating to the continuation of the preceding Notes, No. XLI, which were found among Professor Smith's papers after his death. It is to be understood that they are only unfinished work which he would have greatly altered and extended. The Notes have been placed in what appears to be the most convenient order, and headings have been supplied.] [t These references are to the Memoir on the 'Theta and Omega Functions,' No. XLIII. of the present Volume, p. 415.] Note III.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 369 dJ The values of d.k d.k2 dJ' and d.l taken from the same equations give d. k2 dQ - d.k2 2-K, dQ.K2 2 - dOk2 K'=Od.k ~'.. (5) and eliminating between these equations and the equation (3), we have [writing x for k2] d2Q - Q dX2 4-X( - X)' (6) the complete solution being CQ + C' Q'. The equations (1) and (5) respectively give Q' dK' Q dK' dQ' _K' dQ K'.. (7) of which either is a consequence of the other, because K'Q-KQ'= KJ'-K'J= r..... (8) We have also 1 dQ K dw 1 dQ' K' dw 2 k2 k'2K2 _ _ X ld. k2 -2 do - -- - -I 1 dK 1 dK' 2 2 Q do Q' dwo i, Q'dQ-QdQ', d.k2 dwo - dco Lastly, combining the equations (4) with the equations (vi) of Art. 10, iave K Q =0 k2 cos2 am xdx, } -' no\ we 1 K+iK' Q'= — i Kx Ik2cos2amx;{ k2 cos2 amxdx;t - * * V)J thus the functions Q and Q' do not differ essentially from the functions J=k2K-Q, '=k2K'_-Q' of M. Weierstrass, nor from the functions E=K-J=-k'2K+-Q, E'= J'= k2K' Q' of Legendre; their introduction serves to simplify the formula relating to the transformation of the complete functions of the second species. If mc =. a x Q is any transformation whatever of determinant n, that is, 7, VOL. II. 3B 370 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note III. i a + -Q ' we have a+OPU ( ) = aK(Q)+iK' (Q),. K' (w)=+ (Q) Differentiating with regard to k2, multiplying by -2k2k'2M, and observing that = dX2= x- d k, naM2 ~ I/2 d.X, d.XK' ) Q(Q)(=I2)-2XX2. Q'(Q~)=2X2X'2 d.x2 we find 1 2 ( d.log M M[aQ(Q) +i;3Q'Q)]= Q (()-2kV2k'2(c)) dg, ^[Y Q (Q) + i Q'(Q)] = iQ'() - 2k2k'2iK'(w) d. k In these formulae, which serve to express the transformed functions Q (Q), Q' Q) in terms of the given functions Q (w), Q'(w), K(c), K'(w), the differential coefficient -M dk is to be found from the equation of the multiplier. For example, if co = x 2 =, then, from the equation 1 6 8 (1- 2k) 3= of the multiplier, we find without difficulty dM 16 fM4 - 4 M2 d.k2 (M2 - 1)2 (M- 1)2 + 4k2M' and the formulae then become 8 k 2k'2' M-2 3Q (3w)= 3MQ () + (_ )k2+ 4kM K ), (IQ_ 1)2 + 4kl2M 8 k k'21~2 3 Q'(3c) = 3MQ'()) + ( M- )2 +k M K'(),' where, if w be a pure imaginary, M is the negative root of the foregoing equation'. * Cayley, 'On the Transformation of Elliptic Functions,' Phil. Trans., vol. clxiv. p. 421. Note III.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 371 Some considerations of interest present themselves in connexion with the last-mentioned general formulae. (1) The two equations are not independent: viz. multiplying the first by iK'(w), the second by K(o), subtracting and attending to the relations Q (w) K' () - Q' (w) K(w) = lr, nMK(Q)= AK(@) - (K' (o), inMK' (Q) = - yK(o) + ia K' (), we obtain an identical result. (2) In the case of Jacobi's first transformation n, 0 Q o=0, 1 n the equations become Q( M) 2k2(1 -k2) c dM 11Q1 — I K(c) dk2 Q'() Q 2,k2(1- k2), dM ~nM -— M -K'1 d.k2' of which the first coincides with a relation considered by Prof. Cayley (' Elliptic Functions,' Art. 305), and expressed by him in the form I _ - XM d 'K'....X.. (i) where K A A = K(Q), E = A2 am (u, k2) du, G = 2 am (, X2) du, so that E = k'2K() + Q (w), G = X'2K(Q) + Q (2), and the equation of Professor Cayley becomes Q(W)_ 1 Q( 4) XX'2 dM l K(w) nM LK(Q) M d\]' **.* which is the same as the first of the foregoing equations, because K n(.) ' nlnM"dX dk If instead of the first transformation we consider any reduced transformation 3B2 372 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note III. g, 0 of the type gn the first equation becomes g Q(Q)= Q (W)-2k2(1 k2) K1(1) dM K()) a sbttino which, on eliminating g by the equation g = K and substituting for d XX"2 d MK(q) 2 2'2 dk2 its equivalent nM dX' coincides with equation (ii), i.e. with equation (i). Thus the equation of Prof. Cayley holds for all reduced transformations, i.e. for all transformations of which the second element of the matrix of transformation is = 0. This remark is of some importance, as it enables us to show that the equation d2a Q(w) da __ a~-W+ 22) + ( - ), dx+2 2n K(o) dx +4k2 (1 - k2) d. k which is satisfied by 2 =o [2T()' Q] when w= =-, i.e. when the transformation is the first transformation of Jacobi, is also satisfied by the same function when the transformation considered is any reduced transformation whatever. By combining the foregoing results we can express the elements of the matrix of transformation in terms of the multiplier and of the integrals K, K', Q, Q'; we thus find K(^ ) nM Q (2o)-2nk2(1-k ) d K (e), (i i) 1 ' \ dM ' X -iK' (Q), K (Q) iK' (w), nMiQ (w) - 2 nk (1 - k2) dj- iK' () and the equation of Prof. Cayley is then found by putting 0 = O in the formula. If y = O, we have Q' () 1 (Q) '2 dMN ____ - AL' LK'~2\ + AL dx J.(iv) K' (0) nM' [K' (Q) M........ When /3 and 7 are each = 0, i.e. in the real transformation o = a, x Qa a combination of the foregoing equations gives K () K'c = niMIK (2Q) K(Q), which is true because K (,) i K (W) a a(Q), $jX '(). Note IV.] NOTES ON THE THEO1RY OF ELLIPTIC TRANSFORMATION. 373 IV.-Emnployment of the Transformation of the Function of the second species to obtain expressions rational in k2 and X2 for the coefficients in the formula of Transformation of the nth order. The analytical theory of transformation supplies a direct proof that the coefficients a, b in the developments 1 + a 2 + a2x4... + a(_)x-l = I - sin2 m 4j and 1 + b1x2 + b2X... + b(nl)Xn- = Ilj [1 - k2x2 sin2 am 4ji] of the numerator and denominator functions in the formula of transformation of the nth order are rational in k2, X2, M, and -d; that is in k2, X2 and M, for dM dk, dk 2 is always rational in k2, X2 and M: this follows from the equation of Jacobi combined with the theorem that at a multiple point on the modular curve the branches are all linear, and the tangents all different; so that even in the case where the modular equation has equal roots the coefficients contain no irrationalities other than those involved in the expression of M. To obtain the proof it is convenient to consider the function of the second species defined by the equation Z(u) = k2 sin2 am udu, and expressed by Jacobi in the form Z()0 = K u- 1 ) K iK' where J= Z(K) = k2 sin2 am udu. Employing the usual notation )= K ' q = eirw, and regarding V/k = x (w), K= K (c), J= J(w) as one-valued functions of w defined by the equations lk-' =/ 1 q - qk - 1 +L q2- 1 _2O-l > -:/k=V82q4J Al/k', +qc'=n-' }. (1) K.- d \ K /- +2 q+2q422+..., J k2K-.ir,; we consider the transformation ao= a, f 7+_ 7, d a+n' where mod2, and aS-f3y=n.? a o, 1' 374 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IV. Defining the multiplier M by the equations M K (w) = aK( ) + K' (2) tiK'() = K (Q) + iSKA'(),) we write in the equation of Jacobi ~ for u and Q for w. Dividing by M we obtain 1 I/ t X = J (x) t d rj7r' 1 _ Z( K() M2 du log o0 2 b K(Q)) J(af) u d log [7r(a +3n)..(3) K(Q) tM _du og 2K(w) ( where X2= = (Q). With this equation we combine a known formula for the transformation of the Theta functions, viz. o r(a+0n)U ' -Cx20 7rO J,U^2 x e-i (a + n)4 X t(n-l) [-1 sin2am 4j sin2amu],...(4) tMK+ ivK' in which C= o0 [] a+ [oK], and + = v, v being the greatest common measure of a and f3, and,a being defined by the congruence /y r + as, mod-, in which r and s are any two integers satisfying the equation ar +/3s = v. Instead of, we may take any multiple of n by a number prime to n. Taking the logarithm.and differentiating, we obtain from (4) the equation d lo r(a+ 3)7ru 27r /(a + 3)u d log 7rU d ogK J 2K +- n d- jlog 30 n-1 + S:(n) d log[1 - k2 sin2 am 2 sin2 am u], (5) or finally the equation z(, X2)- nZ(u, k2)+ 2Hu = 2 k2 sin am u cos am uA am u x j=(n-il) sin2 am 4j/. '~ 1 - k2 sin2 am 4j? sin2 am u' where 2i= nJ(w) 1 J(n) _ i3(a+q-) K(w) M2 K(i2) 2K2(w) nJ(_) I J(n) ir/ K(w)- M2 K () - 2MK(w)KK ) (n) () Note IV.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 375 The formulae (5) and (6) are given in fact by Jacobi for the transformation of the elliptic function of the second species. For our present purpose we have to show however that the constant H is a rational function of k2, X2, M, dM; and that the integrals J(w), J(7i), K (w), K (Q) enter into the expression (6) only apparently. To verify the assertion we write - Q (o) + k2K() for J(,), and consequently - Q (f2) + X2 K (f) for J ( S). Similarly for the function J (o) defined by the equation K(o)J' ()-K' (w) J()=.......... (7) we write - Q'(c)+k2K' (o), so that K'(w) Q (w)-K(w) Q'(o) r... (8) Then we have nM[a Q()+i (Q Q (w-2k2(1 -2)) _ d. ' -nM[7(Q)+ i Q' (Q)] = iQ'(,)-2 2k (1 - k2) ) d nM 2K(Q MK d.k, which are the general formulae for the transformation of the complete integrals of the second kind represented by Q(c)= cW k2 cos2 am xdx, E+iK' iQ' () =k2 cos2 am x dx. Introducing the functions Q (w), Q'(w) into the equation (6), we have X2,~ Q (o) O(a) im-b 27-nk-2 x 1 Q (10) 2H=,n- -- M2- () + 2 K(Q) 2 K(co) K() But from (9) Q (Q) _ ~Q(a)) - i3Q'() _ 2nk2 (1 - k2) dM M.K X(K ) MK:Qu M2 d.k2 Substituting this value in (10), we obtain 2H= n X2 2nk2 (1- 2) dM 12 1 d.~k2 K(co) K(Q) [nMKK(n) Q (o0) - K (c) {Q (c) - d i ' (Q)}0 - ( o SI], or finally, 2H=nk2 X2 2nk2(1 -k2) dM 2 iM d.k2'.11) 376 NOTES ON THE THIEORY OF ELLIPTIC TRANSFORMATIOIN. [Note IV. the terms in the square brackets vanishing by virtue of the equations nMK(Q)= JK( i) -i K'(o), K'(c) Q (o)- K(c) Q' (co) = If we now employ the developments sin2amu = u2-(1 +k2)u4+ 1(2+13k2+2k4) 6+... d. sin2 am u d s am = 2u- 4(1 + k2) 3 + 5(2 + 13k2+2k4) 5 +.. du Z(u)= k2u3- k2 (1 +k2) U5 +31k2 (2 +13k2+ 3k4)7+... (of which, however, the general terms are not known, and which are convergent only for values of u not surpassing a given certain limit), we can by equating the coefficients of like powers of u in the two members of equation (5) obtain successively the values of the sums =i(n-l)k2s sin 2 am 4jn, expressed rationally in terms of x2, k2 and H. Thus, for example, 2 k2 sin2 am 4jt7 = H, X2 A zk2 2k4sin4am4jrl 6=] -l 6k + (1+k2)H _1 dill = 2- 22 ( + ) + Ak2(l + 2k) - 4nk2 (1- ) (l + 2) 1 d. But from the theory of transformation 1 + bx2 + b2x4... + b (,_l)X-l = Ij (1 - k2x2 sin2 am 4j7,), so that the coefficients b1, b2,... are rational because the quantities 2k28 sin28 am 4j1' are so. In particular b, = -+1 1 1nk2 Mnb d.k2 2 M 2 b 1 X2(3X2-2) 1 X2 {(3- 2)k-2} nX2k2 ( -k2) did 3- 24 4 12 2+ 2 2iMs d. k2 nk2(1 - k2) (3n - 2)k2- 2 dM n2k4(1 - k2)2 d nk2(3n-4)k2 2} 6111 d.k2 +2 2 kd. kc 24 The last coefficient is known, viz. we have ( () - 2) /k b(n-) = (- )(-) k- I (-)sin2 am 4j =- k (- 2)s/ the expression being rational in k2 and X2. Lastly, the coefficients a are rational because 1 +ax2+a2x4... + a( _)xn-l' = -1 sn - am42 whence it follows that as M k28 = k(n-1)vx b(-1_. Note V.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 377 V.-Relations in the formulae of Jacobi between the elements; and the,7+39 transfornmation represented by the formula w = a+-. The general formulae of Jacobi for any transformation of an uneven order n are ( M ) sinam J(-) 1am- ] U am(, 2= mID — - l-T (n 1) - A1 I(u,X2 cos amUc =,,(n )[1 sin2amu ] cosan a am (1) i l=l sin coam 4j' itu A am u Aam (-, X2) = kD:(-) [1 - k2 sin2 am u sin2 coam 4j7], D = nIIJ("Z-)[1- ksin2am usin2am 4j]; '2= k X ljn('-l)[ 1 2am * * (ii) x A sin4A2aM44 77 1 k T^ (n-1); = (n-1) sin4am 4j n M-= - x Hi_ sin4am 4jn= =n i -~-o - md; AP X J=1 1 3 S cn4Coam4j1)' v7 representing an element of the form pK+ iqK'....... (iii) where p and q are integral numbers having no common divisor with n. It is also known that the modules k2 and X2 are connected by a relation of the form k = 0s ( ), X= 0 (Q), +2_ a,3........(iv) = = x 9, a + i2,, ' where a, is a primary and primitive matrix of integral numbers, having n for its determinant; and q (w) is the function already defined in Note I. The elliptic transformation is completely determined when either (1) the integral numbers (p, q), or (2) the integral numbers a, 3,?, J are given. Thus the two converse problems present themselves, (1) 'given any pair (p, q), to find all the corresponding matrices ai; and (2) 'given any matrix i, to find all the corresponding pairs (p, q).' VOL. II. 3C 378 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note V. The solution of these problems is implicitly contained in the congruences pa+q7y0, p/3+q3- O, mod n,.. (v) which result from the theory of the transformation of the Theta functions; but for the sake of clearness it may be worth while to give the details of the solution, although the arithmetical considerations involved are quite elementary. (a) If p l-P2, q- q2, mod n, the pairs (pi, Iq), (P2, q2), and the elements '1, 2, may be said to be congruous; if hp 1 p2, hq = - q2, mod n, where h is an integral number prime to n, the pairs (p1, 1), (P2, q2), and the elements al, 2,, may be said to be equivalent. Congruous elements are always equivalent; and equivalent elements, when employed in the formulae of Jacobi, give identical results. (b) To find, for any given uneven number n, the number N of nonequivalent elements?, let X (s) denote the number of numbers prime to any given number s, and not surpassing s, and let A, 3' be two relatively prime divisors of n. The number of incongruous elements a, having 8 and A' for the greatest common divisors of n with p, and of n with q respectively, is X () X ( hence the number of non-equivalent elements, having their greatest common divisors, is /n\ n\ X X =X/n\. X() x() ( n) =x );...... (vi) and the whole number N of non-equivalent elements is given by the formula the summation extending to every pair of relatively prime divisors of n, and the pairs, A, 8' and 6', S being regarded as different, except when 6'= = 1. It is easy to simplify the expression thus obtained for N; viz. if n = A x B, A and B being two numbers relatively prime, we find Z X:Zx (, x zx (,); also if n = 0I, 0 being a prime, Hence in general N x,\=n the sign of multiplication extending to all the primes 0 divid n. the sign of multiplieation extending to all the primes 0 dividing n. Note V.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 379 (c) Two matrices a. 1' and a2, 2 are said to be equivalent (or, more 71, i 72, properly, equivalent by primary post-multiplication), when al,, -1 a2, a 2 ||2 1 7I 1 72 82 le I1 being a primary unit matrix. Equivalent matrices, employed in the formulae (iv), give identical values for A2. (d) The number of primitive and primary matrices, non-equivalent by primary post-multiplication, is also nn (l + ); so that there are as many non-equivalent matrices as there are non-equivalent elements n; this we know, a priori, must be the case; because the number of different transformations must be equal to the number of non-equivalent matrices, and also to the number of non-equivalent elements. (e) We may take as representatives of the N non-equivalent elements r the elements of the reduced system - 2 IK+igK' n where g is any divisor of nx, and 21 is any term of a system of residues, even and prime to g, taken with respect to the modulus -. No two of the elements y can be equivalent; for if 2hl1 2 h21, mod n; hg hg2, mod n; we must have in the first place g =g2, because h is prime to n; and in the second place, writing g for g, or g,, hil, mod-; 1_1, mod-; i.e. 1=1,, and the two g1 g elements y, which were supposed equivalent, are identical. Again, any given element pK+iq' is equivalent to one of the elements Y; for, if g be the n greatest common divisor of q and n, let -21 be the residue of -, which satisfies the congruence - 21 -p, mod, and which may be supposed prime to g, because g ) p is prime to g; then the simultaneous congruences,)= -,21 mod n, hq- g are resoluble, because p, q, and n are respectively prime, and because the 3 2 380 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note V. determinant pg + 2 lq is divisible by n; further, the value of h supplied by these congruences is prime to n, because 21 and g are relatively prime; thus the given element is equivalent to the reduced element -2K+ i 581IK+4000i -K' Example. —Let 581 K+400K be the given element. Here n= 4725, 4725 g = 25, n= 189, = 160, -21x 160- 581, mod 189; whence - 21 182, mod 189. g g The congruences 581 h 182, 4000h1 25, mod 4725, give h_ 1147, mod 4725; so 182 K+ 25K b I' that the reduced element 4725 is congruous to 1147 x 1, and consequently equivalent to r. (f) As representatives of the non-equivalent matrices of determinant n, we may take the matrices g, 0 11 ( a lJ21, g g and 21 having the same significations as in (e). g, 0 (g) The reduced pair (-21, g), and the reduced matrix 2 n, satisfy the (211, - congruences (v); and the reduced pairs and matrices cannot be combined in any other way so as to satisfy those congruences. (h) The congruences (v), which may also be written in the symbolic form (p, q) x O, 0,mod n, admit, for any given riiti ati(n)primitive matrixand incongruous solutions (p, q); which, however, are all equivalent: and if we replace the matrix by any equivalent matrix, these solutions remain unchanged. From these considerations it is evident that, to solve the two converse problems, we have only to determine the reduced pair equivalent to the given pair, or the reduced matrix equivalent to the given matrix, as the case may be; either of them being known, the other is known also; and the proposed problem is completely solved. Exaple.-If n = 4725, and (581, 4000) is the given pair, the equivalent reduced pair is, as we have already seen, (182, 25); the corresponding reduced matri i therefo 18289 nd all the cor0 matrix is therefore 18, and all the corresponding matrices are in-18, 1 189 Note V.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 381 25, 0 eluded in the formula 125, x e, - 182, 189x where Il e is a primary unit matrix. Again, if 75, 100 is a given matrix of the determinant 4725, the -168, - 161 equivalent reduced matrix is _ 182 189 hence (182, 25) is the corresponding reduced pair, and all the corresponding pairs are included in the formula (182h+4725a, 25h+4725b), where a, b, h are any integers, positive and negative, of which h is prime to 4725. If we introduce the one-valued functions / = t2(W) Vx = 2(Q), /k' = +'(W), / =- 2(.), we may write the first two of the equations (ii) in the form cos2 am 4j, A2 am 4jn' 42n ()a)a= ii- 2,.n (,) n-8dma.n ' (v~) Similarly the last of the equations (ii) may be written I (- 1)i('-li-7 -t' () 11 sin2 am4ji,, = pm2( am 4j~i..(viii) sin coam 4j q sin2coam 4j ) In the theory of the transformation of the Theta functions it is convenient to suppose that the matrix '? satisfies the congruences B3 0, -0, mod 8, a 1, mod 4. In this case the equations (vii) and (viii) present themselves in the form 2 Aos2 am 4ji 1 /'= k' -Aam 4j 1= in- 1) I An 4 sn m j 4j' = =(-l1~-,) ~iT 4sin-am (-4 1) - -n sin m 4coa At v / /\ ()' sin2coam 4j,' 382 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note V. in which they occur in the 'Fundamenta Nova.' From these special formulae, in which,/x, /, V/k'/, J/k' represent the one-valued functions (2 (Q), 2 (Q), 02(t), 92(W), the general formula (vii) and (viii) may be immediately inferred by linear transformation. n n If = yy', where 7 is the greatest divisor of - which is prime to g, the number of the residues of - which are prime to g (i. e. the number of different numbers 21 in (e), p. 379) is X (y'). Hence, the whole number of nonequivalent pairs is YX ('), the summation extending to every divisor g of n. We have, consequently, /x( ') = n (l+0); these results are easily verified independently (see a Memoir, on the 'Singularities of the Modular Equations and Curves,' 'Proceedings of the London Mathematical Society,' vol. ix., Art. 9). It may deserve notice that the equation (vi) is the first of a series of elementary formulae of the same general type. The next in order is X~n (p p p = ()X) ) X X( XX); the general formula being x( )nx(^)xnx) ( ) X *... =n( I )nX x ( )xT X The s divisors A1, 2,..., *, are not necessarily relatively prime; but, ifp be any prime dividing a- of them, and having a for its exponent in n, and b for the sum of its exponents in 3,, J2,..., we must have either b < a, or else b = a, - < s. If b = a, a = s, the factor X(P) = - 1 must be applied to the left or right-hand member, according as s is even or uneven. * [Vol. ii. p. 252.] Note VI.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 383 VI.-Further Theory of the Functions Q (o), Q' (w)*. The following is a Table of the linear transformations of Q (w) and Q' (we). For brevity the argument w is omitted, and the functions J= Q + k2K, J' = Q' + k2K' are introduced: c+dc n c + cll~ cc + bE2 ac, b [1aQ ()+ Q ()] 1 [c (Q) + (Q)] 11 [| q iQ ' (-l) -(D-z) c, d 1 '1 IV r1 - i J' ') (- 1) +VI [p2 i J ' ( -1)(b-)i * [In explanation of the notation, it is to be observed that every matrix of uneven determinant is, with regard to the modulus 2, of one of the six types 1,0,1 1, 0,1,1 11 1,1 1~ 0, 1 -1,0 0, 1 ' -1,0 ' 1,-1 which are represented by the symbols 1, r, T, p,, p2, and considering these as unit matrices, i. e. supposing d to be equal, instead of only congruent, to the preceding values, they are represented by 11], 1'/, R 1, C l, l 1 p, I p I ~1 Similarly the primitive matrices of any even determinant, considered with regard to the modulus 2, are of one or other of the following nine types: 1, 1 1, 0 0, 1 0, O'0, O 0, 0 1,1 ' I 1,0 ' 0,1' 1, 1 1, 01 0,0 0, 00, O0 ' which are symbolized thus:, C 0 C2 2 C C C C1~) 2,2 2 C3,2 1, 3 C2,,, 3,3 See Arts. 21 and 23 of No. xliii. The functions Q (co) and Q' (o) have opposite signs to the Q (co) and Q' (co) defined in Note III (p. 368). The quantity ra denotes ei"r.] -384 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VI. We add a list of useful particular cases; the formulae are immediately deducible from the Table. The functions P (w) and P'(w) represent J(w) - k'2K(w) and eJ'(w)-k' K'(w) respectively. For convenience, the transformations of K and K' are included in the list, and, as before, the argument w is omitted. II. c=+j|xl= -.' w=K~~=-n K(Q)= K'; (Q) = -J'+K'; P (Q) = - P'; Q (Q) - Q'; III. 0 = K(Q) = 'K; iK'(Q)= iK; iJ' (Q)= - iJ iK; iP' (Q) = - tP; iQ' (Q) = - iQ. 1, 0 1' 0 xQ= l+Q. ) = 1K+ ); iK"'(Q) = k'(+ K+ iX'); J(Q)= ( 1= 0, 1 Q (O)= '; i IV. o =, K(Q) = k (K+ iK'); J(Q)= (J= i'); P(Q)=4(Q+K)+ (Q'+K'); Q(Q)=(-K) + - (J' -K'); iJ'(Q)= k'(Q+ q'); P'() = [(Q - K) + i(Q' - K')] 1 =+Q iK'(Q) = ikK'; iJ' )= J'; iP'(Q) = ('+ K'; iQ'(Q) = (J- '- ). V. o= T -1, K(Q) = kK'; J(Q) k P (Q) = - p() - +- K'; Q(2) = - 1v +1 I = 1 0 -Tl+_ iK'(Q) = ik[K+ iK']; iJ'(Q) = j[- QTiQ']; iP'(Q)- [Q- KT i (Q'+ K')]; iQ'(Q) = [- J+ K i (J'- K')]. Note VI.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 385 VI. &= _ - x =- '2 -1, +1 ~2 K(Q) = k' (K' iK); z(Q) = ik'K; J(Q)=,[K' - J' - t(K- J)]; iJ'(~) = k(K- J); (2)= [K'- Q'i(K- Q)]; iP'(Q)= k (- ); Q(Q) k= - (JT ); iQ'() = -J. The nine typical transformations of the second order (see Art. 34) give the following formulae for Q and Q'. They are derived by differentiation from the corresponding formulae for K and K' in accordance with the equations (1), Note III, p. 368. I. (a) =, xQ-= _-1, x =Q +1 K (Q) =2 V(kk') (K- iK'), K'(Q) = -2V/(kk') (K+ iK'), q (Q) = (2') [ + (k2- ) K], Q(2) = iv(2kk')[Q + (k2 - )K]. i,.,1-2kk') Y=c',~x~=l -1, 1 1~ (b) w=C'1,1xQ= ' 1 x~ l2 -1,- 1 -~' K (~2)= -/(kk') (K+ iY'), K'(Q) = V2/(kk') (K- iK'), Q (~2) = V(2k') [Q + (k2 - 2)K].. (Q2) = [Q+(k2-_)K]. II x 1, 2 K (2) = (k'- ik) K, K(~Q) = (k' - ik) (K' +iK), Q (Q) = 2(k' + ik)[Q + ikk'], Q' (Q) = (k' + ik) [Q' + ikk'K' + i (Q + ikk'K). VOL. II. 3 D 386 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VI. III. 2,x 1_ X Q-2,1 ~ xQ= 2 (Q) = (k' + ik) (K' + iK), K' (Q)= (k' + i) K, Q () = - (k' ik) [Q'+ ik'K + i (Q + ikk'K)], Q' () = - 2 (k' - ik) [Q + ikk'K]. IV. 2,0 -1 Cl'X1,= -1, 1 xQ=2 K (Q) = k' K, K' () = 2Vk'(K' - K), Q (o) = 2Tk [Q + k2K], Q (")) = 2k' [2 (Q'+ k2K') - i (Q+ ki)]. V. 0, 21 =,@ XQ=_-1 xQ=-21 K (Q) = (1 + ') K', K' (Q)= (1 + k')K, Q (f) = - + [q' + k'+(l - k')K'] ' )= 1 + k' [Q + k (1 - k')K]. VI. (o= 3, = 1 x Q= 2 K ()= - (1 + k'), K' (Q) = (1 + k') K', Q ()= + [q+'(l -k')K], Q () = 1 + k [Q' + k'(l - ')W]. Note VI.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 387 VII. -1, 1 2 K (Q2) = kK' K'(n) = 2Jk [ - iK'], 1~k 2 Q (Q)= k 2 -(Q-k'2 K), Q,() -= - 2k [2(Q - k'2K) - i(Q'- - k'K)]. VIII. IX. QW() = 2xk 2- ~ o g x= _&2 W=C2,xa= 0,2 x n=-. K'() )= (1 +k)K, Q ()= - il [Q -k (1 - k)K'], Let h (o) = i; the formulhe of the Table show that if b be a unit Q,(n) 1[Q ' - k (1 - k) K ']. matrix of either of the types (1) or (B), the equation = c, x implies the o~ = C', d equation h() = b dh ), h 3 D 2 3 D 2 388 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VII. or, which is the same thing,: 2a12a2 - 2a.... - 2c-...... (1) the quotients being even and the continued fraction subtractive; then also h():.2a1 2 - 2... — 2a, - h(Q) () The property of the function h (c) expressed by the equations (1) and (2), and the characteristic property of the elliptic function of the second species, may be regarded as having a relation to one another comparable to that existing between the double periodicity of the elliptic functions of the first species and the quasi-periodicity of the modular functions. The plane on which the argument of the elliptic functions is represented is divided into elementary parallelograms, of which the sides are any two simultaneous elliptic periods; and the elliptic function of the first species has the same value at corresponding points in two different parallelograms, while the values of the function of the second species differ by a quantity of the same type as the difference of the vectors of the two points; e. g. Z (x + 2mK + 2m'iK') = Z (x) + 2mQ + 2m'iQ'. c+dQ2 On the other hand, answering to the linear substitutions c = a+, we have a a + b0.' division of the semi-plane on which w is represented into lunular spaces equivalent to the reduced space (B) of fig. 2, referred to in Art. 38. At corresponding points in two such spaces the modular functions D (w) and k (w) have the same value, while the values of h (w) are related to one another as the vectors of the two points. VII. Formulce relating to the Elliptic Functions of the Second Species.* The Formula of Addition. Let x,, X2, x3 be three arguments of which the sum is zero; we have, from the equation of Jacobi [Art. 10, (iii)], Z(x) + Z(x2) + Z(x3)= - d log S x z(=)+z (dx8 +og (0 k2-).... (i) In the equation (i) of Art. 6, let M1 =P2/ = 2/3 = %4 = 5 VI = 2 = 13 =;4 = 5 X + Xz - X3 = 0; * [This Note is to some extent equivalent to IV, but the two Notes could not be united together conveniently.] Note VILI] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 389 differentiate with regard to x,, and write x = 0 in the result. Representing for brevity 4 (0) a (x)) a3 (x2) a3 (x3) by a, and as (0) [a: (Xi) as (X2) as (X3) + as (Xi) 5s (X2) as (X3) + as (XI) a. (X2) a,(X3)] by a,, (s = 0, 2, 3), we attribute successively to Iu, v the values 0, 0; 0, 1; 1, 0; we thus obtain the equations ca + a2 + a, + ao = 0, a1 + a2 - a3 - a0 = 0, a1 - a - a3 + ao =- 0, or, which is the same thing, a - a = - a = - ao. Transforming the right-hand member of (i) by means of the equality a = - a,, we arrive at the equation of addition in its usual form; viz. Z(xl) + Z (2) + Z (x) = - k2 sin am x1 sin am x2sin am X3;.. (ii) x1 + x2 + x = 0. The Formule of Transformation. In the equation of Jacobi [Art. 10, (iii)] we write M for x, and for w; we thus obtain 1 x I_ i 7r 1 Ml =K(i) M^2 - dlg(2K(Q) ) =j(Q) _ d logg o7rX(a+ ).. K(f2) M2 dx 2 K ' '..... where w= ', x b i, X = 04 (Q), and M is the multiplier corresponding to the transformation. For brevity, we suppose that the matrix a, is of the uneven determinant A, and satisfies the congruential conditions b-c0c, a_1, mod 8, assumed to exist in Art. 33. Putting h = 2K in equation (xii) of that article, and designating by C' a quantity not containing x, we find =' [(a+ xb) 2' 0] -i(a'xe= x2 -, x ia(A-I) [1 - k2 sin 2 4j sin2am x], C' X e ) -KX 390 NOTES ON THIE THEORY OF ELLIPTIC TRANSFORMATION. [Note VII. or taking the logarithm, and differentiating, d?oga[(~+ia) gr X dxlogax.(a + b Q)2 Q] 2irb+ A logO [ 2K) j + jlA )-] X log[1 - k sin 4j si2 am x, whence finally ( x)-AZ(x, k2) + 2 Cx = 2k2 sin amxcos am x A amxE2 sin2am 4j 2 (iv) 1 -sin2am4j'sin2amx (v) the value of the constant C being 2C= J( ) 1 J(Q) i7b(a +bQ) K(w) M2 K(12) 2K2 (o0) _) J(Q) __ _ __ (v) K(w) M2 K(Q) 2MK(Ew)K() Dividing equation (iv) by x, putting x = 0, and observing that lim zx)= 0, we also have X C= ksin2am4j.......... (vi) If we denote sin am x by u, the value of sin am (3, x), Art. 33, equation (xvii), is of the form u 1+Alu2 + A2U4 +... + A(A )u1 - A X (k-2) x --- — M 1 + B1u2 + B2u4 +...+ B (A_1)UA - 1- (- ) 1. Isin2am4j) X (U2) ifX(U) = 1 + B U2 + ) + B2 +... = II[1 - k2 sin2am 4j y sin2amx]. Employing this notation, we have for the right-hand member of (i) the equivalent expression B^u + 2B2u3 + 3B3u5 +.... - 2 cosamxAamx B1 + 2B2u+ 3B3u + 1 + BU2 +.B2U4 +... so~~~~~~~~~~C- tha-0-B1~ so that C= -B. Note VII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 391 The constant C is a rational function of k2 and X2; the integrals K(c), K(Q), J(c), J(Qi) entering into the expressions (iii) only apparently: the value is in fact 2 Z2 X\2 2Ak2k'2 d * 2C=fAk 2 ~- d.k. (vii) It may be observed that the coefficients B2, B3,... like B1, are all rational functions of k2 and X2; for these coefficients are linear functions of the quantities lk2s sin28 am 4j, which are rational in k2 and X2, as may be seen by expanding the elliptic functions in equation (ii) in series proceeding by powers of x, and equating coefficients of like powers of x. The same thing is true for the coefficients A, which are linear in the quantities X sin-2 am4j, and which are also connected with the coefficients B by the relation k28 + 1M As k22 xB(^-1_8, where the coefficient of B is rational in k2 and X2, because H sin2 am 4j Y is so. The right-hand member of equation (i) may be transformed by means of the formula of addition; viz. this formula gives Z(x + y) + Z(x - y) - 2Z(x) =Z( + y) + Z(x - y) - Z(2x) - [2Z(x) - Z(2x)], whence 2k2 sin am x cos am x Aam x sin2 am y 1- k2 sin2amx sin2 amy xZ(f X),-aZ(x, k)+2Cx = Z(x + 4jc, k)....(vii) the summation extending to every value of j' from- I(A- 1) to I (A - 1). The formula (viii) may also be obtained by integrating the sum of the squares of the roots of equation (xxv), Art. 33. To complete the preceding theory we add the following list of the transformations of Z(x) by unit matrices. I. For any unit matrix a, b., mod 2, (- 1)(a-)Z[(- 1)(a-) x, k]= Z (x, k). * [C is the same as the H of Note IV (pp. 373-376).] 392 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VII. II. 0,1 XQ= 1 o= -1,0 - i (ix, k')= Z(x, k) - [x - am am x [' L cos am x III. o=I-1 1i xI2=i+, hk +k' = Z'x, k) - \2 [k sin amx cosam x h'Z(exX, Z(x < k;' ' k [ 'am J IV. ~1, Q1 n =I O +x 0,+_1 + 1 _ kZ[x, k] = [x, k]. V. = x1 = --—, -,_1 — Zk'k, Z[x, k] ok2 (x sinam x cos am xa. iZ(iakx, +-=z[x,/ k]- C2(X ---- -- rx k / L - \ A am x VI. 0, 1 _ la (t)= X = —~- --- i'Zx,) =Z[x, =k]- (x- -samam i%' A / L?J \ cos am These formulae may be verified by transforming the function sin amx irn the equation Z (x)= k2 sin2am x dx. For example, iZ(ix, k')- Z (x, k) +x i k'2 sin2am (x, k') dx - 2 sin2am(x, k)dx + x = - J [k'2sin2 am (ix, k') + k2 sin2am (x, k) - 1] dx 4 r[ sin2am(x, k) \1 = 2o [2siam(x, k) + Aam(x, k) dx aocm (x, k) + sin am x A am x cosamx Note VII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 393 The formula IV (which is an immediate consequence of the corresponding formula for sin am x) changes II into VI, III into V, and vice versd. The formulae may also be deduced from the equation (iii) suprct, but the elimination of the complete integrals requires the use of some additional formulae. Taking again as an example the formula II, we have iZ (ix, k) () d log rx~ ' K(=) X- - log- log 2 J(n) i^Q2x d Twx \ K() 2K2 ddx 2l K J(2) iw72x d log/ro x \ d _= _ ___ - - 2K'lg - xlog cos am (x, k) K(J() J) i,2K dx 2K / - Z(x ) - x [K() + K(X) 2K2] - d log cos am (x, k). But J(n) =K'() - J'(w), K(Q2) ='(w) and the coefficient of -x becomes + KJ-KJ +, that is, 1, KK' in accordance with the formula II. Lastly, for the three typical quadratic transformations, writing for brevity s, c, d to denote sin am (x, K), cos am(x, k), A am (x, i) respectively, we have the fbrmulae 1 0 1, (1) ~=, 2 xQ,.......... (1) (k' -ik)Z [('-k', -i (ikk,k)= - ( ii -sd; (l+k')Z(1 + k'), 1+ ] k' 2Z (x,)= - [ x- ];... (2) (1+k)Z[(l + k) x,2k -2Z (x,)=2k[; ) - l (3) which may be verified by either of the methods exemplified in the case of the linear transformation w = - -. Observing that 2Z(x, k) - Z(2x, k) = - k2sin2amxsinam 2x. VOL. II. 3E 394 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII. we may write the equation (3) in the form (1 +k)Z[(1 + k) x, - ki-Z(2x, k) = k (2x - sin am 2x). Employing the notation of Legendre, F(O, k) o (-k-s2) E-(, k)= V(1 - k2sing)d; and writing 2^ q = am (2x, k),, =am[(l +k i) x, + kJ so that Z (2x, k) = F(Q, k)- E ((, k), 2,vk 2, Vk Z[( + k)x, + = F(, - E(, ) + = ~(l+k)F(~,k)-E (*, 2v1) we find ' k'F(, k)=E(~ k)-(1+ k)) E (, 2/+ k sin(4) -t( i^)-(+i + k+ k (4) sin(2 f - ) = k sin+, a celebrated formula, due to Landen, which serves to express an elliptic integral of the first species having a real modulus less than unity by means of two elliptic integrals of the second species having moduli of the same character. A similar, and indeed equivalent, formula is obtained from the equation (2); viz. writing (=am (x,k), =am[(l+k')x, l+k,] we find F'F(, J) = k (1 + k')E (, 1+') - E(~, k) + (1 - k') sin (5) tan (4 - )= tan ). VIII. The Functions Al (x) of Weierstrass. The Abelian functions of M. Weierstrass are defined by the equations [s=0, 1, 2, 3], Al,(x)= -.........(i) where, if s=, a Fim=0 a == hm =oL —J x Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 395 and, if s = 0, 2, 3, a,= (O), and where r J- k2 sin2 am xdx. 7rX These functions which differ from the corresponding Theta functions of 27 -J 2K only by the presence of the exponential factor e 21~, and of a factor not containing x, possess, as is well known, the remarkable property that they can be developed in series proceeding by powers of x, of which the coefficients are integral functions of k2 with integral coefficients, and which are convergent for all values of x. For our present purpose, there is a slight convenience in considering, instead of the functions Al(x), multiples of these functions by the exponential e^k22. These multiples we shall call, in what follows, Abelian functions, and we shall denote them by the symbols A. (x), A (x), A2(x), A (x), so that e.g. Q /7_X\ Ao(x)=e K x........ (ii) These functions respectively satisfy the partial differential equations d-2 + 4k2(1-_ k) ck -k2 (1 - k2)x2A,+gA=0,.... (iii) where go= -k2, gl=l-2k2, g2=1-k2, gs=O; or, which is the same thing, the four functions AoVk', A(l/(kk'), A,2/k, A, all satisfy one and the same equation d2A d) — 2d +4k2(-k2)Cl -k2(_-k)2 A= O..... (iv) The equations (iii) are a little simpler than the corresponding equations satisfied by the functions of M. Weierstrass, viz. d2 A1 dAl dAl dx,+2+2k2x +4k (1-) +4(-) A + k2x2A +g A= (v) where g0=0, g1=1-k2, 2=, g3, g k2; but the gain in simplicity is apparent rather than real, as the determination of the coefficients is not more easily effected in the functions A4 than in the 3 E 2 396 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII. functions Al.. The equations (iii) or (iv), as well as (v), may be derived directly from the partial differential equation of the Theta functions d2a(x, W) 4i da(x, o) dx2 T dw which gives _____ x d(2K dK K2 d(2(K) d K dx d + i7r dx2 and in which we are to put a(s) =eK A x x) x (), and to eliminate dw by means of the equation d.k2 4 = -.. ke (1 - k2)Kf2, the differentiations with regard to k2 being effected by means of the formulae dK_ Q dQ d.k2 2k2k2'" d. k2 2 The Abelian functions are connected by a relation which may be inferred from the corresponding relation between the Theta functions; viz., using for a moment a notation with double suffixes, AO,1=A0, A1, 1=A1, A1, 0=A2, Ao0, =As, and supposing A.,. = A,,, if m -E, n v, mod 2, then from the formula r~ m+/, + (x)-e a+i A"a+*(n+) =.'), ( + Xr + 1 X-t-/^+^)==^ ~~~i"~~Nl Xg ^+ I cr + 1 vr) we obtain the following A,,. (x+ vK + iK) = e(vQ + ixQ')( + IVX + XiI 0 X + K)+ (2n+v) X A(x), where _ + n a{n, n General formulae for the transformation of the Abelian functions are immediately deducible from the formulae for the transformation of the Theta functions. Thus, since in general, whatever be the transformation w = +d, we have 3~, [( l+/_) 2.-'3 (= a + b xT 7~xX ahB].-f ' x T, [() 7r ] - 3B~+ Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 397 where T is a homogeneous function of order n of two of the Theta functions 7C h, w); then, attending to the equation 1 Q(S2) n Q() i7rx(a + 32) _ k2(l k1 dM M2 2K(n2) 2 K(o) 4K2(w) M d.k2 we find 2 dM As X X2)=e Id.k2 xTYs where T, is a homogeneous function of the order n of two of the functions A (x, k2). In particular, if n is uneven and the transformation is primary, we have 1 dMZ2 (~)-X2) - ~- d.l- xAo(x,)x U, where Us may be expressed in terms of the Abelian functions; e. g. U =A amx II [1 - k2sin2 coam 4jn sin2amx] _ A(x)1 [ -k2A (4j/) AI (x)] - A (x) L A](4jf) A2(x)i For the linear transformations of the Abelian functions we find I. If, 1 j mod 2 c=, d 1 (-l)-, l), X~= ~"M = ( -)A = Ao. II. If, = 0, mod 2 c, d = 1, 0 ' ' X2= 1 - k, = Al (ix, 1- k2)= iAj (x, k2), A, (iX, 1 - k2) = Ao (x, k2), A. (ix, 1 - k2)= A. (x, k2), Ao (ix, 1 - k2) = A2 (x, k2). These equations signify that the coefficients of x4+1 and x" in A, and As respectively are functions of k2 (1 - k2), and the coefficients of X4%+ 3 and 4n +2 are functions of k2 (1 - k2) multiplied by 1 - 2k2; while corresponding coefficients in 398 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII. A, and A2 are of the form p ~ (1 - 2k2) q or +~ + (1 - 2k2) q, according as the exponent of x is evenly or unevenly even. III. If |, - mod2, k2 Ill. if 1 mod 2X, 2 = - k k2( k2) d llo = k2 111~=,, ( 1_ ) ckd.k2 2 k2 A (k', k2 l)= 'ek-22A1 (x k2), A2 (k'x, 7c2 _ pk2 )= e-27sz~l2X2, k2) A, (k'x, Y-) = e-i'2 A2, (x, k), A3 (k'x, k2 1) = e- A (x, k2), A (k'x, k2 )= e-2z2A3 (x, k2) a,b 11 12 A (kx, k2) = e-k'22A3l (x, k2) A, (kx, 1) A (kx o) e-k'22A3 (x, k2) A, (kx, )= e-mk'22A0 (x, k2) ab 1, 1 V. If I cd-_ 1, mod2, a= 1mk2 2 A3 (c,d d -1,0 - 22 (x2) A (k, 1 k2) e 2 ( k2) nl=; -,K (i-te) d.~' -" =3(d- "), A, (ikx, 2 - - )=i7e-1 'x2A,(x, kc), 1 - k A, (ikx, e)= <2 - Ao (x, k^, A, (ikx,a ^ )= e- 'A2^, (x,), A, (ikx, _ )= e-, 26" "A_, / k2). Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 399 a,b _ O 1 o VI. If mod2, c, d - 1, -1 1 ' X2= 1 - =l iikf, k2(1-k2) dd.k2 -2 A1 (ik'x, 1 - ) = ik^'e-'2A (x, k2), A, (ik'x, 1- = A3 (ik'x, 1 - ) Ao (k'x, 1 - = eik22A3 (x, k2) e-k2"Ao (x, k2), e-2kxA2 (x, k2). The functions Z=,-,eh1 (M, x2) satisfy a partial differential equation with respect to x and k2. We have from that equation, if A = A(j, X2), d"A dA] ~ dx+ - \I - d.X2J X2(1-X2) x M2 a = where d I\ [dA- dA dA c( d.xJ N2 d.X2 I dx d.\2 Observing that X2 (1 - X2) H1P d d.X~ d =nk2 (1 - k2) d.d2 we find d2A 1 dM dA dx — + 4nk2 (1 - k2) 2 dk'"x dA + 4nk2 (1 - k2) d.2 _X (11- ) x2A = O j}/i or, substituting for A its value V/M.e- ^IZ, and dividing by e -h~'M, dx2 + 4nk2 (1 - k2) d/k - 2 [2 (I ) + 4h2 + 4nk2 (1- k2) Ac2i 2 I =. M4 -+4 But the coefficient of - X2S in this equation is equal to n2k2 (1 - k2). Hence, we have finally dg2 42 (1 2) dk _ 22 (1 -k ) x2 =, d&Z2 +nk ( -k 400 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII. which is satisfied by the four functions XehX2 Ao(j, X2), / h$X2( x \), mr d 2 ')* XX e^2A (Al \2) I/ 1 eXA2 (MA X2). The determinant of the transformation n may have any positive integral value, and the constituents of the matrix of transformation may have any integral values. x 1 VJn If we replace x by - and by, we see that the functions JV -n M A ) -xA. ) 2 (1-k:2) d log M 2 V X~ ~ A~~(-~ X2)d.k e satisfy an equation of the same form as that satisfied by A8 (x, k2). We next proceed to form the equation satisfied by c=- A (x, k2). Substituting for Z and dividing by A", and observing that dE: Adc +nAn_ dA dx dx dx d2 d.2o' dA dT,'dA\2 d2A dx2 = Ad2+ 2nA - + - n (n- 1)An-2 d )+nA-l d dx2 dxd2 dx dl d._ d- dA dk= A d + nA"-l d a, d.k2we find.2 we find +d 2n dX2 d d + 4nk2 (1 k2) d.k A cdx- d+4n - k? + [ A d( Ac)+ I d2A 2- (1 - ]dA kO A dx2 + 4nk2 (1-k) d-k_ k ) =0; or, after all reductions, d2 2nZod (1 n d (n 1)2cn2x dx 2nZ dx + 4nk2 (1 - k2) O~k - n (n - 1) k2cn2x. - = 0. Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 401 Lastly, if in this equation we introduce s = sinamx and k2 as the independent variables instead of x and k2; we have d d dc ds = dd d = d d -cd d k dx ds dx ds' d2' =C2d2 dSs dk2 [d 1_l d- d+ ds da d_- 1 [d = d. k2 T ds d. k2 d.1 k+ ds 2k2 k' sc c where d. k2 is the differential coefficient of a taken on the supposition that a dcr is a function of the independent variables k2 and x, and d —2 is the differential l. k2 coefficient taken on the supposition that a- is a function of k2 and s, and that k2 does not vary in s, and we obtain finally d2o __ (1 - s2) (1 - k2s2) d + [(2n -l)k - 1 - 2 (n - )ks] s d.k =0, () dcr -n(n-l) k (1-s2)+ 4nk2(1 - k2) dk=0,. (a) in which equation the coefficient of d- may be written in either of the two forms (2n - 1) kc2 - d2 and 2 (n - 1) k2c2 - k'2. The equation is satisfied by /x' h1 / kV U0, / XXkU, 1 /~ Mk1 U, /2 1T U For our immediate purpose we add the term n2k2o- to this equation in order to obtain the equation satisfied by ' /X X Uo AV' X x U21, P 3. There is some difficulty in applying the equation d + nk (1 - k) d - n2k (1 - k2) x2 = to the actual determination of the functions U0, U1, U2, U3. The following method serves to exhibit these functions in what may be termed their ' canonical form.' Let the operating symbol d2 d d + 4nk2 (1- k2) d - n2k2 (1 - k) x2 VOL. II. 3 F 402 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII. of this equation be denoted by V; write also for brevity 4k2 (1 -k2) d and, considering first the three even functions eh2A (, X 2 let v = /' /M' vM m in the three cases s=0, 2, 3 respectively. We find, by expansion, X eh X (, x2)= e1 (, X2) = [1+ 2h+ 2 12 /XeTXA3 X X2 -/X2 x V e2A3(,) )1+= 21 +2+h.2 +; so that we have in the three cases alike vehx2A(, X2)=v-nSl12+ x6 Let = 1+a24 a36 +... be a series satisfying the equation V bo = 0, so that the coefficients a2, 3,... are rational and integral functions of k2 with integral coefficients; similarly, let X2 6 x8 = + 2+ 6 - 8! +... be a function determined by the equation Vi= (o; X4 X8 x10 let 02= 4 + '8gl.- -c5! + be determined by the equation V2 =; and so on continually. It will be found that these determinations are always possible; and that, as indicated in the case of the functions 00, t01 02, the first s coefficients in the function j, are zero, and the values of the two following coefficients are unity and zero. All the coefficients are rational and integral WX2s + 28' in k2, and the highest power of k2 in the coefficient of 2 in, is s'. We then have the theorem veh2A(, X\) =- ~q nv. + n22 v..2 - n33v.3 +.... Note V~II.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 403 For if F represent the series on the right-hand side, we have, evidently, VB = vVoo- (, - no) nS + (V2~- )) -282V_,..; X2 i.e. V'= 0; also D= v-nv.. +..; 1.2 so that the series (~ satisfies the same partial differential equation as the function re2A (, X2), while the first two terms of the series coincide with the first two terms of the expansion of the function; and this establishes the theorem, because, given the first two terms of the expansion, it can be continued by means of the partial differential equation in one way only. We next denote the operating factor d2 d d - + 2nZ, (x) d + 4nk2 (1 k2) d. _ ( - 1) k2 cn2 + n2k2 by D; and we observe that we have not only [A, x, (k2) ] but also, separately, D ] =0, D[ ] =.40. 0A0. 'A0 viz. each of these equations is derived from the corresponding equation ViP =, (jprecisely in the same way in which the equation Do = 0 is derived from the equation V = 0. We now expand the functions r in series proceeding by powers of s; we observe that if +2 (s) be this expansion, it follows from the properties of the $2r functions /f that the first term in +,. (s) is 2r! and that S2r 2r + 2 S2r + 4 () = (2r): + pea (2r + 2)! + p2(2r + 4)! +s where pm is a rational and integral function of k2 with integral coefficients of an order not exceeding mi in k2; the coefficient p is however not zero, but k2 (2r + 1) (2r + 2) 2r (2r + 1) (2r + 2) (1+ k2) 1.2 1.2.3 (2r + 1) (2r + 2) [nk2 + r (1 + k) 1.23 2 3F2 404 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IX. We thus obtain for U0 the expression v U= vJo(s) + nSv. x(s) + n22v. 2()+... But U0 is a rational and integral function of order n-1. Hence we may omit in 92(s) all powers of s above s"-l, and consequently all the functions 21v*,2 (S) after (n"-i). xyf_i)(S); the higher powers of s disappearing of themselves. And if we denote by I the operating factor in the left-hand member of the equation (a), p. 401, increased by n2k2, viz. d2 d =(l - s2) (1 - 2S2) d2 + [(2- 1) k2 - 1 -2 ( - 1)k2s2] s d + nk2 [ + (n - 1) s2] + 4nk2 (1 - k2) d. we have the equations o (s) = 0, 11 (s) = + (s), I+2(s) = + (),..., by which the functions xo (s), Al (s),... may be successively calculated; viz. if s2r X2r + 2 (s)= C (2r) + (2r + 2)! +** we have Ca 1 - [4o-2 + (4 2 - 4n- - n) k2] Ca + 2 (2- 1) (2- n- 1) (2- n- 2) k2C_+ n C, = B,, 26 if B, be the coefficient of X in,4_- (s). IX. On Modular Curves *. Let [a, 3, y] or c be a given quadratic point, and let it be required to assign all the modular curves of any species passing through the point X+iY= -+ >(). Let A, B, C be a pair of relatively prime solutions of the equation A', 1', Cb 'yA-23B +aC= 0, so that 7, - 23, a =1 | A), B', C' * [On the back of the last page of the manuscript of this Note Professor Smith has written:'These papers relate to the problem, &c., " Given a quadratic point, to find all the modular curves passing through it," &c., &c. It ought to be worked into a memoir " On the ordinary multiple points of a modular curve; and on the intersections of two modular curves."'] Note IX.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 405 and so that all the solutions of this equation are comprised in the formula 'A - A', x'B - B', X'C- C'. The determinants B2- AC= D, B'2- A'C' = D' are positive, because /2 - ay = - A is negative, and two pairs of conjugate imaginary points cannot be harmonic. Hence the symbols [A, B, C], [A', B', C'] represent real semicircles passing through the point w. The determinants of all semicircles passing through w are included in the formula D'x2- 2J^X' + Dx2, where 2J is the invariant 2BB'- AC'- A'C, and where J2 - DD'= -. The form (D', - J, D) is the duplicate of the form (a, /3, 7y), and is transformed into the product of that form by itself by means of the bipartite linear substitution A,B,B, C A', B', B', C We thus obtain the theorem: 'The modular curves of order D pass through the point [a, 83, y] as often as there are primitive representations of D by the duplicate of the form (a, /, y).' To determine (by means of the corresponding semicircles) the ovals which pass through a given point [a, 3, y] we have the following rule: 'Let (P, Q, B) be the duplicate of (a, /,?y), and let Pi, q1, q1, r P2, q2, q2, r2 be the substitution transforming (P, Q, R) into (a, /3, y)2; let also (P, Q, R) x (=,2)2= D be the given representation of D; the semicircle [8lzP2- P, /1 2-/2 ll, 1 r2 — 2 rl] of determinant D passes through [a,, 3, ].' To determine which of the modular curves of order D passes through [a, 3, y] we should have to distinguish the cases in which a, /3,, 7, 1, u2 have different congruential values for the modulus 2. This discussion, for brevity, we omit. If we apply to the form (D<, - J, D) the reducing substitution of Lagrange, we obtain the orders of the two lowest modular curves, which can pass through the point - + (co), or P. If on the tangents to these two curves at P we measure lengths PT, PT equal to the square roots of the corresponding determinants, and if, completing the parallelogram PTP'T, we construct the 406 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IX. parallelism of which PTP'T' is an elementary parallelogram, the tangents to the modular curves passing through P are the lines joining P to the nodes of this parallelism; and, if N be one of these nodes such that PN contains no other node besides P and N, PN is the square root of the determinant of the modular curve touching PN at P. Gauss has shown that every class of the principal genus of any determinant is the duplicate of as many classes as there are ambiguous classes; i.e. classes producing the principal class by duplication. Thus no point of determinant - A can lie on a modular curve of order D unless D is represented by some class of the principal genus of determinant- A; and, if there be such representations, there are, corresponding to each of them, as many points of determinant- A lying on the modular curves of order D, as there are sub-classes of determinant - A producing the principal class by duplication. The necessary and sufficient condition that a given point [a, 13, y] should lie simultaneously on a modular curve of order D, and also on a modular curve of order D', is that D and D' should both be capable of primitive representation by the duplicate of [a, f3, y]. Let D = (P, Q, R) x (1, ), D' = (P, Q, R) x (V1, 2)2 be the two representations; let 1, 1 =i v, and let (P, Q, R) be transformed by IA V2 ' l: iinto (D,- J, D') of determinant Av2. The form (D,- J, D') is then transformed by 2) - 1 I into v2 x (P, Q, R); that is, it is transformed by the -A2, Al bipartite transformation V2,- V1 P q1l, q11, rl -2, i 1 P2 q2, q2, r2 into v2 x (a, f, 7)2. The semicircles [V2p1 - V1P2, V2 ql-1 q2, V2 rl - V1 r2], [ex2p1 - l1P2 21 - 7 1 q2, Ir2'l - Alxr2], which are of determinants D', D, and which have J for their harmonic invariant and v x [a, 3, y7] for their covariant, are the two semicircles corresponding to the given representation and intersecting at the point [a, 3, r]. The equation DD'= J2 + v2A always subsists (as the preceding analysis implies) whenever any point of determinant A lies on two modular curves of the orders D and D'. But this condition, though necessary, is not sufficient; viz., confining ourselves to the case in which D and D' are relatively prime, J and Note IX.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 407 v must be relatively prime, and the numbers D, D' must have each the characters of the principal genus of determinant - A; when these additional conditions are satisfied, D and D' are necessarily represented by the same form of the principal genus; and the two sets of curves have as many points of determinant - A in common as there are ambiguous classes of determinant A. Let [ca,, y, 7] and [a2, P2, 72] be two quadratic points w,, w2, of which the determinants are to one another as two squares; let A1 = A'0, 2= A 2, A' being the greatest common divisor of A, and A2; and let it be required to assign all the modular curves with regard to which - + D (wl) and - -+ (w2) are inverse. Let (P, Q, R) be a form of determinant A' compounded of (al,,, 71) and (a2, i2, 72). If — +I(w1) and - +( (W) are inverse with regard to any modular curve of determinant D, D is divisible by 0102, and the quotient is capable of primitive representation by (P, Q, R). Conversely, if these conditions are satisfied, a modular curve of determinant D exists with regard to which the two points are inverse. Let (P, Q, R) x (,L, M)2 = D0-; and let 0102 pi) P 1 ) q15,ri P2) 2, q2, r2 be the substitution transforming (P, Q, R) into (al, j1G 7)X (aG2 2, 72); the points o = xl + iy,, W2 = X2 + iY2 will satisfy the equation + I _ q2 - 2 1) l r!-2 r | * ) 22 / x =)P - (2ixii -x + q1yl ) the determinant of the matrix being D; i.e. - + (L,) and - + (2) are inverse with regard to a modular curve of determinant D. If D and D' are two uneven numbers relatively prime, the necessary and sufficient condition that two given points should be inverse with regard to modular curves appertaining to each of those determinants is that the two points should have the same determinant A, and that D and D' should be primitively represented by the same class of determinant A; or, which is the same thing, that D x D' should be represented by the principal class of determinant A. If this condition be satisfied, every point of determinant A is inverse to another point of that determinant with regard to a modular curve (D) and also with regard to a modular curve (D'). If the class by which D and D' are represented is not a class of the principal genus, all the points of intersection thus obtained are 408 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X. imaginary, no two inverse points coinciding. But if D and D' are represented by a class r of the principal genus, the classes which by their duplication produce the class r give real points of intersection. X. On the Quarter Periods K, iK'. The fundamental pair of quarter periods (K, 2iC') is not in general the absolutely least pair of quarter periods appertaining to the doubly periodic function; for it does not follow that the parallelism (K, iK') is absolutely reduced because the parallelism (K, iK') is primarily reduced. Problem. To determine, for any given value of k2, the absolutely reduced parallelism equivalent to (K, iK'). From the preceding discussion it appears that the reduced triangle is formed by the vectors K, iK', -K-i', or- iK', iK', -K, according as the angle from K to iX' is obtuse or acute. It may be proved by the considerations already employed in the demonstration of the theorem (...)*, or by a method presently to be explained, that this angle is obtuse or acute according as the coefficient of i in k2 is negative or positive. We have now to show how, for any given value of k2, the vectors of the reduced triangle can be arranged in order of magnitude. For this purpose we employ the principles contained in a Memoir 'Sur les equations modulaires,' which will be found in the Transactions of the 'Accademia dei Lincei,' vol. i. Ser. iii. (1877)t; it will suffice to consider only one of the propositions to be demonstrated. We understand by [z] the absolute value, i.e. the analytical modulus, of any complex quantity z. 'The inequalities [K] <[K+ ~iK] subsist simultaneously, if [k2] < 1: if [k2] > 1, we have [K - iK] < [K] < [K+ iK'], or [K+ iK] < [K] < [K- iK'], according as the coefficient of i in the imaginary part of k2 is positive or negative.' * [Blanks in the manuscript are denoted by dots enclosed in parentheses, thus (...).] t [Vol. ii. p. 224]. Note X.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 409 iK' Let = X+ i Y; the inequalities [K] [K + iK'] are equivalent to the inequalities 0 2X~+ X2 + Y2. But the quadrant of the circle X2 + Y2 + 2X = 0, which lies within the reduced space defined by the inequalities in question, and which runs from the cusp at 0 to the point - 1 + i, is represented in the plane 2 + x + i y by a semicircle of radius 1, running below the axis of x from the point x= +, to the point x = -; and the two regions containing the points -1 and +1 respectively, into which the reduced space is divided by the quadrant are represented in the plane of xy by the two regions, containing the points infinitely far off on the axis of y in the negative and positive directions respectively, into which that plane is divided by the axis of x from + co to,2 by the semicircle, and by the axis of x from - to - oo. Hence according as k2 lies in the first or second of these regions or on the boundary between them, we have [K] > [K+ iK'], or [K] < [K+ iK'], or [K] = [K iK']. Similarly, if we divide the plane xy into two regions, one lying above the axis of x and external to the circle (x+ )2+y2 1 the other comprising the rest of the plane, it will be found that according as k2 -- lies in the first or second of these regions, or on the boundary between them, we have [K] > [K- iK'], or K] < [K- iK'], or [K] = [K- iK']. These two results taken together are equivalent to the proposition which we have enunciated. To complete the solution of the problem we have to discuss the inequalities [iK'] [K~iK']; their theory depends on the representation of the lines 2X +1= 0, or, rather, of those portions of them which lie within the reduced space, by the semicircles (x - 2)2 + y2 = 1. The final result is perhaps most simply expressed in the following form: The plane xy is divided by the axes and by the circles (x ~ +)2 + y2 = into twelve regions. The regions within both circles are designated by A, those within one circle only by B, those outside both by C; the four regions A are numbered 1, 2, 3, 4 according to the quadrants in which they lie; the regions B and C are similarly distinguished (see fig. 1). The Table indicates the arrangement of the four periods K, iK', KE+iK', K- iK', in ascending order of VOL. II. 3G 410 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X. absolute magnitude, according to the region in which the vector point x + iy = k2 - is situated. (1) iK', (2) K, (3) K, (4) iK', K, iK', iK', K, A. K-iK', K-iK', K+iK', K+iK', (1) (2) (3) (4) (1) (2) (3) (4) iK', K, K, iK', B. K-iK', K-iK', K+iK', K+iK', K-iK', K-iK', K+iK', K+iK', C. iK', K, K, iK', K, iK' iK', K, K, iK', iK', K, K+iK'; K+iK'; K-iK'; K-iK'. K+iK'; K+iK'; K-iK'; K-iK. K+iK'; K+iK'; K-iK'; K-iK'. It will be seen that along the axis of x, [K-iK']=[K+iK]; along the axis of y, [iK'] = [K]; along the upper and lower semicircles of (y + )2 + y2= 1, [K]=[ [K- iK'], [K] =[K+iK'] respectively; along the upper and lower semicircles of (x - )2 + y2= 1, [iK'] = [K- iK'], [iK'] =[K+iK'] respectively; and that when we traverse any one of these six boundaries the quantities, which on it are equal, change places with one another. Cor. 1. The zeros of the Theta functions ~vX 77rx / T X\ ver x\ 7 X\ a, ^ ^^K 2 \92K nK 3 ~2K' Note X.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 411 being respectively 2rK+2siK', (2r +1) K +2siK', (2r+l)K+ (2s+l)K', 2rK+ (2s+1) iK', it follows that the least zeros of a22( ) are always ~Kf, and the least zeros of 0 (7) are always +iK'. But the least zeros of 3(3 ) are (K-+ iK') or (K- iK'), according as the coefficient of i in the imaginary part of k2 is negative or positive; and the least zeros of 3(2-) are ~2K, 2 iK', +2 (K-iK'), ~2 (K+ iK'), according as k2 - lies (1) inside the circle (x+ )2+y2= 1 to the left of the 'axis of y, (2) inside the circle (y- )2+y2=1 to the right of the axis of y, (3) outside the two circles and above the axis of x, (4) outside the two circles and below the axis of x. These determinations assign in all cases the circles of convergence of the developments of sin am - cos am u, A am u, and their reciprocals, in series proceeding by powers of u. Cor. 2. The Table also assigns in every case the absolutely least pair of conjugate periods of the functions sin2 am u, cos2 am u, 2 am U; viz. this pair consists of the least and least but one of the four quantities 2K, 2iK', 2 (K~ iK'). But to obtain the absolutely least pairs of periods appertaining to the functions sin am u, cos am u, and A am u themselves, we have to consider the modular curves of the square determinant 4, instead of the lines and circles of determinant + 1. Problem. To determine, for any given value of k2, the absolutely reduced parallelisms equivalent to (K, iKZ'), ([K - [K i'], [K iK']), and (AK, iK'). Since the reduced triangle of the parallelism (K, iK') is acute-angled, one of the two triangles into which it is divided by the line joining any of its vertices to the middle point of the opposite side is certainly acute-angled, and the half side is always less than the bisecting line. Hence the reduced triangle of the parallelism (K, 1 K') is one of the four, iK T- i,....... (1) K+!iK I iK K ', ', (2) E+2iR 2i'K, -H- iE,....... (2) Ef-FiK \iK, - K,.......... (3) K- iK', iK' - K+-iK,..... (4) 3G 2 412 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X. in each of which the side 1 iK' is less than the side Ki+ iK'. In fact, we have the case (1), (2), (3), or (4), according as iK' (1) amplitude of - > 7r; [K] <[K+ K']; (2) amplitude of -> -r; [K] > [K+iK']; (3) amplitude of K-< r; [K]<[K-iK']; (4) amplitude of <-; [K]>[K-iK']. These inequalities have been already examined; in addition to these we have to consider the inequalities [K] ' [KK] [K ~iK']; [K+ iK] [K + iK']; [K itK] [ iK']; which determine the order of magnitude of the sides in the four triangles; and which are equivalent to the following: X2+ Y2 4; X2+ Y2+ 4X 0; 3(X2+ Y2)+4X=0; 3(X2+Y2)~X+40. In the plane k2= +x+iy, the circle X2+ Y2=4, and the three pairs of circles X2+ Y2+4X=O, 3(X2+Y2)~4X=0, 3(X2+Y2)+SX+4=0, are represented by the four loops of the modular curve (...), beginning with the innermost and proceeding in order to the outermost. If we designate the regions (taken in the same order from within outwards) into which the plane is divided by the curve by A, B, C, D, E, the Table gives the three least quarter periods corresponding to a value of k2- 2 lying within any given region: A. K, iU'; K~+iK+ B. - K', K; K~ iK' C,. -iK', K+~TiK'; K. Note X.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 413 iK', K - iK',; K+ iK. D. -1iiK' K+iK'; K+r iK. E. K~+ iK', iK'; K~ + iK'. The upper signs are to be taken in the region below the axis of x, and vice versd. The region C is supposed to be divided into two, Ci and C2, the first within, the second without the circle (x + )2 + y2= 1. To obtain the reduced parallelism equivalent to (QK, iK') we have only to take the modular curve (...) which is symmetrical to (...) with respect to the axis of y, and to interchange K and K', dividing at the same time by i. We thus obtain the following Table: A. -iK' ',; iK'+ ' K. B. C, K;~ iX' +-IK; iK. K, iK' + K iK'+ K. D. iK, iK'+K; iK'+ - K. 2 E. iK'~K+ K; iK'+ K. The reduction of the parallelism (Q [K- iK'], l [K+ iK']) depends on the remaining modular curve of determinant 4,... which is a symmetrical with regard to both axes. The reduced triangle of this parallelism always has the vectors (K~ iK') for two of its sides, and either K or iK' for its third side: in this enunciation, which is obtained by bisecting the side K ~ iK' of the reduced triangle of the parallelism (K, iK'), the signs of the vectors are neglected. 414 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X. Therefore, besides the inequalities [K] [iK'], [K+ iK'] [K - iK'], which we have already considered, we have to examine the inequalities [K] [K 1 [ _+ i/]; [iK'] 2- [K+ _ iK'] or, which is the same thing, the inequalities (X~1)2+ Y2 4; (X+ 1)2+ Y2 4(X2+ Y2). The four circles X2+Y2~+2X-3=0; 3(X2+Y2)~2X-1=0; or rather the arcs of those circles which lie within the reduced space are represented in the plane k2 - -2- = x + iy by the curve (...), of which the form may be roughly compared to that of a hyperbola having an internal loop at each vertex. We designate by A the central infinite region between the two branches of the curve; by B and B' the infinite regions internal to the two branches of the curve on the right and left of the axis of y respectively; by C and C7 the regions internal to the loops: we then have the following scheme, the upper signs being taken in the upper part of the plane, and vice versd: A. (K+ iK'), ( K++ i'); K, iK A'. -(K~+iK'), ( TK+iE); iK', K. B. (K iK'), i K; 2 (KE i'),K C. (K ~ iK'), iK'; (KF iK'), K. C. K, -(~K+ iK'); (KF iK'), iK. C' - iK', I (K_~ iK'); 1 (K v iK')I K. XLIII. MEMOIR ON THE THETA AND OMEGA FUNCTIONS. Arts. 1-14. DEFINITIONS AND ELEMENTARY PROPERTIES OF THE THETA, OMEGA, AND ELLIPTIC FUNCTIONS. Art. 1.] THE Theta Functions. An exponential series of the type m= + o I an2 + 2bm + c (i) m —oo is termed a Theta series. The necessary and sufficient condition for the convergence of the Theta series is that the real part of the coefficient a shall be different from zero, and negative; subject to this restriction, the coefficients a, b, c, may be any quantities whatever, real or complex. For the purposes of this Memoir it is convenient to employ a notation for the Theta functions somewhat different from that which has been adopted in the Tables.* Writing q = ei"~, we define the function,,, v (x, q) by the equation t (X, q) = iL V ( -)mq (2 m )e(2n+ )i = eir[=+o ~ n= -- 00 - n=+oo 6io{t(2rn+lA)2+ Z (2m +/) (2W + )] le 200 and v denoting any positive or negative integral numbers. 0o, (,q) = e(2K ), 2Kx 0 0(x, q) - 6.(2Kx-), 0 ) = -1k' 3 ( ) ( We thus have (iii) * [This Memoir was written to accompany the Tables of the Theta Functions calculated by Mr. J. W. L. Glaisher.] 416 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 1. For brevity, we shall often write So, 1q(X ) = o(x, q),.1,1(, q) = 3( q).l, o(, q) = 2(x, q), o, 0 (X, q) = 83(X, q); we shall also omit the second argument q, when no ambiguity is from this abbreviation. From the equation of definition (ii.) we infer immediately +2, Y($) =- 2, V(Z) ', +(x) - (-) = ai, V (X),. a. V (A + ) =-(-): ) * *. 9 (, + ): (-)Ve-(2+,).,(),. (.. + ^~ + (X) -= IT( +4I4 '+ ).(X + 2 ['', + v']). likely to arise...(iv). o. (V)... (vi). (vii) (viii)... (ix) Thus there are only four distinct Theta functions: o(, (): = o(X),-l(x) ( = -= 1, (X), 1, o (X) =:2(x), aoo0() = 3(X) (equations iv and v). Of these, a (x) is an uneven function, while the other three are even (equation vi). The Theta functions are singly periodic, having x or 27r for their period according as u is even or uneven (equation vii); the quotient of any one of them divided by any other is doubly periodic, the periods of:3, ^(x) (., (x) being (1) a, 2&7r, (2) r+ w7r, r - wr, (3) 2r, orw, in the three cases (1) -, even, v- v uneven, (2) u - u uneven, v - v uneven, (3) / - uneven, - v even (equations vii and viii); lastly, any one of the four can be expressed as a product of any other by an exponential factor (equation ix); so that, in particular,:.(X) = +r (;+w- 2) (X+1 +(- 1l)),:o ( =) =3 (x + 2w), &^ (,G) = de (r + ) ( + 2 7F) = a9, (X + h:). To obtain the Theta function a,,v(x) from the series 1 I (X) (i), we have only to write a, = iTo, a= i7r&(),++2z), c = -i[ACO + (+ 2 - + f C A1 r I IA co +- 12 +1 4 2 7T )1 Art. 2.] THE OMEGA AND MODULAR FUNCTIONS. 417 Thus, in some sort, the function a3 (x) is the simplest of the Theta functions, the values of the coefficients a, b, c being, for this function, a = iwz7r, b = ix, c = 0. 2. The Omega and Modular Functions. The Theta functions are themselves functions of two arguments x and q; but if we give to x the value zero, or any numerical value, or, again, any value depending on the value of q, we obtain a series of functions containing the single argument q or a. In this Memoir we propose to direct our attention chiefly but not exclusively to these functions of a single argument, which we propose to term the Omega functions. They are important not only in the theory of elliptic functions, but also in other parts of analysis; and they are intimately connected with several interesting questions of arithmetic, algebra, and geometry. An exhaustive account of the researches which have been undertaken by geometers on this subject would exceed our present limits; we propose therefore to give a brief outline of the most essential parts of the theory, and to select for a fuller treatment certain recent investigations which appear to have some interest. We shall usually find it convenient to regard o, and not q, as the independent variable, although for brevity we shall often employ q as an abbreviation for eiT. It will be observed that o may have any complex value of the form x + iy, in which y is positive. If y is negative, the analytical modulus of q is greater than unity, and the Theta series are divergent; the value y= 0 is also excluded, though y may be as small a positive quantity as we please *. The following are the expressions, in terms of w, of the Omega functions which appear at the top of each page of the Tables. We write 3[w] for (0, q). 2[] /2o[]]..... (i) / [] -k ' ~~ ) =K(w) = r,.....[w (ii) K'=K'(o) = K=7[ ]; ) * The collected edition of Riemann's works (Leipzig, 1876) contains some fragmentary notes on the limiting values of the Omega functions, when the analytical modulus of q converges to unity. To these the editor, Professor Dedekind, has added some interesting researches of his own on the same subject. VOL. II. 3 E 418 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 2. r, d _T 1 *[w] ',ir [wO] J J() = 1 K ao Lco] d K1(w) ao []' E = E (o) =K (o) - J(),..... (iii) J= J() = J 2K())+ E'= E'(w) =J'(w). When w is a pure imaginary (i.e. a quantity of the form ia-, - being real and positive), q is real, and k2 is real, positive, and less than unity; the functions K, K', J, J', E, E' are also real and positive. This, in all practical applications, is the only important case, and it is only for such values of w that the values of the Omega functions are given in the Tables; but the theory of these functions requires us to attend to all complex values of W for which the analytical modulus of q is less than unity. The equations (i), (ii), (iii) are to be regarded as defining the functions Vk, Vk', VK, K', J, J', and E. To these we add four other equations of definition, o m) (I 1 + q2 =o 1 i- q2 -- m co Xt,_2_ [l-x24(w-l)3: (0)=/2 q n _q2 T ):n 48( mnl1 -1~ m=l+ ^ /^\ and we observe that these four functions, as well as the seven preceding, are one-valued functions of w. (We understand by,/2 and t/2 real and positive roots of 2; and by q and q1 we understand eli7T and e214i.) Any rational and integral function of p (w), + (o), X (w) is termed a Modular function. We also note the equations ei (7r)() = 3 (0), 1 (17) = 32 (O), } 32 () 31 (0)=, 33 (7) = ( 0);. e4 '3o (2 C) = 3(o ) =~, e4 3 (-2',) =/3o (o), ^e4i - 1 W)= 3(0), ^ e4 3a )=e(0);} e4 i" o0 (2 + -=7) = 2 (0), e4ta31 (ir +- W)= 3(0), ii ewi7Iar (7r+u7rco)= -io(0), ert r+2rc (v o0 which are particular cases of equation (ix), Art. 1, and which serve to express the Theta functions of the half periods in terms of C[o], a2 [)], 3 [w]. Art. 3.] THE THETA AND OMEGA FUNCTIONS AS INFINITE PRODUCTS. 419 3. The Theta and Omega Functions as Infnite Products. The theory of the Omega functions is so closely dependent on the theory of the Theta functions, of which indeed it forms a subordinate part, that we shall find it convenient to give a brief outline of the principal properties of the Theta functions, and to deduce from these, as we proceed, the characteristic properties of the Omega functions. We shall, in the first instance, confine ourselves to those properties which are independent of the theory of the transformation of the Theta and Omega functions. The identity 1+q (v2+v-2)+ q4 (V4 + -4) + (6 +V-6)+.. w= oo m==co m-oo L M i0 =1m=1 m==l = 1 which has been demonstrated by Jacobi and by Cauchy, expresses a fundamental property of the Theta functions. If we replace v by q~v, and multiply by qiv, this identity assumes the form q4 ( + v-l) + q (V + V-3) + q (V5 + -) +... m==oi * ) * ()= = H (1 - q2m) x q4 (v + -1) X x (1 + q2V2) X n (1 + q2-2). m=l m=l m=l Writing successively v = e, and v = ie, in the equations (i) and (ii), we obtain 3(x) = 1 + 2q cos 2x + 2 q4cos 4x + 2qcos 6x +... = 1(1 - q2m) [1 +2 q2-1 cos 2x+q4m-2; (iii) 1 1 30(x) = 1-2 q cos2x + 2q4 x2cos4 6 x +... = (1 - qq2m) I[1 - 2 q2-1 cos 2x+ q4m-2]; (iv) 1 1 2(x) = 2 q4cosx + 2 q cos 3x + 2q os 5x +... 00 00 = 2q 4 (1 - q2m) cosx i [1 + 2q2mcos 2x + q4]; (v) 1 1 (x) = 2q sinx-2qsin3x + 2q sin +5x-... = 2 q lI (1 - q2)sin x HI [ - 2q2 cos 2x + q4]. (vi) 1 1 The zeros of the Theta functions (i.e. the values of x for which these functions vanish) are consequently as follows: 3 2 420 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 4. 0(x) = 0, if x = r7-+J(2s+1l)wx; 31(X) = 0, ifx = r7r+S; (vii) 2 (x) = 0, ifx = (2r+l) 1)s7; + (Vs a3 (x) = 0, if x = ('2r+ 1)r +(2s+ 1)r; i. e. the zeros of 3, y (x) are -(2r+v- 1)7r+ (2s+ u- 1) w7r. Putting x = 0, we obtain from the equations (iii), (iv), (v): a3 (O) = J 2- =l+ 2 q + 2q4 + 2q9+...= l(l q2) ( l+q2-1)2; (viii) v~~~~~~r'r~~~~ 1 3o (0) = /2 = - 2 + 2- 2q2+.... = i - ( - q2,)(i - q11-)2; (iX) 7r I 32 (0) = 2+kK =2qi+2q i+2q24 +... =2q4lI (1-q2m) (1 + q2).. 2 (x) Since 3a (x) is an uneven function, we have L3 (o)=o; but we find 31(o) = 2q I ( -q2m4)3 - 2 6 + 10 14q 4+... (xi) 1 If v denote any uneven number, we have, by an identity due to Euler, y=o m-==o 0 v=O (1 - q^)(+ ) = ( q(1 - qv) (1+ qv) (1 + q2) ( + q4v). =1.. (xii) v=l m=l v=1 Multiplying together the values of 0, (0), 3, (0), 32 (0), and attending to this identity, we obtain o (0) a2 (0) 3 (0) = 2 2kk'K = 2l I (1- q)3 = (0).. (xiii) Dividing (x) and (ix) respectively by (viii), we find 02 () = k, 2(0) = /k'........ (xiv) We also have from (xii), and from Art. 2, (iv), x (W + 1) =e X () = 2 eir-n (1 + ), (w) x *(w)=X3 (=).. (xv) For brevity, we shall often write 'P (w) = k2 = P8(w), ((w) = k= A8(w). 4. Expressions for the Modular Functions (p(w), (wo), X(w) as Infinite Products. The following different expressions for ) (w), for + (w), and for the Eulerian Art. 4.] THE MODULAR FUNCTIONS AS INFINITE PRODUCTS. 421 product II (1 + qm), or, which comes to the same thing, for X (c +1) have been given by Jacobi (Crelle's Journal, vol. xxxvii. pp. 67-77): I. m= + oo m ( - )mq6m2 + 2m (/2 q (1+ q2zM-1) (1- q2n) = q2 q(q)2n(n +l)q~(3n2 +m); )- 1 ( m2)(1-q ) q22 (i +q2m- ) _ lq4m) j~ M - wo 1 4 _ + q- - 1) _ - ) z )m q 2M2 + n e [1] (b (a( / _q4M-2)2(1_ q4m) 2,()q2n2 m 2 /,AM. (d) Pw /2 iTT(-l q,-)q- ) _ i (-)m2m=2+m - II (: + 1)] \(1 + q1(lq2m-1) (1 - q4l ) q2m+m 1 M2[IVM] (^ ^(^)-y~~g~~n^^_^^_= J 3=[s0,] -'^ (a) 1-(o)I 1 - _ - q 2m (a~~ ~ ( + ") - q2)=nm II. ( -)m q 3m2+n) (1 - q2-l\) -(1 q4m) (_) )mq2m2+m -I r (b) q(w)2 -)(1- ) zq2m2+m e (b ^(= (1 + q22 - ) (1 - q ) I I q = e-M 32 [ ( + 1)] 2 []2 "] (C) + (c) = (I -( q24m- 2)2 (1 - q4m) = (-) )qnqM2 (_ )?naq2M2 _3o [)]. o [2 ]' o [2 w] a3 [W] (d) ()) = -(1 - q4m-2)2 (1 - q2m) z ( _ )q2m2 j l 4 2M —1)2(I1- qa2m) - I qM2~ (a) X (+1) (a) 1 -1' 26 6 I- f III. = ^1 [1_q2rn 1 - qm y,( _) q 2 (6 + 1)2. I: ( _ )M q —!-(,M + 1)2 ' (b) x(+1) 1 (1(+ l q2 -1)(1_ q4m) ' Zq(s(+"l 2e2we -iq2" 2(-)q -1i2(+)2 (w + 1) 1 (c)?11 62~ 6 i r = l m 4" ( _ )m (6 In +m 1)2 ( 1-4"1)( qm - q,)Y~l2 4 X to + _i_ _1 (1 + q2 m -1) (I - q2m) 1- (6m-lymOa~) (d) X II I 26e2~ (I - Yq44M-2)2(l _ q4m) - 2: (_ )M2 jn 422 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 4. (e) x(I+)q q n 1-q" v 24 3(1 - q2"-1) (1 - q4m): ( - )In qsg (6 M + 1)2 _ 1(4m+1)2 ' X(-) q3 (f ) x +(nI+q3M2)-(lq3m-1)( 1q3) ( _ (6m+1) 2 2*6e4a^r (l - q 6 -3)) ( - 3(-)q3 '2' (g) X(+ i1) = n ( +"- q6m-) (1 q2) (4+)2 26724 (1 - q6 -5)(1 _ q6in-1)(1 q6 in) ( )nq(3m+ 1)2 We have also for the square and cube of X(w + 1) the analogous formulae IV. X) ( + 1) 12 II (1 + q2,-1) (1 - q4m) (a) X e 1 - q 23e12 1- q" = 1 2^n2m2+rn Zq^s(4m+1)2 -— l12 - -(_ ) q(3s2 +n) i ( _ )m q (6 m +1)2 2( + m) 6 )q(- 2 + 1) 2 1 1 -)mqrn2 1 - ( l )12 x 2( + ) l II 1- q2m_ ) 2.ei' -: (1- q2n"-1)2(l - q2m) V. 3(a l)- I (1 + 2m-l) (1-q4= ) 22(a) X I (1 -q2"m-1)2(I _ -2m).1 Yq2M2M JL(4 m + 1)2 q8 Z, ( ),n7,n - y ( _ )TnqM2 - a o [WI ) (b) X3( + 1) w 2^1 1 - q2M 3 1 98 11 I - qm 1Z(4m + 1)q4m2+2m 7 (4M + 1) q2M2 + m 3(o', q) 3/(0, q) l(0< 22) These expressions may be verified by comparing their type factors with the type factors in the equations defining p (w), + (o), and X(w+ 1); the transformation of the products into sums is effected by employing appropriate particularisations of the formula (i) of Art. 3: among these, besides the formule (vii)-(x), of that article, we may mention the following: VI. ( (a) (a') (b) (b) 1(1 _q2m-g1)l(1- q4m)= - ( _ )mq2m2+, (i + q2m-1) (1 - q4) = Z q2m2+m, 11(1 - qm) = ( - )mnq+(3)2 +) n (I - (- ).q.) -) m( +,q (3m + ), Art. 5.] EXPANSIONS OF THE OMEGA FUNCTIONS. 423 We give three examples of these verifications. In the formula (I. c) we have I (1-q21)m(1 q4m) = Z ( )mq2M2+r by (VI. a), and (1 - q4mn-2)2 (1 - q4M) = 2( - )q2m by Art. 3 (ix); also (1 -q2- 1)(1 q4M) 1 q2m-1 1 = 1 + q2" (1 q42)(1- 24m) - (1 _ q4_n- 2) (1 q2 -1) (1 q4_-2) 1 + q2M-' since by Art. 3, equation (xii), n,- -= 4 (1 + q2m). Again, to verify the equation (III. g) we have (1 + q6m-3) (1 _ql2m) = Zq6 2+3 m by (VI. a'); and, writing q3 for q, and - q2 for v2, in the formula (i) of Art. 3, we obtain n (1 - q6) (1 _ -q6-1) (1 _ 6r,-5) = z (_)Mq32+2; also H (1 + q6m-3)(1 _ l2m) _ H (1 +q6nm-3) (1 + q6) (1 - q6-5) (1 - q61-l) (1 - q6) (1 - qm-5) (1 q6n-1) 1n + T"n1 1 if 1 (1 - q6r-5) (1 q6m-l) (1 _ q6m-3) (1 _ q6m-5) (1 _ q6m-l) 1 -_ qiLastly, to establish (VI. b'), we write ei q] for q, and eiT q~ for v2; we thus find -( _) m(m+1) q(3m2+m) = n [1 - ( -)13m] X H[1 + ( _ )q3 -1l] X [1 (- _)rnqm-2] = II[ l(-)m3]. 