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A TBEATISE ON SOME 
 
 NEW GEOMETRICAL METHODS, 
 
 CONTAININO ESSAYS ON 
 
 TANGENTIAL COORDINATES, 
 
 PEDAL COORDINATES, RECIPROCAL POLARS, 
 
 THE TRIGONOMETRY OF THE PARABOLA, 
 
 THE GEOMETRICAL ORIGIN OF LOGARITHMS, 
 
 THE 
 
 GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS, 
 AND OTHER KINDRED SUBJECTS. 
 
 Kova methoduB, nova eegee. 
 
 IN TWO VOLUMES.— VOL. I. 
 
 BY 
 
 JAMES BOOTH, LL.D., r.E.S., r.R.A.S., &c. &c., 
 
 ~ VICAB OF 8T0KB, BrCKINOHAUSHIBE. 
 
 MTHEMATICS 
 
 LONDON: 
 
 LONGMANS, GREEN, READER, AND DYER, 
 
 PATEHN03TEB BOW. 
 UDCCCLXXIU. 
 
 [All rights reserved.] 
 
PniNTKD BY TAYLOR AND FRANCIS, 
 SED LION COURT, FLEET STREET, 
 
ttmifoii 
 
 TO THE 
 
 PRESIDENT AND COUNCIL 
 
 EOYAL ASTRONOMICAL SOCIETY 
 
 OF 
 
 LONDON. 
 
INTEODUCTION. 
 
 The discoveries and inventions which have enriched and systema- 
 tized every department of Natural knowledge within the last half 
 century, are justly looked upon as the great intellectual conquests 
 of our time. Indeed the rapid succession of brilliant discoveries, 
 and their important applications, would seem at first sight to jus- 
 tify the opinion of many that it is only in the regions of physical 
 inquiry, in the extension of man's dominion over nature, that our 
 knowledge can receive much accession. They believe that mere 
 intellectual speculation, in which the mind is as well the instrument 
 as the subject, is barren of useful results, and that but little 
 remains to reward the labours of inquirers in that field of investi- 
 gation. 
 
 Now, the reality of this distinction cannot justly be admitted. 
 Man's triumphs in the sphere of intellect are as important in our 
 own age as at any former period of this world's history. Let us 
 limit our view to the progress of mathematical science alone ; and 
 it will suffice merely to mention the names of Euler, Lagrange, 
 Laplace, Legendre, Jacobi, Abel, and a host of others to show the 
 advance which this department of human knowledge has made in 
 modem times, depending, as it does, neither on experiment nor 
 observation, neither on the use of ingenious machinery nor of 
 instruments of precision. 
 
 But it may be replied, those discoveries are only to be found in 
 the higher analysis, a comparatively modern invention, while geo- 
 metry remains much as it was when it left the hands of Euclid, 
 Archimedes, and ApoUonius*. But surely there will occur to us 
 
 * " Quelle est la cause de cette direction exclusive dans lea etudes math^ma- 
 tiques ? Quelle sera son influence sur le caractere et lea progres de la science ? 
 Nous n'essayerons pas de r^pondre a ces questions, sur lesquelles on serait peut- 
 etre diificilenient d'accord. Mais quelles que soient les opinions a leur €gard, 
 on ne disconviendra pas du moins, qu'il serait utUe que la m^thode ancienne, 
 suivie jusqn'au siecle dernier, continuat d'etre encourag^e et cultivate concurrem- 
 ment avec la nouvelle." — Cuasles, Aper<^ HMortque, p. 551. 
 
 b 
 
VI INTRODUCTION. 
 
 the names of Descartes and Pascal, and Leibnitz, but above all that 
 of Newton, as original inquirers, and as inventors of methods of 
 investigation of the utmost value and importance. It has well 
 been observed by an English mathematician of some celebrity, 
 that " He who invents a new method, or who renders a mode of 
 investigation intelligible with whatever paucity of mere illustration, 
 does more for the interests of science than he who collects all the 
 illustrative examples of such methods that could be given." 
 
 The addition of a new method of investigation to those already 
 in use, the development of its principles, with illustrations of the 
 mode of its application, are surely not of less value to a philoso- 
 phical appreciation of what that is in which mathematical know- 
 ledge truly consists, than the invention of problems, which, while 
 they embody no general principle, are yet often difficult to solve ; 
 and when solved, frequently afford no clue by which the solution 
 may be rendered available in other cases. 
 
 Again, it often happens that an investigation which, if pursued 
 by one method, would prove barren of results or altogether imprac- 
 ticable, when followed out from a different point of view and by the 
 help of another method, not unfrequently leads by a few easy steps 
 to the discovery of important truths, or to the consideration of 
 others under a novel aspect. Hence the multiplication of methods 
 of investigation tends widely to enlarge the boundaries of science. 
 
 It must, however, be acknowledged that for more than a century 
 and a half geometry made but little progress, especially in this 
 country. The new geometry originated with Monge and his dis- 
 ciples. It may safely be asserted, there is scarcely any branch of 
 knowledge, certainly none of abstract speculation, that has put 
 forth a more vigorous growth, and received a wider cultivation, 
 than the science which, once alone called learning, was the subject 
 matter of contemplation of the exalted genius of a Plato or an 
 Archimedes. 
 
 To hold that no discoveries are to be made in a field which has 
 been repeatedly searched, is an opinion than which there are few 
 more erroneous. To have stumbled upon a new theorem or two 
 was once looked upon as a proof of rare mathematical genius. In 
 the geometry of the ancients the discovery of a new theorem sug- 
 gested no correlative ; it terminated in itself and continued sterile ; 
 while in the modem geometry a new theorem often becomes the 
 prolific source of others that may be derived &om it either by the 
 method of reciprocal polars or central projection, or pedal coordi- 
 nates or other modes of transformation. The discovery of new 
 theorems is reduced to a mere mechanical operation. We have 
 only to take any property of a sphere, suppose. By a 'single reci- 
 procating or polar transformation, as shown by M. Chasles, we may 
 obtain the correlative property of a surface of revolution of the 
 
INTRODUCTION. Vll 
 
 second orderj and then by a second polarization we may derive the 
 corresponding property for a surface having three unequal axes. 
 We have only to make a judicious selection of the pole, and then 
 new theorems may be evolved in a continuous stream so long as 
 we can supply theorems derived from the original figure. A man 
 of some celebrity in his day, as a geometer and natural philosopher. 
 Sir John Leslie, could write some fifty years ago, before the date, 
 or at least before the development of the modern geometry, in the 
 preface to his very elegant work on Geometrical Analysis, published 
 at Edinburgh in 1821*, "The multifarious labours of former writers 
 have left scarcely any room for invention. I have therefore occupied 
 myself in improving the simplicity, clearness, and elegance of the 
 demonstrations." What would be his surprise if it had been given 
 to him to forecast the wonderful progress which the science of pure 
 geometry has made in the interval between that time and the 
 present ? Geometry is stationary no Ion ger . The modem methods of 
 investigation are the most powerful solvents of problems the most 
 complicated and abstruse. Crowds of theorems spring up and pour 
 in upon us. The names of those men of original minds who have 
 shed the light of their genius, within the present century, on this 
 the oldest and most certain of all the sciences, constitute a bright 
 galaxy on the sphere of human intelligence. In proof of tliis 
 assertion do we need to say more than to mention the names of 
 Monge and Dupin, of Poncelet and Chasles, of Gergonne and 
 Steiner, and a host of others scarcely less eminent who have en- 
 riched the Transactions of Continental Societies and the pages of 
 foreign periodicals with their contributions ? They will be found 
 in the twenty volumes of the ' Annales Mathematiques ' of Ger- 
 gonne, a work unrivalled for the originality of matter it contains. 
 Its continuation will be found in the pages of Liouville and Crelle. 
 
 Among the most important methods of the modern geometry are 
 the theory of transversals, the various kinds of projection, parti- 
 cularly that of central projection, the theory of reciprocal polars, 
 that of anharmonic ratio, the elliptic coordinates of Lame, the 
 method of " inverse curves," and, if one might be permitted to add, 
 the theory of tangential coordinates developed in the following 
 pages. There is a method called by its inventors " tnlinear coor- 
 dinates," as also another named " tangential coordinates." These 
 methods have been extensively used by Mr. Whitworth, Mr. Ferrers, 
 Mr. Routh, and other English mathematicians. Though not fami- 
 liar with these methods, I understand them sufficiently to know 
 that they have nothing in common with the method to which more 
 than thirty years ago I gave the name " tangential coordinates." 
 
 To Carnot, a Member of the French Directory, is due the deve- 
 lopment of the theory of transversals ; and it may not be out of 
 • Lkslie, ' Geometrical AnalvsiB,' p. vii. 
 
 62 
 
Till INTRODUCTION. 
 
 place to state here, as an instance, if any ^rere wanted, that mathe- 
 matical genius is not incompatible with the possession of practical 
 common sense, business habits, and untiring energy, how the same 
 Camot was proverbially known as the organizer of victory for the 
 armies of the Republic. 
 
 To Poncelet, an officer of Engineers of the French army which 
 invaded Russia in 1812, and who was made a prisoner in that cam- 
 paign, so disastrous to the arms of France, we must assign the high 
 distinction of being the inventor of the methods of "reciprocal 
 polars " and " central projection." While detained a prisoner at 
 Saratoff, without the use of books and depressed by captivity, he 
 thought out those methods which in the history of the progress of 
 mathematical science will ever render his name illustrious. To the 
 student who has a true taste for excellence and elegance combined 
 in mathematical investigation, and who is tired of the endless mani- 
 pulation of quadratic equations, there is no book I can more ear- 
 nestly recommend than the great work of Poncelet, published at 
 Paris in 1822, on the projective properties of figures. Tt is only 
 in such works that the student will obtain a broad view and com- 
 prehensive grasp of the principles and methods of the science. 
 Nimble dexterity in the management of algebraical symbols in 
 accordance with prescribed mechanical rules is too often taken 
 (mistaken one should say, in this country at least) for a knowledge 
 of the science. 
 
 In pursuing mathematical investigations we must guard against 
 any bias that would lead us to prefer geometrical to algebraical 
 methods, or reciprocally. They have each their proper sphere 
 of action, so that an investigation which is both elegant and 
 simple by the one method may be operose and complicated by the 
 other. Thus to attempt to discuss the theory of Elliptic Integrals 
 by the aid of pure geometry would result in nothing but labour 
 lost and ingenuity thrown away. This is a very instructive illus- 
 tration ; for elliptic integrals of the three orders are nothing but 
 formulae of rectification for those curves of double curvature in 
 which cones of the second degree are intersected by concentric 
 spheres or paraboloids. Thus it often happens that we cannot 
 treat a question of pure geometry by methodb purely geometrical. 
 The essential difference between the two appears to consist in this : 
 that metrical properties are best treated by algebraical methods, 
 and graphical properties by geometrical constructions. 
 
 A wide induction would, I believe, show that all algebraical 
 equations do in fact represent geometrical truths. Thus trigono- 
 metry treats of the relations of circular arcs, parabolic trigono- 
 metry of the arcs of a parabola ; logarithms are nothing more than 
 the relations between the arcs of a parabola, and the corresponding 
 vectors of a logocyclic curve ; elliptic integrals are the algebraical 
 
INTKOSUCTION. IX 
 
 expressions for the arcs of those symmetrical curves in which spheres 
 and paraboloids are intersected by concentric cones, of which arcs 
 the circle and the parabola a£Ford the extreme cases. It is in geo- 
 metry alone that we can find any intelligible explanation of imagi- 
 nary quantities ; and I doubt not that most definite integrals might 
 be shown to be the representatives of certain geometrical magni- 
 tudes. 
 
 The system of coordinates invented by Descartes may be desig- 
 nated as prqj active coordinates, since they are the projections of 
 a point in a plane on two right lines drawn in the same plane, or 
 the projections of a poiat in space on three coordinate planes. This 
 system is primarily applicable to the investigation of the locus of a 
 point*, whether it be constrained to move along a straight line or a 
 curve ; it is only indirectly, and by the help of principles borrowed 
 from a higher calculus, that the system can be applied to a moving 
 straight line or a moving plane. Hence in many cases simple 
 solutions may be found for whole classes of questions, which present 
 almost insuperable difficidties when treated by projective coordi- 
 nates. 
 
 The great advantage which the tangential system of coordinates 
 exhibits over the Cartesian consists in this, that in the transforma- 
 tion of coordinates, whether by rotation or translation, the absolute 
 term continues unchanged in the tangential system, while in the 
 projective system the absolute term changes with every transfor- 
 mation. One other advantage this method possesses, the facility 
 which it aflbrds for the graphic description of cirrves. We have 
 only to assume a set of values along one of the axes of coordinates ; 
 the equation of the curve will enable us to set oflF a series of cor- 
 responding values on the other axis of coordinates. A straight 
 line may be drawn from one of those points to the other. This 
 will be a limiting tangent to the curve. 
 
 Many years ago, after I had taken my Degree, I was much inter- 
 ested in the study of the original memoirs on reciprocal curves and 
 curved surfaces, published in the ' Annales Mathematiques ' of Ger- 
 gonne, and in the works of such accomplished geometers as Monge, 
 Dupin, Poncelet, and Chasles. In the course of my own researches, 
 it occurred to me that there ought to be some way of expressing by 
 common algebra the properties of such reciprocal curves and sur- 
 faces, some method which wotdd, on inspection, show the relations 
 existing between the original and derived surfaces. I was then led 
 
 * " En effet I'^l^ment primitif des corps auquel on applique d'aboid les premiers 
 piincipes de cette science est couune dans la gSom^trie ancienne, le point niath^- 
 matique. Ne sommes-nous pas autoris^s a penser maintenant, qu'en ^renant le 
 plan pour I'^l^ment de I'Stenaue, et non plus le point, on sera conduit a d'autres 
 doctrmes, &isant pour ainsi dire une nouvelle science." — Chasles, Aper^u His- 
 torique, p. 408. 
 
I INTRODUCTION. 
 
 to the discovery of a simple method and compact notation from the 
 following considerations. 
 
 When two figures in the same plane, or more generally in space, 
 are so related that one is the reciprocal polar of the other, then to 
 every point in the one corresponds a plane in the other j to every 
 straight line in the one a straight line also in the other ; to any num- 
 ber of points in the same straight line in the one, as many planes all 
 intersecting in the same straight line in the other ; to any number of 
 points in the same plane in the one, as many planes all meeting in 
 the same point in the other. They may be called correlative figures. 
 Now we know that in the application of algebra to geometry by 
 the method of coordinates, a point is determined in position by ite 
 projections on three coordinate planes, or by three equations — that 
 is, by three conditions. A straight line may in like manner be deter- 
 mined when we are given the positions of two points in it ; and a 
 plane is determined by one condition, which is called its equation. 
 But in the inverse method, a point should be determined by one 
 condition, a straight line by two, and a plane by three. Again, a 
 straight line maybe determined by considering it as joining two fixed 
 points, or as the common intersection of two fixed planes. Now 
 all these conditions may be expressed by taking as a new system of 
 coordinates the segments of the common axes of coordinates between 
 the origin and the 'points in which they are met by a movable 
 plane. Thus, if these segments be designated by the symbols X, 
 Y, Z, the three equations which determine a plane are 
 X= constant, Y = constant, Z = constant. 
 
 Again, the equation in (x, y, z) of a plane passing through a 
 point of which the coordinates are x, y, z, and which cuts ofiT from the 
 
 axes of coordinates the segments X, Y, Z, is =5r+^4-^=:l. Now 
 
 this is the •projective, or common equation of the plane, if we 
 make x, y, and z vary, and consider X, Y, Z constant. But we 
 may invert these conditions, and consider x, y, z constant, while 
 X, Y, and Z vary. The equation now, instead of being the pro- 
 jective equation of a fixed plane, becomes the tangential equa- 
 tion of a fixed point. In this latter case let a, y9, and y be put 
 
 111 
 for X, y, z, and ^, -, y for X, Y, Z ; then the equation may be 
 
 written 
 
 a|+/8.;+yr=l, 
 
 which may be called the tangential equation of a point. 
 