5. Expansions of the Omega Functions in series proceeding by powers of q. Let:f(); L/2. qs Jacobi has given the formula (Fundamenta Nova, Art. 40, equation 27) k=Za (h) logf(q) = h I A=I h where A(h) is [1 + (- )2] times the sum of the uneven divisors of h. Let h= l P )= I[s, h]qh, s being any quantity whatever; since ) (h= h f S(g) =e —i A 424 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 6. we have, if h =, [s, 0] = 1, and, if h > 0, A (h) s2 A (a (b) ) + 3 (a) (b) (c) h]i = 1s —+-^ —+2 + 72 3-Sb +c...+ L S h +i ab 1.2.3 3 ab where 2i extends to every set of i numbers (equal or unequal) which satisfy the equation a + b + c +... = h. But, when s is a positive or negative integer, the coefficients [s, h] are more easily calculated by a recurring formula, similar to that given by Euler for the development of the product 1I (1 + qm), and deduced by him from the equation III. (a) of the last article. Several such formule are supplied by the equations of Jacobi. For example, from equation I. (c), we have f(q) = Z (- l)m q2m2+ M (- lq)q2Ma whence 1n, (- 1)_ x [1, h - 2m2] = (-)h or 0, according as h is or is not of the form 2m2 + m. Similarly, since fs+l(q) =f (q) xf(q), we find, from equation I. (b), E2[s +1, h-2m2 - m] = A [s, h - 4M2 - 2M], by which the coefficients in the successive powers off(q) may be calculated. Formulae, similar to these in their general character, exist for the other modular functions. 6. The Formula for the Multiplication of Four Theta Functions. We next give a formula which enables us to obtain the algebraical relations connecting the four Theta functions, and the differential equations satisfied by them. Let 2s=x +x2+x +x4, 2 c= x1 2 + 2+3 3+ 4, 2 ' =v + v2+v3 + 4, the integral numbers u,, F23, 3,4 and v1, v, v2, 4 being subject to the restriction that a and ' are to be integral. Multiplying together the four Theta functions r, v(Xr), r=1, 2, 3, 4 and transforming, in the general term of the product, the indices of eilrw, of eid7, and of e7i by the identities a2 +b +2 + d2 = (S-a)2 + (S-b)2 + (S-c)2 + (S-d)2 aa+b3+c'y+d =(S-a) (- a) + (S-b) (-3)+(S- c)(S- y)+(S-d)(S- ), where 2S=a+b+c+d, ~wh~ere ~2 i = a + e, + f, 22=a+/9+y+, we find + In 5o_-, +,, (a-v )+ ( -)' +3( ) Il 3~_,,,'-v + l(S- ). Art. 7.] THE ELLIPTIC FUNCTIONS OF THE FIRST SPECIES. 425 Giving in this formula to the symbols /x, ~2, /3 % V4 V1, V.,,., (ii) X1l, X2, X3 4 X first, the values 0, 0, 0, 0, 0, 0, 0, 0, X,I. X2, X3, X4; secondly, the values 1, 1, 1, 1, 0, 0, 0, 0, X1, X2, X3, X4 and adding the results, we obtain the equation Hr3 (X) + T,33 (2 (XI) = 1] I5 (s - x) + 2 (s ) (iii) This is the fundamental formula of Jacobi's Lectures (see Rosenhain, 'Memoires des Savants Strangers,' vol. xi. p. 361). Owing to the relations (Art. 1, ix) which connect the Theta functions of different indices, it is no less general than the equation (i) from which it is derived; it is, however, less easily manipulated. Of the conclusions which may be derived from the formula (i), we shall in this place mention only those which serve to establish the elementary properties of the elliptic functions. 7. The Elliptic Functions of the First Species. Attributing successively to the symbols of the scheme (ii) Art. 6, the values 0, 0, 0, 0 0, 0, 0, 0 0, 0, 0, 0 1, 1, 0, 0 1, 1, 0, 0 X, X, O, 0 0, 0, 0O 0 1, 1,, 0,, x, x, 0, 0 we find (0) + ()= (0),.... (i): (0) 52 (x) = )a2 (o): a (~) + () () (ii): (o) 2 (x) == (0) o (X ) ); e ( ) (iii) VOL. II. A I 426 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 7. or, using the notations of Art. 2, k + k'2 = 8 s()+ (w)=,........ (iv) k3 (x) =- ) + k (x) (,........ (v) k'a(x) = o (x) - k (x)......... (vi) Again, attributing to the same symbols the values 0, 1, 0, 1 1, 1, 0, 0 x -y, x+ y,, 0 0, 1, 0, 1 1, 0, 1, 0 x -y, x+y, O, 0 0, 0, 0, 0 1 0, 0, 1 x - y, x+y, 0, 0 we obtain the three equations, 20 (X - y) 31 (X + y) 3 (0) 32 (0) = -2 ((y) () (X) 30 () + 31 (y) 3O(y),2 () 33(X), (vii) o(X - y) 2(x+ y)0 (0)%2 (0) = 2 (y) (y) f2(X ) ~o(X) - (y) 33(y) x (x) 3(x), (viii) o (x - y) (x +y ) + a (0) 3 (0) = o(y) (Y) o (x) o 3 (X) - al (y) a2(y) i (x)3 (X). (ix) We divide each of these equations by y, and then cause y to decrease without limit; attending to equation (xiii) Art. 3, and observing that 3o(0) = 2(0) = (0) = 0, we deduce the three differential equations d la (x)) 3o2(O (x) 3(). d( - ((X)) = (x) ' '(.(x) d 2 (x)= - 93(0) 3 1() ) (Xi) - 1 k YO _(x) a2(X) P (x) )d X (x,) - 2(o ) - () ) - * * *...... (xii) dx (o (x), a2 (X) If we write - -r X for x, and introduce the elliptic functions sinamx, 2K cosamx, Aamx (i.e. the sine, the cosine, and the Delta of the amplitude of x), Art. 7.] THE ELLIPTIC FUNCTIONS OF THE FIRST SPECIES. 427 which are respectively defined by the equations 1 a x2Kj @,(x) sin amx = - I 0 (x).., rt 3\ (x)' ~/k { / (xiii) cos am x = J\ 2 (/ 2KJ _ 2(x) - (x)'... (xiv) vi3(2K E) 3( Aamx = k 2K - ( t /7rX\ e @( the differential equations (x), (xi), (xii) become. sin amx d = cosama dx x) Z) ' * * *. (xv).. (xvi) *. (xvii) uAamx,. d. cosamx -dx = - sinamxAamx,. dx d. Aamx dx — a= - k2cosamxsinamx. dx e.. (xviii) The equations (v) and (vi) give at the same time sin2 amx + cos2 amx = 1, ( k2sin2amx+ A2amx=l. ) The functions sinamx, cosamx, and Aamx are all doubly periodic (Art. 1), their periods being 4K, 2iK'; 2(K + iK), 2(K- iK); 2K, 4iK'. The zeros of the three functions are respectively (see Art. 3, vii) x= 2rK+2siK,, ( x = (2r + 1)K+ 2siK ',. (x) x= (2r + I)K+(2s+ )iK'. The infinite points are the same for all three, viz. x = 2rK+ (2s + ) iK'. Lastly, in the formula (i) of Art. 6, we successively attribute to the symbols (ii) the two sets of value 0, 0, 0, 0 1, 1, 1, 1, x - y, x+y, O, 0 312 428 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 7. 1, 1, 1, 1 1, 1, 1, 1; X-y5 +y, O, 0 adding the results, we find o(- y) +o(X + y) a(o) =, (x) 0 (y) - (x) 3 (y)... (xxi) Dividing by the terms of this equation the terms of the equations (vii), (viii), (ix), and introducing the elliptic functions, we obtain the formulae of addition sin am (x inamxcosamyAamy+sinamycosamxAamx s am (x sin am x cos am y A amy x sin am y cos am x A am x. xii sin am (x + y) = -,. (XXl) 1 - k2 sin2 am x sin2 am y cosam(x+y)= cosamxcosamy-sinamsinamyAamxAamy...(x ~~~~1cos am (x + y) = - x sin2 am y* (XX) am(x+y) = Aam x Aamy - k sinam x sin am cosam osamy 1 - k2 sin2 am x sin2 am y( ) The quantity K- x is termed the complement of x; the amplitude of the complement of x is the coamplitude of x, and is written coam x. From Art. 2, equations (v) we have sinamK=1, cosamK=O, A amK=k'; and hence cosamx sincoamx = sinam (K- x) = Aamx k;'sinamx cos coamx = cos am (K- x) Aamx k' Acoamx = Aam(K-x)= Aamx Aam The theory of the elliptic functions may be treated in two different ways: (1) We may begin, as we have done in this memoir, with the definition of the Theta functions. We then define the elliptic functions by the equations (xiii), (xiv), (xv), and we show, as has been shown here, that these three onevalued functions are doubly periodic, and that they satisfy the differential equations (xvi), (xvii), (xviii), the algebraical relations (xix), and the formula of addition (xxii), (xxiii), (xxiv). (2) Or we may begin with the definition of the elliptic integral fr( du u2 -...( -.. k(xxv) j0/(1 -2)(1 -k2u'2) Art. 8.] THE COMPLETE ELLIPTIC INTEGRALS. 429 We then define the function u as the synectic integral which satisfies the differential equation du2 dx2 =(1 - u2)(1-k2u2)....... (xxvi) and the initial conditions x=0, u=O, ~ -= 1.... (xxvii) This definition implies the theorem that the equation (xxvi) always admits of one, and only of one, synectic integral satisfying the initial conditions (xxvii). For a demonstration of this theorem, we may refer to the work of MM. Briot and Bouquet*. Assuming the theorem, we evidently have the equation u = sinamx; for sinamx is a synectic integral of the equation (xxvi), and satisfies the initial conditions (xxvii). Hence also v = cos am, w = Aamx, if v and w are two functions defined by the equations 2 = 1 -u2, 2 = 1 - k2S2, coupled with the initial conditions x = 0, v = 1, w =1. In this way the identity of the functions obtained by considering the differential equation (xxvi) with the functions sinamx, cosamx, Aamx, as defined by the equations (xiii), (xiv), (xv), may be completely established. That the functions u, v, w are doubly periodic, as well as synectic, can be inferred directly from the differential equation (xxvi), without employing the expressions of u, v, w by means of the Theta functions; the formulae of addition can also be established in the same manner. 8. The Complete Elliptic Integrals. We have already obtained the equations sin am K= 1, cos am K= O, AamK= k'; we have also (Art. 2, equations vii), I k sin am (K+ iK) =, cos am (K+ iK') = -, A am (K+ iK') =O. * ' Th6orie des Fonctions Elliptiques,' par MM. Briot et Bouquet, ed. 2, Paris, 1875. See livre v. chap. iii. 430 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 9. Hence, if in the equation du /(1 - u2) (1 - k2U2)' we put successively x = K, x = K+ iK', we find that K is one of the values of the definite integral I (1 - u2) (1 -k2u2) ' *. ** (i) and that iK' is one of the values of the integral 1 fl~ du. A /(1 - _2) (1 _ k22** which, by either of the substitutions = (1 -k'2y2)-2, u = (1 k2y2)(iii) " (l-'ky2)^, (l-y)...... (iii) is changed into. 1 du o/(1 _ U2) (1 - k'262) We thus have the two equations rrB r _1 dduu 1 du K= ',... (iV) 0J /(1 - U2 (1 - k2 u2)',/(1 -2)(1 - k'2u2) in which, however, the track of the integration has not been determined. With this determination we shall occupy ourselves hereafter; for the present we observe that, when w is a pure imaginary, k2 and k'2 are real, positive, and less than unity; the quantities K and K' are also real and positive. In this case, therefore, the integrals in the equations (iv) are the rectilineal integrals obtained by causing u to pass from 0 to 1 through a series of real values, the initial value of the radical being in each case +1. 9. The Partial Differential Equation of the Theta Functions. The Theta functions satisfy the partial differential equation da ivr d25 dwo 4 dx2* () which enables us to express their differential coefficients, taken with respect to C, by means of their differential coefficients of an even order taken with respect to x. Thus, from equation (iii) Art. 2, we find Jr' So(O) J (0(ii) 4K o (o)' K = o( (o).. (~) Art. 10.] THE ELLIPTIC FUNCTION OF THE SECOND SPECIES. 431 10. The Elliptic Function of the Second Species. If we differentiate the equation (xxi) of Art. 7 twice with regard to y, and put y= 0 in the result, we find d [o(X)) 0'(o)!2 (0) a21(x) dx 0 (x) *= (0) o0(O) a0(X) or, writing 2K for x, and attending to the equations (ii) Art. 3, and (xiii) of Art. 7, -, / X \ X-I k2sin2amx J 7r d 3 -K L20 (2) K 2Kdx ^(7rx\ Integrating between the limits 0 and x, we obtain =7 -J 7r K~ rr \' Jo x ~~K 2K W ^X2K) k2 sn~axdx x - 2KJ7r of Art. 9, (xiii) of (ii)..... (iii) where the right-hand member is Jacobi's expressions for the of the second species defined by the equation rZ(x) = ksi Z(x) j k2 sin2 am x dx.... elliptic function (iv) The function Z(x) is a one-valued function of x, whatever be the value, real or complex, which we assign to x, and whatever be the course of the integration from the lower to the upper limit in the equation (iv). This may be inferred from the theory of definite integration, since the residue of sin2am x, corresponding to any one of its infinite points, is zero; or it may be proved by considering the simultaneous equations Z(3x)= I ( k2u2du (Jo (- 2)(1 - k2 2) u cdu x = o /(1 - U2)(1 - k22)' (v) in each of which the initial value of the radical is +, and the integrations are to follow any one and the same track. J r — * [Jacobi's function Z (x) is equal to - x - d2sin2 am x dx, and therefore differs by the term as well as by a change of sign, from Z (x) as defined in the text. The Z (x) in the text, which is the same as the Z (x) used by M. Hermite, differs only in sign from the Z (x) of Weierstrass.] 432 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 10. From equation (ix) Art. 1, we find 30' =-,) o; ( ) ~7 + r17r) == - io (7r + 7ro); hence, attributing to x in equation (iii) the values K and K+ iK' in succession, we obtain Z(K) =E +Z(K i K) = Z(K+iK') = k2 sin2 am x dx = J, k2sin2amdxJJ i ^. (vi) k2 sin2am x dx = JJ+ J+ 2-= J+iJf 2'" The equations J= Z (K), iJ=Z(K+iK')-Z(K),..... (vii) are often taken as the equations defining J and J'; we have preferred the definitions which are given in Art. 2, because they exhibit J(w) and J'(o) more directly as one-valued functions of w. A characteristic property of the elliptic function of the second species is expressed by the equation Z(x + 2nK+ 2niK') = Z(x) + 2mJ+ 2niJ'.. (viii) in which m and n are any positive or negative integral may be verified immediately by means of the equation may be inferred from the simultaneous equations (v). Lastly, we may write numbers. This equation (iii); or, again, its truth I l k2U2du J 0 = I/(1 _ U2) (1 - k2u2)' 1 I,, I Ck k2u2 du J) = /(1 - 2) (1 - k22)2 (ix) the course of the integration being the same as in the integrals (i) and (ii) of Art. 8. The second of these equations, by the substitutions (iii) of Art. 8, is changed into l k2du rl,/lj _- k'2,2 JX= d/-=) i * -*, J/( _ u2) (1 2- ku2)3 Jo -u2. (X) where the course of the integration is the same as in the second integral (iv) of Art. 8. It will be noticed that we have also E= r1 k'2u2ddu 2U2A~ku E=K- J = (1 -)(1-du = /1 - du, io ae of h2) (a t i22)3 ( o )_v. which are of the same form as the integrals (x). (xi) Art. 11.] DIFFERENTIAL COEFFICIENTS OF THE OMEGA FUNCTIONS. 433 The symmetrical substitution 1 -v2 1_ tUs. ' = a * ' X. * * (Xii) 1 - k2v..... (xii) which results from combining the inverse of either of the two substitutions (iii) of Art. 8 with the other, changes the integrals (x) into one another, and the second integral (iv) of Art. 8 into itself. 11. The Differential Coefficients of the Omega Functions. Differentiating twice the equations (ii) and (iii) of Art 7, putting x = 0 in the result, and substituting for 3 (0) from equation (xiii), Art. 3, we find 2a(0) o (0) XS(0), a (0) YO (O) 3**. ( '(0) 3'o(0) a3 (0) _ (i)0) whence also, subtracting, and employing equation (i), Art. 7, 3'_(0) I _(0) 3 (A) - 3' (O) = - ra4 (O)..........(iii) a2(0) a3(0) 0o5TM * ** * Attending to the partial differential equation of Art. 9, we obtain from these equations the following: d.logkA/k dlog I lg. -- — L/ =1 = k2 12 cl.log2/k 1. dco i7 do i- ' d, ~= '(r which may be otherwise written d.k2 d.k'2 4 -d — = - d k2k2;...... (V) or again, employing the notation of Arts. 2 and 3, 4 '(c) - 1 - (.)= - - (.)= -;)K2, ()= - (; 1 ( P (,) i, (W) K2 K255() v )K; '(~-) 2= i7, X^ (0 = x[~(Kx) 1- r (vi) Since, by equation (ix) of Art. 3, So= =3@ -o(O) = /2 7T Y1 -7 7r VOL. II. 3 K 434 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 11. k' Substituting in this equation for d- its value from (iv) or (v), we find dKf 2 2 2 = —.K (J-kK).... dao) Z 7r (vii) Again, differentiating twice the equation (i) of Art. 10, and putting x = 0 in the result, we have _ _(0) 2 0(0) - 3[ 3 (o) 4 (0) 3 (0);!2 ao (0) a2-~\I~S I...... (viii) but, by the partial differential equation of Art. 9, o~- (o )=- 2 3 [(], hence the equation (viii) becomes 4i= O ) = - - [oi; 7r _' [0 3 S [] ]2 2k2K4 a0 [a] l ao [W] 2 or, finally, d.KJ_ 2 K2(2 K2)... do, = T2.. ( k (ix) In (v), (vii), (ix), we write for simplicity w = i7rr; we find d.k= 4 d. K = -4k2k/ K2 dT2 = 4K2(KJ- k2K2), d.KJ=2 d= 2K2(J2- k2K2). (x) These equations enable us to form without difficulty the differential coefficients of the various Omega functions defined in Art. 3; the following are useful formule: d In/ W- -r = -2(J-k2KIG~LPi d=J 2k2K2(J _ }), d-r d ( = -2 - 2 2 ( k2j+ Kk2K2), d. logk= 2k'2K2 dT= -2k'K?, d. log =2(22)K2k' d. o 2(2k -_ 1) K2, 1 d.log K(J- k 2d = r(J- ), dr KJ, cidT d. logk' 2k22 d. log () iT. = 2K2 d-r.. (xi) d. logkK dr = (J- K)= - E, 1d.logkk'K - 2. 2cdiT =K(- -- k ). / Art. 12.] DIFFERENTIAL COEFFICIENTS OF THE OMEGA FUNCTIONS. 435 When w is a pure imaginary (in which case T is real and positive), the Omega functions of Art. 3, as has been already said, are all real and positive. Hence d+() is always positive, d () is always negative; so that, as 7 increases from zero to infinity, +(w) continually increases from 0 to 1, and b((o) continually decreases from 1 to 0. On the same hypothesis, K continually decreases from oo to 7r, J from oo to 0, E'= J' from -7r to 1; K' continually increases from 2r to oo, and E'=K- J from 1 to 7,r. The demonstration of these assertions depends partly on the expressions for K, K', J, J', E, E' as definite integrals (equations (iv), Art. 8; (x), (xi), (xii), Art. 10), partly on the differential formulae (x) and (xi) of this article. dn.k2 dn. K2 d". KJ If n > 1, it follows from the formulae (x), that d-, d- n, and dn are respectively of the forms 2n +2kl2k"K"+IV_, 2n+1Kn+2 T, and 2"+Kn+lK T+l; where T Vi, AT, and T,, + denote integral and homogeneous functions of K and J, of the orders n- 1, n, and n+1, in which the coefficients are rational and integral functions of k2, with integral numerical coefficients. 2 2 2 1 d P More generally, if P= ka k'6Ke, p x -n is of the form Kn x TI, the numerical coefficients of the powers of k2 in VT being, in general, fractional; these coefficients, however, are integral if 4, are all integral. (' b' C It will be observed that no new transcendent is obtained by differentiating the Omega functions, since all their differential coefficients of any order can be expressed in terms of k2, K, and J. 12. The Differential Equations satisfed by the Omega Functions. Eliminating dr from the last two of the equations (x), Art. 11, by means of the first, we find CK 1J_ k2K)j d.J k k-E; dwhek by subtrtion k ) whence by subtraction J dE d(J-K) (i k- dk dk* *.(. 3K 2 4:36 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 12. Eliminating in turn J and K from the equations (i), we have dK (1 - 3k2) dK (1-k2) dk2 + k - l d2J 1 + k2 dJ (- k2) k2 k + J= o, which are the differential equations satisfied by K and J of k. Observing that by the definitions of Art. 2, K' =. K, J' -. J + 7 t) _, W, 71, 16 6 2K and that by equation (v) of Art. 11, dk 2 kk'2K2' we find from the equations (i)...... (iii).. * (iv) regarded as functions dK-' 1 dk = k (J' kK'), k = k'2 (K' J').. (v) These equations are of the same form as the equations (i), but contain I', J', instead of K, J. The differential equations (iii) and (iv) are therefore respectively satisfied by K' and J'; and the expressions CK +C'K', CJ+ C'J' are the complete solutions of those differential equations. We have deduced this result from the definitions of K, K', J, J' as one-valued functions of w without making any use of the expressions of these functions as definite integrals. A different demonstration may be obtained by employing the definite integrals; which, however, it must be remembered, cannot be used to define the one-valued functions. Writing, for brevity, Au for V/(1 - i2) (1 - k2a2), let us designate by P and Q any two corresponding values of the integrals P= 'du AUo)u lku22du Q = u - a....... (vi) (i.e. any two values in which the track of the integration is the same). It is not difficult to show that the equations (i), and hence also the equations (iii) and (iv), are verified if we write P for K and Q for J; for this purpose Art. 12.] DIFFERENTIAL EQUATIONS. 437 we have only to differentiate the equations (vi) with regard to k, and to make use of the identities 2 d / 1 \ 1 2 d /u-u. k" dk~t) au du 'Au k'2 d k2u2 1 k22 d /u- u..U3V) -kdkA\?% ) =A duAu * * a* * ) The equations (iii) and (iv) are therefore satisfied by K and J, because these are values of P and Q respectively; but so also are (2n +1) K+2miK', (2n+1)J+2miJ'; i.e. the equations (iii) and (iv) are respectively satisfied by K' and J' as well as by K and J. Since the integral expressions for J and J' become K'-E' and E respectively, when k2 is changed into k'2, we find that the functions E and K' - E' satisfy the differential equation dE 1+k' 2dE (1- k"2) d, k- k d k +E = k' K' -=dkO or d2E 1- k2 dE (1 k) ~+ dk +E=,... (ix) and that the complete solution of this equation is CE+ C'(K' - E'). If V is any algebraical function of k, K, and J (i. e. a function defined by an algebraical equation of which the coefficients are rational functions of k, K, and J), V satisfies a differential equation of the second order containing only V, k, and the first two differential coefficients of V with regard to k, and a differential equation of the third order containing only V and its first three differential coefficients with regard to w or 7. The differential equations satisfied by K and J are instances of the former assertion; as instances of the latter we may take the differential equations satisfied by K and k regarded as functions of r *. These differential equations, which have been given by 1 Jacobi, are as follows (we write c for X, and y for k, and we represent by accents the differential coefficients of c and y with regard to 7): C2 (cc"' + 3CC')2= 16c"2 (1 + ");.. (X) 1 9 25 *Jacobi,'Ueber die Differentialgleichung,welcherdieReihenl + 2q+ 2q4~ 2q9 +..., 2q4 + 2 + 24 +..., Geniige leisten:' Crelle's Journal, vol. xxxvi. p. 97. 438 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 12. 2y'y"' 3y" +(........ (xi) To obtain the equation (x) we deduce from (x) or (xi), Art. 11, the equation cc"= -4k2k'2, and we eliminate k2 between this equation and its first derived equation with regard to 7, viz. C4(CC" + 3c'c") = 16 k2k'2( - 21 k). 2K If we put z = / / -2K, and introduce o instead of 7 as the independent variable, the equation (x) becomes (z"'- 15zz'z" + 30z'3)2 + 32 (zz' - 3z2)3 = z (zz"- 3'2).... (xii) To verify the equation (xi) we write it in the form d'2 |-igd ^1 r~ o 7y d d2 1 +y2. dy,..y2 2d2 [d [logd] - [ logd] + (-1+) 2 = 0,. (xiii) and we substitute in it the values of the first two differential coefficients of dy log dr' deduced from the equations (x) and (xi) of Art. 11, viz. d2 dy j7d i[g~ j d 2K] (2 J.k2K), d2 [log d ] = 8K2(J K) (J+ k2K). The integrals z= /7 x -/ -, y= k, are, of course, only particular integrals of the equations (xii) and (xiii). We proceed to give the general integrals of these equations, which have been assigned by Jacobi; the form of the solution (as we shall see later) is suggested by the theory of the transformation of the Theta and Omega functions. c+d9 Let w = c + b the quantities a, b, c, d being supposed real, and ad - be = n being positive, so that the real part of iQ2 is negative when the real part of ico is negative, and vice versd. Art. 12.] DIFFERENTIAL EQUATIONS. 439 Replacing o by Q, we find from the equations (v), (vii), (ix) of Art. 11: d() 4iQ) X (Q) X (); dK(2 2=i"Xi di(2) = [J(Q) - m(Q)K(Q)] x... (xiv) dJ(Q) 2i d = - q O(Q) [J(Q) - K(Q)] x K2(Q). Let a+bf a+bQ bir I tK1(G-W xK(Q), J(4) = ~zJ(M)+ ^/n ^/n 2,/n Kil)' observing that d= ( n), we find that the equations (xiv) become - () 4 @ (Q)x )x 2(Q) dw doQ 2- 2 d(o c =- [J1 (Q) - ~>(Q) J, (Q)] x K. (),v) do, ^ [.W -^)IW~xK^, ^...-. (xv) dJ(2) = - (Q)[J1(Q) - 1(2)]x (Q) Hence any differential equation, derived by differentiation and elimination from the equations (x) and (xi) of Art. 11, will subsist unchanged if we write simultaneously ~ — ctw\ o~, /\ -( _-+a) for o(w), Y -d+bw^ V /? _-b.,+.x K(_ da for K(w), Ibw(+d -d++bo )' and ^4. C, c-aot\ l ba, c-aow? b exJ d b -w-Kd) +bw^) for J(v). - b + d -d +b 2bw -n d + bwo. When dw is the equicrescent differential, as it is in the equations (xii) and (xi), this observation enables us to assign the general solution of the differential equation when a particular solution is known. Thus the particular solution of the equation (xii), which we have found, is Z- 7(1+2ei7r T +2 + 2i......)4/X7r A /2 V r 440 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 13. If then we put a b c d __+_ — =_ b -i=3, = — =-, _-==a, so that 2=+, a -/ = 1, V/n /n -/n / /n a+o' the general solution of the equation (xii), containing three arbitrary constants, is Z= 2/7r (1 +2ei+2e4i+...e)=_/_X_/ 2 K(_a+ Z /a += 3/ T a +{3 w Again, the equation (xi) or (xiii) remains unchanged when we take dw as the equicrescent differential instead of d r; hence a particular solution of that equation being y = </ (w), its general solution is 1J~(a+ ) 13. The Abelian Functions. Integrating the equation (iii) of Art. 10, between the limits 0 and x, and taking the exponential function of each member, we find ex = e (;.... (i)_ _ ao (0) in which the value of the right-hand side is independent of the track of the integration, because the different values of which f Z (x)dx is susceptible differ only by multiples of 2i7r, The expression for So ( 7i) at which we have thus arrived, is that by which Jacobi originally defined the function 0(x) = 30(o 7(r) in the 'Fundamenta Nova'; the importance of this expression in the theory will be seen from the following observations. The quantity w enters into the right-hand side of the equation (i) only in k2 sin2am x; the equation (xxv) of Art. 7 shows that x can be developed in a series proceeding by powers of u = sin am x, of which the coefficients are rational and integral functions of k2; again, this series gives by reversion a development of sin amx in a series proceeding by powers of x, and having coefficients which are, in like manner, rational and integral functions Art. 13.] THE ABELIAN FUNCTIONS. 441 of k2. The same thing is therefore true of Z(x), of Z(x)dx, and, lastly, of e-f~ Z( () But whereas the developments of sinamx, sin2 am x, Z(x), and f Z(x) dx cannot be convergent for values of x of which the analytical modulus o surpasses a certain limit, the function X= (x) dx gives rise to a development which is necessarily convergent for all finite values of x, real and imaginary. For if, in the cosine-development of o (,r) (see equation (iv), Art. 3), we expand each cosine in a series proceeding by powers of x, we obtain an equation which we may write in the form oo - 2 iv = x 4 2K J 1\X+ a O) (7r4+ o (0) 1.2 o(0) + 1.2.3.4 3o(0) 2*,.K (ii) I T[r] x2 1 Tord] X4+ =1+ +.+ 1.2 To[r]K2 1.2.3.4 To[r]K4 ** if we again put = Z -rr, and denote the series:o [.] = 3o(0, q) = 1 - e-7 + 4- F2T-_ e-972T +... by To[r]. The expansion (ii) is convergent for all finite values of x; so is also ldrg~~ xz ~~~~~~~~~~~~-fx the expansion of e; and so, consequently, is the expansion of e (x) dx which is obtained by multiplying the two together. We shall now (after M. Weierstrass *) represent by Alo(x) the function ALo(x)=e —K (ii i ) A....... (iii) 7" which appears in the left-hand member of the equation (i); we shall write X2 (X4 Alo-(x)- A+ - -*; * A.. (iv) * Weierstrass, 'Theorie der Abel'schen Functionen,' Crelle's Journal, vol. lii. VOL. II. 3 L 442 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 13. and we shall show by a direct method that the coefficients A', AO, A',..., are rational and integral functions of k2 having integral numerical coefficients. For this purpose we transform the partial differential equation (i) of Art. 9 into an equation involving Alo (x) and k2, instead of o (x) and w. The partial differential equation gives immediately.L d.a 72Mx d -r 1 I I I d2( XK) dK 2 dx2 where, on the left-hand side, we have included the partial differential coefficients in brackets, in order to indicate that K is regarded as a constant in the differentiation. Substituting for [ d( 1 d.,a 7rX its value. 2 (2)(,,, ).-r -u~2 \2K d (K we find drT x, r 7X dX 2i 7r x) EK d r dr dx2 Writing, in this equation, for o (2K) the equivalent expression 2k'K x/ e- X2x Al 1(x), effecting the differentiations, introducing (from the equations (x) and (xi) of Art. 11) the values of d, di K d7 ( ) and dividing the resulting dr' d-r ' dr ' equation by,J2k'K xe' x x e x K2, 7i' we obtain the partial differential equation required, viz. dX2. + 2kx d.A(x) +4kk + kxAlo(x) =, dx2 dx d. k2 * *. a (v) This equation gives, for the successive calculation of the coefficients Art. 13. THE ABELIAN FUNCTIONS. 443 A, A,...... the following equation of mixed differences and differentials d.AO A+1 - 4nk2A - 42' 2+ 2n2n -.. i) nd.2 +2n(2n-)lk2A%.=0,.... (vi) whence A=0, A= -2 k, Ao= -8(k2+k4), &c. The equation (vi) shows, (1) that the coefficients of the powers of k2 in A~ are integral numbers; (2) that, if n> O, A~ contains no power of k2 higher than k2("-1) and no term free from k2. Hence, if n>l, we may denote the coefficient of k2(+ 1) in A}, by ac,, where u > 0, v>O; and we then find from the equation (vi), a,V= 4( + 1)a, 1 +4(+1)aI,,~- 2(/xv+ 1)(2, +2v+1)ap_1,. v_.. (vii) This equation of partial differences supplies a formula of reversion by which the values of the coefficients ao,, may be successively calculated. It is symmetrical with regard to the two indices, and v; and, since the value of A = -8 (k +k4) shows that a0, = -8 = a,,,, we have generally ai, =a. This implies that the coefficients AO are reciprocal functions of k2 of the order n- 1, or, which is the same thing, that Al (kx, ) = Alo(x, k). The function Alo (x), of which we have now obtained the development X2/~+2v+4 (2y + 2v + 4)' w +yAlo(x) = 1 + 1(- )/-+a, 1 k2/t+ 21 (21 + 2 + 4), (Viii) is one of the four Abelian functions of M. Weierstrass. The three others are defined by the equations X3' 5 x A 7 - ~2K All(x- All..3+ 21l.2.3.4.5 31.2.3.4.5.6.7 e x 2WK' 7r Alx (x)=-l A +A l - X 3 x~. 2 3 4 7/2K 1.2 2 1.2. 3.4 3 12.3.4.5.6 + =e x,(x) V"-~ x +A2 1.3-4-3 1.2.3.4.5.6+2 =2e v r TT 3L2 444 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 13. so that Al, (x) Al,(x) amxAl,(X) sin amx = Al () cosamxA amA. ( )(x) ^Al,(x)' ^ AIA o(x)' A (a) and, if we use the notation of the Tables, Al(x)=e K x. (xi) 2k'K 7r Since, for values of x which do not surpass a certain limit, the three elliptic functions sin amx, cos amx, A amx are all developable in series proceeding by powers of x, of which the coefficients are rational and integral functions of k2, it is evident a priori that the same thing must be true for the functions Al,(x) = Al0(x) x sin amx, A1 (x) = Alo(x) x cos amx, Al(x) = A10(x) x A amx; while the equations (ix), by which we have defined these functions, show that their developments must be convergent for all finite values of x. To obtain these developments, we first deduce from the partial differential equation of Art. 9, by precisely the same process which we have indicated in the case of Alo (x), a partial differential equation for each of the three functions, containing only k2 and not w. These equations are of the type d2. Al, (x) d Al, (x) + 4 2dAl8x) +GxAl(x) 0,.. (xii) dx2 2kx dx +4 d.k2 + where G1 = k'2 + k2 x2, G= 1 + k2x2, G = k2 + kx2. Treated in the same manner as equation (v) they lead to the developments 2 / + 2v + 1 Al,(x)= s(-).+ v,,b, 2k2 11 (2j + 2 v + 1)' X2/z+2v+2 Al,(x)=l-Z(-)y+,c 2 ) * k* (*iii fI (2A+ 2v+2)' Al, (x) = 1 _ Z ( _ k2 V + 2 X2p+2- 2 where b, V= (4A + 1)b,_ - l+ (41v+l)b_l,V- (2 +2 - 1)(2 +2Y- 2)bA_l_, l i) C,:= (4u + 1)C;,,_ -I+ (4U + l)~c_l, - (2 + 2 i) (2M + 2iP- l)c,_l, v_, Art. 14.] DIFFERENTIAL COEFFICIENTS. 445 so that b, V = by,, or the coefficients of the powers of x in Al, (x) are reciprocal functions of k2, as in the case of Alo(x), the coefficients of the powers of x in A12 (x) and A3 (x) being reciprocal to one another. We thus have the equations Al (kx, ~) =Al (x, k), Al (kx, A) =A3 (, I), Al3 k, ) =A2(X, k) 14. Differential Coefficients of the Omega Functions expressed by means of the Abelian Coefficients. The differential coefficients of the four Omega functions 2k'K /2kk'K /2kK j2K ~/ ----- ----, -—, can be expressed in terms of the coefficients of the powers of x in the expansions of the four Abelian functions. We have To[T]=J 2 kK = o, To[r] = T,] x KJ (equation (xi) Art. 11); also, by a theorem in Art. 11, we may write 1 []- _[7. K2 T0o [] ' K ' (i) where f () is a rational and integral function of - and k2, with integral numerical coefficients, and of the order n in 1. The equations (ii), (iii), and (iv) of Art. 13 now give 1 (+f ) \ X2 f( ) X4 + j J2X4 1 r x 2 X4 (ii) + (-) 2 +2 K 1.2.- + *212.3.4 =[l x+ +^ + 2 -+...]x[l-A~1+A~.234-...], whence, on equating coefficients, Ai(K)=K+10= K, since ~=0; f() 13 2 +A = 1.3X2 - 2, A 2- ff 2 " 1 i~ 446 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 14. and, in general,.()=(2 -1)! -(2 n- 3)!C A~n+ (2n -5)! A n -2 J J -(2n - 7)! C2A~ 3 +... + (-I) -1(2n- 1)A~_l +(-)nA,, where (2r- 1)! is the continued product of the uneven numbers 1, 3, 5,... 2r - 1, and CQ is the coefficient of x' in the expansion of (1 + x)8. If in the right-hand side of the formula (iii) we take successively (instead of A~,... ) the coefficients A ', A2, A...A" A". in the expansions of A12(x) and Al3(x), we obtain expressions for 1 H?[ ] and 1 T([ K2n T2[TJ a K2n T3[7] where T2 [] = 3s2(0, ), and T3 [7] = 3(0, ). The expansion of the uneven function Al, (x) leads to a slightly different 1T()where T(n -/k. ',, 2r K0K, 2k formula. Representing 2K T() where T()= - (0 q) by rK 7r 7r.f-(i), we obtain by the same process as before () = (2n-1)! C2 +I -(2n-3)!Cn+An + (2n-5)!C+1A 2 J (iv) --... +(__)nn-In (2n+l)A_l +(-)nA = 0. It will be seen that (- )"A~ is the term not containing in f( Thus, without using the partial differential equation (v) of Art. 13, we may obtain the equation (vi) of that article by differentiating the equation ) [7]= 2nT, [r] xfn and putting = 0 in the result. A similar remark applies to the coefficients A', AN', A"A'. Art. 16.] ARITHMETICAL THEORY OF BINARY MATRICES. 447 Arts. 15-23. ARITHMETICAL THEOR OF BINARY MATRICES. The theory of the Transformation of the Theta functions and Omega functions depends in great measure upon the arithmetical theory of binary matrices, of which the constituents are integral numbers. We shall therefore give in this place an account of such properties of these matrices as we require for our present purpose. We omit many of the demonstrations, on account of the elementary nature of the subject. 15. Composition of Matrices. a b' |\ a b" If A'I = I, and A"= d" c, dc, ad the matrix a'a + b' c", a' b+ b' d" a ca'+ d'c", c' b"+ d'd" is said to be compounded of the matrices I A I and I A' I; and this composition is expressed by the equation IA I = A' I x A"i. The determinant of the compounded matrix is the product of the determinants of the component matrices. It will be observed that the order of composition is not indifferent; or, which is the same thing, the matrices I A" I x I A' I and I A' I x I A" I are not, in general, identical. In the expression | A' x I A" the matrix l A' is said to be postmwultiplied by the matrix I A' i, and A" I is said to be premultiplied by IA'1. The composition of matrices differs from the multiplication of numbers in the respect just mentioned. But in the composition of matrices the components (as in the multiplication of numbers the factors) may be grouped together in any way we please, provided that the order in which they succeed one another be not changed; for example, I A I x I B I x | C is either I A I x | B postmultiplied by I C, or I A I postmultiplied by B I x C 1. 16. Unit Matrices, Primitive Matrices, Reciprocal Matrices. We shall have occasion to consider only those matrices of which the determinants are positive numbers different from zero. A matrix of which the determinant is +1 is an unit matrix; a matrix of determinant n, of which the four constituents have no common divisor, 448 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 16, is a primitive matrix of determinant n. If n has no square divisors, every matrix of determinant n is necessarily primitive. The matrix compounded of two primitive matrices, of which the determinants are relatively prime, is always a primitive matrix. The matrices, and, - are said to be reciprocal matrices; and C) d -c, a the matrix reciprocal to the matrix A I is sometimes represented by the symbol A 1-1. Reciprocal matrices have the same determinant. The result of compounding two reciprocal matrices is the same in whatever order the composition is effected; in fact n, 0 IAIxIAI -1 =AI-1 x IAI oI= n being the common determinant of the two matrices. The matrix reciprocal to I A I x I B I is I B l- x I A I-; for we have IA Ix BI x IBI-1 x lA -1 = ob, if a and b are the determinants of I A and B. Theorem. If I A and B1 are given matrices, of which the determinants are a and b respectively, the equations \Al = lx x \B..... (i) IAI =I BI x IY.... (ii) are irresoluble when b is not a divisor of a, and may be either resoluble or irresoluble when b is a divisor of a; if either of them is resoluble, it admits of only one solution. For, postmultiplying (i) by (B)-, and premultiplying (ii) by the same matrix, we obtain the equations XI = IA!x BI-1, I Yl = ~ b BI- I X I which completely determine the constituents of the matrices I X I and I Y I but these constituents are not necessarily integral. b Cor. If A =, b the equations (i) and (ii) are both resoluble, and we have IXI= IYI= I I- '. Art. 18.] SYSTEMS OF NON-EQUIVALENT MATRICES. 449 17. Equivalence of Matrices. If A I and B B I are any two matrices connected by the relation IAt = 1elx IB I........ (i) where 1 is an unit matrix, i A i and | B | are said to be equivalent by premultiplication, or, for brevity, pre-equivalent; if, again, I A and I B are connected by the relation IAI =lBJxeI.... (ii) I A I and 1 B I are said to be equivalent by postmultiplication, or, for brevity, post-equivalent. The relation (i) may also be written IBI= eI-x IAI; and the relation (ii) may be written IBI= IAI x eI'-. Equivalent matrices have the same determinant; and the greatest common divisor of their constituents is the same. In any matrix the greatest common divisor of any column is not altered by premultiplication with an unit matrix, nor the greatest common divisor of any row by postmultiplication with an unit matrix. If two matrices are equivalent by premultiplication, their reciprocals are equivalent by postmultiplication, and vice versd. If two matrices are equivalent in the same way to a third matrix, they are equivalent in that way to one another. 18. Systems of non-equivalent Matrices. Let n be any positive integer, and a (n) the sum of the divisors of n; every matrix of determinant n is equivalent by premultiplication to one, and only one, of a system of - (n) matrices. Consider the system of matrices included in the formula aI- g,o[ \G\ h, gI' where g, g' is any pair of conjugate divisors of n (we may suppose these divisors taken positively), and h is any one of the g numbers 0, 1, 2,..., g-1. The number of the matrices G I is - (n); and it can be shown (1) that no two of them are pre-equivalent, (2) that every matrix of determinant n is preequivalent to one of them. VOL. II. 3 M 450 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 18. Similarly it may be proved that every matrix of determinant n is equivalent by postmultiplication to one, and only one, of the a (n) matrices, g, IG'i= h, g the signification of the letters g, g' being the same as before. We may regard all the matrices pre-equivalent, or post-equivalent, to a given matrix as forming a class. For any determinant n, and for either kind of equivalence, the number of non-equivalent classes is r (n); and as representatives of these classes we may take any r (n) matrices we please, one out of each class. Thus the matrices G I form a representative system of the classes of determinant n, non-equivalent by premultiplication; so also, if I a t and I are given unit matrices, do the systems of matrices |a x G, GI x l3, | a | x G I x i |. For each of these systems consists of o-(n) matrices of determinant n; and in the same system no two matrices are pre-equivalent. Similarly the system I G'I, lal x I G'I, I G'lx IS1, \aI x I G' x I31 are representative systems of the a- (n) classes of determinant n, non-equivalent by postmultiplication. Any such system is termed a complete system of matrices of the determinant n; the special systems j G I and I G' I are said to be reduced. If, d denote a complete system of matrices non-equivalent by prec, dmultiplication, the reciprocal system d- is a complete system nonc, a equivalent by postmultiplication; and vice versa. The a-(n) classes are not all primitive, if n has square divisors. Let p', p", p"' be the different primes of which the squares divide n; the number of primitive classes (in either classification) of matrices of determinant n is a(n) = 7(n) - (2) + a)- (p -) +... =n (l +) x~~~~_ / v.Q / \p i/ Vp_2 p^2v P/ the sign of multiplication I extending to all the primes p which divide n. Art. 19.] COMPOSITION OF SYSTEMS OF MATRICES. 451 19. Composition of Systems of Matrices. Let A I and I B I denote complete systems of matrices, non-equivalent by premultiplication, of the determinants a and b, supposed to be relatively prime. Then the system I A I x I B I is a complete system of matrices of the determinant ab. For (1) the number of matrices in this system is -(a) x - (b) = o-(ab); and (2) no two of the matrices in the system are pre-equivalent. For, if possible, let I.4x lB[=lal x All x l, a [ being an unit matrix. Let IBI = x1, x I Gl, B I = I I G1i, | f and I (3 I being unit matrices, I G and (I I reduced matrices of determinant b. Postmultiplying each side of the equation IA I x i x I = la x IA, I x ll x I G by G-1~, we find bO 0 IAlxl|B3x t, = a xIA I x 1 XI Gx\IGj -. Hence the constituents of the matrix on the right-hand side are divisible by b. But the determinant of |a x A I x I 3 is a, which is prime to b; therefore the constituents of ] G x G -1 are divisible by b. It is found, on trial, that this is impossible unless G I and | G I| are identical; hence, finally, B, I B, I and therefore also IA, A1 I are identical; which is contrary to the hypothesis. We may add that if A I and B I denote complete systems of primitive matrices for the determinants a and b, ] A I x I B I will denote a complete system of primitive matrices for the determinant ab. It follows from the preceding theorem that if a = P x Q x R x..., where P, Q, R,..., are powers of different primes, and if IP1, IQ, IR,... denote complete systems of matrices (or complete systems of primitive matrices) for the determinants P, Q, R,..., respectively, IP x IQ I xI I... (where the matrices are to be compounded in any definite order) will denote 3 M 2 452 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 20. a complete system of matrices (or a complete system of primitive matrices) for the determinant a = P x Q x R... When the two determinants a and b are not relatively prime, it will suffice for our purpose to consider the case in which one of them is the power pr' of a prime and the other is that prime p itself. We shall attend only to primitive matrices, First, let u > 1, and let pI and I p denote the reduced systems of primitive matrices of the determinants pt and p, so that p I consists of p4-l(1 +p) matrices, and PI of p +1. It will be found that the system I pf I x i p I contains (i) the primitive matrices (each occurring once) of determinant pp+l1; (ii) the matrices (each occurring p times) which are derived from the primitive matrices of determinant pe1' by multiplying each constituent by p. In all therefore the system Ip Ix l pl contains pp(1 +p) +p x p-2 (1 +p) =p~- (1 +p)2 matrices, as it ought to do. Secondly, let j =1: the system I p x p consists (i) of the primitive matrices I p2 I each once repeated; (ii) of p +1 matrices of determinant p2 having their coefficients divisible by p in all it contains p(p +1) +p +1 =(p +)2 matrices. For brevity we have in this article considered systems of matrices nonequivalent by premultiplication; but the theorems which we have enunciated hold equally for systems non-equivalent by postmultiplication. 20. Reduction of any two Primitive Matrices of the same Determinant to one another. If I Al I and I A2 I are two primitive matrices of the same determinant n, we can always find two unit-matrices | a I and I, such that l A,1 a l x A2 Ix 1II. To establish this it is sufficient to show that we can always satisfy the equation 0n, 1 IA I Ix on, 1 0 with two unit matrices | a I and i 1. Let A I a, b 5 where ad-bc = n, and a, b, c, d have no common divisor. c d hc, d Art. 21.] THE SIX TYPES OF UNEVEN MATRICES. 453 The simultaneous congruences a + b- 0, c'+d-,O0, modn, always admit of solution with relatively prime values of Y and ti (this may be seen by resolving n into a product of powers of different primes, and considering the congruences with regard to each factor separately). Let (i~-)IY =l, a(+b5=nX, cy+d=ny, a l+b6l=Xj, c(,+dn,=yl so that XMul - XIA = 1; we find l 1 x | H i 1 |= n | X, X 1 x, O l 7x, V i n-, nX O, _1 m, l or ^,^ nl 0 lx|, - (! ^1 21. The six types of Matrices of an uneven Determinant. Every matrix of uneven determinant is, with regard to the modulus 2, of one of the six types 1, 0 0,1 1, 0 1,1,1 0, 1 0,1 ) -1,0 I, 1,0 (i I ' which we shall represent by the symbols 1, 4, a, 7, p, p2; giving rise to the system of congruences (mod 2): O2-a2 -72 -1; p3 -1; a-TT=7 E^ fa-p; -+ _AT a_- CA-p2; r _^ (Tp2 _T _pE pr = p2T _E T r - Ta; =ap2 - -2 a Afp _Eip2 -p2-pT E- TT- Tr; _r xf'p2 _ cp =px f p2a - Gu-s fcrxVZ. This system is symmetrical with regard to 4, a, r and p, p2; viz. it is not altered by a cyclic permutation of f, a-, r, nor again by a simultaneous interchange of p, p2 and of any two of the three A, c, r. The relations between the six anharmonic functions x 1 1 x-1 x, 1 -- x, - x-l' x' 1 1-x x 454 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 21. are identical in form with the relations expressed by the foregoing congruences; for, representing these functions, taken in order, by x, +(x), -(X), r(x), p(x), p2(x), we have the equations ~C2 ( =0-2(X)==r2(X)=X1 p3(X)=X, ( [ )fr (x)] = + [C ()] = p (x), &c. &c. &c. If we represent the six unit matrices which we have chosen as representatives of the six types, by the symbols we find I + 12 _ I 111 la 12 = P, I T 12 = Q) I p 13= - I 1 1). (ii) where we write - 11, P, and Q for -1, 0 1,0 and 1,2 0, - 21 0,', 1 We have also 'pl =l xl- xl l= ll-1 xl +1=l-I+XIOx -, - ip2 I=1lj-1 X H1- j I+-X I = I-1X 1 -1oj- - I| + = | - |~ x T | X |-= = | X - X,.., (iii) I o -- = x jX ||1 X-|x I |-I = |7 | X | l -=I1 X I l, 1ra =KJk,-1X / Ox- -lXi- =-OIlxl, xl*I. The powers of I r | and a oare all different; but the matrices + I | are square roots of the unit matrix -I 1 I; and I p I and - p2 are cube roots of the same matrix. Thus each of the equations lXi2=l:l, lXI3=lll, lX14=lll, iXJ6=lll, admits of as many solutions as it has dimensions. No other equation included in the formula IX = 1 admits of any solution other than 1XI= +11. The following congruences are satisfied by the constituents of unit matrices a, b of the six types respectively; the modulus is 16 throughout. C5 6/ Art. 22.] PRIMARY MATRICES. 455 (1) a-d)(a+ b)O, (aa-d)(a+d+c)-O, ( ) (b + c)(b - ca)_O, (b + c)(b -c + d), (C) (a+b)(a-d~b)-O, (iv) () (a+d)(a-d+c)O,.. (p) (b - c) (bC+~d)o, (p2) (b - c) (b+c+a)Eo. 22. Primary Matrices and Primary Equivalence. A matrix of uneven determinant is said to be primary when it is of type (1) and satisfies the congruence a-l, mod 4. Thus 254 is a primary 2, -5 matrix of determinant 27; - 3, is a primary unit matrix if n is any uneven number, the matrices G?= (-)p(~-.g, 0o 2A, (- )(g-)g' (the signification of the symbols g, g', h being the same as in Art. 18) form a complete system of primary matrices non-equivalent by premultiplication. Similarly the matrices \G -= ((-)i('-1)g', 0 2A, (-)(g'-,)g form a complete system of primary matrices non-equivalent by postmultiplication. If I A be any matrix of the uneven determinant n, and of the type (1), it is evident (1) that either I A or - A is primary, (2) that, if n-1l, mod 4, } A I and A l-1 are either both primary or else both not primary, but that if n _ 3, mod 4, one of these matrices is primary and the other not. Theorem I. "Every unit matrix can be represented in one way, and in one way only, in the form x x lal, where r = + 1, | y | is one of the six typical unit matrices, and I a I is a primary unit matrix; similarly every unit matrix can be represented in one way, and in one way only, in the form n xlal x I " Theorem II. "Every primary unit matrix I A can be represented in one way, and in one way only, in the form Pl x Q^2 x p3 X... xQ s,....(i) 456 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 22. where X\, \2),.,X2s are positive or negative integers, of which the first and the last may be zero." For, in the formula (i), if \1 i 0, the two constituents of the lower row of l A are respectively greater in absolute magnitude than the corresponding constituents of the upper row; if Xi=0, the contrary is the case. Hence the representation is possible in one way only, because an equation of the form PXl X Q2 X PX3 X... = P1 x QIz x P3 x... implies the equations X1 = 1, A2 2= 2, X3 = /3. Again, if in I A I the lower constituents are the greater, we can always determine an integer X\, such that in P il x A 1, which has the same upper constituents as IA 1, the lower constituents shall be less than the upper; we can then determine an integer X2 such that in Q-^2x P-^lx X A 1, which has the same lower constituents as p-~l x A I, the upper constituents shall be less than the lower; by proceeding in this way, we shall finally arrive at an equation of the form... P- X Q-X2 xP-^ix IA I = + 1, which coincides with the formula (i), and gives the required representation. Employing the symbols (qD, q2,..., g) and [ql, q2,..., q,] to represent respectively the numerator of the continued fraction ql +- 1 q +a te + fr and the continued fraction itself, we find 1A I } ^2, ***., 2x2_-), (2x?,..., 2x2,) I (2X2,..., 2X_), (2\,..., 2X2) and we may write the equation w = i A x Q (see Art. 24) in the form w=[2X1, 2X2,..., 2X,2, Q]. If the equations A l-=IB x BI, IAI-IBjxIeI, the unit e | is primary, the equivalence of the matrices I A I and I B is said to be primary. Art. 23.] PRIMARY MATRICES. 457 Theorem III. "If Irl represents a complete system of matrices (or a complete system of primitive matrices) of determinant n, non-equivalent by premultiplication, the formula xI(l xlIr.......... (ii) represents a complete system of matrices (or of primitive matrices), nonequivalent by primary premultiplication." For (1) if Z I be any given matrix of determinant n, let Z I = A Ix I r I, IA l being an unit matrix, and IF r being one of the matrices r. Let IA 1= x I a x l, where i a I is primary; then IZlI=xl lxl lxIlrl. Again, (2) if 1 x l x l r1 is primarily equivalent to 2 x \1 2 ' x r 1I, we must have, first of all, I r1 = I| r I, or else the two matrices could not be equivalent at all; then the equation 7 X ll X I rl I=:2x l 1 XI 2X I rl, in which [/3 a is a primary unit matrix, gives by postmultiplication with I r| 11 '1X I I= 2X /3 X 2 l, whence finally | 3 I = 1, i- = 2, (= -; i. e. two matrices included in the formula (ii) cannot be primarily equivalent without being identical. Similarly the formula nxlrIxll,.......... (iii) where I r, I represents a complete system of (primitive) matrices of determinant n, non-equivalent by postmultiplication, represents a complete system of (primitive) matrices of the same determinant non-equivalent by primary postmultiplication. If N be the number of matrices (see Art. 18) in the systems r and r', 12 N is the number of matrices in each of the systems (ii) and (iii). When n is an uneven number these 12 N matrices are equally distributed between the six types; i.e. there are 2 N matrices, or rather N pairs of matrices, the two of each pair differing only in sign, which appertain to each of the six types. This may be seen by taking for r or r' the primary systems G or G' of Art. 18, since each matrix of r or r' gives rise to a pair of opposite matrices of each of the six types. 23. The nine types of Primitive Matrices of an Even Determinant. The primitive matrices of any even determinant, considered with regard VOL. IT. 3N 458 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 23. to the modulus 2, are of one or other of nine different types, which are exhibited in the following scheme: 1, 1 1,1 ' 0, 0 1, 1 l and which we shall symbolize thus: 1, 00, 1 1, 0 ' 0, 1 0,0 0, 0 1, 0 00, 1 1,0 0,1 0,0 0 0, 0' C1, 13 C2, 1 C3,1 (c).... C 2, 2 2,2 C3, 2 Cl, 3 2,3, C3,3 No two of these types are primarily equivalent, either by prerultiplication or by postmultiplication; two types in the same row are absolutely nonequivalent by premultiplication; two types in the same column are absolutely non-equivalent by postmultiplication. This may be seen (Art. 17) by observing that, in any matrix, the greatest common divisor of a column is not altered by premultiplication with an unit, nor the greatest common divisor of a row by postmultiplication with an unit. The types C,,,,,, C C3, 3 are their own reciprocals; C,, is the reciprocal of C,,,. The types (C) are unchanged by premultiplication with the corresponding unit-types in the scheme n, A, A +, +, 7 and by postmultiplication with the unit-types in the scheme, C-, T AI, 0-, 7, 07 T Again, the type matrices equivalent by premultiplication of the schemes which lie in the same column of (C) are with the corresponding unit-types in either 1, 7, 0r, 7, - 1, 1, x, P2 x,~, I p P, P2 1, p p2, 1 Art. 23.] NINE TYPES OF EVEN MATRICES. 459 and the type matrices which lie in the same row of (C) are equivalent by postmultiplication with the same unit-types; thus, for example, X C2,1 - E2, 2 X C2,3, C1, 1= C2, x = C3, 1 x P2. If we select any one of the types (C), for example C, s, and denote by (n, '2 any two of the six unit-types, the expression ( x C, s x '2 (which has 36 = 9 x 4 different values) represents the nine types (C) indifferently, i. e. each of them four times. If n be any even number, and N the number of primitive matrices non-equivalent by premultiplication, the 12 N matrices non-equivalent by primary premultiplication are equally distributed between the nine types (C). To establish this, we first consider the case in which n= 2A is a power of 2. The primitive matrices G of determinant 2D (see Art. 18) divide themselves into three groups: 2F, 0 (i)h 1,. h an uneven remainder of 2'; (ii) I..... 2- O<s<.u, and h an uneven remainder of 2 -S, except when s= u, in which case h = 0; 2~, 0 (iii)..., h, 1 ' h an even remainder of 2t. Each of these groups contains 2 -1 matrices; hence the N= 3 x 2~matrices of G are equally distributed between the three types C1, 2i Cs, 2, C3, 2 except only that the matrix 2 is of the type C2, instead of C 2 Next 0,21A V L C2, 22Nx let n=2uxm, m being uneven, and let IM1 represent the general term of a system of primitive matrices of determinant m, of type (1), and non-equivalent by premultiplication. The matrices Ml x I G are a complete system of primitive matrices of determinant 2 x m; these matrices are equally distributed between the types of the first, second, and third columns of (C), because I MI x I G and I G i appertain to types occupying the same column of (C). Lastly, to obtain a system of primitive matrices, non-equivalent by 3N2 460 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 24. primary premultiplication, we have to premultiply each of the matrices I M I x I G I by the twelve unit matrices?1 x I; and it will be found, by means of the properties of the type-system (C), that four of these units do not alter the type of I MI x l G, and that this type is changed into each of the other types which occupy the same column by four of the remaining eight. Hence the 12x.3x 2/-xm (l +)1 matrices contained in any system of primitive matrices of determinant n= 2z x m, non-equivalent by primary premultiplication, consist of nine groups, each containing 2txmxll (1+) pairs of opposite matrices, and appertaining respectively to the nine types (C). There is a corresponding theorem for primary equivalence by postmultiplication. Arts. 24-33. THE TRANSFORMATION OF THE THETA AND OMEGA FUNCTIONS. 24. Enunciation of the Problem of Transformation. Let w and Q be two complex quantities connected by the relation c+dnQ = a b&2.(i) a+b2'...( the coefficients a, b, c, d being integral numbers relatively prime, and the determinant ad - be = being positive; so that, as we have already observed (Art. 12), the real part of iQ is negative when the real part of iw is negative, and vice versd. We shall frequently write the equation (i) in the symbolic form a, b or in the equivalent form a, b -1 d, -b c, d -c, a.. The problem of Transformation is "to express the Theta and Omega functions containing Q by means of the Theta and Omega functions containing o;" and the general nature of the solution of which this problem is susceptible is indicated by the theorem: Art. 25.] THE PROBLEM OF TRANSFORMATION 461 "Any Theta function of the arguments (a + b Q) 7 and Q can be expressed in the form i7X2 e )2x T, where, if A >1, T is a homogeneous function of the order A of two of the Theta functions of the arguments 7rh and w; and if A= 1, T is a multiple, by a coefficient not containing x, of one of the Theta functions of the argu7rX ments -, and ao." The demonstration of this theorem and the determination of the forms of the functions T may be obtained by a method which is due to M. Hermite, and which we proceed to explain. 25. General Solution of the Problem of Transformation. Method of M. Hermite. Lemma. "Let the values of the complex variable x be represented in the usual manner by the points of a plane; and let F(x) be a function of x, synectic throughout the whole plane, and having the period 1, so that F(x+ )=F (x); F (x) can always, and in one way only, be expressed by an exponential series convergent for every finite value of x, of the form + 00 2miTrx A,1e.......... (i) - 00 2 mi rx This is, in fact, the theorem of Laurent; for, if z=e I, F(x) is a function of z, synectic throughout the whole plane upon which z is represented, except at the point z =; viz. the values of x corresponding to any given value of z are all included in the formula x + sl, where s is an indeterminate integer. But F(x + s = F(x); i.e. F(x) is a one-valued function of z, and is finite and continuous when z is finite and different from zero. Hence, by the theorem of Laurent, F (x) is developable, and in one way only, in a series proceeding by ascending and descending powers of z, convergent for all values of z which lie between two circles having their common centre at the point z= 0, the radius of the inner circle being as small and that of the outer circle as 462 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 25, great as we please; i.e. F(x) is developable, and in one way only, in a series of the form (i), convergent for all finite values of x. Let us now represent by F (x) a function of x, synectic throughout the whole plane, and satisfying the two equations F(x +h) =(-)"nF(x)......... (ii) F(x + h) = (-)ne F(x),...... (iii) rn and n denoting any two integers (which, however, we may suppose, without loss of generality, to be either 0 or 1), and the symbols h, w, A retaining the meanings attributed to them in Art. 24. We proceed to show that by the equations (ii) and (iii) F(x) is determined as a function of x containing A arbitrary constants, which enter linearly into its expression. From the Lemma we infer that, to satisfy the condition (ii), we must have S= - 00 +) F(x)= z Ae (iv) s o; and again that, to satisfy the condition (iii), we must have A, x e(28 + Mn)ir = (- )nAs +Ae^i7.... (v) Let j be one of the numbers 0, 1, 2,..., A-1, and let j' be any number included in the formula rA+j, where r is an indeterminate integer; we find from (v) -TW [(2j' + m)2_-(2j + m)2] X(2j + M)2 A, = ( - x Aj( e ) r xa Cj, {froe where aj e- 4A( x A. If then for brevity we write ( (2 + 2j M)2 c + (2rA + 2j + mm) (LJ = I (_-) re 4 A, r — o we obtain, finally, the equation -F(x)=aoZo(x)+~a {(x)+... +~a^_-_(x); ~ *. (vi) which shows that any function, synectic throughout the whole plane, and satisfying the equations (ii) and (iii), can be expressed as a linear function of the A functions Ij(x). Art. 25.] THE PROBLEM OF TRANSFORMATION. 463 From this result we infer that if F0, F1,..., F_- are any A synectic functions of x, satisfying the equations (ii) and (iii), and independent of one another (i. e. not satisfying any linear homogeneous relation of the type CoFo + ClF, + C2F2 +... + C_ F, = 0, where Cl, C2,..., do not contain x), all the functions F, synectic throughout the whole plane, which satisfy those two equations, may be expressed as linear and homogeneous functions with constant coefficients of F0, F1, F2,..., FFA. If we suppose that F(x), besides satisfying the equations (ii) and (iii), also satisfies the equation F(x) = (-) F(- )...... (vii) (i.e. if we suppose that F(x) is either an even or an uneven function), the number of arbitrary constants in the general expression (vi) is reduced by about one half. Substituting in the equation (vii) for F(x) the equivalent expression (vi), and denoting by ji, j2, two of the numbers j (the same or different), we find that if i +j2 + M 0, mod A,... (viii) we have necessarily aj=(-)+aj,,.. 2. (ix) where C - A (jl +j2 + M). We now distinguish between the cases in which A is uneven and in which A is even. (i) Let A be uneven: the equation (ix) shows that the coefficients acj are equal in pairs-one of them, of which the index j is determined by the congruence 2j+m_ 0, mod A, pairing with itself. If nm+ o is uneven, the equation (ix) shows that this coefficient is zero, and that there are only (A - 1) independent coefficients; if nm + is even, the odd coefficient is not determined by the equation (ix), and there are (A +1) independent coefficients. We shall presently see that the case in which nm + a- is uneven does not present itself in the theory of the transformation of the Theta functions. (ii) Let A be even. And (a) let m = 1 be uneven: we then have j +j2 +1 =A, o-'= n, al = (- 1) + ~'aj; i.e. the coefficients +aj are equal in pairs, and there are A independent coefficients. (3) Let n=O0: of the coefficients aj, A -2 are equal in pairs, each of the 464 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 25. two coefficients a0 and a l pairing with itself; for these two coefficients the equation (ix) becomes a =(-)o, a= (-) +aX; and thus there are A + 1, or 2A or 2A- 1 independent coefficients, according as a and n + - are both even, or one even and one uneven, or both uneven: the last case does not present itself in the theory of transformation. To apply the preceding theory to the Theta functions, we introduce an auxiliary function II defined by the equation i^m, n() = V([a+b20] 7h n (, x e.(x). where m and n are integral numbers defined by the equations m= au+bv+ab, ) (xi) n=c, +dv+cd.. From the elementary properties of the Theta functions (see Art. 1, equations vi, vii, viii) it follows that the function L verifies the equations Im;x h+ ( ) -n (+, nX ) H m, n( +c)=(-)ne (7 + )xI/,(I), * * (Xii) mT-, n = = (, )v -m n (); which, if we put C- ==u, coincide in form with the equations (ii), (iii), and (vii). We now distinguish between the cases in which the determinant is uneven, and in which it is even. I. The determinant is uneven. In this case the equations (xi) give rise to the congruences n- 1 (, - 1) + (v - ) mod 2;..(xiii) n-lI=c{^-l}+d(P-l) from which we infer that if Au, v are both uneven, m, n are both uneven, and vice versd. Hence mn + ~fv = mn + a- is always even; and every synectic function which satisfies the equations (xii) can be linearly expressed by means of any 2 (A + 1) independent functions which satisfy that system. Let p represent any one of the - (A + 1) even numbers 0, 2, 4,...A - 1; and let 3 (r n ) denote one of h Art. 25.] TRANSFORMATIONS OF AN UNEVEN ORDER. 465 the three Theta functions other than ", (p-, w); the I (A +1) functions [( X,( )]P h ' [(j.)]..(xiv) are independent, and satisfy the equations (xii). We have therefore the equation X2.([a+fib] r.) =e-ib(a +b)hx X n(, ) n x T,...(xv) where T is a homogeneous function of the order I (A- 1) of the squares of,3, and 3. This equation establishes the theorem of Art. 24, for the case in which A is uneven. The following Table gives, for each of the six types (Art. 21) to which an uneven matrix may appertain, the values (mod 2) of m and n corresponding to given values of Au and v. 1 x0 T p p 0 p2 rn — V +v uv.J+v+l v n = / + A+V I v + v +1v+l II. The determinant is even. In this case the theorem of Art. 24 may be established in the same manner; and the general character of the result is the same, though there is a little more variety in its form. The number of functions linearly independent of one another which satisfy the equations (xii) is always either I A +1 oor A, but never A - 1; for, since by hypothesis the matrix a, b is primitive, we infer from the equations (xi) that we cannot have simulc, d taneously m n-n-0, - v 1; i.e. these equations exclude the case in which the solution of the system (xii) contains only - A -1 arbitrary constants. According as that system (xii) admits of A +1, or 2 A linearly independent solutions the expression for,, ([a + b5] I-h, &~) is of the type X2 -irb(a +b. e h2 Ax,. (xvi) or X2 -i 7r(a + b n) X2 7rX \ T x e b k x Bx ( ), ). (xvii) VOL. IL 30 ~ ~ ~ ~ ~ VOL. II. 3 0 466 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 25. where A and B are homogeneous functions of the squares of any two of the Theta functions of the arguments 7h and w, and are respectively of the orders 2 A and I A -1; a and a are two Theta functions of which the indices depend on ju, v and on the type Cr, (see Art. 23) of the even matrix of the transformation. The following Tables show how to determine, for any transformation of an even determinant, the form of the expression for %, v ([a + bn] T L ). TABLE I. C1, 0 C2, 1 3, 1 m _ I + v + 1 J v fl V C1,2 C2,2 C3, 2 mEE 0 0 0 C1, 3 02, 3 03, 3 n _ M + v + 1A V n _0 0 0 TABLE II. uA~/^ m n 'a a3 B 0 1 1 I'l,o; o0,1 i B 1 1 1,; o,o ii B 0 1 0 a,o; o,0 ii B 1 1 0, 1; o, iv B 0 0 1 vo; o, B 1 0,; lo vi A 0 0 0 Art. 25.] TRANSFORMATIONS OF AN EVEN ORDER. 467 TABLE III. C!, 1 lC2,1 C3,1 So A A B,i 31 B, ii B, ii B, ii 32 A B,i A 33 B, i A A. 1,2. 2, 2., 2 So A A B, v a3 B, vi B, vi B, vi 32 A B, v A 33 B, v A A C1, 3 2, 3 3, 3 So A A B, iii a3 B, iv B, iv B, iv 32 A B, iii A ~33 B, iii A A Table I. gives the values, for the modulus 2, of the indices m and n, corresponding to given values of, and v for each of the nine types of the matrix of the transformation. Table II. gives the form of the expression equivalent to,Qv([a+ b]-, ) for each admissible combination of the values (mod 2) of the indices, x v, m, and n. Thus, if, x v= 0, m=n 1, mod 2, that expression is of the type (xvii); and the Theta functions designated by Sa and as are 3, o and So,. This assertion is proved by observing that the I A functions included in the formula '51, 0 ( a)]W X [0, I ( (where p is any one of the A uneven numbers 1, 3,..., A- 1) are independent, 302 30 2 468 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 26. and satisfy the equations (xii), if we attribute in those equations to a x v, rn, and n values satisfying the congruences,u x v -O, m n _, mod 2. Table III. refers throughout to Table II., and is founded on the Tables I. and II. It gives the form of the expression for s ([a + b Q], ), answering to each of the four values of the index s, and to each of the nine types of the matrix of transformation. Thus, if a, b e of the type C2 C, d.e. if a, b 1,0. e. if d 1, 0, mod 2, the expression of C) d 1, 0 ) 2 ([a + bn] h 7') is of the type (B, i) in Table II., so that we have the equation 7i7r~X -z-ixb(a+bnb)- 7X 2 q 7rX\ h X T h) 32(\+bQ] W'Q)=e 3(J>( h n 6X3( ), where B is a homogeneous function of the order A - 1 of the squares of any two of the Theta functions 2 (i ') ) 26. The Multiplier. Since w=i _ (_), K ( == K the equation (i) of Art. 24 may be written in the form.K' () cK (Q) + diK'(2) K(w) K(Q) + biK()'....... (i) or, which is the same thing, M x K() = aK (Q) + b.iK'(Q), 1 (ii) x iK'(w) = cK(Q) + d.iK'(). Either of these equations may be regarded as defining, for any given matrix c, d, the quantity M (which is termed the multiplier) as a one-valued function c, d Art. 27.] COMPOSITION OF TRANSFORMATIONS. 469 of w. We observe that if the sign of the matrix be changed, the sign of the multiplier is also changed. If in the equations xv-xvii of Art. 25 we put h = 2K(w) = 2iK, and write for brevity A = K(Q), these equations assume the form x2 where T is a homogeneous function of order A of two of the Theta functions A / TT \ 4t2K'' ) We have supposed in Art. 24 that the transformation b is primitive; C, d but, by introducing the multiplier, we can adapt the formulae to the case of a non-primitive transformation. Let al, b, a, b - ga, gb c,, dl x c, d gc, gd where a, b is primitive and g is any positive or negative integer; and let M, M1 be the multipliers appertaining to the transformations a,' 1,, b1 c, d CD, dl From the definition of the multiplier we have Mi- gMl; and, hence, if we write gx for x, the equation (iii) becomes l ( 2jA MIL~)= T1 xe 44KAMli, where T1 is deduced from T by writing gx for x-the indices of the Theta functions which occur in the expression of T1 (equations xv-xvii, Art. 25) being still determined by the equations (xi) of Art. 25, and not (at least when g is even) by the equations m = aC + blv + aCb, n =c + d, + a2b2. 27. Composition of Transformations. Theorem I. "If the primitive matrix a, b is compounded of any number c, d of other matrices, so that a, _b a1, 1 a2, P2... a) c, d 2 72 1 2 2 s I, 470 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 27. the transformation w =, x Q results from the successive application of the c, d r transformations a., [s al, 1 a I _-i= |. Q8 |X, ~-2= X Ws_,..s., f —= | X 01. 7s8 ~S 7s-1, Ss-l 7l, J1 Thus we may first express the Theta functions containing Q by means of the Theta functions containing s-_,, these by means of the Theta functions containing s-2, and so on continually until we have expressed the Theta functions containing Q in terms of those containing w. Since the equation a, b W = x Q can be written in the form c, d ' Sn SX JS — — AS-1 X X..., -- t,X -7Sa, 7s-i' a -1 - 71, al we can also by a converse process express the Theta functions of w in terms of the Theta functions containing Q. Theorem II. "If any number of transformations be compounded, the multiplier appertaining to the resulting transformation is the product of the multipliers appertaining to the components." f al, 01, a2, 02, If, for example, w= 1 x= a2, 2 X w2 and M1, M, M are the 7', x,l 72x x2,, multipliers respectively appertaining to these transformations, and to the transformation compounded of them, we have KEd)^ = a, K(w{) +, 'iK'Q(l), iK'o) = 71K(wo) + 1 iK' (j,), and K() aK(w) + 2iK'(W,) 1 2) K(2) + 2iK'(2); whence, eliminating K(w) and iK' (o,), we find f (H = (al a2 + 31 2)K(2) + (al2 + 3,1 2) iK' (02), iK'(w) = (7a2 + S1/2)K(2) + (7^1/2 + SI 2)iK(w2); and these equations imply that M= M, x Mi. Art. 27.] COMPOSITION OF TRANSFORMATIONS. 471 In the first theorem of this article we have expressly supposed the resultant matrix (and consequently the component matrices) to be primitive. In the second theorem the matrices compounded may be any whatever. When we compound two primitive matrices, of which the determinants are not relatively prime, the resultant matrix is not necessarily primitive. Let ga, gb a a1, a2, gc, gd 7, ~.1 72)' ' ' where g is some integral number different from unity; and let A, Al, A2 be the determinants of a, b al, i a c, d ' 71, '3S 72, d The transformations = ga, gb ge= 9agclb xQ.......... (ii) gc, gd X and a1, ai 2, 31 P2 7Si '1 72 S2 give different results; viz. if M be the multiplier appertaining to the transa, b formation w=, x Q, the transformation (ii) expresses % ( M 2 ) as a homogeneous function of order A of the Theta functions (g7[, w); but the compound transformation (iii) expresses g (27'rx, Q) as a homogeneous function of the order g2A = ^A A2 of the Theta functions 7 (-r x ). If, in particular, al, 0, a2, P2 a=, b ~1,0 1, 1 and 2 2 are reciprocal, we have g = A= A, A== 1, b 72,? c2, 72 7 2 C, d 01 W=Q, A =K, M = 1: the transformation (ii) becomes an identity; while the transformation (iii) expresses (g-x,) as a homogeneous function of order g2 of the Theta functions < (, w). We are thus led to the theorem of Jacobi: "The composition of two reciprocal transformations of determinant g serves to expresThs the Theta functions of the argument gx by means of the Theta functions of the argument x." This theorem is, in fact, a corollary from the Theorem II., which shows 472 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 27. that the product of the multipliers appertaining to two reciprocal transformations of determinant g is - g Since every matrix can be expressed as a product of matrices of which the determinants are primes (Art. 19), it appears from the Theorem I. that in the theory of transformation it is sufficient to consider transformations of a prime order. Again, if we regard the theory of linear transformations (i. e. of transformations of which the matrix is an unit), as known, we need only consider, for any determinant A, a-'(A) different transformations. For (Art. 18) every primitive transformation of determinant A is included in the formula w = G' x a X nt, where I a I is an unit matrix and I G'! is a system of A-' (A) primitive matrices of determinant A, non-equivalent by postmultiplication. Or, again, we might employ the formula = a l x G I x, where I G I is a system of cr'(A) primitive matrices non-equivalent by premultiplication. If a,3 is an unit matrix of type (1), the equations (xi) of Art. 25 show that m _,=, mod 2, n = v, mod 2; and we shall presently see that M= + 1; hence by a linear transformation of type (1) the Theta functions are unchanged, and only acquire an exponential factor. It is therefore convenient, instead of the a'(A) transformations I G or G' I, to consider the 6 o-'(A) pairs of transformations included in either of the formulae o = r/ x a I x Q, ( = a X rx n, where [a is a primary unit, and Ir, I' are systems of primary matrices non-equivalent by primary postmultiplication, and by primary premultiplication respectively. Lastly, since every primitive matrix of determinant A can be exhibited in the form a I x A, x (Art. 20), it is possible, in the theory of the transformations of order A, to confine Art. 28.] LINEAR TRANSFORMATION OF THE THETA FUNCTIONS. 473 ourselves to the consideration of the single matrix, and to obtain the 0, 1 results relative to the other transformations of determinant A by compounding this transformation with linear transformations. 28. Linear Transformation of the Theta Functions. When A = 1, T is a constant, and the formula (xv) of Art. 25 becomes <^. r/. rz L^\"^X r^~] / -iirb(a+bn)X T7rX \ a,, (a+bn), hQ=Cxe =Cex,Y ( a i (1) where c+ds W = a+dbQ ad- be = 1, a+b ' m = a + by + ab, n = cu + dv +cd. The determination of the constant C, which depends on w, on the indices A and v, and on the integral numbers a, b, c, d, has been effected by M. Hermite in the following manner *. We shall at first suppose that b is different from zero, and positive. Putting h =1, we have from (i) r= +- oo CX i mn (_ )nreir(2 r+mn)x+ (2r+m+)2] r= -oo -ivy (_-)vreir[b(a+bfl)X2+(2r+m)(a+b.)x+i(2r++)2fl] y"= -co Denoting the right-hand member of this equation by i" x S, multiplying by e-miirxdx, and integrating from x = 0 to x =1, we find r1 C= i~"-mn^^i m^^ x e-reilxSdx. Let f(x, r) = (a + b Q) 2+ [(2r + ) (a + bQ) - m]x + a(2r +u )2n - _r, so that r= + co e-mirx xS= ] eir (r) r= -oo * Liouville's Journal, 2nd series, vol. iii. p. 26. VOL. II. 3 P 474 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 28. we find immediately f(x + r) -f(, r + b) = 2ra, whence we infer eirf(x + 1, r) - ei'rf(x, r + b) rl s b — r + oo e-min rx Sdx=2 / eif( s)dx O 50 = -o By a formula due to Cauchy, P 7ti(Px2+2Q~) - _l _ eT P the coefficient of i in the imaginary part of P, and the real part of the radical / - iP, being each of them positive and different from zero. Evaluating the integrals + 00 eirfy(x, s)dx by this formula, we obtain for C the expression 1 C= S", x H x,/_- ib(a + b ).(ii) where, '- (a X2 - 2 /- m + dm2 - abj) A, M= - CX- 4X e miv -mn e - i7r(ac + 2 + 2 bcav +bdv + 2abc1 + 2abdv + ab2~ ( _)a tbcd X e ir (aci2+2 e+ 2 b + bcd2 + 2accdI + 2bedv - abc' s=b-l ait r s - b -1 6/t (H I b)2 the real part of the radical (a ) being positive. the real part of the radical 2/ - ib(a + b6^) being positive. To sum the series (iv) we employ the formulae of Gauss: s=n- 22izrm e = (2 ) 4(in-)2 -/n when n is uneven; sO rb/ c) ) 1 (iii) * * * (iv) ) s=n-1 2 2i m, e =0 - O when n is unevenly even; (v) s=O S8 =, — _2i eve. 2 9 2e -7-:)m(I + / I>./n,: = - x i x( s/2n when n is evenly even. s=O =m' + m * The symbol ilm stands for e"i", and therefore presents no ambiguity of sign. A similar abbreviation is occasionally used in the sequel. Art. 28.] LINEAR TRANSFORMATION OF THE THETA FUNCTIONS. 475 In these formulae n is a positive integer, m a positive or negative integer prime to n; the symbols (-), (-), (-) are the quadratic symbols of Legendre, as generalized by Jacobi;,/n and 1/2n are the positive square roots of n and 2n. To apply these formulae to the evaluation of the series (iv), we consider separately three cases. (1) Let a = 2 a' be even, so that b is uneven; we have s -- b —1 2 ir' _ He e-4abir x b e- - abirX (- )xi4(b-l)2 'b s= b = ( ) x i(b-1) x i-o2bx lb, observing that ( —)=((-1 b-i) +(21) If a=O, b=1, the method by which this formula has been obtained is inapplicable; but if we attribute to the symbol (-) the value +1, the formula gives the true value of the series (iv), viz. H= 1. (2) Let a and b be both uneven; we find s= b-1 Vfib a 1 si(b-a) ~s=-l,1 8s=b-1 HI=e-eabiv x x (-)se b =e —abirr x e b = () x(b-) x -a s9O =s=0 = Xi the form of the result being the same as in case (1). (3) Let a be uneven, b even; then s=b-1l _2ir a s=2b-1 2s2ir =e- "~ b x 2e (x- )~ e * X 2 -e^(b-a) AH= eX e b be-4 e 2b s=0 s=O IIbi ex -a ) / But ' - - ( e-a-ix (b 1a) X —(ba) b But (b )-(a-)i) b)X -a 4 a as will be found by applying the laws of quadratic reciprocity; hence b la -- H= i-2 3P 2 476 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 29. This formula is also applicable to the case (2); in fact when a and b are both uneven ( (_1-(-l)-(a-l).(b-.!)j-a( ) = (a The value of C is therefore C A, x H(a, b). (i) / -. (a + b)' where the real part of the radical is positive, and where S,,, and H (a, b) =,/l are eighth roots of unity, of which S,, is determined by the equations (iii), and H (a, b) by the formulae H(a, b) = (-) j-a, if a is uneven, (vii) = (ab) i-i x q-i(a-i)(b-) if b is uneven. If b is negative, and different from zero, the value of C is given by the equation C=, xH(- a, -b) (viii) /i (a + b) the real part of the radical having a positive sign. If b is zero, we have a =d= +1, = + c + = ac +; and the determination of $,. (ah, t) is, in this case, obtained immediately from the equations defining the Theta series; we have, in fact, (, ar7r'X ) x e427 x 7 ( (ix) all, (a n h x xeDxi,c,... (ix) This result is in accordance with the general formulae (vi), (vii), if in them we attribute to the indeterminate symbol (-)= (-1) the value +1. 29. Linear Transformation.-Deternination of the Multiplier. Putting in the formule (i), (iii), (vi), (ix), x =, = 0, we have Art. 30.] LINEAR TRANSFORMATION OF 9 (co) AND + (Co). 477 /2A /2K(n) S,= ox H(a, b)a, V i.- ~ (a + b a)ab ', (d) if b is positive, and different from zero; and if b = 0, a = d = ~ 1, then A/ = c [ ].*........ (ii) Hence - = (a + bQ) = i x 0, x H2(a, b) x.-, ] (iii) 0 O if b is not zero; and 2(a 1) 0_,. [oil Of - (- 2 ( 1) [].. (iv) if b is zero, or any even number. 30. Linear Transformation of (o() and,(w). Since (Art. 2, equation (i)) 2 a [W1]o[.....2 (.) 0 (0) -2 [ 2( ) [S] (i) aj-s-3 [ '^ ''] we can employ the general formulae of linear transformation (Art. 28, equations (i), (iii), (vi), (vii)) to express ~(2(Q) and 92(Q), in terms of b2(W) and J,2(w). By extracting the square root we obtain, but with an ambiguous sign, the corresponding expressions for (Q2) and,(Q2) in terms of 0 (w) and (w): this ambiguity can be avoided by employing the equations of Jacobi (Art. 4, I (b), (d); II (c), (d)): <)((a) 2 2,9 [ c] /. (ii)] oh[]I_ o [2WI O o[2w] [] 1 The resulting formulae are contained in the following Table; they were first given in a complete form by M. Hermite. The symbol r is written for e i'. 478 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 30. TABLE OF THE LINEAR TRANSFORMATIONS OF (co) AND + (o). a, b 1 (z. ~ ). 1 a. X2. X'. 1 c,0r f(Q) (H W)' (H). + (0). @ A 2 cd ) 1,0 (2) ->ac() (2),acp(&2) 2 ()abp (C) (2)n-abJ,(s) 0, 1 k() 2 k 2 or 1 (2)1-1( q:() |()2derd(Q) | (2)rd(co) ()r- (d ) 09 1 q- - ^-"( &72? I'' - 1,1 k'2 k2 |o r2, | 2 ) ae (dc| () ( 2 )Tac **-]|...,-a,, | (a k — a(( _o 0(r p )-,(w) ac 2 ) ) bd ( _ - _b c~ <(a) 0,~~~~~~~~~~~~~~~P 1 |-1,0 2 12 1 (-) k'2 k'2 (2\db 2 bd()............ (2), 1 ( _ ),- q.( li - 2\ c_ _____ 1 db 2 - 1) d)__ / "'iZ '(co) _1 k'2 (2)r/ )ac2,jd) ( )acl..... ~) () () "~'.... Art. 30.] LINEAR TRANSFORMATION OF ) (co) AND + (co). 479 The matrix of the linear transformation being a, so that c, d 'sa c+dln O= d- I ad-bc= l, a+bi' the Table gives, for each of the six types of the unit matrix, the values of p (t2) and, (2) in terms of p (w) and (co); and also conversely the values of? (o) and 4 (o) in terms of (() and + (r). This second set of determinations is implicitly included in the first, and is given for convenience only; for the same reason, a column is added giving the values of 1 (K2) = X2, j (02) = X'2 = 1 - X2, in terms of d) (o) = k2, and I ()c) = 2 =1 - k2. The value of the multiplier, corresponding to each transformation, is also given (Art. 29, equations (iii) and (iv)). To demonstrate the formulae of the Table, we first consider the case in which c, b is of the type 1. We have C, d 2 []21 and c + d_ c + d x 12 a+b ' 2 a+2b x 2 Hence, by the formulae of Art. 28, 1,,n 0] =e_4i2,r a([O. 1i-ax /V[ —i(a+bn)] 0 1l,, ] - e fiiac~( )a-.ax3, l, 0[2] The combination of these equations gives 2 28ac ~(o = ( 2e 8ildc (o),.... (iii) O(n)=( )e-8. Again, Againo [f] =c+fdQ 2c+d x 2nf '+( 'j ])' 0a+0b ' O 2 a+ bx2 -whence (n)= e*i"x (- ) 0] or a,(_( 2 )Cela x( ).....(iv) a t 480 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 30. Next, let a, b be of the type '; we have c, d 2eo= Cf cl x IEl = c +i-d x 2fQ! 2 c~2dxc^Q 1 c+_dx2_2 - a+b x IQ 2afb x 2,1' 2bx2Q 2< 2ca+bx2f' and hence 1,, 0 []() e-rab x - o,1 [] ()= (vb )e-iax( );........ (v) and also + ( = So I -= 2 i b) e-.A x, 0 [CO] or ^ (-i) =()e x ().......... (vi) Each of the formulae (v) and (vi) involves the other. The cases in which, b is of one of the types a-, r, p, p2 may be treated c, d similarly, but the formulae relating to them may also be obtained as follows: Putting a, = 0,1 = J in (v) and (vi), we find C, Ci -A1, 0 pQ(1) ( )(-,)=~(W ).... ( )(vii) a, b 10 Again, if' d = 1 1 = a 1, we have from the equations of definition |c1 d I I 1, of p (X) and c (o), Art. 2, equation (iv),, (n)= (o-l)= -i", (Q)= (-l )=... (viii) 4r (o)' = ' The formulae relating to the unit-matrices 17 I, I p V, I p2 | may be obtained by combining the formulae (vii) and (viii). If, for example, a, b = r I X ~c, d letcc=II=ax|jxal, let c)= I-crI x W 1, = I P I| X o2g, 0)2= I a- I x Q, Art. 30.] LINEAR TRANSFORMATION OF 9 (co) AND +(co). 481. we find successively x+ (W2) q5 (W1) (CO)', I.. (ix) -() = I() + (02) 1 9 (*1) ~ (*) Similarly, since IPI =I IXIo, p2=jI l-i X + we have, if a, = c, d and, if a, b 2 c, d, (2) =- ( +- (co),, () =.. + () = eWi + (c) (X) (xi) Combining the special formule (viii) to (xi) with the general formulae (iii) to (vi), we obtain the formulae of the Table. A single example of the process will suffice. Let a, l be of the type p2, and let us write C) d (: I p2 X 1i, a,b a,, b' Ia, b' c, d = I pl- I x d' c dc d, ' c, d 'i ' b' the matrix /, being of type 1. C', d' From (iii) and (ix) we have (Qut-(,e-i x ()) But O (0 1) = -- + (W) a b' a, b - a-c, -b-cl C'I, da p c,d a, b whence, by Art. 21, equation (iv), ()e-ia =- ( ) e- i e i r ab and finally ()=( ac ()r 1t = ) (a )b in accordance with the Table. VOL. II. 3 Q oq 482 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 31. 31. Linear Transformation of x(o). The formulae for the linear transformation of the functions X(o), which were first given by M. Hermite, have been presented by M. Schlaefli in the following form. Let e= e2il', and distinguishing the six cases in which the matrix of transformation a, is of the type 1, 4, -, o, p, p2 respectively, let C be an integral c, d number, defined for the modulus 48 by the congruences C (a + d) (b - acd); cases +, p2, C-(b- c)(bcd-a); cases 1, -, C (b- c)(abc- d); cases 1,, T C (a + d) (abd - c); cases +, p. We then have x (o) = c x x(); cases,, +, ( ) (C X X ( cases p,... ** (i ) X (o)= c x (n) cases P2,. To establish these formulae we have from Art. 3, equation (xv), x(A ). X (1+ o> = 'X and X3( ) (()x (o) =( ) X (o)= x3() Hence X( -) X(o) is a cube root of unity; but this cube root is +1; for, since X (ao) and X (- 1) are one-valued and continuous functions of w, the cube root of unity must be the same for all values of c, and cannot be imaginary when the real part of co vanishes, in which case X (w) and X (-) are both real. From the two equations X^= (^) @ O= I | I xQ, Q )I ( -I x...... (iii) (to = X x X ~(O A - llx Art. 31.] LINEAR TRANSFORMATION OF X (o) 483 we find, attending to the relations l|=||xl|+X|X|, | p|= |+|lx 1 pj l=l-'1 x I, x (l). X(w)= - xQ) x ()= P x x (Q); x((): ~-~ (* Xx (n); x(")= -'xx("). X (CO) = Y2 X X (n); X(o)) = -2 X X (Q); W=171 Xf2, ) =IP x n, (iv) (A= p 12 X. We have also Let, as in Art. 22, = I (- 2 X I, ) = I T 12X. }... (v) a, b =I 2,1XT c, d where I X is one of the six unit matrices tI, 1 1m, (v1, I the formulae (ii) are verified, if in them we C-=2/x-Er; C=,- I+l1; C=Z -V - -I; X oI2 1 2x... xl l, 71, IPI, Ipl2 attribute to the cases 1, I,, cases p, a, cases p2, 7. exponent C the values...... ~....(vi) These values of C coincide with the congruential values given by the formulae (i) when a, b is one of the six simple unit matrices; to establish the coincidence generally it is sufficient to show that C' C+ 2u, C C- 2, mod 48,....(vii) C' and C" denoting the values assumed by the expressions in the right-hand members of the congruences (i), when I a 21 xa, and I 12V, reC, d c, d spectively are substituted, The congruences (vii) are readily verified in the several cases; for example, if a, bis of the type J, so that 3 Q 2 3Q2 484 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 32. a, d are even, b, c uneven, we have C-(a+d)(b-acd)_ b(a+d ),mod16, C- (a + d) (abd - c) -- c (a + d), mod 16, the two values coinciding because (b +c) (a + ) 0, mod 16. Hence C' _ b(a d + 2i b) C+ 2(, mod 16, C"_ -c(a +2vc +d) -C-2r, mod 16. Again, the four values of C in the congruences (i) are all congruous to one another for the modulus 3; as may be seen by distinguishing the three cases adl, b cO, mod 3; ad-O, bc - 1, mod 3; ad- -bce -1, mod 3. And using either of the last two values of C we may immediately verify the congruence C'= C+ 2, mod 3, while either of the first two values will serve to establish the congruence C" C- 2 v, mod 3. It will be noticed that the formulae (vi) are obtained by direct reasoning; but that the congruential values (i) (which may be exhibited in many different forms) are established only by an a posteriori verification. 32. Linear Transformation of the Elliptic Functions. For the linear transformation of the elliptic functions the determination of the constants C of Art. 28 is not necessary; and it is sufficient to assign the ratio of any two of them to one another. This may be done by means of the formula (see Art. 33, equation (x)) l, +,, l[(a+b&l)7A, i] = ix eiir(nl —,iv l) x e gir(mln-blV-) x m+n, n( +nl h o) x e b (n) in which au, v, are any two integral numbers, and m,, nl are determined by the equations ma = a, + b vl, n = cm + dvl, the other symbols having the same signification as in equation (i) Art. 28. In the following tables, however, we have given, for convenience, the values of the transformed Theta functions, as well as those of the transformed elliptic functions; the latter set of values being derived from the former by writing Art. 32.] LINEAR TRANSFORMATION OF THE ELLIPTIC FUNCTIONS. 485 a+ bQ _ 1 h = 2K(o), -h = 2 iK()' and forming the quotients in accordance with the equations of definition (Art. 7, equations (xiii), (xiv), (xv)). We consider the general transformation of type 1, subject only to the restriction that b is not to be negative, but only the simplest transformations of the types +, c, 7, p, p2; the real parts of the radicals V(a + b l), / (2, V( + 1) are supposed to be positive, and ) denotes, as before, e7r. I. a, b 0 10, od, c, d = 0,1 mod 2, c+dn a+bf'.-i(a-l) ir6(a+bn)x /(a + b —n) j-bd X P X o rT)/ 3o((a+ bQ) 7,n)= h- I)= S,((afn) h.=(_)I(a-l) it~bd-ac)X PX WX( 1L> 24((a+bn) hj, )= ((a +b ) 'h a, )= (n) = -(a 1- ('), tixPxx23Q), + (n)_= ( ) iab, (), a(( W 1 X2 k=,2 - ( )I(aThe elliptic functions are unchanged; viz. sin am (M' x2) = sin am (( -)(a-l)x, k2) = (- )t(-l) sin amx, cos am, X2) = cos amx, Aam(,, X2) =Aamx. a, b_ O, 1 c, d - -1, 0 II.,2 1 D e h (0= --- P = i' lIt ao(C, -)= n2Px a( h), MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 32. al 7 Xfl = n 2X l7X I h h>=42p~~( > r-(, QX)= M?~) "P x ao r( X,'Px3.G-),.Y~x5.(), (n) = +(>), + (n) >= ~ (005 X=2_k2 M= L? A.sin amx sin am (ix, k'2) = --—, cos amx cos am (ix, k'2) - cos amx ',2 Aamx cos amx III. a,b 1, 0 c, d ~1,1 ' (7rX' -)= ( "h ') = ) = _ l + +,? 3 (7 )' I r ), hX f) = T2r2(Thj) 2 (h, )= o ( h) pq() h Q) = ~ (n) = I(~) + (Q),k(&)' q (O) k2 1 X2 =- F2 A k', si am, I 2 sin am s kz) = k. Aamx k. Aamx ' cos am (k'x, ki) cos amx A amx A am (k x, k 1 A amxa Art. 32.] LINEAR TRANSFORMATION OF THE ELLIPTIC FUNCTIONS. 487 IV. a, b c, d ~ 1, 1 0, ~1 ' )+ + 2' e-,~ (ln ~). P-= -, N/(Q + 1) -vsz t l) ( j- (~ 1 +l), 2)= 2P X (-h ) np 3 ( o i (7 (+ +1 +n), ) = +)= Px (h) h - -h T 3a 7(+1+Q) l= yrz2pX3( ah )n 1 ~ (n) - 4p (0),, (n) = -++1 +) () ~ (0j)' 1 1 X2 = M- +k, sin am (kx, = k sin amx, ( l)p cos am (kx, I) = A amx, A am (x, -) = cos amx. V. c, b +1, 1 c, d -1, 0 ' -1 1 + Q' I.2 e )P P=. ( 4(n + 1) al 7r X( I + f), n) = q q12P X a3 (7h >.2T2P X,,m-X) 2((~1+),n>)= 2+2Po(-), (~(+~ 1 +,2)= n Xa2 7x pF > 488 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 33. q (a) = ri ]1 + (0) k'2 X2= - - ]C2 ' sm am *zm k'2. sin am x a sin am (ikx, i sin amx 1 + (r -, 'l - =ik, cos am (kx, - k ) = A am V F2 Aamx' Aam ikx, k2'= A-amx VI. a, b O, 1 c, d -1, T 1 ' 1+~ (0)= —, xn P- =e s/ wx ~ = x2+./ r X l(x7rxn Q) =i q2+2P x r) X s7rx2 \ /,Trxn D = t >h ' 2 (Q) = ) = %= (o)' (X2 ) YV2 x(. h ) + (n) =. + I (c)I 1i I = ik' sin am (ik'x, k) isi am x cos am x' /.7, 1 Ao amx cos am (z ik, ) = A am c k'2/ cosamx A am (ik'x, t7) = I. m k -= cos am-x 33. Transformation of any uneven Order.-Development of the Solution. We have now to determine the function T in the equation (xv) of Art. 25. The zeros of a3, [(a + bf) h'x f ] are, by Art. 3, equation (vii), -~ Art. 33.] TRANSFORMATION OF ANY UNEVEN ORDER. 489 h x x [2r+v 1 +(2s+- 1)]. (i) where r and s are any two integers whatever. For given values of r and s, the equations pa+qc= (2r+-1)A, }... pb + qd = (2s + u - 1)A,... are always resoluble in integral numbers p and q, in such a manner that to every different pair of values of r and s there corresponds a different pair of values of p and q. When we attribute to r and s all pairs of values in succession, the pairs of values assumed by p and q are, in their totality, the same as the pairs of values satisfying the simultaneous congruences pa +qc -1, _ mod 2, (ii) pb+qd- i- 1, J pa+q q 0, mod A......... (iv) pb+qd- 0, The solution of the congruences (iii) is (see Art. 25, equation (xiii)) p -n-, q-m-l, mod 2;... ( v) the congruences (iv) admit of A sets of incongruous solutions p-O, qO; p - pj, q-+qj, modA,.(vi) where j= 1, 2,..., - (A - 1), and where we may suppose that pj, qj satisfy the congruences (v). Thus all the numbers p and q satisfying the simultaneous congruences (iii) and (iv) are included in the formule [(2r'+ n -1)A, (2 s'+m - 1) ], [2r' A p+, 2s'A+ qj],. (vii) r' and s' denoting any two integral numbers whatever. Since a, b, c, d have no common divisor, a certain number of the pairs [pj, qj] (viz. as many as there are numbers less than A and prime to it), have no common divisor with A. If [p, q] is such a pair, the A- 1 pairs may be all represented by the formula ~+[jp, jq], in which j is any one of the uneven numbers 1, 3,..., A- 2; or, again (when,- - v =, mod 2) any one of the even numbers 2, 4, 6,..., A - 1. The zeros of a3, v [(a + b2) h, ] are accordingly V. VOL. II. 3 R 490 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 33. (a) h [2r'+n- + (2s'+m- 1)c], +O(b).q) (viii) (b) h [2 r'+2 s'~j P+ ] By Art. 3, equation (vii), the formula (a) represents the zeros of h,,n, ( )Again, if? =p+ q- the -(A -1) formulae (b) represent the zeros of the -(A- 1) 2A 2 functions 2? X 2 2 7rx,2 ('7 at (' ) 1 '( ^)- '(h), at( 32, a. and 2,., denoting any two different Theta functions of the arguments h a h and co. For the periods of the even, uniform, and doubly periodic function a,,( X) P, ',h ) being h and ho, and one solution of the equation 32 N XT _'_) 2f, (7rjq) being x =jhi, all the solutions of that equation are included in the corresponding formula (b). Thus the equation (xv) of Art. 25 becomes i, [(a + )h ), 7rX, 7rX ] X - -2CxX, nC ^j X [,a ( $)2,), 2)(7)X,(7.)] (ix), ~L1 ~., _ ~_)fI, fi(T - - a., (~X ~, X2 -i7r(a+n) X e C denoting a constant which can always be determined by putting x = 0, when the matrix of the transformation and the indices,u, v; a, a'; 13, A' are given. The constants C, appertaining to the same matrix, but to different values of these indices, are not independent of one another. To establish the relations between them, let,/, vr be any two integral numbers; let also l == a/AI + bv, n, =-c,+dvl; Art. 33.] TRANSFORMATION OF ANY UNEVEN ORDER., 491 and let us write in the equation (ix) x - Ih(mico - ni) for x, and consequently (a+bQ)) - (1Q a- v)A for (a+bQ)2) Changing the indices of the Theta functions in (ix) by the equation (ix) of Art. 1, we find, after some reductions,,++A,0, +a1,[(a++ b l),x ] = Cx ei1r(Anll-lvl)xeitr(mln- lv')x (-l) 2(-l)mnla' xq + a(l7r h^, ) (x)) h~f1)21, pi +nl a2 h MX) X I[J3a+m1,a'n(lt )S, '(7,)-(- )m(a-') x +,,3+(h)s (J )] — irb (a + bn) h xe A2 The formulae (ix) and (x) complete the general solution of the problem of transformation of the Theta functions when A is uneven. The following special cases are convenient in the theory of elliptic functions. In the matrix a, b c, d let b =c-.Q, =mod 8, acc l, mod 4; this supposition is admissible if we regard the theory of linear transformations as known (see Art. 27); also, in the formulae (ix) and (x) let =v= 1, a=a =1, 1 =0, /' =l, whence m=-ca, n-d, mod8; pq-O mod2; so that we may attribute to the index j the series of even values 2, 4, 6,..., A - 1. T-inrbca+ bn)Writing Q for e we obtain from (ix),[(a+ b Q), 2]=(-) 2 ^-1)CQ X ( ) x ~ ^[ Xl( x) 0 (rj ) _ (7) ' ( ) ]; ) ' and from (x), in which we write successively, (A), = 1, vl =, n1a —=a, n,1O, mod 8, (B) =0, v=, n= -O, n1-d,, mod8, (C) lx=vl=l, m1-a, nl_-d, mod8; 3R 2 492 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 33. s~'X 7-r X So[(a+bQ)h, Q]=( V-)i^-()CQx2o( Cx) 1 x.[~ (-h) o(7rj) -: (h) l(aj)], I 2[(a+bn), h G]=(-)~(a-1)C&X^( h) h x [32 ( h )3~(rj )_33(TS)32 (7~j)] ). (xii). (xiii). (xiv) rr= ( - ) X ' X x i,[ (_ )( 2(7j )_- (7- ) 2(j,)]. I. Let h=2K; = -PK qK'= Kn; introducing the elliptic and modular 2A functions, and attributing to j the values 1, 2,.., (A- 1), we find from (xii), (xiii), (xiv), /X = 9\2(Q)2 = x H2 cos2 am 4j JX = cp'(m) = kLj x k Axsin2 coam 4j '; = ~9~ [S2] =xA~' am 4j ~ (xv) =,(- )'[42] k. IA2coam4j' \=+2(Q) = Y- = 2x 3[Q] - n.a2am4j x k Again, dividing (xi), (xiii), (xiv) respectively by (xii), (xv) and (xvi), we have.... (xvi) and attending to sin am (,) = sin2am x 1 -sin am x 1 sin2am 4j' M 1- k2sin2 am 4j 'sin2 amnx (xvii) Cos am X A am (2, x), /x = cos am x x H = A amx xII 1 2 sin2 am x sin coam4j( 1 - k2 sin2 am 4jsin2 am x 1 - 2sin2coam 4jysin2amx 1 - k2 in2amsin a jsinamx ' (xviii) (xix) [ (bK(f)=) 'lix n flsin2am4j( ( K + b(2 ) JlsinZcoam y) 1V (w) v^x in am4j = (-)'( IAsincoam4-' (xx) The equations (xvii), (xviii), (xix) have been transformed by Jacobi as Art. 33.] TRANSFORMATION OF ANY UNEVEN ORDER. 493 follows. The equations for the addition of the elliptic functions (equations (xxii)(xxiv) of Art. 7), give immediately sin2 amx -sin2 amy sin am (x +y) x sin am (x - y)= 1k2 sin sin2amy' v l \ l-ksn2am sm2amy l cos2amy- A2am ysin2am x cosam(x+y) x cos am(x - y) =- sin2a si.. (xxi) 1 - k2 Sin2 amxsin2am y A2amy + k2 co2 amy sin2 am x aam (x+y)x Aam(x-y)= —. ----. Aam (x+y)x Aa(1 - k2 sin2amxsin2am y Employing these formulae to transform the type factors in (xvii), (xviii), (xix), and attending to the values of /X, /X', M, we obtain the equations Ix \ k^ sin amp, X) = - x II sin am (x + 4j''),. (xxii),x ) X'4 1, c. cos am, \) = 1,i x II cos am (x + 4j'),... (xxiii) Aam ( \)= k'i x Aam (x+4j'),.... (xxiv) in which j' is to receive all values from 0 to A- 1 inclusive. A third set of expressions for the transformed elliptic functions has also been given by Jacobi. Comparing the equations (xvii) and (xxii) we see that the roots of the equation z [z2 - sin2 am 4j(] - Msin am ( x) x I[z2s-i -1= 0,. (xxv) which is of order A and in which j = 1, 2,..., 2 (A- 1), are all comprised in the formula z = sin am (x+ 4j), j = 0, 1, 2,..., A- 1. Hence, equating the sum of the roots to the coefficient of - zA-l, we find sin am, ) = M sinam(x+4j).... (xxvi) and similarly, from a comparison of (xviii) with (xxiii), and of (xix) with (xxiv), 494 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34. X \ kM cos am,) = (- 1)i(a-l) k cos am (x + 4j'/),... (xxvii) am(,x)=(-l( )(A-l) AM Aam(x+4fj),...(xxviii) the sign of summation Z in the three equations extending to the values 0, 1,..., A-1 ofj. 34.] Quadratic Transformations of the Elliptic, Theta, and Modular Functions. The matrices of determinant 2 (see Art. 23) are of nine different types; and if we regard matrices differing only in sign as identical, two classes of matrices non-equivalent by primary postmultiplication, and only two, appertain to each of the nine types. There are thus nine pairs of different quadratic transformations; but, so far as the elliptic functions are concerned, the difference between the two transformations of any one pair is not essential. As representatives of the eighteen classes of matrices we take the matrices Ca,, ra, 3 exhibited in the annexed schemes; the first index referring to the columns, the second to the rows. (C) (r) 1, 1, 0 0 1 1 -1, 1 1, 2 2, 1 -1, 1, 2 -2, -1 -1, -1 -1, 0 0, 1 0, 2 1 2, 2 2 2 1, 2 2, 2 — 1, - 1 1 -1, -1, -2' -2, -1 1, -1 1, 2 2, 1 1, 1 1, 0 0 1 0, 2 ' — 2, -2 -2, 0 -2, 0 0, 2' -2,-2 It will be seen (1) that Ca, p and ra, 3 are of the same type, (2) that Ca, and rF,a are reciprocal, (3) that the matrices C, and also the matrices F, are transformed into one another (irrespectively of sign) by premultiplication and postmultiplication with the matrices of the scheme 1, p, p2 P2, 1p P, P2, 1 in the manner explained in Art. 23. Art. 34.] QUADRATIC TRANSFORMATION., 495 In the following Table I. of the eighteen quadratic transformations of the elliptic functions, M is the multiplier, X is the transformed modulus, s, c, d are respectively sin am (x, k2), cos am (x, k2), A am (x, k2); the columns (M x S), (C), (A) exhibit the numerators of the expressions for Mx sin am (, X2), cos am (, X2), am(, X2) respectively; the common denominator being placed in the column (D). Thus the first line of the Table supplies the three equations sin am (2/ 'x [ k'+k ikx \ ]= 2 /ikk'sin am x x A am x a -2' 2 -ikk' / 1- k(k - ik') sin2 am x '- FIc+ ik 2 ~ 1- k (k/ + ik') sin2 am x cos am (2/itkk x, L[2-v'/ / 1 - k (k'- k') sin2 am x' am (21/" x ' [~i/c+ 12\ cos am x amA(2 ', L2i2, k',J) =I 1-k (k- ik') sin am x -1 + - and indicates that they arise from the transformation c= +- or, more generally, from any transformation of the type 1,1, - 1,1 x Ilx, where a I is a primary unit matrix. It will be observed that if in the Table I. we substitute for one of the matrices C or r any other matrix equivalent to it by primary postmultiplication, the signs of M and X may change; but this does not affect the elliptic transformation. When the index 3 is 1 or 2, cr2x ra, is primarily equivalent by postmultiplication to Ca,; the same relation holds between T2 X a, s and Ca, when = 1 or 3. This observation explains why it is that the formulae for ra, may be obtained from those for Ca, by changing the sign of either k or k' when d = 1, by changing the sign of ' when 3 = 2, and of k when 3= 3. Again, the matrices of the nine pairs C,, 3, r; C,, F2,; C2, 3, ', 3; S = 1, 2, 3 are connected by a relation of the type l|x C| = |rl x aI 496 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34. where 1 a I is a primary unit matrix. Thus the transformations corresponding to the two matrices of any one of these pairs may be reduced to one another by means of the linear transformation IV. of Art. 32. The two reciprocal transformations C3, 2 and I, 3 are of special importance, because they alone, of all the eighteen, transform real elliptic functions having moduli less than unity into functions of the same kind. The former, which is the transformation of Landen, is also termed the transformation "a majori ad minoren," because it diminishes the modulus; the latter, which increases the modulus, is the transformation " ai minori ad majorem," and is sometimes called the transformation of Gauss, though it is apparently due to Lagrange. If we employ the notation of Legendre for elliptic integrals of the first species, viz. F(p, ( = Hd J1 - ksin2x the transformations of Landen and Gauss respectively give rise to the equations 1~~n= i~"( ~1 ~k1 F(Q, k) + 'F(I+ k)' tan (-) = k'tan,. (A) F(, k) = l Fk sin (2-) = ksin,... (B) which supply expeditious methods, known as the descending and ascending modular scales of Legendre, for computing the values of elliptic integrals by successive transformations. In these formule k and k' are positive and less than unity, and only the real value of the integral is considered; (A) is derived from C3, 2 in Table I. by writing p= am ( ), ^=am[(l +k'), l k; and (B) from (F2,3) by writing ( = am [2x, k], or sin - kscd 4 and I =am n(l+k)x, 2 +k iThe formulae A and B are not essentially different; in fact, if in either of them we call the transformed modulus c, and interchange p and 4, we obtain the other. As before )i denotes eli. Art. 34.] QUADRATIC TRANSFORMATION. 497 TABLE I.-QUADRATIC TRANSFORMATIONS OF THE ELLIPTIC FUNCTIONS. Matrix. 1 M A M At MxS D C 1,1 FC, 1 rlz r2,1 C3,1 c1, 2 F1, 2 C2 2 Cls r1,3 r 1,2 C2, C3 2 r2,3 C,3 2r 2 Vkk' 2 -2 12kk k'- ik k'+ ik k - ik' k + ik' 2 i Vk' 2 Vk' i(1 -k') i(l +k') 1+ k' 1-k' 24k 2i Vk 1-k 1+k i(1 + k) i (l-k) k' + ik k'-ik 2,7- 2 /Pk, 2 l kk' k + ik' k-ik' (i 1 +') i (1-k') 2 k' 2 i /k' 1-k' 1 + k' l+k 1-k 2 i ^k 2Vk i(1-k) i(1+k) sd sd sd sd sd sd sc sc sc sc sc SC s s S s s S 1-k (k + ik') S2 1 -k (k-ik') S2 c c 1-k (k-ik') s2 1-k(k + ik') s2 1-(1 -k') 2 1- (1+ k') s2 d d 1-(1 +k')s2 1-(1-k')s2 1 -ks2 1 +ks2 cd cd 1 - ks2 1 - ks2 c c 1-k (k - ik') s2 1 -k (k + ik') s2 1-k (k + ik') s2 1-k(k-ik') s2 d d l1 1 + k') s2 1-(1-k')sa 1 - (1 +-k') S2 1- (1 +k') 2 cd cd 1 + ks2 1 - ks2 1-ks2 1-ks2 1 + ks2 1-k (k-ik') S2 1-k(k —ik')s2 1- k (k + ik') sZ 1-k (k+ik') s2 I-k (k-ik') s2 1-(1-k')s2 1-(1-k')s2 1- (1- k') s d d 1 + ks2 1-ks2 1 -ks2 1 +ks2 cd cd Every matrix of determinant 2 is comprised in one of the eighteen formulae I C || XI | a, I ||r l X | a 1i,........ (i) where 1 a |I is a primary unit matrix; or, again, in one of the three formulae 1 2 ~~~! ~ I) 0, 2 0, 1 1,'2 x lIal. 0 2 x llf 20 x all, x ll.... where 11 a 11 is any unit matrix whatever. The quadratic transformations of the Theta functions are of eighteen different types, corresponding to the eighteen formulae (i); but the formulae (ii) VOL. II. 3 S 498 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34. are more convenient for the determination of the transformed Theta functions corresponding to any given matrix of determinant 2. For, if we were to use the formulae (i), it would be necessary, whenever the second element is different from zero in each of the two compounded matrices, to multiply together two radical quantities and to determine the sign of the product; viz. if the transformation C+Da A -+ B A+Ba2 is regarded as compounded of the quadratic transformation c + do1, a + bw, and the linear transformation o1 =-Q we have to consider the product V(A + B) = V/(a+bw) x / (a + Q), where the signs of the radicals V(a + bw,) and V(a + j3n) can be determined by the rule that their real parts are to be positive; but the sign of V (A + B&), which alone appears in the final result, depends on the signs of its two factors, and cannot be determined by inspection, or (as it would seem) by any simple rule, without first resolving it into its factors. The inconvenience thus arising is obviated if we use the formulae (ii) in which the second coefficients of the matrices of determinant 2 are zero; and it is accordingly preferable in the theory of the Theta functions to consider only the three transformations n n r C2, 1 3, 2 2, 3' In the following Table II. ei'" is still represented by a, and 9 is written forS(Y, ) TABLE II.-QUADRATIC TRANSFORMATIONS OF THE THETA FUNCTIONS. (i.) 1 ), w=C2,1 xx = 1,2 xl=lx2f2. \h'"j+ Art. 34.] QUADRATIC TRANSFORMATION. 499 al T-Xn)=Aaxa A (1 h, Q) -M _ _ _x 2 a2 ( X)E2 = A 2 X a," A(n= ^- ^(@]f )(),() +,.(o) B1- -= 1 x >9 (Q a = r7- 1/2 (C)r) tt (r)( -,,, 2^i2q)2(co)*2(').,2(u) = )+ 4(() (w ) = 2 K * ' (ii.) 2, 0 (O-,32XQ 0,1 | X 0-20. (2h ) = A x ( x 3), A = ]3 [] - 1x -' 2(2) ~()AX) (,2 ) f(- ),, 2 3 2 )4(<),,. 2 + +(~% ) 3 S 2 500 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34. (iii.) < = r, 3X = 1h xQ = 2Q, co = F2, 3 0 22 o (-X -) = B3 x ( 3 2), h 0 (r,Q =A3 X K' X 3 ()([n)] / /2 () (2) A. R- ^[c] * 7) () __(__)_"(o) 2#(/2^(l) )( o)), The equations in the preceding Table connecting the given and transformed modular functions are included in a complete system exhibited in Table III. TABLE III.-QUADRATIC TRANSFORMATIONS OF THE MODULAR FUNCTIONS. A. Matrix. C1,1; rlI, C2,1; r2,1 C3,1; r3,1 C1,2; r,1,2 C2,2; r2,2 C3,2; r3,2 '1,3, rl,3 C 2,3; r2,3 C.,:; F3,3 Equation. 1 =,/2 (c) (O)( (() i (n), #2 (n) = -'1 /2(o) ) *(&) 9(Q), ~)2(n)=,/2q(/) -(c.) (&2l), ~2 (') = (-12/2 (C) +(~) 4 (n), 02 (0) 02 (Q) = ~?-2 ) ( ), 2 ) (a ) 2 () = 2/2 * ( ) (), 2) () = *2/ () (n) (Q), 42 (Qn) 92 (0)= 2/2(n) (o)) +2 (<o) 92 (Qn) = >2,2 + (<) + (Q). Y. a (c - b) - a(b +2c) a (2b + c) a (c - b) - a(b + 2c) a(2 b + c) a (c - b) - a (b + 2 c) a(b+2c) a(2b+c) Art. 34.] QUADRATIC TRANSFORMATION. 501 Matrix. C1,1 1 1 c2,1 03,1 P3,1 01,2 P1,2 l2,2 2,2 C3,2 3,2 F3, 2 r1,3 1,3 02,3 I B. Equation. +2g 1= - 2 [44(C) + i+4 (o). b (JI) = - 2 [04 () + 4 ()1, - 4 [4 (C) +.4 -2( = 2 -4 ([) + 4(w), p(a) ________ 4(n) [+ t + )' 2 (n) r204(o) 1 _ __+4 () 42() 2 1 + *4(R )' 1 +14(W) +2(n) - +4 () I X. 2a (b + c) - 2a(b + c) -2ab 2ab - 2ac 2ac 2a(b+c) - 2 a (b + c) -2ab 2ab -2ac 2ac 2a(b + c) - 2a(b - c) -2ab 2ab 1?7 44(CO) 1 + +4 (CD) +' (n) - 2,4(W) 2(fl) 1+<4(&) ' +2(n) - {4 (C) 92(Q) 1 + 4(0o))' 1 = -4(4(0) I ) - 4+,4 (co) 2(f) - I+ 4() I 1 + IP4 ((O) ' P2,3 I 502 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34. Matrix. Equation. X.,~ %4(.) r3,3 2- 3 2 () 1 + +4( )' To obtain the corresponding equations between p (n) and q (X), proper to any transformation c = 1 ClI x l a 1x, co= i r II x 1 a II x a, where a =, da, b is a primary unit matrix, we have to multiply the right-hand side of the equations in the Table by r. Table III. contains the complete theory of the quadratic transformations of the modular functions p (co) and + (w); viz. given any equation whatever of the type ( = c+ ad- bc = 2, no rational relations subsist between ( (a), (co), and p (Q), +, (Q) which are not included in the formulae furnished by the Table. The two formulae (A) and (B) corresponding to any given matrix are not rationally deducible from one another; but they coincide with one another when rationalized, so as to contain only 1 (w) and d (Q). To effect this rationalization we raise each member of the equations in Table A to the power 8; in Table B each member of the equations is to be raised to the power 4, and the two equations (Cij) and (ri,j) are to be multiplied together. The nine equations thus obtained are the nine modular equations of order 2. It only remains to show how the formulae of the Tables I., II., and III. are to be verified. In Table II. the form of the expression of the Theta functions containing Q, in terms of the Theta functions containing w, is determined by the general theory of Art. 25; the reduction of the six coefficients, indicated by that theory, to only two, A and B, is effected by substituting x+~h, x+hw, x +~h+-Ihw for x, and employing the formulae (ix) of Art. 1; the values of A and B, which are found by putting x= 0, are conveniently expressed in terms of the modular Art. 34.] QUADRATIC TRANSFORMATION. 503 functions, and A/ by means of the equations of Jacobi (Art. 4, I. and II.), which give immediately (1) 2[- ] =2(- 1) x 12[ - 1 l] = /22 22[W- i1] or, by Art. 32, III., 11 = r'ft1I-11.- " (2) [3 [IW -] o[c - 1] ( ) 3 02 2] (I (D-S.); or, by Art. 32, III., a3 r3 B-i - ^ M 3L( [2- 1] = -2 ) (3) o,[2c] = _[; (4) r20[2] -- ^2 J2 (2c)' (5) a2 [2] =A/2 ~ (O) a3 []; (6) 3[2] = ] The equations of Jacobi (Art. 4, I., d, and II., d) also give by multiplication (c),2 (1) +)2()) = 2 (c) q (co), whence, writing 2c for o, (b).2(C) p2(2) =V /2 (2 o) (); and again, writing co - 1 for co, and substituting for (co - 1), (co - 1) their values in terms of p (co), + (co), (a) 2( -O_)=-2() - (O)(1 )) ~ -). The equations (a), (b), (c) are included in the Table II.; the values of 2 (o10) -), of 02 (2co), and of 92(2CO), which are given in the same Table, for the cases (i), (ii), and (iii) respectively, are obtained by forming the quotient of the two equations which contain the coefficient B, and putting x = 0. The values of M in the same three cases are, by the definition of the multiplier,:2 [W] '23 [] W []. 504 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 35. and then, attending to the equations (2), (4), (6) of this article, become 1 1 4(0) 4 4(w)(C) ( = (1 + '4( ), 92 ( - )- ~(), (2c) p2, + CO()), in accordance with Table I. The formulae in Table I., relating to the transformations C2,1, C3,2, r2,3, result immediately from the equations defining the elliptic functions (Art. 7) and from the formulae of Table II. Lastly, the remaining formulae of Table I. and the corresponding formulae of Table III. are to be deduced by linear transformation from the formulae appertaining to the three fundamental transformations C2, 1, C3, 2, 2,3. To facilitate these verifications, and the determination of the transformed Theta functions corresponding to any of the matrices C or r, we add a scheme of the postmultiplying unit matrices, which serve to reduce the matrices C and r to the three fundamental matrices C2,1, C3,2, F2,3. (i.) Cs,1 = 2,1 X Xs,1, C8,2 =3,2 X Xs,, Cs,3 = Ir2,3 X XS,3, 1, 1 1, 0 0, 1 -1, 0 ', 1 -1,-1 0,1, 1, 1 1, 0 Xs, t = -, A.,'-1-i,-i ' -.1, 0 ' o, 1 ' 1,-1 1, 2 2, 1 0, 1 -1, -1 -' 1, 0 (ii.) rs,l= 2,1 X Y8,1,,2- = 3,2x y,2, rS,3 2,3 x YS,3, -1, 1 1, 2 2, 1 0, -1 -1, -1 -1, 0 1, 0 0, 1 1, 1 YS1, 1 Y>'t= -1, 1 ' -1, -2 -2, -1 1, 1 1, 0 0, 1 -1, 0 0,1 ' -1 11 Arts. 35-45.] GEOMETRICAL REPRESENTATION OF BINARY QUADRATIC FORMS. Art. 35.] Quadratic Forms of a Negative Determinant. Adopting the usual conventions, we represent the complex quantity o = x + iy by a vector in a plane; the extremity of this vector we term the point w. As y Art. 36.] BINARY QUADRATIC FORMS. 505 is essentially positive, we have only to consider the region which lies above the axis of x; by the plane (xy) we shall understand this region only. Every complex quantity o is the root of a quadratic equation of which the coefficients are real and the determinant negative; the ratios of the coefficients are given when co is given; for, if c = x + iy satisfy the equation a + 2 bo + co2 = 0, we have a b c x2+y2 -= *.* *.X* Vice versd every binary quadratic form of a negative determinant is represented in the plane (xy) by a point; for we may regard the equation a + 2bco + cco2 = 0 as corresponding to the quadratic form (a, b, c) of determi- b - i,/A nant - A (viz. we have A = ac - b2), and the point = --- as representing the quadratic form. This representation is admissible whatever real values the quantities a, b, c may have. In the general case the ratios only of the coefficients are given when the representing point is given; but, if we suppose the coefficients to be integral numbers, and the form to be positive and primitive, the three coefficients themselves (and not merely their ratios) are given when the representing point is given. _ _ _ _n a,/3 If o = 7+ i, where is an unit matrix, the two points co and Q are a + fg' where?,/ equivalent; and the equivalence is primary when the matrix a, is primary. When two points are equivalent the corresponding quadratic forms are also equivalent; viz. if (a, b, c), (A, B, C) are the quadratic forms corresponding to co and Q respectively, (a, b, c) is transformed into (A, B, C) by a, 36. The Reduced Space. A positive quadratic form (a, b, c) of negative determinant is said to be reduced when the absolute values of its coefficients satisfy the conditions (1) aC[2b], c>[2b], a_<c; of which the second is a consequence of the other two... (ii) (2) If a=[2b], b>0; ifa=c, b>o. If co = x + iy is a reduced point, i. e. a point corresponding to a reduced form, the conditions (ii) give VOL. II. 3T 506 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 36. (1) [2 x] 2 + y2, [2x] 1, x2 + y2 < (2) If X2+y2=[2X], x<0; ifX2+y2=1, X<o.J In these formulae the symbols [2b] and [2x] denote the absolute values of the quantities 2b and 2x. The geometrical interpretation (see figure 1) of these conditions is that the reduced point w lies within the space AA' external to the semicircles x2 + y2 = + 2x, and internal to the semicircle 2 + y2 = 1; we include in the space AA' that part of its contour which lies to the left of the axis of x, and we exclude from it the other part of its contour. Since every binary quadratic form of a negative determinant is equivalent to one, and only one, reduced form, we have the corresponding theorem, "Every point in the plane (xy) is equivalent to one and only one point in the reduced space AA'." Again, if we regard two substitutions as identical A IA' which differ only in sign, we have the arithmetical theorem, "Every quadratic form is equivalent to its reduced form by one substii/ i \ tution only; except when the form appertains -+ to the class (1, 0, 1) or (2, 1, 2); the number Fig. 1. of reducing substitutions in these excepted cases being respectively two and three." We therefore infer that "any given point is equivalent to its reduced point by one substitution only, except when the reduced point is + i or (- 1 + i V3), in which cases there are respectively two or three reducing substitutions." For our present purpose we have to modify the preceding enunciations by limiting ourselves to the consideration of primary equivalence. A form (a, b, c) is primarily reduced, when it satisfies the conditions (1) [b]<a, [b]<c. (2) If[b]=a, b>O; if[b]=c, b>O. Similarly, the point w = x + iy is primarily reduced when it satisfies the conditions (1) [x]~x 2\+y2, [X]1.. (iv) (2) If[x]=X2+y2, x<O; if[x]=l, x<O. It may be proved, by a slight modification in the demonstration of the Art. 37.] BINARY QUADRATIC FORMS. 507 theorem of Lagrange, (i) that "any quadratic form of negative determinant is primarily equivalent to one and only to one primarily reduced form;" (ii) that "the primary reducing substitution is also unique." If, therefore (see figure 2), we designate by Z the space included between r S the parallel lines x = + 1, and external to the semicircles x2 + y2 = + x, we obtain the theorem, "Any given point in the plane wy is primarily B B equivalent to one, and only to one, point in the space /; and the two points are primarily -- equivalent by one substitution only" *. We reckon the line x= -1, and the semi- / ' D E circle x2 + y2 = - x as appertaining to the space I, but we exclude the line x = 1, and the semicircle x2 + y2 = x. The lines x = - 1 and x = + 1 we denote by SL and S+, respectively; the Fig. 2. semicircles 2 +y= - x and x2 + y2= by T_1 and T+r. The space Z we shall henceforward term the reduced space. The cornicular points T 1, 0, oo, we denote by ~p, q, r. 37. The Circular Affinity of Moebius. y+~12 If c and n are two vectors connected by the relation co= 3+ 2 where a,' are any quantities whatever, real or imaginary, but a - P3y is not zero, any two corresponding loci described by the points c and 12 are connected by a relation known as the circular affinity of Moebius. If / = 0, the relation between the two figures is one of similarity; in every other case the relation is one of inversion. For if f3 be different from zero, the equation connecting w and Q may be written in the form (cAi) (12 + a) y aS * See a Note on the primary periods of the elliptic functions in the ' Messenger of Mathematics,' new series, vol. xii. p. 73, Arts. 1-3 [vol. ii. p. 342]; where, however, the point corresponding to the form (a, b, c) is taken to be - - instead of co, or, what is the same thing, the point is taken to be _+_i__. -b+iV/A -b + i instead of b a c 3 T 2 508 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 37. which shows that the vectors joining w to - and - to - are inversely pros a )G 13 portional. The fixed points and -F, which, in each of the two figures, answer to the infinite in the other figure, are termed the centres of inversion. Figures related to one another by circular affinity possess the following properties: (1) Circles in either figure which pass through the centre of inversion of that figure are transformed into straight lines, and vice versd. All other circles are transformed into circles. (2) The anharmonic ratio of four points on a circle or straight line in the one figure is equal to the anharmonic ratio of the corresponding four points in the other figure. (3) The two figures are similar in their infinitesimal parts; except at the a points - and - which (as has been said) correspond in the two figures respectively to the infinite of the other. The truth of this theorem may be inferred from the observation that to an infinitesimal circle (c) in one figure there corresponds an infinitesimal circle (C) in the other figure; and that to the centre of (c) there corresponds a point distant from the centre of (C) by an infinitesimal of the second order, the radii of the two circles being regarded as infinitesimals of the first order. But the theorem is only a particular case of the general proposition of Lagrange and Gauss, that if X+iY= f(x +iy), where X, Y, x, y are real quantities, the points X + i Y and x + iy describe figures which are similar to one another in their infinitesimal parts, the only cases of exception presenting themselves at points where the derived function off is zero, infinite, or indeterminate. (4) To any simple contour in either figure, not passing through the centre of inversion, there answers in the other figure a simple contour, not passing through the centre of inversion; and the two contours are similarly described; z. e. positively or negatively in both figures alike*. It is sufficient, in order to * By a simple contour we understand a closed and finite contour which does not intersect itself. Such a contour is said to be described positively or negatively according as the describing point has the enclosed space on the left or on the right; or, which is the same thing, according as the element of the contour and the element of the internal normal are situated with regard to one another as the vectors 1 and i, or as the vectors 1 and — i. Art. 39.] BINARY QUADRATIC FORMS. 509 establish this assertion, to verify it in the three cases o = a+ 12, o = afl, co= - a being any constant quantity. 38. The Subdivisions of the Reduced Space. The twelve spaces into which Z is divided by the lines and circles of figure 2 form, when taken in pairs, six regions which are in circular affinity with one another in the manner indicated by the equations A B C D E F 11I 11 1,1 7= Ip =IP lI (i) A' B' C' D' E' F' I = ' = 1 - |( ' = I| =p -p 2[ The matrices I 1, 1 [, I I, i, i P I, I p2 are those of which the values are given in Art. 21; the matrices denoted by the accented symbols are derived from these by changing the signs of the second and third constituents; thus, for example,! _1, 11 I 1, -1 1 Ipl= -,o [ip'1= o; and we have 1 '1 = -1*1, | 'I1 = It- 1, I1'1 = I ~1-1 Ip'I= -I pl x j, -2) I p2 _ p2 X l The equations (I.) are to be interpreted as follows:-The space A is transformed into the space C by the equation C= 1I1 x A, or C= 1+A; the space C is transformed into the space D by the equation Io'i-lXC=Ir-lxD, or C= o- Ixl1-xD 1,0 1,-1 1,1 D &c. &c. 11 x 0, 1 xD= 1, 0xD D 1-D &C. 39. Quadratic Forms of a Positive Determinant. A form (a, b, c) of positive determinant D (= b- ac) is represented in the plane (xy) by the semicircle a+2bx+c(x2+y2) = 0,. (i) 510 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 39. which we shall designate by the symbol [a, b, c]. The equation of the semicircle may be written in the form -x+ y a + b (x + iy) )b,x(c+ y)...... (ii) The point x + iy (but not, except when b = 0, or x = 0, the point - x+ iy) is a point on the semicircle. If c = 0, a supposition which implies that D is a square, [a, b, 0] is a straight line; for convenience, however, we shall regard this straight line as a semicircle of infinite radius. Theorem I. " If (a, b, c) is a form of positive determinant, and (a, /3, y) a form of negative determinant - A (where A = ay - /32) connected with (a, b, c) by the invariant relation ac - 2 b/3 + ya = 0, the point representing (a, (3, A) lies on the semicircle representing (a, b, c)." For, if x+ iy be the point representing (a, 3, y), we have = -, x2y2 = 7 7 and the invariant relation becomes a+2b +c (2+y2) = 0. Thus the semicircle [a, b, c] may be regarded as representing the quadratic form (a, b, c), because the points of the semicircle represent quadratic forms of negative determinant which are harmonically associated to (a, b, c). If 7 and c are of the same sign, ac-2 b/3+ ya o0, according as the point (a, /3, 7) lies outside or inside the semicircle [a, b, c]. By equivalent semicircles we understand semicircles which are related to one another by a circular affinity, the coefficients of the transformation being integral numbers, and the determinant a positive unit. We then have the following theorems: Theorem II. "Equivalent points lie on equivalent semicircles; and equivalent semicircles correspond to equivalent quadratic forms." Or, which is the same thing, "If the form (a, b, c) of positive determinant is transformed into (A, B, C) by the linear substitution a' d, the semicircle [a, b, c] is transformed into the semicircle [A, B C] by the circular transformation semicircle [A, B, C] by the circular transformation w =a - + O l' a+/i0' Art. 39.] BINARY QUADRATIC FORMS. 511 To establish this theorem, let w = x + iy, = X + i Y, and let w be a point on [a, b, c], so that -x+iy = x+iy)b,;c a, b (x+i); changing the sign of i in the equation x+iy == a' x (X+iY), we have also -x+-iy = ' -) - =+-y = ' x(-X+iY); whence -X+iY=, A x (-X + y) == a 3 x b' x( iy 7, a a, bab x (i,, abc a, 3 (xiY B, C)7, x == I x X |NlX X x(X+AY) x (X+1Y): Y, a a, b %~ i ' A, B or X + i Y lies on the semicircle [A, B, C], and [a, b, c] is transformed into [A, B, C]. This Theorem is the geometrical expression of the invariantive character of the equation ac- 2b +7a = 0. Theorem III. " If -x 1+ iy= C b x (x+iy2), or, which is the same thing, if -x2+iy2= a x ( 1+iy1), the two points x1+iy' and x +iy2 are inverse points with regard to the circle [a, b, c]." For, equating real and imaginary parts, we find a + b (x1 + X2) + c (1X2 + yy) = O, b (Y1 - 2) + c (yX2 - y ) 2 X; and of these equations the first expresses that x + iy,, x +iy are conjugate points with regard to [a, b, c]; the second expresses that the line joining them is a diameter. 512 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 40. 40. Automorphics of a Quadratic Form. The automorphics of the quadratic form (a, b, c), the coefficients of which we now suppose to be integral numbers, are included in the general formula 1 T-bU, -cU mx aU, T+bU' (ii where T, Uare any integers satisfying the equation T2 - D U2 = m2, and m = 1, or 2, according as the form (a, b, c) is properly or improperly primitive. The meaning of course is that if a,3 _ 1 T-bU, -cU ~, - mX aU, +bUT then (a, b, c) (x, y)2 = (a, b, c) (an + fy, yn + ~y)2, By these automorphics the rational semicircle [a, b, c] is transformed into itself, and the transformations are homographic (Art. 37, 1 and 2). Consequently the chords of the homography (i. e. the chords joining the pairs of corresponding points) all touch a conic of which the major axis is the diameter of the semicircle. This conic is an ellipse of which the equation is a+2bx+c(x2+ -y2)= 0, m2 and of which the eccentricity e is given by the equation e= U,D or +e I (T+U/D)2. To establish these assertions we write the equation of transformation, I T-bU, -cU rnm aU, T+bU X in the form T-bU T- bU (m + + b)(Q T -... (iV) and, denoting by Z, z the points U - T (which lie on the axis of x, outside the semicircle, and at a tangential distance Ufrom it), we draw any two lines ZPQ, zpq, making equal angles with the axis, and meeting the semicircle Art. 40.] BINARY QUADRATIC FORMS. 513 in the points PQ, pq respectively (see fig. 3). The equation (iv) shows that the pairs pP, qQ are pairs of corresponding points. Hence the tangent at the extremity of the minor axis of the ellipse passes through the points of contact of tangents drawn to the semicircle from z and Z. Thus the foci of the ellipse are the points inverse to Z and z with regard to the semicircle; from this conclusion the equation of the ellipse and the expression for its eccentricity are immediately derived. M Z M2 0 Ml z Fig. 3. If pP are any two points on the semicircle connected by the automorphic (iii), _M,P _MP 1+e the constant anharmonic ratio - ' M2 is equal to1 +, M M2 being the diameter of the semicircle. As all the automorphics of (a, b, c) are powers (with integral exponents positive or negative) of the fundamental automorphic (i. e. of that automorphic in which I' and U are the least positive integral numbers satisfying the equation T2- D U2 = m2), so the homographic transformations into itself of the semicircle [a, b, c] may be all obtained by repeating one of them continually in either direction. Let Z be the fundamental automorphic, E the ellipse touched by the chords of the corresponding homography; also let PP,, PP2, P2Pa,..., PP-, P- P-, 2P3,,... be the two series of chords of the semicircle which, beginning with any given point P on the semicircle, can be drawn to touch E; we may suppose that PP1 are connected by 2, PP_ by L-1; then the homographic transformations (PP), (PP),.... (PP 2), (PP 3),..., which arise from the duplication, triplication, &c. of the fundamental transformation (PP) or (PP-1), correspond to the automorphics of which the symbols are 2, 3,.... 2-2, 2-3,.. For all semicircles of the same determinant and of the same order (the semicircle appertains to the properly or improperly primitive order according as m = 1 or m = 2) the automorphic transformations are similar, since the eccentricities of the regulating ellipses depend only on T, U, D, and m. VOL. II. 3 514 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 41. Every point on the semicircle is equivalent by some one automorphic transformation to a point on the arc of the semicircle cut off by any tangent to the ellipse E; and no two points of such an arc can be equivalent by any automorphic transformation. 41.] The Subclasses of Primarily Equivalent Quadratic Forms. The introduction of the conception of primary equivalence renders it necessary to divide each class of equivalent quadratic forms (as defined by Gauss) into subclasses of primarily equivalent forms. The number of subclasses in each class is six, three, or two, as will be seen from the following enumeration of the different cases: We consider first under the headings I. and II. the case where the determinant is a positive number not a square; secondly the case where the determinant is a square or a negative number other than -1 or - 3; and thirdly the case where the determinant is = - 1 or - 3. First: I. Properly Primitive Classes. Every form (a, b, c) of a properly primitive class belongs to one or other of the three sets: (A) a=-c1, mod 2; (B) aO0, c 1, mod 2;.. (i) (C) a 1, c 0, mod 2; and the result of applying unit transformations of the six types of Art. 21 to forms of these three sets is indicated in the following scheme: A B C A= P1, 4 p2 p, 7 ----------------------.. (ii) so that, for example, (A) = (B) x p2 = (B) x a-, the symbols (B) x p2, (B) x a-, repre Art. 41.] BINARY QUADRATIC FORMS. 515 senting the sets obtained by applying unit transformations of the type p2, and of the type a, to forms of the set (B). We must now distinguish the cases in which the least number U1, satisfying the equation T2- D U2= 1, is even and in which it is uneven. (a) Let U1 be even; then U2, U,... are all even, and all the automorphics of any given form f are primary. Hence the forms fx, fxx, fxo-, fxT, fxp, fxp2.... (iii) are all primarily non-equivalent; since if any two of them were primarily equivalent, f would have a non-primary automorphic. Thus, in this case, each class contains six sub-classes of forms primarily non-equivalent. (b) Let U, be uneven; then U3, Us,... are uneven, U2, U,... are even; i. e. the automorphics of an even order are primary, and the automorphics of an uneven order are of the type k, r, or a- according as the form is of the type (A), (B), or (C). In this case the six forms (iii) are primarily equivalent in pairs; for example, iff is of the type (A), f andfx + are primarily equivalent, because f has automorphics of the type; i. e. f can be transformed into fx \{l, not only by transformations of the type A, but also by transformations of the type + x + or 1. Each class therefore contains three subclasses of forms primarily nonequivalent. II. Improperly Primitive Classes. (1) Let D- 1, mod 8; the values of T, U in the equation T2 - D U2 = 1 are all even, and the automorphics are all primary. The forms are of three types: (A') a c 0, mod 4, (B') a 2, c 0, mod 4... (iv) (C') a O, c 2, mod 4, which are related to one another as the types (A), (B), (C) in the scheme (ii); and each class contains six subclasses of forms non-equivalent primarily. (2) Let D _5, mod 8. The forms are of one or other of the two types (A") b + 1, mod 4, (B") b- 1, mod 4, e * (v) 3 2 516 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 41. which are related to one another in the manner indicated in the scheme A" p B"J A"'= 1, P p, +, a7 T.... (vi) B" = + (T, I7, T 1, p, p2 Here, again, we must distinguish between the two cases in which the equation T2 - D U2 = 4 admits, and in which it does not admit, of uneven solutions. (a) Let T2- D U2= 4 admit of no uneven solutions. In this case all the automorphics are primary; and each class contains six different subclasses-viz. iff is any given form, f,fx p,fx p2 are of the same type asf, andfx *, fx cr, fx - of the other type. (b) Let T2 - D U2 =4 admit of uneven solutions: then the least solution is uneven, and every third solution is even. There are thus automorphics of each of the types 1, p, p2, and each class contains but two subclasses, viz. one of each of the types (A") and (B"). There is no known criterion for distinguishing a priori between the two cases (a) and (b). When D 1, mod 8, the criteria (v) as well as the criteria (iv) apply to the improperly primitive forms; and thus, if (a, b, c), (a', b', c') are two equivalent improperly primitive forms of such a determinant, they are or are not primarily equivalent according as the congruences a =,a', b b', cc', mod 4. (vii) are or are not satisfied. When D_5, mod 8, the conditions (vii) are necessary, but not sufficient, for the primary equivalence of two equivalent improperly primitive forms. The conditions a a', c c, mod 4 (viii) are necessary (but, of course, not sufficient) for the primary equivalence of any two equivalent forms (whether properly or improperly primitive). Secondly: If the determinant is a square, or a negative number other than -1 or - 3, each class contains six subclasses of forms primarily non-equivalent. Art. 42.] BINARY QUADRATIC FORMS. 517 Thirdly, for the determinants - 1 and - 3: there is but one class of forms of determinant - 1; this class contains three subclasses, viz. one belonging to each of the sets (A), (B), (C); and each of these subclasses (besides the identical pair of automorphics + 1) has a pair of automorphics which are respectively of the types ++, ~+, ~+-. The class of improperly primitive forms of determinant -3 contains two subclasses, viz. one belonging to each of the sets (A") and (B"), and each subclass has three pairs of automorphics, viz. a pair of each of the types + 1, ~ p, + p2; the properly primitive class of determinant - 3 contains six subclasses. Returning to quadratic forms of a positive determinant, two quadratic forms of a positive determinant, such as (a, b, c) and ( - a, - b, - c) correspond to one and the same semicircle: in certain cases these two forms are equivalent, but it follows from the conditions (vii) and (viii), that they can never be primarily equivalent. Hence the number of subclasses of semicircles of a given determinant primarily non-equivalent to one another is always half the number of the subclasses of forms of the same determinant. Let H and _H' be the numbers of subclasses of properly and improperly primitive semicircles; h and h' the numbers of classes of properly and, improperly primitive forms of the same determinant. Let v = 1 or 2, according as the equation T2 - D U2= 1 admits or does not admit of solutions in which U is uneven; and let v' = 1, or = 3, according as the equation T2 - D U2 = 4 admits or does not admit of solutions in which T, U are uneven; we then have H=-xvxh, H'=v'xh'........ (ix) 42.] Reduction of Quadratic Forms of a Positive Determinant. A semicircle which enters the reduced space (figure 2, p. 507) is a reduced semicircle; its reduced arc is the arc lying within the reduced space. The boundaries of the reduced space, viz. the semicircles x2 + y2 + x = 0 and x+ 1 = 0 are improperly primitive semicircles of the square determinant +1; of these, x + x2 + y = 0, and 1 + x = 0 are reduced. The semicircle x = 0 is also an improperly primitive semicircle of determinant + 1; and the three semicircles + x2 + y2 = 0, 1 + x = 0, x = 0 represent the three subclasses of improperly primitive semicircles of determinant 1, which exist in accordance with the general theory. The semicircles (a) - 1 +x2+y2 = 0, (3) ~ 2x 2 + y2 = 0, (y) + 1 + 2x= 0, represent the three subclasses of properly primitive semicircles of determinant 1. No other semicircle of determinant 1, besides those which we have named, can enter the reduced space; for the improperly primitive semicircles this is evident, 518 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 42. because the whole of each reduced semicircle lies within the reduced space, and any point on an improperly primitive circle of determinant 1 must be equivalent to a point on one of them, and cannot therefore be equivalent to a reduced point not lying on one of them; in the case of the properly primitive semicircles + 2 + x2 +y2 =, +1+2x=0, the reduced arc of either semicircle of each pair is equivalent to the unreduced arc of the other semicircle of the same pair; and thus the semicircle -1 + x2 + y2 = 0, and the reduced arcs of + 2 x + x2 + y2 = 0, and of + 1 + 2 x are the only properly primitive reduced arcs of determinant 1. We may observe (i) that the reduced semicircles of determinant 1 form the boundaries of the twelve regions considered in Art. 36; (ii) that the only semicircles which lie wholly within the reduced space are the two boundaries 1 + x = 0, x + x2 + y2 = 0, and the semicircles x = 0 and - 1 + x2 + y2 = 0; (iii) that every rational semicircle which passes through one of the cornicular points p, q, r is a semicircle of square determinant. Any given semicircle of determinant D is equivalent to a certain series of reduced arcs. For if w be any point on the given semicircle, and Q be the corresponding reduced point, as w describes any arc of the given semicircle, Q will describe the corresponding arc of the equivalent reduced semicircle: but when Q reaches a boundary of the reduced space, the reducing substitution of w changes, and Q reenters the reduced space at the opposite boundary; viz. if before the arrival of Q at the boundary, the reducing substitution is w = a x Q, we shall have immediately after the passage of the boundary w = I a I x 1 31 x Q, where I s I is | r 1-2, a1 12, I7 r -2, 1 r 2, according as 52 arrives from the interior of Z upon the boundary S 1, Si, Ti,, or T1,. If Q arrives at one of the cornicular points, co arrives at the same time at the extremity of the given semicircle, and vice versa. The number of reduced rational semicircles of a given determinant is finite. For every reduced semicircle must intersect one at least of the lines x = 0, + 1 + x = 0; but the number of rational semicircles [a, b, c] of a given determinant D, which intersect the line x = 0, is evidently finite, because, a and c being of opposite signs (except when one of them is zero, as may happen in the case of a square determinant), the Diophantine equation b2 - ac = D admits of only a finite number of solutions; and, if we shift the semicircles obtained by its solution to the right or to the left through the distance of a unit, we obtain the semicircles which intersect the lines + 1 + x = 0. Art. 43.] BINARY QUADRATIC FORMS. 519 We have now to consider separately the cases in which D is, and in which it is not, a perfect square. 43.] Case when the Determinant is not a square. No reduced arc can pass through one of the points p, q, r; thus the reduced arcs are of twelve different types, distinguished from one another by the boundaries which they traverse, [S-1, S+], [S+1, S-I], [T-1, T+I], [T+i, T-1], [S, -_D, [T-, [SL_, rT+ ], [T+1, S_], (i) [S+1, T_,], [T_, s+l, [S+1, T+], [T+1,, S+1]; so that, for example, [Sl, T+,] is an arc entering the reduced space at Sil, and quitting it at T+,. Two types such as [S_1, S+,] and [S+1, S_,], or again such as [S_1, T+J], [T+1, S_1], are not essentially different; but it is convenient to distinguish between them in order to indicate the direction in which the reduced arc is described. If A1 is a reduced arc having SE, or TE, for the second constituent of its type-symbol, and if A2 is the reduced arc immediately succeeding Al (i. e. the reduced arc primarily equivalent to the continuation of A1 immediately beyond the boundary in which A1 terminates), the continuation of A1 is transformed into A2 by | - 12E, or | 126; A2 has for the first constituent of its typesymbol S_,, or T_,, and enters the reduced space at the point on S_e, or T_,, symmetrically situated (with regard to the axis of y) to the point on S,, or XT, at which Al quits the reduced space; lastly, the corresponding directions on A1 and A2 make equal angles with the corresponding directions on the boundaries SE, S_,, or TI, T_. The series of reduced arcs is periodic; for, if a point c describe a given semicircle, the reduced point Q will describe a series of reduced arcs. But we have just seen that a reduced arc quitting the reduced space at a point on S_, S+D, T_1, T+1, is necessarily succeeded by an arc entering the reduced space at a point on S,,, SX_, T+, TIL, symmetrically situated to the former point with regard to the axis of y. Thus the series of reduced arcs can never terminate, and as there is only a finite number of different reduced arcs, the same arcs must recur; further, each reduced arc determines the reduced arcs which immediately precede and follow it; hence the reduced arcs must arrange themselves in a periodic series in which they succeed one another in a definite order. All the reduced arcs must be included in this period, because as w describes the given semicircle, the reduced point Q either describes a reduced arc continuously, or passes discontinuously from the end of one reduced arc to the beginning of 520 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 43. another perfectly determinate arc; i. e. Q can never leave the period of reduced arcs, and this point must therefore comprise every reduced arc equivalent to the given semicircle. To determine the law of succession of the arcs of the period, we shall term arcs of any of the eight types (ST) or (TS) transitive arcs; and arcs of any of the four remaining types (SS) or (TT) intransitive arcs. A succession of intransitive arcs cannot form a period; and, as the transitive arcs must be alternately of the types (ST) and (TS), there must be an even number of them in any period. Let us take as the first arc of the period a transitive arc Al, which we may suppose to be of the type [T_,, Se,, co, e1 representing positive or negative units; let A1 be followed by, - 1 intransitive arcs of the type (S_,,, Sej); let these be followed by a transitive arc (S_,,, T2), and this in its turn by 2 - 1 intransitive arcs of the type (TL,, T,); let the next arc be a transitive arc of the type (S. 2 S,,3) and let the series be continued until we arrive again at the transitive arc A1 with which we began; the last transitive arc (that immediately preceding A,) will be of the type [S_T,_, TJ], the suffix 2s- 1 being uneven because arcs of the types [ST] occur in even places in the series of transitive arcs. Writing e62 for e0, we find that the matrix I A = C 1211 X 1AX J 8sX... X r 12j2 2...... (ii) is an automorphic of the transitive semicircle A1. For the transformations which change A1 into the successive semicircles of the period are respectively I -r 1261 1 0- 14 1,..., I 1 12,1,1,1 I - 12ely, X I Tr 12"2, | C- 122"2 X I T | 2,.... so that (ii) is a transformation which changes A1 into the arc immediately succeeding the last arc of the period, i. e. into itself. Let A- = [aO, /a, a], and let 0 represent either root of the corresponding equation ao + 2 0 + a02 = 0;... (iii) the formula (ii) gives rise to the equation 0-IA1l x 0 =[2/lei1, 2U262,..., 2/2s2s, 0];..... (iv) and hence if 02 be the greater of the two roots of the equation (iii), (one of the roots of that equation is, in absolute magnitude, greater and the other less than unity, since [a0, 0o, a1l is a transitive semicircle) the quotients 2/, e, 2,2 C2... may Art. 44.] BINARY QUADRATIC FORMS. 521 be obtained by developing 0, in a continued fraction in which the integral quotients are the even integers, positive or negative, which approach most nearly to the complete quotients. The roots of the equations corresponding to the successive reduced semicircles are 01, 0- 2q,, 0 -4,..., 0l- 2(U2- 1), 1 1 1 1 02 ~2-22 02,-4E2"1 02-2(,2-1)e2' 03, 03- 23 3-43, 0..., 3-2 (3- 1)63, 02s 28 -26e2 02s -4e28 02 -2 (2s- 1) 2 where 1 1 1 01= 2eC+-, 02 =2f 2e2 + 0.2,, 02s =2828 + - 072' 03 l1 0i, 02, 03, being all greater in absolute magnitude than unity, and the roots of the equations which correspond to the transitive semicircles being 011 1 ), 0,3) * * Thus the succession of the reduced arcs in the period is regulated by the development in a continued fraction with even quotients of either root of the quadratic equation corresponding to any transitive semicircles of the period. According as we develope the greater or the lesser root of the quadratic equation, the transitive semicircle is of the type [TS] or [ST]; we may thus describe the period either backwards or forwards, obtaining two continued fractions which consist of the same quotients in reverse order. 44.] Comparison of the Geometrical and Arithmetical Reduction. (1) Given any reduced arc, we can by geometrical construction only, and without calculation, trace the complete period of reduced arcs; for the point of entrance of the arc succeeding any given arc is known, and also the tangent at that point. Again, if in the diagram exhibiting the complete period of reduced arcs we begin with the transitive arc A, or [a0, io, a,], we can, as we have already seen, obtain the equation (iv), Art. 43, by counting the reduced arcs, and VOL. II. 3 X 522 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 44. attending to the order in which arcs of different types succeed each other. Lastly, if we write that equation in the form O= A'BI 0, Dx ' the integral numbers, ao, 0o, a, are determined by the equalities C D-A -B a0 2 30 al Thus when a single reduced arc is given, we, can not only complete the period geometrically, but can also ascertain the equations of the semicircles of which it is composed. (2) On any given semicircle (see figure 3, p. 513) let PP1 be an arc cut off by any tangent of the ellipse corresponding to the fundamental primary automorphic. While the point w describes the arc PPI, the point Q describes the complete period of reduced arcs, and describes that period once only. For as soon as w, after travelling from P to PI, passes beyond P, it begins describing an arc primarily equivalent to the arc PPf; therefore, as soon as w passes Ai, Q must begin the period over again; and, until w has passed Pa, this repetition cannot begin, or we should obtain an automorphic of the given semicircle transforming points of the arc PIP into points of the same arc, which is impossible if PPJ is an arc of the fundamental homography. But, while w describes the arc PP1, Q may occupy the same position more than once, because points on the arc PP1 may be primarily equivalent, though not by an automorphic of the semicircle. Whenever this happens, the chain of reduced arcs intersects itself; and a certain number of such intersections does in general occur in every period. (3) We have here assumed that all the primary automorphies of a quadratic form are powers of one of them. But this theorem may itself be established geometrically; for this purpose it is sufficient to consider reduced forms only. Let r be the reduced arc of the reduced semicircle A; and let there be an automorphic V of A by which r is transformed into?, y being an arc of A which lies wholly outside the reduced space. If a point c setting out from any point pr of r travel along the circumference of A until it arrives at the equivalent point 71 of y, the equivalent reduced point 2 will in the mean time describe the period of reduced arcs a certain number of times iJ in the positive or negative Art. 44.]1 BINARY QUADRATIC FORMS. 523 direction, returning at last to the point r,. Thus the equation connecting 71 and Pr will be 7?= A|+~x;rl i. e. the automorphic V is a power of the fundamental automorphic (A). (4) If we fold the reduced space over itself so as to bring the boundary S, on S_1 and T1 on T-L, we may regard the chain of reduced arcs as forming an unbroken curve. We may, in fact, regard the reduced space as forming a quasicylindrical surface, having the singular point p at an infinite distance, and closed at its lower end by the semicircular curve qr joining the two singular points q and r; we may imagine the surface inflated so that qr is not a singular line upon it. (5) To obtain the period of semicircles primarily equivalent to a semicircle of which the arithmetical symbol [ai, b,, aJ] is given, we may employ an algorithm resembling that of Gauss for the reduction of a quadratic form of positive determinant. We compute the series of equivalent (but not primarily equivalent) forms, (ao, bo, a), (al, b, a), (a, b, a), (3, b3), *.... where a8 + is determined by the equation b8 - D and b6, which lies as near as it can to I/D, by the congruence b + b8_- 0, mod 2 a8. From this series we derive the series of primarily equivalent semicircles [ao, bo, a], [a2, 2-b, a], [a, b, a3], [a4, -b,..... which eventually becomes a periodic series of transitive semicircles, the first period beginning with the first transitive semicircle. If [a,, 3o, a] = [a2h, b21, a2 h+1] is a transitive semicircle occupying an uneven place in the series, we obtain the formulae of the last article by writing A = -o)-,/~D be2+j- b2+j-l_ j a a2h+j In the period of transitive semicircles the coefficients f3 are alternately positive and negative; and two consecutive semicircles are of the types (TS), 3x2 524 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 45. (ST), or of the types (SIT), (TS), according as they have the same last coefficient, or the same first coefficient. If we use -/D instead of + /D throughout the process we obtain the same period of transitive semicircles, but in a reverse order. If the semicircle which we have to reduce is given, not by its arithmetical symbol, but geometrically, the preceding algorithm indicates the following series of operations:-Let C, be the given semicircle, cl either of its terminal points. (i) We substitute for C, an equal equivalent semicircle C2, such that the point c2 equivalent to cl shall lie between -1 and + 1. (ii) We substitute for C2 an equivalent (but not a primarily equivalent) semicircle C3, inverse to C, with regard to x2 + y2 = 1, so that the point c3, equivalent to c2, lies outside of the segment (- 1, + 1). The alternate repetition of these operations gives eventually the period of transitive semicircles; the alternate semicircles of the period not appearing themselves, but being represented by their inverses with regard to x2+y2= 1. To the algorithm for the reduction of a quadratic form of negative determinant there corresponds (in a similar manner) a geometrical process for the reduction of a point; and the substitution which reduces a form / of negative determinant is also a reducing substitution for every quadratic form f of a positive determinant harmonically related to p; and, vice versd, every reducing substitution off also reduces an infinite number of forms p of negative determinant harmonically related to f 45.] Case when the Determinant is a Square. Let D = A2; we may suppose A positive; and, as we have already discussed the reduced semicircles of determinant 1, we may also suppose A > 1. An extreme semicircle is a reduced semicircle passing through one of the points +p, q, r; two of these points cannot lie on the same semicircle, since A > 1; we may therefore regard the extreme semicircles as being of three different types, (p), (q), and (r). Theorem. "The reduced semicircles, equivalent to a given semicircle r of a square determinant, comprise two, and only two, extreme semicircles; the remaining reduced semicircles form a chain of reduced arcs connecting the reduced arcs of the extreme semicircles." "If r is properly primitive, the extreme semicircles are of the types (p,p), (2, q), (r, r), according as r is of the type (A), (B), (C)." Art. 45.] BINARY QUADRATIC FORMS. 525 "If r is improperly primitive, the extreme semicircles are of the types (q, r), (r, p), (p, q), according as r is of the type (A'), (B'), (C')." Let r = [a, b, c]; and let us consider the quadratic formf corresponding to r, f = (a, b, c) (x, y)2 = m (px +p'y) (qx + q'y), where m =, or 2, according as r is properly or improperly primitive; p, p', and again q, q', are relatively prime; and pq'- qp = To demonstrate the theorem, we observe (1) That iff is transformed into F= (A, B, C) (x, y) = m (Px + Py) (Qx + Q'y) by the matrix a, we have the equation PP a,3 Pi PP' q, q X| Q, Q and hence, if a' be primary, the congruence IS -I Q~ Q P|I mod 2. q, Q, Q' (2) That the form corresponding to an extreme semicircle is of one or other of the types m( x+~ 'y) (X + X'y), x -'X = nA m (X x + A Y) (q X + X'Y), \Y - en =m(Xx+X'y)(rvx+v'y), Xr{-X =.(i where t and i' do not surpass unity in absolute magnitude, and \ and X' are 2A different from zero, and do not surpass in absolute magnitude. (3) That if r is properly primitive, ' P is of the type q, q, | o 1,1 0,1 1,0, mod2, 1,1 0,1 1,0 526 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 45. according as r is of the type (A), (B), (C); and that if F is improperly primitive,, P, is of the type lor,; por n; p2 or; according as r is of the type (A'), (B'), (C'). Hence the extreme semicircles, if they exist at all, are of the types assigned in the theorem. It only remains to show (1) thatf can be transformed by a primary matrix into one, and only one, form of each of the two types (i); (2) that the two forms thus obtained give essentially different extreme semicircles. Consider the equation qq 7, ~, ' in which. is a primary unit matrix, and;, X', X ' are subject to the in equalities stated above. We regard p, p', q, q' as given numbers; a, 3, ', S, a, a', x, x' as numbers to be determined, and we have to show that the equation admits of one solution, and one only. For brevity, we attend only to the case in which 1 and r' are both units. We first determine a and 7 by the conditions pa+p'=7, a-l1,mod4, 0, mod 2, 2A [qa+ q ] < -m; which are always satisfied by one, and only one, set of values for a, y, and 7; the equations pa/ +P'~ 70+A = 1, then give r^/S= a-p', l = y +p. We have also X=qa+q?, X' =q3+q'= ( 'x+ 2A) The last of these equations determines the unit a', for n'X must be negative 2A in order that X' may be less in absolute magnitude than. Thus the form f m (] x + 'y) (\x + X'y), equivalent tof, is completely determined; similarly a form Art. 45.] BINARY QUADRATIC FORMS. 