 Moreover, as the continuous motion of a point, in a plane 
 suppose, subjected to move in accordance with certain fixed con- 
 ditions expressed by a certain relation between x and y may be 
 conceived to describe a curve, so the successive positions of a 
 
INTKODUCTION. XI 
 
 straight line cutting off segments from the axes of coordinates 
 having a certain relation to each other may be imagined to wrap 
 round or envelop a certain curve, just as we may see a curve 
 described on paper by the successive intersections of a series of 
 straight lines. Hence there are two distinct modes according to 
 which we may conceive all curves to be generated, namely by the 
 motion of a tracing point, or the successive intersections of straight 
 lines — by a pencil or straight edge, as a joiner would say. These 
 conceptions are the logical basis of the methods by which the prin- 
 ciples and notation of common algebra are generalized from the 
 discussion of the properties of abstract number to those of pure 
 space. The former view gave rise to the method of projective coor- 
 dinates ; the latter suggests the method of tangential coordinates. 
 
 It is sometimes very easy to express both the projection and tan- 
 gential equations of the same curve or curved surface ; it is fre- 
 quently a matter of extreme difficulty. 
 
 Thus, if the projective equation of an ellipsoid be 
 
 its tangential equation will be 
 
 a, b, c being, as in the preceding equation, the semiaxes. 
 
 The chief object kept in view in the following pages is to develop 
 the principle of Geometrical Duality, a principle apparently one of 
 the most obvious, and singularly fruitful in results, beyond any 
 other geometrical method hitherto discovered. 
 
 That the principle of Duality should not have been discovered 
 by the great geometers of Ancient Greece is the more remarkable, 
 as the five regular solids, the Platonic* bodies as they were called, 
 Were with them a favourite subject of speculation. In a later age 
 Kepler believed them to be the archetypes of the planetary motions f. 
 
 * " II n'est pas ^tonnont qu' Aristae ait €ciit sur leg cinq corps i^guliers ; car 
 cette th^oiie a €\j6 fort cultiy^e, et en grand honneur des la plus haute antiquity 
 des sciences chez les GiecB. Fythagore en avait fait le principe de sa cosmo- 
 gonie, dans laquelle les cinq corps r^guliers r^pondaient aux quatre ^l^mens et a 
 rimiyers, ce qui a fait qu'on les appelait les cinq figures mondianes {Jiguree mun- 
 dante) Platon adoptait ces id^es, et avait aussi cultivd cette thiSorie, sur laquelle 
 Theatete, I'un de ses disciples, passe pour avoir ^crit le premier. Ensuite, on 
 trouve done Aristae, puis Kuclide, ApoUonius et Hypsicle. Ces cinq corps r€gu- 
 liers ont jou^ un si grand role dans I'antiquit^, par suite des id^es pythagonciennes 
 et platoniciennes, qu'on les regardait comme 4tant le but final auquel ^taient 
 destinies et I'^tude et la science des g€ometres." — Chasles, Aper^a Hiatorique, 
 p. 514. 
 
 t " Repetat lector ex Mysterio meo Cosmographico, quod edidi ante 22 annos : 
 numerum planetarum seu cuniculoruni cii-casolemdesumptiunesse a i-apientissimo 
 Conditore ex quinque iiguris rogularibus solidis.'' — Kefleii, JIarmvnkes Mutuli, 
 lib. v. p. 276. 
 
Xll INTRODUCTION. 
 
 But the five regular solids afford a prominent iUustration of this 
 principle of geometrical duality. The cube and the octahedron 
 have each the same number of edges, while they interchange their 
 faces and solid angles. Their edges are conjugate polars, and are 
 consequently the same in number. In the same way the dodeca- 
 hedron and icosahedron are correlative solids ; they have the same 
 number of edges, while they interchange their faces and solid 
 angles ; and, finally, the tetrahedron is its own correlative solid. 
 When Euler, after many bootless efforts, had succeeded at last in 
 establishing by elaborate proof* the singularly simple relation 
 which connects the number of faces, comers, and edges of any 
 polyhedron, he did not perceive how by the application of this 
 principle of duality the number of his conclusions might have been 
 doubled. Had Euler known this universal relation, he would have 
 seen that, whatever be the formula connecting the solid angles, 
 faces, and edges of any polyhedron, it must be satisfied by the 
 interchange of the solid angles and faces, while the number of 
 edges continues the same. Had Legendre been acquainted with 
 this principle he would have known that when any polyhedron, regu- 
 lar or irregular, has any given number of edges, there must be pos- 
 sible another polyhedron having the same number of edges, but 
 interchanging the numbers of its faces and solid angles with those 
 of the correlative polyhedron. 
 
 To restrict the application of this principle to geometrical curves, 
 surfaces, or solids would be to mask the exceptionless universality 
 of this ail-pervading principle. It not only holds good as regards 
 curve lines and surfaces, whether they be algebraical or transcen- 
 dental, whether they be continuous or discontinuous, whether they 
 be described in accordance with some geometrical law or capri- 
 ciously {libera manu, so to speak), but is true for every form of 
 bounded space. The rounded pebble on the beach, the angular 
 fragment in the quarry, each has its own strictly defined correlative 
 polar figure, just as well as the most symmetrical solid of the ancient 
 geometry. So universal is the principle of duality that to some it 
 appears as a general law of nature f- In Physics we find it in 
 
 * " On a vu daus le III"n« volume du present recueil (p. 169) que ce n'est 
 qu'apres des tentatives r^it^r^s qu'Euler est parvenu a ^tablir d'une maniSre a 
 la foiB complete et g^n^rale, son curieux th^oreme but la relation constonte entre 
 le nombre des faces, celui des sommet8,et celui des aretes d'un polyedie quelconq ue. 
 
 " On salt que, dans ces demiers temps M. Cauchy a d^montr^ d'une mamere 
 beaucoup plus simple un autre th^oreme dont celui d'Euler n'est qu'un cas par- 
 ticulier."— Gergonnb, Annates Mathimatiqties, vol. xii. p. 333. 
 
 A very elegant and simple demonstration of this curious theorem which had 
 so long baffled that illustrious geometer Euler, will be found at page 333 of the 
 XIX.tu volume of the ' Annales Math^matiques ' of Gergonne, based on the rela- 
 tions of a group of reticulated polygons. 
 
 t " On peut croire pourtant qu'une unit6 absolue n'est pas le principe de la 
 nature. Les dualismes nombreuz qui se remarquent dans les ph^nom^nes natu- 
 
INTRODUCTION. XUl 
 
 attraction and repulsion^ in the motions of rotation and translation j 
 in the emotions of the mind, good and evil, virtue and vice ; in the 
 double origin of our ideas ; in the sensations of the body, pleasure 
 and pain. The principle of duality meets us everywhere. But 
 this is not the place to follow out such a train of inquiry. 
 
 The idea or conception of duality or correlation had long been 
 admitted into and developed in spherical trigonometry, and the 
 properties of polar triangles investigated by geometers with abundant 
 labour and much acuteness. But it would seem never to have oc- 
 curred to those accomplished mathematicians that the correlation 
 which may be developed in spherical triangles and polygons was 
 only a particular case of a general principle which binds together, 
 so to speak, all the properties of space without exception. 
 
 For those who have not yet mastered the great principle of 
 duality and the results which flow from it, one or two simple illus- 
 trations of it may not be here out of place. The well-known the- 
 orem of Pascal on the hexagon inscribed in a conic section, and its 
 dual the hardly less celebrated theorem of Brianchon, are subjoined 
 side by side. This latter is otherwise remarkable, as being the 
 first result of the application of the principle of duality. See XILI. 
 Cahier du Journal de I'Ecole Poly technique, p. 301. 
 
 PuscdCa Theorem. Brianchon's TTteorem. 
 
 The opposite sides of a hexagon m- The opposite angles of a hexagon 
 
 scribed in a conic section being pi-o- ctVcumecribed to a conic section being 
 
 duved to meet, two by two, in three Joined two by two by three straight 
 
 points; these three points range along lines, these tlu:ee lines meet in the same 
 
 the same straight line. point. 
 
 Maclaurin's Theorem for the organic Its Seciprocal Polar. 
 
 description of conic sections. The three angles of a triangle rest 
 
 The three sides of a triangle pass on thiee fixed lines; two ot its sides pass 
 
 through three ^ed jjoiiUs ; and two of through ^ed points; the third nV2e will 
 
 its angles move along fixed straight envelop a conic section, 
 lines; the third angle will describe a 
 conic section. 
 
 No reader, who is capable of understanding the subject, will 
 imagine that this new system of coordinates is proposed with a 
 view to supersede the old. The Cartesian system cannot be snper- 
 
 rels, comme dans les differentes parties des connsissances humaines, tendent au 
 contraire & nous faire supposer qu'une dualiti constante, ou double unit^ est le 
 vrai principe de la nature. Cette dualitiS, nous la trouyous dans I'objet meme 
 de la g^om^trie, ainsi que nous venons de le dire ; dans la nature des propri^t^s 
 de r^tendue ; dans le double mouvement des corps celestes, ou sa Constance 
 reconnue la fait admettre comme principe ; et dans miUe autres ph^nomenes ; et 
 I'on sera conduit, je crois, a regarder qu un duaUsme universd est la grande loi de 
 la nature, et regno dans toutes les pai'ties des connaissances de I'esprit humain." 
 — Chasleb, Aperpi Hittoriqite, p. 290. 
 
XIV INTRODUCTION. 
 
 seded. As the properties of space are dual^ so must the systems of 
 investigation be dual also. The one method supplements the other^. 
 " Ita utrumque, per se indigens, alterum alterius auxilio eget." 
 Where the one may with ease be applied the other will be found to 
 failt. ThuS; while in the solution of problems of rectification the 
 tangential method holds out many advantages on the score of faci- 
 lity and simplicity, in quadratures it will afford but little help, and 
 recourse must be had to the familiar integral \ydx. 
 
 It would be a very inadequate conception of the power of this 
 method to suppose that the systems of tangential and projective 
 coordinates are merely supplemental one to the other. IVom their 
 combination whole classes of problems may be evolved and deter- 
 mined without dif&culty. For example, it will be shown that, the 
 projective and tangential equations of the same curve being given, 
 we may at once write down the equation of the locus of the vertex 
 of a given angle, one of whose sides passes through a fixed point 
 while the other side envelops the given curve, as also the equation 
 of the locus of the foot of the perpendicular let fall from the origin 
 on the tangent. As an instance, if 
 
 Aw^+Aff^+2Bxij +2Cx+2Cff=l, .... (a) 
 a^ + a,v^ + 2^^v + 2y^+2y,v = l, (b) 
 
 * " On se tromperait en croyant que la g^om^trie dans ses moyens de pioc^der 
 a la lecherclie de la v^rit^, doit avoii des bomes pos^s seulement par la nature 
 de cette science, et non pai la nature meme des choses. On se tromperait ^gale- 
 ment en croyant ces bomes moins recul^es que celles on I'analyse pent atteindre 
 en marchant vers le meme but. Ces deux m^thodes sous des formes diif^rentes, 
 Bont les d^veloppements identiques d'une seule et meme science qui soumet a 
 la fois toutea les grandeurs i ses combinaisons, a ses rapprochements. L'une, 
 sans jamais perdre de vue les choses memes qu'elle doit considerer, porte partout 
 I'^vidence avec elle ; elle rend sensibles toutes ses conceptions, toutes ses opera- 
 tions, et les grandeurs graphiques sont pour elle un moyen de peindie dans I'espace, 
 et sa marche et ses r^sultats. L'autre substitue aux quantity dont elle s'occupe 
 des signes purement abstraites, elle d^pouille les grandeurs de tout ce qui n'est 
 pas inherent aux relations qu'elle envisage ; elle ramSne tout a des lois g^n^rales ; 
 
 tivement & la parole. .... En consid^rant ainsi I'analyse et la g^om^trie dans 
 leurs rapports, ces deux sciences s'^laireront mutuellement, et chacune d'ellea 
 s'acroitrai de tous les progres et l'autre. Ne rejetons done aucun de ces moyens 
 pour proc^der a la lecherdie de la y6nt6," &c. — ^Ditfin, Oiw^ofpementt de Gio- 
 mStrie,ji. 236. 
 
 t " The exercise of the mind in understanding a series of propositions, where 
 the last concluMon is geometrically in close connexion with the first cause, is 
 very different from that which it receives from putting in play the long train of 
 machinery in a profound analytical process. The degrees of conviction in the 
 two cases are very different.^ To the greater number of students, therefore, I 
 conceive a popular geometrical explanation is more useful than an algebraical 
 investigation.' —Sir G. B. Aiby, the Astronomer Royal, On Gravitation, p. 7. 
 
INTRODUCTION. XV 
 
 be the projective and tangential equations of a curve of the second 
 
 ' A?* + A,w« + 2B^w + 2 (Cf + C,v) [^ + v^) = (J^ + „2)2 
 
 will be the tangential equation of the curve enveloped by one side 
 of a right angle which moves along the curve (a) while the other 
 side always passes through a fixed point, and 
 
 aw* + agf* + 2^xy + 2 {yx+yg/) {a^ + y*) = (a:« + y*)* 
 
 will be the projective equation of the point in which a perpendicular 
 from the origin meets a tangent to the curve whose tangential 
 equation is (b) . 
 
 It may appear to some that examples are needlessly multiplied 
 in the following pages ; but it may be replied that in the treatment 
 of a very abstract subject, especially if it be a novel theory, exam- 
 ples, if judiciously selected, may throw light u])on the obscurities 
 of imperfect explanations and defective discussions. It is granted 
 but to few to grasp a theory as its development proceeds, even in 
 the hands of a master. Examples are required pour fixer les idees. 
 They assist us to incorporate new truths with that older knowledge 
 which we have made our own. In many cases the author has been 
 satisfied with laying down the principles of the method as applied 
 to a few particular instances, without following up the investigations 
 into all their details. To have done this would have swelled the 
 bulk of the volume without any equivalent advantage. Something 
 must be left to the ingenuity and industry of the reader to develop 
 and to amplify. The book is intended to be suggestive, not ex- 
 haustive. Where the claims of so many new and important theo- 
 rems to recognition are so continuous and pressing, one cannot stop 
 to draw out the deductions that follow from each into a cluster of 
 corollaries. So far from any exercise of ingenuity being necessary, 
 the discovery of new theorems by the help of this and other kindred 
 methods does not call for the exercise of much patient thinking. 
 They spring up so spontaneously, as it were, that the difSculty 
 is to keep under restraint the imagination as it courses along those 
 trains of thought sure to end in the discovery of some new, and 
 it may be unexpected, geometrical truth. It would have been an 
 easy task, a fascinating labour, to have swelled the pages of this 
 volume with diversified researches; but I have kept within the 
 limits which at the outset I prescribed to myself, and have been 
 satisfied to point out the way to others. 
 
 A somewhat detailed account of the subjects investigated in this 
 volume win be expected by the reader. In the first place I have 
 endeavoured as far as practicable to adopt a uniform and consistent 
 notation. I have striven to use the same symbols in the same 
 sense. Wherever possible I have made the absolute terms of the 
 equations equal to unity. In this way the constants are reduced 
 
XVI INTKODUCTION. 
 
 to the smallest number, and the orders of the constants, whether 
 they be lines, surfaces, or solids, may be inferred from inspection, 
 being always of the same or the inverse order of the variables of 
 which they are the coefficients. 
 