527, equivalent tof, may be found from the equations PP al, 1, x, |, P x |1 1 And these two forms correspond to essentially different semicircles; for the equation, = - q is impossible, and the equation e/ = -, or ^ _ ~~J, '? Ah, A' =, +- I,Y] 2A implies the equation = 2; or A = m =1, contrary to the hypothesis that A > 1. We may obtain the reduced arcs equivalent to a given semicircle r of a square determinant by the following geometrical construction:-Determine a reduced semicircle equivalent to r (Art. 44); and continue its reduced arc in both directions (or in one only if it is an extreme semicircle) by a chain of reduced arcs. The same considerations which we have already employed in the case of determinants which are not squares show that every reduced arc must be included in the chain. But when we arrive at the extreme semicircles (or at. the extreme semicircle, if the semicircle from which we set out be itself extreme) the chain must terminate; for an extreme semicircle cannot be continued (in the direction of its extreme point) beyond the boundary of the reduced space; in fact, the extreme points of the two extreme semicircles answer to the extreme points of I; and while to describes the circumference of r from one of its extreme points to the other, the reduced point Q describes the complete chain of reduced arcs running from the extreme point of one of the two extreme semicircles to the extreme point of the other. When r is given arithmetically by its symbol, we may obtain the symbols of the chain of reduced semicircles in the following manner:-We first determine, by the arithmetical process given above, one of the extreme semicircles primarily equivalent to r; and then by the same process we obtain the substitution which transforms this extreme semicircle into the other. Let this substitution be an I | | a 1 I2,1l X I 7.|2"22 X | C I 23 3 *** 7, where the first and last exponents may be zero; the substitutions ac s26E1 | 461... 1C 12, o-2EII x |7 2, I..., |*C2E11 X | 12 2,2 528 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 45. applied successively to the first extreme semicircle, give the complete chain of reduced arcs taken in order. When the chain of reduced arcs is given graphically, the matrix a ' is found by counting the reduced arcs of the different types; the equations of the extreme semicircles are then also. known; viz. one of the factors of the form (a, b, c) answering to the first extreme semicircle is x + y, y, or x, according as that semicircle is of the type + p, q, or r; and the other factor is (7+ S)x-(a +3)y, yx-ay, or Sx-/y, according as the second extreme semicircle is of the type _p, q, or r. Example. Let the proposed semicircle be [8, 9, 7], D = 81 - 56, = 25. We have (8, 9, 7) (x, y)2 = (2x+y) (4+ 7 y); and we find successively 2, 1 1, 0 0, 1 4, 7 -2, 1 I -10,7 0, 1I - 7, -2 _ -10, - 3 -10, 7 -10, -3 0, -1 - 7, -2 12X -2x 14 -10, -3 -3" xI Hence the chain of reduced circles equivalent to [8, 9, 7] is [0, -5,7], [8, 9, 7], [8, -7, 3], [-8, -1, 3], [0, 5, 3]. Again, if we suppose the circle [8, 9, 7] to be given geometrically, but its equation (or its symbol) not to be given, we have first to construct, as in Art. 44, the chain of semicircles; the aspect of the chain gives us the matrix l 12 x1i-2x 114 I- 7,-2i which transforms the first of the extreme semicircles into the second; whence, finally, observing that the extreme semicircles are each of the type (q), we find that the symbol of the first of these semicircles is [0, - 5, 7]. Art. 46.] THE MODULAR FUNCTIONS. 529 Arts. 46-51. GEOMETRICAL REPRESENTATION OF THE MODULAR FUNCTIONS 4 (w) AND l (o) [ = 8 (X) AND S8 (,)]. 46. Discussion of the Equation,I(o) = A. Let A be a given complex quantity having any value whatever except 0, + 1, oo. The following theorems (of which the third results from the combination of the first and second) are of fundamental importance in the theory of the modular functions. (A) 'A complex quantity Q (having the coefficient of i different from zero and positive) can always be assigned satisfying the equation (8 (Q) = A.' (B) 'If (p(Q) = 08s(o), the complex quantities Q and o (in each of which the coefficient of i is different from zero and positive) are primarily equivalent.' (C) 'The equation (8s() = A always admits of one, and only of one, reduced solution.' These propositions are immediate consequences from a general theorem of Riemann (' Inaugural Dissertation,' Art. 21), relating to the representation of one plane surface upon another; but they may also be established by means of known properties of elliptic functions. (A) The differential equation du2 d 2= (1-2)(1-A),......... (i) taken with the initial conditions du X = 0, u= 0, 1,... (ii) defines u = (x) as a uniform doubly periodic function of x, perfectly determined at every point of the plane upon which the complex variable x is represented. It is always possible (and, indeed, in an infinite number of different ways) to take as conjugate periods of X(x) a pair of primary periods, i. e. a pair of periods, 4L and 2 iL', verifying the equations (a) X(L)=+1, (b) X(L+iL')= - and also satisfying the condition (c) that the coefficient of i in the quotient Lshall be positive; this coefficient cannot be zero, because the quotient of any two conjugate periods of a uniform doubly periodic function is always imaginary. VOL. II. 3 Y 530 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 46. For if we represent by L1 and iL1' the rectilinear integrals L,=~1 d) (1, r (1 du VA du L1= I (l u2) A2) L J /(1 -U2) (1 - Au2) the initial sign of the radical in the former integral being positive, and the sign of the radical in the second integral being so taken as to satisfy the condition (c), while the sign of1- is the same as in the equation (b), it will be found that N/A 4Li and 2iL1' are conjugate periods of X (x), and that the equations (a) and (b) are satisfied, i. e. that [4L1, 2iL1'] is a pair of elliptic periods of X(x)*, There is an infinite number of pairs of primary periods, for if a, a be any primary unit 7, matrix, the equations (a), (b), and the condition (c) are satisfied by the quantities L= a L1 + iL1', iL' = L1+ SiL1, (the sign of V/A in the equation (b) changing when 7 is unevenly even). If * For the method by which this result is obtained, see MM. Briot et Bouquet, 'Theorie des Fonctions Elliptiques' (Paris, 1875), pp. 351-368. By a pair of conjugate periods we understand a pair of periods forming an elementary parallelogram of the doubly periodic function (ibid. pp. 231-234). The primary periods defined in the text differ from the elliptic periods of MM. Briot and Bouquet, only because we have left the sign of the radical 4/A undetermined in the equation (b). This enables us to enunciate the theorem (D): 'If [4 LI, 2 iL1'] is any given pair of primary periods, all pairs of such periods are included in the formula [4L, 2 iL'] where (L, i')= x (L, l') and a is a primary matrix.' y, b If we adhere to the definition of MM. Briot and Bouquet, who suppose the sign of V/A to be fixed, we must add the condition that y is to be evenly even. The theorem (D) is equivalent to the theorem (B) of the text; for both [4K(co), 2iK'(o()] and [4(2(Q), 2aiK'(&2)] are pairs of elliptic periods of one and the same function u = (x). In the Report on the Theory of Numbers (Reports of the British Association for 1865), Art. 125, pp. 330 et seq. [vol. i. p. 295], the theorems (A) and (B) were enunciated; and the principle of the demonstration here given was indicated. M. Hermite has recently called attention to the importance of the theorem (B) in a note on a Memoir by M. Fuchs ('Borchardt's Journal,' vol. Ixxxiii. p. 29); and a demonstration of this theorem, depending on the theory of elliptic functions, has been given by M. Dedekind (ibid. pp. 266-269), who has also observed (p. 274) that the properties expressed by the theorems (A) and (B) follow from the principle of Riemann. See also the Note on the primary periods of the elliptic functions, cited in the footnote, p. 507. Art. 46.] THE MODULAR FUNCTIONS. 531 [4L, 2 iL'] is any pair of primary periods, the zero points of X (x) are 2 mL+ 2niL', and its infinite points are 2mL+ (2n+1)iL', mn and n denoting any positive or negative integral numbers; and the zeros and iL' infinities being all simple. Let - =, and let K and K' be determined by the L' K' equations K= K(o), K' = K (w); since = we may write K K' L = -, L' - and we shall find that the doubly periodic function o7r X x -- = sin am [xu, 8 ()] has the same periods 4L and 2iL' as X (x); it has also the same zero points and the same infinite points; hence, by the principles of the Theory of Functions, X () = C x sin am (cx), C denoting some constant multiplier; but X (L)= 1; and sin am (,L) = sin am (K) = 1; i.e. C= 1, and X(x) = sin am (u x). Lastly, (L + iL') = sin am (,LL + itL') = + VA ' but sin am ( LZ + i/L') = sin am (K+ i') =; whence we infer 04(C)= +2A, 48(O)=A. We may add (though this is not required to complete the preceding demonstration): (1) that u = + 1; for, when x= 0, li sin am ( _ux) M X but also lim sin am ( x) = lim X () x x 3Y2 532 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 46. (2) that the equation (b) is always satisfied when the equation (a) is satisfied; viz. if, of two conjugate periods 4L and 2 iL', one satisfies the relation X (L) = 1, the equation X (L+iL')= =+ ~ is always satisfied; for any two conjugate VJA periods may be expressed by the formula 4L=4aL1 +2 3 il', 2iL' = 4 y L1 + 2 J iL', where a~ - 3y = + 1. But if X (L) = 1, we must have a = 1, mod 4, and f3 0, mod 4. Hence 3 is uneven, and (L + i') = (- ) [L + iL] = + ( ) (B) If 8 (co)= (8(Q) = A, we form the two elliptic functions /rx \ <\ 7r X 0 ^(-). x 2 K(Q) ' P 2 (C) It, X tMa=r2)5 28(Q), Each of these functions satisfies the differential equation (i) and the initial conditions (ii). Hence the two functions are identical, and their zero points and their infinite points are the same. Considering the zero point 2K(c), and the infinite point iK'(c), of the former function, we must have K(o) = aK(Q) + ifK'(Q), iK'(w) = 7K(Q) + iK'(Q);... (iii) where 7 is even and ~ uneven. Similarly, we should find K(Q) = aK(c) + i3K'(o), iK'(Q) = 1K(o) + i K'(o),... (iv) where 71 is even and Jx uneven. But, if a8 - y7 = m, we may obtain from (iii) K(Q) =-K() - i K'(.), K'(Q) =-7 K(o) + i K'()... (v) rn/ m /m m v/ The systems (iv) and (v) must be identical; for otherwise the quotient i(co) = W K(co) would be real. Hence m divides a, /3, 7,; i.e., m= + 1. The negative sign must be rejected, because the sign of i in 2 and in o, +, is the same; a + fp2'i therefore m = ad - /, = + 1. Since a = Si, a is uneven: and since sin am [a K(Q) + il3K'(Q)] = + 1, Art. 47.] THE MODULAR FUNCTIONS. 533 f3 must be even, and a =1, mod 4. Thus the matrix a, is primary, and w, Q are primarily equivalent. 47. Limiting Values of I (co). We shall now suppose that the point o is confined within the reduced space (see figure 2, p. 507), and we shall examine the values assumed by the function s8(w) in the vicinity of the cornicular points ~ p, q, r. Let r be a constant quantity included between the limits - 1and + 1, so that - 1 _ r < + 1; and let o- be a positive quantity increasing without limit. (i.) Let =co=+ i, so that o moves along the straight line x=T, and approximates to the cornicular point oo, or r. It is evident from the equation of definition of /) (w) that Lim 8(co) = 16e'"X(cos r7r+isin 7r); i. e. 08(w) approximates without limit to zero, travelling in the direction indicated by the vector ei. (ii.) Let w = -, so that as C increases without limit, o moves along the semicircle 7 (X2 + y2) + x = 0 toward the cornicular point 0, or q. Then V (c) = ( 3(-!)=1- s(- ()=1- a-+ir) =1 - 1-6 e- "(cos 7rT + i sin -rr) ultimately; i. e. 08(co) converges to +1 in the direction indicated by the vector eiT; that direction depending on the curvature of the circular path pursued by o. (iii.) Let r be positive, and let co 1 - + -, so that co converges to the cornicular point + 1 or -p, moving along the semicircle 7 [(-1)2+y2] +x-1 = 0. Here ^8 (co) = 1 - a(-+i-) = 2 - 16 (CosTTr - i sin 7rn) ultimately, omitting quantities of the order e-I; so that d) (co) becomes infinite, and travels, as C increases, in the direction of the vector - e-"'i. If r is negative, we write co = - 1 - 1 so that co converges to the cornicular point - 1, or, moving along the semicircle r [(x +1)2+ y2] + x +1 = 0, and we obtain the sameresult as before. 534 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 48. The function 08s(o) cannot attain the values 0, 1, oo at any point of the reduced space other than the cornicular points oo 0, ~ 1, respectively. For if a be finite and different from zero, the infinite products II (l+q2-1 ) and II (1 + q2m) are finite and different from zero; again, if cr converge to zero, so that w approximates to the cornicular point 0 or +1, 8 (Cw) converges (as we have just seen) either to 1 or to oo; hence /8 (W) cannot be zero, unless a- be infinite, i. e. unless w approximate to the cornicular point oo; and by using the trans1 1 formations wv = - S. = + 1 - [, we may prove the corresponding assertions for the other cornicular points. 48. Transformation of the Beduced Space by the Modular Functions b(w) and (o). Denoting by X and Y real quantities, we shall now write X+ i Y= D (a>)- = I - (co); and taking a pair of rectangular axes X, Y, in a new plane (XY), we shall regard the point X+iY as answering to the point co in the'reduced space. We are thus in fact mapping the reduced space upon the plane XY; so that by a wellknown theorem to which we have already referred, the transformation is homeomeric, z. e. the infinitesimal parts of the two figures are similar except at points at which the first derived function of the mapping function is infinite or zero. The derived function of (Xco) is (Art. 9) h (o) = 4 (c) x t(^) x K2(C), which is always finite and different from zero, so long as the real part of zco is finite and different from zero; i. e. so long as co does not approximate to one of the cornicular points. If (=-T+iT approximate to the cornicular point ao, we find Lim V'(c) = i7r Lim D(co) = 16iz7e-~ (cos rr + i sin 7rr); i. e. V'(c) converges to zero. Writing Q = r + iT, and employing the substitutions 1 1 C= — = +1 -which give @ (a) = - ('(Q) x Q2, a'(C) = C(Q) 2 4)2 (2) Art. 49.] THE MODULAR FUNCTIONS. 535 we cause Q to approximate to the cornicular point oo, and we infer that if o approximate to the cornicular point 0, Lim 4'(o) = - 16z7r( + iT)2e-'~(cos 7rT + i sin rr7) = 0; and that if c approximate to the cornicular point + 1, Lim 4' (co) = I- ir '6 (cos T- - i sin rr) (r + ir)2 =. Thus the infinitesimal similarity of the two figures holds at all points of the reduced space except at the cornicular points oo, 0, ~ 1, to which there correspond in the plane (XY) the points (-, 0) or A_,, (+, 0) or A +, and the point at an infinite distance; spaces in the vicinity of the cornicular points oo and 0 being infinitely contracted in the plane (XY) and spaces in the vicinity of the cornicular points + 1 being infinitely expanded. The correspondence between the reduced space and the plane (XY) is a correspondence one to one.. For to every point in the reduced space there answers one, and only one, point in the plane (XY); and vice versa to every point in the plane (XY) there answers one, and only one, point in the reduced space; this results either from the general theorem of Riemann, or from the proposition (C) of Art. 46, and from our examination of the values of J((w) in the vicinity of the cornicular points. 49. Lines answering to the Semicircles oJ Determinant + 1. We shall now determine the lines which in the plane (XY) answer to the reduced arcs bounding the twelve spaces into which the reduced space is divided (Art. 38). See figure 2 (p. 507) and figure 4 (p. 536). (A) To the line x= 0 considered as described from 0 to oo, there answers the finite rectilinear segment A - A +1, described from A - to A + 1. For if o = Bi, and the real quantity a- decrease from oo to 0, 4(co)- - remains continually real, d4 (co). and the differential quotient d ( is real and negative; i. e. E (co) - increases continually from - 2 to +. Again, to the two equivalent lines x= + 1, considered as described from + oo to + 1, there corresponds the line [S], i. e. that part of the axis of X which lies between - - and - oo; for if o = 1 + -i, (co) is real and negative, and d da is positive, so that as a decreases from oo to 0, D (o)- l decreases from - to - oo. Lastly, to the equivalent circles x2 + y2 = + x considered as described from 0 to + 1, 536 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 49. there corresponds the line [T], i. e. that part of the axis of X which lies from + t to + oo; this result is immediately deduced from the last by employing the transformation 1 o =-, Q:=,, <v( )- g: (Q)- = -1 ((Q). In the equation (c) =X+iY, Y has the same sign as x in the equation = x + iy. For very great values of y this is evident from the equation defining Fig. 4. < (w); it is therefore true for all positive values of y, because, if x changes its sign, o must either traverse the line x = 1, or must quit the reduced space at one of the boundaries to enter again at the equivalent boundary; in either case, by what has just been proved, X +i Y must traverse the axis of X; i. e. Y must change its sign with x. (B) To the semicircle 2 +y2 1, considered as described from -1 to +1, there answers the axis of Y from - co to + oo. For if w= x+ iy lies on this semicircle, we have -1 -x+izy or <(x+iy) = * (-x+ y) = 1 - (-x+iy).. (i) But by a principle which we shall frequently employ, I((x +iy) and Art. 49.] THE MODULAR FUNCTIONS. 537 ) (-x +iy) are conjugate imaginaries, so that if ~(x+iy)= +X+iY, then (-x+iy)= - +X-iY; and the equation (i) becomes X=0. Hence to any point on the semicircle x2 + y2= 1, there answers a point on X= 0. Conversely to any point on X = 0, there answers a point on the semicircle. For if 1 (x + iy) = + iY, then,(-x+iy)= 1 _iY= 1 -' (+iy) = (x+iy) = ( + Y) But, because x + iy is reduced, -x + iy and -— + are also reduced. Therefore x+ zy the equation ( -xz+iy)= (- x+y) implies the equation - x + iy =, or x2 + y = 1. x+ iy It is evident that to the spaces lying within and without the semicircle x2 + y2 = 1 there answer in the plane (XY) the regions to the right and to the left respectively of the axis of Y. (C) To the reduced parts of the lines +1 +2x=0 there answers in the plane (XY) the circle (X- 1)2 + y2 _ 1; viz. to the line - 1 + 2 x = 0, considered as drawn from Xc to (1+ i), the an there answers the semicircle drawn from to through the upper region of the plane XY; and to the line 1 + 2x = 0, considered as drawn from I(1 -i) to co, there answers the semicircle drawn from 2 to - 2 through the lower region of the plane XY. For, if x = A -, we have x + iy = - 1 + (- x + iy); whence ' i) - 'p(-x+iy),4 (x + iy) ( - + iy) or +X+iY= 2 1- +X-iY' i.e. (X —)+Y2 =l Conversely, it is readily seen that the equation (X- )2 + Y= 1, implies the equation 4(x+iy) = '(+~-x+iy); VOL. II. 3 z 538 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 50. but if x + iy be reduced, + 1 - x + iy is also reduced, the ambiguous sign being that of x; hence x + ty = 1 - x + iy, or 2x= + 1. (D) Lastly, to the reduced quadrants of the semicircles x2 + y2 = + 2x, considered as described from 0 to 1 +i, and from -1 + i to 0, there answers the circle (X+ 1)2 + Y2 = 1, considered as described from - to - 2, through the upper region, and from - to - through the lower region of the plane (XY). We might establish this directly; but it is simpler to deduce it from the preceding result. As -- describes the line x=- from oo to 1 (1 +), o describes the quadrant x2 + y2 + 2x = 0 from 0 to - 1 +i. But <()- =-^[e(- )-i]; hence, while w describes the quadrant, ( (w)- describes in the plane XY a locus centrally opposite to the locus corresponding to the reduced part of x= 2; i. e. D (w)-2 describes the semicircle (X+ )2 + Y2 1 from I to - 3 through the lower region of the plane X Y. The figure 4 shows the distribution of the spaces which in the plane XY answer to the twelve regions into which the reduced space is divided. The symmetry of the figure with regard to 0, the middle point of the segment A_1 A + renders it convenient to take that point as the origin; i. e. to write, as we have done, F (w)= + X + i Y, instead of writing (w) = X + i Y. If P be any point in the plane XY, we have A_-P = Q AP(o ), A = ), OP=2 [ () - )]. Jacobi ('Fundamenta Nova,' Art. 29) found that the modular equations of the 3rd and 5th orders assumed their simplest forms when expressed in terms of the quantities 1 - 2k2, 1 - 2X2. 50. Discussion of the Correspondence between the Reduced Space and the Plane XY. (A) The values of 1 at the points of intersection of the boundaries are as follows: (0)= 1, (1)= _a, )( GO)=O; ~(l+;)=-(-l+;)=-1, <(;)=, } ~+ (I+2;)= ( - + }V3, ( i )= I * 3, j (+ 2"2/ = 2 2 D/ 2 f f/ Art. 50.] THE MODULAR FUNCTIONS. 539 These are the critical values of the anharmonic ratios of four points, corresponding to the cases in which (1) two of the points coincide, (2) the system is harmonic, (3) the system is equianharmonic. (B) The broken line, made up of the straight lines qr, rp, and the semicircle pqr (which forms on the tricuspid surface a closed curve passing through the three singular points) is transformed in the plane (XY) into the straight line A + A_l oo A+,, the evanescent angles at r and q (which are the halves of the cornicular angles at those points) being each changed into two right angles; a distortion which is not inconceivable because the infinitesimal similarity does not exist at the points p and q. In the same way the reduced parts of the straight lines 2x 1 =0; and again the reduced arcs of the semicircles X2 + y2 + 2 x = 0 are represented by closed curves in the plane XY, not presenting any singularity at the points answering to r and q. (C) The relations between the 12 spaces in the plane (XY), which answer to the 12 spaces of Art. 38, are as follows: B=1-A, C-= A D= 1 - ' A - A A, B, C,..., denoting vectors drawn from A_,. These equations, which hold for the accented as well as for the unaccented symbols, result from the formulae (i) of Art. 38, compared with the formulae of Art. 21. They show that the six pairs of spaces are related to one another as the six anharmonic ratios of four points, the points of intersection of the boundaries corresponding (as we have already seen) to the critical values. (D) If (for brevity) we say that two points which are symmetrically situated with regard to a straight line are inverse to one another with regard to that straight line, we may enunciate the theorem: ' Two points which are inverse to one another with regard to any one of the boundaries in either figure, have corresponding points in the other figure which are inverse with regard to the corresponding boundary.' This is a particular case of a general theorem to which we shall refer later, but it may also be verified independently. For example, in the reduced space, F is inverse to D' with regard to X2 + 2 1, and C' is inverse to A with regard to x= -; hence, in the plane XY, F and D' are inverse with regard to X= 0, A and C' are inverse with regard to (X- 1)2 + Y2 = 1. 3 2 540 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 50. (E) The lines [S] and [T] are lines of discontinuity in the correspondence between the reduced quantity w and the vector X+ iY= (w)-. Let the point X+ i Y traverse the line [S], passing (for example) from the upper to the lower part of the plane. Immediately before the passage, w is close to the boundary S +1; and immediately after the passage w is close to the boundary S-1; hence, when X+iY traverses [S], (w changes discontinuously from the value w to the value c - 2 = 1 C 1-2 X W. Similarly, if X + i Y traverses [T], passing from the lower to the upper part of the plane, w, which immediately before the passage is close to the boundary TL1, is immediately afterwards close to the boundary T; and thus changes abruptly from the value w to the value 1 = 12 x o. Let us now suppose the point X+iY, setting out from a given initial position P, to describe any closed contour whatever (C) in the plane (XY); and, still denoting by w the reduced argument which satisfies the equation (wa)- =X + i Y, let us represent by 0 an argument, which satisfies the same equation, and which initially coincides with w, but which, instead of remaining always reduced, varies continuously with X+i Y; it is required to find the relation between 0 and w, when X+ i Y, after having described the given contour, returns again to P. If X+iY traverses neither [S] nor [T], we have 0= w along the whole contour; if (C) be a simple contour including A_1, but not including A,, we have, when X+iY returns again to P, 0 = I j+2x W, the lower or upper sign being taken according as the revolution round A_1 is positive or negative (i. e. left- or right-handed); if (C) be a simple contour, including A +,, but not A_i, we have, when X+ i Y returns to P, 0= [T 2 x a, the ambiguous sign being determined as before; if X + i Y traverses [S] or [T] any even number of times, no passage across [T] or [S] intervening, the relation between 0 and W is the same after the last of these passages that it was before the first. Thus, without altering the ultimate value of 0, we may continuously deform the contour (C) in any manner we please, provided only that in so doing we never allow it to cross either of the points A,_ or A +r. By such continuous deformation we may substitute for (C) a series of elementary contours, alternately surrounding A_ and A +, and succeeding one another in a perfectly definite order. If (C) be thus reduced to a series of a, contours round A_1, followed by fi contours round A +, followed by a2 contours round A _, and so on, the numbers al, a2,..., 1?, 2,..., being positive or negative according as the elementary contours to which they refer are described in the positive or negative direction, we obtain Art. 51.] THE MODULAR FUNCTIONS. 541 for the ultimate relation between 0 and w, when X + i Y finally returns to P, the equation 0 = | a- lI2a X | T |-21 X | f 12a2 X i T | 202 X... X; where, however, the first exponent, a1, is zero, if the set of elementary contours which come first in order surround A+, and not Ai. A particular case of the preceding result is occasionally useful. If (C) is a simple contour, including both A_1 and A +,, and described in a negative direction round those points, (C) may also be regarded as a simple positive contour surrounding the point at an infinite distance in the plane (XY); and the relation between 0 and w is expressed by the equation 0=I|K2x 2x ^1, 2 0 = I C 1-2 X 17|12 X = I 2 | -2, -3 or by the equation -3, 2 0 = |7 T 2 X | C |2 X W = |, - 2, 1 according as, setting out from P, and following the contour in the proper direction, we first traverse [T] or [S]. 51. Limiting Values of (Q). Let n be any positive uneven number let a, b be a primitive and primary c, db c +dcC matrix of determinant n, and let Q =+ * We proceed to examine the values a+ bco acquired by 1 (Q) when the reduced point w lies in one of the cornicular angles r, q, p; i. e. when J (w) approaches one of the values 0, 1, so. (i.) As in Art. 47, let w = r + i0, where - 1 7r < + 1, and where a is positive and increases without limit; also let a, be equivalent, by primary premultiplication, to 2h, g so that (g)-(2h+g); it will be found that g' is the greatest common divisor of b, d, and that h is determined by the congruences b d 2h-, - a, 2h - c, mod g. 9g g 542 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 51. Since 1 (Q) = D ( -h- + 9-ia0), (Q) converges to zero at the same time with 4 (w), and we find (cf. Art. 47) g' 4 g' 2h. 4-4- -i' Lim 'D (Q) [((w)] =2 x eg The numbers g and g' may be negative, because g1l, mod 4; but g x g', = n, and-, =, are positive. g g (ii.) Let =-, so that () (o)= 1 - ( + i o); also let - + T ~ x c, d x - 1, 0 =-I[ 2h, glg ' where I vJ I is a premultiplying unit of the type *; it will be found that g' is the greatest common divisor of a and c, and that h, is determined by the congruences a C 2ha,_ b, 2h=,- d, modgl. 1l gl Since 4(Q)=1- I (2h- + r + gl i), 1 (Q) converges to +1 at the same time with ) (o), and we have g1 91 / 4-4g 2 Al ir Lim [1 - (Q)] - [l - (co)] = Lim I (Q) + [ (>) ] ' =2 x e (iii.) Leto =-, where 7 = +1 or -1 according as w lies in the cornicular angle + 1 or - 1. We then have () =l - and a, b 0, c=,a |x 1 x| [r+o-]. c, d -1,, - Let a, b 0,l1 =, x g2, 0 c, d - -1, 2h2, 2' I V2 I being a premultiplying unit of the type p2: g2' is the greatest common divisor Art. 51.] THE MODULAR FUNCTIONS. 543 of a + by and c + dr, and h2 is determined by the congruences 2h a+ - b, 2h2 + d, mod g2 S2 g2 We then have 4(2hA g2 +9 ) g2 g2 so that 4)(Q) increases without limit at the same time with 4)(w), the ultimate values of ~4 (w) and 4 (Q) being respectively g2 2,2 +.2'' Vo' — i7.... 1- i6exe i"7, and i- I e2 xe g2 g' 4 -Y4 -i22 whence we have Lim 4e(Q) [4(@)]9 2 xe The values of g2, g2', h2 depend on the unit; hence the relation between the limiting values of 4 (Q) and 4) (w) is different when 4 (w) is infinite in the upper, and when it is infinite in the lower region of the plane (XY). If in the equation Q = |, x x, V w is not reduced, let wc be the reduced vector equivalent to o, and let c= ' x; then 4 (co) = ' (c), and Q = c, X X c 1. c, d? Hence in this case also the ultimate relation of (Q) to 4 (w), when (w) approaches one of the values 0, 1, oo, may be ascertained by the preceding formulae, the numbers g, g', and h being determined from the matrix a, x a, and not from the matrix a, b c, d We may therefore enunciate the theorem: 'If Q and w are any two vectors connected by the relation 9= a,b whera, is a p e ad where, b is a primitive and primary matrix of determinant n; and if one of C d 544 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 52. the two quantities 1 (Q) and ( (w) approximates to one of the values 0, 1, oo, the other approximates to the same value." Again, if in the equation ~(, d Xo, the matrix a, is not of the type (1), let W= a, x w1, we being reduced, and c, d? a, ' being a unit matrix of the t i t ' the type inverse to n,, c, d; th en a, a, /3 alC, bd,, the matrix |, x an being primary. The ultimate relation between I (Q) and (wi) may be found by the preceding formulae; and thus the ultimate relation between D (Q) and 1 (c) is known. Arts. 52-58. THE MODULAR EQUATION. 52. Definition of the Modular Equation. Theorem. If I At represent a system of primitive and primary matrices of an uneven determinant n, non-equivalent by post-multiplication, every rational and integral symmetric function H of the a' (n) quantities (see Art. 18) (Q) = ( A I- x co) is a rational and integral function of FI(w). To establish this theorem we have to show (a) that H is a function of () = - + X+ i Y, one-valued throughout the whole plane (X Y); (b) that H is finite for every finite value of 1 (w); (c) that, when 4 (w) is infinite, H is infinite, and comparable to a power of 1 (w) of a finite and integral exponent. (a) If ())= + X + i Y describe any closed contour whatever in the plane XY, and if w vary continuously with 4 (w), the value of w, when P (w) returns to the point from which it set out, is e I x w, where I e I is a primary unit matrix, depending on the form of the closed contour (Art. 50), and the values of Q are at the same time I A I-~ x e x w. But the matrices I A I-1 x e [ are, in some definite order, equivalent by primary premultiplication to the P., A lip It Art. 53.J THE MODULAR EQUATION. 545 matrices I A I-1. Hence the -' (n) quantities (I\A I- lx Ixe o) are in the aggregate the same as the quantities d (I A -1 x o); i. e. the symmetrical function H remains unchanged. (b) If any one of the quantities P (Q) is infinite, ) (o) is also infinite (Art. 51). Hence H is finite at every finite point of the plane. (c) If F (w) is infinite, each of the quantities d(Q) is infinite, and comparable to a finite power of ' (X). Hence H is also infinite, and comparable to a finite power of 1 (c). The exponent of this power must be integral, because the value of H does not change when D (w) describes a simple contour round the point co. From this theorem it follows that the equation F(X2, k2)= n (x2 - 1 (Q))= 0, of which the roots are the squared modules resulting from the a-' (n) primitive and primary transformations of order n, has for its coefficients rational and integral functions of k2 = p8 (o). This equation is termed the modular equation of order n: a discussion of its properties will occupy the following articles. The case of an even determinant is considered in Art. 64. 53. Theorem i. The modular equation is irreducible. If possible, let F (X, k2) = H (2, 2 ( 2, 2),..x ( X.. (i) each of the factors Hi, H2 being rational and integral functions of X2 and k2. When k2 = 1 (w) = 2 + X + i Y describes a closed contour in the plane X Y, the roots of H1= 0, and of H2= 0, may change inter se, but each of these two sets of roots must, in the aggregate, remain unchanged, because the equations themselves remain unchanged. But by properly selecting the contour described by F) (W), we can cause any root of F= 0 to change into any other. Let Ql=IAll-lx1X, Q2=1A2l-1xW so that d) (Q2) and 1 (Q,) are two of the roots of F= 0. Let IA,1 -'=lal xA IA-lx l 1 the units I a and I I being primary; if 1 (co) describe the contour (Art. 50) corresponding to I /3, the value of P (Q2) after will be that of 1 (2,) before. Hence VOL. II. 4 A 546 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 53. the equation (i) is impossible, unless the roots of H =0 and H = 0 are the same for all values of c. And this supposition is inadmissible, because the equality of two values of d? (Q) implies that c is the root of a quadratic equation having integral coefficients. Theorem ii. The equation F(X2, k2) =0 is symmetrical with respect to X2 and k2. If k2 = 1 (I Al I-1 x o), one of the corresponding values of X2 is J (c). For one of the matrices I A \-1 is equivalent by primary premultiplication to + | Al; if A2 j-1 be this matrix, we have ~ ( A21-lx IAI-lxco) = P(lAlxlA,-lxco) =- (:n, | X )=q(= ), i. e. (c) is one of the values of X2. If therefore a, b is any pair of values satisfying the equation F(X2, k2) = 0, b, a is a pair of values also satisfying it; i. e. the equation is symmetrical with regard to X2 and k2. Theorem iii. The equation F= 0 remains unchanged, if we write in it for 2a 1-X2 ( k2 X2 1 1 k2 and X2: (1) 1 - and -X2, (2) kk 1 and X2, (3) k andX2, 1 1 k2-1 X2-1 (4) k2 and 1 (5) k and X Let e be any given unit matrix, if k2 = (i e x cO), the values of X2 are comprised in the formula ( ej xQ)= ) (e I x A I x w). For elxQ EI I A I x I xel- xlelxW, and the matrices e I x I A I x e I-1 are a complete system of primitive and primary matrices of determinant n (Art. 17). To prove the theorem we take in succession for | e the matrices I / i, 1 C 1, T|, P I, I p2, and employ the formule of Table A. Theorem iv. If k2=0, 1, oo, the corresponding values of X2 are 0, 1, cc respectively. (See Art. 51.) From these theorems it appears that the function F(X2, k2) F(k2, X2) is of the form i=N-l j=N-1 k2N+ - Z ai, j k2i +X2, i=1 j=1 Art. 54.] THE MODULAR EQUATION. 547 where N= o'(n) and the coefficients a, j, if arranged in a square, are symmetrical with regard to both the diagonals of the square, so that ca, j = aj, i = a N-i, N-j; the former of these equations resulting from the interchange of k2 and \2, the latter from the simultaneous substitution of - for k2 and - for X2. Since F(X2,1) (X2-_1)N, the sum of the coefficients in any row or column is equal to the corresponding coefficient in the expansion of (\2- 1)N. The identity F(1-x, 1-y)=F(x, y), which implies that F (l + x, I + y) is an even function of x and y, gives another set of linear relations connecting the coefficients aj, j. And these are all the linear relations between the coefficients which are deducible from the theorems of this article. 54. Theorem. Let P and Q be two numbers relatively prime; and let F(P, x\22, )=o, F(Q, 2, k2)=o, F(PQ, 2, k2)=0 be the modular equations of the orders P, Q, PQ respectively; the resultant of the elimination of z from the equations F(P, \2, z) = 0 and F (Q, z, k2) = 0 is F(PQ, \2X k2)= 0. Let IP and I Q I represent complete systems of primitive and primary matrices for the determinants P and Q respectively. If k2 = P (w), we have F(Q, z, k2)= nQ[z- ( QI x)]. Again, if z = I (I Q I x o), we have F(P, 2, z)=p [x2 (l PI x I Qx)]. The resultant of F(Q, z, k2)= 0 and F (P, X2, z) = 0 is evidently n lH [X2 - ( P I x I Q x )] = O. But I P l x Q is a complete system of primitive and primary matrices for the determinant PQ (Art. 16 et seq.); hence the resultant is F(PQ, 2, k2) = 0. The theorem requires some modification when P and Q are not relatively prime. It suffices to consider the case in which Q is a prime p, and P is a power py of p. 4A2 548 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 55. (a) Let u > 1. The system |p I| x Ip 1, resulting from the composition of the primitive and primary systems \IpL I and \p I, consists of the primitive and primary matrices Ip11 +, each taken once, and of matrices, each taken p times, derived from the primitive matrices 1 p-1 I by multiplying each constituent with p. Hence the resultant of F (pl, X2, z) 0 and F(p, z, k2) = 0 is F(p\+1 X2, k2) X[F(p-1, X2, 2)] = 0. (b) Let L = 1. The matrices IP 1 x Ip I consist of the primitive and primary matrices lp2 1, each taken once, and of p +1 matrices derived from primary unit matrices by multiplying each constituent with p. Hence the resultant of F(p, X2, z) = 0 and F(p, z, k2) = 0 is F(p2, X2, k2) (X2 k2) +1 = 0. It follows from the theorems of this article that if we have obtained the modular equations for primes, we can obtain the modular equations for composite numbers by elimination. 55. Theorem i. The coefficients ai, j are integral numbers. It suffices to consider the case when n is a prime, since the theorem may be proved by elimination when n is composite. As in Art. 5, let f(q) = ( ~) + 2q so that k2 = 24qf8(q). The equation F(n, X2, k2) = is satisfied identically (i. e. whatever q may be, provided only that its analytical modulus is inferior to unity) by the values k2= () = 24qf8 (q), X2 = (nco) = 24qf8 (q). We may therefore substitute these expressions for k2 and X2 in the modular equation, and may determine the coefficients aj successively by equating to zero the coefficients of the powers of q; this is, in fact, the method employed by Sohncke in his memoir 'Aequationes modulares pro transformatione functionum ellipticarum' (Crelle, vol. vi. p. 97). Let the terms of F be arranged according to descending powers of X2 and k2; i. e. let aij X2i k2j precede a, j,X2i' k2', if i > i' or, if i=i', j >j'. The lowest power of q arising from the term ai,j X2ik2 is qi +j, and this power is lower than the lowest power arising from any term which precedes aij X2i k2j. Writing for brevity Aj = 24(i+j-n-l). aj,j, then A,, =1, Art. 56.] THE MODULAR EQUATION. 549 and denoting the coefficient of qh in the expansion off85 (q) xfPS'(q) by C [h, s, s'], so that (Art. 5) a=I C[h, s s'] = z [8s, a] x [8s', h - na], a=we find, on equating to zero the coefficient of qni+j, c[(i-s)n+j-s', s, s]xAss==,...... (i) the sign of summation extending to all the coefficients As, ' which do not precede A,j. This equation serves to calculate successively the coefficients Aj; and, since C[0, i,j]=1, it shows that they are all integral numbers. Hence also, if i +j~ n + 1, the coefficients aj3, = 24(nf+ l-i-i)Ai,j are integral; i. e. all these coefficients are integral, because the matrix I ai, j is symmetric with regard to its diagonals. We find from the formula (i) aC, = -24(n-1), a,2= 24(n-2) xSn, C1 3 = - 24n-10 x (8n2- 13n), a,= 24n-11 x n(n - 2) (8n - 23),...; and similarly, for any given index j, the value of aj may be determined. But a general expression of these coefficients cannot be obtained in this manner. Theorem. When n is a prime, the coefficients aj, except al,,, =an, = -24n-4, are divisible by n. For, assuming that the theorem is true for the coefficients which follow aj, j, and observing that A,,= -1, we have, from the equation (i) Ai,j - C[(i- 1)n +j - 1,, 1]+ C [(i- 1)n +j- 1, O, n + 1]-0, mod n, i.e. A,j= 0, mod n; since by Fermat's theorem f8(q) xf8(q) f 8n + 8(q), mod n. The coefficient a, j, where we may suppose i+j n+ 1, is divisible by 24(f+ -i-j); but this is not always the highest power of 2 which divides a, j. 56. The Sums of the Powers of the Roots. Let z = i,)+ (n z s( Q), where n is still an uneven prime, and Q is any one of the n quantities w h~b 550 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 56. The formula of Newton shows (1) that Es is of the form As, 1 )A (A,2() + A... + A, (.4 ns(.;. (i) the coefficients A8, h being integral numbers; (2) that to obtain the coefficients aij of the modular equation, it is sufficient to know the n2 coefficients comprised in the matrix s=1, 2,...n 8,hl, h=1, 2,...n, or, having regard to the relations which subsist among the coefficients a, j, the ((n+ 1)2 coefficients which lie on or between two semidiameters of the matrix. Since Os (nos)= 4 (nT -1) (bn(s) + terms containing higher powers of 1 t(w), the series (i), as far as the term A8, nn8Ilnsl(w) inclusive, is the development of 2V (Q) in a series proceeding by powers of 1 (a). Hence, if h ns - 1, we have A 1, =I(L)x f(~)d4(....d (ii) AS'J2-JtJ )h+l(), ~ ~ the track of the integration being such that d (w) describes a closed contour round 0; so that, for example, we may integrate in a straight line from w = - 1 + ir to W= 1 + i, 7 being any constant greater than a, so as to bring the track of w within the reduced space. But, employing the notation of Art. 5, and supposing s ~ n, 's(Q) 2 48 x n x [s, nx-s] q x=1 and V(h+l) d h <()) 24h d(qf8h(q) =-24h X X X ( - A)[ 8, X 1 dq 1 Hence, since A,, is the absolute term in the coefficient of q within the sign of integration, we have finally n a-h A, h X 24(8-h)X Ca[8sna-s]x[-8h,h-a... (iii) a=1 Art. 57.] THE MODULAR EQUATION. 551 Since - A s,... +, As, -s ' () + 2 (a) + ' ns() e we obtain by a similar process n a-s A, h= 24(h-)X [8h, na-h] x [-8s, s- ],.. (iv) a=1 so that hAs, h is symmetrical with regard to s and h. The equations (iii) and (iv) serve to express the coefficients A8,h, and consequently the coefficients a,, j, in terms of the coefficients of the series 2: (iM)q, and its powers up to A(m) -(n2-) and its powers up to 2[ (ha qg]d inclusively. (See Art. 5.) 57. Development of the Roots of the Modular Equation. Employing the method of the last article we find 4 (n ) = 2 (-,) (W) x [ao+, al(w)+... +a8S() +...], a=I. (i) 1 a= as= 24x(n+ ) x Z (a+l)[8, a]x[-8(+s),s-na]; 248 X (n + s) a=O n-1 4 - 2 1 1 4 s x Z (s+l-an)[8, s-an] x [-8+ a; 2 n(s +1) +( ) 2h the n values of the right-hand member of (ii) corresponding to the n quantities ^(W2h-), h=O, 1,...,n-1. Writing x={Q(co), we represent the right-hand n n member of (i) by A, and of (ii) by 1 n-i 1 2O+Xn1,+...+ +X 2n, the infinite series 20,:,,... 2_ being rational; if X2 =y we have I n-,F (X, Ay)=(2- y)x [oII-2/+..2+.+ 2 [_ y].. (iii) 552 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 58. 1 the sign of multiplication extending to the n values of the radical xn (see a paper on 'The Higher Singularities of Plane Curves,' Proceedings of the London Mathematical Society, vol. vi. p. 153*). The continued product II is equal to the determinant A(y, x) = 1 n-1 n-2,xn. n-,2..... (iv) 1 2 x n ZX, x * *2 Zo - y thus, if Sr (x) is the sum of the principal minors of order r in A (0, x), then the coefficient of yi, (i = 1, 2,...n), in the modular equation, or ai,x + a,2 2+... + ain xn, is equal to (-l)iSi+1 (x) taken as far as the term involving xn. There is, 1 however, one exception; the coefficient of Xn in - S (x) exceeds a, by 2(- 1 Instead of the developments (i) and (ii) we may employ the series ___ ^ = -24(n-i)n1-n(w) x [yo+7yl (o)+... +7 S( o)+...], 4F (nco) a =I. (V) a=O s =^48 X. X I (a - l)x[-8Sa]x[8{(n-s),s- na]. — 11 1 s = 24 n X 1 X n ( at() f O.. (vi) 4- a=I2 w n r - I GS = - X z (s -1 -an) x [-8, s - ai] x -8-, a * S - a=O 58. The Analytical Parallelogram. Returning to the case of any composite uneven number, let 1, yn, 72... n be the divisors of n in ascending order of magnitude, and let n, ~, 2... 1, be the conjugate divisors corresponding to them. If the terms of the equation be * [Vol. ii. p. 101.] Art. 59.] THE MODULAR EQUATION. 553 arranged in an analytical parallelogram in the usual manner, the inferior limiting polygon (inside which the squares are empty) is obtained by joining in order the points (O, N)......... 1 (l,N- n).......24 (N-1) (1+y, N-n-1).....24 (N-1 +81 —) (1+71+72 N n- 1- ).. - 24(N-1++82-7l-72) (N- n, 1)...... -24(-1) (NO).........1 The coefficients of the terms corresponding to the vertices of the polygon are given in a parallel column, and N is written for '(n). The coefficients are obtained by causing k2 to decrease without limit, and examining the initial terms of the expansions of the different values of X2 in series proceeding by powers (integral or fractional) of k2. The exterior limiting polygon, outside of which the squares are empty, is of course symmetrically situated on the other side of the diameter joining (0, N) to (N, 0). Arts. 59-62. THE EQUATION OF THE MULTIPLIER. 59. Theorem. The reciprocals of the multipliers corresponding to the -'(n) primary and primitive transformations of an uneven determinant n are the roots of an irreducible equation of order a-' (n): the first coefficient of the equation is unity, and the other coefficients are rational and integral functions of k2 with integral coefficients. Let A [ represent a system of primary and primitive matrices of determinant n, and let the multiplier corresponding to the transformation co= I Ai I x Q (see Art. 26) be denoted by M (o, I A I) = Mi. (i.) If ) (c) describes any closed contour in the plane XY, and if w at the close of this contour becomes x e I X c, where e I is a primary unit matrix, the multiplier M(co, I Ai l) becomes M( e I x co, i Ai 1), or, which is the same thing, MI(LC, I e1- x li). Let 1E-lxIAi=AAjix\^i, I being a primary unit matrix; then M(c, e El-Ax I ) )= 1 (, Aj lxl i)-=Ii(, Ajl); VOL. II. 4B 554 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 59. or Mi becomes Mj. Hence any symmetrical function of the multipliers remains unchanged, when 1 (o) describes any closed contour whatever in the plane X Y; 1 i. e. the o' (n) quantities y/ are the roots of an equation ( 2 )=0,..... (i) of which the coefficients are one-valued functions of k2. (ii.) To complete the demonstration of the theorem, it is convenient to examine the effect, upon the system of multipliers, of substituting for w one of the expressions I x w, I I x, I 1 X X, I p I x c, I p 2 x <. The results are given in the following Table, in which | Y | represents successively each of the five unit matrices, and the index j is that of the matrix A dj cation with a unit matrix of type (1), to the matrix + 1~1- x A I x I. The Table gives, for each of the five substitutions, the form of the equation of the multiplier, and the relation of its roots to the roots of the equation (i). TABLE OF THE LINEAR TRANSFORMATIONS OF THE MULTIPLIER. Art. 59.] THE EQUATION OF THE MULTIPLIER. 555 Of the five formulae contained in the Table it will suffice to demonstrate one. Let LT7-1xIAl x 1I = (-1)ciXAjlxlil,..... (ii) where I1 a \ ',n' is a primary unit matrix; so that 7, a, | Ai x 1|'1 x )o =T I X | -1 X x Aj -'1 x co= T X |1'1 X j; then (- I) "-.j= _M(, | xAi I x A x | ) -=4() xM( 7- xo, A, Ii) x(Ai1-lx 1T X, ITl), XM (I Ail M(Tx l-xQ T)I =4 (r)' + 4 AI)x (17 X I l-1 X 2j)' = () xM(, A) 1)Y4(j). But, from equation (ii), ci = ( - 1) x (cja + dj), or ci, Cj +, mod 4. We have therefore, finally, ( 1, =(- ) (4Cj() _ 1 and these o'(n) quantities are the roots of the equation ( T V - ))=0- or A (,=0. (iii.) We proceed to determine the values of the multiplier when o is at one of the cornicular points, or when k2 = 0, 1, oo. (1) If k2 = 0, we have / (% o) = ( -(a - 1)i )(gl) g and g' being conjugate divisors of n. For, taking for I AI the system (-1)(9-1) x g' ~ - 2h, ' 4B2 556 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 59. we have K t2h+gw 1~_- l) ( -i), 9'...... (iii) MK(co) and, when o approximates to the cornicular point, lim K (w) = 7r, lim K(2h+g( ), limM=( l)1(and there are (g') roots of which this is the limiting value, because there are S (g') values of h corresponding to any given value of g. (2) If k2 = 1, we have (M 1^)= ^( -(-l)2('-l 1 ) since /1 2 = lU-) ( -1-) (3) To find the limiting values of the roots when k2 is infinite, we have iir& g —g ' lim ' k) =lim T1 -(-1) - x e X) In this equation, which is derived from (iii) and from the formula (T) of the Table, we have supposed that h is even, and that k2 = p (o-i), o- being real, and increasing without limit. Thus, when k2 increases without limit, the roots are partly zero and partly infinite, except when n is a square, in which case there are as many finite roots as there are numbers prime to Vfn and not surpassing it. The product of the roots is finite, and equal to + Ilga(g'), the sign being (- 1)("-1 when n is a prime, and + in every other case. (iv.) Since the multiplier cannot become infinite except when o approximates to one of the cornicular points, we infer from the preceding discussion that the coefficients of the powers of 1 in the equation (i) never become infinite except when k2 is infinite; and that, when k2 is infinite, they either remain finite, or become infinite as a finite power of k'. These coefficients are therefore rational and integral functions of k2. Art. 60.] THE EQUATION OF THE MULTIPLIER. 557 60. Case when n is a Prime. -Determination of the Coefficients. When n is a prime, let Aj (k2) = Aj be the coefficient of M, in the equation of the multiplier, and let zO, Zh be the reciprocals of the multipliers appertaining to the transformations n,0 1,0 (-.l)(-)x 0, 1 ' - 41h, n respectively. (-1) 7-~)x 0, 1 -4h, n If in the equation IA (M, -2 = 0, we put w = ai, and, allowing a to increase without limit, regard 4(co) as infinitesimal of order 1, the root zx x )(4 is infinitesimal of the order (n - 1), and each of the roots Z x P4(n is infinite of the order 2 (1 — ) Thus the coefficient Aj( ) cannot contain k2 to any power higher than (n + 1-j) x (1- ); in particular, An and A,_, are of the order zero, and A, is of the highest order possible, viz. (n - 1). Let Aj(k2) = s8ajs k2s; if in the equation I (%, k)= 0 we put 1 _(_O) _ =Z X -0-, =r, = % x ()0 ()), =, then the term aj, M k........ (iv) is infinitesimal of the order (n - 1)j- s; and no two terms (except the absolute term (-l)"(n1)n and the penultimate term a, (n1) 7,kn 7 1) are of the same order. If therefore we substitute for z and - their developments in series proceeding by powers of q, we can successively determine the coefficients aj,s by equating to zero the coefficients of like powers of q. Or, again, we may put 1 44 (~2h).I. = z x ( h): the term (iv) is now infinite of the order 1 (I - )j + s; and no M Z pX p4(W) two terms (except I + and the penultimate term) are of the same order. Since the coefficient of q-(ln)j-s in (iv) is 24 i8, we infer that a,, is integral and divisible 558 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 61. by 248. All the coefficients aj, are divisible by n, except when j= 1; because the coefficients of M- M - in the expression u.(1 ( (c e ^2K1o) 1 - l - i ( a) = M K (n) j' when developed in series proceeding by powers of q, contain n as a factor in every term; similarly it may be shown that al,o + a,,,.2 +... a,, ( lI)k4-1 a,0 k2+... {1+ 3a, (+ )+ )kn-} modn2 1 21.3 1 3... n-22 the series within the brackets on the right-hand side consisting of the first numbers 1, 2, 3,..., (n - 1). Since - -- m=+OO m2 b=(-1) m==+o 1 numbers123nb E- ( an + b)2r V-axA, 2 a MA, we have terms of the (n +L 1) quantities A. 62. Expression of the Multiplier as a Rational Function of k2 and X2. We have (Art. 11) cdT[(o=) _ 4(icmr~+b) do 4 id (O) x( =-)K2()) dem (Q) _ dQ) dQ 4i (Q)n (ai+ bQ)2 adc -( (d) d* (o) xd dc o = d Xd-= n Art. 62.] THE EQUATION OF THE MULTIPLIER. 559 Hence, by division, 1~= (K + b)2 x (Q) M2 = (a bg7) xK2 (C) n (w) J(C) d T(Q),dF k(l - k2) _ k2(1 -. 2)kd 2) X(1 -X2)dk X-2(1_ -X2) ( F * This theorem holds for every transformation of any order whatever, even or uneven; F = 0 representing the modular equation between k2 and X2. 1 But can itself be expressed as a rational function of k2 and X2. Attending only to the case when n is an uneven number, we write the equation of the multiplier in the form + -2. 2 1. and we substitute for 7j2 in i, and L2 the value given by equation (i). We thus obtain for an expression of the form 1 - 1 ( g ).2 2 v the numerator and denominator being rational functions of k2 and X2, which may simultaneously vanish for particular pairs of values of k2 and X2, but which do not vanish by virtue of the modular equation; since, if they did so vanish, - 1 would be a root of the equation of the multiplier as well as M, which is not in i general the case. The expression of thus obtained may be put into an infinite number of different forms, because X2 and k2 are not independent, but are connected by the modular equation. In particular the equation 1 fn(X2, 2) 3M A(k2) (when f. is a rational and integral function of X2 and k2 of the order n in X2 560 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 63. and having integral coefficients, and A (k2), which contains only k2, is the discriminant of the modular equation) may be deduced by the following method, which is due to Professor Cayley. It may be shown as before that the a' (n) functions B. defined by the equations X "2-=0, 1,..., (n)-, jX. 1, j=0 1, 1,...,c'(n)-1, are rational and integral functions of k2 with integral coefficients. Solving these equations for the a'(n) multipliers which enter linearly into them, and denoting byf, and f rational and integral functions of X2 and k2 in which we may suppose that the highest exponent of X2 is -'(n) - 1, we find 1 _ fi(^x), ^k2)_ f(Xj2 k2) MjF'(x, A() 2) The theory of the equation of the multiplier in the case of an even determinant presents no special difficulty. The results are of the same general character as in the case of an uneven determinant; but the discussion involves many points of detail, and cannot be attempted within our present limits. Arts. 63-73. THE MODULAR CURVES. 63. T17e Modular Curves of an Uneven Order. We have seen (Art. 51) that, if we represent by e(x) any one of the six anharmonic functions X)1~X, l -, - —, -, -, X 1- x X-1 X - the modular equation is unchanged by the simultaneous substitution of e (k2) for k2, and e(X2) for /\2. Hence if e (x), e2(x) denote any two, the same or different, of the six functions (i), the thirty-six substitutions F[e1 (k2), E2 (X\)] give only six different equations, viz.: (1) F(k2,- X2) =0, (4) F(k2, x2) =0, (2) F(k2, =) 0, (5) F(k, X2 =... (ii) (3) F(k, 2 -l) =0 (6) F(kr, 1 0) Art. 63.] THE MODULAR CURVES. 561 for these we shall employ the notation F8(k2, 2)=o, s=1, 2,3, 4, 5, 6, the left-hand members being supposed integral; so that e. g. F2(k X2)=X2x F (k2 ), F4(k2, X2) = F(k2, 2). If k2 -= I(X), c = A i x Q, IA I representing a system of primary matrices of determinant A, and of the type x, a, or, 1, p, p2 in the six cases respectively, the roots of the six equations are comprised in one and the same formula \2= (Q). The first three equations (as well as the fourth) are symmetric with regard to k2 and X2; the fifth and sixth are changed each into the other by the interchange of k2 and X2. If then we denote by X and Y rectangular Cartesian coordinates, the equations rs (2+X+iY, L+X-iY)=0, s=1, 2, 3,4 are real, and, as we shall presently see, represent real curves, which we shall term the first, second, third, and fourth modular curves; we observe, however, that when A_ 3 mod 4, the fourth modular curve reduces itself to the two conjugate points [ + 1, 0]; the fifth and sixth equations represent a pair of conjugate imaginary curves. The first and fourth curves are each of them symmetric with regard to both axes; the fourth curve is its own inverse with regard to each of the two real circles (X+ )2+X2=, and the first curve with regard to each of the two imaginary circles X2+ (Y+ i)2 = The second and third curves are symmetric with regard to the axis of X, and symmetric to one another with regard to the axis of Y; the fifth and sixth (imaginary) curves are symmetric with regard to the axis of Y, and symmetric to one another with regard to the axis of X. The second and third curves are VOL. II. 4 C 562 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 63. respectively the inverses of the first curve with regard to one of the circles (X-_)2+y2=1, (X+2)2+ Y2=1; each of these curves is also its own inverse with regard to the other of these circles. Similarly the two imaginary curves are the inverses of the fourth curve with regard to the two imaginary circles X2 + (Y+ )2 = -. Lastly, the substitution X = i Y', Y= - iX', changes the first curve into the fourth, the second into the fifth, the third into the sixth, and vice versd. These assertions are the geometrical equivalents of the Theorems ii. and iii. of Art. 53. Let g and g' be any two conjugate divisors of A, and let I be the greatest common divisor of g and g'. Resolve g into the product of two factors 71 and 72, of which y,71 contains only those prime divisors of g which do not occur in a, and 72 contains only those prime divisors of g which do occur in '?. Representing byf(z) the number of numbers prime to any given number and not surpassing it, we write f'(g) = 7f (72), and we define the numbers v, A, B, by the equations 2v = f (,),........... (iii) A= f'(g), g ^/a,........ (iv) B = zf'(g), g VA......... (v) In (iii) the summation extends to every divisor g of A: and in general each termf(i7) occurs twice in If(i), because v is the same for each of any two conjugate divisors; but if A = 02 is a perfect square, the term f'(0) =f(0) occurs only once in f(t?); in the same case we divide the term f'(0) =f(0) equally between A and B, so that in every case A + B = f '(g)= '(A). The first and fourth modular curves are of the order 2A; when A is not a square they are each of them completely and parabolically cyclic, having at each cyclic point v branches of the aggregate order A - B and class B, all touching the line at an infinite distance. When A= 02 is a perfect square, there are at each cyclic point only v - -0' branches, of the aggregate order A + B and class B - -2 Art. 64.] THE MODULAR CURVES. 563 in this case each of the curves has 0' real points at an infinite distance, the asymptotes being Ycos - u-X sin u=Oj i 2 s + 1 2 2 2s+1 (vi) Ysin u +Xcos1 o u u= 0 ' u = 0* * (.v) for the two curves respectively. The points (~+, 0) always belong to the fourth modular curve, but only as isolated points when N is not a square; when N is a square this curve has 0' real branches at each of these points, the tangents being Y sin 1u + (X + I) cos u = 0. The points (~+, 0) and (0, + ) are foci, and indeed the only foci, of both curves; of these the points (0, ~ i) lie on the first curve, and the two real points (+, 0) are its two foci (properly so called), the axis of Y being the only corresponding cyclic axis, or directrix. The fourth modular curve has, in the strictest sense of the term, no foci at all, as the cyclic lines passing through the points [ ~ - 0] touch the curve at those points only. For a demonstration and fuller discussion of the preceding results, the reader is referred to a paper' On the Singularities of the Modular Equations and Curves,' in vol. ix. of the Proceedings of the London Mathematical Society * 64. The Modular Curves of an Even Order. Let I C 1 represent a system of primitive matrices of an even determinant 2' x A, non-equivalent by primary premultiplication, but all of one and the same of the nine types Cij of Art. 23; let also Q = I C I x w. If I a I is a given primary unit matrix the matrices C I x a [ are equivalent by primary premultiplication to the matrices I C I taken in a certain order. It may therefore be proved, as in Art. 52, that any symmetrical function of the 2' x -'(A) quantities (nQ) =I( C x W) is a rational function of ( (w). There are thus nine modular equations of any even order; which we may represent by the formula Fi, j (2A, k2;, x,) = 0. * [Vol. ii. p. 242.] 4C2 564 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 65. Each of these equations is changed into each of the other eight, by four of the thirty-six anharmonic substitutions, and is transformed into itself by the remaining four; viz. if a and b are any two of the six typical units, it may be inferred from the Tables in Art. 25 that a 1 x C, j I x f is of the type Ci,,j for four of the combinations a, i, whence it follows that Fij is changed into Fij,, when we write simultaneously cE (I a | x LQ) for (Q1), and <(| f3 x o) for J (w). In all the equations alike the numerical coefficients are rational, as may be shown by an application of Sohncke's method; for brevity we omit the discussion of the relations subsisting among the coefficients, and of the methods of determining them. Of the nine modular equations three, and only three, are symmetrical with regard to k2 and X2, because the matrices C,7i, C2,2, C,3, and these only, have reciprocals which are of the same type as the matrices themselves. Thus there are three modular curves Fi,,(-+X+Y, I +X-iY) = O, = 1, 2, 3, which we shall designate as the first, second, and third respectively, and which represent (exactly as when the determinant is uneven) the three sets of properly primitive forms of the determinant 2DA. The real relations subsisting between these curves are the same as in the case of an uneven determinant, viz. the first curve is symmetric with regard to the axis of Y, the first and second curves are inverse to one another, and the third inverse to itself with regard to the circle (X-)2+ Y=O; the first and third curves, and the second curve, have the same relation with regard to the circle (X+ )2+ Y2 = 1. The imaginary properties are different; for example, the first curve is transformed into itself by the substitutions X= + iY', Y= iX'; i.e. it is symmetric with regard to each of the imaginary cyclic lines intersecting at its centre. 65. Representation of the Rational Semicircles by the Modular Curves. The following theorem establishes a remarkable connexion between the modular equations of order A and the quadratic forms of the positive determinant A. Art. 65.] THE MODULAR CURVES. 565 Theorem. If the reduced space Z be mapped on the plane XY by the equation ~(^) 2+ X+iY, the reduced arcs of the properly primitive semicircles of determinant A are represented by the first, second, or third modular curve of order A, according as these reduced arcs appertain to the subclass (A), (B), or (C); and when A _ 1, the reduced arcs of the improperly primitive semicircles are represented by the fourth modular curve. Let [a, b, c] be any rational semicircle of determinant A (even or uneven); and let the equation of [a, b, c] be written in the form - a + b (x + iy) + b + c (x + iy) If 2 = (Z+iy), X2 = (-_ x+iy), we may write simultaneously ck2-_+X+iY, X2- +X_Y,because k2, X2 are conjugate complex quantities, Q (c) being a real function of io. b c (i) Let [a, b, c] be properly primitive; when A is uneven the matrix b, a, b is of the type a, a, or r; and when A is even that matrix is of the type C,,, CQ2,, or C3,, according as [a, b, c] appertains to the set (A), (B), or (C) of Art. 41. Hence k2 and X2 satisfy the equation F (k2, X2) =0; or, which is the same thing, the point X, Y lies on the curve Fi (+X+iY, +X-iY)=O, where i is 1, 2, or 3, according as [a, b, c] appertains to the set (A), (B), or (C). (ii) Let [a, b, c] be improperly primitive; A is uneven, the matrix b is of the type 1, and the point X, Y lies on the fourth modular curve. Thus every point which in the plane (xy) lies on a rational semicircle of determinant A is represented in the plane (XY) by a point lying on one of the modular curves of order A. Conversely, the points w, which in the plane xy answer to any given point of one of the modular curves, lie on rational semicircles of determinant A. For if X and Y are real quantities satisfying the last-mentioned equation, + X-iY is one of the modules derived from 566 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 66. - + X + iY by a transformation of order A. Hence, if as before +X+iY=- (x+iy), +X-iY=P(-x+iy), -x + iy and x + iy are connected by an equation of the form a, 6 -x+ty=, ' x (x + y), a,f3 the quantities x and y being real, and a being a matrix of determinant A. Separating in this equation the real and imaginary parts, we find a =, + 2aX +/(X2 +y2) = 0, or x + iy lies on the semicircle [y, a, /3] of determinant a2 - / = A. When two modules, either of which can be transformed into the other, are conjugate complex quantities, the transformation is said to be symmetrical. It is evident that all modules represented by points appertaining to modular curves of order A, and these modules only, admit of symmetrical transformation by a matrix of determinant A. 66. Case when the Determinant is not a Square. Let A be not a square; and let a point co describe in the reduced space 2 the period of reduced arcs appertaining to any given subclass of semicircles of determinant A; the corresponding point X, Y will describe a continuous closed contour in the plane (XY) forming one of the ovals of which the corresponding modular curve is composed. Since no semicircle of determinant A passes through any one of the points p, q, r, no oval of the modular curve can pass through either of the points A_1, A +, or can have a point at an infinite distance. The portion of an oval terminated at each extremity by one of the boundaries (-oo, A_1), (A+1, + o) we term a ilcament. The filaments are of six types, four transitive and two intransitive, corresponding to the six essentially different types of reduced arcs (Art. 43). (1) An intransitive filament [S_&, S+1] lies wholly on the left-hand side of the axis of Y; it begins at a point on the boundary [- o, A_,], and enters * By an oval of a modular curve we understand a distinct and separate portion of the curve forming a continuous and closed series of real points. The term oval is suggested by the form of the portions of the curve in the simplest instances; but an oval may intersect itself any number of times, and (when the determinant is a square) may have points at an infinite distance. Art. 66.] THE MODULAR CURVES. 567 the lower region of the plane XY (i. e. the region in which Y is negative); it afterwards bends upwards and traverses the segment A_10, curving round so as to terminate in a point (in general different from that at which it began) upon the boundary [- so, A_1]. The intransitive filaments [Ti, T+,] are symmetrical with regard to the axis of Y to filaments of the type [S-1, S+,]. (2) A transitive filament [S_1, T_j] begins at a point on the boundary [- c, 7 A _] and enters the lower region (which it never leaves) of the plane XY; it traverses the axis of Y, and terminates in a point on the boundary [A,, + oo]. A transitive filament [S_1, T+1] begins at a point on the boundary [- cx, A_.] and enters the lower region of the plane; it afterwards traverses the segment A_1, A+,, and the axis of Y, and terminates in the upper region of the plane at a point on the boundary [A +,, + co ]. Such a filament always has a point of inflexion; it may traverse first the segment A_1, A +1 and afterwards the axis of Y, or vice versa; or it may cut them both simultaneously, passing through the point 0. The transitive filaments [S+4, T+i], [S +, T_ ] are symmetrical with regard to the axis of X to the transitive filaments [S_1, T_,], [S_1, T+1] respectively. As in the case of reduced arcs, if we change the order of the letters in any type symbol we do not change the type of the filament, but only the direction in which it is drawn. Every oval contains at least two transitive filaments, and its general form is as follows:- Beginning, as we may suppose, with a transitive filament of the type TS, it winds a certain number of times round the focus A_,; it then crosses the axis of Y by a second transitive filament, and winds a certain number of times round the focus A +1: it thus keeps crossing from one focus to the other, and winding round each of them alternately until it finally closes. Combining this discussion with preceding results, we see that given any oval, we can, from a consideration of the filaments of which it is composed, construct the complete system of reduced quadratic forms appertaining to the subclass represented by the oval. Beginning with a transitive filament (TS), let, - 1 be the number of non-transitive filaments immediately following it; let the next transitive filament be followed by non-transitive filaments, and so on continually. Attribute to a, u3,... a positive or negative sign according as the revolution round A_ - is positive or negative; and to x21, 4, I... a positive or negative sign according as the revolution round A +1 is negative or positive. The fundamental automorphic of the reduced form, represented by the first 568 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 67. transitive filament, is I2 x 2 j2Xx 2 t3Xr 12l4X..., and the determination of this automorphic is equivalent to a determination of the period of reduced forms represented by the oval. We may therefore say that the modular curves of order A present us with a species of chart of the reduced forms of determinant A, each oval representing a subclass, each filament a reduced form, and the periodic continued fraction corresponding to each subclass exhibited by the aspect of the oval. The number of ovals contained in each of the first three modular curves is evidently (Art. 41) 3 H, and H' in the fourth. 67. Case where the Determinant is a Square. Let A = 02 be a perfect square; and let J represent the number of solutions of the congruence a2+1-, mod20, so that J =0, if 0 is divisible by 2 or by any prime of the form 4m+ 3. As in the former case each modular curve consists of a certain number of ovals; but, in certain cases, an oval corresponds to more than one chain of reduced semicircles; viz. the oval may be divisible into two, three, or six sections, each section answering to a single chain of reduced arcs. Each oval is subject to the condition that it must pass either through the point A_1, or through the point A_2, or else have a point Po at an infinite distance. If the oval satisfies this condition only once, it is a simple oval, and represents but one chain of reduced semicircles; if it satisfies the condition more than once, it is a multiple oval, and corresponds to as many different chains of reduced arcs as it has branches passing through A_1, A,,, or points at an infinite distance; each section of the oval beginning and ending at one of the points A_1, A+1, Am. Each of the first three modular curves has S simple ovals and (H- 3 ~) double ovals; the ovals of the first curve having one or two infinite branches, and those of the second and third curves having one or two branches passing the points A_1 and A_2 respectively. Lastly, the fourth modular curve (which exists only when 0 is uneven) contains ~ triple ovals and 7 (H'-3 ) sextuple ovals; each oval having, according as it is triple or sextuple, one or two branches of each of the types A_, A +, A,. The two, three, or six subclasses represented by different sections of the same multiple oval always appertain to the same Art. 67.] THE MODULAR CURVES. 569 Gaussian class; and it is to be observed that if the extremities of a section lie at the same point A_, or A +, they have different tangents, and if they both lie at an infinite distance they belong to branches having different asymptotes. It will suffice to establish the preceding results in the case of the fourth modular curve. Consider the pairs of extreme semicircles [2a, b, 0] and [2a, b, o] x I o-1 =[2a - 2b, b, O], [0, b, 2a] and [0, b, 2a] x i| 1-1 = [O, b, 2a - 2b], [2a- 2b, b-2a, 2a] and [2a- 2b, b-2a, 2a] x I = [2a, 2a - b, 2a -2b], which are all improperly primitive, if b is uneven, and reduced, if a is less in absolute magnitude than b. It appears from a preceding investigation, that if ac describes either semicircle of one of these pairs up to its extreme point, and then passing through the extreme point describes the other semicircle of the same pair, the corresponding point (XY), determined by the equation - + X + i Y = i) (co), will describe a line of continuous curvature passing through A_1, A +, or a point at an infinite distance, as the case may be. Thus a semicircle P1 of the type (r), ending a chain P of the type (qr), is continued by a semicircle Q0 of the same type (r), which begins a chain Q of the type (rp); the chain Q ends in a semicircle Q1 of type (p), which in its turn is continued by a semicircle Ro of the same type (p) beginning a chain R of the type (pq). Lastly, the semicircle R, of type (q), which terminates the chain R, is continued by a semicircle P' of the same type, beginning a chain P' of the original type (qr). The chain P' is equivalent to the chain P by a transformation of the type - x * x r = q,; but the two chains may also be primarily equivalent to one another. If they are primarily equivalent the oval closes, and is triple; if they are not primarily equivalent we begin with the chain P', and perform the preceding process over again; we thus arrive at a chain P", which is certainly equivalent to P, because,2= - 1 in this case the oval is sextuple. To determine whether P, P' are primarily equivalent or not, we have only to examine whether P0 is or is not equivalent to itself by a transformation of type \. Now P0 is of the type [4, 0, 0]; and this is transformed into -24, 0 -[40, 0, 0] by 1+4 42; i.e. there are as many triple ovals as there are solutions of the congruence VOL. II. 4 D 570 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 68. 4 2 +1 -0, mod 0, or, which is the same thing, of the congruence 2+1 =0, mod2 0. It will be observed that two sections of an oval which meet at either of the points A_, or A + are analytically inverse with regard to the other of those two points; and that two sections which have the same asymptote have the origin for a common centre. In fact, when A is an uneven square, each oval of the modular curve taken separately has the properties of symmetry and anallagmatism which in the general case characterize the curve as a whole. 68. Geometrical Construction of the Transformed Modulus by means of the Modular Curve. Let P be a given point in the plane of a geometric curve C of order n; each of the imaginary cyclic lines passing through P meets the curve in n - e imaginary points at a finite distance, if the curve has branches of the aggregate order e passing through each cyclic point. The n - e real lines on which these n-e pairs of conjugate imaginary points are situated, are the cyclic axes of P with regard to the curve; and the n -e points inverse to P with regard to these lines are termed the points inverse to P with regard to the curve; or, which is the same thing, the points inverse to P with regard to the curve are the points which, taken one by one with P, are the antipoints of pairs of imaginary points lying on the curve. If P is a focus of the curve, or, again, if P is one of the antipoints of a pair of conjugate imaginary multiple points of the curve, some of the cyclic axes of P, and some of the points inverse to P, may become coincident. Theorem. Let f (4, n)= 0 be an algebraic equation between ~ and; of the order /u in,: and let,=x8+iy8, s=1, 2, 3,...,ux, be the u values of r corresponding to the value x + iy of i; if the equation f(x + iy, x - iy) = C= 0 is real, the points xs - iys are the inverses of x + iy with regard to the curve C. The condition that C= 0 is real implies that f(~, v) is of the same order, in ~ that it is in; and that = n - e. Again, the point T, of which the coordinates are 2 (x + xS) + 2i(y + Ys), =2 (y - y) - i (X -), is an imaginary point on C; the conjugate point T' is therefore also a point on C: but the antipoints of T, T' are the points x + iy, X8 - iya. Art. 68.] THE MODULAR CURVES. 571 The condition that C=0 should be real is that f(4, n) should be of the form f + i( - )f2, f, and f2 denoting symmetric functions of ~ and. If this condition be not satisfied, the theorem still subsists if we replace the curve C by two conjugate imaginary curves, each of which has to be regarded as cut by one of the cyclic lines passing through the point (x, y). We obtain from the theorem the following construction for the transformed modules corresponding to any given modulus P. Let Q1, Q,... be the points inverse to P with regard to one of the modular curves; and Q_, Q-,... the points inverse to Q,, Q,... with regard to the axis of X; the transformed modules determined by the modular equation corresponding to the curve considered, are represented by Q_,, Q_2,.. The following general theorem, which, however, must be taken as subject to certain limitations, may serve to show the importance of inverse points in the theory of functions. 'If the plane (xy) be mapped upon the plane X Y by means of the equation X+ iY=f(x + iy), points inverse with regard to any curve are transformed into points inverse with regard to the corresponding curve.' Let x = a(0), y =- (0), where 0 is a variable parameter, be the equations of a curve in the plane (xy); let also X+iY=f[a(0)+if3(O)]=A(0)+iB(0),......(i) so that X = A (0), Y= B(0) are the equations of the transformed curve. If 01, 02 are a pair of conjugate imaginary quantities, a(0,), 3(0,); a(02), 3(02); or (al, l), (a, /32), are a pair of conjugate imaginary points on y; the corresponding points on r we may represent by (A,, B,), (A2, B2). The coordinates of the antipoints of these two pairs of imaginary points are obtained by equating real and imaginary parts in the equations xl+ 1 iyl = al + i; x2+ iy2=a2 + i32; X,+iY1=Al+iB,; X,+iY2=A2+iB2. For each of these pairs the equation (i) gives X+iY=A +iB, which establishes the theorem. 4D 2 572 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 69. In the demonstration the functions a(0), f3(0) are supposed to be real when 8 is real; the three functions a(0), p3(0), f(x+iy) are also regarded as monodromic throughout the portion of the plane xy considered. The theorem may be regarded as defining for any Gaussian transformation X + iY=f(x + iy) the relation between the imaginary points appertaining (in the sense of coordinate or projective geometry) to any figure in the given plane, and the corresponding imaginary points in the plane (XY): viz. this relation may be expressed by saying 'corresponding pairs of imaginary points have corresponding antipoints,' or, again, by saying 'the imaginary cyclic lines correspond to one another in the two planes.' By virtue of the theorem, the construction for the transformed modules given above is an immediate consequence from Art. 39, Theorem III. 69. Theorems relating to the Multiplier. Theorem I. The multiplier corresponding to a symmetrical transformation of determinant n, has Vn for its analytical modulus. Let k2 =(o))= +X+iY, X2= q(Q2)=+X-iY, o=\A I x Q, where IA I is a matrix of determinant n; if w = - x + iy, then Q is equivalent to x + iy, and the equation between co and Q2 assumes the form - x+y= a + b (x + iy) b + c (x + iy)' 'a, b, c being integral numbers satisfying the equation b2- ac = n. Let M=p+i q, K=A+iB, iK'=(-x+iy)(A+iB)=A'+iB', the quantities p, q, A, B, A', B' being real; then A= K(x+ iy)= A-iB, iA' = - A' + iB'. Equating real and imaginary parts in the equations (Art. 26, ii) (p+qi)K =bA+ciA'. (p + qi) K'= aA +biA', we obtain the system (p - b)A + qB -cA'=O, -qA -(p+b)B -cB'=O, -aA + (p+ b)A'-qB'= o, aB+ qA'-bB'=O, Art. 70.] THE MODULAR CURVES. 573 of which the determinant is (p2+q2-n)2. Since A, B, A', B' cannot be all equal to zero, the determinant must vanish, and we have,/p2 +22 = 2/n. The same result might be obtained from the equation of Jacobi (Art. 61) which, in the case here considered, assumes the form 1 'b(-x+iy)k(-x+iy)q('( x+iy)d( x+iy) _2 =n>( x+iy)k( x+iy)P'(-x+iy)d(-x+iy)' P-Qi P+Qi' when P and Q are real quantities. Thus, while the point representing - 2 + k2 describes an oval of the modular curve, the point representing the corresponding multiplier describes an arc of the circle X2 + Y2 = n; this point, however, does not always move in one direction on the circumference of the circle, and the stationary points, at which the direction of its motion is reversed, answer to values of -- /nei, for which two roots of the equation g (, k2) = 0 become equal. Theorem II. The square of the multiplier appertaining to the transformation of any given modulus into its conjugate is n x e2i(a + '-0) when n is the order of the transformation and (the given modulus being represented by the point P on the modular curve) a, a', and 0 are the angles made with the positive axis of X by the focal radii vectores of P, and by the tangent at P respectively. This is the geometrical interpretation of the equation of Jacobi in the case in which the transformation is symmetrical. 70. Symmetry of the Modular Curves. The first and fourth modular curves are symmetric, each to itself, with regard to both the axes; the third and fourth modular curves are symmetric, each to itself, with regard to the axis of X, and each to the other with regard to the axis of Y. In certain cases, which we proceed to enumerate, ovals appertaining to the modular curves are symmetric with regard to one or both axes, or have the origin for a centre. 574 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 70. I. Ovals symmetric with regard to the axis of X. If [a, b, c] is a semicircle appertaining to the period of an oval symmetric with regard to the axis of X, [a, - b, c] is also a semicircle appertaining to the period; i. e. either (1) the forms (a, b, c) and (a, - b, c), or else (2) the forms (c, b, c) and (-a, b, -c) are primarily equivalent. The former hypothesis is inadmissible in the case of improperly primitive forms, the latter in the case of properly primitive forms. Theorem. If any one subclass of forms contained in a given class is primarily ambiguous, every subclass contained in that class is also primarily ambiguous' For if (a, b, c) is transformed into (a, - b, c) by the primary matrix I 0 and into (A, B, C) by any unit matrix a,,then (A, B, C) is transformed into (A, - B, C) by the primary matrix | *d jx x |0 x1 -7, a -7, Hence each of the first three modular curves contains the same number of ovals symmetric with regard to the axis of X. (A) Properly Primitive Classes. (1) Let U, the least number which satisfies the equation T2- DU2 = 1, be uneven. Then every ambiguous class contains three subclasses, each of which is primarily ambiguous. For every ambiguous class contains forms of one or other of the types (a, 0, c) or (2a, a, c). Of these (a, 0, c) is identical with its opposite, and therefore primarily equivalent to it; and (2 a, a, c) is transformed into (2c, -a, c) by T- aU, -cU, 1, -1 2aU,, T+ aU, 0, 1 '** (i) which is primary. (2) Let U, be even. In this case no ambiguous class can contain forms of each of the types (a, 0, c) and (2 a, a, c). Ambiguous classes containing forms of the first of these types are, as before, primarily ambiguous; but the matrices (i) are no longer primary, and hence ambiguous classes containing forms of the type (2 a, a, c) are not primarily ambiguous. When D _=1, 2, 4, 5, 6, mod 8, there are no forms of this type; when D 0, 3, 7, mod 8, one half of the ambiguous classes contain forms of each of the two types. Art. 70.] THE MODULAR CURVES. 575 Combining the two results, we find that each of the first three modular curves contains as many ovals symmetric with regard to the axis of X, or half as many, according as D 1, 2, 4, 5, 6, mod 8, or D =0, 3, 7, mod 8. The symmetry is of the first species*; for if [a, b, c] is a transitive semicircle of the period, so is also [a, - b, c], which is symmetrical to it. Then two semicircles occur, one in an even, the other in an uneven place in the period; i. e. one is described in the direction [TS], the other in the direction [ST]. (B) Improperly Primitive Classes. If (a, b, c), any form of determinant D, is equivalent to (-a, b, -c), we have T2-Du2 = -1; and conversely, if T2- Du2= -1, every form (a, b, c) of determinant D is equivalent to (- a, b, - c); and, if the forms are improperly primitive, the equivalence is primary, for T-bu, cu Im.. au, - T- b*u *....... (i) which transforms (a, b, c) into (-a, b, -c), is primary. Hence when the equation T2-D2=- 1 is resoluble, every oval of the fourth modular curve is symmetric with respect to the axis of X; but, except in that case, no oval of the curve possesses this symmetry. The symmetry is of the second species; for the semicircles [a, b, c], [-a, b, -c] occur either both in even places, or else both in uneven places. II. Ovals symmetric with respect to the axis of Y. If [a, b, c] is any semicircle of a period, the oval corresponding to the period is symmetric with respect to the axis of Y, (1) if (a, b, c) is primarily equivalent to (c, b, a), (2) if (a, b, c) is primarily equivalent to (- c, - b, - a). In either case the extreme coefficients must be either both even or both uneven; so that ovals symmetric with regard to the axis of Y exist only in the first and fourth modular curves. (A) Properly Primitive Classes. (1) If (a, b, c) is primarily equivalent to (c, b, a), the class containing (a, b, c) must be ambiguous, and (a, b, c) must be equivalent to (a, - b, c) by a trans* When any closed line is symmetric with regard to an axis, a point describing the line continuously may either describe corresponding portions in corresponding directions, or it may describe them in opposite directions. In the latter case the symmetry is said to be of the first species, in the former of the second. The symmetry of an ellipse with regard to either axis is of the first species; the symmetry of a hyperbola, or of a lemniscate, with regard to one axis is of the first species, and with regard to the other of the second. 576 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 70. formation of the type x. It may be inferred that (a, b, c) is equivalent in this manner to (a, - b, c), (a) when U1 is uneven (see I. (A) (1) supra, and observe that (a, b, c) has automorphics of the type \4); (I) when U1 is even, and the class (a, b, c) contains a form of the type (2 a, a, c); viz. in this case (a, b, c) is not primarily equivalent to (a, - b, c) (see I. (A) (2) supra), and must therefore be equivalent to it by a transformation of the type X. (2) If (a, b, c) is primarily equivalent to (- c, - b, - a), then (a, b, c) is equivalent to (-a, b, -c) by a transformation of type +, and the equation T2- D2=- 1 is resoluble, the values of U being necessarily uneven. Conversely, if this equation be resoluble, every form (a, b, c) of determinant D, with its extreme coefficients uneven, is primarily equivalent to (- c, - b, - a). Combining these results we find (1) that if D)0, 3, 7, mod 8, the first modular curve has one half as many ovals symmetric with respect to the axis of Y as there are properly primitive ambiguous classes of determinant D; and (2) that if the equation T2- Du2- 1 is resoluble, all the ovals of the first modular curve are symmetric with respect to the axis of Y. The symmetry is of the first species in the first case, and of the second species in the second case; the two cases cannot coexist, because D 0, 3, 7, mod 8 in the first case, and D 1, 2, 5, mod 8 in the second. (B) Improperly Primitive Classes. The forms (a, b, c) and (-a, - b, -c) cannot be primarily equivalent. If (a, b, c) is primarily equivalent to (c, b, a), the class containing (a, b, c) is ambiguous, and therefore contains a form of the type (2a, /, 2a). If U1 is uneven, the class contains only two subclasses; these we may represent by (2 a, /3, 2 a) and (2 a, -, 2 a), since these two forms are certainly not primarily equivalent. If U1 is even, the class contains six subclasses; these we may represent by the forms (2a, ~+3, 2a), (2a, ~13, 2a)xp, (2a, ~+3, 2a)Xp2. If (a, b, c) is one of the four last of these forms, (a, b, c) is not primarily equivalent to (c, b, a); for example, if (a, b, c)= (2a, {, 2 a) x p, then (a, b, c) is transformed into (c, b, a) by transformations of the type p, and of no other type. Hence every ambiguous class (whether U1 is even or uneven) contains two and only two subclasses (a, b, c), such that (a, b, c) is primarily equivalent to (c, b, a). We infer therefore that the fourth modular curve contains as many ovals Art. 70.] THE MODULAR CURVES. 577 symmetric with respect to the axis of Y as there are ambiguous classes. The symmetry is of the first species. III. Ovals symmetric with respect to both axes. Such ovals exist only in the first and fourth modular curves, and in them only in the following cases: (1) When the equation T2 - D2 = 1 is resoluble, every oval in the first or fourth curve respectively is symmetric with regard to the axis of Y or of X. Hence all ovals of the first curve which are symmetric with regard to the axis of X, and all ovals of the fourth curve which are symmetric with regard to the axis of Y, are symmetric with regard to both axes; the symmetry with respect to the two axes being dissimilar. (2) If in the equation l'2-Du2= 1, U1 is uneven, every oval in the first modular curve which is symmetric with regard to either axis is also symmetric with regard to the other axis. The symmetry is of the first species with regard to both axes. It will be observed that, when U1 is uneven, the equation 2 - Du2 = - 1 cannot be resoluble. IV. Central ovals. The ovals considered in III. are of course central; we need therefore only consider here those ovals which are central, but symmetrical with regard to neither axis: central ovals can, of course, only exist in the first and second modular curves; and the oval is central, if any form (a, b, c) belonging to it is primarily equivalent either to (c, - b, a) or to (- c, b, - a). (A) Properly primitive Classes. (1) If in the equation T2-Du2=, U1 is uneven, every form (a, b, c), of which the extreme coefficients are uneven, is primarily equivalent to (c, - b, a); for every such form (a, b, c) has automorphics of the type +J. (2) When D is the sum of two squares prime to one another, a certain number of classes of determinant D are such that (a, b, c) and (- c, b, -a) are equivalent; viz. the number of such classes is equal to the number of ambiguous classes: the two sets of classes being entirely distinct when the equation T2- Du2 = 1 is irresoluble, and coinciding when that equation is resoluble. It will be observed that, in the case here considered, D 1, 2, 5, mod 8, and that U, is therefore even. Thus, if U1 is uneven every oval of the first modular curve is central, and of the first species; if D is the sum of two squares relatively prime, there VOL. II. 4 E 578 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 71. are half as many central ovals as there are ambiguous classes. The symmetry is of the second species: if the equation T2 - Du2- 1 is not resoluble, the central ovals are entirely distinct from the ovals representing ambiguous classes; but if this equation is resoluble, the two sets of ovals coincide. (B) Improperly Primitive Classes. Here the forms (a, b, c) and (c, -b, a) cannot be primarily equivalent; hence the only case in which the fourth modular curve contains central ovals is that in which the determinant is uneven and is the sum of two squares relatively prime. In this case the number of central ovals is the same as in the first modular curve (viz. half as many as the improperly primitive classes); and (as in the case of the first modular curve) the central ovals are simply central, or are symmetric with respect to both axes, according as the equation T2 - Du2 - 1 is not or is resoluble. For a fuller explanation of the arithmetical principles upon which the preceding discussion of the symmetry of the ovals is based, the reader is referred to a memoir, ' Note on the Theory of the Pellian Equation and of Binary Quadratic Forms of a Positive Determinant,' in the Proceedings of the London Mathematical Society, vol. vii. pp. 196-208 *, where, however, the equivalence considered is general, and not as here primary. 71. Transformation of a Modular Curve by Inversion with regard to another Modular Curve. Let IHI be a given matrix of determinant h, and let = HI x 0; if describes a semicircle C of determinant A, then Q describes the semicircle C x H, derived from C by applying to it the transformation I H 1. Thus any transformation of a quadratic form by a linear substitution of which the determinant is greater than unity, gives rise to a transformation of the modular, curve answering to the quadratic form. We have in fact the general theorem, 'The inverse of any modular curve with regard to itself or any other modular curve is a modular curve, or is composed of modular curves.' For the arithmetical theory itself we refer to the 'iDisquisitiones Arithmeticae,' section 5; to a memoir by Professor Lipschitz (Crelle, vol. liii. p. 238); and to the Report on the Theory of Numbers (Reports of the British Association for 1862, p. 503, Art. 106 et seq.) t; confining ourselves in this place to a few examples in illustration of the geometrical theorem. * [Vol. ii. p. 148.] t LVol. i. p. 231.] Art. 71.] THE MODULAR CURVES. 579 To the symbol (-) we attribute the value +1, -1, or 0, according as A is a quadratic or a non-quadratic residue of p, or is divisible by p. Let the modular curves of any given determinant n be denoted by Fs (n), s= 1, 2, 3, 4; let A and h be both uneven, and h 1, mod 4; in the equation T2 _A U2 = 1, let U, be the first of the numbers U1, U,... which is divisible by h; and let p be any prime divisor of h. Let P be a point on F1 (A), and let the points inverse to P with regard to the curve F4 (h) = 0 be denoted by Q1, Q2... There are in all h (l + -) points Q; of these m = hI 1 - (- describe the curve F1 (ah2) while P describes F1 (A); viz. F1(Ah2) has m times as many ovals as F1 (A), and while P describes any oval of F1 (A), the points Q describe adifferent ovals of F1 (A h2), a points describing each oval simultaneously. When A is a non-quadratic residue of every prime p, F (Ah2) is the complete inverse of F1 (A) with regard to F4 (h). In every other case the complete inverse includes certain modular curves F1( \2 ), 2 being a divisor of A h2 and containing only primes which divide h. For brevity we confine ourselves to the particular case in which h=p is a prime and () = +1. In this case there are p + 1 points Q, of which p - 1 describe the curve F1 (Ap2), and two, Q1, Q2, the curve F1 (A) itself. Thus the inversion of F1 (A) with regard to F4 (p) gives a 'one to two' transformation of F1 (A) into itself. The prime p is always represented either by an ambiguous form of determinant A, or by each of two opposite and non-equivalent forms. (1) If p is represented by an ambiguous form of determinant A, p2 is represented by the principal form, and while P describes any oval S of F1 (A), the two points Q1 and Q2 both describe another oval S:, and both describe it in the same direction, never meeting one another, although they may cross at a double point. In particular, if p is itself represented by the principal form of determinant A, S1 is the same as S. Whether S, coincides with S or not, the transformation may be repeated ad infinitum; viz. if the points inverse to P are P_1 and P1, then the points inverse to P1 are P and P2, the points inverse to P2 are P1 and P3, and so on continually, the points P, PI, P2... forming a series which never closes, and lying alternately on S and S. when those two ovals are different. (2) If p is not represented by an ambiguous form, the ovals described by Q1 and Q2 4E 2 580 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 72. are different, except when the equation T2- AU2= -1 is resoluble; in which case Q1 and Q2 describe the same oval, but describe it in opposite directions. Let p be represented by a fundamental form f of the principal genus; i. e. by a form which compounded with itself gives all the forms of the principal genus (we suppose the determinant A to be regular); and let v be the number of forms in the principal genus; the points P, P1,..., P,-1 will all lie on the v different ovals appertaining to one and the same genus, and the point P, will lie on the same oval as P. In fact if that oval represent the form r, the points P,... P,,- lie on the ovals answering to the forms xf, P xf2,..., rxfn-l. We may observe (3) that the number v of ovals which are thus cyclically changed into one another is always the number of forms in the composition period of the form f; i. e. the index v for which fv becomes the principal class; (4) that if instead of F4 (p) we use F4 (p2), or any modulus curve of an uneven square determinant prime to A, as the transforming curve, only ovals of the same genus can be transformed into one another. The preceding discussion may suffice to show that the arithmetical theories of the division of the classes into genera, and of the composition of classes, have a certain geometrical signification in the theory of the modular curves. But, so far as is yet known, there is nothing in the aspect of the ovals to indicate the order in which they are related to one another by composition, or their distribution into genera. 72. Multiple Points and Points of Intersection. A point w, defined by a quadratic equation, of which the determinant is negative, and the coefficients integral, may be termed a quadratic point; and the same designation may, for brevity, be applied to the corresponding point - + (w) in the plane (XY). We do not regard the cornicular points in the plane (xy), or the points corresponding to them in the plane XY, as quadratic points. The determinant of a quadratic point w, or - - + I (w), is the determinant aTy-_2 of the primitive quadratic equation a+23co)+T7y2=0 by which ) is defined. The theta and elliptic functions of which the modules are quadratic admit of complex multiplication; and, conversely, if the functions admit of complex multiplication, the modulus is quadratic. Let aT - j2 = A, r2 + V2A = n, the transformation of order n, I - va, - +v Ia, T + V,3 Art. 72.] THE MODULAR CURVES. 581 by which (a, /, y) is changed into n x (a, P, y), does not alter the value of o, and consequently leaves K (co), iK'(w), (I(co), F'(() unchanged: the multiplier is the complex number defined by the equation 1 1 1 + M=-[1 +3 O + V7r] [T + i VA]; i. e. the transformation is equivalent to a multiplication of the argument by a complex number. For an account of the researches to which the theory of complex multiplication has given rise we must refer to a note of Abel (' (Euvres,' vol. i. p. 272), to the Memoirs of MM. Kronecker and Hermite, and to the Report on the Theory of Numbers (Arts. 126 and 130-138*). Theorem. 'Every real point of intersection of a modular curve with itself; or with another modular curve, is a quadratic point; and an infinite number of modular curves can be drawn through any quadratic point in the plane (XY).' For if Z be a point at which two branches of a modular curve, or of two different modular curves, intersect, the corresponding reduced semicircles C6 and C2 intersect in a quadratic point z lying in the reduced space; through this point there pass an infinite number of rational semicircles X\C +X2; i. e. there pass through Z an infinite number of modular curves. Theorem. 'The inverse of any quadratic point in the plane XY with regard to any modular curve is a quadratic point; and the determinants of the two points are to one another as two squares.' If - + (2 + iy2) - 2 + (x1 + iy) of the determinants A2, A, are the two points, we have -x + y2 c, d +(xl+iyl); whence, if either of the two points is quadratic, the other is so too; and if D2 ad -be = D, A =- -A2, 2 being a square divisor of D2A2, containing only primes which divide D. Theorem. 'If two points are inverse to one another with regard to two different modular curves, or are doubly inverse with regard to the same curves, they are quadratic points.' + a, b a', b' For if - x2 + iy=, d x (xi + iy1) = ', x (x, + iy), the two matrices Cy d C d C [* Vol. i. pp. 303 and 321.] 582 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 72. being different, we have d,b b a, b ', Xi +it = x C, ' x x(+ iy); Xi i -, a c, d z. e. xl + iyi is a quadratic point, and therefore also x2 + iy. Theorem. 'If the determinants of two quadratic points are to one another as two square numbers, the two points are inverse with regard to an infinite number of modular curves.' For if the determinants of D (x + iyi) and (x 2+iy) are as two square numbers, we can always, and in an infinite number of ways, establish between - x2 + iy2 and x1 + iy, an equation of the form a, b x2 + iY2 = c, d x (x + i ) X2+2"2= C 6^ di); viz. if A,, A21 are two matrices satisfying the equation, it is also satisfied by the matrices X A1 + X2A2\. For brevity we omit the demonstration, which depends on the theory of the composition of quadratic forms. A complete discussion of the intersections and multiple points of the modular curves, involving a systematic examination of the properties of the quadratic modules, is not exempt from difficulty, and would surpass our present limits. We confine ourselves to a few observations which may serve to show the general character of the theory. (i.) The modular curves have no superlinear branches, except those at the cyclic points, and at the points (~, 0) or (0,+ ~ ), which arise from the cornicular values of c. (See the 'Memoir on the Singularities of the Modular Equations and. Curves,' Proceedings of the London Mathematical Society, vol. ix. pp. 242-272.) Again, except at the points (+~, 0) no modular curve can have a conjugate point (i. e. a real point) with pairs of conjugate imaginary branches passing through it. For if the point o corresponding to the real point - + 1 (w) be not a cornicular point, it may be shown that as many real semicircles (appertaining to the modular curve) pass through o as there are branches of the modular curve passing through - + $ (c). In speaking of the intersections and multiple points of the modular curves we shall henceforward exclude the cyclic points and the two pairs of points (~12 0), (0, ~2 i) [* Vol. ii. p. 242.] Art. 72.] THE MODULAR CURVES. 583 (ii.) Two branches of a modular curve, or of two different modular curves, cannot touch. For real points this is evident, since two rational semicircles cannot touch, except when the determinant is a square; even in this case, semicircles passing through one of the cornicular points correspond to curves in the plane XY which do not touch. But the same thing is true for imaginary points. At a point of intersection, whether real or imaginary, we must have a,, bi. c I, hd, nc, d2 Q-|,| x co= 'dxw; the two matrices not being equivalent by permultiplication, or else the two branches would be the same. If there be contact,,() x -d, and consequently 4 (W) dco d, must have the same value for each of the two determinants of 2. We dw thus obtain D1 _ D2 (a+ b )2O- (a2+f bCO)2' )D, D2 being the determinants of the two matrices. But this equation implies that w is real. The same conclusion might be obtained by considering the multipliers appertaining to the two transformations. (iii.) Let D1, D2 be two uneven numbers relatively prime, and let us consider for simplicity the modular curves of the fourth species only. The equation equaiF(+X, 1+X, D1xD2)=0 determines the intersections of F(Dj x D2) with the axis of X; viz. the real values of X give the real intersections of F(Dj x D) with the axis, and the imaginary pairs of values give the real antipoints of the pairs of imaginary intersections. The system of points (X) thus obtained, consisting of points on the axis of X, and of pairs of points symmetrically situated with regard to the axis of X, is the same as the system (Y) consisting of the real intersections of the curves F(D1), F(D), and of the antipoints of their imaginary intersections; i. e. if P1 is any point of (X), P1 has, with regard to F(D1) and F(D2), a common inverse Q1 which is itself a point of (X); and Pi, Q, are the antipoints of an imaginary pair of intersections of F(D1) and F(D2). 584 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 72. If P2, Q2 are the points symmetrical to P1, Q1 with regard to the axis of X, then P2, Q2 belong to the system (X), and are the antipoints of another pair of imaginary intersections; if P1 coincide with Q1, and P, with Q2, we have two real intersections instead of two pairs of imaginary intersections: if Pi coincide with Q2, and P2 with Q1, we have a single pair of imaginary intersections on the axis of X. To establish this theorem we observe that F(X2, k2, D1xD2)=0 is the resultant of the elimination of z from F(D2, X2, z) = 0 and F(Dj, z, k2) = 0. Hence F(D1 x D2, k2, k2) =0 is the resultant of the elimination of X2 from F(D,, \2, k2) = 0 and F(D1, X2, k2)= 0. And, in accordance with this result, it can be shown by a method due to M. Kronecker, that, if Di1, I D2, and I -D x D2 denote complete systems of primitive and primary matrices for the determinants D1, D2 and D1 x D2, the two equations n [(c) - (l D xx D2> x)] = o....... (i) n [( D 1 x co) - ( Dx2 )] =....... (ii) (of which the first is F(DxD D, k2, k2)=0, and the second is the resultant of F(D1) and F(D2)) give precisely the same values for I(co); and give each value with the same multiplicity, the cornicular values being always excepted. It can also be shown that if 0 is a value of co satisfying the two equations, so that one of the quantities D1 x 0 is equivalent to one of the quantities | D2 x 0, either of these equivalent quantities is itself a value of c satisfying the two equations. These considerations suffice to establish the identity of the system (X) with the system (Y). But the way in which the points of the system (X) are paired so as to form the system (Y) depends on the theory of the composition of quadratic forms; viz. if the system (X) contains a group of points (A), having the determinant -, then Di and D2 can be represented by one and the same form -7 of that determinant, and the pairing of the points (A) in the system (Y) depends on the composition of the forms corresponding to the points (A) with successive powers of 7. The same principles are applicable to the case in which D1 and D2 are not relatively prime. But in this case the system (Y) includes, besides the system (X), the systems (X'), (X"),... representing the intersections of the curves, F(DJ2), F( /...;, "...;, being common divisors of D1 and D2, and each of the systems (X'), (X"),... having in (Y) a certain multiplicity. Art. 73.] THE MODULAR CURVES. 585 To find the multiple points of the modular curve F(D)= O, we have to compare the discriminantal equation II [E (I D I x wo)- ( D' Ix c))]2 = 0, where ID, D'I are any two different matrices of the complete system, with the system of equations where d is any divisor of D; and the system of the discriminantal points and antipoints is the same as the aggregate of the systems representing the intersections of the curves F(d2) = by the axis of X, each such system being taken with a certain multiplicity. (iv.) When two modular curves intersect at a real point, the angle of intersection is the same as for the corresponding semicircles. Let [A, B, C], [A', B', C'] be two semicircles of determinants D and D' intersecting at the point [a, f3, y]; let =AC' -2BB''+A'C; and let v be the greatest common divisor of the matrix A B, C', so that AA,' BL>, C uX,.y, A, B, C v X |7, -a l= A', B', C' lastly, let ' be the external angle of intersection of the semicircles; we then have cos / DD sin If therefore the determinant A of a point of intersection of two modular curves of order D and D' is known, a measurement of the angle of intersection gives the decomposition DD' =72+ vA, the values of the indeterminates r and v being relatively prime when D, D' are relatively prime. 73. Intersections of the Modular Curves with the Axis. As an example of this theory let D'= 1, and let us consider the real intersections of the first modular curve of an uneven order D by the axis of X. The number of these intersections is 2 2P x [Dr T2 + v2A, where m is the number of primes dividing A, and [D = 72 + V2 A] is the number of solutions of the equation D =72+ 2A, in which r and v are relatively prime, and v is positive and uneven; the sign of summation 2 extending to every value VOL. II. 4 F 586 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 73. of A from 0 to D. This result may be obtained by counting the number of the reduced semicircles, each of which traverses twice the boundaries of the reduced space, while some of them in addition traverse the line x = 0. For any uneven value of A, let A =p x q, p and q being relatively prime; then F1 (D) cuts the axis of X at the two points -a+(/ ) and - I+p / ) lying upon the line AA + at equal distances on either side of the centre. For any even value of A, let A = 2' x p x q, p and q being uneven and relatively primes; then F1 (D) cuts the axis of X between - D and A in the points 1 1 2 (/2 ) 2 and in the symmetrically situated points between A + and A+. When 7= 0, and consequently v= 1, the point of intersection is single, and the curve cuts the line A_1A + at right angles 2d times, if D has d prime factors. When 7 is not zero, there is always a double intersection of the axis of X by two symmetrical branches, the angles of intersection being given by the equations 7 vJA cos '=,D' sin =~ + D Thus for each equation D=72+v2A in which 7 is not zero (as there are two solutions, since T may be either positive or negative) we have 2k double points with parallel tangents; and if for the same value of A, there are h sets of values of r2, v2 satisfying the equation, there are h pairs of branches at each of the 2k points, the tangents at corresponding pairs being parallel. It thus appears that if the modular curve be traced, we have only to measure the angles at which it cuts the axis of X to obtain all the representations of which D is susceptible by the principal class of any negative determinant, the value of the second indeterminate being uneven. The following are special cases. (a) The modular curve F1 (D) passes through the origin as many pairs of times as D is the sum of an even square 72 and an uneven square v2: each pair of symmetrical branches cutting the axis of X at angles of which the trigonometrical tangents are + -. This theorem may be expressed without any reference to the geometrical Art. 74.] THE MODULAR FUNCTIONS p (co) AND * (co). 587 considerations by which it has been obtained; viz. if we consider the modular equation (as Jacobi has done) between q = 2k2 -, = 2X2 -1, the lowest terms in q and I are A x II[D(q2 + 12) + 2(r2 _ 2)q, A being an integral number; we have thus a direct method for decomposing a number into the sum of two squares relatively prime, all the decompositions being obtained simultaneously. (b) The curve F1(D) passes through each of the points X= +-1/2 as many pairs of times as D can be decomposed into an uneven square and the double of an uneven square, the tangents of the angles corresponding to the v=/2 decomposition D =2+ 2v2 being + v2* (c) The curve F1(D) passes through each of the points + 1/3 as many pairs of times as D can be decomposed into the sum of an even square and vV/3 the triple of an uneven square; the tangents of the angles being +, if D = 2 + 3 2. Arts. 74-82. THEORY OF THE MODULAR FUNCTIONS (w) AND +(c(). 74. Theory of 5 (o). Primary Matrices. a, b The unit matrices, d which satisfy the equation c, d A(a + ~d * (1) form a group, characterized by the congruences b-0, mod 2; c d2- 1, mod 16...... (2) In this article we shall designate as primary those matrices of uneven determinant, and those only, which satisfy the congruences (2) and also the congruence a 1, mod 4; by equivalent matrices we shall understand matrices which can be reduced to one another by (pre- or post-) multiplication with a primary unit matrix. We then have the theorems: (i.) Every unit matrix can be expressed in one way, and in one only, in each of the forms +iA Ixll2 xll; ~+llxl2|xI XAlI; 4F 2 588 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 74. where A, IA' are primary units;,x, t' have values included in the series 0, ~ 1, + 2, ~ 3, 4; and I| j is one of the six unit matrices of Art. 21. In particular every unit matrix of type (1) is equivalent, by premulti. plication, and also by postmultiplication, to one of the matrices +~1!2, the exponents u, u' being in this case the same. (ii.) If a unit matrix of type (1) be expressed in the form I i' 2a X I j21 x | 12 32 xl 2 X......... (3) it is primary or not according as the congruence a 0, mod 8, is or is not satisfied. And two unit matrices such as (3) are or are not equivalent according as the congruence 2a _ 2a', mod 8, is or is not satisfied. (iii.) If I X I is any unit matrix, the least exponent for which I X \ is primary is always a divisor of 48 (other than 48 itself). (iv.) Let U = lI X l -2 l- 2 ( = 0, ~+1, + 2, + 3); U = a X I r 1-2 X 18; UO = C 16; every primary unit matrix can be expressed in one way, and in one way only, as a product of powers of the nine matrices U,, U4, U0. (v.) Every prime matrix of an uneven determinant A is equivalent to one of the 48 x r'(A) pairs of matrices included in the formula 0', 0 where gg'= A; g' 1, mod 4; h =- (g2 - 1), mod 2. (vi.) If, is a matrix of determinant A and of type (1), the exponents C, d 2 u and 2X, for which the matrices a,b a,b H,,!x c, d ' c, d xI a2 are primary, are respectively determined by the congruences c+2ixdA d2-1, mod 16, c+2Xd -d2-, mod 16. If {A represents a complete set of primary and primitive matrices of Art. 75.] THE MODULAR FUNCTIONS (co) AND (co). 589 determinant A, and if [ a [, I are given primary unit matrices, the formulae a! xlA[x|Il, HIxlAlxlol-2z HI'i xIA I x TI, also represent complete sets of primary and primitive matrices of determinant A. 75. Theory of (4). The Reduced Space. Every point co in the plane xy is equivalent to one point, and only one, in a certain reduced space lying between the lines x= +8, and outside the sixteen semicircles g2+ - (2 +l)x+x2+y2=0 (= -8, -7,..., 6, 7).... (i) For in the reduced space of Art. 36 there is always one point Q, and only one equivalent, in the sense defined in that article, to any given point w. Let 2x = lal x o, so that Ial is of type (1), and let io-l2x aI be positive, M, being one of the numbers 0, + 1, ~2, + 3, 4e, where e is a positive or negative unit according as the real part of 21 is negative or positive; then the point I = 1- 12P X IC= - o 12 X I a X w lies in the reduced space and is the only point in that space equivalent to co. The sixteen semicircles (ii) are equivalent in pairs; viz. the semicircle U = - [8v2- 4v, 1 - 4v, 2], = 2v - 1 is transformed into the semicircle U^=[8v2+4v, -1- 4v,2], L=2v by the transformation | U7|| |2V l 2X 1T7|12 X Ia-2,I 1-24v, -2 I'I -I1vILMcKv 8 vX, 1-4- V for the seven values of v from - 3 to + 3 inclusive; and the extreme circle US-= -(112, -15, 2), = 7, is changed into the extreme semicircle 4 = (112, 15, 2), u= -8, 590 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 76. by the transformation - 15, - 2 1 = U I- 8 X |- T|2 X 1| C = 112, - 15 -112, -.15 Similarly the semicircle U-= -[16, -1, 0] is changed into [16, 1, 0] by the transformation I U I 1=, 1= 0 We may form the reduced space into a closed surface by folding it on itself so that the two boundaries x= +8 coincide, and also the two semicircles of each equivalent pair; the eight points x = + 1, ~ 3, ~ 5, + 7 will then coincide and form a singular point on the surface from which the eight pairs of coincident semicircles diverge; the surface has also nine other singular points; viz. one at an infinite distance, and eight at the points x=O, ~2, ~4, ~6, 8, which are the extremities of the semicircles remote from the extremity common to them all. 76. Transformation of the Reduced Space by the Modular Function p (w). We now map the reduced space on the plane XY by the equation (w)= X + iY. The seven pairs of equivalent semicircles [8 V2 +4v, +T1-4v, 2], r=0, +1, ~+2, +3, are represented by the lines Re~i, R being real and varying from 1 to oo as w passes from x=2v to x=2v+1 along one or other of the equivalent semicircles; the equivalent pair of semicircles [112, + 15, 2] from x =- 8 to x= -7, or from x = 8 to x = 7, are represented by the axis of X from -1 to o; lastly, the equivalent boundaries x= + 8 from y = o to y= 0 are represented by the axis of X from 0 to - 1. If c and Q are the reduced values of co immediately before and immediately after the passage by 1 (c) = X + iY of any boundary U,, the equation of passage is w=I U I x, or w= U -1 x according as X+iY traverses the boundary from right to left or from left to right as seen from 0. The seven lines x = 2v (none of which are boundaries) from y = so to y =0 are represented by the lines Re4i", from R=0 to R=1; the eight lines x=2v-1, v=0, ~+l, 2, ~3, 4 from y=oo to y=O are represented by the lines Re (2v 1)^ from 0 to oo; the eight semicircles [4v2-4v, 1-2v, 1] from x = 2v to x = 2v - 2 are represented by the eight arcs of the circle X2 + Y2 = 1 Art. 77.] THE MODULAR FUNCTIONS + (w) AND + (co). 591 from e"4vi to e (v-1)ihr respectively. The seven semicircles [4v2-1, -2v, 1], v=O, ~1, ~2, +3, from 2 v- 1 to 2+ 1 are represented by seven of the infinite branches of the curve R8cos 80=, lying respectively between the asymptotes 0= -(4v -1)7r, 0= 1 (4+l1)7r, and drawn from the former of these lines to the latter: the two quadrants [4v2-1, -2 v, 1], v= +4, from x=7 to the boundary x=8, and then from the boundary x= -8 to x= -7, form the remaining loop from 0 = 1 7r to 0 = U 7-. Lastly, the pairs of lines [2 v + 1, 10] v 0, = 1, + 2, + 3 and the pair [+15, 1, 0] are represented by the eight loops of the inverse curve RS=2cos80; these loops lie in the same angles as the loops of the curve R8cos80=; each of them represents the reduced parts of the pair of straight lines answering to it; viz. if the point w describe the line x = 2 v -1 from 0 to 2v- + i, X+i Y describes the loop from 0 on the tangent 0= 1(4/v-1)7r to the point R=2, 0= v7r, and then as 0 describes the line x=2v+l from 2v+l+1 i to co, 0 continues the loop to the point 0 on the tangent 0 = (4v + 1)7r. All the semicircles, properly and improperly primitive, of determinant + 1 are included in the preceding enumeration. 77. Theory of p (w). The Rational Semicircles. Given any semicircle we can always find a reduced semicircle C, equivalent to it. Consider this semicircle as described by a point w in a definite direction. For brevity, we attend only to the case in which the determinant is uneven and not a square. Let U' be the boundary at which C1 quits the reduced space, and let | U'l be the transformation by which U' is transformed into the equivalent boundary; then if C2 is the semicircle by which the continuation of CQ is represented in the reduced space, C7 is transformed into C2 by | U' |. Continuing this process we shall certainly arrive at Ci again, because the number of reduced semicircles equivalent to a given rational semicircle is finite, and because each reduced semicircle determines the reduced semicircles which precede and follow it. The reduced semicircles equivalent to C( therefore form a period including C, itself; and T= U' I x U" x U"' I x... is an automorphic of C7. It can be shown that T is one of the two primary fundamental automorphics of C7; i. e. that every primary automorphic is a power (with a positive or negative exponent) of T. Hence, conversely, if the quadratic form Cl be given, and also T the fundamental automorphic of C, we have only to 592 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 78. express T as a product of powers of the unit matrices U, in order to obtain analytically the series of forms (or of semicircles) equivalent to C1. 78. Theory of +(c). We shall state very briefly some of the corresponding results relating to /])( eti, x p(- 1 + co). the functions +(w) = ( -) and ( = e x ( + ) If a, d is a unit matrix satisfying the conditions b c-O, mod 2, c, d b ca2-1, mod 16, we have (co)= ( j ( ); and conversely, if a, b l2xTl2vlx 2H2x1Tl^X... the condition b = 2- 1, mod 16, is equivalent to the condition v -0, mod 8. If | A is a unit matrix appertaining to the group of q (X), I + x IAl x i+\is a unit matrix appertaining to the group of +(o): we have in fact I +\ X I Xal2,Xl_2X x.. X2 X l-= Irl-i2LX [l-c12^x1 X... For the reduced space of j (w) we may take the inverse of the reduced space of ( (); viz. the space bounded by the semicircles V;1= -[2, -1+4v, 8v2_-4],, =- [2, 1+4v, 8v2+4v] (v=-3, -2,..., 2, 3), Vd-= -[2, 15, 112], V4=[2, -15, 112], VW6= [0,1,16], VO=[O, -1, 16]; these eighteen boundaries being equivalent in pairs, viz. V;j1 x I VV = -v, where VY= llxl U I XII-1; so that 7,1-e 1 —16 ^_,,3,,,, -15, 112 Vo=l~1_~o= 1,-161 y,_j,/-sx,4 -rX-1 Vo 0 1' V4=I ' I8xllI2ll 2, -15 7- | 1 | 2V X |C12| X| |1 - 4 2 - -8v2 1 2When the reduced, 1 4vth When the reduced space is mapped by the equation j(o) = X+ iY, the Art. 79.] THE MODULAR FUNCTIONS (co) AND (co). 593 boundaries V, are represented by the same lines as the boundaries U, in the theory of the function (co); and the 'equation of passage' is co = Vy x Q, and 2 being the reduced values of c immediately before and immediately after the passage X+ i Y from right to left (as seen from 0) across the boundary Th. 79. Theory of (. The function,(') is unchanged by the substitution of + d, if J(W) a+bcb' an b = I c- 12H1 X I r _2vl X... ca, b I 1 ' ' is a unit matrix of type (1) satisfying the conditions b+c-O, mod 6, or,U+yvO, mod 8. If I A I, BI are unit matrices appertaining respectively to the groups of (f)(), +((~), |cIx|AIxIol-l, lal-lxlAlxlc~I, lTlxlBlxlT-1-, ITl-1xlBI\x11 appertain to the group of,() For the reduced space of we may take the reduced space of 'P (X). The sixteen semicircular boundaries [2/U2 +2/, -2,A-1, 2], Al= -8,..., + 7, are equivalent in pairs; viz. W;1 = [8v2 + 4v, - 4v- 1, 2], - =2 v, is transformed into W, =[8v2+12v+4, -4v-3, 2], t=2v+l, by I W = a=lv 12vxITI2xlKl-2 X 2, yv -4,..., +3. Similarly the rectilineal boundary W- = [16, 1, O] is transformed into W=-[16, -1, 0] by I Tr1= l.16 In the figure defined by the equation X+iY=-s = er x p (- +), VOL. II. 4 G 594 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 80O W, from oo to - 8, is represented by the axis of X from 0 to oo, and W, from 2v+1 to 2v+2, or W,-, from 2v+1 to 2v, is represented by Be (2v+)ii from R = 1 to R = oo: the 'equation of passage' when X + i Y traverses the boundary from right to left, as seen from 0, being o = I W\I x Q. 80. Theory of ) (w). The Subclasses of Primarily Equivalent Quadratic Forms. The number of subclasses of quadratic forms contained in any given Gaussian class is ascertained as follows: I. Properly Primitive Classes. Let v. be the least number U which satisfies the equation T2- A U2= +1, v2 the least even number satisfying that equation, so that v2 = v if v, is even, V2= 271T1 if v1 is uneven; let also a = 2 or 1 according as v, is even or uneven; let v= 2, 4, 8, according as vU2 is uneven, unevenly even, or evenly even. Then any Gaussian class contains a x v subclasses of each of the types (A) and (C) of Art. 41, I.; 4r subclasses of the type (B), if A-l, mod 4, v=2; and 8a subclasses of the type (B) in every other case. When -= 2, the subclasses here considered are distributed equally between the two subclasses defined in Art. 41, I. (b) as appertaining to each of the three types. II. Improperly Primitive Classes. (1) If A 5, mod 8, each Gaussian class contains 48 or 16 subclasses according as the least numbers satisfying the equation T2-AU2 =4 are even or uneven; viz. each subclass of Art. 41, II. (2), contains eight of the subclasses considered here. (2) If a 1, mod 8, each Gaussian class contains sixteen subclasses of the type (CO) of Art. 41, II. (1), and sixteen or eight of the types (A') and (B'), according as v-. 0, or 4, mod 8: viz. each of the subclasses there considered contains eight of the subclasses here considered, if it is of type C', and eight or four (as the case may be) if it is of either of the types (A') or (B'). The demonstration of these assertions may be obtained by considering separately each subclass of Art. 41, II., and examining the character of its automorphics. For example, let (A, B, C)=f be a properly primitive form T-Bu, -Cu of type (B). If vu is even, the automorphics A B T+ CB are all primary because T- +1, mod 8, u=0, mod 4, B-1, mod 2, A 2, mod 4, whence Au (r +Bu)2 -1, mod 16. The sixteen substitutions [| 12, 1 I x a- \2, which Art. 81.] THE MODULAR FUNCTIONS ( (w) AND (cO). 595 alone transformf into forms of the same type (B), give in this case non-equivalent forms; if for example fx 1 r Ix I A 12FC is transformed into fx I r x a- 1 2V by a primary matrix Iv, 171 x la 12 x I VI x I(~ -2v x I ir-1 is an automorphic off: this matrix is therefore primary, whence also lo12xivlxl l - 2 is primary, or 2 A 2v, mod 16. If vl is uneven, the fundamental automorphic is of the type 7, and may be represented by the formula I Ix I C 12Xx r -1, I v I being a primary unit and X an uneven number determined by the congruence A - 2XB B2 - 1, mod 16, or A +6XB=B2-1, mod 16, according as A is _3, or -5, mod 16. In this case the eight forms fx C! 12t are non-equivalent to one another, but are respectively equivalent to the forms fx I T I X 12-2X; because the form fx Ir x Ix Il 12-2X is the same as the form fx I v I x I a,12, which is transformed into fx I o 12L by the primary matrix r [-2 x v 1-1 x o- - 12A. 81. Theory of, (w). The Modular Equations. Let I A I represent any term of a system of primary and primitive matrices of determinant A; and let Q = I A x w; any integral and symmetrical function Z of the a' (A) quantities (P (Q) is a rational and integral function of. (w). For if p (w) describe any closed contour in the plane X Y, traversing in succession the boundaries U, U', U",..., u, ',,u" times, we have,= x U'X U' X U'" "... X WDla 1-1 X cll wc denoting the final value of w, and la ~ a primary unit. Hence the o(A) quantities (2) = (A I x I a I x o) are the same as the quantities p(Q) = p(IAI x w); i. e. 2 is a one-valued function of 0 (co) throughout the whole plane; the demonstration may be completed as in Art. 52. This theorem establishes the existence of a modular equation of the order N= d (A) between v = (2) and u = p (o); this equation, which we shall represent by f(V, u) = n (V- (Q2)) = 0, is characterized by the following properties, (i.) It may be written in the form VN+ Zai, u + ()uN=0, the limits of i and j being 1 and N-1, and the coefficients a, j satisfying the relations 42 2 596 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 81. Thus the matrix a, j I is symmetrical or skew symmetrical with regard to both its principal diagonals according as \ _ +1, or ~_ 3, mod 8; and the equation is not changed by interchanging u and (x)v, or by writing for u and v their reciprocals; viz. fv, ) = ()f(u, (2v) = () vf(, (ii.) The equation is not altered if we write esilru for u, and e8i^rv for v: viz. f (e^Asi v, e4 i u) = e4^si f(v, u). Thus the coefficients a j are zero, except when i+Aj-A i +j-N, mod 8. The equation between e'(2s+l)i x u and e(28+l)Ai^ x v is obtained by changing inf(u, v) the signs of those terms in which - (Ai +j- N) is uneven. If u=e4si8 the values of v are all eighth roots of unity: viz. if g, g' are conjugate divisors of A, there are f'(g') values of v (see Art. 63) equal to (2) IAsi7r. g Of the 48 x a' (A) values of ( (Q) determined by the equation a =, bx 0J c, d X' where a, b is any primitive matrix of determinant A, there are 8x -'(A), which correspond to matrices of the types C 1 2, - =0, 4, ~ 1, ~2, 3, and which satisfy, as we have seen, the eight equations f(u, e-~ =0v) =0. Again, 8x '/(A) of the matrices ac, b are of the eight types Ia-2Ixlil; and the C, 4 corresponding values of p(Q) satisfy the eight equationsf(, e ) = 0. But the values of p (Q), corresponding to each of the remaining sets of 8 x c'(A) matrices, satisfy equations of the order 8 x a' (A) which do not decompose into factors of the order a' (A): viz. the values corresponding to matrices of the types I p, Ip, ] 1, Ip 2, respectively satisfy the equations F (8(), 1-,8(Q)) =0, '(qs(o), 1 - Q082 ))=0, Art. 82.] THE MODULAR FUNCTIONS + (c() AND +(c). 597 ( P8 (2)- i F (PI (o() 'a 1) =,. 82. Theory of / (w). Representation of Rational Semicircles by the Modular Curves. We have next to consider the representation of the semicircles of determinant A by the modular curves. I. The modular curves representing improperly primitive semicircles. Theorem. If a=eii7r, and f(u, v)=O is the modular equation between u-= (co) and v = p (Q), The eight equations f(x +y, -)=0 X =, 1, _2, +3,4 are real, if A =1, mod 8; and the curve [ + 2] is the curve [X] turned through a negative angle ~r. The four equations f (x+iy, - )= 0, X=0, 1, 2, are real if A_ 5, mod 8; and the curve [2 X +1] is the curve [2X - 1] turned through a positive angle 4r. The two equations F( + iy, ) =0 if A 3, mod 8, are real; but the corresponding curves are imaginary. The two equations F(x + iy, -+Y) =0 are real, if A _7, mod 8; the corresponding curves are imaginary. (i.) Let A =1, mod 8. The sixteen subclasses of semicircles of type (C') contained in any Gaussian class are represented by the four curves ( +i4 y,' a )= 0, X=l, 3, 5, 7; viz. if C be a semicircle [A, B, C] satisfying the congruence A-2B B2-1, mod 16, the semicircles c, CX |o C, CX I7-1 X I|o12, Cx r1 |-1 X I r1-6; Cx Icr 1, CX I (r-6,, Cxrl- x 1 4X, Cx Ti-1 X I 1-4; Cx |l 4, Cx |(-4, Cx T-j1 X 16, Cx r-1 X -2; CX I r 16, Cx I a 1-2, CX T-|1 X I a- 18, CX Tr -1; 598 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 82. are respectively represented by the curves [X=1], [\=3], [\=5], [X=7]. For any given semicircle [A, B, C] of the type (C), the index of the equation by which the semicircle is represented is given by the congruence A- 2XB-B2 1, mod 16. The subclasses of semicircles of type (A), which are eight or sixteen in number according as 4v is uneven or even, are represented by the curves [x = 2v] in the following manner:-Let (A, B, C) = C be a semicircle of type (A), satisfying the congruences AB2-1C, mod 16; then the semicircles C, CxI+l, CxlclR, Cxl|lIxIx 8, Cxla"2, CX I+xlcr12, Cxll-6, CXI Ix I -6, CxI r 14, Cxl\ xla-l4, 1 7CX I 1-4, Cx I + I X r i-4, CXII6 1,, CCxIIxI, Cxa1-2, Cx +lx I C1-2, are respectively represented by the curves [ = 0], [ =2], [= 4], [X = 6]. Lastly, let C be a semicircle of type (B), satisfying the congruences A B2- 1mod 16, 2B+ C0, mod 16, the semicircles C, Cx IC, Cx I, l8, C xlo-7, CxI C 1, Cx o 2, Cx Ior-6, Cx l- 1-5, Cx o-4, Cx oI5, CxIo-1-4, Cx Io-3, CX I|o6, Cxlo-,7, CxIIo-2, CxI<o-1, are respectively represented by the curves [X= ], [ =2], [ =4], [ =6]. In these Tables the semicircles on the right of the vertical line are identical with the semicircles on the left, where Iv, is uneven. In both cases the index of the equation satisfied by any semicircle [A, B, CG is given by the congruence A-2XB-B2- 1. (ii.) Let A-5, mod 8. Art. 82.] THE MODULAR FUNCTIONS q (co) AND d (45). 599 (1) Let the equation T2 - Du2 = 4 admit of uneven solutions; and let C be a form satisfying the congruences A+6gB B2-1, A+C=-, mod16; then the subclasses C, Cx I\, CxlIcjs, Cx ilxI-8, CXla,-2, CXIIX Io-(-2, CX I|I6, Cxl 4I X Il 6, Cx - I-4, Cxl xI xo-4, CxIo-I4, Cx|I x o-4, CX| CI, Cx I ix I |X 6, CxI 12, CX | +Xl|lC2, are represented respectively by the curves [ = 1], [ = 3], [X = 5], [= 7]. (2) Let the equation T2 - Du2 = 4 admit of no uneven solutions; and C be a form satisfying the same congruences as in (1); also let Xl=IplxI-2.xl 112 or p I X 12 X IT -2 according as A =5, or 13, mod 16; then the forms symbolically represented by Cx [1 + ]x [+X+ X2] [1 + c8], Cx [1 x + X+ x2] x [1 + 8] x r-2, Cx [1 + +] x [1 + x +2] x'[1 + ] x -4, C x[1+ ]x [1 +X+X2] X[1+ C8] X 0-6, are respectively represented by the curves [X = 1], [X = 3], [X = 5], [ = 7]. In both cases the index of the equation satisfied by any semicircle [A, B, C] is given by the congruence A + 6XB B2- 1, |h |p! 1 n i; Mt l X X -2 X T\2- 2; | p x x r[- p= I _ -2 P- p,1, 2' -1, 1. II. The modular curves representing properly primitive semicircles of type B, viz. - [0,, 1], mod 2. Theorem. The equations f(x+iy, ) =0 x+ty 600 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 82. are real (i.) if A — 1, X 4, 0, mod 8; (ii.) if - 5, X -2, 6, mod 8; (iii.) if A 3, X 1, 3, 5, 7, mod 8; (iv.) if A- 7, then for all values of X, mod 8. (1) Let A -3, mod 8. Let [A, B, C] be a semicircle satisfying the congruence A+6BB2-1, mod 6; then the semicircles Cx ITI xI-2, C, CXI'7TxIo16, CXICI8, CX 17| X | C 14, Cx -I6", Cx 7-x l-4, CX C 1-2, CXIT XllI-, Cxl l-4, Cxlr lxI! -l, Cxl- 4, Cx T, Cx | T, CX CXr I 1I Cx], are represented by the curves [X= 1], [X = 3], [X = 5], [X = 7] respectively, the semicircles on the right of the vertical line being equivalent to those on the left, if the least solution of T2- U2= 1 is uneven; since in this case C has automorphics of the type or -^-3 x I 1-l (2) Let A _7, mod 8, and let C or [A, B, C] be a semicircle satisfying the congruence A-2B=-B2-1, mod 16; then the semicircles C, CX TIX | 12, Cx | l8, CX 1 X T| |6, CX I o 1, Cx1T lx |I4, Cx |l 6, C X l C1-4, Cx\C14, CX\|T X |-5, CX|7|-4, Cx| T|X |C|2, CxI Cx |6 IXJ C x 1|,-t, 1 CX i C 1-2 Cx I- 1, Cx|(76, CxlI-x o.|8, Cxl|-j2, Cx\T\, are represented respectively by the curves [A=1], [x=3], [ = 5], [= 7]. The four remaining curves (though their equations are real) are imaginary; they have only a certain number of real conjugate points. (3) Let A-1, mod 8. The curve [X=0] is real, and the curve [X= 4] imaginary, or vice versa, according as Al1, or =9, mod 16; the semicircles of type (B) all satisfying Art. 82.] THE MODULAR FUNCTIONS (w) AND +(co). 601 the congruence A =B2- 1, mod 16, in the former case, and the congruence A- 8B-B2- 1 in the later; in both cases alike they are all represented by one and the same curve. If C be any semicircle of a given Gaussian class, the others are symbolically given by the formula C x [1 + T] X [1 + 02 + 4 + 6] X [1 + 8], the factor 1 + -8 being omitted if I4v is even. (4) Let A 5, mod 8. If C satisfies the congruence A+4B B2-1, mod 16, the semicircles C x [1 + rT2] X [1 + 04] X [1 + s8] satisfy the same congruence, and are represented by the curve [X = 2]; the semicircles Cx [T + -2] X [1 + 04] X [1 +.8] satisfy the congruence A-4B-B2-1, mod16, and are represented by the curve [X = 6]. III. The modular curve representing properly primitive semicircles of the type (A). All these semicircles are represented by the equation f(1 - (+iy)8, (x+y)8)= 0, which is derived from f(1-(X+iY), X+iY)=0, by the transformation R=r8, 0=80. The above equation is irreducible, but breaks up into 8 factors if we adjoin the radical /1 - (x + iy)8. IV. Lastly, the modular curve f [( + iy)8, (x iy)8 1] 0 represents all the properly primitive semicircles of type (B). The limits of the present Memoir preclude any detailed description of the VOL. II. 4 H 602 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 83. geometric theory of the modular function X(w). The reduced space corresponding to this function is the reduced space of Art. 36 repeated forty-eight times; and the general character of the theory closely resembles that of the functions ()w) and +(w). Arts. 83-88. THEORY OF THE MODULAR FUNCTION T() =[1-X24())3]- X48(W). 83. Transformation of the Reduced Space. In the theory of the primary unit matrices of Art. 21 the reduced space consists of six regions (Art. 38); and any one of them may be taken as the reduced space when the unit matrices considered are no longer subject to the condition of being primary. We take as the reduced space the region BB' inverse to AA' (see fig. 2); we thus substitute the condition c a for the condition a< c in the definition (Art. 36) of a reduced form of negative determinant. We include in the reduced space BB' the boundaries which lie to the right of the axis of y; and we exclude the equivalent boundaries which lie to the left of the axis of y. The equation T( a Ix )= T(w),......... (i) where I a I is any unit matrix, shows that the reduced space is mapped homceomerically (see Art. 48) on the plane XY by the equation T(w)=X+iY,........ (ii) the correspondence between the two spaces being one-to-one. For it follows from the theory of the functions 4)(w) and +(wo) that for any given values of X and Y there always exists one, and only one, reduced value of co satisfying the equation (ii). It will be found that rT() = T(1)= T(o)= oo, Lim T() x [1 =, Lim T(1) 2(1)= 1, Lim T(c) x Q2(o)= 1; (1ii) 27 34 T(i), T(i), T) = T - K4(i);.... (iv) 4 26 3 T(p) =T'(p)=T"(p)=0, T"'(p) = - x K();... (v) where p = ( ~1 + iV3). Since T'(w) _ (2k2 - 1) (2 - k2) (1 +k2) 4, Since T( —o() - = 1k2+k4 x - K2(), (vi) Art. 84.1 THEEORY OF THE MODULAR FUNCTION T(w). 603 we infer that T'(o) is nowhere infinite in the reduced space, except at the point X = oo, and nowhere zero, except at the points o = i, o = p. Thus the only points at a finite distance in the plane XY which present any singularity in the homceomeric transformation are the points 2-7 and 0; angles at these points being respectively double and triple of the corresponding angles at i and p. The arc Q1, from i to p, is represented on the axis of X by the segment from 7 to 0; the line P1, from p to oo, by the segment from 0 to - oo; the axis of y, from i to oo, by the segment from -7 to + oo: the regions of the reduced space to the right and left of the axis of y are represented respectively by the regions below and above the axis of X: this is readily seen by considering values of w lying near the cornicular point at an infinite distance. Hence, if in the plane XY we carry a slit along the axis of X from - o to 274, the plane so divided corresponds with continuity to the reduced space; but the continuity ceases if the point X+ iY traverses the slit. Thus, if X+ i Y describes (either in the positive or negative direction) a closed contour round 247, 1 traversing the segment (0, 27), we have w —, varying continuously with (t)1 X + i Y, and w1, W2 being the initial and final values of w; if X + iY describes a closed contour round 0 and 27 simultaneously, traversing the segment (- cc, 0), we have w2= +1 +l+w, the upper or lower sign being taken according as the contour is positive or negative with regard to the infinite region of the plane XY; lastly, if X+iY describes a closed contour round 0, traversing both the segments (- m, 0) and (0, 2-), we have 0,-1 1, 0 — 1 02 - 1, 0 CW2 = X X o = X C)l, '2e 1, 0 Jx, 1 l O, or 1,0 0 - 0, -1 0, - 02= 1 e, 1 1, 0 1, - according as the segment first traversed is (- oo, 0) or (0, 2e7); e denoting in each case T 1 according as the contour is positive or negative round 0. 84. The Modular Equations and Curves. If Q = I A I-~ x o, where I A I represents any term of a system of primitive matrices of determinant n, non-equivalent by postmultiplication, we can prove, as in Art. 52, that any rational and integral symmetrical function of the -'(n) 4 H 2 604 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 85. quantities T(Q) is a rational and integral function of T(w). Thus, if x= T(w), the expression F(y, x)= n [y- T(Q)] is rational and integral in x, and F (y, x) =0 is the modular equation between T(co) and T(Q). This equation is irreducible; it is symmetrical with regard to x and y (see Art. 53); and the numerical coefficients are integral. For, if H, is the coefficient of yr in F(y, x), it is evident, from the theory of the modular equation between k2 and X2, that H can be expressed as a rational fraction in terms of k2; and that the numerical coefficients in the numerator of this fraction are integral numbers, while the denominator is a power of k2(1 -k2), because X2 = k2'n", 1 (1 _X2) = (1 - k2)"6'; whence it follows that when H,. is expressed in terms of T(w) the numerical coefficients of the powers of T(co) are integral. If in the modular equation F(T(Q), T(w), n) =0 we write T(o) = X+ iY, T(Q) =X-iY, we obtain the equation of a curve in the plane XY, which is the image in that plane of the rational semicircles of determinant n in the plane xy (see Art. 65). 85. The Rational Semicircles. As in Art. 42, every rational semicircle is equivalent to a series of arcs lying within the reduced space: these reduced arcs are of six different types, eight transitive, and two intransitive, characterized by the symbols [P-l, Q-], [P1, Q1], [P1, Q-1], [P, Q1], [Q-1, P -], [Q1, P-], [Q-1, P1], [Q1, i]; and [P_1, 1], [P, Pi ]. The reduced arcs equivalent to any properly primitive semicircle of determinant +1 are the boundaries P1, Q1; similarly the arc equivalent to any improperly primitive semicircle of determinant +1 is the axis of y from i to oo. Let the series of reduced arcs, equivalent to a given semicircle of determinant D, begin with a transitive arc C7 of the type [Q_0, P1J; let C0 be followed (1) by ax- 1 intransitive arcs of the type [P-_1, P1J] (2) by a transitive arc C' of the type [PiE, Qj1, (3) by a transitive arc C2 of the type [Q-1, P'21, and so on continually. For brevity, we shall attend only to the case in which D is not a square, so that none of the reduced arcs can pass through the cornicular Art. 85.] THEORY OF THE MODULAR FUNCTION T(w). 605 point at oo. In this case the series of reduced arcs must form a period repeating itself for ever and beginning with C,. For (1) the series can never terminate, (2) the number of reduced arcs appertaining to a given determinant is finite, (3) each reduced arc absolutely determines the reduced arcs which precede and follow it. We thus arrive at the automorphic equation C,= Ic i"1'x l\ftx I crl22x! IXx... Xl\ xC.. (i) or, which is the same thing, at the development 0=[f~le, 2, * e, ]......... (ii) the continued fraction being subtractive, and 0 being that root of the equation ao + 2 b0+a,02 = 0, associated to the semicircle Cl=[ao, bo, aj], which is the greater in absolute magnitude. To determine arithmetically the period of reduced arcs, when the symbol of one of them, which we may suppose to be the transitive semicircle C = [ao, bo, a], is given, let el be a unit of the same sign as - -, and let T L T+1 be the chord a, intercepted by C, on the indefinite line p-ip+,; if 2X is the absolute length of this chord, we have The figure shows that C will not cease to be reduced if we shift it through The figure shows that C will not cease to be reduced if we shift it through as many unit spaces as there are units in the segment, the absolute value of which is 1 bo i-.e, +X: hence =E Xe ]* (1)a, hence [=E x-r 1i,. - b......=1 (1) E denoting the nearest integer. If C2 = [a,, b, a2], we have - bl= l el a, + bo, whence b, + bo-0, mod...... (2) The centres of the semicircles C, and C[ lie on opposite sides of the origin; thus 606 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Alt. 85. bo and b, have the same sign; again, the equation (1) shows that b, lies as near as it can to >/(4 D - 3 a2), consistently with the congruence (2), the sign of the radical being the same as that of bo or b,. We thus obtain the algorithm b, + b, - 0, mod ax; e611- _ bo + b,; b2+bOmod 2; E22 = -b+ bi b, + bo- 0, mod a,; e g = - —; a2 I -_. b2-D b1 b2-D a3= -; a2 the numbers b,,, b,..., b, having all the same sign, and b, lying as near as it can to V,(4 D- 3 a) taken with that sign. Example. Let D = 66, = [5, 6, -6]. We find C1 L5, 6, -6], [11, 0, -6], [5, -6, -6]; C2=[-6,6,5], [-13,1,5], [-10, -4,5], [3, -9,5]; C3=[5, 9, 3], [-10, 6, 3], [-19, 3 3 [-[22, 0, 3], [-19, -3, 3], [-10, -6, 3], [5, -9, 3]; 4=[3, 9, 5], [-10,4,5], [-13, -1,5], [-6, -6,5]; C5 = C1; and the continued fraction is 6+/662_ 1 1 6 -3- 1 3 -2-... - 2 -o 1. 3 -+ 1 -6+ + ol 3-2+ the reduced arcs being of the types indicated in the following scheme: Cl=[Q1P1 ], C2=[ Q1P_ 1], 03 =[QP_,], 4 —[Q_-1P], 2x[P 1 1],7 5x[ PJ,1 ] 2 x [E PP, ], c =[P_ Q-_]; C;=[ P Q1 ]; c=[- Pl ^-l ]~ The algorithm is of the same form as that of Gauss (Art. 43, supr&a, Disq. Arith. Art. 86.] THEORY OF THE MODULAR FUNCTION T (o). 607 Art. 184), but the determination of the coefficients b is less simple; viz. in the algorithm of Gauss these coefficients are all positive and less than VD, lying as close to it as they can consistently with the congruence b + bi — 0, mod a,. The Gaussian period of forms equivalent to (5, 6, - 6) is (5, 6, -6), (-6, 6, 5), (5, 4, -10), (-10, 6, 3), (3, 6, -10), (-10, 4, 5), (5, 6, -6),... giving the continued fraction 6+/66 =[2, 2, 1, 4, 1, 2]. We may begin the period of reduced arcs with any arc of the period, and may trace it geometrically in either direction; in the arithmetical theory there is the same freedom of choice. In the algorithm of this article we have begun, for simplicity, with a transitive arc, and have considered it as described in the direction QP. One half of the whole number of transitive arcs (viz. the arcs CQ, C2, C3,... one of which comes after each matrix I| in the automorphic (i), or at the beginning of each integral quotient in the development (ii)) are described in the same direction QP; the other half (viz. the arcs CO, C',..., which are the inverses of C2, C,... and which immediately precede them) are described in the direction PQ. If we begin the arithmetical development with one of these inverse arcs C', C2... and consider it as described in the direction QP, we obtain the same series of semicircles in the reverse order, and a continued fraction having the same period of quotients, but also in the reverse order. 86. Semicircles passing through a Singular Point. Some special considerations are needed in the cases in which one of the singular points p and i lies on a reduced arc. (i.) Let [a, b, c] be a semicircle passing through p,, then a+eb+c=O0, and D=a2+ac+c2: there is therefore a primitive representation of 2D by the improperly primitive form 2x2 + 2xy + 2y2. The only numbers D, of which the doubles admit of such representation, are of the type R or 3R, R being divisible only by primes of the form 3 n +1; if R is divisible by aC different primes of that form, there are 608 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 86. 2T sets of representations of 2D by (2, 1, 2), each set consisting, except when R = 1, of six representations. Let then D = a2 + ac + b2 be a given representation of 2D by (2, 1, 2); the six representations of the set are (+~, ~c), (+(a+c), Ta), (Tc, +(a+ )). In one of these representations both the indeterminates are positive; we may therefore suppose that a and c are positive. With these six representations we shall also consider the six representations of the opposite set, viz. (~c, +a), (Ta, ~(a+c)), (+(a+c), Tc). These twelve representations give rise to two systems of semicircles of determinant D, each containing six semicircles, viz. three passing through p and three passing through p_-, the two systems being symmetrically placed with regard to the axis of x, and the six semicircles of each system being equivalent. The semicircles of one of the two systems are [a + c, a, -c], [-a, a+c, -c], [-c, -a-c, -a], [a+c,-c,-, -a], [[-, -a, a+], [- c, c, a+c]; we shall denote them by the symbols Fr, F2, F3, F4, rF, r6, taken in order. The semicircles of the other system are derived from these by changing the signs of the middle coefficients. The semicircles Fr and T6 have radii less than unity; they do not enter the reduced space, and do not belong to the period: F2 and F3 do not enter the reduced space, but they have radii greater than unity, and appear in the period as two consecutive transitive circles. Lastly, Fr and F4 actually enter the reduced space, and are to be regarded as intransitive circles of the period, respectively preceding rF and following rF; the equations connecting these semicircles being rl=1+rF, r2=1, r3=1+r,. 3 Whenever the determination of b., in the algorithm of Art. 85, is ambiguous, i.e. whenever the radical,V/4D)-3a 2 lies evenly between two values of b, satisfying the congruence bs + bs- 0, mod a., the ambiguity is to be removed by taking for b, the greater in absolute magnitude of the two values, and the resulting semicircle will pass through Art. 86.] THEORY OF THE MODULAR FUNCTION T(co). 609 the singular period p. Conversely, whenever the chain of reduced arcs passes through that singular point, the case of ambiguity presents itself in the determination of one of the circles C,, C,.... The four arcs rI, rF, r3, r4 are to be regarded as forming a chain which passes but once through the singular point p. Thus when these reduced arcs are mapped by T(c) on the plane XY the evanescent arcs r2 and r, contribute nothing to the trace in that plane, and the arcs Fr and F4 are mapped in one continuous line. For, if 71, y4, q-i, q+1 are the tangents to rF, r4, Q_-, Q+1 at the point p, or p_-, we find from the equations of these semicircles 7i plql +74p-1 q-1= 600, the angles considered being internal to the reduced space; whence, since angles at the points p are tripled in the plane XY, it follows that the tangents corresponding to y71 and 74 make at the point 0, with the line (0, 27) and on opposite sides of the line, supplementary angles; so that these tangents are in a straight line. Example. The transitive semicircle [3, 8, -5] of determinant 79=72+ 3.7 + 32 gives rise to the period [3, 8, -5], [-5, 7, 6], [6, 5,-9], [-9, 4, 7], [7, 10, 3], the corresponding values of the radical being 2/241, 2 /208, 73, 6,1 8.2 Here 6- lies evenly between the values 3 and 10, which alike satisfy the congruence b+4 0, mod 7. Thus [7, 10, 3] is a transitive semicircle passing through p _; and the four semicircles [-10, - 3, 7], [3, - 10, 7], [7, 10, 3], [-10, 7, 3] form the chain r1, r2, F3, i4. There are evidently as many chains of four semicircles passing through p as there are sets of representations of 2D by (2, 1, 2). More than one of these chains may belong to the same period, and in particular a symmetrical period must contain an even number (if any) of such chains. Example. Each of the periods (-4, 5, 3), (3, 7, 4), (4, 5, -3), (-3, 7, -4) VOL. II. 4 I 610? MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 86; of determinant 37 = 42 + 4.3 + 32, and (1,4,-3), (-3, 5, -2), (-2, 5,-3), (-3,4,1) of determinant 19=32+3.2+22 contains two chains, symmetrical to one another, of the type (F). The latter of these periods is ambiguous, the former appertains to a determinant for which the equation T2 - D U2 = - 1 is resoluble. The case D=3, which has been excluded in what precedes, forms no exception to the general theory. There is but one set of representations of 3 by x2 + xy + y2; and there is but one class of semicircles of determinant 3. The period of this class contains only one semicircle C, viz. [1, 2, 1], the complete series being [1, 2, 1], [-2, 1, I], -3, 0-, ], [-2, [1,-2, 1], which includes the chain [-2 -1, -, -1], [, 2, ], [,2, 1], [-2, 1, -1]. This chain is symmetrical to itself, and the only chain (F) possessing that property. (ii.) Every semicircle passing through the point i is its own inverse: hence the arc of any transitive semicircle C' which arrives at this point is continued in the reduced space by itself described in the opposite direction. Thus every period which contains one semicircle passing through the point i also contains a second, but cannot contain more. The number of semicircles of determinant D which pass through the point i is 20, if 0 is the number of compositions of D by the addition of two squares prime to one another; viz. if D = + b2, the two compositions implied by this equation give the four semicircles [ab,a], a, - b], [-, a,,, -b], [-b, a, b]. Hence the number of periods containing semicircles which pass through i is 0, of which, when D is uneven, 20 are properly and as many improperly primitive. The semicircles [ + a, b, r a] respectively intersect the semicircles [ T b, a, + b] at right angles; and as angles at the point i are doubled when the figure is mapped by T (w) on the plane YZ, the trace of the semicircle [+ a, b, T a] forms one continuous line with that of the semicircle [ b, a, b]. But the two semicircles do not in general belong to the same period; viz. when D is uneven, one is properly and the other improperly primitive; and again, whether D is even or uneven, whenever there is an ambiguous period con Art. 87,] THEORY OF THE MODULAR FUNCTION T(co). 611 taining a pair of semicircles passing through i, these are of the type [+~a, b, T a], and do not intersect at right angles when D is greater than 2. When, however, D is even, and the equation t2 - Du2 = - 1 is irresoluble, the two extreme forms of the same period may be at right angles to one another, and indeed must be if D is the double of a prime. 