 Hence the projective equation of the surface of the second order 
 will be written 
 
 in which A, A,, A,„ B, B„ B„ are inverse rectangles, and C, C,, C„ 
 inverse straight lines. 
 
 In like manner the tangential equation of the surface of the 
 second order may be written 
 
 in which a, a,, a,,, /3, /3„ jS^, are rectangles, and 7, j,, y,, straight 
 lines. 
 
 The first twenty-four Chapters of this volume treat of the trans- 
 formations of tangential coordinates, and of the application of this 
 system, by one uniform method* to the discussion of theorems and 
 problems. The ever-recurring analogies between the Cartesian and 
 the tangential system of coordinates are continuously indicated as 
 the work proceeds. The method is first applied to develop the 
 properties of curves and curved surfaces of the second order, next 
 to the genesis and rectification of curves of higher orders, their 
 evolutes and involutes, &c. 
 
 In Chapter XXI. another system of coordinates is established, 
 which I have ventured to call pedal tangential coordinates, and 
 their properties developed. 
 
 In Chapter XXV. the peculiar notation of tangential coordinates 
 is laid aside, and the principles of geometrical duality are esta- 
 blished on the simplest elementary conceptions of pure geometry, 
 and then applied to the investigation of the properties of surfaces 
 of the second order having three unequal axes. It is shown that 
 every such surface, with two exceptions, has four directrix planes 
 parallel to the circular sections of the surface, and four correspond- 
 ing foci, which directrix planes and foci coalesce when the surface 
 becomes one of revolution. It is moreover shown that for every 
 property of a sphere there exists its correlative on a surface having 
 three unequal axes. These theorems will be found in Chapter 
 XXVI. 
 
 In Chapter XXIX. metrical methods are applied to the theory 
 of reciprocal polars, and new classes of properties of the conic 
 sections established, more particularly those which are connected 
 vrith the lines called the minor directrices of these curves. 
 
 * " Le vrai secret d'un systeme est dans sa m^thode. Mettez une m^thode 
 dans le monde, vousy mettez un syst^iue que I'aTenir se chargeraded^velopper." 
 — Victor Cousin, Court de thuHaire de la philosophie, voL i. p. 84. 
 
INTEODCCTION. XVll 
 
 In Chapter XXX. the properties of the curve which I have named 
 the Logocyclic Carre are discussed, a curve which has singular 
 analogies with the circle. If the vectors of this curve be drawn 
 from the origin and be taken to represent all the natural numbers 
 from to CO , the corresponding arcs of a conjugate parabola will 
 represent the logarithms of the numbers. 
 
 In the same Chapter the geometrical origin of logarithms is 
 established ; and it is shown that every system of logarithms may 
 be represented by the arcs of a corresponding parabola, the pecu- 
 liarity of the Napierian parabola being that the distance from its 
 focus to its vertex must be assumed as equal to unity. 
 
 In Chapter XXXI. the principles of the trigonometry of the 
 parabola are investigated. It is universally true to state that 
 there is and can be no relation established between the arcs of a 
 circle for which we cannot find a correlative for the arcs of a para- 
 bola. The transition may easily be made by changing cos 6 into 
 sec ■&, ■*/ — 1 sin 6 into tan ■&, + into J- , and — into -r . 
 
 It is not a little remarkable that while these mysterious imaginary 
 expressions connected with the circle have been thoroughly inves- 
 tigated, the corresponding reciprocal theorem has entirely eluded 
 discovery. Volumes have been written on the development of the 
 imaginary theorem 
 
 {cos6-\- V — lain ^)"= cos m^-f- V — Isinn^, 
 while nothing has been known of the real theorem 
 
 (sec ^+ tan 5)»=sec(^-L^-i-e&c.) 4- tan (5-^0-1-0 &c.). 
 
 In the same Chapter the principles of parabolic trigonometry are 
 applied to the investigation of the properties of the Catenary and 
 the Tractrix. Curious relations are established between the arcs of 
 a catenary whose abscissae are in arithmetical progression. It is 
 also shown by the same method that the catenary is the evolute of 
 the tractrix. 
 
 The last Chapter is devoted to the investigation of the projective 
 equation of a cone whose vertex is at the intersection of three con- 
 focal surfaces, and which touches a fourth confocal surface. 
 
 I propose, if declining years and failing strength permit me, to 
 complete this work, and to embody in a second volume my researches 
 on the geometrical origin and properties of Elliptic Integrals, and 
 te-^ply them to the investigation of the free motion of a rigid 
 body round a fixed point, together with other collateral inquiries. 
 
 The tone of thought demanded by subjects such as these falls 
 dull upon the public ear, and excites no responsive sympathy. It 
 may therefore be proper to say that this volume is sent forth to the 
 world with the anticipation of a very limited circulation, because 
 it must be admitted that a cultivator of abstract science, without 
 any view to practical results or profitable returns, has no reason- 
 
XVlll INTRODUCTION. 
 
 able ground of expectation that his labours will be recognized or 
 appreciated in this country. With us the pursuit of knowledge 
 for its own sake or indulgence in scientific research, unless it 
 may be made to minister to some practical result (that is, to 
 some paying result), is looked upon as little better than intellectual 
 trifling*. WiU it pay ? is th? test of all mental labour. It was 
 very Afferent in the schools and agorae of that nation we are so 
 prone to hold up for admiration as exhibiting models of intel- 
 lectual greatness hitherto unequalled. Nor is this exclusive devo- 
 tion to the adaptation of science to money-making so universal in 
 other countries as amongst ourselves. Yet it was not always so. 
 One might appeal to the age of Newton and Locke, the age of 
 deep thinking and profound learning, in proof of this position. ' 
 The causes of this degradation in the objects of intellectual pursuit 
 are many, and some of them deeply seated. Not the least of these 
 is the influence which the philosophy of Bacon has exerted on the 
 tone and tendency of public opinion in this country. No doubt 
 the author of the ' Novum Organon ' conferred great benefits on 
 mankind by laying down so clearly the true principles of physical 
 investigation. He has marred his philosophy, however, by the 
 motives he presents to us for its cultivation. He who could pro- 
 pound the maxim, worthy of Epicurus, that the true object of 
 science isf to make men comfortable, had no very exalted con- 
 ception of the dignity of man's understanding. 
 
 It is plain firom his tone of thought that the philosophical Chan- 
 cellor had a very clear prsenotion, to use his own phraseology, of 
 that emphatically English idea, comfort. There is little doubt that 
 he would have valued more the invention of an efiicient kitchen- 
 range, or an ingenious corkscrew, than the Ideas of Plato or the 
 discoveries of Archimedes j;. 
 
 This may to some appear a philosophical heresy ; but yet it is 
 qtdte certain that Lord Bacon's powerful influence, based on the 
 soundness of his notions as to the true mode of procedure in con- 
 ducting experimental inquiries, has had a depressing effect on the 
 views of his countrymen, whose highest intellects are now devoted 
 to the production of sensation novels, or to the discovery of inge- 
 nious contrivances to subserve the unbounded luxury and promote 
 the material enjoyments of a self-indulgent people. 
 
 And if we turn aside from the paths of commerce and the busy 
 haunts of men to the quiet cloisters of academical retirement, we 
 shall find the same motives equally powerful and all but universal. 
 
 • "In primis, hominis est propria veri inquisitio atque investigatio." — Cicero 
 ' de Officiis, lib. i. c. 13. 
 
 t " Meta autem Bcientiarum vera et leg^tima non alia est quam ut dotetur 
 
 vita humana novia inventis et copiis." — ^Bacon, Novum Organon, lib. i. aph. 81. 
 
 % Lord Bacon, like Hobbes, knew but little even of the elements of Mathematics.' 
 
INTEODUCTION. XIX 
 
 Mathematical studies, and indeed one might say all studies, are 
 pursued not for their own intrinsic worth, but with reference to 
 the universal competitive examination. The inquiry is not, is it 
 true ? but will it teU ? not, is it important as a principle ? but is 
 it likely to be asked as a question ? In what profession but the 
 law are profound research and extensive learning valued ? and these 
 are so because they pay. 
 
 Our universities are admirable institutions for the development 
 of the intellect and the formation of habits j but they are not 
 equally adapted to enlarge the boundaries of knowledge*. It is 
 with regret that one is compelled to admit the fact that the great 
 discoveries of modem times have been made by men who con- 
 ducted their researches and worked out their discoveries far away 
 from the theatres and libraries of our great centres of learning. 
 
 There is no reason in the nature of things why it should be so. 
 Is the water that is drawn from the stagnant pool or spreading 
 lake fresher or purer than that which rises from the gushing 
 spring ? Who ought to be so ready to draw from his stores of 
 knowledge as he who spends his life in acquiring them? The 
 commimication of knowledge to youth is an important function 
 of a university ; but it is only one, an important or most important 
 one if you will; but there are others hardly less important. 
 Were Eton and Harrow and Rugby and the other great public 
 schools of this country to be translated in all their integrity to 
 Oxford or Cambridge, they would not constitute a university, in 
 the highest and best sense of the word, because while teaching is 
 the exclusive province of the public schools, our imiversities have, 
 or ought to have, a higher function to discharge't. We ought to 
 have not only the standing army but the pioneers of knowledge. 
 Our universities do not encourage the Livingstones of science. 
 Here, too, the same blighting influences prevail |. It is useless to 
 
 * " Rursus in Moribus et institutis Scholarum Academiarum, Collegiorain et 
 similium Conventuum, quae doctonim hominiun sedibua et eraditioms cultuiae 
 destinatse sunt, omnia piogreasui scientiaruiii adversa inyeniuntuT. Studia enim 
 hominum in ejusmodi locis in quomndam auctorum scripta, veluti in caiceres, 
 conclusa simt ; a quibus ai quia dissentiat, continuo ut homo turbidus et rerum 
 novarum cupidus corripitur. — Bacon, Novum. Organon, lib. i. aph. 90. 
 
 t "... Sit denique alia scientias colendi, alia inveniendi ratio. Atque quibus 
 prima potior et acceptior est, ob festinationem, yel vitse civilis rationes, vel quod 
 illam alteram ob mentis infirmitatem ctmere et complecti non possint, optamus 
 ut quod sequuntur teneant." — ^Bacon, Preface to the Naiium Organon. 
 
 \ The decline of science in England was long ago commented upon by one who 
 was himself a brilliant luminaiy in his day. . . . "Here whole branches of con- 
 tinental discovery are unstudied, and indeed almost unknown even by name. It 
 is vain to conceal the melancholy truth. We are fast dropping behind. In 
 Mathematics we have long since drawn the rein, and given over a hopeless 
 race ; in Chemistry the case is not much better. Nor need we stop here. There 
 
XX INTRODUCTION. 
 
 deny it. In this country profound acquirements in literature or 
 science are not held in the same estimation they once commanded 
 even amongst ourselves. In times gone by, for those denied the 
 
 are indeed few sciences which would not furnish matter for similar remark." — 
 Sir J. F. W. Heuschel, Article on Sound, Eiusyclopadia Metropolitana, p. 810. 
 
 And to the same effect, another high and still moi-e recent authority : — 
 
 "Now, as to this important subject, the spirit in which we pursue education, 
 the degree in which we turn our advantages to account, I must say of us here 
 in England that we do not stand well. Our old Universities, and the schools 
 above the rank of primary, have as a class the most magnificent endowments in 
 the world. It may, however, be doubted whether the amount of these endow- 
 ments, in England alone, is not equal to their amount on the whole continent 
 of Europe taken together. Matters have metaded, and are, I hope, mending. 
 We have good and thorough workers, but not enough of them. The resulte 
 may be good as far as tbey go ; but they do not go far. But in truth this 
 ' beggarly return,' not of emptf but of ill-filled boxes, is but one among many 
 indications of a wide-spread vice — a scepticism in the public mind, of old as well 
 as young, respecting the value of learning and of culture, and a consequent slack- 
 ness in seeking their attainment. We seem to be spoiled by the very facility 
 and abundance of the opportunities around us. We do not in this matter stand 
 well, as compared with men of the middle ages, on whom we are too ready to 
 look down. For then, when scholarships and exhibitions, and fellowships and 
 headships, were few, and even before they were known, and long centuries 
 before triposes and classes had been invented, the beauty and the power of 
 Knowledge filled the hearts of men with love, and they went in quest of her, 
 even from distant lands, with ardent devotion, like pilgnms to a favoured shrine. 
 
 " Again, we do not stand well as compared with Scotland, where, at least, the 
 advantages of education are well understood, and, though its honours and rewards 
 are much fewer, yet self-denying labour, and unsparing energy in pursuit of 
 knowledge, are &r more common than with us. And once more, we do not 
 stand well as compared with Germany, where, with means so much more 
 slender as to be quite out of comparison with ours, the results are so much more 
 abundant, that, in the ulterior prosecution of almost every branch of inquiry, it 
 is to Germany, and the works of the Germans, that the British student must 
 look for assistance. Yet I doubt if it can be said with truth that the German 
 is superior to the Englishman in natural gifts, or that he has greater or even 
 equal perseverance, provided only the Englishman had his heart in the matter. 
 But Germany has two marked advantages : a far greater number of her educated 
 class are really in earnest about their education ; and they have not yet learned, 
 as we, I fear, have learned, to undervalue, or even in a great measure to despise, 
 simplicity of life. 
 
 "Our honours, and our prizes, and our competitive examinations, what for the 
 most part are they, but palliatives applied to neutralize a degenerate indifiference, 
 to the existence of which they have been the most conclusive witness ? Far be 
 it from me to decry them, or to seek to do away with them. In my own sphere, 
 I have laboured to extend them. They are, however, the medidnes of our infir- 
 mity, not the ornaments of our health. They supply from without inducements 
 to seek knowled^, which ought to be its own reward. They do something to 
 expel the corroding pest of idleness, that special temptation to a wealthy country, 
 that deadly enemy in all countries to the Dody and the soul of man. They get 
 us over the first and most difficult stages in the formation of habits, which, in 
 a proportion of cases, at least, we may hope will endure, and become in course 
 of time self-acting. 
 
 " One other claim I must make on behalf of examinations. It is easy to point 
 
INTRODUCTION. XXI 
 
 gifts of fortune, the only access to the Temple of Fame was through 
 the portals of the learned professions, as they were called, or the 
 services of their country, whether military or naval. The essential 
 elements of success in these were high intellectual and moral en- 
 dowments, unflagging labour and enduring perseverance. 
 
 Qui studet optatain cursu contingere metam 
 Multa tulit fecitque puer, sudavit et alsit. 
 
 But all this is changed. There are so many ways now, and some 
 of them very questionable, of attaining to high social position, and 
 to the possession of enormous wealth with a very attenuated garb of 
 shreds and patches of trite information picked up anyhow, inaccu- 
 rate and vague, that men do not care to undergo the study and the 
 toil required to "plate themselves in the habiliments" of knowledge. 
 With many, the maxim, to sell in the dearest and to buy in the 
 cheapest market, would seem to comprise the whole duty of man. 
 
 It is no satisfactory answer to these remarks to say that there 
 are exceptions to be found. No doubt there are, and brilliant 
 exceptions. It might seem invidious to particularize, where varied 
 excellence is so abundant. But this only adds strength to the 
 argument ; for it might with truth be said, what development of 
 genius might we not witness were not its genial current frozen by 
 the cold and chiUing maxims of a spurious political economy ? It 
 would be blame misplaced to find fault with the Government for 
 not encoujaging profound learning or scientific research. Did the 
 country desire (which it does not) that such should be honoured 
 and rewarded, there is little doubt that the Government of the 
 day, consisting mostly of men of high attainments themselves, 
 would be glad to foster and advance the cultivators of that learn- 
 ing which is popularly considered no better than pedantry, and of 
 science which is held to be of scant utility. 
 