87. The number of Ovals of the Modular Curve. Let h, h' represent, as in Art. 41, the numbers of classes of properly and improperly primitive forms of determinant D; let H, TH' be the numbers of classes of properly and improperly primitive semicircles; and let 0 be the number of compositions of D by the addition of two squares prime to one another, so that 0 = 0, if D is divisible by 4 or by any prime of the form 4n+ 3, and 0 = 2 if D = R or 2 R, where R is uneven and divisible by a different primes of the form 4n + 1. Then, if D is uneven, H=+ 1h+10 H' = 2h' + I7, and, if D is even, H= I h + I 0. For, in general, every class of semicircles [a, b, c] corresponds to two classes of forms (a, b, c) and (-a, -b, -c). But if (a, b, c) and (-a, -b, -c) are equivalent there is only one class of forms corresponding to [a, b, c]. There are, when D is uneven, I0 properly primitive classes, and as many improperly primitive classes, which are equivalent to their own negatives. Hence H= (h- _ 1)+ I= +, H': = (h'- )+ 0 = 1 hi+ - 0. Similarly, when D is even, there are 0 properly primitive classes which are equivalent to their own negatives; so that H= (h- 0)+O= h + 0. Thus (1) when D is uneven, the whole number of distinct ovals comprised in the modular curve is Ih+ h'; (2) when D is even, and the equation t2-Du2 -1 is resoluble, the number is ~h; (3) when D is even, and the equation t2 -Du2 _1 iS irresoluble, the number is + h a, if a- is the number of periods of semicircles containing orthogonal semicircles passing through i. In the two former cases 1 0 of the ovals have a double point at (427); in the last 4 I 2 612 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 88. case, (O0-a-) have a double point at at (247), and a- pass through that point once and only once. 88. The Cases of Symmetry. In certain cases the periods of semicircles are symmetrical with regard to the axis of y, and consequently the corresponding ovals symmetrical with regard to the axis of X. The principles upon which the discussion of these cases rests will be found in the memoir already cited in Art. 63. I. Let the equation t2 - Du2 = -1 not be resoluble. If the class of forms (a, b, c) is ambiguous the period of semicircles has a symmetry of the first species with regard to the axis of y. For if [a, b, c] is a semicircle of the period, [a, - b, c], which is equivalent to [a, b, c], is also a semicircle of the period; i. e. the period is symmetrical. Let [a, b, c] be a transitive semicircle C; then [a, -b, c] cannot appear in the period as a semicircle C; for if it did so appear the form (a, b, c) would be equivalent to (- a, b, - c), and the equation t- Du2 = - 1 would be resoluble. Hence [a, - b, c] must appear in the period as a semicircle C'; i. e. the two symmetrical semicircles [a, b, c] and [a, - b, c] are described in opposite directions. Conversely, when the period of [a, b, c] has a symmetry of the first species, the period of forms (a, b, c) is ambiguous. For, taking any two symmetrical semicircles of the period, and continuing simultaneously the chain of reduced arcs in opposite (and therefore symmetrical) directions from them, we shall arrive ultimately either at a semicircle [a, 0, c] which is symmetrical to itself and also ambiguous; or at two consecutive circles, symmetrical to one another; in which case also it is evident that the period is ambiguous. II. Let the equation t2 - Du2= - 1 be resoluble. In this case the ambiguous periods of semicircles, and these only, pass through the point i. Then (1) all the periods have a symmetry of the second species; the ambiguous periods have also a symmetry of the first species, since each period consists of a chain of circles described first in one direction and then backward in the other. Conversely (2) whenever any period has a symmetry of the second species all the periods of that determinant have it, and the equation t2 - Du2 = - 1 is resoluble. For (1) if [a, b, c] is a transitive circle C, [-a, b, - c] is also a transitive circle C, because the forms (a, b, c) and (- a, b, - c) are properly equivalent. Hence these two symmetrical circles are described in the same direction; or the symmetry is of the second species. It Art. 89.] DIFFERENTIAL EQUATION OF MODULAR EQUATIONS AND CURVES. 613 will be observed that when the period is ambiguous the half-period or chain running from one of the two extreme semicircles to the other has a symmetry of the first species, since the two extreme semicircles are certainly described in opposite directions. (2) Conversely, if two semicircles [a, b, c], [a, - b, c] described in the same direction both belong to the same period, the equation t2-Du2 = - 1 is resoluble. For, as before, we may suppose [a, b, c] to be a transitive semicircle C; [a, - b, c] is also a transitive semicircle, and as it is described in the same direction QP as [a, b, c] it is a semicircle C; i. e. the two forms (a, b, c), (-a, b, -c) are properly equivalent, and t2-D2 - 1 is resoluble. Thus symmetry of the first species is the mark of ambiguity; symmetry of the second species is the mark of the resolubility of the equation t2 - Du2 = - 1. Arts. 89-90. THE DIFFERENTIAL EQUATION OF THE MODULAR EQUATIONS AND CURVES. 89. Differential Equation of Jacobi. We have seen in Art. 12 that k = Q4(W) satisfies the differential equation d-k dk 3d2 k 1 k2 2( d)4 CO3 Q \ 1 l+ 2 d 4, 2do"-~-3-/ - _ c..... (i and that the complete solution of this equation is q04(7+t), a,, 7, representing any constants whatever. Hence if X be any transformed modulus whatever, we have also 2d Add- -3(d d- _ I + ) )4;... (;; d(03d c d 2 1+ i 2 dc and if we eliminate w from (i) and (ii) we shall obtain a differential equation between k and X. To effect the elimination, we must write the equations (i) and (ii) in the form which they assume when dw is not equicrescent; i. e. we must add the terms dk2 [r d3 3(d20oW2] dX2 [ d3 dW_23 2 do2L d2 w23 2 ' d- L2 dC1O3 to the left-hand members of these equations respectively. On multiplying 614 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 90. (i) by d\2, (ii) by dk2, and subtracting, the differentials of co disappear, and we find 2dk dx(d3kd - d3 dk) - 3{dX(d2k)2 -dk(d2X)2} +k2 d2 (1 + k2 2 1+ X\2 dX 0iii 1 - k2,) 1 - X2 }0 which is the differential equation between k and X given by Jacobi. 90. Transformations of the Equation of Jacobi. If z is any function of y, and y any function of x, we have identically dzd-z 2 dzd z 2 3 2^ ---3zd2) 2- -3 3d ) 2 — -3(clY dx dx3 dX2 dy dy3 +dy2 d+ dxd3 dX2 1dz Z ddz2 dy2 ()\7) () dx dy dx as may be verified without difficulty by means of the identity, already employed in Art. 12, dz d3z 3( d2z 2 2 kd x2 _ d2 rd 2z Zd 2 -2 2[log j ][ -logjddx) Let x= o, y =k, z =,:x being a given function of k; and let,', JA", //"' be the derived functions of ~ with respect to k. Combining the identity (iv) with the equation (i), we find that pA satisfies the differential equation 2d3A d d...... 2 _ 4 2dudd 3dd) _f((w ) ( f ( ). (ii) where f(u) represents the quantity 1 1[( ) k2 '2 I (-i) k 2_ + 3'"2].....((iii) which is given as a function of k, but which we may conceive to be expressed in terms of a. Again, writing in the identity (i), x =, y=, z = v, v being the same Art. 90.] DIFFERENTIAL EQUATION OF MODULAR EQUATIONS AND CURVES. 615 function of X that,y is of k, and combining the result with the equation (ii) of Art. 89, we find d3v dv 3(d2v)2 f()()dv4 (i) dco do... Eliminating co from (ii) and (iv) in the same manner as in Art. 89, we obtain a differential equation between, and v, of which the type is (, v) +d2 V2 [f()d2 -f()^]=0,..... (v) (v, v) representing the differential expression 2d, dv (dvd3e - df d3v) - 3[d v 2(d)2 - di2 (d2v)2]. The following are the forms of f i() in a few important cases (a) u = k; f (A) f42. ~c1 +,U2 1 1 (' t-1 - -).; / +1 ' (b) = k2; (c) u = k' (1 - 2); 1_ - + 2 1_ I () = 2 (A - 1)2 2 1 3 1 f(~)= 4 (M -) 1 /(- 1+ 3 1 4 (m-)2' 1 (A -.1)2 (1 - IkC + k4)3. (e) M= -_/k; (f) = /kk'; 8 1 23 1 3 1 9 A 2 36, (-24) + 4 (- 27)2' f () = i1 + (1- - )2' 1 3 242.46 3 242 22 - - + 4 (A24_ I)2 4 4/24 1, If in the equations (v) we write X+iY for u and X-i Y for v, we obtain the differential equations satisfied by the modular curves. These equations are, in the six cases just enumerated, d () p COS a COS a, (a) ds R A — - COSa2,\ sina sin a, sin a2 AR ^ R RR R2 )R d (1 ( d ps sin a, cos a 1I sin (a, + a2) sin a2cos a _ ~ a'ds + ~- 2 R~R~ ' -; 616 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 90. d(p sin a cos a 3 sin(a+a') 3 sin a' cosa' (c' ds + 2 8 RR' 4 +'2- 0; d( p) 8 sin a cos a 23 sin (a + a') 3 sin a'cos a' ds+ 9 R2 72 RR' + = R0; e +sin a cosfC a j17 COSa j=7 sin fa (e P + (- 1)V. X 2(-1)J; 3 j=O.0 ) ds + 2 +4 j=0 / x. — ds 4 4j= o A j=o A 0 Rj^+12 In these formule 1 is the curvature and ds the element of the curve at P any point P; the curvature at P is positive or negative according as the tangent at P revolves in the positive or negative direction (i. e. in the direction from 1 to i, or from i to 1) when we travel along the curve in the direction in which ds is measured; R, R', R,,... are the absolute lengths of the radii vectores drawn from certain fixed points to P; if B is produced beyond P to Q, and if PT is the tangent at P drawn in the direction of ds, a is the angle QPT, and is positive or negative according as the rotation from PQ to PT is positive or negative; the angles a', a1,... are similarly defined; R is drawn from the origin in all the formulae; BR and RB are drawn from the points + 1 in (a'), and from the points A_., A,+ of Art. 48 in (bt); R' is drawn from the point 4 in (c') and from the point 927 in (d'); lastly, in (e') and (f'), the points, from which the lines Rj are drawn, are respectively eaJi, and 2-L~ i2jiE. The formulae (b) and (e) may be obtained immediately from (iii); but (c) is more easily deduced from (b), and (d) and (f) from (c). For example, to obtain the formula (d) we write y=k2(1-k2), _ = 2, and we observe Y2 that by virtue of the formula (c) y satisfies the differential equation 2dy d3y dy 2 1 3 1 3 1 dy do dW3 dw2 L \dW2) 2y2 4 y(y-~) 4 (y- ~)~ y-) *Y 4) y (vi) =:- H (c) (do Art. 90.] DIFFERENTIAL EQUATION OF MODULAR EQUATIONS AND CURVES. 617 Again, combining the identity (i), in which we write x=w, z =x, with the equation (vi), and denoting by /',,u",,"' the differential coefficients of,u with respect to y, we find d3d_ d2A82 1[f 2/"'_3Ma'2](dm\4 "(d4 But (y-l)2(y2), and 2, 2 [log/' d log YBut A= -_ 'ad - -2 -- log -log 12 (1 + 2y + 3y2) y2(y- 1) (y + 2)2 also (y + 2)2 (- 4y) = 4 ( - y)3-27y, (1+2y+3y2) (1 - 4y)2 + y2(y 1)2(y +2)2 (1 -y)6_ 8(1 y)3y2+54y4. Hence Irt /-X~y~ 2~2 /// r- 3 //o2 f(e-) = i(-) 2 '2- y )y6+ 2(1 -y)(l- 4y)+12y2 12( +2y+3y2) -(y 1)4 (y + 2)2L y2 (1 - 4y)2 y2 (y_ -)2 (y + 2)2 -1 r[(l —y)3(1-4y)(y+2)2+12y2l(y-1)2(y+2)2+12(1+2y+3y2)(l-4y)2] A2 (y + 2)4 (1 4y)2 1 16 (1 - y)6 _ 123 (1 - y)3y2 + 648 y4 -~2 [4 (1 - y)3 - 27 y2]2 16 A2 - 123 A + 648 A2 (4 - 27)2 81 23 1 3 1 9 2 ~ 36 A (A,- 27) 4 (, - 7)2 To explain by an example the way in which the equations (a'), (b'),..., are deduced from (a), (b),... and (v), we consider in particular the equation (f'). Here we have (1) (, v)=4i xds5x d (), d(dv= ds; 2,dAd2 / v\ v k if 2- =-Red, ) = 4Re- xd if - = Rei~, v =.-i. VOL. II. 4 K 618 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. 24A 23 j-23 1 24r23 j=23 1 (3) 24- I1 f24_4 j=oy-a~oP' 2-4 j o- aO-j' where a =2- -, 0 = eTI, so that 242. A46 d2 24246d2 j23( d' d j=z2 d _ d (A'24 - ~,) ( 24,)2 - _o "-~+ - o /dR. Rji =4/iz-R x d%, if,-aoj = Rj eij, v-a0ej = R: e- fj; 24u 22 1 j=2 O-j j=1 1 24 22 j 1 1 (4) 24 2 2 = 2 2 2~_1 aj= 0 -aOj j=0 /2' -a202j' V24- 1 or_ a220-2 ' =- j=O -- so that 24/,22 d _ 24 22 dV2 = 2 1 Rj.dRj+12. df, +R +12. dRj. d,12 l24 1 24l - R2 dR Rd s. (5) ds-= c a, -ds na. And by these substitutions the equations (v) and (f) are transformed into (f'). CONTENTS TO THE MEMOIR ON THE THETA AND OMEGA FUNCTIONS. ARTS. 1-14. Definitions and Elementary Properties of the Theta, Omega, and Elliptic Functions. ART. PAGE 1. The Theta Functions............. 415 2. The Omega and Modular Functions.......... 417 3. The Theta and Omega Functions as Infinite Products....... 419 4. Expressions for the Modular Functions q)(w), *(co), X(w) as Infinite Products... 420 5. Expansions of the Omega Functions in series proceeding by powers of q.... 423 6. The Formula for the Multiplication of Four Theta Functions...... 424 7. The Elliptic Functions of the First Species........ 425 8. The Complete Elliptic Integrals......... 429 9. The Partial Differential Equation of the Theta Functions...... 430 10. The Elliptic Function of the Second Species........ 431 11. The Differential Coefficients of the Omega Functions....... 433 12. The Differential Equations satisfied by the Omega Functions...... 435 13. The Abelian Functions........ 440 14. Differential Coefficients of the Omega Functions expressed by means of the Abelian Coefficients............ 445 ARTS. 15-23. Arithmetical Theory of Binary Matrices. 15. Composition of Matrices............ 447 16. Unit Matrices, Primitive Matrices, Reciprocal Matrices....... 447 17. Equivalence of Matrices........... 449 18. Systems of Non-equivalent Matrices......... 449 19. Composition of Systems of Matrices.......... 451 20. Reduction of any two Primitive Matrices of the same Determinant to one another.. 452 21. The Six Types of Matrices of an Uneven Determinant....... 453 22. Primary Matrices and Primary Equivalence....... 455 23. The Nine Types of Primitive Matrices of an Even Determinant..... 457 ARTS. 24-34. The Transformation of the Theta and Omega Functions. 24. Enunciation of the Problem of Transformation........ 460 25. General Solution of the Problem of Transformation.-Method of M. Hermite... 461 26. The Multiplier............ 468 27. Composition of Transformations.......... 469 4K 2 620 CONTENTS TO THE MEMOIR ART. PAGE 28. Linear Transformation of the Theta Functions........ 473 29. Linear Transformation.-Determination of the Multiplier...... 476 30. Linear Transformation of q (co) and j (co)......... 477 31. Linear Transformation of X (o)........ 482 32. Linear Transformation of the Elliptic Functions........ 484 33. Transformation of any Uneven Order.-Development of the Solution... 488 34. Quadratic Transformations of the Elliptic, Theta, and Modular Functions... 494 ARTS. 35-45. Geometrical Representation of Binary Quadratic Forms. 35. Quadratic Forms of a Negative Determinant....... 504 36. The Reduced Space............ 505 37. The Circular Affinity of Moebius. 507 38. The Subdivisions of the Reduced Space........ 509 39. Quadratic Forms of a Positive Determinant...... 509 40. Automorphics of a Quadratic Form.......... 512 41. The Subclasses of Primarily Equivalent Quadratic Forms......514 42. Reduction of Quadratic Forms of a Positive Determinant.... 517 43. Case when the Determinant is not a Square..... 519 44. Comparison of the Geometrical and Arithmetical Reduction.....521 45. Case when the Determinant is a Square...... 524 ARTS. 46-51. Geometrical Representation of the Modular Functions 4P(Xo) and @(co) [= 8(co) and 8(co)]. 46. Discussion of the Equation (co) =......... 529 47. Limiting Values of @(co)........... 533 48. Transformation of the Reduced Space by the Modular Functions 4(co) and @(co). 534 49. Lines answering to the Semicircles of Determinant + 1..... 535 50. Discussion of the Correspondence between the Reduced Space and the Plane XY. 538 51. Limiting Values of 1(&2)............ 541 ARTS. 52-58. The Modular Equation. 52. Definition of the Modular Equation.........544 53. Theorems relating to the Modular Equation. 545 54. Ditto......... 547 55. Ditto................ 548 56. The Sums of the Powers of the Roots.......... 549 57. Development of the Roots of the Modular Equation.......551 58. The Analytical Parallelogram.......... 552 ARTS. 59-62. The Equation of the Multiplier. 59. Theorems, and Linear Transformation.......... 553 60. Case when n is a Prime.-Determination of the Coefficients......557 61. Theorem of Jacobi......... 558 62. Expression of the Multiplier as a Rational Function of k2 and 2..... 558 ON THE THETA AND OMEGA FUNCTIONS. 621 ARTS. 63-73. The Modular Curves. ART. PAGE 63. The Modular Curves of an Uneven Order......... 560 64. The Modular Curves of an Even Order.......... 563 65. Representation of the Rational Semicircles by the Modular Curves..... 564 66. Case when the Determinant is not a Square.....566 67. Case when the Determinant is a Square........ 568 68. Geometrical Construction of the Transformed Modulus by means of the Modular Curve.570 69. Theorems relating to the Multiplier.......... 572 70. Symmetry of the Modular Curves........... 573 71. Transformation of a Modular Curve by Inversion with regard to another Modular Curve. 578 72. Multiple Points and Points of Intersection........ 580 73. Intersections of the Modular Curves with the Axis........ 585 ARTS. 74-82. Theory of the Modular Functions q (o) and * (c). 74. Theory of 4 (co). Primary Matrices.......... 587 75. Theory of q(co). The Reduced Space.......... 589 76. Transformation of the Reduced Space by the Modular Function q(oZ).... 590 77. Theory of qb(c)). The Rational Semicircles......... 591 78. Theory of ( )............. 592 79. Theory of ()- (co)............ 593 80. Theory of 4b(ow). The Subclasses of Primarily Equivalent Quadratic Forms...594 81. Theory of q(co). The Modular Equations....... 595 82. Theory of + (oX). Representation of Rational Semicircles by the Modular Curves..597 ARTS. 83-88. Theory of the Modular Function T(o) = [l-X24(o)13-X48(o). 83. Transformation of the Reduced Space.......... 602 84. The Modular Equations and Curves......... 603 85. The Rational Semicircles............ 604 86. Semicircles passing through a Singular Point........ 607 87. The number of Ovals of the Modular Curve......... 611 88. The Cases of Symmetry............ 612 ARTS. 89-90. The Differential Equation of the Modular Equations and Curves. 89. Differential Equation of Jacobi.......... 613 90. Transformations of the Equation of Jacobi...... 614 XLIV. MEMOIRE SUR LA REPRESENTATION DES NOMBRES PAR DES SOMMES DE CINQ CARRES. [From the M6moires prOsent6s par divers savants A l'Academie des Sciences de l'Institut National de France, vol. xxix.] Art. 1. SOIT fi, Z ai, j..j,.. (1) une forme quadratique des n indeterminees xi, x2,....,,, les coefficients a, j = aj i etant des nombres entiers. Nous representerons la matrice de cette forme (c'est-k-dire le systbme de ses coefficients rangds en carre) par nxn i=12,.,, = a | aj| -1 ' 2' ' '. (2) Nous rappelons que, dans la notation des matrices, une equation telle que equivaut aux n2 equations ai = =3ij, tandis que 1'equation aij = I [ij I exprime seulement 1'egalite des deux determinants. Pour abr6ger, nous supposerons que f soit une forme definie et positive, et que son determinant soit different de zero. Soit A=Iajl.......... (3) ce determinant, et designons par A, le plus grand diviseur commun des deter 624 MEMOIRE SUR LA REPRESENTATION [Art. 1. minants mineurs de l'ordre s, qu'on peut former avec les elements de I j 11: en supposant (comme nous ferons toujours) que la forme f, soit primitive, on aura A = 1. On sait que les nombres A0o=, A1, A2, A3,..., A,,. (4) ne changent pas quand les indetermin6es se transforment par une substitution lineaire k coefficients entiers et au determinant + 1*. On peut done les regarder comme les invariants arithm6tiques def1. Mais il y a quelque avantage k consid6rer, au lieu des nombres A,, la suite suivante: A,... A,. 1 =.12 I=A2.-l n-2 Les n-1 nombres Ir, I2,..., In,_, sont entierst; nous les appellerons desormais le premier, le second, le... (n-l)i ee invariant de f D. D'ailleurs on aura A =I-1 Xl I -2... XI1, A,=I s-xI2 -2xI-_3x... xI-x.. (6) s-1l Definition des formes adjointes.-Soit 11 a(,)j 1 la sime matrice derivee de * Si le determinant de la substitution est un nombre autre que l'unit6, mais premier a A,, les invariants de la transformee s'obtiennent en multipliant les invariants de la proposee par les carres des invariants de la matrice de la substitution. Mais si le determinant de la substitution a un diviseur commun avec A,, le resultat devient plus compliqu6. + Philosophical Transactions, vol. cli. p. 317 [vol. i. p. 396]; Dans ce memoire, nous ferons usage des m6thodes employees dans les 6crits suivants: 10. On Systems of Linear Indeterminate Equations and Congruences (Phil. Trans. vol. cli. p. 293), 1861 [vol. i. p. 367]. 20. On the Orders and Genera of Ternary Quadratic Forms (Phil. Trans. vol. clvii. p. 255), 1867 [vol. i. p. 455]. 30. On the Orders and Genera of Quadratic Forms containing more than three Indeterminates (premibre notice, Proceedings of the Royal Society, vol. xiii. p. 199; seconde notice, ibid., vol. xvi. p. 197), 1867 [vol. i. pp. 412, 510]. On aurait surtout desir6 de reproduire les r6sultats gen6raux renferm6s dans la notice de 1867, en les soumettant A un examen attentif et en y ajoutant des d6monstrations rigoureuses. Mais le temps ayant manque pour un travail si 6tendu, on s'est restreint, autant que possible, dans les limites de la question proposee par l'Academie. Toutefois on a trait6, des le commencement, des formes quadratiques de n indetermin6es, parce que les proprietes des formes quaternaires, dont depend effectivement la solution du problem e, sont plus faciles a saisir quand on les enonce d'une manibre parfaitement generale. Art. I.] DES NOMBRES PAR CINQ CARREPS. 625 II aj, j I, c'est-t-dire la matrice dont les elements sont les determinants mineurs de l'ordre s, appartenant d la matrice a],|j II, et pris dans leur ordre naturel. ll.n Soit S= II s n-; la matrice |i a() 1 est symetrique de 'ordre S. II. sf. n - s " 'J l La forme i=s j=S f8=- Z Z a(.xjixj i=1 j=1 est la (s- 1)i*me forme adjointe de fl. On remarquera (1) que fn,, la derniere des formes adjointes, est la contravariante, (2) que si f, se transforme en F1 par nx n une substitution lineaire l|ijl, fs se transforme en F, par la substitution s xs derivee I| j I] Les coefficients de f sont divisibles par A,, et la forme 0 = f, est primitive (proprement ou improprement). Nous nommerons les formes 02, 0),.3, 0n-l es formes adjointes primitives; nous ecrirons quelquefois 1 =fl. Definition des ordres.-Deux formes f, et fl, ayant les m6mes invariants, sont censees appartenir au meme ordre, quand, pour toutes les valeurs de s, les formes 0Q et 08' sont en m6me temps paires ou impaires*. Definition des genres.-Soit F1 une forme quadratique de r ind6terminees, F2 la premiere forme adjointe de F1; les caracteres gen6riques du systeme 01, 02, 0** On-1' se deduisent de 1'equation connue F, (X x2 e...,,)xFy y,..., x Yr) 2 Y F2 (X, X,..), (7) dans laquelle XI, XY,..., sont les determinants de la matrice Xl) X21 rXr YX, Y2,., Yr pris dans un ordre convenable. * Pour abrdger le discours, nous nous servirons assez souvent de ces expressions au lieu des 'formes improprement et proprement primitives' ou des 'formes de seconde et de premitre espece' de Lejeune-Dirichlet. VOL. II. 4 L 626 ME'MOIRE SUR LA REPRENSENTATION [Art. 2. Soit 0S la premiere forme adjointe de 08, q un diviseur premier impair de I; on verifie sans peine que est une forme entiere et primitive, et que les nombres premiers a q, qui sont representes par 0, sont ou tous residus quadratiques de q, ou tous non residus; et l'on attribuera "a 0 le caractere generique particulier (6)=1 OT (S)=- 1 selon que le premier ou le second cas a lieu. On peut appeler caracteres generiques principaux les caracteres que nous venons de d6finir, et qui dependent des diviseurs premiers impairs des invariants. Ces caracteres ne sont pas, en general, les seuls qui existent. En effet, (1) quelques-unes des formes 0s peuvent avoir des caracteres (dits supplernentaires) par rapport aux nombres 4 et 8; (2) nous verrons ci-apres qu'il peut exister, dans le systeme des 0, une suite de formes consecutives ayant un caractere simultang. L'ensemble des caracteres generiques particuliers (soit principaux, soit supplementaires, soit simultanes) constitue le caractere complet du systeme 0. L'etude des caracteres generiques est beaucoup facilit6e par l'emploi d'une forme canonique pour representer la forme proposee. Nous allons nous occuper de la recherche d'une telle forme dans les articles suivants. 2. LEMME I.-Toute forme primitive peut representer des nombres premiers B un nombre donne ou les doubles de tels nombres, selon que la forme est impaire ou paire. LEMME II.-Soit, dans une matrice donnee de l'ordre n, pis la plus haute puissance du nombre premier p qui divise A,; tout determinant d'un ordre s< n qui appartient a la matrice et qui n'est pas divisible par pis +1 contient au moins un determinant de lordre s - 1, qui n'est pas divisible par pis-, + 1'. Theoreme.-Soit A un nombre compose avec des puissances quelconques des nombres premiers impairs qui divisent An; on peut trouver des formes qb, en nombre infini, equivalentes a fl, et qui satisfont aux congruences x a x + al x + a I I x +... + a I I.. I,_ x2, mod A, ala '-1 oiA * a~ a~ 3... a,-l, mod ~ * Phil. Trans., vol. cli. p. 320 [vol. i. p. 399]. Art. 2.] DES NOMBRES PAR CINQ CARRES. 627 Demonstration.-Puisque 0_-= - G_ est une forme primitive, on peut n-1 supposer que le nombre = A x [a,n]*, qui est represent6 par 0_n, est premier a A. Done les congruences simultanees an, k, + ai, 2 k2 + ai, kn - j, ost, lkl+i^2k2+... +atxqlkn-~2 it z=1,2,3,...,n-,.... (ii)1 an,... + a, 2 2 * * * +,, k 7n ni- 1 sont resolubles pour le module A; et l'on aura, en particulier, kn 1, mod A. Cela pose, la substitution 1,,..., k, 0, 1, 0,... k2, 0, 0, 1,..., k3, 0, 0, 0 kn0, 0, 0,..., 1, nxn n xn transformera la matrice aI. j 11 dans une autre 11 bij il, dans laquelle on aura (- )x (n-l) (n -1)x(nl-l), i= 1, 2..., n-1, bl =1 t,lj= 1, 2,..., -j 1, (iii) et, en outre, bi^=0, m, im=, 2,... n-l, 1 An.... (iv) bnnE- =_ modA. Zn 7 n- _1 (n-l) x (n-l) En appliquant a la matrice I aij I, dont les invariants sont 1 I Z2,..., I-n-3) 7YIn-2) * 1Nous employons la notation [a,^], [aS,j, a9,,... pour signifier les determinants d d2 d.,, a d. a, d. a,.,la,j. 4L2 i 628 MEMOIRE SUR LA REPRESENTATION [Art. 3. une transformation analogue a celle que nous venons d'employer, on parvient i (n-1) x(n —1) une matrice I ci,j, dans laquelle c;,, —0, mod A, i= 1, 2,..., n- 2, n-1, -1 - 'A- mod A, 7n1 An-2 Cn-1, n-l 7 ~lnl An_2 m 1 (n-2)x(n-2) le mineur principal _ x jc1 etant premier k A. Cette seconde transformation peut bien changer les valeurs de bl, n ), b2, *... bi -1, n mais les congruences (iv) subsisteront toujours. En continuant de la sorte, on n x n parviendra enfin k une matrice ]j mij 11 qui satisfera ' la congruence 1, 0,..., 0 71 0 I A2 12, 0,..., O0 72I1 II j I o0 1A3 mod A, 0 0, o, e3 2 o 0, 0, 0,... 7n n -l-1 et l'on aura ainsi une forme 1= ijx.ix equivalente f, et satisfaisant aux congruences (I). 3. LEMME I.-Soit fi une forme impaire; soient aussi impairs j, la premiere forme adjointe de f, et AI son premier invariant, on peut toujours trouver, si non dans la matrice de fl, du moins dans une matrice equivalente, deux coefficients principaux*, tels que aii et aii aj-aj, dont l'un contient l'autre, et qui sont tous les deux impairs. * Les coefficients des carres dans les formes adjointes sont des mineurs principaux de la matrice, c'cst-i-dile des mineurs symetriques par rapport a la diagonale principale. Art. 3.] DES NOMBRES PAR CINQ CARRES. 629 Reciproquement, si On-, n-_2 sont des formes impaires, le dernier invariant etant aussi impair, on peut toujours trouver, soit dans la matrice donnee, soit (n- l) x(n-1) dans une matrice equivalente, deux coefficients principaux, tels que abj | et (n-2) x (n-2) | aj 1, dont le second est contenu dans le premier, et qui sont tous les deux impairs. LEMME II.-Si 01 est une forme paire, I1 et 02 sont impairs; et reciproquement, si 0_n- est paire, 0_-2 et In_i sont impairs. Theoreme.-Supposons que Ie soit pair, I +, I, + 2,..., In-i impairs. Sin - s est un nombre impair, les formes 0,, 0,1+, *.., O,-, sont toujours impaires; si n - s est pair, ou ces formes sont toutes impaires, ou bien elles sont alternativement impairs et paires, 0, etant impaire et 0n_ paire. Demonstration.-Soit 8 la plus haute puissance de 2, qui divise Ai; S une puissance quelconque de 2, qu'on peut supposer plus grande que Sn. A. Soit, en premier lieu, 0,_- une forme impaire; nous ferons voir que l'imparit de In_ et I,_=, c'est-a-dire l'egalite des trois nombres - ' S-1' -_' ean-d,n-2 in-3 entraine l'imparite de 0,-2. Supposons (ce qui est toujours admissible) que - x [a,,,] soit un nombre impair, que nous designerons par n,. Les congruences (ii) de l'article 2, en y ecrivant au lieu de A, seront resolubles pour le module S, et nous conn-1 n-1 nXn -' X n duiront, comme dans larticle cite, a une matrice [1 bij 1l, equivalente a 1 a,j [1, et satisfaisant aux congruences b, 1, mod S; b 0, mod, i=, 2,..., n-1; -ii nYn mn -, (n-1)x(n-l) (n-l)x(n-l) on aura d'ailleurs 1| bi,j j = I1 aj, j et les puissances de 2, qui divisent les determinants mineurs des differents ordres dans cette matrice, seront S1,..., *,,-2 Soit,_ 2X,_ 2 la forme contravariante de cette meme matrice, n,,_3 X,n_ l'avant-derniere de ses formes adjointes; si +n-2 est une forme impaire, il est evident que _n-2 est aussi impaire. Supposons done que n n-2 soit paire; dans 630 MEMOIRE SUR LA REPRESENTATION [Art. 3. ce cas 'n-3 sera impaire (lemme II); par consequent, on peut supposer que 1 (n-3)x(n3) i=l 2.n. n-3 "j-=l, 2,..., -3, n xn soit un nombre impair. Mais, dans la matrice II bij II, un des coefficients principaux de lordre n- 2 est congru, pour le module 3, au nombre 1 S, h h _ x -3 = - x n-2_7Yn n-1 /7n Done 0,_2 est une forme impaire, puisque-est congru, pour le module n,' 7n ~n-2 a un nombre impair. Maintenant, on conclut du lemme I qu'avant d'entreprendre la transfornxn axn mation de la matrice |,j dans la matrice I b, j transformation que nous designerons ci-apres par (A), on peut supposer que la premiere de ces matrices ait ete choisie parmi les matrices 6quivalentes, d'une telle maniere que les nombres (n-1) x ) (-2)(n- (2)(n2) 1 I et 1 ~n-l Sn-2 soient tous les deux impairs. Cela pose, on pourra employer la transformation nxn (n-1) x (n-1) (A) deux fois de suite, en operant premierement sur [ ai j 1, et ensuite sur |[ aaj I, nxn qui fait partie de 11 b,j Il. En supposant que In-3 soit impair, cette seconde transformation fera voir; 10 que O_-3 est une forme impaire; 20 qu'on peut (n-3) x (n-3) (n-3) x(n-3) 1 1 supposer - | aj impair, en m6me temps que,- _ aj, j, de sorte que l'on puisse Jn-S Sn-2 (n-2) x (n-2) encore operer la transformation (A) sur la matrice I1 i, j ]. De cette maniere, en supposant impairs les nombres I_4, I,_-o, +..., I+, on demontrera successivement que 0n-4, 0_-5,..., 0,s+ sont des formes impaires. Quant a 0s, elle est certainement impaire, puisque IJ est pair. On voit que les applications successives de la transformation (A), dont nous avons fait usage, conduisent k une forme 0l, equivalente a f, et satisfaisant k la congruence + +1 X[/sx2x+~s+sX 1z 1j xj23,+ x S,+1S+1+s+2s *, (i) i=lj=l -s.. Art. 3.] DES NOMBRES PAR CINQ CARRES. 631 1 sxs dans laquelle -x i. l, +I +, s8 F+,+2.*, n sont des nombres impairs, et z8+1 s8 B. Soit en second lieu 0,_i une forme paire; dans ce cas I_1 est necessairement impair; en supposant que I,_2 soit aussi impair, nous allons demontrer que 0n_2 est impaire, 0,_3 paire, IZ_3 impair. On pourra supposer (lemme I) que les nombres 7n = 2, [ an, n ] et. [tn-', ] 21 1] soient simultanement impairs. Les congruences (ii) de l'article 2, dans lesquelles on ecrira au lieu de ~ n-1 An-I seront encore resolubles pour le module S; mais on aura k, E 2, mod J, tandis que k,,_- sera impair. Soit kn_- 2 M - 1, mod; la substitution 1 0, 0, 0,..., 0, kl 0, 1, 0, 0,..., 0, k2 0, 0, 1, 0,..., 0, k3 0, 0, 0, 0,...,, k,_ 0, 0, 0, 0,..., 1, 2 nxn nxni conduit a une matrice 1j cij, j quivalente a 11 aij, jqui satisfera aux congruences b,,_0; i=1, 2,..., n-2; bi,_-, 2, 6^b - mod, 7'n 7 l 632 MEMOIRE SUR LA REPRESENTATION [Art. 3. et dans laquelle on aura aussi (n-2)x(n-2) (n-2)x(n-2) lbj II- =, IIl (n-1) x(n- 1) (n-2)x (n —2) 1 Puisque 0,_l est une forme paire, - x e a,j [ est pair; mais - x j a,j | est -1 n xn impair; car x bi,j I etant impair, la valeur, mod,S de (n-1) x (n-1) (n-2) x (n-2) 2 x I - j 17 x 1I 2_ _ _ b ---— X /b Ij, j- _9~, I j,l et, par consequent, de (n-2) x (n-2) Sn X Sn-2x, x I bj /n — 1 -- doit 6tre impaire. On conclut de l1 que 0_-2 est une forme impaire; pour (n- ) x (n-li) demontrer que n-3_ est paire, soit I Cij = 1 x S1 X -1 x 7_, o1 X _n etant la plus grande puissance de 2 qui divise le determinant, et operons une transformation (n-l) x (n-1) nxn (A) sur la matrice 1 ci,j I1. On trouvera ainsi une transform6e de I1 C|, 11, que nxn nous d6signerons par 1I di,j 11, dans laquelle on aura (n-2) x (n-2) (n-2) x(n-2) I di,j = II 1,J 11 -dij-O, mod S; z=n —, n; j=l, 2,..., n -2, d 2 n d o An n, n.= s n-1, n-1-= j 7n ~n-1 7?i-1S ~-1l d -__ mod d,, n-i — 7n n-l' (n-2) x(n- 2) De plus, la contravariante rdduite n-a de la matrice lt a, j sera une forme 1 (n)x(-3)x(paire. En effet, s'il en etait autrement, on pourrait supposer que. x I di,j fAt impair; mais alors x [dn_, n.-], et, par cons6quent, 0_,, serait impaire, car n-1 1 1 (n-3)x(n-3) x [d_-1,._] est du m me ordre de parite que - x dij 1, ainsi qu'il resulte 7-_ In-3 Art. 4.] DES NOMBRES PAR CINQ CARRES. 633 de la congruence (n- 3)x(n - 3) 2 2w I [dn-1, n-l]=| 1 xIx (I- CI, — ) mod X - 7n —1 nn -2 1) n et de l'equation 23_3=3_-. On voit facilement que la parite de!'-3 entraine la parite de fj3, et par consequent l'imparite de In_. Designons par (B) une transformation analogue a celle par laquelle nous nxn n x, avons reduit la matrice I1 aij | a la forme I di,j |. Si In 4 est impair, la (n-2) x (n-2) matrice cII dj|l admettra elle-m6me une transformation (B), puisque sa forme contravariante ',-3 est paire, et ses deux derniers invariants sont impairs. On aura done,-_4 impaire, a-5 paire, I-_. impair; d'oh l'on conclut que 0n,_4 sera paire, 0,_5 impaire. En r6eptant le meme raisonnement, on parvient a d6montrer que si 0_n- est une forme paire, les formes paires et impaires se succedent alternativement dans la serie des formes adjointes, et qu'il y a toujours un nombre impair d'invariants impairs avant d'arriver k un invariant pair. Si, comme ci-dessus, I, est le dernier des invariants pairs, n - s sera un nombre pair, et, apres avoir oper6 la transformation (B) (n - s) fois de suite, on sera conduit a une forme q1, equivalente af,, qui satisfera a la congruence i= j= s + - '2 xi x + [(x+1, X,+2 X, + - [( +3, X+4)2 +... + (Xn-l, X,)], mod 3, (ii) i=lj=l s 1 sxs S 'S ou l'on aura, comme dans la congruence (i), x I,B I impair, -+, > -, et ou les symboles (x,+,, x+ 2)2 representent des formes binaires paires, dans lesquelles on peut supposer que le premier coefficient est impairement pair. 4. Du theoreme de l'article precedent, on deduit immediatement cet autre plus general. Theoreme.-Soit I + 1, I+, *... Ia+ une suite d'invariants impairs, precede et suivie par des invariants pairs I, et I-T+,+,. Si a' est pair, les formes aI 0( + 1) 80+2 2..., 0 + + + seront toutes impaires, ou alternativement impaires et paires, la premiere et la derniere etant impaires. [Dans l'enonce de ce theoreme, il faut regarder Io=,=0 comme des invariants pairs, et 00 = 0 = 1 comme des formes impaires.] VOL. II. 4 M 634 M3 MOIRE SUR LA REPRESENTATION [Art. 4. En effet, il resulte de l'inegalite - -1 > > —, qu'on suppose satisfaite dans les congruences (i) et (ii) de l'article precedent, que la parite ou imparite des formes s xs 01, 02,..., 0,_ depend uniquement des matrices fij 11 qui figurent dans ces congruences, et nullement des termes multiplies par - Ainsi, pour avoir la demonstration du th6oreme enonce, il suffit de traiter ces matrices par les procedes deja developpes (A et B). On remarquera que l'emploi successif des transformations (A) et (B) conduit k une forme 0, equivalente a f, et qui satisfait a la congruence -= '(1, a) + (a + 1, ) + ( + 1, )+...+ +(s + 1, n), rod, (i) 0 ~a ${ Ss dans laquelle I, I,,,,..., I,, a</< 7<... <s sont les invariants pairs, et les expressions (1, a), (a+ 1, 3),... representent des sommes de a, 3- a, 7-/3,..., carres avec des coefficients impairs, ou (lorsque quelques-uns des nombres a, /3- a, 7- f3,..., n - s sont pairs) des sommes de formes binaires impaires. Soit, par exemple, 1, 1, 1, 2, 4, 2, 1, 2, 1, 1, 4, 1, la suite des invariants; ), satisfera a une congruence de la forme q1E_ (1, 4) + 213 65 + 23^ X6+ 24(7, 8) + 25 (P93O + o3O x-o + x1l xl) + 27 (12, 13), mod ^ en representant par l'expression (1, 4) soit la somme (x1, x)2 + (x3, x4)2, soit la 4 somme 2 3i x,; et par (7, 8), (12, 13), soit les sommes f7 x + f x2, 312 x2 + f313 x soit les formes (x7, X8)2, (12, X13)2. Ajoutons qu'on peut trouver des formes Q> e4quivalentes afi, et qui satisfont en m6me temps 6 la congruence (i) de cet article et ~ la congruence (i) de l'article (2). C'est ce qui resulte des propositions suivantes. nxn LEMME.-Soit I ai, j II une matrice donnee, symetrique ou non, dont les elements soient des nombres entiers; soit aussi, dans cette matrice, D le plus grand diviseur commun des determinants mineurs de l'ordre s; on peut toujours trouver deux matrices du n "me ordre, \\u\\ et \\v \, dont le determinant est Art. 4.] DES NOMBRES PAR CINQ CARRES. 635 l'unite positive, et qui satisfont l'equation* -Dn -Dn-1 DI1,aiI1-I= [I x L, D_ **" 0 X II *.. (ii) On remarquera (1) que les nombres entiers Js= D, D sont les D~ ' DS-l invariants arithmetiques de la forme bilineaire ai, j Xiyj, (2) que l'equation (ii) admet une infinite de solutions I u 11 et I I v. Pour abreger, nous designons par II a1, a,... |[ une matrice, dont tous les elements sont zuro, excepte ceux (a1, a2,...) qui se trouvent sur la diagonale principale. nxn Probleme.-Soit II aj une matrice donnee, qui satisfait a la congruence nXn xn xn I I jlI 1, pour un module quelconque P; trouver une matrice II,,j Ij qui, nx L nxn nxn satisfasse i l'equation | xj || = 1 et k la congruence,j l - I aij 11, mod P. nxn Solution.-Remplapons 1| a,,j| I par l'expression 6quivalente donnee par la formule (ii). Si D._1= 1, la matrice l x,,jll= l lu x vll satisfera aux conditions du probleme. Si D,_- > 1, ecrivons D - P a la place de D dans la formule (ii), et designons la matrice resultante par I| 1ij. Le plus grand commun diviseur des mineurs d'ordre n - -1 dans |],j || est Dn_2, puisque les invariants de cette matrice sont les m6mes que les invariants de la matrice Dn p Bn-D-1 D Dn-_ Dn-2' Do Mais l'in6galit6 D_, > 1 entraine celle-ci, Dn_ >D,2; donc on aura remplace la matrice donnee |ai, jl par une autre 3,ji, 1 qui sera toujours congrue oa | a j ri pour le module P, mais dans laquelle on aura diminue le plus grand diviseur commun des determinants mineurs d'ordre n -1. En continuant cette suite d'operations, on parviendra enfin a une matrice dans laquelle ce diviseur sera l'unite, et qui donnera, comme ci-dessus, la solution du probleme. * Voir, pour la demonstration de ce lemme, les Philosophical Transactions, vol. cli. p. 314 [vol. i. p. 391]. 4 M 2 636 ME3MOIRE SUR LA REPRESENTATION [Art. 5. n-fn n-fn n-n Corollaire. — tant donnees des matrices I a, j li, llj I, II j Il,..., en nombre quelconque, qui satisfont aux congruences laj l, modP,,,j -l1, mod Q, Iy,j l1, modR,..., les nombres P, Q, R,..., etant premiers entre eux deux a deux, on peut toujours trouver une matrice 1 xi, j I qui satisfasse aux congruences Il,,xj l- l a,,j |, mod P, 11,i,j 1, mod Q, -I l, mod R,..., et dont le determinant soit l'unite positive. En effet, on determinera les n2 nombres mi,j de sorte qu'on ait mi, j-a,j, mod P, -ij, mod Q, -yij, mod R,...; il en resultera n Xn mi,j i-1, mod Px QxRx..., et la matrice cherchee sera determinee par les conditions?nxn nxn x,|llx, jll-mi, j, mod P x Q xR...; xi, jl=l1, qui sont precisement celles du probleme precedent. Maintenant, soient I I et I a II des substitutions qui transforment fl en des formes qui satisfont respectivement a la congruence (i) de cet article et a la congruence (i) de l'article 2. Pour avoir une transformde de f, qui satisfasse simultanement aux deux congruences, on n'aura qu'a prendre une substitution de determinant + 1, dont la matrice soit congrue j ai 1I pour le module A et a I i,J II pour le module S. 5. Dfinition.-Soit I Ir, I une matrice equivalente h | aij I; nous dirons que les nombres r, 1 2x2 1 3x3 I (n-l x(n-l), - ',2 8,n-l que, pour abreger, nous designerons par i n2, n23..., -1, sont representes simultanement et primitivement par les formes 0i, 02,..., n0. Si 11 r 5ij 11 etait derive de I] ai,j I par une substitution dont le determinant fut plus grand que l'unite, la representation serait simultanee mais non pas primitive. Art. 5.] DES NOMBRES PAR CINQ CARRES. 637 Si les nombres mi sont impairs ou impairement pairs, selon que les formes 0 sont paires ou impaires, et si, de plus, deux nombres consecutifs mi et mi+ sont toujours premiers entre eux, nous dirons que le systeme des nombres mi est un systeme caracUteristique de nombres represent es simultanement par les formes O1, 02,..., O,_, et que la matrice 11 ri,j I est une matrice caractteristique. On peut toujours trouver en nombre infini des matrices caracteristiques, equivalentes a une matrice donnee, m~me parmi celles qui satisfont aux congruences (i) de l'article 2 et (i) de l'article 4. n xn Soit i rij ]1 la matrice d'une forme F equivalente k fl et satisfaisant a ces deux congruences. Designons par M1 1 M1 M,113..., Mn-I les nombres 1 1 2x2 a-r~, - I r, jl,...; AI111f 2j ij{9 *** ces nornbres seront representes simultanement par 01, 02, *.., On-; ils seront premiers a tout diviseur impair de A.; de plus, ils seront impairs ou impairement pairs, selon que les formes correspondantes sont paires ou impaires. Mais le systeme (M) ne sera pas necessairement un systeme caracteristique, puisque deux nombres consecutifs peuvent avoir un diviseur commun. Considerons la (n-1) x(n-1) matrice llrij dont le determinant est Anl x Mn_, et supposons que 21t — 1 (n-2)x(n-2) et Mn-_2= - x rij aient un diviseur commun,a. Ce diviseur est impair, puisque M.n_ et Mn 2 ne sauraient 6tre pairs en meme temps; il est aussi premier AD, et par consequent a A _2, qui est le plus grand diviseur commun de tous (n-i) x (n-I) les determinants mineurs d'ordre n- 2, qui appartiennent a la matrice I rj [I. (n-1) x (n-1) Done on peut transformer cette matrice par une substitution lineaire II aij II telle que, dans la matrice transformee, la valeur de Mn-2 soit premiere h M,_. Apres cette transformation la forme r pourrait, a la verite, ne pas satisfaire aux deux (n-i) x(n-i) congruences citees; pour eviter cet inconvenient on se servira, au lieu de aj 1[, (n-1) x(n-1) d'une substitution 3i,j ill, qui satisfait aux congruences ||fi,j -- ll a, j, mod M,,_,- ou mod - Jlr _> si 3Mn est impairement pair, et 11,i, ||I 1 - 1, 1, 1... I, mod x A. 638 MEMOIRE SUR LA REPRESENTATION [Art. 6. On aura ainsi une forme r dans laquelle Mf_ et M,_2 seront premiers entre eux; si ]n_2 et 1-3 ont un diviseur commun, on s'en debarrassera en operant (n-2) x (n-2) une transformation analogue sur la matrice I[ ri, | |. On arrivera done finalement a une forme r, dont la matrice sera caracteristique, et qui satisfera en m6me temps aux deux congruences. Une telle forme sera pour nous desormais une forme canonique. 6. ltnumeration des caracteres supplementaires.-II sera facile maintenant de faire une enumeration complete des caracteres supplementaires d'un systeme donne. Soit Is le plus grand diviseur impair de I,, I, l'exposant de la plus haute puissance de 2 qui divise IS, augment6 d'une unite quand une des formes Os _, 0,s+ est paire et de deux unites quand ces formes sont paires toutes les deux. Alors on a les regles suivantes pour former les caracteres supplementaires. I. Si, >_ 2, 0, aura le caractere II. Si, >= 3, 08 aura le caractere (_l)^(02-1) III. Si Iy =1, -__1 > 2, As+lI 2, 08 aura le caractere (-1)1(s — 1) OU (_ l)-(-S-l1)+ ( 2-1) selon qu'on aura (-l:*,-(O -1)++I(O -)=( ) +) o(_l)^2(8-1 -1)2 8+1-1 ) _2 (8 1)OU = IV. Si P,=0, Ms.__ 2,, +_ 2, et si, de plus, Q0 est impaire, cette forme aura le caractere ou n'aura aucun caractere supplementaire, selon qu'on aura (- 1) (_,-l+(-1): (- l)(_)-1) ou =(-l1)(8+* ). La demonstration de ces regles s'obtient immediatement, en se servant des formes canoniques. Ainsi, par exemple, pour demontrer la quatrieme regle, observons qu'on a la congruence 8=/s-[x+I+x+, mod4. O s [3=l[+1X82x1], mod 4. s~ Art. 7.] DES NOMBRES PAR CINQ CARRES. 6393 Done 08 aura le caractere (-1)2(s-) ou n'aura aucun caractere, mod 4, selon que I.si s + 1I- ou 1, mod 4, ce qui s'accorde avec la regle, puisque 08s- X 08 +1 l- x is +1, mod 4. Les suites de formes adjointes alternativement paires et impaires meritent une attention speciale. Soit 1, I +1,..., h,+20, une suite en nombre impair d'invariants impairs, IJ-, et Ia + 2a +1 etant pairs; soient aussi paires les formes 0, Oa +2, *.** 0 + 2,T' Aucune de ces formes n'aura un caractere supplementaire, tandis que chacune des formes impaires 0 _-, 0O+D, 0a+3,...5 0 +2,'+1 aura un caractere par rapport a 4 (premiere regle). Mais ces caracteres ne sont pas independants les uns des autres. En effet, en se servant de la forme canonique, on etablit l'equation (_ - 1) ~(0+2s-1- 1) +2 - 1) Is + 1) d'ou l'on tire cette autre (-1) 1)(+2s-1 -1 (_ 1)s X (- 1) (Ix +2 X... x,+2 s-2-1) X (- 1)1- (a —) s designant un quelconque des nombres 0, 1, 2,..., '+ 1. 7. Definition des caracteres simultanes.-Soit 0 c+1, 0+2,..., 0O+ a/ une suite de -' formes impaires; supposons aussi qu'on ait 2, +a,+al =i 2, +1 < 2, /Ar +2<2,..., /* 1+r, < 2; alors 0O et 0 + r'+, auront des caracteres supplementaires; mais aucune des formes 04 +1, S+, 2...,, +., ne pourra avoir un tel caractere. Designons par * (a-, ') l'expression s=a+r s=a+f s-a+at ~[ r(0S-1)(0sl+-1)+ 2(IJ:+1)(0~-1)+ -1 X,(02 -1)],.. (i) s=a s=al s=a+I dans laquelle 0, 0,+1,..., 0a,, +a+ sont des nombres representes simultanement par ces m6mes formes et faisant partie d'un systeme caracteristique pour les formes 0,..., n-l. L'unite (_ l)a(ma ') m aura la meme valeur pour tous les systemes de nombres O. * O** 0a +, +1, 640 MEMOIRE SUR LA REPRESENTATION [Art. 7. et cette valeur sera pour nous desormais le caractere simultane des formes 0o+1 0G,+2, 0, 0f+ar. Pour demontrer que l'equation (_ ~l)'a,') = + 1 jouit de la propriete invariantive que nous' avons enoncee, considerons d'abord un systeme de nombres caracteristiques, 1 n, O) 1,..., _I,, ' _n-ln- 1, mn etant impair, et w, = 2, ou 1, selon que 08 est paire ou impaire. Ces nombres sont lies entre eux par les equations suivantes: c1 m1 x m = J2 + I1w2m2 xl, W2 m2 x m2 = J2 + 12w3m3 X x MlI w m3 x m3 = J3 + I3w4m4 xw2n2, n —lmn-1 X mn —= J-1 + In-l 1 X c mn_-2, qui se d6duisent de l'Nquation (7), article er, et dans lesquelles ma' represente - [a, ], le symbole [a,,] se rapportant a la matrice i= 1,2,...,s+ 1, De ces equations on tire les suivantes: 22 n /-2 (~1 m )( 3m3l a -n I2 (-3.3 n-3 nl'n-l (-In-2 n-2 nm2 n- -2 -(_n-~2 n-2) ^ ( ln-1) 1i n- 1 m n-1 Art. 7.] DES NOMBRES PAR CINQ CARRES. 641 qu'on peut ensuite remplacer par celles-ci: km1- - X m (ml )= m ) >M3 2 - 2 ( ) x ( ) ( 2) x ( _2 n - 2 M2 - 2 (tM33 ) > ( mrl-1 2_Xn-2 )X( mi-2 Mn-2 Mn-2 n-2 n - 1 - rni nf, -2) = (23nX-1 >) Mn-l Mn-l Mn-1 Multiplions ensemble les equations de ce dernier systbme et transfbrmons les deux nombres par la loi de reciprocite; il en resulte s=n-2 s-n-l I 2 (ms- 1) (ms+i - 1) 2 (I+ 1) (-1) (Ms(-1) s1 - fl+ ) 2MS-l) sn-i m S s8 —1 = 1 relation qu'on peut aussi ecrire s=n-1 fS (_ l)(o,n-) ( )'....... (ii) en observant que 0o= 1, 0f= 1, et en admettant que lorsque 0s est une forme paire, on remplace 0 par 0'= 0s. II importe d'observer que les symboles 0' disparaissent d'eux-m6mes de l'expression (-1) (o, n- ). En effet, les termes de 4 (0, n- 1), oh figure 0, sont les suivants: [2 (08 )+1 (08 - -1)+ 2(I +1)]X 2 (0s- )+ +S 8_(02- ). Mais 0s = 0, et nous avons deja Btabli que le coefficient de 2 (0 - 1) est pair. On conclut de ce que nous venons de dire que l'expression (_-1)(,n-i) depend uniquement des caractbres supplementaires et simultanes des formes 0, et qu'il n'y entre aucun autre element. Supposons maintenant que les formes 01, 02..., 0 _n soient toutes impaires, et que les invariants soient tous impairs ou impairement pairs. Dans ce cas, VOL. II. 4 N 642 MEMOIRE SUR LA REPRESENTATION [Art. 7. aucune des formes 01, 02,..., 0 '_ n'aura un caractbre supplementaire; mais ' equation (ii) fait voir que l'ensemble de ces formes aura un caractere simultane puisque, en effet, la valeur de cette unite coincide avec celle de II (f-) Done, dans ce cas particulier, l'existence d'un caractere simultane est demontree. Pour etendre cette conclusion au cas general, revenons a la d6finition de a (a-, '), et remplagons les formes 0 par un systeme canonique equivalent. On aura ainsi 0e+l I(arl+la+2I+lz+2 (~[~(r~l~+/3-~'+21I I x 2 mod 4 -+fi +a,+1/7+1 X I-+2x.. x Ir+, +ff'+i], mod 4, 8=0 -en designant, pour abreger, par a le produit I,s. Dans l'expression k droite, s=l substituons,, I +2,.1i+2..., des nombres b +1, b5 +2,..., positifs et premiers entre eux, respectivement congrus k + 1, O+2,.., pour le module 8. Soit Xl = ^(r+l~z2o*+l"+ b7+2-/(r+lJX2+2+. + ba +< (+ I +l xI X +2x... x... +o, X2 —r+a XI = bo, a+,+ b,+1~1l+2~~ I+2 2 les invariants de XI seront I.-, I, +..., I,+ a-, Cx +,,, en designant par C s=a +- ' +1 le produit II P,. Soient encore X2, X3,..., les formes primitives adjointes a XI; s=a+l on se convaincra sans peine que les congruences 0,+-ax Xs, Xs-a x 0 m, mod 4,..... (iii) ont lieu pour toutes les valeurs de s depuis 1 jusqu' a-'+1; et, de plus, que lorsque I,+, est pair, la congruence correspondante subsiste encore pour le module 8. Done les nombres de tout systeme caracteristique pour les formes XI1 X2, "'* Xa' seront congrus, pour le module 4 ou 8, apres avoir etd multiplies par a, aux nombres correspondants d'un systeme caractdristique pour les formes 0. Mais il a ete demontre que les formes X ont un caractere simultand; soit Art. 7.] DES NOMBRES PAR CINQ CARRES. 643 (- 1)P ce caractere; on aura S = (Xs — 1 8 = ' s=l 8=1 4+-(C 1)(X,.-1)+ (-1) 8 (x]-) mod 2. s=l Substituons, pour XI, X2,..., leurs valeurs donn6es par les congruences (iii), et faisons b -C, a ab C, mod 4. En observant que O-C=a, +, + ~ -b, mod 4, on aura la congruence 8- = ( p=*(,'+ a-1) ( (- 1-l)+ 1(a-1)+i(a~- -), s=l oih nous avons design6 par 7 le nombre des invariants f +,...,,,r qui ont une valeur paire. Donc, enfin, (-l)P(,T') aura la m6me valeur pour tous les systemes caract6ristiques appartenant aux formes 0. c. Q.F.D. Remarquons que si IFA est > 3, 0q qui a certainement un caractere, mod 4, aura aussi un caractere, mod 8. Dans ce cas, l'unit6 (-1)p est elle-m6me invariantive; la m6me chose a lieu si le nombre y est pair. L'Uquation (ii) est d'une grande importance dans la th6orie qui nous occupe. Elle exprime une condition de possibility pour l'existence des genres, et fait voir que de touS les caracteres complets qu'on peut former pour un systeme donne d'invariants, la moitie correspond k des genres qui n'existent pas. Nous verrons plus tard que les genres, dont les caracteres satisfont k la condition de possibilite, existent toujours. La consid6ration des caracteres simultanes est tres utile en beaucoup de questions, qui regardent les formes quadratiques; cependant, il faut admettre qu'on pourrait ^ la rigueur s'en passer dans la theorie des formes ternaires, quaternaires et quinaires. En effet, pour ces formes il n'y a jamais qu'un seul caractere simultane, dont par consequent la valeur est d6termin6e par celle des autres caracteres. Mais deja, pour certaines classes de formes senaires, il y en a deux dont le produit est determin6 par la condition de possibilit6. 4N2 644 MEMOIRE SUR LA REPRISENTATION [Art. 8. 8. Theoreme.L-Soit p un nombre premier impair, qui ne divise aucun des nombres a,,..., a,,; soit aussi K= (-1)ia2...an, n etant = 2i + 1 ou = 2i; et designons le nombre des solutions de la congruence 2 a X.2,u, mod p, s=1 par p" X {); on aura: si n=2i+1,....... (i) (x)=- X 1+( p p; sin= 2i,..... (ii) (x) =- x 1 + ( ) - -- ( ). on selon que,u est congru, ou non, i zero pour le module p. Ce th6orbme est compris, comme cas tres particulier, dans les resultats obtenus par M. Camille Jordan*. Theoreme II.-Soit f1 une forme quadratique donnee, p, un nombre premier qui divise I,, mais ne divise aucun des nombres I, 12,..., IS-1; le nombre p"x (u) des solutions de la congruence f,-, modp, est donne par les formules suivantes, dans lesquelles on a fait Qo= 1, 21 = -I,, Q2 = - 12, * Traite des substitutions, p. 156-160; Journal de mathematiques, 2e serie, vol. xvii. p, 368-402. Art. 8.] DES NOMBRES PAR CINQ CARRIS. 645 Q,3= - x - Ix - 1, aI. = pr emier ~- I.p. I.,u premier a p. n=2i, _un - 2 Pn -3' - [1+( - _,- ____ 1 _pi-_ 1 Qn3 fn2 2i-\ 1 1 pI \ p rp-r [1 p + pt I - 1 p -1 pL v p p-J n=2i+1, () )= lr i - ()~"-2 On-1 1 p L p/jp p' 1 rF + (,i / n-3on-2\ 1 1 pL v, p piJ-l 1 r' -Q4-3\ 1 1 - l p p'J 1 r- F (~ -20-l2n-2r r _' 1 -pL + p J/p0 - pL k-Jp)Y' P: - 20'> H. U-O, modp; p=ps. 1 iP (0) = 0() si s est impair, (O)- [+( -p - p_ - ( p1 p- P si s est pair. La demonstration de ces formules se deduit immediatement du theoreme (I), en considdrant, au lieu de la forme f, une forme canonique dquivalente. On voit que dans tous les cas le nombre ~ (,u) s'exprime par les caracteres quadratiques des invariants et par les caracteres generiques principaux du systbme de formes adjointes i fJ. Si, au lieu de p, on prend ulne puissance 646 MIMOIRE SUR LA REPRESENTATION [Art. 8, quelconque de ce nombre, le nombre des solutions s'exprime toujours par les memes caractBres. Lorsqu'on prend pour module une puissance de 2, les caracteres supplementaires et simultanes figurent dans les resultats, qui deviennent par consequent plus compliques. Mgme pour les modules 4 et 8 (qui sont les seuls qu'il nous est necessaire de considerer ici) il faut distinguer plusieurs cas. Pour abrager, nous ne nous occuperons que de quelques cas particuliers, qui nous seront utiles plus tard. Theoremne 1II.-Soit f, une forme impaire, dont tous impairs; soit aussi (- 1)(=( —l)(~,n-l) le caractbre simultane des formes adjointes f,; c une unitd negative; A, l'unite definie par la congruence A — xIx _x L._ x..., mod 4, les invariants sont donnde positive ou s=l, 2,..., n-I; enfin soit w le residu pour le module 2 du nombre 4 (A,- +1) (A-2-1) + 4(A, _,+l)(A.3-2 )+ +l(A3+ 1) (A2 1)+ (A2+ 1) (A- 1); le nombre 22n x 1 (e) des solutions de la congruence fi-, mod 4, sera donne par les formules I. n =4i, II. n=4i+l, ( (6) -1 [1 + (- 1) ++ c _ 1 ]o7 @ =4 [ ( )i22i j () = [1 + (- 1) i++6A 1] 4 22i.1 III. n=4i+2, ()=-4[l+(-l) t+ - + i ff+(i\l I) ] IV. n=4i+3, D(e)=~[l+(-l)4++' 22+] Demonstra4ion.-Soit x 1 - xi +3 Ixl+3 Lx+..+. In- LLI..., Art. 8.] DES NOMBRES PAR CINQ CARRES. 647 une forme canonique 6quivalente fi, le module etant une puissance quelconque de 2; ecrivons aussi X-X, mod 4, X1= I1 Ax + 2 X2 +... + 7n, Xn, i, 12, '**,,, Btant des unites positives ou negatives. Soit,A + v =n, v 6tant le nombre des unites negatives r; h = 2 (v - 1) ou = (v - 2), selon que v est impair ou pair. En observant que le nombre des solutions de la congruence s=S x2 e, mod 4, s=1 est donne par la formule 228 x 4[1 + - (1 + i) + 1 (1 - i)], i etant l'unit6 imaginaire, on trouve sans peine que les formules du theoreme sont verifiees lorsqu'on y ecrit (-1)h a la place de (- 1) + 6. Il faut donc demontrer l'egalit6 de ces deux unites. Soit z - [(1 - 1) (n12 - 1) + (l 2 - 1) (1 2 13 - 1) + + (n —7112 * — 71n-2 1) ("1"2 "' %n-L 1)] +2 [(qh- 1) + (r11i2- 1)-+(1 ]2 73 - 1) +.. + (rl' 72 --- 7- 1) -'+ 1 (nl2,*:n * + 1) (I1 *'' *n* -1- -1)] -4 [(11 _ 1) (11]2 +J 1) + (nl" 2 - 1) (nl 72 "3 + 1) + - + ('.12 —..- -l-1) (1 2... in + 1)], mod 2*. On trouve d'abord = -h, mod 2; car la valeur de 2, (mod 2) ne change pas lorsqu'on y permute entre elles deux unites consecutives qui ont des signes * On peut regarder (- 1): comme le caractere simultan6 des formes adjointes a X; 71/, 11 X /22,?h X 7/2 X 773., X 72 X * X... X7 n-l etant des nombres represent&s simultanement par ces formes, et tons les invariants etant des unites positives, excepte le dernier, qui aura pour valeur 771 X 7/2 X..X,n - An-1. 648 MEMOIRE SUR LA REPRESENTATION [Art. 8. contraires; de plus, la congruence = h, mod 2, se verifie immediatement si l'on suppose que les,u premieres unites soient positives, les v dernieres negatives. I1 reste done a demontrer que - + C-, mod 2. l]crivons, dans l'expression congruentielle de 2:, pour r1,,2...* rn les valeurs equivalentes 7l =/1, 23-33 I3 I2,...,, n-n 1f3l I 1 2.., -1, mod 4, on aura 2,-[(0-l1) (A +l 02 + 1) + (Al 0 - 1 ) (A2 03+ 1) +... + (A,-3 60n-2- 1) (An-2 _-l + 1) + (A,2,On_ - 1) (An l + 1)], mod 2; d'ou l'on tire f -1-4 [(An-3 On-2 1)(A 0 +1) -4 (An-2-1) (An-1 + 1) - + ( -2 - 1) +[(0n-2-1) (f-1-1) + (In + ) (0-l - 1)], mod 2; et, par consequent, en ajoutant les valeurs de 2n - Z,_, - n- n-2,... 1 =n-o + c, mod 2. c.Q. F.D. Voici des formules plus g6ndrales. Soient ae, mod 4, un nombre quelconque impair, 28 une puissance quelconque de 2, 2n ( (a) le nombre des solutions de la congruence fl -a, mod 21. On aura, dans les memes cas que ci-dessus: I. (a)= [1+(-1)+ — 1)]II. Si e-=A_,, mod 4. (P (a) = 1( I )+w+~[-a. ( 2) [ ( ) 22[ (J 22i-]] Si e-A-_, mod 4. (): (-(-)'+i+ ) nII (a (,[)= -1 [ +(- i)*^- +.22 Art. 8.] DES NOMBRES PAR CINQ CARRES. 649 IV. Si E A-_,, mod 4 (I,(a)=2 1+( 1)'i-)+f22i+l Si e= A,_-, mod 4 r 1/A._i - 2 i]. () =- I ( 1 )+ + [ 22i+1 (V 24i+11] I1 existe des formules analogues pour les valeurs paires de a. Ainsi, par exemple, en ne considdrant que le module 4, on aura les formules I. z (2ua) = [l- (-1) +a++ (A+ 1)x 2] II. q)(2a)= [1 +(-il)i+a+*+*x 21]. III. 4 (2a)= I[+(- 1)'+ ) +a+-+(A-1)x2,i IV. (2a)= [ l+(-1) +a+~~+' Ax2i, le nombre des solutions de la congruence 1 2 a, mod 4, etant toujours designle par 22n $ (2a). Pour les puissances plus elevdes de 2, il faudrait distinguer entre les solutions primitives (celles dans lesquelles il y a au moins une des indeterminees dont la valeur soit impaire) et les solutions derivees. Mais ces formules ne nous etant point necessaires, nous pouvons les passer sous silence ici. Theoreme IV.-Soit f, une forme paire, dont tous les invariants sont impairs, on aura pour le nombre 2n x 1 (1) des solutions de la congruence =fi _1, mod 2, l'expression (1)= 2 [1 o(2) -n], le nombre n des indeterminees etant necessairement pair. On tire la demonstration de cette formule de lobservation qu'une forme binaire improprement primitive represente des nombres impairement pairs pour VOL. TT. 4 o 650 MEMOIRE SUR LA REPRESENTATION [Art. 8. trois des quatre systemes de valeurs (mod 2) des indetermin6es, ou pour un seulement, selon que le determinant de la forme est _ 3, ou = 7, mod 8. Supposons, par exemple, qu'on ait n _0, mod 4; A = 1, mod 8. Consid6rons une forme canonique equivalente f, et supposons que les congruences aa' - b2 3, mod 8, aa - b2 7, mod 8, soient satisfaites pour i, et v respectivement des form s binaires improprement primitives, dont la forme canonique se compose. Les deux nombres u et v seront pairs, et le nombre des solutions de la congruence proposee sera [(1 + 3)- (1 - 3)Y] x [(1 + 3)v + (1 - 3)v] +Fi [(1 + 3)/ + (1- 3)t] x [(1 3) - (1 3)] =[2 - 2In] ce qui s'accorde avec la formule du theorhme. On demontre sans difficulte que la congruence f 1a, mod 28 admet le meme nombre de solutions pour toute valeur impaire de a. Done, en designant, comme ci-dessus, par 2n8 i1 (a) le nombre des solutions de cette congruence, on aura (I (a)= 2 1-( l 2]* Theore'me V. -Soit f1 une forme dont tous les invariants sont impairs, excepte le premier '1, qu'on suppose impairement pair; soient aussi impaires les formes J2, f3,...; en d6signant par a un nombre impair et conservant les m6mes notations que ci-dessus, on aura pour le module 2a (3, 3) I. ~ (a)= [l+(-l (a+l) ) -) ]. (a) 21[+( - ) - (A+(-1l)i(a- )) x A] III. 1(a)= [1 +-1)i+()w+c(2 )(a-1)x 1] IV. (o) = [1+(-1) i + + ( 2)( a 2 (_l) - )) Art. 9.] DES NOMBRES PAR CINQ CARRES. 651 I1 est bien entendu qu'on doit remplacer 1 par - 1 dans les definitions des unites (-1)a et A, et que (-1)~ est toujours le caractere simultane def,f, f.... La demonstration de ce theoreme peut s'obtenir par le moyen des formules prdcedentes, en consid6rant la forme canonique fi -=a, + xI [a + 1a [a2 2x +...], mod 21, et en determinant le nombre des solutions de la congruence a2 2+ a32x3+... u, mod 4, dans laquelle il faut remarquer qu'on a 2 a3...an al, mod 4. 9. The'oreme L.-Soit 'qp une forme quadratique de n - 1 indeterminees, repr6sentee primitivement par fi; soient aussi Dni le determinant et P_ -2 la contravariante de /, divis6e par A-_2; la forme -Jn-l X On-2 sera congrue k un carr6 pour le module D-1. En effet, en supposant que la matrice de ( soit (n-l) x(n-1) I,,J 11........... (i) il y aura toujours une forme dquivalente kf, dont la matrice sera nxn i,,jl,.......... (ii) et l'on dtablira facilement l'6quation - A x [a, j, a, = [ai, n] x [aj, n]- [ai, j] x [a,, J,... (iii) qui subsiste pour toutes les valeurs de i et j depuis 1 jusqu'a n - 1, et dont les deux membres deviennent identiquement zero, si l'un de ces deux indices est egal a n. En observant que [an, n] = Dn-, on en conclut la congruence DnA —1 dans laquelle nous avons derit n-1 402 652' MMMOIRE SUR LA REPRlSENTATION [Art. 9. I1 faut remarquer (1) que si D- est premier - A\,-, la forme 'pn- est primitive, (2) que m6me quand D-I est pair, la congruence (iv) subsiste pour les coefficients des doubles rectangles 2x, x2, 2x, x3,.... Theoreme II.-Soient f une forme primitive, p un nombre premier dont la puissance p8 divise le premier invariant de f, M un nombre premier h p, qui satisfait k la condition M( ) f la congruence Mxf- o, mod p8, sera resoluble, et admettra deux solutions. Theoreme III.-Soient f une forme proprement primitive, 2" une puissance de 2 qui divise le premier invariant de f Si s = 1, la congruence M xf fo, mod 2s, sera toujours resoluble, et admettra une seule solution. Si s = 2, cette congruence admettra deux solutions, ou sera irrdsoluble, selon que la condition ( 1) (- i) = (_ 1) (/-) est satisfaite ou non. Si s = 3, les conditions de resolubilite seront _)(M^1) = (_1(f-(M2-l(-) () = ( (f12-1) et lorsqu'elles sont satisfaites il y aura quatre solutions. Probleme.-Soit 0, une forme quadratique de n- 1 indetermindes, ayant pour invariants I,, I2,..., MIn_2, et dont la contravariante primitive _nsatisfait a la congruence -In- x q n-2-, mod M, Ini aussi bien que I'_2, In_3,..., I1, etant premier a M; trouver toutes les Art. 9.] DES NOMBRES PAR CINQ CARRES. 653 formes de n indeterminees, ayant pour invariants I1, 12,..*, I-2, In-_ qui contiennent J comme partie. La solution de ce probleme pour n=3 a et6 donn6e par Gauss dans les Disquisitiones Arithmeticae. Dans un memoire sur les formes ternaires insere dans les Transactions de la Societe royale de Londres (vol. CLVII, 1867, p. 269), cette solution est presentee dans une forme qui admet une extension facile au cas gdndral. Soit comme ci-dessus n-lxn-1 11 ai,j 11 la matrice de /,, et posons -In-I x _ =-2X [On, 1 l +/n,2 2+ +... +, _ x_1?,]2, mod i.. (iv) Cette congruence indeterminee equivaut a I (n2 - n) congruences numeriques, qu'on peut cdrire comme il suit: -In-i X [ai,j, a, ] = 0i, j M- i, n i3n, n, z=1, 12,..., n-1, 1^ *..(v) j=l, 2,...,n-l, J et qui fournissent un systeme de nombres entiers, avec lesquels nous formons la matrice symetrique nxn en attribuant a f3n, n [qui ne figure pas dans les equations (v)] la valeur M. Cela pose, designons par a, j x An_ les determinants mineurs de la matrice de q,, de sorte que An-2 X ai, = [ ai, j, a, n], et determinons les nombres a,,, an,2,.., a,,, n- par les equations al, j an, 1 + a2, j an, 2+ * * a + an-1,j an, n-l= 7 - j 2 j=l, 2, 3,..., n-1. n-1 xn-1 Cette determination est toujours possible, puisque le determinant 11 aij [ est different de zero; quant au nombre a,,, nous supposerons que sa valeur soit 654 MEMOIRE SUR LA REPRESENTATION [Art. 9. donnee par l'equation PI, n an, 1 + n, a 2+.* * * + Pn, n n,, (vii) = I1x I x... X -23 X I-xl: An, I n-I dans laquelle f,, =M est different de zero. Les nombres a,, a,,..., an,, nous serviront pour completer la matrice nxn 11ai,jl1;. * *...... (viii) mais il reste a faire voir: (1) que ces elements nouveaux sont entiers; (2) que les invariants de la matrice ainsi compl6t6e sont I1, 2,..., I,_. Pour cela, observons d'abord que la matrice reciproque de la matrice (viii) est precisement n xn i An-lx xi,,j II En effet, les equations (vi) peuvent s'ecrire [an, j] = An- Rn, j, j =l, 2, 3,..., n - 1; de plus, on a evidemment [n, n] = An- x M= An-_ n,.; et, par cons6quent, en substituant les valeurs trouv6es dans l'equation (vii), nxn | cai, j I= AnII reste encore a d6montrer les 2 (n2 - n) egalit6s [a,, ] = An-_, j, i=1, 2,..., n-1, j=l, 2,..., n-1 on y parvient aussit6t en comparant les deux systemes d'equations (iii) et (v), auxquels satisfont respectivement les nombres [aij] et [/ij]. Maintenant, on conclut de la congruence (iv) que les nombres /n,i, z=1, 2, 3,..., n, sont premiers entre eux, puisque, s'il y avait un diviseur commun de tous ces nombres, I,_1 et M=,,, n ne seraient pas premiers entre eux. Art. 9.] DES NOMBRES PAR CINQ CARRES. 655 Cela pose, les n equations ai, 1 Gj, I + ai, 2 /j, 2 + * + ai, 2 a j, n = 0, ai,1 Pi, 1 + ai, 2 Pi, 2 + * * * + ai, /, n = A — font voir que ac, (i < n) est entier, puisque tous les nombres qui entrent dans ces equations (hormis a, n) sont entiers; et, de plus, les coefficients de ai, dans les diff6rentes equations sont premiers entre eux. La meme demonstration reussit pour i = n, quand on suppose que la proposition est d6jv demontr6e pour. les valeurs de i inf6rieures a n. Nous avons dejk vu que le determinant de la matrice (viii) est A,; on salt aussi que le plus grand diviseur des mineurs du ni'me ordre est An-1_ puisque [ai, j]= A_ x 3i,, les n nombres 3i, etant premiers entre eux. Mais par hypothese le determinant 1 n-1xn-1 A-X I a,j, n-1 dont la valeur est M, est premier k An_. Donc, par un resultat connu (lemme II, art. 2), le plus grand diviseur des mineurs d'un ordre quelconque dans la matrice de JD est egal au plus grand diviseur commun des mineurs du m6me ordre dans la matrice (viii); d'oW l'on conclut que les invariants de cette matrice sont I,, -2,..., I.n, ainsi que nous l'avons enonce. Il est bon de remarquer que si l'on supposait que le nombre M fAt premier 5 If_1 seulement, sans Wtre premier aux autres invariants I1, I2,..., In-2 on pourrait determiner comme ci-dessus la matrice (viii), dont les coefficients seraient entiers, et qui aurait An pour determinant et An-1 pour le plus grand diviseur commun de ses mineurs de lordre n- 1. Mais il ne s'ensuivrait pas que An2_ An_3,..., fussent les plus grands diviseurs communs des mineurs des ordres inferieurs, et par consequent les invariants de la matrice (viii) pourraient 6tre diff6rents de I1, I2,..., In-. Pour faire voir quelle est la relation entre les diff6rentes formes fl qu'on peut construire, par le procede que nous venons de d6crire, en partant d'une m6me forme 40b, remarquons que si l'on prenait comme solution de la congruence (iv) le systbme fn3 1 -k1MM,, 2-k2 M, 3.,3.k3M., 656 MI]MOIRE SUR LA REPRI]SENTATION [Art. 10. au lieu du systeme Pn, 1, Pn, 2 On, 3) ' la nouvelle forme fi serait equivalente a l'ancienne et s'en deduirait par la substitution 1, 0, 0,..., kl 0, 1, 0,..., k2 0, o, 0, 1...,. (ix) 0, 0,,..., k 0, 0, 0,..., 1 R6ciproquement, si f' est une forme quelconque contigue k fi [c'est-h-dire equivalente a fi par une substitution telle que (ix)], on d6montrerait sans peine que ces deux formes conduisent a des solutions identiques (par rapport au module M) de la congruence (iv). On conclut de la qu'on peut construire un systeme de 2 + formes f1 diff6rentes entre elles, dont aucune n'est contigue a une autre, et ayant cette propriet6 que toute forme qui contient 0p comme partie sera contigue a une d'entre elles. 10. La forme (1 aux invariants [I1, 12,..., MI_2] 6tant represent6e primitivement par la forme f, aux invariants [I1, 12,..., I, —], et le nombre M etant toujours assujetti k etre premier aux invariants, les caracteres ordinaux et gen6riques defi (sauf un cas d'exception dont nous ferons mention plus tard) sont determines par ceux de q1. (1) Soit 1 impair; il s'ensuit que 0i_- est une forme impaire; si _.-2 est une forme impaire, ou si, n-2 etant paire, I'_, est aussi pair, on voit sans peine que tout systeme de nombres, caract6ristique pour les formes (1, P2,.., devient un systeme caract6ristique pour 01, 02,..., en y joignant le nombre M. Mais si.n-2 est pair, In-h impair, on a d'abord I_ 2 impair, puis 0_-2 impaire, parce qu'une suite de formes alternativement paires et impaires ne peut pas se terminer par une forme impaire 0n-, ayant un invariant impair I,_. Done, dans ce cas, les nombres caract6ristiques pour..., _n3, n-2, ne peuvent pas, k la rigueur, 6tre regard6s comme caracteristiques pour les formes..., _n-3, On_2. Cependant ces nombres suffisent pour determiner les caracteres gen6riques de..._, 0n _, f20n82,n, puisque, les invariants..., I_2? Il-\ etant impairs, il Art. 10.] DES NOMBRES PAR CINQ CARRES. 657 n'est question que des caracteres principaux; le caractere simultane de..., n_-2,n-_ se ddduisant de la condition de possibilite. (2) Soit M pair, ce qui implique que tous les invariants sont impairs; si n est pair, les formes (I, p2),..., 01, 02,... sont n6cessairement impaires, puisque dans les deux cas il y a une suite d'un nombre pair d'invariants impairs; la m6me chose a lieu si n est pair et /l impaire. Mais si n est pair et la forme f1 aussi paire, il peut y avoir une ambiguite dans la determination de lordre de 01, puisque les formes 01, 02,..., peuvent 6tre ou impaires ou alternativement paires et impaires. Dans chacun de ces deux ordres comme dans les cas ou il n'y a qu'un seul ordre admissible, les caracteres generiques des formes 01, 02,... sont absolument determines. Le cas d'ambiguite que nous avons signale merite un examen plus attentif; cependant nous le laissons de cote ici, puisqu'il ne nous est pas necessaire pour la theorie de la decomposition des nombres en cinq carres. R6ciproquement, le caractere generique de p1 est determine par celui defi. On a; en effet, (^)' (-8), s-, 2,... n-2, si q est un diviseur premier de I; de m~me, si I est divisible par 4 ou 8, les caracteres de 0, et (P par rapport a ces modules sont coincidents; enfin, les caracteres de n -2 par rapport aux diviseurs de M se determinent d'apres la congruence -I_1 x 1n-2= -1, mod M. Quant aux caracteres simultanes, ceux qui se rapportent k des suites de formes qui se terminent avant d'arriver a On-2 ou Pn-2 ont la mme valeur dans les deux systemes; tandis que la valeur du caractere simultane qui peut exister pour les dernie-res formes de chaque systeme est determinee par la condition de possibilite. Cependant, il faut bien remarquer qu'ici encore il y a un cas d'ambiguite. Si Mll est impair, et si la serie...,, -I,_3, In-, se termine par une suite d'un nombre pair d'invariants impairs, les formes..., On-3) fOn-2, n-, seront necessairement impaires, mais les formes..., _3, pn-2 pourront Wtre ou impaires ou alternativement impaires et paires. Done, dans ce cas, il y a deux genres de formes des invariants [I1, I,..., MI_ 2], qui peuvent etre representes par un VOL. II. 4P~~~~~ VOL. II. 4P 658 MEMOIRE SUR LA REPRESENTATION [Art. 11. genre donne de formes des invariants [I], 1,,..., In_, I_-]; mais ces deux genres appartiennent a des ordres diff6rents. Ce cas est tres important pour notre but actuel, puisqu'il se presente dans la theorie de la decomposition des formes quaternaires en cinq carres. 11. D'apres la definition d'Eisenstein, la densite d'une classe de formes quadratiques est la fraction qu'on obtient en divisant l'unite par le nombre des substitutions lineaires, a determinant + 1, qui transforment en elle-m6me une forme quelconque de la classe. La densite d'un genre est la somme des densites des classes qui appartiennent au genre. Ajoutons h cette definition que la densite de la representation primitive d'une forme de n - s indeterminees par une forme de n indeterminees est le produit des densites des deux classes auxquelles ces formes appartiennent; en particulier, la densite de la representation d'un nombre par une forme est la densit6 de cette forme, ou plus exactement de la classe t laquelle la forme appartient. Soit nxn-1 i1, 2,..., n-1,) II, j=1, 2,, n...... (i) une matrice de n - 1 colonnes verticales qui sert a representer primitivement la forme /1 par f, et designons par [Sn,1, [n, 2 *],... [S,,] les determinants de cette matrice, qui seront premiers entre eux, puisque la representation est primitive. Alors M sera repr6sente par On-1, en attribuant aux indeterminees les valeurs [Sn, 1], [S, 2], *.-. Nous dirons que ces deux reprdsentations sont contravariantes ou simnultanees. ltant donnee une representation de 0q par 01, il n'y a dvidemment qu'une seule representation de M par 0,_,. Mais si ~ est la densit6 de 01 ou de,,_, 5 une representation donnee de M par 0n,,- il correspond S representations differentes de f, par 01. En effet, les matrices (i) qui ont [S,,],,[], *... pour determinants sont comprises dans la formule nxn-1 n-ixn-1 la seconde matrice ayant +1 pour son d6terminant; d'oi l'on voit qu'il y en a precisement S qui transforment fi en f. On conclut que la densite d'une Art. 11.] DES NOMBRES PAR CINQ CARRES. 659 representation donnee de M par 0_,- est toujours egale $ la densit6 des repr6 -sentations contravariantes de /1 par 01. On peut se servir des considerations pr6cedentes pour demontrer la proposition importante, 'que tout genre dont le caractere complet satisfait a la condition de possibilite, existe effectivement.' Pour cela, considerons un caractere generique donn6, qui satisfait h la condition de possibilit6, pour les invariants I1, 1I,..., I_-,. Soit M un nombre, dont les caracteres par rapport aux diviseurs de In,_ coincident avec les caracteres g6neriques proposes; il faudra que M soit premier aux invariants et impair ou impairement pair, selon que la forme contravariante On- est impaire ou paire. En supposant que la proposition soit vraie pour les formes de n- 1 indeterminees, imaginons un genre auxiliaire des invariants [Ix, I-,..., Mln_2], dont les caracteres, par rapport aux diviseurs de I, I2,..., In-2, soient les memes que les caracteres proposes, et dont les caracteres, par rapport aux diviseurs premiers impaires de M, soient les m6mes que ceux de - IT,. Pour completer ce caractre generique, on lui assignera les memes caracteres simultanes qui font partie du caractere generique propos6, exception faite du caractere simultane des dernieres formes adjointes, si toutefois il y a un tel caractere. Or, le caractere qu'on a suppose au genre auxiliaire satisfait k la condition de possibilite. En effet, si w = 1 est cette condition et Q2=1 l'equation correspondante qui par hypothese est satisfaite par le caractere generique propose, les facteurs de Q qui ne se presentent pas en w sont /-_ x ( (-2 - l) (-~ 1) X (-1) (I- + 1)(0_- ) X(-)n-l1 (2-1) et les facteurs qui se presentent en w sans 6tre facteurs de Q sont (-) x ( -l)I(M-1)(_2-l)..... (iii) En faisant attention t l'equation (, - 2 -( -, )=(-) (- ) (M-) X -ll(M- l) ( M ) et en observant que, dans les formules (ii) et (iii), on peut remplacer 0,_- par M et 2- ((n-2-l) par 2 (0_-2 -1), on voit que les deux unites exprimees par ces 4P2 660 MEMOIRE SUR LA REPRESENTATION [Art..12. formules sont egales, et que, par consequent, l'une des equations Q= 1, w= 1, entralne n6cessairement l'autre. Done le genre auxiliaire existe actuellement, et une forme quelconque de ce genre servira pour construire une forme du genre propose, dont l'existence est ainsi etablie. Pour abreger, nous avons suppose M impair; mais la demonstration n'exige que de legers changements pour 6tre applicable au cas oui M serait pair. 12. Theoreme.-Soient f1, f[, f',..., des formes representant les classes differentes d'un genre donn6 (F) aux invariants [J1, 12,..., I,, -; soient de m6me (P, pq', q',... les formes representant les classes, aux invariants [I1, I2...,MI,_2], qui admettent une representation par les formes de (F); en supposant toujours que M soit premier a tous les invariants de (F), soit A le nombre des diviseurs premiers impairs de M; soit j = 0, 1, 2, selon que M est (1) impair ou impairement pair; (2) 0, mod 4; (3) 0, mod 8; soit enfin 7 la densite totale des classes (q1), (1), *...; la somme des densites des representations primitives, soit des formes (91k), ((1),... par les formes (fi), (f),..., soit du nombre M par les formes contravariantes (fn-_), (fn_),..., sera egale 2/+j x 7. On se rappellera que les formes (4), (41),..., representent les classes, soit d'un genre, soit de deux genres qui appartiennent t des ordres diff6rents. Mais cette remarque, quoique tres importante pour les applications, n'a aucune influence sur la demonstration du theoreme. Pour abreger, nous supposerons qu'on ait exclu le cas dans lequel n et M sont tous les deux pairs; cette exclusion faite, on peut s'assurer que toutes les formes (qk), ((1),..., sont representees par un seul genre (F). Pour le cas n = 3, la demonstration du theoreme se trouve dans le memoire Sur les formes quadratiques ternaires (Philosophical Transactions, vol. CLVII). Le meme raisonnement, avec de legers changements, s'applique au cas general, et il suffira de la reproduire tres succinctement ici. Par le procede de l'article (9), de chaque forme (P on deduit 2L+j formes F1, qui contiennent (p comme partie et qui appartiennent au genre (F); ces formes sont toutes differentes entre elles et il ne se trouve pas parmi elles deux formes contiguis. Considerons une quelconque des formes representatives f1; on voit facilement qu'il y a autant de representations de 01 par f, qu'il y a de transformations de f dans des formes F1. Soient - la densite de f, s le nombre des formes- F, qui sont equivalentes af'; le nombre des transformations de f en des Art. 13.] -DES NOMBRES PAR CINQ CARRES. 