 It would not be hard to prove that views such as these are short- 
 out tlieir inherent imperfectiuna. Plenty of critics are ready to do this ; for in 
 the case of first employment under the State, they are the only tolerably efficient 
 safeguard against gross abuses, and such abuses are never -without friends. But 
 from really searching and strong examinations, such as the best of those in our 
 Universities and schools, there arises at least one great mental benefit, difficult 
 of attainment by any other means. In early youth, while the mind is still natu- 
 rally supple and elastic, they teach the practice, and they give the power, of con- 
 centrating all its tbrce, all its resources, at a ^ven time, upon a given point. 
 What a pitched battle is to the commander of an army, a strong examination is 
 to an earnest student. All his faculties, all his attainments must be on the 
 alert, and wait the word of command ; method is tested at the same time with 
 strength ; and over the whole movement presence of mind must preside. If, in 
 the course of his after life, he chances to be called to great and concentrated 
 efforts, he will look back with gratitude to those examinations, which more 
 perhaps than any other instrument may have taught him how to make them." — 
 Address delivered at the Liverpool College by the Might Hon. W. E. Gladstone, 
 Dec. 21, 1872. 
 
 c 
 
XXa INTRODUCTION. 
 
 sighted and low. It has been somewhere observed that had not 
 the ancient Greek geometers investigated the properties of the 
 plane sections of a cone, Kepler would have had no curve to fall 
 back upon when he was obliged to abandon the circle as an accurate 
 type of the orbits of the planets, and Newton would have lacked 
 the profound and laborious arithmetical calculations of Kepler, 
 the work almost of a lifetime*, on which to base the connexion 
 of the three great laws of planetary motion with the principle of 
 universal gravitation. The true mechanism of the heavens would 
 still be to us an inscrutable problem, and navigation would even 
 now be guided by the stars, as in the time of Palinurus. 
 
 But for the wonderful discoveries of Newton in the science of 
 Astronomy, and the consequent improvements in the art of Navi- 
 gation, with the influence of this latter on the extension of com- 
 merce,, it might have been that these transcendent truths would in 
 our time have come to be looked upon as no better than Verba 
 otiosorum senum, as Dionysius the tyrant of Syracuse sneeringly 
 said of the speculations of Plato. 
 
 No man can forecast the consequences that may follow from any 
 discovery in abstract science or physical research. What important 
 results in aerial locomotion were at one time anticipated from the 
 invention of the balloon ! How little was once thought of the 
 experiments on steam by the Marquis of Worcester ! or who 
 attached any importance to the investigation of the nature of 
 those feeble forces elicited by the attrition of a bit of sealing- 
 wax or a lump of amber ? The consequences of any truth so dis- 
 covered may well be held, using the language of Bacon, to be 
 " partus temporis, non partus ingenii." 
 
 * Kepler, who lived in advauce of his a^, who appealed to the verdict of 
 posterity, and not in vain, in a striking passage of the Jfarmomce Mundi which 
 breathes a tone of saddened enthusiasm, thus records the grand discovery of his 
 life, after seventeen long yeais of unremitting labour, the prerequisite of a still 
 grander discovery, the law of Universal Gravitation. 
 
 " Rursum igitur hie aliqua pars mei Mysterii Cosmogiaphici suspensa ante 
 22 annos quia nondum liquebat, absolvenda et hue inferenda est. Inventis enim 
 veris orbium intervallis per observationes Brahei plurimi temporis labore con- 
 tinue, tandem, tandem, genuina proportio temporum periodicorum ad propor- 
 tionem orbium — 
 
 Sera qmdem respexit tnertem, 
 Respexit tamen et longo post tempore ventt ; 
 
 eaque, si temporis articulos petis, 8. Mart, huius anni millesimi sexcentesimi 
 decimi octavi animo concepta, sed infeliciter ad calculos vocata eoque pro falsa 
 rejecta, denique 15. Maji reversa, novo capto impetu expugnavit mentis meae 
 tenebras tanta comprobatione et laboris mei septendecennalis in observationibus 
 Brabeanis et meditationis huius in unum conspirantium, ut somniare me et prte- 
 sumere qussitum inter principia primo crederem, sed res est certissima exactis- 
 aimaque, quod IVoportio qua est inter binorum quorumcungue planetarum tempora 
 periodica, sit prmcise sesquialtera proportionis mediarum dikantiarum, id est orbium 
 ipsorum." — Keplrb, Harmoniees Mundi lib. v. cap. iii. p. 279. 
 
INTEODUCTION. XXUl 
 
 I know not that apologies disarm criticism ; it is said they more 
 frequently provoke it. But still it may be proper to say that this 
 book is the result of the meditations of the better part of a life- 
 time. Desultory and rare they have been, intermitted for years, 
 drawn away to other subjects, taken up again at different times 
 and lengthened intervals. It has been to me a heavy drawback 
 and deep discouragement, that I have had no fellow-workers to 
 share in these researches. Neither have I entered into the labours 
 of any. Without sympathy and without help I have worked upon 
 those monographs now presented to the public. Nor let any one 
 imagine that this isolation of the understanding is but a little loss 
 or trivial hindrance. From the sympathetic contact of mind with 
 mind truth is elicited. The electric spark of thought in its passage 
 often flashes light on that which was clouded or obscure before. 
 That which to one intelligence may appear clear as crystal, to 
 another intelligence, nowise inferior, may seem distorted or con- 
 fused by the media through which it is transmitted. Thus the 
 concentrated and patient thinking of many intellects, some of a 
 high order perhaps, combined and cooperating to one end, develops 
 and expands a principle or a theory which the silent efforts and 
 unaided labours of the solitary worker would have failed to accom- 
 plish. Nor have I had the help of those who have gone before 
 me; for these researches, as here presented to the reader, are 
 entirely original. I may add, that this work was not written to 
 improve the text-books in use ; nor is it published now to lighten 
 the labours of tuition, or to supply the requirements of official com- 
 petitive examinations. While this renunciation of scholastic uti- 
 lity may contract its circulation, on the other hand it has left me 
 at liberty to follow out any train of thought to whatsoever conclu- 
 sions it might carry me. I have waited long in the expectation 
 (at shall I say hope ?) that some of the many accomplish^ mathe- 
 maticians of the present day would take up those subjects and 
 expand them (for they admit of great development), and so produce 
 a treatise from which any student of moderate ability might have 
 gleaned enough to enable him to extend those researches still 
 further. But I have waited in vain. 
 
 Had I seen any likelihood that those results, of which from time 
 to time I have given abstracts in the proceedings of learned Soci- 
 eties, would be developed and published in a connected form, I 
 should have shrunk from the toil of compilation, and left it to 
 fresher and more elastic minds to expand and deliver to the world 
 those discoveries which I would not willingly leave to dull forget- 
 fulness or " to lie in cold obstruction." 
 
 J. B. 
 
 Stone Vicarage, 
 
 June 10, 1873. 
 
 c2 
 
ADVERTISEMENT 
 
 TO THE 
 
 FIRST EDITION OF THE 
 
 ESSAY ON TANGENTIAL COORDINATES. 
 
 I FEAK that brevity and compression hare been but too much stndied in the 
 following essay ; but the necessity of comprising the whole matter in a small 
 compass, and the pressure of other avocations, will plead, I hope, a sufficient 
 apology. 
 
 From the same cause I have been obliged to omit altogether subjects which 
 might have been with propriety introduced, for example the general theory of 
 shadows, and have only touched upon others which would require perhaps further 
 development. 
 
 Among other applications of the method, that to the theory of reciprocal 
 polars will, I trust, be found simple and satisfactory. 
 
 My attention has just been directed by a friend to a letter from M. Chasles, 
 dated December 10, 1829, published in the ' Correspondence MathSmatique ' of 
 M. Quetelet, torn. vi. p. 81, in which the writer asserts his claim to the invention 
 of a system of coordinates, noticed by M. Plucker in one of the livraisons of 
 Crelle's Journal, to which work I have never had an opportunity of referring. 
 After some preliminary observations, he states his system as follows : — " Pour 
 cela, par trois points fixes a, b, c, je m^ne trois axes parall&les entre eux, un plan 
 quelconque rencontre ces axes en trois points dont les distances aux points a, b, c, 
 respectivement, sont les coordonnSes x, y, i, du plan," &c. ; and then goes on to 
 apply his system to a few examples, using the principles and notation of the 
 differential calculus. To any one consulting the letter from which the above 
 extract is taken, it will be apparent that the method there proposed, however 
 excellent and ingenious it may be, bears not the least resemblance to the one 
 developed in the following pages. 
 
 It must have often appeared an anomalous fact in the application of algebraic 
 analy^s to geometrical investigations, that while the locus of a point could be 
 found from the simplest and most elementary considerations, the envelope of a 
 right line or plane could be determined only by the aid of principles, artificial 
 and obscure, derived from a higher department of analysis. 
 But this is not the only or the greatest objection to the method at present 
 
XXV 
 
 universally followed ; it is in most cases operose, and in some impracticable, to 
 
 dV 
 reduce the equation V=0 to the form -=- =0, and then eliminate the auxiliary 
 
 variable a between these equations — a difficulty which becomes far more for- 
 midable in problems of three dimensions, where we are obliged to eliminate the 
 auxiliary variables a and /3 between the three equations 
 
 -* £-». %-"■ 
 
 As it follows A priori from the principle of duality*, that for every locus of a 
 point there exists a corresponding envelope of a right line or plane, it would seem 
 that the comparative paucity of theorems of the latter species generally known 
 can be owing to nothing but the want of a simple and direct mode of investi- 
 gation. 
 
 From these considerations I have been led to the discovery of a method simple 
 in principle, and easy of application, analogous to, but different from, that of rec- 
 tilinear or projective coordinates — as for distinction they may be called — in which 
 the reciprocals of the distances of the origin from the points where the axes of 
 coordinates are met by a right line, or plane, touching a curve or curved surface, 
 are denoted by the letters i,v,(; an equation established between them may be 
 called the tangential equation of the curve or curved surface. 
 
 By the help of this equation we may elude the necessity of differentiating the 
 equation.y=0, and discover the envelopes of right lines and planes with the 
 same facility as the locus of a point by projective coordinates. 
 
 But it is not alone in inquiries of this nature that the method is chiefly 
 valuable ; there is a large class of theorems relating to curves touching given 
 right lines, and surfaces in contact with given planes, which may be treated by 
 the method proposed with the greatest facility, whose solution by projective coor- 
 dinates woidd lead to exceedingly complicated and unmanageable expressions. 
 
 J. B. 
 
 Tkinity College, 
 March 2oth, 1840. 
 
 * See various memoirs on this subject by MM. Ger^onne, Poncelet, and others, 
 dispersed through the volumes of the ' Annales de Math^matiques.' 
 
TABLE OF CONTENTS. 
 
 [The numben on the left hand denote the sectionB, the numbers on the 
 right hand the pages.] 
 
 CHAPTER I. 
 
 ON THE TANGENTIAL EQUATIONS OF A POINT AND A STRAIGHT LINE IN 
 
 A PLANE. 
 
 1.] Tangential equations of a point and a plane 1 
 
 2.] On the transfonnation of coordinates 2 
 
 3.] On the translation of coordinates 2 
 
 4.] When the angle of ordination is changed 3 
 
 6.] Length of perpendicular from a siren point on a given line in terms of 
 
 its tangential coordinatefi and the prcg ective cooidinates of the point . . 3 
 6.] Expression for the value of the angle between two given straight lines 
 
 whose tangential coordinates are given 3 
 
 7.] The tangential coordinates of a straight line passing through two fixed 
 
 points whose projective coordinates (o/S) and (a,fi,) are given 4 
 
 8.] to 10.] Discussion of the forms which the expressions for the perpen- 
 dicular assume in projective and tangential cooidinates 4 
 
 CHAPTER II. 
 
 ON THE TANGENTIAL EQUATIONS OP THE CONIC SECTIONS. 
 
 12.] Tangential equations of the central conic sectiona referred to centre and 
 
 axes 7 
 
 Formulse of transformation when the rectangular axes pass through 
 
 an; point in the plane of the section, when the section is an ellipse . . 8 
 
 13.] "When it is an hyperbola 9 
 
 14.] When the origin is at a focus 9 
 
 1.5.] When the ori^n of coordinates is on the curve 9 
 
 16.] Expression for the angle which the asymptote of the hyperbola makes 
 
 with the axis of X 10 
 
 17.] Relation between the constants when the tangential equation of the 
 
 curve breaks up into two linear equations 11 
 
 18.] Values of the constants when the curve touches the axes of X or Y . . 11 
 19.] To find the values of | and w when they are tangents to the curve 12 
 
TABLE OF CONTENTS. XXvii 
 
 20.] Tangential equations of the polea of the axes of X and Y 12 
 
 21.] To find the species of the conic section when the reciprocal of the per-' 
 pendicular from the origin on the tangent is a rational function of 
 the tangential variables 13 
 
 22.] General expressions for the values of | and v in terms of x and y and 
 
 conversely 14 
 
 23.] Relations between the partial difFarentials of the projective and tan- 
 gential equations of a given curve 14 
 
 24.] Application of these formulae 15 
 
 2t5.] On Asymptotes 16 
 
 2*3.] On the tangential equation of the curve referred to oblique axes of coor- 
 dinates 17 
 
 27.] General expression for the distance between the point of contact of a 
 tangent and the foot of the perpendicular let fall on that tangent from 
 
 the origin 17 
 
 28.] Value of a semidiameter in terms of the tangential coordinates of the 
 
 tangent through the extremity of the diameter 18 
 
 CHAPTER III. 
 
 THE PRINCIPLES ESTABLISHED IN THE PRECEDING SECTIONS 
 APPLIED TO EXAMPLES. 
 
 29.] The product of pairs of perpendiculars let fall from two points on a 
 
 straight line is constant ; the line envelopes a conic section 19 
 
 30.] The vertex of a right angle moves along the circumference of a circle ; 
 one side passes through a fixed point ; the other envelopes a conic 
 section 19 
 
 31.] Tangents are drawn to an ellipse from any point of a concentric circle 
 whose radius isVa^+6^; the line joining the points of contact enve- 
 lopes a confocal conic 20 
 
 32.] The sum of the perpendiculars let fall from n given points in a plane to 
 a straight line in the same plane is constant. The straight line enve- 
 lopes a circle 21 
 
 33.] Perpendiculars are drawn to the extremities of the diameters of an 
 
 ellipse. They envelope a curve ; to determine its tangential equation 22 
 
 34.] Two semidiameters of a conic section and the chord joining their extre- 
 mities contain a given area. The curve enveloped by this chord is a 
 
 similar conic section 22 
 
 K a polygon of n sides be inscribed in a conic section, the sides being 
 inversely as the perpendiculars let fall upon them from the centre, the 
 polygon wUl circumscribe a conic section similar to the given one . . 23 
 
 35.] Envelope of the chord joining the extremities of two focal chords of a 
 
 conic section 23 
 
 36.] A straight line revolves in a conic section, having always a constant 
 
 ratio to the parallel diameter ; it will envelope a similar conic section 25 
 
 37.] The product of the sides of a right-angled triangle diminished by fixed 
 
 quantities, is constant; thehypotenuse will envelope a conic section. . 26 
 
 38.] Let the sides of a rectangle be produced, and cut by a transversal ; to 
 find the tangential equation of the curve to which this line is always 
 a tangent, under certain conditions 2G 
 
XXVin TABLE OF CONTENTS. 
 
 39.] The vertex of an angle of constant magnitude moves along the circum- 
 feience of a circle ; one side passes through a fixed point ; to deter- 
 mine the curve that will be enveloped by the other 28 
 
 40.] An angle of given magnitude revolves roiind a fixed point, intersecting 
 by its sides two given straight lines ; the line which joins the points of 
 intersection envelopes a conic section 29 
 
 41.] An angle of ^ven magnitude 6 revolves round a point in the plane of a 
 conic section, cutting the curve in two points ; the line joining these 
 points will envelope a curve whose tangential equation is of the fourth 
 order 30 
 
 42.] A right-angled triangle has its right angle at a focus of a conic section, 
 while the hypotenuse envelopes the curve ; one acute angle of the 
 triangle moves along a given straight line, the other vrill describe a 
 conic section 32 
 
 43.] Form of the tangential equation of a conic section when the origin is 
 
 anywhere in a concentnc circle passing through the foci_ 33 
 
 44.] A triangle is inscribed in a conic section ; two of its sides always pass 
 
 through two fixed points, the third side envelopes a conic section . . 34 
 
 45.] A series of central conic sections have the same centre, and their axes 
 in the same direction, but such that the difierence of the reciprocals 
 of the squares of their axes is constant ; the tangents drawn to a point 
 on each, their intersection with a common diameter, envelope a con- 
 centric hyperbola, if the intersected curves be a series of elbpses, and 
 an ellipse if the intersected curves be hyperbolas 37 
 
 On Polygons inscribed and circumscribed to Conic Sections. 
 
 46.] Conic sections are inscribed in the same quadrilateral, the polar of any 
 
 point in their plane envelopes a conic section 38 
 
 47.] A series of conic sections are inscribed in the same quadrilateral, their 
 
 centres range on the same straight line 39 
 
 48.] The centres of conic sections inscribed in the same quadrilateral all range 
 on the straight line which joins the points of bisection of the two dia- 
 gonals of the quadrilateral. (Newton's theorem) 40 
 
 CHAPTER IV. 
 
 ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 
 
 49.] Tangential equation of the parabola derived from its projective equa- 
 tion 41 
 
 50.] Relation between the constants when the curve touches the axes of 
 
 coordinates 43 
 
 This theory applied to examples 43 
 
 51.] The sum of the sides of a right angle is constant. The hypotenuse en- 
 velopes a parabola 43 
 
 52.] An angle of given magnitude moves along a fixed straight line, one side 
 always passes through a fixed point, the other side will envelope a 
 parabola _ _ 44 
 
 53.] Parabolas are inscribed in a triangle ; the locus of their foci is the cir- 
 cumscribing circle 44 
 
 54.] An angle of given magnitude revolves round the focus of a parabola ; to 
 determine the curve enveloped by the chord which joins the points 
 in which the parabola is intersected by the sides of the angle 45 
 
TABLE OF CONTENTS. XXIX 
 
 CHAPTER V. 
 
 ON THE TANGENTIAL EQUATIONS OF THE POINT, THE PLANE, AND THE 
 STRAIGHT LINE IN SPACE. 
 
 56.] On the tangential equations of a point and a plane in space 49 
 
 On the transformationa of tangential coordinates in space 51 
 
 57.] Expression for the perpendicular from the origin on a plane whose tan- 
 gential coordinates are given 51 
 
 53.] On the translation of the axes of coordinates in parallel directions .... 52 
 69.] On the tangential equations of a plane passing through the origin of 
 
 coordinates 53 
 
 CHAPTER VI. 
 
 ON THE TANGENTIAL EQUATIONS OF THE STRAIGHT LINE IN SPACE. 
 
 CO.] Two methods of defining the position of a straight line in apace 54 
 
 Gl.] Tangential method 5-3 
 
 62.] To express the cosines of the angles which a straight line makes with 
 the axes of coordinates in terms of the constants /i, v, a, /3 of the tan- 
 gential equations of the given straight line 56 
 
 63.] To determine the conditions that two straight lines may meet in space 56 
 
 65.] To investigate the conditions that a given line may be found in a given 
 
 plane 57 
 
 60.] To find the tangential coordinates of a plane which shall pass through 
 
 a given point and a given straight line 57 
 
 67.] To determine the angle between two given planes 58 
 
 (i8.] A straight line is perpendicular to a given plane, to determine the rela- 
 tions t>etween the coefficients of the given straight line and the given 
 plane 58 
 
 09.] To determine the angles which the straight line, in which two given 
 
 planes intersect, ma£es with the axes of coordinates 59 
 
 70.] To determine the conditions in order that a given strught line and a 
 
 given plane may be parallel 60 
 
 71.] A straight line is parallel to a given straight line, to determine the rela- 
 tion between the constants 60 
 
 72.] To investigate an expression for the angle between two g^ven straight 
 
 lines whose equations are given 60 
 
 73.] To find an expression for the angle between a given plane and a given 
 
 straight line 61 
 
 CHAPTER VII. 
 
 ON THE TANGENTIAL EQUATIONS OF SURFACES OF THE SECOND ORDER. 
 
 74.] On the tangential equations of surfaces of the second order referred to 
 
 their centres and axes, as axes of coordinates 62 
 
 75.] Translation of the ori^n to any other point in space, the axes of coordi- 
 nates continuing parallel 63 
 
XXX TABLE OF CONTENTS. 
 
 76.] Values of the tangential ordinates |, v, ( in terms of the partial differ- 
 ential coefficients of the projective equation of the given surface 64 
 
 77.] Tangential equations of the poles of the three coordinate planes 66 
 
 Tangential equations of the polar plane of the origin 66 
 
 Condition in order that the origin of coordinates may he on the given 
 surface 67 
 
 78.] Vivlues of certain coefficients when the surface touches one or more of 
 
 the coordinate planes 67 
 
 79.] Determination of the point of contact 67 
 
 80.] Formula of transition from the projective to the tangential equation of 
 
 any sur&ce, and reciprocally 68 
 
 CHAPTER VIII. 
 
 ON THE UAONITUDE AND POSITION OF THE AXES OF A SURFACE OF 
 THE SECOND ORDER. 
 
 81.] Transformation of the tangential equation of a surface of the second 
 
 order 69 
 
 82.] A general expresdon for the distance hetween the point of contact of a 
 tangent plane to a surface, and the foot of the perpendicular let fall 
 from the origin on this tangent plane 70 
 
 83.] Definition of axis of central surface 71 
 
 Tangential ctdnc equation of axes 71 
 
 84.] To determine the angles which one of the axes of the surface makes with 
 
 the axes of coordinates 72 
 
 85.] On the particular case when the surface is one of revolution 73 
 
 CHAPTER IX. 
 
 ON THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 86.] The tangential equation derived from the projective equation of the 
 
 paraholoids 75 
 
 87.] On the transformation of the axes of coordinates in the case of the para- 
 holoids 76 
 
 88.] Given the general equation of the paraholoid to any set of rectangular 
 axes passing through the vertex, to determine the magnitude and 
 
 position of the parameters of the principal sections 77 
 
 89.] On the hyperholic paraholoid 78 
 
 90.] Limits of the surface 80 
 
 91.] Tangential equation of this surface derived from its projective equation 80 
 92.] To ascertain whether in a tangent plane to this surface there can exist 
 
 any linear generatrices 80 
 
 93.] No two successive generatrices of this surfsuse can meet, or be in the 
 
 same plane 82 
 
 94.] To determine the sections of this surface made by tangent planes in 
 
 certain positions 82 
 
TABLE OF CONTENTS. XXXI 
 
 CHAPTER X. 
 
 ON THB APPLICATION OF ALOSBBA TO THE THEORY OF 
 KECIFROCAL FOLARS. 
 
 95.J Line of contact of a cone touching a surface of the Recond order is a 
 
 plane curve 83 
 
 96.] Case when the surface is a central ellipsoid 84 
 
 97. j Pciar equations of surfaces of the second order 84 
 
 Illustrations and instances of the principle of Duality 85 
 
 98.] Application of the principles established in the foregoing sections to 
 
 the investigation of certain theorems 85 
 
 The sum of the perpendiculars let fall from » given points on a plane 
 is constant. To determine the envelope of the plane 89 
 
 99.] The sum of the squares of the perpendiculars let fall from n given 
 points on a plane whose tangential coordinates are f , v, ^ is con- 
 stant, and equal to nl^ ; the plane envelopes a surface of the second 
 
 order 89 
 
 100.] A series of surfaces of the second order touch seven fixed planes ; the 
 poles of any given plane relative to these surfaces ore also on a fixed 
 plane 90 
 
 101.] A surface of the second order touches seven given planes, to find the 
 
 locus of its centre 91 
 
 102.] If two surfaces of the second order are enveloped by a cone, they may 
 
 also be enveloped by a second cone 91 
 
 103.] Let a plane cut off from three fixed rectangular axes, segments the 
 sum of which, multiplied by a constant area, shall be equal to the 
 tetrahedron whose faces are the three coordinate planes and the 
 limiting tangent plane ; to determine the surface enveloped by this 
 latter plane 92 
 
 104.] If a series of planes retrench from a cone of the second degree a con- 
 stant volume, they will envelope a discontinuous hyperbofoid, or one 
 of two sheets 94 
 
 105.] On the cubic equation of axes, when the surface is one of revolution, 
 
 and the origin at a focus 96 
 
 106.] Three straight lines, constituting a right-angled trihedral angle, 
 revolve round a fixed point in space,meetmg asut&ce of the second 
 order (S]| in three points. The plane which passes through these 
 three points envelopes a surface of revolution (2) of the second 
 order, whose focus is at the given point, and whose directrix plane 
 relative to this focus is the polar plane of the fixed point relative to 
 the given surface (S) 97 
 
 107.] To show that the continuous hjrperboloid admits of linear gene- 
 ratrices 99 
 
 108.] A Biuface of the second order is cut by a given secant plane ; to deter- 
 mine the tangential equation of the section of the surface made by 
 this plane 100 
 
 109.] The reciprocal polar of any surface of the second order, the centre of 
 
 the directrix surface being on the given surface, is a paraboloid . . 101 
 
XXxii TABLE OF CONTENTS. 
 
 CHAPTER XI. 
 
 ON CONCYCLIC iURrACES OF THE SECOND ORDER. 
 
 110.] Concyclic surfaces are the reciprocal polars of confocal surfaces .... 102 
 
 111.] Examples of the analogies between concjclic and confocal surfaces — 
 
 dual relations 103 
 
 112.] Through a given point three central confocal surfaces may be de- 
 scribed — an ellipsoid, a continuous and discontinuous hyperboloid 104 
 
 113.] A series of concyclic surfaces of the second order touch a riven. plane 
 whose tangential coordinates are |, v, f. To determine the equation 
 of the axes of these surfaces 105 
 
 114.] A common tangent plane is drawn to three concyclic surfaces of the 
 second order, the three points of contact two by two subtend right 
 angles at the centre 105 
 
 lis.] Let there be two concyclic ellipsoids, and any point on the external 
 one be assumed as the vertex of a cone enveloping the other, the 
 plane of contact will meet the tangent plane to the first surface 
 through the vertex of the cone in a straight line, such that the dia- 
 metral plane passing through this line will be at right angles to the 
 diameter which passes through the vertex of the cone 106 
 
 116.] Let a cone envelope an ellipsoid, so that the plane of contact shall 
 touch a second surface confocal with the former. The line drawn 
 from the vertex of the cone to the point of contact of this tangent 
 plane vdll be at right angles to it 107 
 
 117.] Parallel planes are drawn to a series of confocal ellipsoids; to deter- 
 mine the locus of the points of contact 109 
 
 118.] To a series of concyclic surfaces tangent planes are drawn touching 
 the surfaces in the points where they are pierced by a common dia- 
 meter ; to find the surface enveloped by these tangent planes .... 110 
 
 CHAPTER XII. 
 
 ON THE SURFACE OF THE CENTRES OF CURVATDRE OF AN ELLIPSOID. 
 
 119.] Tangential equation otthe surface of centres of an ellipsoid 112 
 
 120.] Investigation of the shape of this surface 113 
 
 Any two parallel tangents being drawn to the surface of centres and 
 to the ellipsoid, the difiference of the squares of the coincident per- 
 pendiculars let fall upon them from the centre is always equal to the 
 
 square of the coinciding semidiameter of the ellipsoid 114 
 
 121.] On the umbilical lines of curvature of an ellipsoid 116 
 
 123.] The areas of the umbilical parallelogram, of the ellipse, and of the 
 evolute circumscribed by and inscribed in the four umbilical nor- 
 mals, have certain reciprocal relations wMch are independent of the 
 
 axes of the ellipsoid 118 
 
 124.] On the projective equation of the surface of centres of an ellipsoid . . 120 
 125.] Three concyclic surfaces of the second order are touched by a com- 
 mon tangent plane in three points; these points, two by 'two, will 
 subtend right angles at the centre, and the locus of all the points of 
 contact with the two variable hyperboloids will be a surface which 
 may be called the " surface of contacts " 1J2 
 
TABLE or CONTENTS. XXXIII 
 
 CHAPTER XIII. 
 
 ON THE APPLICATION OF THE METHOD OF TANGENTIAL COORDINATES 
 TO THE INVESTIGATION OP THE PROPERTIES OF TRANSCENDENTAL 
 AND OTHER CURVES OF A HIGHER ORDER THAN THE SECOND. 
 12G.] On the tangential equation of the caustic by reflection of the circle . . 124 
 127.] Two cases of the general theoiem which may repay discussion .... 127 
 128.] Determination of the points in which the limiting tangent, when 
 
 vertical, cuts the axis of X 128 
 
 CHAPTER XIV. 
 
 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 129.] The theory of tangential coordinates applied to investigate the pro- 
 perties of epicycloids and hypocycloids generally 129 
 
 130.] to 132.] The theory applied to selected cases 130 
 
 133.] On hypocycloids 133 
 
 To determine the equation of the hypocycloid, when the radius of the 
 
 rolling circle is one half that of the base circle 133 
 
 134.J On the hypocycloid whose radius is one fourth that of the base circle 133 
 
 135.] To determine the involute of the quadrantal hypocycloid 134 
 
 136.J On the projective eqviation of the involute of the quadrantal hypo- 
 cycloid 135 
 
 137.] On the tangential equation of the trigonal hypocycloid 137 
 
 138.] On the tangential equation of the hexagonal hypocycloid 138 
 
 139.] On the tangential equation of the cycloid 139 
 
 141.] On the tangential equation of the logarithmic curve 140 
 
 142.] On the tangential equation of the cissoid 141 
 
 143.] On the tangential equation of the lemniscate 141 
 
 144.] On the tangential equation of the cardioid 142 
 
 145.] On the projective equation of the curve whose tangential equation is 
 
 oV+6=|'=fl''6"(|'^+«»)* 143 
 
 146.] On the tangential equation of the curve inverse to the central ellipse, 
 
 or the curve whose equation is o'ar'+6^i/°=(.i'^+y")' 1J4 
 
 147.] On the reciprocal polar of the evolute of an ellipse 145 
 
 148.] On the tangential equation of the semicubical parabola 147 
 
 149.] On the tangential equation of the cubical parabola 147 
 
 150.] On the involute of the cycloid 147 
 
 CHAPTER XV. 
 
 ON PARALLEL CURVES. 
 
 151. J Definition of parallel curves 149 
 
 152.] Tangential equation of the curve parallel to the ellipse 150 
 
 153.] The normals of a parabola are increased by the constant line A; to 
 
 iind the tangential equation of the curve 151 
 
XXxiv TABLE OP CONTENTS. 
 
 Tangential equation of the curve parallel to the evolute of an ellipse 152 
 
 To find the parallel curve of the quadrantal hjrpocycloid 152 
 
 154.] To find the equations of the parallel curves of the involute of the 
 
 quadrantal hypocycloid 152 
 
 Tangential equation of the curve parallel to the semicubical parabola 152 
 Tangential equation of the curve parallel to the cycloid 153 
 
 CHAPTER XVI. 
 
 ON THE TANGENTIAL EQUATIONS OP EVOLTJTES. 
 