661 formes F1 sera d x s; par consequent, si est la densite de <^, la densite des representations de /b par f, sera d x sx =. tendons cette conclusion toutes les formes f1, f,...; la densite des representations de /iP par les formes de (F) sera - x 2s; mais Zs = 2A +J, parce qu'il y a 2A +J formes F1, dont chacune est equivalente k une, et a une seulement des formes ft. Done enfin la densite des representations de toutes les formes P1, ~,..., par les formes de F sera 2A+j x 2 1 = 2+j x 7. C. Q. F. D. 13. Soient I1, 12, I les invariants d'un systeme de formes de quatre indeterminees x1, X2, x3, x4; (F) un genre donne de ces formes. Nous nous proposons de determiner la densite W de (r), en supposant d'abord que les formes 01, 02, 03 et les invariants I1, I, 13 soient tous impairs. Nous designerons par q les nombres premiers qui divisent les invariants; qj ne divisera qu'un seul de ces nombres, q,j en divisera deux, q1, divisera tous les trois. Soient L un nombre positif qu'on peut supposer aussi grand qu'on voudra, ML un nombre impair et premier a II q qui ne surpasse pas L, qui est represente (primitivement ou non) par 03, et qui en outre satisfait aux conditions M 1, mod 4, (-) _1, mod qj, pour toute valeur de q/ (on verra plus tard l'avantage qu'il y a a introduire ces restrictions). Soit enfin T la somme des densites des representations de tous les nombres Ml par les contravariantes 03, 0, 0,..., des formes 01, 60, ',..., comprises dans (r). On parvient a determiner la densite de (r) en egalant deux expressions differentes qu'on T peut trouver pour la limite de L, lorsqu'on fait croltre L indefiniment. I. Les valeurs positives ou negatives de x, x, x o, x4, pour lesquelles une forme donnee 03 acquiert une des valeurs designees par M, sont evidemment distribuees dans un certain nombre de progressions arithmetiques x,= A y, +, = Ay + (i) gX= Ay2+X, 2 =4=, +Ay4+X,), dont la difference commune est 4 H q =. Nous designerons par 4 X /(A) le nombre de ces progressions; d'apres les 662 MEMOIRE SUR LA REPRESENTATION [Art. 13. th6oremes de l'article (8), on aura pour / (A) l'expression (A> )=xrxn(i-)xn i[+(- 1 1) ] ] xII[1-( 3 l,2 3 _1,2-] X -] _ _ 2] q1, 2 q1, 2 q2 2 ou l'on a 7=1 si J1xI-1, mod 4, et T= [2 +(- 1) (+1) (I3- )+ ] si I1 x I3 -1, mod 4, en designant par (- 1)" le caractere simultan6 des formes 01, 02, 03, c'est-h-dire de (r). Soit vi le nombre des systbmes de valeurs de x1, x2, x3, x4 comprises dans la ieme des formules (1), qui satisfont a l'inegalite 03 (X1, x2, X3, X4) < L; on a, d'apres les principes connus, A4 i 1 7T2 Lim_ L2- -2 -[-==3= 3 et, par consequent, /g1 72 2 1 7'2'3 '1' L2 2 X 1717TX^^, 2- x X 3s- I ~ 13 ). Multiplions chaque membre de cette equation par la densite de la classe 01 ou 03, et prenons la somme des produits pour toutes les classes de (F); on trouve immediatement T 172 Lim = W x xI()x 12' 2.31 X J1I3 II. On obtient une seconde determination de la meme limite en se servant du theoreme de l'article (10). Soit (7) le genre de formes ternaires aux invariants (I1, MlL), qui admet une representation primitive par les formes de F; ce genre est certainement impair, puisque les deux invariants sont impairs. Soient 'q une forme ternaire appartenant au genre (7), et qui peut etre repr6sentee primitivement par 0,; j2 la contravariante de pi. Les caracteres principaux de (y) se determinent par les equations suivantes, dans lesquelles x designe un diviseur quelconque impair de M: Art. 13.] DES NOMBRES PAR CINQ CARRES. 663 (bql) (ql) (q2 (q2) 1 (OIL (_ (__ 0 \ ( 2l, 2) 1, / 2 2 (1, 2, 3 (q, 2, 3 3 (0) = (rj3).J \l 2U!71 2 P.1 2, 3 ' 2 II n'y a aucun caractere supplementaire. Soit (- 1)' le caractere simultan6 de 01, P2; on aura En effet, les valeurs de a' et a sont donnees respectivement gruences - ~[( 21 - 1) + (I1 + 1) ( - 1) + (2 + 1) (2 -1)] =- [(01 - 1) (02 - 1) + (- 1) (03-1) + (I1 + 1) (01 - 1) + (I, + 1) (60 - 1) + (3 +1) (03 - 1)], mod 2; par les con* (4) J d'ailleurs, si (mn, mn2) est un systeme caracteristique pour les formes (01, 12), (mi, m2, M) est un systeme caracteristique pour (0k, 02, 03), de sorte que dans les congruences (ii) on puisse ecrire min, mi2 la place de (1, (12, et m1, izn, MM la place de 01, 02, 03; cette substitution met en evidence la congruence a = -, mod 2. Cela pose, on a, d'apres une formule connue*, l'expression suivante pour la densite w du genre (7): W= 1 2X 1I T x I [2 +(-_) 4(Al), (M l)+2] XII^(1 I If(1- 1 - (, 2 1, 2, 3 X 2[t q1- qI 2[+ q1,3 )qI3 2) X I [[1 -i+ ( - MIt7, 3 2, 3 X H1 e[1+ (' —r I1~2)] * Phil. Trans., vol. clvii. p. 291 [vol. i. p. 499]. 664 MEMOIRE SUR LA REPRESENTATION [Art. 13. En multipliant par 2/ et faisant attention aux equations (3) et aux conditions satisfaites par M, cette formule devient 2Pxw=i. w I x 12 x [2+ (-1)- (l+l)( I2+1l)+~(] 2xw=*x/1112 )x tI, ql 1. xnI[1+( -2 )]>x n2[1+~ (-1200) 1 ] qL v g'i /J " '-1,3 / I,3 X +(_IIxnr + (t) 1 2 ' * *(5) 2!7 2'a, / 42 2 [ < ^2,3 ^2,3 -xMxTn[i+(' li2)], p Btant un coefficient qui reste le meme pour tous les nombres M et pour tous les genres ternaires (y), qui ont (i1, M1I2) pour invariants, et qui peuvent etre representes primitivement par les formes de (F). D'apres le theoreme de l'article (10), la formule (5) exprime la densite des representations primitives du nombre donne M par les formes de r. Done la densite totale des representations, tant derivees que primitives, de M par ces m6mes formes est exprimee par la formule px ^[ X n [i+( y ) ] ], dans laquelle le signe de sommation se rapporte a tous les diviseurs carres de. M M,,a designant maintenant un diviseur premier de c On en d6duit 1'equation LT XM- [2 n[ ( 1/2) 2 ] L L MiL Lr2 L /A qu'on peut aussi 6crire p MI (II h M -T L X dz ~dI2 le nombre d designant, dans cette derniere formule, un diviseur quelconque Art. 13.] DES NOMBRES PAR CINQ CARRES. 665 de M. Pour passer aux limites, posons d =, et effectuons en premier lieu la sommation par rapport a S. Le nombre d peut avoir une valeur quelconque premiere 411nq et comprise entre 1 et L; tandis que, pour une valeur donnee de d, J peut avoir seulement des valeurs comprises entre 1 et d, et appartenant a certaines progressions arithmetiques dont la diff6rence commune est A = 41 q. Ces progressions ne sont pas les m6mes pour toutes les valeurs de d; toutefois, leur nombre est toujours le m6me; en effet, ce nombre est represente par la formule A x X (A), si l'on ecrit, pour abreger, xn i(l4) xn(i-l ) Hi 1 x rI (1-1)xII 1 (1 - 1 I q1 1 q2 xn fl X —, 21, 2, 3 oi l'on remarquera que les facteurs 2 se presentent ou non, selon que S x d = M doit avoir un caractere quadratique par rapport a q ou doit seulement etre premier par rapport a ce nombre. Ii est vrai que si d > d-, quelques-unes de ces progressions ne contiendraient qu'un seul terme; mais on peut sans erreur sensible supprimer les termes qui se rapportent a ces valeurs de d. Soit a, a+A, a+2A,..., a+(x-l),..... (6) une des progressions dans lesquelles se trouvent les valeurs de S correspondant a une valeur donnee de d <, et posons L b= a — (z-1), b dtant positif et moindre que A. On aura l'equation L2 [ a+ (_ +A) +... + (a+- l)] _ 1 + 1 27b-A (a- b)(A-a-b) 2Ad2 +2 2AL d + 2AL2 VOL. II. 4 Q 666 MEMOIRE SUR LA REPRESENTATION [Art. 14. T d'oh l'on voit que la partie de - dans laquelle d a la valeur donnee sera comprise entre les limites dP (d XOn aura done T I, 12 1 LimP P )) 2L- PX(A)S( I ) 2 d2 la sommation se rapportant a tous les nombres d premiers a A, depuis 1 jusqu'a o, ou, ce qui revient au meme, T __ 2 ( 1 Lmz -zy - x(a)xX I-(^ Xr( 2) iT2 =pX(A) x [1_( - 1 )2/ - m -2 2 la sommation se rapportant a tous les nombres impairs. T En egalant cette valeur de la limite de l a celle que nous avons deja trouvee, on a finalement w=V - x x x X ( -), x (1) "3 Ia[ e ~> - 1 10 X [ [( q 2) + ( 3 02 )] 1 xri[l+[( -32) 1 ]f1+( 102) 1 ] 4 [ (q41,2 ) J 2 [ (q2, 3 q 22 3 |1 4 I (-I7m),) On remarquera que cette expression est sym6trique par rapport a IL, 01, et I1, 03, comme cela devait Btre. La valeur de 3 est + 1, si I x 1- - 1, mod 4, et X-[2+(-1) 14(1 +)(I2 l) f], si 1 x I1l, mod 4. J1 est le nombre des nombres premiers q/ et J3 a une pareille signification. 14. La formule que nous venons de trouver, pour le cas oh les formes et les invariants sont impairs, s'applique avec de 16geres modifications a tous les autres cas; et, en effet, il n'y a que la valeur du coefficient ' qui doit etre changee. Art. 14.] DES NOMBRES PAR CINQ CARRE]S. 667 Soient 23x ' le nombre des formes lindaires (mod 8) dans lesquelles sont contenus les nombres auxiliaires M, qui, dans chaque cas, sont employes dans l'investigation; 247C x 7 le nombre des systemes des indetermninees (mod 2") qui satisfont k la congruence f M ou A,-M, mod2k, l'exposant k etant 1, 2, 3, selon les cas; enfin, X le coefficient* propre au genre de formes ternaires, qui peut 6tre represente par le genre donne de formes quaternaires; on aura c., Y A' 972 - 0 y _X O U - d X _-, selon que la forme 03 est impaire ou paire. Voici le tableau de ces coefficients: A. [1, 12, 13]-[1, 1, 1], mod 2. (a) [0,, 0, o3]-[1, 1], mod 2; =1I, si I x I — 1, mod 4; 2 [2 +(-1) +4 ( )(i-l)+ ] si I x I3_ + 1, mnod 4, (- 1)' etant le caractere simultan6 de 0Q, 06, 03. (b) [0,, 0,, 03] [0, 1, 0], mod 2; [4 + 2 I)] B. [J,, 7^ ] _ [1, 1, o], mod 2. Dans ce cas, les formes 06, 02, 03 sont impaires. (a) I4-2, mod 4; = +1. (b) I-4, mod 8; =l [2+(-) rl(e+l)(I 2 3, l)+,. (- 1) etant le caractere simultane de 0i, 0,. * Phil. Trans., vol. clvii. p. 291, ou ce coefficient s'appelle C [vol. i. p. 499]. 4Q 2 668 ME MOIRE SUR LA REPRESENTATION [Art. 14. (c) I3-0, mod 8; =1 [2 +-(- 1) 4 (I1+1)(2 03+1) + ]. Ici encore (- 1)Y est le caractere simultane de 0,, 02. B'. [I,, 2, 13]- [0, 1, 1], mod 2. On echange entre eux I1 et 13, 01 et 03 dans les formules B. C. [li, 2, 13] -[1, 0, 1], mod 2. (a) [01, 02, 03] [1, 1, 1], mod 2. (1) 12-2, mod 4; y_ 3 S-4 (2) 1I2-4, mod 8; (= 64[3 +(-1) (I 2+ l)][3 + (-1) (132+ ) (3) I2-O,mod8;.=_.[3+(-_1)2(1 02+l)]X[3 + (-) 12 (13 02+ )] (b) [0, 02, 03] [0, 1, 1], mod 2. (1) 12-2, mod 4; _= [3 + (- ) (3f2 + 1). (2) 12 0, mod 4; 12[3 +(- 1) (32 + 1)] X [2(- 1) (If 2 -1)] (b') [0,, 02, 03]-[1, 1, 0], mod 2. Ce cas est reciproque de (b). (c) [01, 02, 03]-[0, 1, 0], mod 2; = ~-[2 +(- ) 8 (12s- 1)]x[2 +(- 1) 8 (2- )] D. [I,, 2 I 3] [0,, 1], mod 2. (a) [0,, 02, 03][1, 1, 1], mod 2. (1) I 2, mod 4, 3 Art. 14.] DES NOMBRES PAR CINQ CARRIS. 669 oh i = 0, 1, 2, represente le nombre des congruences I1 1 4, mod 8, - = 0, mod 8, qui se trouvent satisfaites. (2) I1,O, mod 4, - +i3[3+(- l)(I3f2+l)]1 oh i = 0, 1, 2, represente le nombre des congruences 1, 0, mod 8, 12- 0, mod 8, qui se trouvent satisfaites. (b) [0,, 02, 03-[1,, 0, ], mod 2. (1) 12-2, mod 4, 2 _ 2-+i (2) 2_ 0, mod 4, =21+ [2 + (- 1)(I3-1)]. Dans ces deux formules (b), (1) et (2), i= 1 ou = 0, selon que a congruence 12 —0, mod 8, est satisfaite ou non. D'. [I1, 12, I3]; [1, 0, 0], mod 2. C'est le cas reciproque de D. E. [I, 12,, Ij[0, 1, 0], mod 2. (a) [0, 02, 03]-[1, 1, 1], mod 2. (1) Si les congruences 1-0, 13-0, mod 4, ne sont pas satisfaites en meme temps, 3 2y ^+i 670 M7EMOIRE SUR LA REPRESENTATION [Ait. 15. i = 0, 1, 2, etant le nombre des congruences JI-0, mod 4, I_ 0, mod 8, 12-0, mod 4, 12 0, mod 8, qui sont satisfaites en m6me tenps. (2) Si l'on a a la fois I-=I,-0, mod 4, =2+3 i[3+(-1)2(i2~03+1)], ou i = 0, 1, 2, est le nombre des congruences I-0, mod 8, I30-, mod 8, qui sont satisfaites en meme temps. F. [I,, 12, 13] [0, O, 0], mod 2. (1) Soit 122, mod 4, 3 Y= 22+i z= 0, 1, 2, 3, 4, etant le nombre des congruences I- 0, mod 4, I1 0, mod 8, I3 0, mod 4, I3 0, mod 8, qui sont satisfaites en m6me temps. (2) Soit I2- 0, mod 4, 3 S=24+(i = 0, 1, 2, 3, etant le nombre des congruences I, od 8,, od 8, 1, mod 8, mod8, 0, 8, qui se trouvent satisfaites en m6me temps. 15. Parmi ces formules, il suffira de demontrer celles dont nous aurons besoin pour la theorie de la decomposition des nombres en cinq carres. JDemonstration de laformule (A, b).-On peut prendre pour M tout nombre Art. 15.] DES NOMBRES PAR CINQ CARRES. 671 impair; done '= -, et par le theoreme (IV) de larticle (8), u' —A[1- (/-I3)]. La valeur de X etant 3, il vient t 5[44 +(R )] Demonstration des formules B.-(a). Ij 2, mod 4. Nous prendrons pour M tous les nombres congrus, pour le module 8, 5 un seulement des r6sidus 1, 3, 5, 7; onauraainsi = X [2 + (- 1) (1 + 1) (MI2 + 1)+ ], en designant par (- 1)01 le caractere simultane des formes ternaires (i1, MI2), qui peuvent etre representees par les formes du genre quaternaire donne. D'ailleurs on a, d'apres le theoreme (V) de l'article (8), 1 = i [2 + (- 1)P +], en designant par (- 1 ) le caractere simultane du genre quaternaire et par p le nombre -I,+ 1) (I2 + 1) + (M- 1) (I1 I3 - 1) + S (M 1) Les indices c- et a sont donnes par les formules 1 - 4 (01 - 1) (02 1) + I (i1 + 1) (01 - 1) + (IM+ 1) (02 1), mod 2, = (0 - 1) (02 - 1) + (02- 1) (03 - 1) + (I1 + 1) (0- 1) + 4 (12 + 1) (02 - 1) +4 (I + 1)(03-1)+ I(02- 1), mod 2. Mais si (,11, Im2) est un systeme caracteristique pour une forme ternaire representee par 0,, (min, m,2 M) est un systeme caracteristique pour 0,; on peut done ecrire mn1, In2, Mi pour 01, 02, 03 dans les deux formules precedentes. On trouve ainsi 4 (I1 +1) (MI2 +1) + p + -, mod 2, et, par consequent, 4=1. (b). I3 4, mod 8. La forme 03 aura le caractere (- 1) (:- ), qui appar 672 MEMOIRE SUR LA REPREPSENTATION [Art. 16. tiendra aussi h tous les nombres M. En effet, la forme canonique equivalente satisfera k la congruence 03 Mx2, mod 4. I1 y aura aussi pour (01, 02) un caractere simultane que nous designerons par (- l1), et qui appartiendra en m6me temps aux formes ternaires representees par 03. Donc on aura 1 = 1' -X =1 [2+ ((- 1) +1) (12 3 + 1) + ] =4 [^2+(-1)(-4'+1) (12 +l) + O] (c). 13-0, mod 8. Ici l'ona 03 l-Mx, mod 8. Tout est comme dans le cas precedent, seulement il faut ecrire n'= -. On trouve par consequent =- 1[2 + (-L 1 (I+ l)(I2 03+ 1) + 16. Puisque la forme s=5 s=l qui est elle-meme sa contravariante, represente la seule classe qui existe de forme quinaire de determinant +1*, il est evident qu'on peut d6duire des theoremes precedents une expression pour le nombre des decompositions d'un nombre donne M en cinq carres. Pour cela, on determinera la densite du genre unique, ou des deux genres, des formes quaternaires (1, 1, M), qui peuvent 6tre representees par cinq carres. Soient y cette densite, I le nombre des diviseurs premiers impairs de M, j= 0, 1, 2, selon qu'on a (1) M impair ou impairement pair; (2) ME 4, mod 8; (3) M= 0,.mod 8; * Cette proposition peut se demontrer, soit par des considerations tout a fait elementaires, soit en determinant la densite des classes quinaires de determinant + 1. I1 n'y a 6videmment qu'un genre, et la,. densite de ce genre, par une application tres simple de la methode que nous avons employee pour les formes quaternaires, se trouve etre egale A I * 24X 115 Art. 16.] DES NOMBRES PAR CINQ CARRES. 673 N le nombre des representations primitives de M par cinq carres. La densite de la classe quinaire etant 4x 115' on aura pour N l'expression suivante: N=24 II5 x 2+j x, -24 X 5 x X2.xMx x 1 M )1 dans laquelle "'= 2j, 2 Btant un des coefficients de la table ou la somme de deux de ces coefficients. Tout se reduit donc a la determination du coefficient /'. I1 y a six cas a consid6rer: I. M=-3, mod 4; II. M-=5, mod 8; III. M-Il1, mod8; IV. M 2, mod 4; V. M= 4, mod8; VI. M= 0, mod 8. I. Soit M- 3, mod 4. II n'existe pas de formes paires ayant (1, 1, M) pour invariants, et il n'y a, par consequent, qu'un seul genre qui admette une representation par une somme de cinq carres. On a donc '=, c'est-a-dire <'=1, puisque 1 pour tous les genres. II. Soit M_ 5, mod 8. II existe un ordre pair aussi bien qu'un ordre impair de formes quaternaires aux invariants (1, 1, M), et, dans chacun de ces deux ordres, il y a un genre de formes capables de representation par une somme de cinq carres. Dans l'ordre pair, la condition de possibilite, a laquelle tout caractere generique doit satisfaire, s'exprime par l'equation ou -4 1-) (02 - 1)+ 4 (02-1) (031) + (0-1) + (02-1) + (03-1), mod 2; mais on a 2 (02 - 1) 1, mod 2, parce que 01 est une forme paire dont le premier VOL. II. 4R 674 MEMOIRE SUR LA REPRESENTATION [Art. 16. invariant est - 1, mod 4; done on a -r 1, mod 2, les formes 0, 03 disparaissant d'elles-m~mes de l'expression de a- (ce qui d'ailleurs est d'accord avec la theorie generale de l'article 7), et la condition de possibilite devient (M)=-1 oubien ( ) 2-( +1 D'autre part, la congruence - 03-, mod M, qui exprime la condition necessaire et suffisante pour que 01 soit capable de repr6sentation par une somme de cinq carres, conduit a la m6me valeur de ( -; d'ot l'on conclut qu'il existe un genre pair qui admet une telle representation. Pour ce genre la valeur du coefficient ( sera;. Quant a l'ordre impair, le genre defini par les equations ()03 (-), (- 0)3=( ) —_, oti q est un diviseur premier de M, satisfera h la condition de possibility et sera en m6me temps capable de representation par une somme de cinq carres; le coefficient y sera 2. Done on aura, en ajoutant, r +2 -1o.0 III. Soit 11M 1, mod 8. L'ordre pair existe toujours, mais aucune forme de cet ordre ne peut 6tre representee par cinq carres; en effet, puisque (-) + 1, )= + 1, la congruence - 03-=, mod Mi est incompatible avec la condition de possibilit6 / =-1 _, =-1 Dans l'ordre impair il y aura, comme dans le cas II, un genre capable de Art. 16.] DES NOMBRES PAR CINQ CARRES. 675 representation par cinq carres. Pour ce genre on aura (-1)=-1, = +12, et, par consequent, IV. Soit M =2, mod 4. Ici il n'y a qu'un seul genre impair qui soit capable de representation, et l'on a 'r== 1. V. Soit M1-4, mod 8; M1= 4M'. De la congruence -03- 2I, mod M, on tire (-1)(03-)=-1, (3,) =(-1) ) done, en d6signant par (-1) le caractere simultane des formes 01, 02, on a (-1= -1,_ puisque, par la condition de possibilite, (-l)~X( —l)-}(M'+I)(3-I)x( 3)= +1 On a ainsi (=. Mais il faut doubler cette valeur, parce que j = 1. Done on aura = r VI. Soit M-0, mod 8. On aura, pour le genre de formes capables de representation par une somme de cinq carres, l(-1)(03-1)=_-! ((l)}i(o3-l)= +1, (03)=(- )M1), (-1)=1, en designant toujours par M' le plus grand diviseur impair de M. La valeur de ' sera -; mais puisqu'il faudra la quadrupler, le resultat sera le meme que dans le cinquieme cas. 4R 2 676 ' MEMOIRE SUR LA REPRESENTATION [Art. 17. 17. On obtient sous forme finie les sommes des suites infinies qui figurent dans les formules des articles 13 et 16, en employant les menthodes de Dirichlet et de Cauchy". Comme on sait, on peut donner aux resultats un assez grand nombre de fbrmes differentes. Celle que nous avons pref6rde est loin d'6tre la plus simple, mais elle se presente comme cas particulier d'une forme qu'on peut donner, la somme de la serie 1 m(D) 1 7rv m'I mf pour toute valeur de v t. Soit D = co2 x J, ~ n'etant divisible par aucun carre; soit aussi q un nombre premier impair qui divise w mais ne divise pas J; on aura les expressions suivantes pour la valeur de la serie IS= D 1 7r2 \m/im2 I. SiD 1l, mod 4, 2 s= s s S [1-()]X )( -1) x II[1- ) ] s etant suppose premier a 3. II. Dans tous les autres cas, s- _ s _s ) x [-( 1 ] le nombre s etant premier 4 4J. La premiere formule est en ddfaut lorsque = 1; mais dans ce cas 1 7r2 m2 8 * Lejeune-Dirichlet, Recherches sur l'application de l'analyse infinitesimale a la theorie des nombres (Journal de Crelle, vol. XXI); Cauchy, Memoire sur la theorie des nombres (Memoires de l'Academie. vol. xvii, note 12). t Voir la seconde note sur les formes quadratiques dans les Proceedings de la Societe royale, annee 1867 [vol. i. p. 510]. Art. 17.] DES NOMBRES PAR CINQ CARRES. 677 En employant ces formules, on trouve les expressions suivantes pour le nombre N des representations du nombre o2 x 3 par une somme de cinq carres: T. Si =1, mod 4, 3 X I1] a N 5 x x n[1 -()2 ( 8)s(s - ); -=12, si - 1, mod 8; =28, si 6 5, mod8; -w 1, mod2; -;20, si -5, mod 8; =0, mod 2. Si = 1, il faut remplacer les facteurs xn l-( - ) 2] par 2. II. Dans tous les autres cas, N-5x X<[1 1 -( x )s (s-43); d 7 4" 1 S r=1, si )-1, mod 2; -=, i o =-0, mod 2. En faisant = 1, on deduit facilement de ces formules les expressions plus simples donn6es par Eisenstein. A. Soit en premier lieu =- 2m; on aura N= -x --- (- 8- ) ). 2 rm 1 1 En faisant successivement s 4m+p, les limites de p etant 1 et 4m; p=2m+_r, les limites de r Btant 1 et 2m; enfin r=2m-t, les limites du nombre impair t etant 1 et m; on obtient N= 40 (-l)-(t-l)(+l) (2)(t ) x [t+(-1) (t-1) (2m -t)], ce qui est bien la formule d'Eisenstein. B. Pareillement, si = m 1, mod 5, nous tirons de la formule ci-dessus N= 140 x -- m 678 MEMOIRE SUR LA REPRESENTATION [Art. 17. en designant par a et b les nombres moindres que m qui satisfont respectivement aux Equations ()=1, ()=- 1. Si a et 3 sont des valeurs de a et b qui ne surpassent pas 2 m, on aura, en /2 observant que (-) — 1 \m 5 Ea2= cca2 + (2a)2, = $a2+ (2a)2+, (2m - 2a)2, = a2 + (2a)2 + (m - 2 a)2 + 3 m2 - 4m a, = -a2 +b 3b2 + 3, m2 - 4mla, le coefficient,u etant le nombre des residus a ou 3. Le m6me raisonnement donne aussi 5b2=X a2 + b2 + 3m2- 4 m:, d'oh 1'on tire:4 a2 l b2 N= 140 b =112 (S3- Za), d'accord avec le resultat d'Eisenstein. C. Si ME 1, mod 8, on a 62'2 - _ b2 N=60, mais on trouve facilement 3a2= (2a)2 -Ca=2 - a2- t (2a)2 + (m-2a)2 + 3um2- 4ma = 3um2 - 4mSa; 3lb2 = 3 m2 - 4ml3. Done N=o 80(S - Sa) D. Si A- 3, mod 4, il s'agit de transformer la somme S=5x -I (_l)l(s-l)(s. m 1 m Art. 17.] DES NOMBRES PAR CINQ CARRES. 679 Faisons s = 2m +p; on aura d'abord 1 2 m ^/ S= -10 x (_ i)- (-1) p2 1 puisque 2m _( — 1)<(p-) (;P) = o ensuite ecrivons p = mn+ 2 t; on aura aussitot la formule d'Eisenstein (+2 1 40x ((rn (-ily ). Eisenstein a aussi indique pour le m6me cas les formules suivantes: (m —l) (m-3).S ^ (-1)(l-)s 4 7m (1), (_l)s(-)S= -4 (-)S S 1 T... (2) -~4 x ( ); m 3,mod8. 1 Puisque les transformations de cette espece, quoique tres elementaires, sont quelquefois assez difficiles a retrouver, nous en donnerons une demonstration ici. Les lettres a, b, a, f3 ayant la meme signification, designons par A le nombre des a et par B celui des 3. Etant donne un des nombres b, il y aura toujours parmi ces m6mes nombres un autre b', qui satisfera k la congruence 2b-b'-O, mod rm; de plus, si b< rm, on aura 2b-b'=O, si b'> mn, done Yid ~ b' 2b - b' 2 b - V = m; done Pidentite d = b donne immndiatement m m lb On trouve pareillement S c -=B, et, par consequent, b-a ma= A - B... (3) m 680 MEMOIRE SUR LA REPRIESENTATION DES NOMBRES PAR CINQ CARRES. D'un autre c6te, tout nombre b satisfait, soit a l'6quation b = m - 2 a, soit h 1'quation m = 2 3; done Z b = mA + 2 (1 - a), et pareillement c a = mB + 2 Z (a - (), d'ou l'on tire (b-a)=m(A-B) + 4 (3-a);.. (4) et ayant egard a (3), (a — 3) = O.e......... (5) On a done, en designant par 2a' et 2 ' ceux des nombres a et 1 qui sont pairs. (m- 1) /3 (_ ) () = [(- -( 1)] + (a — 1) 1 m 4 (m-3) / 8 =4S(.'-l 2 ^) ~ s. C. Q. F. D. Pour demontrer 1'equation (2), on se sert des formules - (A - B),..... (6) x(a-b)=m(A-B) + 4:(-a),..... (7) qui donnent (/-a) + S (A - B) =0, et qu'on etablit de la meme maniere que les equations (3) et (4). Les equations (3) et (6) avaient etd deja donnees par Lejeune-)irichlet (Journal de Crelle, vol. xxi, p. 152). APPENDIX. I. ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIO N OF THE BRITISH ASSOCIATION. [Bradford, Sept. 18, 1873, Report of the British Association for 1873, Transactions of the Sections, pp. 1-11.] FOR several years past it has been the custom for the president of this Section, as of the other Sections of the Association, to open its proceedings with a brief address. I am not willing upon this occasion to deviate from the precedent set by my predecessors, although I feel that the task presents peculiar difficulties to one who is by profession a pure mathematician, and who, in other branches of science, can only aspire to be regarded as an amateur. But, although I thus confess myself a specialist, and a specialist it may be said of a narrow kind, I shall not venture, in the few remarks which I now propose to make, to indulge my own specialty too far. I am well aware that we are certain, in this Section, to have a sufficient number of communications, which of necessity assume a special and even an abstruse character, and which, whatever pains may be taken to give them clearness, and however valuable may be the results to which they lead, are nevertheless extremely difficult to follow, not only for a popular audience, but even for men of science whose attention has not been specially, and recently, directed to the subject under discussion. I should think it, therefore, almost unfair to the Section if, at the very commencement of its proceedings, I were to attempt to direct its attention in any exclusive manner to the subject which, I confess, if I were left to myself, I should most naturally have chosen-the history of the advances that have been made during the last ten or twenty years in mathematical science. Instead, therefore, of adventuring myself on this difficult course, which, however, I strongly recommend to some successor of mine less scrupulous than myself, I propose, though at the risk of repeating what has been better said by others before me, to offer some general considerations which may have a more equal interest for all those who take part in the proceedings of this Section, and which appear to me at the present time to be more than usually deserving of the notice of those who desire to promote the growth of the scientific spirit in this country. I intend, therefore, while confining myself as strictly as I can to the range of subjects belonging to this Section, to point out one or two, among many, of the ways in which sectional meetings, such as ours, may contribute to the advancement of science. VOL. II. 4 S 682 APPENDIX. [T. We all know that Section A of the British Association is the Section of Mathematics and Physics; and I daresay that many of us have often thought how astonishingly vast is the range of subjects which we slur over, rather than sum up, in this brief designation. We include the most abstract speculations of pure mathematics, and we come down to the most concrete of all phenomena-the most every-day of all experiences. I think I have heard in this Section a discussion on spaces of five dimensions, and we know that one of our committees, a committee which is of long standing, and which has done much useful work, reports to us annually on the Rainfall of the British Isles. Thus our wide range covers the mathematics of number and quantity in their most abstract forms, the mathematics of space, of time, of matter, of motion, and of force, the many sciences which we comprehend under the name of astronomy, the theories of sound, of light, heat, electricity; and besides the whole physics of our earth, sea, and atmosphere, the theory of earthquakes, the theory of tides, the theory of all the movements of the air, from the lightest ripple that affects the barometer up to a cyclone. As I have already said, it is impossible that communications on all these subjects should be interesting, or indeed intelligible to all our members; and, notwithstanding the pains taken by the committee and by the secretaries, to classify the communications offered to us, and to place upon the same days those of which the subjects are cognate to one another, we cannot doubt that the disparateness of the material which comes before us in this Section is a source of serious inconvenience to many members of the Association. Occasionally, too, the pressure upon our time is very great, and we are obliged to hurry over the discussions on communications of great importance, the number of papers submitted to us being, of course, in a direct proportion to the number of the subjects included in our programme. It has again and again been proposed to remedy these admitted evils by dividing the Section, or at least by resolving it into one or more sub-sections. But I confess that I am one of those who have never regrettedat athis proposal has not commended itself to the Association, or indeed to the Section itself. I have always felt that by so subdividing ourselves we should run the risk of losing one or two great advantages which we at present possess; and I will briefly state what, in my judgment, these advantages are. I do not wish to undervalue the use to a scientific man of listening to and taking part in discussions on subjects which lie wholly in the directioinin which his own mind has been working. But I think, nevertheless, that most men who have attended a meeting of this Association, if asked what they have chiefly gained by it, would answer in the first place that they have had opportunities of forming or of renewing those acquaintances or intimacies with other scientific men which, to most men engaged in scientific pursuits, are an indispensable condition of successful work; and, in the second place, that while they may have heard but little relating to their own immediate line of inquiry which they might not as easily have found in journals or transactions elsewhere, they have learned much which might otherwise have never come to their knowledge of whiht is going on in other directions of scientific inquiry, and that they have carried away many new conceptions, many fruitful germs of thought, caught perhaps from a discussion turning upon questions apparently very remote from their own pursuits. An object just perceptible on a distant horizon is sometimes better descried by a careless sideward glance than by straining the sight directly at it; and so capricious a gift is the I.J APPENDIX. 683 inventive faculty of the human mind that the clue to the mystery hid beneath some complicated system of facts will sometimes elude the most patient and systematically conducted search, and yet will reveal itself all of a sudden upon some casual suggestion arising in connexion with an apparently remote subject. I believe that the mixed character and wide range of our discussions has been most favourable to such happy accidents. But even apart from these, if the fusion in this Section of so many various branches of human knowledge tends in some degree to keep before our minds the essential oneness of Science, it does us a good service. There can be no question that the increasing specialization of the sciences, which appears to be inevitable at the present time, does nevertheless constitute one great source of danger for the future progress of human knowledge. This specialization is inevitable, because the further the boundaries of knowledge are extended in any direction, the more laborious and time-absorbing a process does it become to travel to the frontier; and thus the mind has neither time nor energy to spare for the purpose of acquainting itself with regions that lie far away from the track over which it is forced to travel. And yet the disadvantages of excessive specialization are no less evident, because in natural philosophy, as indeed in all things on which the mind of man can be employed, a certain wideness of view is essential to the achievement of any great result, or to the discovery of anything really new. The two-fold caution so often given by Lord Bacon against overgeneralization on the one hand, and against over-specialization on the, other, is still as deserving as ever of the attention of mankind. But, in our time, when Vague generalities and empty metaphysics have been beaten once, and we may hope for ever, out of the domain of exact science, there can be but little doubt on which side the danger of the natural philosopher at present lies. And perhaps in our Section, as at present constituted, as many parts as we include-I will not say sciences-but groups of sciences. Perhaps there is something in the very diversity and multiplicity of the subjects which come before us which may serve to remind us of the complexity of the problems of science-of the diversity and multiplicity of nature. On the other hand, it is not, as it seems to me, difficult to assign the nature of the unity which underlies the diversity of our subjects, and which justifies to a very great extent the juxtaposition of them in our Section. That unity consists not so much in the nature of the subjects themselves, as in the nature of the methods by which they are treated. A mathematician, at least-and it is as a mathematician I have the privilege of addressing you-may be excused for contending that the bond of union among the physical sciences is the mathematical spirit and the mathematical method which pervade them. As has been said with profound truth by one of my predecessors in this chair, our knowledge of nature, as it advances, continuously resolves differences of quality into differences of quantity. All exact reasoning-indeed all reasoning-about quantity, is mathematical reasoning; and thus as our knowledge increases that portion of it which becomes mathematical increases at a still more rapid rate. Of all the great subjects which belong to the province of this Section, take that which at first sight is the least within the domain of mathematics-I mean meteorology. Yet the part which mathematics bears in meteorology increases every year, and seems destined to increase. Not only is the theory of the simplest instruments of meteorology essentially mathematical, but the discussion of the observations 4 s 2 684 APPENDIX. [I. -upon which, be it remembered, depend the hopes which are already entertained with increasing confidence of reducing the most variable and complex of all known phenomena to exact laws-is a problem which not only belongs wholly to mathematics, but which taxes to the utmost the resources of the mathematics which we now possess. So intimate is the union between mathematics and physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create 'for each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature.' Sometimes, indeed, the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armoury of the mathematician the weapons which he has needed ready made to his hand. But much oftener the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma. Perhaps I may be allowed to mention an example of each of these two ways in which mathematical and physical discovery have acted and reacted on each other. I purposely choose examples which are well known, and belong, the one to the oldest, the other to the latest times of scientific history. The early Greek geometers, considerably before the time of Euclid, applied themselves to the study of the various curve lines in which a conical figure may be cut by a planecurve lines to which they gave the name, never since forgotten, of conic sections. It is difficult to imagine that any problem ever had more completely the character of a ' problem of mere curiosity,' than this problem of the conic sections must have had in those earlier times. Not a single natural phenomenon which in the state of science at that time could have been intelligently observed was likely to require for its explanation a knowledge of the nature of these curves. Still less can any application to the arts have seemed possible; a nation which did not even use the arch was not likely to use the ellipse in any work of construction. The difficulties of the inquiry, the pleasure of grappling with the unknown, the love of abstract truth, can alone have furnished the charm which attracted some of the most powerful minds in antiquity to this research. If Euclid and Apollonius had been told by any of their contemporaries that they were giving a wholly wrong direction to their energies, and that, instead of dealing with the problems presented to them by nature, they were applying their minds to inquiries, which not only were of no use, but which never could come to be of any use, I do not know what answer they could have given which might not now be given with equal, or even greater, justice to the similar reproaches which it is not uncommon to address to those mathematicians of our own day who study quantities of n-indeterminates, curves of the nth order, and, it may be, spaces of n-dimensions. And not only so, but for pretty nearly two thousand years the experience of mankind would have justified the objection; for there is no record that during that long period which intervened between the first invention of the conic sections and the time of Galileo and Kepler the knowledge of these curves possessed by geometers was of the slightest use to natural science. And yet, when the fullness of time was come, these seeds of knowledge, that had waited so long, bore splendid fruit in the discoveries of Kepler. If we may use the great names of Kepler and Newton to signify stages in the progress of I.] APPENDIX 685 human discovery, it is not too much to say that without the treatises of the Greek geometers on the conic sections there could have been no Kepler, without Kepler no Newton, and without Newton no science in our modern sense of the term, or at least no such conception of nature as now lies at the basis of all our science, of nature as subject in its smallest as well as in its greatest phenomena, to exact quantitative relations, and to definite numerical laws. This is an old story, but it has always seemed to me to convey a lesson, occasionally needed even in our own time, against a species of scientific utilitarianism which urges the scientific man to devote himself to the less abstract parts of science, as being more likely to bear immediate fruit in the augmentation of our knowledge of the world without. I admit, however, that the ultimate good fortune of the Greek geometers can hardly be expected by all the abstract speculations which, in the form of mathematical memoirs, crowd the transactions of the learned societies; and I would venture to add that, on the part of the mathematician, there is room for the exercise of good sense, and, I would almost say, of a kind of tact, in the selection of those branches of mathematical inquiry which are likely to be conducive to the advancement of his own or any other science. I pass to my second example, of which I may treat very briefly. In the course of the present year a treatise on electricity has been published by Professor Maxwell, giving a complete account of the mathematical theory of that science, as we owe it to the labours of a long series of distinguished men, beginning with Colomb, and ending with our own contemporaries, including Professor Maxwell himself. No mathematician can turn over the pages of these volumes without very speedily convincing himself that they contain the first outlines (and something more than the first outlines) of a theory which has already added largely to the methods and resources of pure mathematics, and which may one day render to that abstract science services no less than those which it owes to astronomy. For electricity now, like astronomy of old, has placed before the mathematician an entirely new set of questions, requiring the creation of entirely new methods for their solution, while the great practical importance of telegraphy has enabled the methods of electrical measurement to be rapidly perfected to an extent which renders their accuracy comparable to that of astronomical observations, and thus makes it possible to bring the most abstract deductions of theory at every moment to the test of fact. It must be considered fortunate for the mathematicians that such a vast field of research in the application of mathematics to physical inquiries should be thrown open to them at the very time when the scientific interest in the old mathematical astronomy has for the moment flagged, and when the very name of physical astronomy, so long appropriated to the mathematical development of the theory of gravitation, appears likely to be handed over to that wonderful series of discoveries which have already taught us so much concerning the physical constitution of the heavenly bodies themselves. Having now stated, from the point of view of a mathematician, the reasons which appear to me to justify the existence of so composite an institution as Section A, and the advantages which that very compositeness sometimes brings to those who attend its meetings, I wish to refer very briefly to certain definite services which this Section has rendered and may yet render to Science. The improvement and extension of scientific education is to many of us one of the most urgent questions of the day; and the British Association has already exerted itself more than once to press the question on the public 686 APPENDIX. [I. attention. Perhaps the time has arrived when some further efforts of the same kind may be desirable. Without a rightly organized scientific education we cannot hope to maintain our supply of scientific men; since the increasing complexity and difficulty of science renders it more and more difficult for untaught men, by mere power of genius, to force their way to the front. Every improvement, therefore, which tends to render scientific knowledge more accessible to the learner, is a real step towards the advancement of science, because it tends to increase the number of well qualified workers in science. For some years past this Section has appointed a committee to aid in the improvement of geometrical teaching in this country. The report of this committee will be laid before the Section in due course; and without anticipating any discussion that may arise on that report, I think I may say that it will show that we have advanced at least one step in the direction of an important and long-needed reform. The action of this Section led to the formation of an Association for the improvement of geometrical teaching, and the members of that Association have now completed the first part of their work. They seem to me, and to other judges much more competent than myself, to have been guided by a sound judgment in the execution of their difficult task, and to have held, not unsuccessfully, a middle course between the views of the innovators who would uphold the absolute monarchy of Euclid, or, more properly, of Euclid as edited by Simpson, and the radicals who would dethrone him altogether. One thing at least they have not forgotten, that geometry is nothing if it be not rigorous, and that the whole educational value of the study is lost if strictness of demonstration be trifled with. The methods of Euclid are, by almost universal consent, unexceptional in point of rigour. Of this perfect rigorousness his doctrine of parallels, and his doctrine of proportion, are perhaps the most striking examples. That Euclid's treatment of the doctrine of parallels is an example of perfect rigorousness, is an assertion which sounds almost paradoxical, but which I, nevertheless, believe to be true. Euclid has based his theory on an axiom (in the Greek text it is one of the postulates, but the difference for our purpose is immaterial) which, it may be safely said, no unprejudiced mind has ever accepted as self-evident. And this unaxiomatic axiom Euclid has chosen to state, without wrapping it up or disguising it, not, for example, in the plausible form in which it has been stated by Playfair, but in its crudest shape, as if to warn his reader that a great assumption was being made. This perfect honesty of logic, this refusal to varnish over a weak point, has had its reward; for it is one of the triumphs of modern geometry to have shown that the eleventh axiom is so far from being an axiom, in the sense which we usually attach to the word, that we cannot at this moment be sure whether it is absolutely and rigorously true, or whether it is a very close approximation to the truth. Two of those whose labours have thrown much light on this difficult theory are present at this meeting-Prof. Cayley, and a distinguished German mathematician, Dr. Felix Klein; and I am sure of their adherence when I say that the sagacity and insight of the old geometer are only put in a clearer light by the success which has attended the attempt to construct a system of geometry, consistent with itself, and not contradicted by experience, upon the assumption of the falsehood of Euclid's eleventh axiom. Again, the doctrine of proportion, as laid down in the fifth book of Euclid, is, probably, still unsurpassed as a masterpiece of exact reasoning; although the cumbrousness of the forms of expression which were adopted in the old geometry, has led to the total exclusion of this part of the elements fromn the ordinary course of geometrical education. A zealous I.] APPENDIX. 687 defender of Euclid might add with truth that the gap thus created in the elementary teaching of mathematics has never been adequately supplied. But after all has been said that can be said in praise of Euclid, the fact remains that the form in which the work is composed renders it unsuitable for the earlier stages of education. Euclid wrote for men; whereas his work has been used for children, and it is surely no disparagement to the great geometer to suppose that after more than 2000 years the experience of generations of teachers can suggest changes which may make his Elements, I will not say more perfect as a piece of geometry, but more easy for very young minds to follow. The difficulty of a book or subject is indeed not in itself a fatal objection to its use in education, for to learn how to overcome difficulties is one great part of education: Geometry is hard, just as Greek is hard, and one reason why Geometry and Greek are such excellent educational subjects is precisely that they are hard. But in a world in which there is so much to learn, we must learn everything in the easiest way in which it can be learnt; and after we have smoothed the way to the utmost of our power there is sure to be enough of difficulty left. I regard the question of some reform in the teaching of elementary geometry as so completely settled by a great concurrence of opinion on the part of the most competent judges, that I should hardly have thought it necessary to direct the attention of the Section to it, if it were not for the following reasons:First, that the old system of geometrical instruction still remains (with but few exceptions) paramount in our schools, colleges, and universities, and must remain so until a very great consensus of opinion is obtained in favour of some one definite text-book. It appears to me, therefore, that the duty will eventually devolve upon this Section of the British Association, of reporting on the attempts that have been made to frame an improved system of geometrical education; and if it should be found that these attempts have been at last successful, I think that the British Association should lend the whole weight of its authority to the proposed change. I am far from suggesting that any such decision should be made immediately. The work undertaken by the Association for the improvement of geometrical teaching is still far from complete; and even when it is complete it must be left to hold its own against the criticism of all comers before it can acquire such an amount of public confidence as would justify us in recommending its adoption by the great teaching and examining bodies of the country. Secondly, I have thought it right to remind the Section of the part it has taken with reference to the reform of geometrical teaching, because it appears to me that a task, at once of less difficulty, and of more immediate importance, might now be undertaken by it with great advantage. There is at the present moment a very general agreement that a certain amount of natural science ought to be introduced into school education; and many schools of the country have already made most laudable efforts in this direction. As far as I can judge, there is further a general agreement that a good school course of natural science ought to include some part or parts of physics, of chemistry, and of biology; but I think it will be found that while the courses of chemistry given at our best schools are in the main identical, there is the greatest diversity of opinion as to the parts of physics and of biology which should be selected as suitable for a school education, and a still greater diversity of opinion as to the methods which should be pursued in teaching them. Under these circumstances it is not surprising to find that the masters of those schools into which natural science has hardly as yet found its way (and some of the largest 688 APPENDIX. [L and most important schools in the country are in this class), are doubtful as to the course which they should take; and from not knowing precisely what they should do, have not as yet made up their minds to do anything of importance. There can be no doubt that the masters of such schools would be glad on these points to be guided by the opinion of scientific men, and I cannot help thinking that this opinion would be more unanimous than is commonly supposed, and further that no public body would be so likely to elicit an expression of it as a Committee appointed by the British Association. I believe that if such an expression of the opinion of scientific men were once obtained, it would not only tend to give a right direction to the study of natural science in schools, but might also have the effect of inducing the public generally to take a higher and more truthful view of the objects which it is sought to attain by introducing natural science as an essential element into all courses of education. All knowledge of natural science that is imparted to a boy, is, or may be, useful to him in the business of his after-life; but the claim of natural science to a place in education cannot be rested upon its practical usefulness only. The great object of education is to expand and to train the mental faculties, and it is because we believe that the study of natural science is eminently fitted to further these two objects, that we urge its introduction into school studies. Science expands the minds of the young, because it puts before them great and ennobling objects of contemplation; many of its truths are such as a child can understand, and yet such, that, while in a measure he understands them, he is made to feel something of the greatness, something of the sublime regularity, and of the impenetrable mystery, of the world in which he is placed. But science also trains the growing faculties, for science proposes to itself truth as its only object, and it presents the most varied, and at the same time the most splendid examples of the different mental processes which lead to the attainment of truth, and which make up what we call reasoning. In science error is always possible, often close at hand; and the constant necessity for being on our guard against it is one important part of the education which science supplies. But in science sophistry is impossible; science knows no love of paradox; science has no skill to make the worse appear the better reason; science visits with a not long deferred exposure all our fondness for preconceived opinions, all our partiality for views that we have ourselves maintained, and thus teaches the two best lessons that can well be taught-on the one hand the love of truth, and on the other sobriety and watchfulness in the use of the understanding. In accordance with these views I am disposed to insist very strongly on the importance of assigning to physics, that is to say to those subjects which we discuss in this Section, a very prominent place in education. From the great sciences of observation, such as botany or zoology or geology, the young student learns to observe, or more simply, to use his eyes; he gets that education of the senses which is after all so important, and which a purely grammatical and literary education so wholly fails to give. From chemistry he learns above all other things, the art of experimenting, and of experimenting for himself. But from physics, better as it seems to me than from any other part of science, he may learn to reason with consecutiveness and precision, from the data supplied by the immediate observation of natural phenomena. I hope we shall see the time when each successive portion of mathematical knowledge acquired by the pupil will be made immediately available for his instruction in physics; and when everything that he learns in the physical laboratory will be made the subject of mathematical reasoning and calculation. I.] APPENDIX. 689 In some few schools I believe that this is already the case, and I think we may hope well for the future both of mathematics and physics in this country when the practice becomes universal. In one respect the time is favourable for such a revolution in the mode of teaching physical science. During the past few years a number of text-books have been made available to the learner, which far surpass anything that was at the disposal of former generations of pupils, and which are probably as completely satisfactory as the present state of science will admit. It is pleasant to record that these text-books are the work of distinguished men who have always taken a prominent part in the proceedings of this Section. We have Deschanel's Physics, edited, or rather re-written, by Prof. Everett, a book remarkable alike for the clearness of its explanations and for the beauty of the engravings with which it is illustrated; and, passing to works intended for students somewhat further advanced, we have the treatises of Prof. Balfour Stewart on Heat, of Prof. Clerk Maxwell on the Theory of Heat, of Prof. Fleeming Jenkin on Electricity, and we expect a similar treatise on Light from another of our most distinguished members. These works breathe the very spirit of the method which should guide both research and education in physics. They express the most profound and far-reaching generalizations of science in the simplest language, and yet with the utmost precision. With the most sparing use of mathematical technicalities, they are a perfect storehouse of mathematical ideas and mathematical reasonings. An old French geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street. This is of course a brilliant exaggeration, but it is no exaggeration to say that the eminent writers to whom I have referred have given something of this clearness and completeness to such abstract mathematical theories as those of the electrical potential, the action of capillary forces, and the definition of absolute temperature. A great object will have been attained when an education in physical science on the basis laid down in these treatises has become generally accepted in our schools. I do not wish to close this address without adverting, though only for one moment, to a question which occupies the minds of many of the friends of science at the present time-the question, What should be the functions of the State in supporting or in organizing scientific inquiry? I do not mean to touch on any of the difficulties which attend this question, or to express any opinion as to the controversies to which it has given rise. But I do not think it can be out of place for the president of this Section to call your attention to the inequality with which, as between different branches of science, the aid of Government is afforded. National observations for astronomical purposes are maintained by this as by every civilized country. Large sums of money are yearly expended, and most rightly expended, by the Government for the maintenance of museums and collections of mineralogy, botany, and zoology. At a very recent period an extensive chemical laboratory, with abundant appliances for research as well as for instruction, has been opened at South Kensington. But for the physical sciences-such sciences as those of heat, light, and electricity-nothing has been done; and I confess I do not think that any new principle would be introduced, or any great burden incurred, capable of causing alarm to the most sensitive Chancellor of the Exchequer, if it should be determined to establish, at the national cost, institutions for the prosecution of these branches of knowledge, so vitally important to the progress of science as a whole. Perhaps, also, upon this VOL. II. 4 T 690 APPENDIX. [T. general ground of fairness, even the pure mathematicians might prefer a modest claim to be assisted in the calculation and printing of a certain number of Tables, of which even the physical applications of their science are beginning to feel the pressing need. One word further on this subject of State assistance to science, and I have done. It is, no doubt, true that for a great, perhaps an increasing, number of purposes science requires the assistance of the State; but is it not nearer to the truth to say that the State requires the assistance of science? It is my conviction that if the true relations between science and the State are not recognized, it is the State, rather than science, that will be the great loser. Without science the State may build a ship that cannot swim, and may waste a million or two on experiments, the futile result of which science could have foreseen. But without the State science has done very well in the past, and may do very well in time to come. I am not sure that we should know more of pure mathematics, or of heat, of light, or electricity than we do at this moment if we had had the best help of the State all the time. There are, however, certain things which the State might do, and ought to do, for science. It, or corporations created by it, ought to undertake the responsibility of carrying on those great systems of observation which, having a secular character, cannot be completed within the lifetime of a single generation, and cannot, therefore, be safely left to individual energy. One other thing the State ought to do for science. It ought to pay scientific men properly for the services which they render directly to the State, instead of relying, as at present, on their love for their work as a means of obtaining their services on lower terms. If any one doubts the justice of this remark, I would ask him to compare the salaries of the officers in the British Museum with those which are paid in other departments of the Civil Service. But what the State cannot do for science is to create the scientific spirit, or to control it. The spirit of scientific discovery is essentially voluntary; voluntary, and even mutinous, it will remain: it will refuse to be bound with red tape, or ridden by officials, whether well-meaning or perverse. You cannot have an Established Church in science, and, if you had, I am afraid there are many scientific men who would turn scientific nonconformists. I venture upon these remarks because I cannot help feeling that the great desire which is now manifesting itself on the part of some scientific men to obtain for science the powerful aid of the State may perhaps lead some of us to forget that it is self-reliance and self-help which have made science what it is, and that these are qualities the place of which no Government help can ever supply. II. ARITHMETICAL INSTRUMENTS. [South Kensington Museum Handbook to the Special Loan Collection of Scientific Apparatus, 1876, pp. 22-33.] OF all those branches of human knowledge which are comprehended under the name of Science, Arithmetic is that which has the most abstract character, and which, at the same time, is of the most universal application in the study of natural phenomena. The art of counting, or of numeration, is one of the earliest, if not the earliest product of nascent civilization; and, in the case of the savage races of mankind, the greater or less progress which has been made towards the acquisition of this art affords no unfair measure of the degree of culture and of intellectual development which has been attained. It is said that there are races whose scale of numeration is limited to two or three; others can go to five, or ten, or twenty. And we may be sure that no tribe of men, untaught by a superior race, ever acquired the art of counting by hundreds or thousands, without possessing a high average of mental capacity, and without sharing in the privilege, accorded only to certain nations, of occasionally producing men of inventive genius, and real leaders of thought. The more favoured branches of the Semitic and Aryan families-the Jews, the Egyptians, the Greeks, the Sanskrit-speaking nations of India-must have reached this, comparatively speaking, advanced standard of culture at a very remote period. But it is remarkable that the real extent of the domain of arithmetic-a domain in a certain sense coequal with that of exact science-was not perceived till a much later epoch. The Greek philosophers, at least as early as the time of Aristotle, had learned to distinguish between discrete, or discontinuous, and continuous quantity. All counting, properly so called, is of discontinuous quantity; all measurement is of continuous quantity. To use a simple illustration: if we are counting points or dots on a line we can say, 'two dots and one dot make three dots'; if we are measuring inches we can equally say, 'two inches and one inch make three inches.' But in the latter case we can, if we please, pass by insensible degrees, and through every intermediate gradation of magnitude, from two inches to three inches: in the former case we can only pass abruptly from counting two dots to counting three dots; there is no such thing as half a dot, and no intermediate stage is conceivable. But while this important distinction was clearly seen in very ancient times, being indeed of a nature to commend itself specially to the philosophical spirit of classical antiquity, there was not an equally distinct apprehension of the truth that continuous quantity, no less than discontinuous, appertains to the domain of arithmetic. By whom 4T2 692 APPENDIX. [II. the first dim perception, or by whom the first vivid realization, of this truth was attained, we have no means of ascertaining with precision. It must have been gradually impressed on the minds of men by the growth of science. It is, perhaps, hardly discernible in the writings of Plato and Aristotle: it underlies, but is carefully excluded from, the fifth book of the Elements of Euclid. It must have been present to the mind of Archimedes when he measured the proportions to one another of the sphere, the cylinder, and the cone; it must have forced itself on the notice of the Greek astronomers, whose business it was to record numerically at discontinuous intervals the phases of continuous phenomena; and it became firmly established as an axiomatic principle by the development of that mode of arithmetic which is called algebra; by the great invention of Descartes which reduced geometry to algebra; and, last of all, by the creation of those arithmetical methods which are briefly described as the infinitesimal calculus. Although this conception of the absolute continuity of arithmetical magnitude is of a very abstract character, it has exercised a prepondering influence over scientific thought. That, on the one hand, all natural phenomena take place by a continuous process, and that they are all measurable quantitatively: that, on the other hand, the law of any continuous process can be expressed by an arithmetical formula, and the amount of any quantitative measurement can be stated in arithmetical figures, are propositions which are admitted by every one who understands them, and which, indeed, are in some instances believed with a more unlimited faith than is warranted by the evidence, strong as it is, which can be brought in support of them. Nor is this all; for if there be any one opinion concerning nature at the present time universally accepted by scientific men, it is that the minutest as well as the greatest phenomena are subject to a 'reign of law.' And if we ask for the strongest reasons which can be given for this belief, they may be summed up by saying that, so far as our measurements are exact, and so far as our arithmetic has been able to cope with the arithmetic of nature, we have uniformly found our observations of continuous phenomena to be in strict accordance with our deductions from the abstract science of continuous number. We proceed to offer a few observations with reference to each of the two branches of arithmetic-that of discontinuous and that of continuous quantity. The course of these remarks will make it clear why it is that a science of incalculable importance to other sciences does not, nevertheless, make any considerable display of its pretensions in an exhibition of scientific apparatus. (i) The simple operations of counting, and of recording numbers counted, and of comparing them with one another, which constitute the main business of practical arithmetic, have been so facilitated by the two great inventions of the decimal system of notation, and of logarithms, that, in many cases, but little inducement has existed to supersede the labour involved in such calculations by means of mechanical appliances. Counting machines, however, for certain purposes have been found indispensably necessary. A clock is defined by Sir John Herschel as a machine for counting and recording the number of the oscillations of a pendulum; though to this definition we are obliged to add that every clock must also contain a mechanism adapted to maintain the state of oscillation of the pendulum against friction and the resistance of the air. A pedometer is an instrument for counting and recording the number of steps taken by the person carrying it. Distances along a road are approximately measured by rolling II.] APPENDIX. 693 a wheel along the road, an apparatus being annexed to the wheel which counts and records its revolutions. In the same way a turnstile may be made to record the number of its own revolutions, i.e. the number of persons admitted through it. The above are simple instances of counting machines employed for the common purposes of life; but the construction of calculating engines, adapted to more varied and complicated purposes than that of simple counting, is to be reckoned among the great achievements of mathematical and mechanical skill. The first idea of such a machine appears to have been due to the celebrated Blaise Pascal; the apparatus constructed by him was arranged for the addition and subtraction of sums of money. Two calculating machines, constructed in 1775 and 1777 by James Bullock for Viscount MIahon, are included in the Exhibition. But the idea of a difference engine, which should serve to calculate tables of analytical functions, was first successfully realized by Charles Babbage; the analogous contrivances which had previously been proposed having been designed merely for the performance of single arithmetical operations, such as addition, subtraction, multiplication, and division. The later years of Babbage's life were devoted to the construction, or rather to the design, of a great analytical engine, which was intended to possess a range of calculating power, far exceeding that of the difference engine, and, in fact, extending over the whole field of arithmetical analysis. An article on Babbage's Difference Engine, in the EdiublrgSl Review for I834, suggested to George Scheutz, of Stockholm, the idea of constructing a machine for simultaneously calculating and printing arithmetical tables. After many discouragements, which were overcome by the indefatigable perseverance of George Scheutz and his son Edward, this machine was at last completed in October, I8.53. The 'Specimens of Tables, calculated, stereo-moulded, and printed by Machinery,' published by them in London in I857, afford a convincing proof of the completeness and utility of their invention. Its originality was gladly recognized by Babbage; and indeed two things only are common to the engines of Babbage and Scheutz; the principle of calculation by differences, and the contrivance by which the computed results are conveyed to the printing apparatus. Several arithmetical machines, on a smaller scale and of simpler construction, have been produced in recent years. Some of these are actually in use in the public offices of this country. We may mention especially the calculating machine of M. Thomas, of Colmar, and the panometer of Edward Grohmann, of Vienna. In the ancient world, and before the invention of the decimal notation, the common operations of arithmetic were carried on with the aid of a 'counting board,' or abaclus, the units being represented by counters, or pebbles (calculi, whence the word calculation). The authorities are not entirely agreed as to the precise arrangement of the ancient abacus, which, probably, was not always the same in all instances. It would seem certain, however, that the principle of decimal arrangement was to some extent adopted; counters in one compartment being valued as units, in that to the left of it as tens, and so on. It may seem strange that this partial introduction of a decimal system should not have led sooner to the invention of a decimal notation such as we now employ. The transition would probably have been instantaneous if the idea of employing a distinct symbol for zero had occurred to those who used the abacus. But it is precisely the introduction of this symbol which forms the central point of the whole decimal notation; and it may be admitted that in the abacus itself there was nothing to suggest its introduction. The 694 APPENDIX. [II. nearest approximations in the modern world to the ancient abacus are the bean-tables, multiplication boards, and other similar appliances employed in elementary education; * and the marking boards in use in certain games. The monetary transactions of the ancient world were occasionally on a scale approaching those of our own times. When Vespasian became emperor he found that, after the profligate expenditure of Nero, and the subsequent civil wars, the indebtedness and pressing requirements of the imperial and public treasuries amounted to no less a sum than about three hundred and thirty millions sterling. However rudely the accounts of these vast liabilities may have been kept, they must have required an enormous amount of calculation; and all this calculation must have been performed with the abacus; for it would have been almost impossible with the written characters of the Roman system of numeration. Perhaps no single instance could better serve to show the great saving of human labour which has been effected by the use of a decimal notation. The arithmetic of whole numbers, of which we are here speaking, has its theoretical as well as its practical part. This theoretical part is called the Theory of Numbers, and is perhaps the only branch of pure mathematics against which the charge of uselessness has ever been seriously alleged. Nevertheless, at all periods of the history of mathematical science it has excited a keen interest, and to it, rather than to researches of more obvious utility, we owe the development of the practical branch of arithmetic. As early as the second or third century before Christ, the Indian ritualists were led to the problem 'To find two square numbers of which the sum shall also be a square' by the existence of a religious feeling which required that altars of different shapes should have the same superficial area. By these and similar inquiries they were brought into contact with many questions of mensuration, and learned to solve them by approximate methods of considerable exactness; the value, for example, which they obtained for the side of a square equal to a circle of given diameter is correct as far as the third decimal place inclusively. Contemporary records of these researches still exist, and though they tell us of a time when science was in its infancy, they bear emphatic testimony to the genius and patient industry of the ancient workers. They are further characterized by that predominance of the arithmetical above the geometrical spirit, which forms so marked a contrast between the mathematical tendencies of India and of Greece. But while in these earlier treatises we can watch the growth of mathematical conceptions, called forth and fostered by the practical requirements of the old Vedic ceremonial, the purely scientific study of geometry and arithmetic in India belongs to a later period, probably to the fourth century after our era. Even then, the Hindus were the first to discover the method of solving indeterminate equations of the first degree, a method which was not known in Europe till the seventeenth century, and perhaps not demonstrated till the eighteenth. But the crowning achievement of Indian mathematical genius was the solution of the problem known as the Pellian Equation, upon which the analysis of indeterminate equations of the second degree may be said entirely to depend. The Indian mathematicians gave no demonstration of their solution. That demonstration was first given, at least fourteen hundred years later, by Lagrange, one of the greatest of European mathematicians, and the memoir in which he has recorded this discovery has always been regarded as one of the principal monuments of his genius. The * A series of these appliances have been contributed to the Exhibition by the Committee of the Russian Pedagogical Museum. ,I.] APPENDIX. 695 indeterminate equation of the first degree, to which we have above referred, is specially deserving of mention in this place, because it admits of mechanical applications to the theory of wheel-work, and also because it can be represented by a simple geometrical illustration. We may mention, for a similar reason, another important research connected with the theory of numbers, viz. the calculation of Tables giving for each number the least number by which it is divisible; or, if it is a prime number, indicating that it is so. Such tables (which considerably abbreviate certain computations) have been constructed for the first nine millions; the tables of the fourth, fifth, and sixth millions exist, however, in manuscript only, and have never been published. The first attempt to form a Table of Primes was made by Eratosthenes, and the partly mechanical method adopted by him (and called after its inventor, 'the sieve of Eratosthenes') has been adopted in principle, though with appropriate modifications, by his successors. There is in general so little appearance in those laws of nature with which we are acquainted of any adherence to integral or whole numbers, that we may be allowed to call attention to two important classes of phenomena which form an exception to this remark. We refer to the laws of chemical combination, and to the laws of crystallography. If we imagine chemical substances existing in the ideal condition of perfect gases, the law of chemical combination may be expressed in its most abstract form by saying that if two perfect gases combine chemically, and form a compound which is also supposed to exist in a perfectly gaseous condition, the volumes of the two gases before combination and of the gas resulting from their combination are to one another as three whole numbers. The law of integral numbers to which the faces of a crystal are subject is sufficiently illustrated by the models in the Section of Mineralogy; and it would be out of place to discuss it here. It may, however, be proper to remark that the whole numbers which present themselves in the formulae whether of chemistry or of crystallography are never very considerable. In the case of some organic bodies the number of equivalents that enter into the formula has to be counted by hundreds; but in all such instances, owing to the imperfection of the methods of chemical analysis, the determinations that have been given must be regarded as open to correction. The 'indices' of any face of a crystal actually occurring in nature rarely exceed ten. (2) The geometrical and mechanical appliances for aiding in the operations of arithmetic, as applied to continuous magnitude, are not very numerous, and possess in most cases a theoretical rather than a practical importance. A very ingenious instrument of this kind, and one that has been extensively used, is the slide rule, which may be described as an apparatus for effecting multiplications and divisions by means of a logarithmic scale; the requisite additions and subtractions being performed without calculation by a proper adjustment of the instrument itself. The principle on which it depends admits of being applied in various ways, and thus there are slide rules of very various forms, and adapted to very different purposes. But the card of four figure logarithms is a formidable competitor to any logarithmic scale, and it may be doubted whether at the present time these really beautiful contrivances are in as common use as they deserve to be. The Exhibition contains a complete series of them by Messrs. Aston and Mander; besides the Estimator of Dr. F. M. Stapff, and the Pocket Calculator of General de Lisle. 696 APPENDIX. [II. Instruments for solving triangles and for finding the roots of quadratic and higher equations may next be noticed. Some of them are remarkable for their ingenuity; some are useful as educational appliances, because they serve to illustrate, in a very beautiful way, the connexion between arithmetic, or algebra, and geometry. Others again are of great interest from the difficulty of the problems which they propose to solve, and the profound character of the principles which they employ in the solution. To this last class belongs the interesting application by Professor Sylvester of the Peaucellier movement to the extraction of the roots of numbers. With regard to all these arrangements it must be observed that the solutions which they afford are only approximate, and that the degree of the approximation cannot be carried beyond a certain point. This arises, not from any imperfection in the theory of the instruments, but from the circumstance that the solution of the problem is given by measurement; and that all measurements are necessarily approximate, and subject to errors which cannot be reduced beyond a certain point. In this respect the analytical solution of a problem possesses a great theoretical advantage above a solution obtained by geometrical or mechanical means. The analytical solution is indeed in general approximate, no less than the geometrical or mechanical one; but the degree of the approximation is no longer limited; for if we are dissatisfied with the degree of approximation we have obtained, we can go back and repeat the process over again, retaining small terms which we before omitted, until we arrive at a result as near to the truth as we please. Of course this theoretical advantage ceases to have any practical importance whenever the degree of approximation attainable by the mechanical appliance is sufficient for the purpose in view. At an earlier stage of the development of analytical science, graphical methods for the solution of analytical problems were of more importance than they are at present. When Descartes showed that the solution of a biquadratic equation could be made to depend on the determination of the intersection of a parabola by a circle, it is possible that, at least in certain cases, the very best method of finding the roots of a proposed biquadratic equation, which the resources of mathematics could then supply, was to describe the parabola and the circle, and actually to measure the ordinates of the points common to the two. But the continual progress of improvement in analytical processes, coupled with the greatly increased facility in calculating the arithmetical values of analytical expressions, which was obtained by the invention of logarithms, has completely outrun the present capabilities of geometrical methods; and these methods are now seldom used, except for obtaining rough first approximations. Thus it is that the degree of perfection which has been given to analysis has enabled it to dispense with mechanical aids; although instances are not wanting which may serve to show that the mechanical methods may yet receive a great future development. In addition to the applications of the Peaucellier movement, to which we have already referred, we may also mention that Sir William Thomson has recently planned an integrating machine which will integrate mechanically any differential equation, or set of simultaneous differential equations, containing only one independent variable. In the important case of the linear equation of the second order with variable coefficients, the actual construction of the requisite mechanism would, in the opinion of Sir William Thomson, present no insuperable difficulty. The kinematical principle employed in this integrating machine is due to Professor James Thomson, and consists in the transmission II.] APPENDIX. 