 155.1 Values of the variables in the tangential equation of the evolute in 
 terms of the partial differential coefficients of the projective equa- 
 tion of the curve whose evolute is required 153 
 
 156.] To determine the tangential equation of the evolute of the ellipse . . 153 
 
 157.] Tangential equation of the evolute of the parabola 154 
 
 158.] On the evolute of the semicubical parabola 154 
 
 159.] On the evolute of the cubical parabola 154 
 
 160.] On the evolute of the quadrantal hypocycloid 155 
 
 161.] On the tangential equation of the evolute of the lemniscate 156 
 
 162.] On the evolute of the curve whose equation is e^3f'+Vy'=(x'+y'y 167 
 
 CHAPTER XVII. 
 
 ON REVOLTING ANGLES, PROJECTIVE AND TANGENTIAL. 
 
 163.] Definitions of the curves termed tangential and projective pedals . . 158 
 
 164.] A-f^lication of this theory to examples 169 
 
 (a) Tangential pedal of a straight line. 
 
 (/3) Tangential pedal of a point. 
 
 M Projective pedal of a conic section. 
 
 (d) Becomes a cardioid, when the origin is on the curve. 
 
 165.] Tangential pedal of a circle is a conic section 160 
 
 166.] Projective and tangential pedals of the semicubical parabola 161 
 
 167.] Projective pedal of the cycloid 161 
 
 Simple geometrical proof of this theorem 161 
 
 168.] A light angle moves along a cycloid; one side passes through the 
 centre of the base, the other side will envelope a curve whose tan- 
 gential equation may be found 161 
 
 169.] Projective pedal of the cardioid. 162 
 
 170.] Projective pedal of the quadrantal h3rpocycloid 162 
 
 171.] Tangential pedal of the logocyclic curve 163 
 
 172.] Tangential pedal of the evolute of an ellipse 163 
 
 Projective pedal of the evolute of an ellipse 163 
 
 Tangential pedal of a cissoid is a parabok 163 
 
 Projective pedal of a cissoid 163 
 
 173.] Projective pedal of the curve parallel to an ellipse 163 
 
 174.] Projective pedal of the curve parallel to a parabola is a conchoid 164 
 
TABLE OP CONTENTS. XXXV 
 
 CHAPTER XVIII. 
 
 ON RIGHT ANGLES REVOLVING ROUND FIXED POINTS IN THE PLANE 
 
 OP A CURVE. 
 
 175.] General projective equation of the locus in terms of the partial dif- 
 ferential coefficients of the tangential equation of the given curve. . 165 
 176.] Projective equation of the locus when the g:iven curve is an ellipse. . 166 
 
 177.] When the given curve is a cuhical parabola 166 
 
 178.] When the given curve is a semicubical parabola 166 
 
 179.] When the given curve is a parabola 167 
 
 180.] When the revolving angle is no longer a right angle. General for- 
 mulae in this case for the locus 168 
 
 CHAPTER XIX. 
 
 TO INVESTIGATE THE LOCI WHEN THE REVOLVING ANGLES IN THE 
 TWO PRECEDING CHAPTERS ARE OTHER THAN RIGHT ANGLES. 
 
 181.] Transformation of formulae on this hypothesis 169 
 
 182.] Application of modified formulae to the ellipse 170 
 
 183.] To the trigonal hypocycloid 171 
 
 CHAPTER XX, 
 
 ON THE LOCI OP RIGHT ANGLES REVOLVING AROUND GIVEN CURVES. 
 
 184.] A right angle revolves so as to touch with its sides a given curve, to 
 
 determine the projective equation of the vertex of the right angle 171 
 
 185.] Application of this method to examples 172 
 
 When the ^ven curve is an ellipse 172 
 
 186.] When the curve is a parabola 172 
 
 187.] To investigate the equation of the vertex of the right angle which 
 
 envelopes the involute of the quadrantal hypocycloid 173 
 
 188.] A right angle revolves, touching with its sides the curve parallel to 
 
 the paraM)la ; to determine the locus of the vertex 173 
 
 189.] A right angle revolves, touching with its sides a cycloid ; to determine 
 
 the locus of the vertex 174 
 
 190.] A right angle revolves round a fixed point, whose sides meet a curve 
 
 in certain points ; to determine the tangential equation of the curve 
 
 which is enveloped by the line joining these points 175 
 
 191.] Illustration of this method 175 
 
 192.] Another illustration 176 
 
 193.1 Recapitulation of the principles established in Chapters XV., XVI., 
 
 XVn., XVIII., XIX., and XX : 177 
 
XXXvi TABLE OF CONTENTS. 
 
 CHAPTER XXI. 
 
 ON PEDAL TANGENTIAL COORDINATES. 
 
 194.J Definition of this system of coordinates 178 
 
 195.] Illustrationa of this method 179 
 
 Pedal tangential of a straight line 179 
 
 Pedal tangential of a circle 179 
 
 Pedal tangential of an ellipse 179 
 
 196. J, 197.] Properties of the evolute of an ellipse analogous to those of an 
 
 ellipse 18" 
 
 198.] On the pedal tangential cmre of a semicubical parabola 181 
 
 199.] The pedal tangential of the curve whose equation is o'y"+6'.r'=;tV 
 
 is an ellipse 181 
 
 CHAPTER XXII. 
 
 ON THE RADIUS OF CURVATCRE AND THE RECTIFICATION OP PLANE 
 
 CURVES. 
 
 200.] Definitions of the rectification of plane curves and of the radius of 
 
 curvature, 181 
 
 Primary expression for the radius of curvature 182 
 
 201.] Primary expression for the arc of a plane curve 183 
 
 How the ambiguity of sign is to be determined 184 
 
 202.] Explanation of the critical point in Fagnani's theorem on the arcs of 
 
 an ellipse 184 
 
 203.] Expression for the radius of curvature in terms of the partial differ- 
 ential coefficients of the tangential equation of a plane curve .... 185 
 Expression for the arc of curve in terms of the same coefficients .... 185 
 204.] On the rectification of the circle by the method of tangential coordi- 
 nates, the origin being taken any where in the plane of the circle . . 185 
 
 205.] On the radius of curvature of the ellipse 186 
 
 206.] On the radius of curvature of the parabola 186 
 
 207.] On the rectification of the ellipse 187 
 
 208.] On the rectification of the parabola 187 
 
 209.] On the radius of curvature and the rectification of the cycloid 187 
 
 The square of the radius of curvature at any point of a cycloid, toge- 
 ther with the square of the arc measured from that point to the 
 vertex,, are equal to the square of the diameter of the generating 
 
 circle 188 
 
 210.] On the re«tification of the evolute of the ellipse 188 
 
 211 .] On the rectification of the semicubical parabola, and its radius of cur- 
 vature 189 
 
 212.] On the rectification of parallel curves 190 
 
 213.] On the radius of curvature, and the rectification of epicycloids and 
 
 hvpocycloids 190 
 
TABLE OP CONTENTS. XXXVll 
 
 214.] Application of these formulge to examples 193 
 
 (a) The cardioid. 
 
 (P) The semiciTcalar epicycloid. 
 
 (y) The quadrantal epicycloid. 
 
 (h) The trigonal epicycloid. 
 
 216,] On the cuTrature and rectification of hypocycloids 193 
 
 216.] Relation between the radius of curvature and the arc of any epicy- 
 cloid 194 
 
 217.] On the radius of curvature of the cubical parabola 194 
 
 218.] On the rectification of the involute of the quadrantal hypocycloid . . 195 
 219.] On the rectification of the curve whose tangential equation is 
 
 6»|»+aV=«»6»(f+w»/ 196 
 
 220.] On the rectification of the inverse curve of the central ellipse 196 
 
 CHAPTEE XXIII. 
 
 ON THE RELATION BETWEEN TANGENTIAL EQUATIONS, AND THE 
 SINGULAB SOLUTIONS OV THE DIFFERENTIAL EQUATIONS OF 
 PLANE CURVES. 
 
 221.] Relation between the tangential and projective equations of the same 
 
 curve 200 
 
 (a), (j3) Illustrations of this principle 201 
 
 CHAPTER XXIV. 
 
 ON THE GEOMETRICAL THEORY OF RECIPROCAL POLARS. 
 
 222.] Elementary principles on which the geometrical theory of redprocal 
 
 polars is established 202 
 
 223.-] Definition of polarizing sphere 202 
 
 224.] If any point n-, be assumed in a plane (n), the polar plane (n,) of tiiis 
 
 point vrill pass through v the pole of (n) 204 
 
 When any number of straight lines are parallel to one another, their 
 
 conjugate polars will all he in the same plane 204 
 
 The polar plane of any point assumed in a straight line will pass through 
 the conjugate polar of this straight line ; and the polar plane of any 
 point assumed in a plane will pass through the pole of this plane . . 204 
 225.] The reciprocal polar of any plane curve is a cone whose vertex is in 
 
 the pole of the plane of the curve 204 
 
 6.] From the centre a of the polarizing circle (Q) a perpendicular a>P is 
 
 (S)at^ - '• ~ 
 
 let Ml on a tangent drawn to the ciirve (S) at T. The radius oiT 
 produced will be perpendicular to the corresponding tangent drawn 
 to the reciprocal polfur curve (T) ; and the perpendicular oiP on the 
 tangent to the curve (S) at T will pass through the point of contact 
 
 r of the corresponding tangent to the reciprocal polar (2) 205 
 
 227.] To find the conjugate polar of the normal to a tangent plane applied 
 
 to the primitive surrace (S) 206 
 
 d 
 
XXXviii TABLE OP CONTENTS. 
 
 CHAPTER XXV. 
 
 THE GEOMETRICAL THEORY OF RECIPROCAL POLARS APPLIED TO THE 
 DEVELOPMENT OF A NEW METHOD OF DERIVING THE PROPERTIES 
 OF SURFACES OP THE SECOND ORDER WITH THREE UNEQUAL AXES 
 FROM THOSE OF THE SPHERE. 
 
 231.] Fundamental theorem.— If a point talien in the plane of a ciide (S) 
 he made the centre of a polarizing circle ^Q), the reciprocal polar 
 (S) of the circle (S) will be a conic section having a focus at the 
 . centre of (O) 208 
 
 232.] The polar of the centre of the circle (S) is the directrix of the conic 
 
 section (2) 209 
 
 233.] If in the transyerse axis of a surface of revolution (S) we assume a 
 point V as tiie common vertex of two cones of revolution circum- 
 scribing the sphere (8) and the polarizing sphere (O) whose centre ■ 
 is at ID the focus of (S), if the base of the former cone pass through 
 the focus of (2), iJie base of the latter wiU pass through the centre 
 of (2) 209 
 
 234.] The properties of surfaces of the secraid order having three unequal 
 axes &rived from those of surfaces of revolution of the second 
 order 211 
 
 235.1 The properties of two surfaces of the second order mav not be so de- 
 rived . 212 
 
 236.] Relations established between a^ b, and e the semiaxes of the reciprocal 
 surface (2), A and C the semiaxes of the primitive surface (S), and 
 R and D, the former being the radius of the polarizing sphere (Q), 
 while D is the distance between the centres of (S) and (b) 212 
 
 237.] Conversely, the semiaxes of the primitive sui&ce (S) and D may be 
 expressed in terms of a, b, and c,the semiaxes of (2). Eccentricities 
 of the three principal sections of (2), expressed in terms of A, C, 
 and D 216 
 
 238.] Definition of the polar focut 215 
 
 239.] Definition of the conjugate umbilical directrix planes 216 
 
 240.] Portion 'and inclination of the conjugate umbilical directrix planes . . 216 
 
 241.] Angle between an umbilical diameter and an umbilical directrix plane 217 
 
 242.] In every central surface of the second order having three unequal axes, 
 
 there are /our foci. Position of these umbiliciu foci defined 218 
 
 243.] Definition of the Principal Parameter of a surface of the second order 
 
 ' having three unequal axes 219 
 
 244.] The umbilical focus is the pole of the corresponding umbilical direc- 
 trix plane with respect to the redprocal siir&ce (2) 220 
 
 245.] In the axis of the surface (S) there are seven remarkable points. 
 
 Positions of their correlative seven polar planes 220 
 
 246.] The principal semiparameter of the surface (2) expressed in terms of 
 A, C, D and R, found to be independent of A and D. Remarkable 
 results which foUow from the independence of L the principal semi- 
 parameter of (2), a surface having three unequid axes on A, the 
 radius of the principal circular section of the primitive surfiuM (S) 
 and D the distance between the centre of (S) and the polar focus 
 
 of (S) 221 
 
 Principal parameter of the elliptic paraboloid 222 
 
 247.] Position of the polar plane of the extremity of the major axis of (S) 222 
 
TABLE OF CONTENTS. XXXIX 
 
 248.] A Table of the valnes of the magnitudea and poutiona of the several 
 planes, lines, and points connected with a central surface (S) of the 
 second order haying three unequal axes 223 
 
 249.] On the hyperboloid of two sheets and its asymptotic cone 22o 
 
 250.] The polar plane of a the centre of (O) with resjiect to (S) is the polar 
 
 plane of C, the centre of (S) with respect to (Q) 226 
 
 251,] The reciprocal polar of a continuous hyperboloid is also a continuous 
 
 hyperboloid 226 
 
 252.] On the oblate spheroid 227 
 
 CHAPTER XXVI. 
 
 ON THE APPLICATION OF THE THEORY DEVELOPED IN THE PRECEDING 
 CHAPTER TO THE DISCUSSION OF SOME THEOREMS AND PROBLEMS. 
 
 253.] Lemma L — To express the base A B of a triangle A B C in terms of 
 the perpendiculars on the polars of the points A, B, and the distance 
 
 of the pole of A B from C. Poncelet's theorem 228 
 
 Lemma II. — ^From a given point R let a perpendicular R P be let fall 
 on the given straight line C V, and let the point O be assumed as 
 the cenbe of the polarizing circle whose radius is R ; let S be the 
 pole of V, and B T the polar of the point R. The peipendiculars 
 from the points R and 8 on the straight lines C V and B T will have 
 the same Talio to each other as the hnes OR and OS 228 
 
 254.]*If from any point on an umbilical surface of the second order perpen- 
 diculars be let &11 on the two conjugate umbilical directrix planes, 
 the rectangle under these perpencuculai's will have to the square of 
 the distance of this point from the polar focus a constant ratio .... 229 
 
 (a), (i8), (y), (8), («). Particular cases of this theorem 231 
 
 Algebraical proof of this theorem (note) 231 
 
 2.55.] When the surface becomes an oblate spheroid 232 
 
 256.] The sum of the products, taken two by two, of the perpendiculars let 
 Ml from the four umbilical foci on a tangent plane to (S) is equal 
 
 to2J»Mn»i»+^cos»i' 233 
 
 257.] Through the polar focus of (S), a surface of the second order, a plane 
 and a straight line being drawn at right angles to each other, the 
 line meeting the sur&ce in the point r, and the umbilical directrix 
 plane (A) in the straight line (0), the plane which passes through 
 the pomt r and the straight line (d) will envelop a surface of revo- 
 lution (2,) whose focus will coincide with the polar focus of (2), 
 and whose directrix plane will pass through the polar directrix of 
 the polu focus a in tne principal section in the plane of X Y .... 234 
 
 * This theorem, founded on Lemma IL, was published in the Philosophical 
 Magazine for the year 1840, p. 432. 
 
 " n est une preprints principale des coniques, qui se retrouve dans les cdnes, 
 et dont nous n'avons point encore fait mention relativement aux surfaces du second 
 degr£ ; c'est que : ' la somme ou la difference des r^ons vecteurs men^s d'un point 
 d'une conique aux deux foyers est constante.' Nous avons fait, pendant long- 
 temps, des tentatives pour trouver quelque chose d'amdogue dans les surfaces; 
 mais sans obtenir aucun succes." — CShasles, Aper<;u Historique, p. 391. 
 
 d2 
 
Xl TABLE OF CONTENTS. 
 
 258.] Algebraical proof of this theorem 236 
 
 259.] If in any Burface (S) haying three unequal axes a straight line be drawn 
 meeting the surface in two points r and t, and the umbilical direc- 
 trix planes (A) and (A,) in two points also, t and d„ the angles which 
 these pairs of points subtend respectiTely at the polar focus a will be 
 equal to each o^er, or the angle rufi will be equal to the angle r^caS, 237 
 A straight line touches the surface (2) and meets the umbilical direc- 
 trix planes (A) and (A,) in two pomts S and i, ; the se^ents of this 
 line between the point of contact r and the umbilical directrix planes 
 will subtend equal angles at the polar focus <a 238 
 
 260.] K through any point d assumed on one of tlie directrix planes (A,^ a 
 secantbe drawn parallel to the other directrix plane (A), and meeting 
 the surface in the points r and r,, the rectangle under uiese segments 
 will be equal to the square of the distance of 8 from the polar focus a> 238 
 
 261.] Let the principal parameter C C, of a surface of the second order (Z) . 
 having three unequal axes be the base of a triangle whose vertex is 
 the point Gt anywhere on the surface. If the sides of this triangle 
 be produced to meet the umbilical directrix plane (A) in two points 
 i and i,, the points 8 and 8, will subtend a right angle at the polar 
 
 focus a 240 
 
 If the shorter axis of an oblate spheroid be iaken as the fixed base of 
 a triangle C C, G inscribed in the spheroid, whose sides are produced 
 to meet one of the directrix planes (A) in two points Q and Q„ these 
 pointe, Q and Q,„ wiU subtend a r%ht angle at the centre of the 
 
 oblate spheroid (2) 240 
 
 If the parameter of a surface of revolution (S) be the base of a triangle 
 inscribed in it, and the sides be produced to meet the directrix 
 plane (D) in d and <2,, the points d, d, will subtend a right angle at 
 
 the focus F of (S) 240 
 
 If a fixed chord be taken in a surfEuse (2), and this fixed chord e c, be 
 made the base of a triangle whose vertex G is anywhere on this 
 surface, the sides of this triangle Gc, Gc, being produced to meet 
 the umbilical directrix plane (A) in two points 8 and 8„ the points 
 8 and 8, wiU subtend a constant angle at the polar focus m 241 
 
 263.] If through a straight line (d) in the umbilical directrix plane (A) of a 
 surface (2) of the second order, two tangent planes (6) and (6,) be 
 drawn meeting a third tangent plane (e„) in the straight lines (3,) 
 and (d„), the planes through these two lines and the umbilical focus 
 V will cut the umbilical directrix plane (A) in two straight lines, 
 through which and the polar focus a if two planes be drawn, they 
 will be at right angles to each other 242 
 
 264.] If two tangent planes (6) and (e,) are drawn to a surface of the second 
 order (2), having three unequal axes, meeting in the straight line 
 (d), and if r and r, be the pomts of contact of these tangent planes, 
 let the chord r r, meet the umlnlical directrix plane (^ in v; and 
 let the plane drawn through v, the umbilical focus v, and (&) the 
 intersection of the tangent planes (e) and (ei,) meet the umbuical 
 directrix plane (A) in the straight line (8) ; the line <uv will be at 
 right angles to the plane drawn through a> the polar focus and (8) 242 
 
 205.] Let two tangent planes (e) and (e,) be drawn to an oblate spheroid, 
 and their chord of contact be produced to meet the directrtt plane 
 in v; and if a plane be drawn through the umbilical focus v and the 
 line (S), in which these two tangent planes intersect, and meeting 
 the directrix plane in the line (8), the diametral plane C(8) will be 
 at right angles to the diameter Ck 243 
 
TABL£ OF COMTDNTS. xli 
 
 266.] If a tingent plane (e) be diawn to (2), touching it in the point t, 
 and cutting the umbilical diiectriz pume in the line (3), and if 
 through the umbiliod focus v and we point r a stiaight line be 
 drawn meeting the directrix (A) in the point 8, the line ad drawn 
 from the polar focus <» to d will be at right angles to the plane drawn 
 
 from a through the straight line (ft) 243 
 
 If a tangent plane be drawn to an oblate spheroid cutting one of the 
 direcliix planes (A) in a straight line (d), the diametral plane drawn 
 through (5) will be at right angles to the diameter Or drawn throvigh 
 the point of contact r 243 
 
 267.] Two tangent planes, (e) and (e,), are drawn to a surface (2) haying 
 three unequal axes, cutting the umbiUcal planes in two straight 
 lines (d) and (d,). A stra^ht line drawn through the points of 
 contact meets the directrix plane (A) in X. The line drawn from 
 the polar focus oi to X is equallT mclined to the planes uCd) and 
 
 <»(8,r 244 
 
 Two tangent planes are drawn to an oblate spheroid, cutting the 
 directrix plane (A) in two straight lines (8) and (8,), while the 
 chord through the points of contact meets it in X, the diametral 
 planes 0(8) and 0(8,) are equally inclined to the diameter OX 244 
 
 268.] Two tangent planes, (e) and (S,), are drawn to a surface (S) meeting 
 the directrix plane (A) in two straight lines (8) and (8,) ; and if 
 through the intersection of these tangent planes (3) and tiie umbi- 
 lical focus V a plane (U) be drawn cutting the directrix plane (A) 
 in a straight line (x)t ^^ planes a>(8) and a(8,) will be equally m- 
 dined to the plane a>(x) 244 
 
 269.] If two tangent planes, (e) and (6,), to a surface of the second order 
 be cut by another plane (n) passing through the points of contact 
 TT„ and cutting the tangent planes in two straight lines rn- and 
 T,ir, if the three sides of the triangle rr,, rn-, and r,ir be produced 
 to meet the directrix plane in the points X, 8, and 8„ the angles 
 XoS and Xa>8, will be equal 245 
 
 270.] Two tangent planes, (6) and (e,), are drawn to a surface having three 
 unequal axes. A third plane (n) is drawn through the chord of 
 contect cutting the tangent planes in the lines (v) and (w/). These 
 lines and the line' (Si), in which the tangent planes intersect, are 
 produced to meet the directrix plane (A) in the three points n-, n-,, 
 and 8. The angle «raidss7r,<ii>8 246 
 
 271.] Two tangent planes, and a secant plane through the chord of contact, 
 are drawn to an oblate spheroid ; the diameter drawn through the 
 point in which the common intersection of the two tangent planes 
 meets the directrix plane (A) is equally inclined to the diameters 
 which pass through the points in which the common intersections 
 of the secant plane with uie tangent planes meet the directrix plane 246 
 If through a surface (2) with three unequal axes a chord be drawn 
 meeting this surface in the points r and t, and the umbilical direc- 
 trix plane (A), and if through a the polar focus planes be drawn at 
 right angles to or and wt, cutting the second directrix plane (A,) in 
 («r) and (w,), and planes be drawn through r(i!r) and T,(vr,) inter- 
 secting in the strai^t line (X), the plane «(X) will be at right angles 
 to the straight line <a» 246 
 
 272.] Let there be a triangle whose vertex is at the umbilical focus, and 
 whose base is a chord of the surface (2). If the three sides be pro- 
 duced to meet the umbiliceJ directrix plane (A) in three points v, 8, 
 and i„ the line drawn from the polar focus u to i> wiU be equally 
 inclined to the focal lines o8 and uS, 247 
 
Xlii TABLE OF CONTENTS. 
 
 273.] Any pwi of conjugate tangents to the surface (2) being pxoduced to 
 meet the umbilical directrix plane (A) in two points a and 0, the 
 lines drawn from the polar focus a to these pomts will be at right 
 angles 248 
 
 274.1 Conjugate tangents are parallel to a pair of conjugate diameters of the 
 central {done of the surface (S) parallel to the tangent plane con- 
 taining the conjugate tangents 248 
 
 276.] If a cone, whose vertex k is on the umbilical directrix plane ^A), be 
 circumscribed to a sur&ce (2), the plane of contact (Gq of this cone 
 with (2) will pass through v-, the umbilical focus, and will cut (A) 
 in a straight line (y). The line drawn from the polar focus » to 
 the vertex k of the cone will be at light angles to the plane drawn 
 from a through the line (y) in which (G), the plane of contact of 
 the cone circumscribing (2), meets the umbilical directrix plane (A) 261 
 
 277.] If a tangent plane (e) be drawn to (2) touching it in the point r, and 
 cutting the conjugate umbilical directrix planes (A) and (A,^ in the 
 lines (d) and (fi,), the planes drawn througn these strtught lines and 
 the polar focus a will be eq^nally inclined to the line or, drawn from 
 
 the polar focus a to the pomt of contact r 252 
 
 When the surface becomes an oUate spheroid it will hence follow 
 that, If a tangent plane be drawn to an oblate spheroid cutting the 
 pivallel directrix planes (A) and (A,) in the straight lines (JS) and 
 (8,), the diameter through the point of contact wul be equally in- 
 clmed to the diametral planes 0(8) and 0(8,) 262 
 
 278.] K a secant plane be drawn cutting the sur&ce (2) in a plane section 
 (K), and the conjugate umbilical directrix planes (A) and (A,) in 
 the stnught lines (8) and (8,), the planes drawn through tiie polar 
 focus a> and these straight lines (8) and (8,) will be parallel to the 
 circular sections of the cone whose' vertex is at a> and whose base 
 
 is (K) 252 
 
 The circular sections of a cone whose vertex is at the centre of an 
 oblate spheroid, and whose base is any plane section of this snrfiice, 
 ore parallel to the two diametral planes which pass through the 
 straight lines in which this base intersects the minor direcctrix 
 
 planes (A) and (A,) of the oblate spheroid (2) 263 
 
 The cone whose vertex is at the polar focus a> of a sur&ce (2) having 
 three unequal axes, and whose base is a plane section of uus suiibce 
 passing through the polar directrix, is a right cone whose circular 
 section is parallel to the plane of XY •. 253 
 
 279.] Algebraical proof of this theorem 263 
 
 280.] If through any straight line on the umbilical directrix plane (A) two 
 tan^nt planes be dravm to (2), and if through the thr«e perpen- 
 diculars let &U from a on these tiiree planes a secant plane (Q) be 
 drawn cutting the tangent planes in lines which produced meet the 
 plane drawn through a> parallel to the umbilical directrix plane (A) 
 m the points t and t,, and if an-=A, ayT,=h, and utK=r in the tri- 
 angle TUT,, the base tot, will have to the line uk a constant ratio, or 
 
 When the sur&ce (2) becomes a sm&ce of revolution (8) round the 
 transverse axis, c=.h and »=«, hence 
 
 ^=2e. (g) 265 
 
 Hence also this other theorem in the conic sections. 
 
TABLE OF CONTENTS. xliii 
 
 If from Any point in the directrix a pair of tangents l)e dravs to the 
 curre, they will cut off equal segments from the ordinate passing 
 
 throufih the focus , 265 
 
 281.] The reciprocal polar of the theorem, that in a sui&ce of revolution 
 the sum of the focal vectors is constant, is as follows. 
 
 To a surface (S), having three unequal axes, let a tangent plane (e) 
 tte drawn cutting the umbilical planes (A) and (A,) m tiie straight 
 .lines (A) and (8,),and the line joining the polar focus and the polar 
 direcmx into the segments A and h,. Let the planes that are drawn 
 from CD through the straight lines (A) and (8,) make the angles <(>, <j), 
 with the vertiual ordinate pasdng through the polar focus o, the 
 resulting expresdon becomes 
 
 ^(sec^+sec<^,)=— 267 
 
 CHAPTER XXVII. 
 282.] Some properties of surfaces of revolution of the second order, esta- 
 blished Dy M. Chasles, extended to surfaces having three imequal 
 axes 257 
 
 CHAPTER XXVIII. 
 
 ON THE BECIPROCAL POLARS OF CONFOCAL SURFACES. 
 
 284.] If there be a series of txmfocal primitive sur&ces (S), (S,), (S„),&e., 
 their reciprocal surfaces (2), (2,), (2„), &c. will be caacydK. They 
 will have a common polax locus ; they will have the same umbilical 
 directrix planes ; and tiie graphical properties of one set of surfaces 
 may easily be transfoimed into their reciprocals on the polarized 
 
 surfaces 266 
 
 Examples 265 
 
 CHAPTER XXIX. 
 
 ON METRICAL METHODS AS APPLIED TO THE THEORY OF 
 RECIPROCAL POLARS. 
 
 286.] If a fixed point O be taken, and a number of fixed straight lines BT, 
 B,T^ &c. be drawn in the same plane, and another current point S 
 be assumed from which and from the point penendiculars are let 
 fall in pairs on these fixed lines, if the reciprocal of the vector OS, 
 multipUed bf the sum of the ratios of each pair of perpendiculars 
 on the fixed lines, be constant, the point S will describe a conic 
 section having its focus at O 266 
 
 287.] Geometrical proof of the theorem thatj if a plane be drawn through a 
 focus of a surface of revolution, this pomt will also be a focus of 
 the section 268 
 
 289.] The common directrices and foci of the equHaterdl hyperbola possess 
 two distinct classes of properties — ^those which belong to them as 
 being the common or ortunaiy directrices and foci, as t£o that other 
 new and equally extensive class to which they are in like manner 
 related as being the minor directrices and foci of a central conic 
 section 270 
 
 290.] The properties of the minor directrices and their corresponding fixd 
 derived by the help of the method of reciprocal jpolars, from the 
 known properties of tiie common focus and 'directrix^ 270 
 
 291.] If a tangent be drawn to a conic section meeting the minor directrices 
 in two points M and M„ the sum of the distances of these points from 
 
xliv TABLE OP CONTENTS. 
 
 the minoi axis is to. the sum of their distances from the centre in a 
 constant Tstio e, the eccentricily of the section. In the hyperbola 
 the differences ore to be taken 272 
 
 292.] The product of the focal perpendiculars let fall from the minor foci F 
 and F, on a tangent to ue curve is to the square of the perpendicular 
 from tiie centre on the same tangent as the square of the semidia- 
 meter a, passing through the pomt of contact Q is to the square of 
 the semi- major axis a 272 
 
 294.] The ratio of the focal perpendiculars on the tangent is the same as 
 that of the distances from the centre of the points in 'which the 
 tangent cuts the minor directrices 273 
 
 295.] If a tangent be drawn to a central conic section meeting the curve in 
 the point Q, and the minor directrices in the points M and M„ the 
 rectangle under the distances of these points from the centre \nll be 
 to the rectangle under the segments of this tangent as the square 
 of the perpendicular from the centre upon it is to the square of the 
 semi- minor axis 273 
 
 296.] The ratios of the focal perpendiculars on a tangent, of the distances 
 to the centre of the pomts M and M, in which this tangent meets 
 the minor directrices, of the. segments into which this tangent is 
 divided between the point of contact Q and the minor directrices, 
 as also of the perpendiculars from this point Q upon them, are the 
 same for all 273 
 
 297.] If from any point M in the minor directrix of a conic section we draw 
 two tanraits to it meeting the transverse axis in the jioints X and 
 X„ the distance between tiiese points will be to the distance of the 
 point M from the centre as the distance between the foci is to the 
 semiparameter 274 
 
 298.] The distances from the centre of the points in which a tangent to the 
 curve meets the minor directrices may be expressed m rational 
 functions of the projective coordinates of the pomt of contact .... 274 
 
 300.] If a chord be drawn through a fixed point d in tiie plane of a conic 
 section meeting the curve in the pomts r and r„ the product of the 
 tangents of half the angles AFr and SFr, wiU bia constant, F being 
 a focus 276 
 
 301.] If a straight line FQ be drawn in the plane of a conic section, and 
 from any point P in it two tangents be drawn to the curve meeting 
 the minor directrix in two points T and T^ while the given straight 
 line meets it in Q, the product of the tangents of half the angles TOQ 
 and T,OQ will be constant 276 
 