697 of rotation from a disk or cone to a cylinder by the intervention of a loose sphere, which presses by its weight on the disk and cylinder, or on the cone and cylinder, as the case may be; the pressure being sufficient to give the necessary frictional coherence at each point of rolling contact. Sir William Thomson proposes to apply this principle to the construction of a machine adapted to calculate the harmonic constituents of any given function; and he believes that by employing such a machine in the analysis of the tides a single operator will be enabled to find, in an hour or two, any one of the simple harmonic elements of a year's tides recorded in curves by an ordinary tide gauge in the usual manner, a result which hitherto has required not less than twenty hours' computation by skilled arithmeticians. As another indication of the same tendency to substitute (wherever it may be found possible) mechanical or graphical contrivances for abstract calculations, we may refer to an excellent German treatise, the 'Graphische Statik' of Professor Culmann, which has already exercised a powerful influence upon the course of technical education in Germany, and of which it is the object to solve important engineering problems, relating to the stability of constructions, by mere geometrical drawing without the use of analytical formulae. VOL. II. 4u IlT. GEOMETRICAL INSTRUMENTS AND MODELS, [South Kensington Museum Handbook to the Special Loan Collection of Scientific Apparatus, 1876, pp. 34-54.] NEXT to the science of number, the science of space is that which is at once the most abstract, and admits of the most universal application to the study of natural phenomena. Everything that takes place takes place in space; and thus Geometry, or the science of space, necessarily intervenes in all exact observation of events. When we begin to think about space at all, the properties of it which first impress the mind are its continuity, and its apparently indefinite extent, our imaginations being perhaps unable to conceive the absence of either of these two properties. Probably we next notice the existence of three dimensions of space (as seen in the length, breadth, and height of any object), and we cannot conceive it to possess more or fewer. We further observe, (i) that at any two different points space is exactly similar to itself, and (2) that in all the directions which exist at any one point it has identical properties. These general assertions, if not really of themselves evident, are at least readily admitted as being in accordance with universal experience. They are all assumed in, and may be said to form the basis of, that analytical representation of space which we owe to Descartes, and which justly entitles him to be regarded as the founder of modern geometry. In accordance with this representation we regard space as a complex (if we may use this word as a translation of the German lMannigfaltigkeit) of three indeterminate quantities corresponding to its three dimensions; the surfaces, lines, and points which exist in space, being, in technical phraseology, the 'loci' obtained by imposing one, two, or three restrictive conditions upon these indeterminates. As often happens in similar cases, the mode of representation thus introduced is capable of being extended so as to apply to other objects or conceptions beside that for which it was first employed; and thus mathematicians have been led to consider complexes of more than three indeterminates, or, again, complexes not possessing the properties which we have enumerated as characteristic of space. This is the origin of such phrases as 'a space of four dimensions,' or of such assertions as 'it is conceivable that a space may not be exactly similar to itself at all its points.' These speculations are perhaps not calculated directly to promote our knowledge of the space in which we live and move, and to which they seem entirely inapplicable; but they have had the effect of advancing our knowledge of the relations of quantity, and have thus had an indirect, but not unimportant, influence upon the recent progress of geometrical science. So great has been the influence of the Cartesian mode of representation upon geometrical speculation that it has perhaps, to a certain extent, and in certain cases, III.] APPENDIX. 699 unduly led away the minds of geometricians from that direct intuition of space upon which geometry must after all be founded. And there can be no doubt that an Exhibition of models such as those included in the present Catalogue is calculated to render a great service to geometrical science by calling attention to the concrete shapes of objects, which are too apt, even in the mind of the serious student, to exist only as conceptions very imperfectly realized. We may for the purposes of this introduction adopt a threefold classification of the properties of space, as being either, i. Properties of Situation; or, 2. Graphical Properties; or, 3. Metrical Properties. Of each of these three classes of properties we shall here say a few words to illustrate their importance and meaning. I. The Properties of Situation of a figure in space are those which exist irrespectively of the magnitude and even of the shape of its parts, depending solely on the connexion of the parts, and on their situation with reference to one another. As neither the term 'properties of situation,' nor the description which we have just given of these properties, can be regarded as conveying a distinct image, a few very simple examples of what is meant may not be out of place. If we draw, upon any surface such as a plane, two closed contours of however complicated an outline, it is quite possible that they may never meet one another, or that they meet in one or more points, and do not traverse one another. But if they traverse one another at all, they must do so an even number of times; i.e. twice, or four, or six times, &c. The truth of this proposition will be easily admitted, and it will be seen that, to understand the assertion made, we require no conception of magnitude, nor even the conception of the straight line or plane. All that we require is the idea of a continuous closed curve, and of a surface upon which it is drawn. Again: conceive of two bodies, one a hollow sphere, the other a hollow anchor ring, hollow bodies. The two closed spaces in which he will thus successively find himself differ from one another at least in one remarkable respect. There is but one way of travelling from one point A inside the sphere, to another point B, also inside it; we might, of course, trace any number of routes we please from A to B, but all these routes are really reducible to one and the same route; and an elastic thread connecting A and B might be stretched so as to assume the shape of any one of them. But now take two points A and B inside the hollow anchor ring, and it will be seen at once that there are two different ways, irreducible to one another, of travelling from A to B. We have thus before us an example of a singgly connected space (the interior of the sphere), and a doubly connected space (the interior of the anchor ring). The distinction depends entirely on the properties of situation of the two bounding surfaces; and it is one which has been found to be of some importance in the theories of the motion of fluids, and of electricity. As a third example, take an oblong strip of paper, and fasten its two ends together so as to form a portion of a cylindrical surface. Take another similar piece of paper, and again fasten its two ends together, but give the paper a half twist so as to bring the upper surfaces of the two ends in contact with one another. Between the surface thus formed, and the cylindrical surface at first obtained, an important distinction will be found to subsist; viz. the cylindrical surface has an outside and inside surface, and there is no way of passing from one to the other except by penetrating the paper or crossing its edge: 41u 2 700 APPENDIX. [mT. whereas the two sides of the second surface form one perfectly continuous sheet; so that by travelling once along the whole length of the oblong strip, we should pass from a point on the surface to the point exactly corresponding to it on the other side of the surface; and we should not return again to the point from which we set out, until we had completed the tour a second time. The distinction which we thus learn to draw between surfaces which have two sides, and surfaces which have but one, is fundamental, and depends solely on the properties of situation of the figure, as we have now defined them. No complete corps de doctrine has yet been formed of the properties of situation of figures. This is partly owing to the great difficulty of the inquiry, partly to the fact that it is only in very recent times that the attention of mathematicians has been called to the subject, by the unexpected light which researches into it have been found to throw on some of the most obscure questions of the integral calculus. We cannot, therefore, expect to find this part of the science of geometry extensively illustrated by models, or by drawings expressly prepared for the purpose. But any great collection of geometrical objects cannot fail to supply examples of such properties; and, what is of more importance, may be expected to suggest entirely new points of view in a branch of inquiry, which, more than almost any other within the range of pure mathematics, is dependent on direct observation. 2. The Graphical Properties of space are those which involve the conceptions of the straight line and plane, but do not involve any conception of magnitude, or of measurement. The Elements of Euclid will be searched in vain for an example of a purely descriptive theorem, though it would seem that one of the lost treatises of that great geometer-the 'Porisms'-was devoted to this part of geometry. In modern times researches into the Desargues. By a strange fatality, the purely geometrical works of these two eminent men were lost, or wholly neglected, for more than a century, and it is only in comparatively recent times that they have received the attention which they merited. We may take as a simple instance of a graphical theorem the proposition of Desargues: 'If two triangles lying in the same plane are such that the lines joining their vertices taken in pairs meet in a point, the three intersections of the pairs of sides opposite to these vertices lie in a straight line; and conversely.' 3. Lastly, the Metrical Properties of space are those which involve, implicitly or explicitly, the consideration of magnitude. Thus the old proposition of Pythagoras, ' The square of the hypothenuse in a right angle is equal to the squares of the sides containing the right angle': and the theorems of Archimedes, 'The surface of a sphere is equal to the curved surface of its circumscribing cylinder; the volume of the sphere is two-thirds of the volume of that cylinder,' are metrical propositions. They could not be made intelligible to a person who had not the conception of the equality of geometrical magnitudes; nor verified by any one who had not the means of making exact quantitative measurements; whereas the proposition of Desargues, above quoted, is intelligible to any one who knows what a straight line and a plane are, and may be verified by any one who has a sheet of paper, a pencil, and a straight edge. Comte proposed to define geometry as 'la science qui a pour but la mesure des grandeurs.' As a scientific statement this definition is probably insufficient, because a great part of geometry consists, as we have seen, in propositions which have no immediate III.] APPENDIX. 701. connexion with measurement. It must, indeed, be admitted that by far the most important applications of geometry to natural science, and to the business of life, turn on the metrical properties of figures. But, in a purely theoretical point of view, there is reason to believe that the graphical properties of space are the more universal, and deeply-seated in the nature of things, notiora naturce, as Lord Bacon would have said; and that the metrical properties are, in a certain sense, secondary and derivative. As an example of the character of universality, which we thus attribute to graphical properties, we may take the general principle of the dlality (as it has been termed) of geometrical figures. This principle asserts that all purely graphical theorems are twofold; i.e. that any graphical proposition relating to points and planes in space give rise to another, which is correlative to it, but in which the points have been replaced by planes, and vice versa': the line joining two points being replaced by the line of intersection of two planes. We proceed to indicate the principal classes of material appliances which are of use in geometrical investigations, or in the applications of geometry to the arts, or lastly, in its employment as a means of education. We shall mention successivelyA.-Instruments used in geometrical drawing or mapping, and in copying geometrical drawings or maps. B.-Instruments used in tracing special curves. C.-Models of figures in space. D.-Modes of representing figures in space by means of plane drawings. A.-INSTRUMENTS USED IN GEOMETRICAL DRAWING. The Ruler and the Compasses-the two great instruments of geometrical drawing and construction-are of a very remote antiquity. Probably a stretched string-such as is still used by carpenters-was the earliest form of apparatus for obtaining a straight line; and a string attached to a peg (a contrivance still adopted by gardeners in laying out a flowerbed) afforded the earliest means of describing a circle. Compasses such as we now use, and indeed several of very different forms, have been found in the excavations of Pompeii. But it is probable that the use of the compasses, which is now universal, for transferring with exactness measured lengths from a scale to a drawing, or from one drawing to another, was hardly practised in ancient times. Had this practice prevailed, it is difficult to suppose that it would not have superseded the second and third problems of the First Book of Euclid, in which lengths are transferred by means of the actual description of circles. Among more recent improvements in the construction of compasses we may notice (1) the arrangements adapted for very fine work, and known as hair-compasses, needlepoint compasses, and spring-dividers; (2) the proportional or reducing compasses, by which we are enabled to reduce or augment in any given ratio the distances which we transfer from one drawing to another; (3) the triangular compasses, by which the position of three points forming a triangle can be transferred from one drawing to another, and which thus serve as an instrument for transferring angles; (4) the beam compasses, consisting of a beam or 702 APPENDIX. [III. bar, along which the two points of the instrument may be moved backwards and forwards, the distance between them admitting of adjustment with great precision, by means of a micrometer screw. Next in the universality of its employment in all geometrical plan drawing is the scale of equal parts. Let one pair of opposite sides of a rectangle be divided, say, into ten equal parts, the points of division on each being numbered i, 2, 3,.. 9, and let the lines II, 22, 33,.. 99, be drawn parallel to the sides of the rectangle. Let the other pair of opposite sides be similarly divided into ten equal parts, but let the points of division be joined in a slanting direction by the parallels oi, I2, 23, 34,...; it will be found that the first set of parallels are divided into hundredths by the second set. Such a diagonal scale is placed on every so-called plane scale, and serves to divide one of the primary divisions into hundredth parts. With a fine pair of compasses, we may succeed in taking off from the scale any required length with an error perhaps not exceeding one five-hundredth part of a primary division of the scale. Besides the scale of equal parts, the plane scale usually has engraved upon it a scale of chords, and a protracting scale. These are the simplest known contrivances for setting off an angle, given in degrees and minutes, or for approximately measuring an angle already laid down. With a good scale of chords, an angle can, it is said, be set off true to the nearest minute. But, for the best and most convenient solution of the problem, 'to construct an angle equal to a given one,' we have recourse to the divided circles, or parts of circles, known as circular, or semicircular, or quadrantal, protractors. Less elementary in their theory than the preceding simple instruments, are the arrangements called Pantographs, which enable us to copy any given plane figure upon a different scale. Of this instrument there are two principal forms, known as the older pantograph and the Milan pantograph. In each of these there is a linkage movement in which only one point is absolutely fixed. The linkage is so arranged that two points on different bars always remain in the same straight line with the fixed point, and at distances from it which are to one another in a constant ratio. It follows from this that if one of these two points be made to describe any figure, the other will describe a similar and similarly situated figure, the centre of similitude of the two figures being the fixed point. In the older pantograph the similarity is direct, in the Milan pantograph it is inverse; 'i.e. in the first case the figures are on the same side of their centres of similitude, in the second they are on opposite sides. B.-INSTRUMENTS FOR TRACING SPECIAL CURVES. Geometrical drawings consist very mainly, but not exclusively, of straight lines and circles. And the same limitation is observable in all the ordinary constructions of theoretical geometry. It has been ascertained that every problem which admits of one solution, and one only, can, if the data of it are given graphically, be solved with the ruler only, i.e. by drawing straight lines only, and without using the compasses; and, again, that every problem which is quadratic, i.e. which admits two solutions, but not more, can be solved with the ruler and compasses. Indeed, it is further known that, for this purpose, one circle, traced once for all, would, as a matter of theory, be sufficient. These considerations are perhaps sufficient to account for the great preponderance of III.] APPENDIX. 703 importance which attaches, in theoretical geometry, to the straight line and circle. On the other hand, the straight line and circle possess in common a property which is peculiar to them among all plane curves, and which is invaluable in all the practical applications of geometry. They are the only plane (or untwisted) lines any part of which can be applied exactly to any other. In very many mechanical arrangements this property is indispensable, and it is advantageous in all cases where accuracy of form is required, because it offers a simple means of verifying that accuracy has been obtained. There is but one twisted curve which has the same property, viz. the helix, or screw-curve, and it is precisely because any part of a screw-curve can be superposed upon any other that the screw and nut arrangement is possible, which renders this curve of so much use in mechanics. But notwithstanding these prerogatives of the straight line and circle, the tracing of other curves is occasionally indispensable both in theoretical and practical geometry. It is by no means an easy matter to invent a good method of tracing a curve. Even when the theory of a curve is pretty well known, that theory may fail to suggest any mode of describing it mechanically; and not every mode which theory suggests can be made to work accurately in practice. Of all curves, after the circle, the ellipse would seem to be the simplest and easiest to draw, but some authorities on the subject recommend the draughtsman not to attempt a true ellipse, but to put together an imitation semi-ellipse, by means of six or seven arcs of circles with centres and radii appropriately chosen. It is said that such an imitation will impose even on a well-trained eye, although it is plain that, whereas the curvature of an ellipse changes continuously, the curvature of the imitation curve changes abruptly at the points of junction of the circular arcs. The discovery of M. Peaucellier that a linkage can be constructed capable of describing a straight line may perhaps hereafter revolutionize this part of geometry. Already it is known that any conic, section, and several of the more important curves of the third and fourth orders, can be described by linkage movements, or compound compasses, as M. Peaucellier has called them, not too complicated to work steadily. Theoretically, the results obtained are of a far wider scope, and Professor Sylvester has shown good reason for believing that every geometrical curve is capable of being described by a link-movement. We may briefly mention some other mechanical arrangements which exist for describing certain plane curves. (I) The ellipse can hardly be described with great accuracy by means of a thread attached to its two foci and stretched by a pencil, because of the extensibility of the thread. It can be described as an hypocycloid, as in the apparatus of Mr. A. E. Donkin; or we may use the old-fashioned elliptic trammels, or some other form of elliptic compasses of more recent invention. If none of these arrangements are at hand, perhaps the best way to obtain easily a really good ellipse is to take an oblique section of a carefully turned cylinder. All the three conic sections can be described by means of the conograph of Dr. Zmurko, of Lemberg. (2) The cycloid is the curve traced by a point in the circumference of a circle rolling on a straight line; the epicycloid and hypocycloid are traced in the same way, only that the rolling circle rolls on the outside or inside of a circle instead of on a straight line. These definitions have suggested various modes of describing these curves mechanically; an interesting cycloidograph is exhibited by Dr. Zmurko. 704 APPENDIX. [III. (3) Mr. A. G. Donkin has constructed a beautiful apparatus for tracing harmonic curves. This machine will draw as many different forms of the curve y = a sin (mx + a) +- sin (nx + 3) as there are means for varying the constants a, b, m, n, a, 3: the number of variations being practically unlimited, except in the case of m and n, which are the numbers of teeth in certain wheels of which only a limited number of changes can be provided. Thus the machine exhibits to the eye the effect of the composition of two harmonic curves of any different intensities and phases, and of different intervals. (4) Several forms of spirals (or volutes) are of use in the arts, and appropriate modes of describing them have been given. Among these curves we may select for special notice the involute of the circle, which gives the proper form for the teeth of toothed wheels. (5) We may refer, lastly, to the epicycloidal curves of Mr. Perigal; and to the beautiful diagrams, not properly epicycloidal, but of a more complicated type, obtained by his compound geometric chuck. C.-MODELS OF FIGURES IN SPACE. Models of certain geometrical solids, for example, of the so-called three round bodiesthe sphere, the right cone, and the right cylinder; of the five regular solids-the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron; and of the various forms of prisms, pyramids, and parallelepipeds, have been long in use as educational appliances*. All these geometrical forms have been well known to mathematicians from very ancient times; surprising as it may appear that researches into the nature of such solids as the dodecahedron and icosahedron should have preceded other inquiries of a more elementary and at the same time of a more important character. In very recent times the great development of geometrical speculation has led to the effort to reproduce in a material shape a much greater variety of geometrical conceptions. We proceed to enumerate some of the more important classes of objects of this kind. (I) Models of surfaces of the second order. Among these, two of the central surfaces, the ellipsoid and the hyperboloid of one sheet, have occupied a great place in the recent history of geometry. We may mention a few important theoretical considerations which models of these services have served to illustrate. The curvature of a surface at any point is either such that the surface at that point is entirely concave on one side and convex on the other, or else it is such that the surface on each side alike is partly convex and partly concave. A sphere is an example of a surface of the first kind; the upper surface of a saddle may serve as an instance of curvature of the second kind. But the ellipsoid and the hyperboloid furnish perfect typical examples of these two kinds of curvature. At any point of a surface there are always two directions at right angles to one another, along one of which the curvature is the greatest, and along the other the least. At special points, called umbilics, the greatest and least curvatures (and therefore all the curvatures) are equal to one another. The sphere has the peculiarity that every point on * A complete collection of such models, made by Stronkoff, is exhibited by the Committee of the Russian Pedagogical Museum. III.] APPENDIX. 705 it is an umbilic; on the sphere, therefore, there are no directions of greatest and least curvature; but on every other surface two series of curves can be drawn, cutting one another at right angles, and indicating for each point of the surface the directions of the greatest and least curvature. These lines are called the lines of curvature; they are, it may be said, detected by the eye itself on any surface. The ellipsoid has four umbilics; if a thread, attached to any two of these, be strained along the surface by a moving pencil, the pencil will describe a line of curvature, so that when the ellipsoid has once been modelled, and its umbilics determined, it is easy to draw its lines of curvature with sufficient approximation. It was suggested long ago by Monge that a semi-ellipsoidal vault would form the most appropriate covering for an oval room, that the natural lines of the vaulting would be the lines of curvature, and that the umbilics would be the proper centres of illumination. Every surface of the second order is either umbilical or rectilineal; i.e. it either possesses umbilics, or it is capable of being generated by the motion of a straight line: it cannot unite both properties. The ellipsoid, as we have just seen, is umbilical; the hyperboloid of one sheet is rectilineal; and the two systems of straight lines which lie upon this surface are very conspicuous in any model of it. It will be seen that one line of each system passes through each point on the surface; but that no two lines of the same system ever meet one another. The hyperbolic paraboloid, which may be regarded as a variety of the hyperboloid of one sheet, is characterized by similar properties: the Exhibition includes a beautiful series of models of these two surfaces made by M. Fabre de Lagrange, in I872, for the South Kensington Museum. A series of cardboard models of surfaces of the second order (the cone, the cylinder, the ellipsoid, the hyperboloids, and both the paraboloids) is also exhibited by Professor Henrici, of University College, and by Professor Brill, of Munich. These models exhibit very clearly the circular sections of the various surfaces, having, indeed, been constructed by means of them; with the exception of the hyperbolic paraboloid, in which (as well as in the hyperbolic and parabolic cylinders) such sections (strictly speaking) do not exist. (2) Models of surfaces of the third order. Nothing like a complete series of models of surfaces of the third order has as yet been attempted; and indeed it may be said that our knowledge of these surfaces is still too imperfect to justify such an attempt. N evertheless, models of certain of these surfaces have been made, and we may mention a few important properties which they serve to illustrate. (a) As we have already said, the two curvatures of an ellipsoid at any point upon the surface are turned the same way, whereas the two curvatures of an hyperboloid are everywhere turned opposite ways. But, generally speaking, a curve surface consists of two regions, on one of which its two curvatures are turned the same way, and on the other opposite ways. These two regions are separated by a bounding line, technically called the parabolic curve. Surfaces of the third order offer typical examples of these general geometrical facts. But an example, though of a less perfect kind, is afforded by the figure of a ring (such as an anchor ring, or a wedding ring); the parabolic curve here consisting of either of the two circles on which the ring would rest if placed lying on a horizontal plane. (b) Wherever the two curvatures of a surface are opposite to one another, there VOL. II. 4 X 706 APPENDIX. [III. always exist upon the surface two sets of lines along which the surface is inflected; i.e. at any point of the surface these two lines separate the directions in which the curvature is turned one way from those directions in which it is turned the other way. On the hyperboloid these 'curves of principal tangents,' as they are called by the German writers, are represented by the rectilinear generators; on surfaces of the third order we have the simplest examples of the general case in which they are not straight, but curved. The eye can just recognize these curves upon a surface, though of course in an approximate manner. The two which pass through any point are equally inclined to the lines of curvature at that point; they have a theoretical importance, even greater than that of the lines of curvature, because their definition is graphical, and not metrical. (c) As a general rule, three lines of curvature pass through an umbilic. On the surfaces of the second order this property of the umbilics does not exist; and the first example of it occurs on surfaces of the third order. It is clearly seen in Professor Henrici's model of the cubic surface xyz = k3 (x+y z - )3. (d) A model of a surface of the third order ought to exhibit to the eye one of the most characteristic properties of these surfaces; i. e. that of containing a certain finite number of straight lines. The maximum number of these lines is twenty-seven; and the model exhibited by Dr. Wiener has this maximum number. A model, showing the distribution in space of the lines themselves, unaccompanied by the surface on which they lie, has been constructed by Professor Henrici. In the model () to which we have just referred, the twenty-seven lines are all real, but are coincident in sets of nine; so that there appear to be three only. (3) Models of ruled, or rectilinear surfaces. Ruled, or rectilinear surfaces are those which may be generated by the continuous motion of a straight line in space, and which therefore may be may be said to consist of an infinite number of straight lines. Models of these surfaces exist in great variety, because they can be constructed with threads or wires, instead of being carved out of a solid material, or moulded out of a plastic one. They are, of course, only approximate or diagrammatic, the generating lines being represented in such number only as may suffice to convey to the eye an accurate impression of the form of the surface. Rectilinear surfaces are of two very different kinds, being termed skew, or developable, according as the successive generating lines intersect or not. Of skew surfaces the hyperboloid may be taken as an example: it may be defined as the surface generated by the motion of a straight line, which moves so as always to intersect three fixed straight lines which do not meet in space. In the hyperbolic paraboloid the generating line always intersects two fixed straight lines, and is always parallel to a fixed plane: this surface is the simplest example of the family of skew surfaces called conoids. The series of M. Fabre de Lagrange contains several models of conoidal surfaces; they are all generated by the motion of a straight line, which (i) continually remains parallel to a fixed plane, and which also (2) continually intersects a fixed straight line, and (3) some other fixed line in space. The surface of the thread of a square cut screw (or, more precisely, the surface formed by drawing lines from all the points of a screw curve perpendicular to the axis of the screw), affords a familiar instance of a conoidal surface. In the 'Skew Helixoid' of M. Fabre de Lagrange the lines drawn from the points of the screw curve to the axis are not perpendicular III.] APPENDIX. 707 to the axis, but are inclined to it at a constant angle. The recent progress of geometry has led to a careful study of the skew surfaces of the third, fourth, and fifth orders: of some of these, models have been already made; one of the cubic surface, called the cylindroid, is exhibited by Dr. Ball, Royal Astronomer of Ireland. Developable surfaces form a class of surfaces entirely sui generis. They are called developable because, if such a service consist of a flexible and inextensible membrane, it can be 'developed,' or flattened out, upon a plane, without any tearing or crumpling. Cones and cylinders are the simplest instances of developable surfaces, but they are far from giving a complete idea of the general character of these formations. A more typical instance may be obtained by considering all the lines tangent to any twisted curve. It will be found that these tangents all form a developable surface, and that the twisted curve is an edge-curve or cuspidal line upon the surface. This edge-curve is characteristic of a developable: in the cone it dwindles to a point, and in the cylinder this point lies at an infinite distance. Since a developable surface, when made to roll on a plane or on another developable surface, has a line of contact with it, whereas in general two surfaces made to roll on one another have only a point of contact, it is easy to see that these surfaces are of great importance in the arts of construction. But, in a theoretical point of view, it is even more important to notice what is rendered visible by any model of such a surface:-(I) that at each point of the surface one of the two curvatures is infinite, the generating line being always one of the lines of curvature; (2) that, whereas in other surfaces each tangent plane has an infinite number of tangent planes lying near to it (because we can travel from any point on the surface to an adjacent point in an infinite number of different directions), in the case of developable surfaces this is not so, but each tangent plane is preceded and followed by only one other tangent plane; these planes, in fact, forming a singly, instead of a doubly, infinite series. It follows from this, that in the duality of space there answers to a developable surface a curve line, whereas to any surface not developable there answers a surface not developable. As an example, easy to understand and to remember, of a developable surface, we may mention the 'Developable Helixoid' of M. Fabre de Lagrange. Most of the models of ruled surfaces to which we have referred are so arranged as to be capable of deformation; i.e. their shape can be changed by altering the form, or the relative position, of the director curves or straight lines, which serve to regulate the motion of the generating straight line. Thus the same model is rendered capable of assuming in succession the forms of several different surfaces; and the study of the transformations by which we pass from one of these forms to another is of great interest and importance. The cardboard models of Professor Henrici admit of deformation in a similar manner, since the angle at which the two planes of circular section are inclined to one another may be changed. (4) Models of certain Special Surfaces. (a) The Surfaces of Pliicker. In very recent times the properties of systems of lines in space have been studied with great ardour, chiefly by the German geometers. The consideration of such systems is suggested by many geometrical problems (for example, the study of the straight lines which cut a given surface at right angles, the so-called normals of the surface), but 4x 2 708 APPENDIX. [III-. it is also forced upon our notice in all those optical researches in which the properties of systems of rays form the subject of inquiry. A further interest attaches to the study of systems of straight lines in space, inasmuch as an examination of their properties would seem to be an indispensable preliminary to the more general inquiry into the properties of systems of curves; such systems of curves meeting us at every turn in Physics; as stream lines in Hydrodynamics, as lines of flow in the theory of the conduction of Heat, as lines of force in the theories of Electricity and Magnetism. The present Collection contains a beautiful series of models of a class of surfaces which are found to play an important part in the theory of rectilinear systems. A certain historical interest attaches to these models; they are copies of those exhibited in 1866, at the Nottingham Meeting of the British Association, by the celebrated mathematician, Julius PlUcker, to whom, more than to any other single person, we are indebted for our knowledge of the geometry of systems of lines. They were made by Epkens, of Bonn, and were presented by Dr. Hirst to the London Mathematical Society. Professor Hennessy, of the Royal College of Science for Ireland, also exhibits a series of models illustrative of the researches of Plicker. (1) The Wave Surface. The importance of this surface in the undulatory theory of Light forms its principal claim to attention. But, even apart from any physical interpretation, its geometrical properties entitle it to a place in a collection of geometrical models. It furnishes us with an instance of a closed surface of two sheets, one of them lying inside the other; it is an apsidal surface, and its reciprocal surface is a surface of the same nature as itself; finally, it offers typical examples of the singularity termed a conical node: and of the correlative singularity of a tangent plane touching a surface along a conic section (in the case of the wave surface the conic section is a circle). The geometrical study of these singularities led Sir William Rowan Hamilton to his celebrated discovery of the optical phenomena of external and internal conical refraction. (c) The Surface of Steiner. Professor Cayley exhibits a rough model of this surface, which has attracted considerable attention among mathematicians, from its being the polar reciprocal of the cubic surface with four conical nodes, and from its having the property that every one of its tangent planes cuts it in two conic sections. (d) The Amphigenous Surface of Professor Sylvester. This surface is of great importance in the theory of equations of the fifth order. A model of it has been prepared by Professor Henrici. (e) Surfaces of constant curvature.. The total curvature of a surface at any point is the product of its two principal curvatures at that point; and the total curvature is positive or negative, according as the two principal curvatures are in the same direction or in opposite directions. It is an important geometrical theorem, that if two inextensible and flexible surfaces have at corresponding points the same total curvature, either of them can be 'developed' upon the other without tearing or crumpling. Thus every surface of constant positive curvature can be developed upon a sphere. Surfaces of constant negative curvature III.] APPENDIX. 709 cannot, of course, be developed upon a sphere; but they possess a great theoretical interest of their own, since it has been ascertained that the geometry of figures traced upon these surfaces is precisely that which the geometry of Euclid would become, if we were to erase from it the assumption known as the eleventh axiom. Models of both these classes of surfaces are contributed by Professor Henrici. (5) Crystallographic Models. These are either models of those geometric polyhedra (solids bounded by plane faces) which actually occur in nature; or they are intended to serve as illustrations of the theory of crystallography, and to exhibit the relations of the plane faces of a crystal to its crystallographic axis. D.-REPRESENTATIONS OF FIGURES IN SPACE BY MEANS OF DRAWINGS ON A PLANE. We have lastly to refer to modes of representation by drawings on a plane of figures in space. We have here at our disposal the ordinary methods of perspective; but the practical use of these is too dependent on the hand and eye of the artist to satisfy the requirements of geometrical rigour. The method known as 'Descriptive Geometry' is therefore preferred for the purposes of geometrical drawing. This method consists in representing an object in space, by means of its orthogonal projections on two planes at right angles to one another. It therefore can hardly be said to be of recent date, as, in principle, it comes to representing a figure in space by a plan and elevation. But to Monge is perhaps due the idea of placing the plan and elevation on one sheet, and treating them (for the purpose of geometrical construction) as one plane figure; and to him is certainly due the development of the principles of the method into a complete and scientific system. The volumes of plates which accompany works on descriptive geometry (of which, since the time of Monge, there have been a considerable number) offer copious and varied illustrations of the method. But models have also been constructed, which serve to exhibit to the student the relations of the object represented to its two projections, and of these to one another. We may refer, among others, to the diagrams and models exhibited by Professor Franz Tilser, of Prague, by Professor 0. Reynolds, of the Owens College, Manchester, by the Committee of the Russian Psedagogical Museum, and by Professor Pigot, of the Royal College of Science for Ireland. The epures of descriptive geometry, however accurate, and however useful for constructive purposes, do not offer much assistance to the imagination in conceiving complicated geometrical figures. Such assistance, however, is abundantly afforded by stereoscopic representations; and it is earnestly to be hoped that the applications of stereoscopy to geometry may hereafter receive a much greater development than has been the case as yet. Any polyhedron can (as is well known) be represented with extraordinary beauty by stereoscopy; the edges only of the polyhedron being drawn on the two faces of the stereoscopic slide. It ought in the same way to be possible to represent any twisted curve; and further, any developable surface, by representing first the twisted curve which forms its cuspidal line, and then a sufficient number of the straight lines tangent to that curve. Similarly to represent a skew rectilineal surface, it would be sufficient to exhibit stereoscopically a certain number of its generating lines. Surfaces which are not rectilinear 710 APPENDIX. [iII. could, theoretically at least, be represented by a sufficient number of their lines of curvature, or by means of their curves of principal tangents, when those curves exist. It must be admitted, however, that the accurate tracing of the plane diagrams which would be required for such representations would be subject to very serious practical difficulties, which it would be desirable to avoid by using special methods adapted to each particular surface; for example, in the case of the ellipsoid and the other umbilical surfaces of the second order, by employing the two systems of circular sections. IV. INTRODUCTION TO THE MATHEMATICAL PAPERS OF WILLIAM KINGDON CLIFFORD. [London, 1882, pp. xxxiii-xlviii.] IT will be generally admitted that the publication in a collected form of the works of the eminent men, who have moulded the mathematical sciences into their present form, has become little less than a necessity to those who desire to follow in their footsteps, and to advance, if possible, beyond the limits attained by them. And mathematicians will gratefully acknowledge that no inconsiderable progress has already been made towards satisfying this requirement. To the Academy of Sciences of Paris we are indebted for magnificent editions of the complete works of Laplace and Lagrange; the government of Norway has given to the world the collected memoirs of Abel; the Academy of Goettingen has fulfilled the same duty towards the great names of Gauss and Riemann; and the Academy of Berlin has followed the example by undertaking editions of the works of Steiner and Jacobi. In our own country we have collected the works of Green, of Mac Cullagh, of Gregory, of Leslie Ellis, and of Macquorn Rankine; not to mention the volumes of reprinted memoirs which we owe to living writers; for example, to Sir William Thomson and to Professor Stokes. Such collections, we may hope, will form an increasing portion of every scientific Library. At the present time the results of mathematical research almost always appear in the Transactions of Societies, or in periodicals specially devoted to the mathematical writings; the contents even of the most original treatises being generally anticipated by their authors in memoirs which are often not wholly superseded by the works themselves. But the number of the periodical repositories of mathematical literature has become so great, that papers consigned to them, although preserved, as we may hope, for all time, are in imminent danger of passing out of sight within a few years after their first appearance. They are preserved from destruction, but not from oblivion; they share the fate of manuscripts hidden in the archives of some great library from which it is in itself a work of research to disinter them. This 'mislaying,' if it may be so termed, of important memoirs is not only a loss to the history of science, but interferes seriously with the discovery of new knowledge. For notwithstanding the ardour with which mathematical investigation is at present pursued in every direction, a much longer time than is perhaps sometimes supposed elapses before a mathematical work of genuine originality, be it a brief note, or an elaborate treatise, becomes antiquated. It would be out of place in this connexion to mention the Principia of Newton, which stands apart by itself, and of which not only the methods and results, but even the very words have become the common 712 APPENDIX. [Iv. property of all men of science. Nor need we even refer to works which have marked the beginning of a new epoch in their respective departments, such as the MWcanique Celeste, the Mecanique Analytique, the Disquisitiones Arithmeticse, the Traite Analytique de la Chaleur, the Fundamenta Nova, or the Systematische Entwickelung der Geometrischen Gestalten-the freshness of which time has hardly impaired, while the superstructures which have been based upon them have added incalculably to their importance. But leaving out of count these and other great classics of the science, the trains of thought hidden in the opuscula of Euler, of Lagrange, of Gauss, of Poisson, of Cauchy, of Abel, and Jacobi, are still unexhausted; and, far from having lost their value by the lapse of time, have in many cases acquired an increased suggestiveness from the light which the more extended knowledge of recent times has thrown upon them. Such a prospect of future and long-continued usefulness, we may venture to hope, awaits many of Clifford's memoirs; and, even more than the immediate interest attaching to them, justifies, if any justification be needed, their appearance in their present collected form. It might be interesting to inquire why it is that mathematical writings retain a scientific (as opposed to a merely historical) value for a longer time than memoirs recording researches in the sciences of experiment and observation. Among many partial answers which might be given to this question, one is suggested by the character of many of Clifford's papers, and has its foundation in the nature of the subjects with which they deal. Speculation in pure mathematics resembles metaphysical speculation in this, that the whole universe of thought to which it refers is so closely interdependent, that a clear-sighted and powerful thinker cannot fix his mental vision (however keen his effort after concentration may be) on any one region in it, without catching glimpses of something that lies beyond, and without discerning, more or less dimly, new relations to be examined, and new lines of research, which may perhaps have no immediate relevancy to the particular inquiry in which he is engaged. And these glimpses, if recorded, or even if only half unconsciously indicated, in the account which he afterwards gives of his work, are not unlikely to suggest a wholly new departure to some kindred spirit in a future time. On this ground, more strongly perhaps than on any other, we may venture to commend the present volume to the rising generation of English mathematicians. The collection includes papers-some of them youthful efforts-some suggested, one might say casually, by the researches of scientific friends-which relate to special problems, and which nevertheless would be sufficient of themselves to establish a considerable mathematical reputation. There are others, again, the work of a maturer time, and planned with a wider scope, which are models of artistic perfection, in respect both of the clearness and depth of the thought, and of the manner of its presentation. Lastly, besides these finished pieces there are others, rough-hewn and imperfect in execution, but conveying a still stronger impression of the fertility of the invention; and of the far-reaching power of mental vision, with which Clifford was endowed. Some of these fragmentary records are full of great ideas, shadowed forth in outlines, not always free from indistinctness, but always suggesting long vistas of future discovery, the path to which seems for the moment to lie clear before his mind. Their very incompleteness reminds us how much the world has lost by losing him; and brings home to us the melancholy feeling that, however highly we may estimate the work IV.] APPENDIX. 713 which he actually accomplished during his brief lifetime, he must nevertheless be counted among ' the inheritors of unfulfilled renown.' But if the republication of Clifford's papers stands in no need of any justification, some apology is wanted for an Introduction which can offer but little interest to those who do not intend to study the volume itself, and which to those who do, must seem at the best superfluous. Perhaps, however, even in these days of increasing specialization, there may still be found an intermediate class of readers, who are not mathematicians by profession, who nevertheless do not regard analysis and geometry as volumes sealed except to the initiated few, but as belonging, in their results at least, to the whole world of science. Some persons, one is willing to believe, partly from a recollection of their own early studies, and partly from a general sympathy with all branches of intellectual activity, are disposed to follow with an appreciative, or at least an indulgent curiosity, the exposition of new mathematical ideas. For such friendly readers these pages are intended; and it would be strange if the class of persons, among whom they are to be found, has not been considerably increased by the admirable lectures in which Clifford has himself analyzed, in popular phraseology but with the utmost scientific precision, the fundamental principles of geometry and arithmetic, as they appear in the' fierce light' which has been turned upon them by a controversy in which both metaphysicians and mathematicians have taken part. All then that can with any propriety be attempted here is, in the first place, to characterize some one or more of the principal trains of thought which seem to have exercised an abiding influence on Clifford's mind; and then to classify his memoirs in a few main groups, pointing out the central ideas of each group, and showing, as far as possible, the interdependence of these ideas upon one another. Clifford was above all and before all a geometer. Nine-tenths, and more, of the contents of this volume, including nearly all the systematic researches recorded in it, are geometrical. It is true that in the treatment of geometrical questions he shows a marked preference for symbolical methods; and, as might be expected, a marvellous command over analytical expression. It may even be true that the limitations involved in a scrupulous adherence to the methods of pure geometry would have been distasteful to him. Of his skill in the use of these special methods the 'Problems and Solutions' so liberally contributed by him to the Educational Times afford abundant proof. But among his more elaborate papers there is perhaps but one, the 'Geometry on an Ellipsoid,' which would satisfy purists of the school of Poncelet and Chasles, as being wholly free from the contamination of analytical methods; and even in this beautiful application of the method of the stereographic projection-in the generalized sense in which that method is used in modern pure geometry-the 'imaginary circle at infinity' occurs in the first sentence. But, whatever the method employed-and in variety of method Clifford takes an evident pleasure-the properties of space remain the perpetual subject of his contemplation. Purely analytical researches, undertaken without any impulse from or reference to geometry, are few and far between. For even the Elliptic and Abelian functions were approached by Clifford from the side of Geometry. His early note 'On some Porismatic Problems' relating to the theorem of Poncelet, which asserts that 'given two conies, a polygon of a given number of sides, all whose vertices shall lie on one of them, and all whose sides shall touch VOL. II. 4 Y 714 APPENDIX. [Iv. the other, can either not be drawn at all, or else can be drawn in an infinite number of different ways' led him to the study of the connexion, established by Professor Cayley, between this theorem and the addition of elliptic functions of the first species; and thus to the discovery of a geometrical theory of the transformation of the elliptic functions, which forms the subject of one of of his most brilliant investigations (XXII). But in his further prosecution of the subject the elliptic functions again disappear from view, and he returns to the geometrical doctrine of correspondences, and to the theory of the polyhedra, of which the faces osculate a skew cubic curve. In like manner it would appear that he was attracted to the consideration of the Abelian Integrals by their relation to another and widely different part of geometry, the Geometry of Situation, as it has been termed. His memoir on the 'Canonical Dissection of a Riemann Surface,' founded on the researches of Clebsch and Liroth, contains the simplest account which has yet been given of this important chapter of a great theory; and the reduction of a Riemann surface to the surface of a solid having a certain number of holes through it, presents to the mind what is perhaps the clearest image which it is possible to obtain of the space of two dimensions upon which a many-valued algebraical function can be mapped with the same distinctness with which a one-valued function can be mapped on a plane. But the study of the Abelian Integrals-however geometrical may have been the form in which he first envisaged this theory-led him by an inevitable sequence of ideas to the Theta functions, which form the indispensable basis for a study of the relations subsisting between the upper limits of a system of algebraical integrals, and the values of the integrals themselves. His posthumous and unfinished memoirs on the 'Double and Multiple Theta functions' (XXXVIII, XXXIX, and XL) form a not unimportant contribution to pure analysis; and, though now published long after their proper date, are, it may be hoped, neither too late, nor too incomplete, to exercise some influence on the development of this rapidly growing theory. The ' Algebraical Introduction to Elliptic functions,' which is in fact a treatise on the single Theta functions, probably took its rise in connexion with these ulterior researches; Clifford desiring to obtain a complete command over the manipulation of these series in the simplest cases, before proceeding to apply them to the general problem of the inversion of algebraical integrals. Enough has been said to show that Clifford's predilection for geometry lay deep. But to this his favourite science he attributed the widest imaginable scope, and at times regarded it as so co-extensive with the whole domain of nature. He was a metaphysician (though he would only have accepted the name subject to an interpretation) as well as a mathematician; and geometry was to him an important factor in the problem of 'solving the universe.' Thus he was a geometer of a type peculiarly his own; and his dealings with the science were characterized by an amount of scepticism and an amount of faith which one would hardly expect to find combined in a mathematician. He had early read and translated Riemann's celebrated discourse (IX) 'On the Hypotheses which lie at the Basis of Geometry,' and had imbibed the views set forth in it as a part of his intellectual nature. Some men who have an ardent love for new knowledge find it difficult to maintain an unflagging interest in geometry, because they regard it as a purely deductive science, of which the first principles (axioms, postulates, and definitions), whether derived from experience or not, are unquestionable, IV.] APPENDIX. 715 and contain implicitly in themselves all possible propositions concerning space. Thus the unknown, or at least the. unforeseen, seems to be excluded from geometry, because whatever may be found out hereafter must be latent in what is already known. But upon the view put forward by Riemann and adopted by Clifford, the essential properties of space have to be regarded as things still unknown, which we may one day hope to find out by closer observation and more patient reflection, and not as axioms to be accepted on the authority of universal experience, or of the inner consciousness. These speculations had so much influence on a great part of Clifford's work that it may not be out of place to pursue the subject a little further. In this lecture, 'On the Postulates of the Science of Space,' he has stated his own views on the question with singular clearness and brilliancy; and the pages in which he has expressed them are likely to be remembered, as marking an important moment in the controversy concerning the nature of space and the origin of our knowledge of it, which is likely to last as long as metaphysical inquiries have any interest for mankind. In this lecture he enumerates four fundamental postulates on which the ordinary conception of space is founded, (i) its continuity, (2) its flatness in its smallest parts, (3) its similarity to itself at every point, or, which is the same thing, the possibility of the existence of the same figure in any two different places, (4) the possibility of the existence of figures similar to one another, but on different scales of magnitude. The second of these postulates requires some comment to make it intelligible. Perhaps the simplest account that can be given of a space which is flat in its smallest parts is, that if anywhere in it we take three points very near to one another and join them by the shortest lines that can possibly be drawn, the triangular figure so formed willlie very nearly in a plane; the mathematical equivalent of this statement being that the square of the distance between any given point and any other infinitely near to it can always be expressed as a homogeneous quadratic form, in which the indeterminates are the infinitesimal differences between the co-ordinates of the two points, and the coefficients are functions of the co-ordinates of the given point. If we go further and join one of the infinitely near points by the shortest lines possible to every point on the line already joining the other two, the assemblage of these lines will form a triangular surface-element: if this surface-element is absolutely plane, whatever be the three infinitely near points which we have taken, the space is flat; if the surface-element has a finite curvature, the space, while retaining the property of elementary flatness, is said to have curvature; and this curvature is measured, for the surface direction determined by three points, by the curvature of the surface-element which we have constructed. As to the first postulate, Clifford indicates his readiness to adopt either of the two opposite hypotheses that space is continuous or that it is discontinuous, while admitting fully that no phenomena have yet been observed which point to its discontinuity. Of the second postulate, in this respect following Riemann, he speaks in the same general terms; we must not shrink from rejecting it, if its rejection should be found to assist us in the explanation of natural phenomena. The postulate is not inconsistent with a hypothesis which at one time was a great favourite with him, and which he has described in a remarkable communication (V) presented to the Cambridge Philosophical Society in i870. In this brief note comprised within a single page, he appears 4Y 2 716 APPENDIX. [IV. to. adopt the hypothesis (for his language on the point is not quite free from ambiguity), that space has everywhere a finite curvature, but that this curvature is continually changing, and that all the phenomena of the universe may possibly consist in changes of the curvature of space. A finite curvature, it will be remembered, is consistent with, and indeed implies, elementary flatness. Unfortunately, Clifford, though in earlier days he was fond of discussing this theory, no doubt as one possible mode of 'solving the universe,' has left no memoranda relating to it, perhaps because the efforts which he made to work it out in detail led him to no satisfactory conclusion. In the note of 1870 he speaks of it with a confidence which must not be taken too literally. He would probably have allowed that Lord Bacon's criticism on Gilbert, 'postquam in contemplationibus magnetis se laboriosissime exercuisset, confinxit statim philosophiam consentaneam rei apud ipsum prmepollenti,' admitted of an application, mnutatis mutandis, to his own effort to resolve all philosophy into geometry; though he would no doubt have maintained with the utmost depth of conviction that, for aught we know to the contrary, the properties of space may change with time. But whatever importance he may have temporarily attached to the opinion that space may not be independent of time, this idea has left no other perceptible traces in his mathematical writings. Very different is the case with another hypothesis as to the nature of space, which is somewhat less widely divergent from ordinary conceptions, and to which Clifford appears at all times to have turned with peculiar favour. This hypothesis admits the first three of the postulates enumerated above, as expressing true properties of space, but rejects the fourth, substituting for it the new postulate that space has a finite but very small curvature, which is approximately the same for any two points, and for any two surface directions at the same point. Admitting this postulate we find ourselves in the presence of two alternatives, between which we have to choose. For we may imagine either that the curvature of all surfaceelements, constructed in the manner above described, has the same positive value, or that it has the same negative value; understanding by a positive curvature a curvature such as that of the outer portion of the surface of an anchor-ring, where the tangent plane at any point just meets the surface and does not cut it; and by a negative curvature a curvature such as that of the inner portion of the same surface, where the tangent plane cuts the surface at the point of contact. To the hypothesis that space has a constant negative curvature considerable historical interest attaches. For this hypothesis was first arrived at, not by following out such general views as those indicated by Riemann, but in a more elementary manner. The celebrated twelfth axiom, as is well known, is the basis of Euclid's theory of parallel lines; and the assertion made in it is in fact equivalent to an assumption of the fundamental proposition of plane geometry, that the three angles of a triangle are equal to two right angles. It is now universally allowed that all efforts to demonstrate Euclid's axiom have failed; but the Russian mathematician, Lobatchewsky, appears to have been the first person to whom the idea occurred of dispensing with the axiom altogether, and trying to see what would become of geometry without it. The idea was obvious, but it was also profound; and Lobatchewsky was rewarded by the discovery that it is possible to construct a consistent and complete system of geometry upon the hypothesis that the three angles of a triangle are less than two right angles. Till the discovery of IV.] APPENDIX. 717 Lobatchewsky, the only substantial addition that had been made to Euclid's theory of parallel lines was a demonstration by Legendre, that the angles of a triangle cannot be greater than two right angles. As a matter of fact the demonstration of Legendre depends on the assumption that space is infinite; an assumption which, from the point of view taken by Riemann, cannot be regarded as justified by experience: but the considerations upon which the demonstration rests decided Lobatchewsky, as between the two alternative hypotheses that the angles of a triangle are less, and that they are greater, than two right angles, to adopt the former. This was in effect to adopt the hypothesis (though it does not appear to have occurred to Lobatchewsky in that light) that a plane has negative curvature. It was reserved for an Italian mathematician, Beltrami, to show that the plane geometry of Lobatchewsky is identical with the geometry of a pseudo-spherical surface, i. e. of a surface of constant negative curvature. What Clifford thought of the philosophical importance of the work of Lobatchewsky the following quotation may serve to show:'Each of them [Copernicus and Lobatchewsky] has brought about a revolution in scientific ideas so great that it can be only compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the Cosmos. Before the time of Copernicus men knew all about the universe. They would tell you in the schools, pat off by heart, all that it was, and what it had been, and what it would be... In any case the universe was a known thing. Now the enormous effect of the Copernican system, and of the astronomical discoveries that have followed it, is that, in place of this knowledge of a little, which was called knowledge of the universe, of Eternity and Immensity, we have now got knowledge of a great deal more; but we only call it the knowledge of Here and Now... This then was the change effected by Copernicus in the idea of the universe. But there was left another to be made. For the laws of space and motion implied an infinite space and an infinite duration, about whose properties as space and time everything was accurately known. The very constitution of those parts of it which are at an infinite distance from us, 'geometry upon the plane at infinity,' is just as well known, if the Euclidean assumptions are true, as the geometry of any portion of this room..., so that here we have real knowledge of something at least that concerns the Cosmos; something that is true of the Immensities and the Eternities. That something Lobatchewsky and his successors have taken away. The geometer of to-day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach...; but he knew this as of Here and Now; beyond his range is a There and Then, of which he knows nothing at present but may ultimately come to know more. So, you see, there is a real parallel between the work of Copernicus and his successors on the one hand, and the work of Lobatchewsky and his successors on the other.'-Lectures and Essays, vol. I, pp. 298-300. But in spite of this eulogium, the conception of space which has left the deepest traces in Clifford's writing is not that of Lobatchewsky, but that founded on the alternative hypothesis (rejected by the Russian geometer) of a constant positive curvature. This conception lies at the bottom of Clifford's theory of biquaternions, to which he 718 APPENDIX. [IV. devoted much continuous thought, and which was the origin of his researches into the classification of geometric algebras. A space of constant positive curvature is most easily represented to the mathematician (in the absence of any possibility of imaging it to the mind) as the locus of an equation of the form X2 +y2 + z2 + w2 = constant in a flat space of four dimensions in which xyzw are rectangular co-ordinates. It is related to the two dimensional surfaces of a sphere, just as in ordinary geometry space of three dimensions is related to a plane surface. The following description of a space of this kind is taken from the lecture ' On the Postulates of the science of Space.' It can hardly be necessary to point out, that in the last sentence Clifford is half laughing at himself. 'I cannot perhaps do better than conclude by describing to you as well as I can what is the nature of things on the supposition that the curvature of all space is uniform and positive. 'In this case the universe, as known, becomes again a valid conception; for the extent of space is a given number of cubic miles. And this comes about in a curious way. If you were to start in any direction whatever, and move in that direction in a perfect straight line according to the definition of Leibnitz; after travelling a most prodigious distance, to which parallactic unit 200,000 times the diameter of the earth's orbit would be only a few steps, you would arrive at-this place... Upon this supposition of a positive curvature the whole of geometry is far more complete and interesting, the principle of duality, instead of half breaking down over metrical relations, applies to all propositions without exception. In fact I do not mind confessing that I personally have often found relief from the dreary infinities of homaloidal space in the consoling hope that, after all, this other may be the true state of things.'-Lectures and Essays, vol. I, pp. 322-3. A third line of thought, different from those followed by Lobatchewsky and by Riemann, had no doubt a large share in determining Clifford to regard the hypothesis of constant positive curvature with special favour. One of the earliest geometrical inquiries of wide scope which interested him was the connexion between the descriptive and metrical properties of figures. In two unfinished memoirs,' On Analytical Metrics' (XI) and ' On the Theory of Distances' (XVI), he applied himself to work out the conception, which he justly attributed to Poncelet, that the metrical properties of any figure are in reality descriptive properties of the figure considered in relation to certain fixed geometrical elements, which Professor Cayley has termed the Absolute. In the ordinary geometry of a plane, the Absolute consists of two fixed imaginary points, and of the real straight line containing them [the imaginary circular points, and the straight line, at an infinite distance]. For these two imaginary points, in the geometry of a spherical surface, we have to substitute the imaginary circle in which the sphere is cut by the plane at an infinite distance. In ordinary space of three dimensions the Absolute is the same imaginary circle and the plane at an infinite distance in which it lies. Professor Cayley in his celebrated 'Sixth Memoir of Quantics,' generalized this conception by substituting any quadric whatever for the imaginary circle at an infinite distance (which may be regarded as a quadric surface of which one dimension IV.] APPENDIX. 719 has vanished). The effect of this substitution is to change the metric properties of space, the nature of the change depending on the nature of the quadric chosen as the Absolute. If we wish the space which we thus bring under contemplation to possess one of the most obvious properties which we know by experience to characterize the space in which we live (viz. that the rotation round a fixed axis, which bring a body from any given position back into the same position again, is a finite and not an infinite operation), our choice of the form of the Absolute is limited to three hypotheses, (i) the Absolute is an imaginary quadric, (2) the Absolute is a real umbilical quadric (i.e. a quadric not having real right lines on it) and the space considered is internal to the quadric, (3) the Absolute is an imaginary quadric which has degenerated into a conic section by losing one of its dimensions. Of these three hypotheses the last corresponds to the ordinary conception of space: the spaces characterized by the suppositions (I) and (2) have been termed elliptic and hyperbolic respectively by Professor Klein, who succeeded in showing that in each of them the curvature is constant, being positive in the elliptic, and negative in the hyperbolic space. Thus the geometry of Lobatchewsky is the geometry of hyperbolic space; and Professor Klein's discovery of the identity of the two has thrown a wholly new light upon the researches of the former geometer. Of Clifford's study of the details of the system of Lobatchewsky only one brief note is preserved (Appendix, p. 531). Indeed he seems to have quickly abandoned hyperbolic for elliptic geometry, influenced no doubt by the reason indicated in the passage which we have quoted-the perfect duality of the properties of elliptic space. In the geometry of Lobatchewsky every straight line has two real points on it at an infinite distance, viz. the two real points in which it intersects the Absolute. Again, among the planes which pass through a given straight line there are two which belong to the Absolute, and which therefore are to be regarded as planes at an infinite distance. But these two planes are imaginary, being the two planes which can be drawn through the given line to touch the Absolute. Thus in the hyperbolic geometry there is no perfect duality, because when we compare the points which lie along a line, and the planes which pass through it, the absolute elements are real in the one case, and imaginary in the other: in fact, the space which is the dual correlative of an hyperbolic space is not itself a similar space, but is analogous to the space outside the Absolute of the hyperbolic space. On the other hand, in elliptic geometry all the elements of the Absolute, whether points or planes, are imaginary, and the duality is as perfect as it is on the surface of a sphere. It follows at the same time that all distances as well as all rotations are finite, and that a point moving on a straight line (or more properly on a shortest line) will come round after a finite journey to the point from which it set out, just as a plane revolving round a straight line returns after describing a finite angle of 360~ to its original position. [The rest of the introduction, pp. xlviii-lxx, which relates to the separate memoirs, is not reprinted.] THE END.