 303.] If two_ diameters are drawn at right angles in a conic section, one 
 meeting the curve in Q, the other meeting the minor directrix in 
 G, the line QQ- envelops the drde whose diameter is the minor axis 278 
 
 306.] If a secant be drawn to a central conic section cutting the curve in the 
 prants P and P„ and the minor directrices in the points Q and Q„ 
 and if through the pcnnte P and P, tangents be drawn meeting the 
 miiior directrices in tiie points MM, and NN„ the sum of the angles 
 wUch the straight lines MM, and NN, subtend at the centre wiU be 
 equal to twice the angle which QQ, subtends at the same centre. . 280 
 
 307.] If through any two pdnts P and P, on a central conic section a secant 
 be drawn, and two tangents through the points P and P, meeting 
 the naiaor directrices in the four pointo M, M„ N, and N„ and per- 
 . . pendicnlars be let fidl from these four points on the secant, namelv 
 feP,M,P,NP,„N,P,.„ the ratios »~ ' ^ 
 ^ M^ NP„ N.P„, 
 MO'M,0' NO '"NTT' 
 will all be equal , 282 
 
 308.] Algebraical proof of this theorem ,',.', 282 
 
TAIILE OP CONTENTS. xlv 
 
 309.] If two fixed jtointB A and B be aasnmed on a conic section, and a thiid 
 point C variable in podtion, and if the cbords CA, CB be pioduced 
 to meet the minoi ouectriceB (^) and (A,) in the points M, M, and 
 M, N,, the line MM, will subtend a constant angle at the centre O ; 
 so also will the line NN, on the other directrix (AJ ; and the sum 
 of these angles will be constant and equal to the angle AOB .... 285 
 
 310.] If a triangle be inscribed in a conic section whose base is the minor 
 axis, and whose vertex is variable along the curve, the sides being 
 produced will meet the minor directrix in two points MM,, which 
 will subtend a right angle at the centre 285 
 
 311.] Let the sides of a quadrilateral inscribed in a conic section meet one 
 of the minor directrices in the points A., B„ C,,, D, ; the sum of 
 the angles which the segments AgBg and GJ)„ subtend at the centre 
 wiU be equal to two rignt angles 286 
 
 313.] Simple method of finding the curvature of a conic section 287 
 
 314.] On groups of conic leettom having the same minor directrices. 
 
 Let a series of concentric conic sections, all having the same minor 
 directrices, be cut by a transversal, the portions of this line inter- 
 cepted by any pair of these curves will subtend equal angles at the 
 centre ; and if, through every pair of points in which this transversal 
 intersects the sections, tangents are drawn intercepted botii ways by 
 the directrices, the sum of the angles which any pair of these tan- 
 gents subtend at the centre is constant, being equal to twice the 
 angle which the common transversal intercepted both ways by the 
 directrices subtends at the centre 288 
 
 315.] Two diameters at right angles revolve round the centre of two conic 
 sections having the same minor directrices, each diameter meeting 
 one of the curves ; the line joining these points will envelop a circle 289 
 
 316.] A series of concentric conic sections, having the same minor direc- 
 trices, is cut by a common diameter ; the tangents drawn through 
 the points where this diameter intersects the curves envelops a con- 
 centric conic section 290 
 
 3I7.] On certain properties of the equilateral hyperbola 291 
 
 CHAPTER XXX. 
 
 ON THE LOGOCYCLIC CURVE, THE TRIQONOHETRY OF THE PARABOLA^ 
 AND THE GEOMETRICAL ORIGIN OT LOGARITHMS. 
 
 319.] On the equations of the logocyclic curve 295 
 
 320.] to 340.] On the properties, quadrature, and rectification of the logo- 
 cyclic curve 296 
 
 341.] On certain properties of inverse curves .7^ 309 
 
 342.] On the relation between the central forces by which any curve and its 
 
 inverse may be described 311 
 
 CHAPTER XXXI. 
 
 ON THE TRIGONOMETRY OF THE PARABOLA. 
 
 343.] Definition of eotgugate amplitudes 313 
 
 344.] Table of analogies between the formulgs of circular and parabolic tri- 
 gonometry 315 
 
 345.] Expression for the length of an arc of a parabola 318 
 
zItI table of CONTSNT8. 
 
 346.] Relations between tiie conjugste arcs of a parabola 319 
 
 348.] Relation of the oidinates of tiuee conjogate parabolic arcs 322 
 
 349.] Oeometrical npiesentation of the functions 
 
 8ec(<^-i-x)andtan(0J-x) 323 
 
 350.] Relations between multiple parabolic aica 323 
 
 353.] Solution of cubic equation hj parabolic trigonometry 328 
 
 CHAPTER XXXII. 
 
 ON THE GEOHETKICAI. ORIGIN OF LOGAEITHHS. 
 
 354.] Value of e the Napierian base in terms of functions of the angle which 
 a focal perpendicular on a tangent makes with the axis, 
 
 or sec fl+tan 0=e 328 
 
 356.] To represent numbers and their logarithms by the logocydic curve and 
 
 its conjugate parabola 331 
 
 362.] To represent the decimal or any other system of logarithms by a cor- 
 responding parabola , 337 
 
 366.] The quadrature of the hyperbola depends on the rectification of the 
 
 parabola 343 
 
 367.] On the relations between focal perpendiculars on the n sides of a poly- 
 gon circumscribing a parabola : 345 
 
 370.] On some analogies between circular and parabolic trigonometry .... 347 
 371.] On the roots of s^-2(k»+1=0 when a >1 348 
 
 372.] On parabolic trigonometry as applied to the investigation of the cate- 
 nary and the tatctriz 349 
 
 373.] On some properties of the catenary 360 
 
 376.] On the tractrix 352 
 
 CHAPTER XXXin. 
 
 ON SOME PROFEKTIES OF CONFOCAL SURFACES. 
 
 379.] Three confocal sur&ces of the second order intersect in a common 
 point Q, the vertex of a cone which envelops a fourth confocal sur- 
 iBce ; to determine the projective equation of this cone referred to 
 the normals of the tiuee Bor&cee, at the common point Q, as axes 
 of coordinates 35g 
 
 382.] Two cones having their common vertex on a sui&ce of the second 
 order, an ellipsoid sumKwe, (a/>/!,) envelop two confocal surfiwee. 
 The diametral plane of the surface conjugate to the diameter pass- 
 ing through the common vertex of the two cones will cut off from 
 their common side a constant length, independent of the position of 
 the common vertex of the two cones on the surface (afi/i ) 363 
 
 383.] A cone whose vertex is on a saxbee of the second order envelops a 
 confocal sur&ce. To determine the length of the axis of the cone 
 between the vertex and the plane of contact 364 
 
 385.] Along a line of curvature tangent planes are drawn to a sur&ce of the 
 second order. The perpen^culars from the centre on these planes 
 generate a cone of the second order, whose focal lines coindde with 
 the optic axes of the sur&ce or with the perpendiculars to its rar- 
 cular sections 3Qg 
 
BY THE SAME AUTHOR. 
 
 Examination the Province of the State. Being 
 
 an attempt to show the proper function of the State in 
 Education. 8vo. 
 
 " . . . . The first suggestion of this system seems to have been in an able 
 pamphlet, published 07 the Rev. Dr. Booth, addressed to the Marquis of 
 Lansdowne " — Thoaghtt on National Education, Inf Lord Lyttetton, p. 10. 
 
 " Dr. Booth, in his pamphlet, ' Examination the Province of the State,' pub- 
 lished some years ago, laid down the general outlines of the system of promoting 
 education by means of examinations, which now meets with such genenu 
 acceptance." — DaUy News. 
 
 How to Learn and What to Learn. Two Lectures 
 
 advocating the System of Examinations established by the 
 Society of Arts, and delivered, the former at Lewes on the 
 24th of September, and the latter at Hitchin, on the 16th of 
 October, 1856. Published by the Society of Arts. 
 
 " Among the many pamphlets, speeches, and addresses, with which the press 
 has this year teemed, on the all-engrossing subject of education, these lectures 
 by Dr. Booth are far the best in our estimation. They are more liberal and 
 more comprehensive ; they are marked by sounder sense ; and, what will weigh 
 still more with most men, they are evidently the production of a man who has 
 thought much and deeply on the subject of which he speaks, and who brings to 
 the aid of a mind at once vigorous and capacious the benefit of an extensive 
 experience. Dr. Booth is the Treasurer of the Society of Arts, which has done 
 more than any other body of men to promote the general improvement and 
 extend education among the yeoman classes of this country, or rather among 
 those who hold a position in society akin to the ancient yeomanry, whether 
 found in town or country. We have no better name by which we can distinguish 
 them : they are not the very poor ; they are not strictly the mid^e classes ; but 
 they range indefinitely between these two pales of society. • « • 
 
 " In the success of so good a cause we feel the deepest sympathy. We believe 
 that these two lectures cannot fiul in exciting that sympathy where it is not now 
 felt ; and in that persuasion we recommend them to those who are deeply inter- 
 ested in the cause of education, and who believe, as we do, that it is the great 
 and absorbing question of the day." — Morning HerM. 
 
 " Worthy of the high reputation of the author." — DaHy News. 
 
 « We should be glad to see these lectures of Dr. Booth very extensively circu- 
 lated among the clergy and laity. We agree with much that he says; but what 
 we especisjSy desire to commend as an example is, the very lucid and spirited 
 style in which his lectures are written." — English Churchman. 
 
 "We recommend to general notice two lectures by Dr. James Booth, entitled 
 How to Learn and What to Learn, in which the subject here slightly touched on 
 is folly and ably treated." — Chambers^ Journal. 
 
 In one Volume, 8vo, price 7s. 6d. 
 
 The Theory of Elliptic Integrals and the Properties 
 
 of Surfaces of the Second Order applied to the investigation 
 of the Motion of a Body round a Fixed Point. 
 
 LONGMANS, GEEEN, READER, AND DYER. 
 
In One Yolume, crown 8to, price 5s. cloth. 
 
 The Lord's Supper, A Feast after Sacrifice. 
 
 With Inquiries into the Doctrine of Transubstantiation^ and 
 the Principles of Development as applied to the Interpretation 
 of the Bible. By James Booth, LL.D., F.R.S., F.R.A.S., &c., 
 Vicar of Stone, Buckinghamshire. 
 
 " This is a csrefiil and scholarly attempt at a vuz mtAia between the merely comme- 
 moratiTe theory of the Eucharist and the doctrines of Transubstantiation and Con- 
 substantiation. Dr. Booth evidently regards the former as bald and defective, and both 
 of the latter as eztntTsgant and superstitioiis. The nature of the Holy Bite preferred 
 by the author is the BSpniimn Saeryieiale of Made and Cudworth, answering to the meal 
 of the Jews after, and upon parts of, their sacrifices. We commend the treatise as a 
 valuable contribution to this discussion, which never vras more rife amongst polemical 
 divines than at present, and which may grow in heat and range within a few years." — 
 Englisi Churchman, June 9, 1870. 
 
 " This volume will well repay perusaL It is the work of a clear thinker and well- 
 informed man. Dr. Booth is weU known to mathematicians as one who is at home in 
 the most abstruse problems. When we state that, our readers will know they are in the 
 bands of a man with powers of continuous thought, who is able to trace his way throngb 
 ■11 intricacies and obsoureness, if a route be possible to human powers. But the orm- 
 nary reader (we mean non-nuUbematical reader) will observe nothing of the mathema- 
 tician in our author's manner of handling his present subject. His style and method 
 are distinguished solely by their cUamess, simplicity, and orderliness. And the book 
 consists mainly of quotations from Mb divines of the past. Quotations from such acute 
 and learned tmnkers as Cudworth, and Waterlond, and Mede, with other divines of lesser 
 note, form the staple of a large portion of the volume. This remark, however, does not 
 apply to the latter half of tlM volume, which consists of two chapters, the one entitled 
 ' On the Principle of Development as applied to the Interpretation of Qm Bible,' and the 
 other ' On Transubstantiation.' Token as a whole, the volume brings together much 
 that is valuable and suggestive, and in the main thoroughly sound, on the sacraments, 
 and specially on the Lord's Supper ; and the doctrine of Transubstantiation is handled 
 OS might have been expected by so able and profound a mathematician. The history of 
 the rise and progress and final result of the doctrine is given briefly, yet truly. It is 
 traced to a false philosophy long since buried out of sight and forgotten. It would be 
 profitable work for some of the author's co-religionists to read, mark, and inwardly 
 digest the chapter on Transubstantiation, that not cunningly but chmvriltf devised fikUe. 
 —WeeJcly Seviea, June 18, 1870. 
 
 " This is a learned and well-written attempt to establish, in a logical manner, the true 
 nature of the Lard's Sapper, reliance being mainly placed on the brief narratives of the 
 Gtispels and of St. Paul, further elucidated oy a reference to the ancient Jewish language, 
 histoiy, and customs. Dr. Booth's position embraces the view once (he says) ^rnost 
 univerrally held in the Church of England, ' That the Lord's Supper is a Feast upon a 
 Sacrifice ;" and to set it forth he has combined and expounded the views of snob men as 
 Joseph Mede, CudwortJi, Potter, Worbuiton, Waterlond, Hampden, and others. This 
 gives to the treatise a somewhat fragmentary air ; but, taken as a whole, it is clearly, 
 intelligently, and devoutly written, and will doubtless be sooeptoble to some disciples of 
 those mmous men. On a subject of such subtlety — ^where the widest diversity of opinion 
 still fiercely prevails — it cannot hope to please the many, though it is wdl woitiiy of 
 careful examination. Dr. Bootli has studied his sulgect with care, and brought to his 
 difficult task the fruits of extensive reading." — Standard, Junt 23, 1870. 
 
 " Dr. Booth's modest volume is avowemy not so much an original production as on 
 attempt to recall by selected citations what he thinks tlie too much nwlected learning 
 of the fathers of the Church of England. The volume is divided into nur chapters, in 
 the first of which he adduces authorities to prove that the Lord's Supper is not a mere 
 service of commemoration ; in the second he adduces authorities to prove that it ought 
 to be regarded as a fesst of thanksgiving, implying a preceding sacrifice ; in the tSid 
 he treats of the principle of development as appUed to the interpretation of the Bible ; 
 and in the fourta he discusses and dismisses the doctrine of transubstantiation, ind- 
 dentolly^ treating at some length of the influence of tlie philosophy of Aristotle. GPhe 
 most original thoughts and iUustrationB occur in the third chapter, and the reasoning 
 seems to us most conclusive in the fourth. The quobitions have evidently been selected 
 with thought and core, and evince much reseoroh ; and the author's own writing is finished 
 and good. The volume is the careful ^production of a thoughtful scholar, though it 
 conveys the impression to us that the mind of the writer has been somewhat over^d bv 
 scholastic learning, so as to be in on artificial state, and partiallr disabled from receiving 
 in their freshness and simplicity the truths which we conceive to be really revealed in 
 the scriptures to the human heart." — Theoloffioal Review, October 1870, p. 591. 
 
 LONGlllNS, GBEEIT, I^ADEB, AND DYEK. 
 
TANGENTIAL COORDINATES. 
 
 CHAPTER I. 
 
 ON THE TANGENTIAL EQUATIONS OF A POINT AND A STBAIGHT LINE IN 
 
 A PLANE. 
 
 The projective equation of a straight line in a plane^ referred 
 
 to two intersecting lines in the same plane, is -+^=1. In this 
 
 equation a and h denote the intercepts of the axes cut off by the 
 straight line, while x and y are the variable current coordinates of 
 any point moving along this line. 
 
 Now if we fix this point, thus making w and y constant, and sup- 
 pose a and i to vary instead, we shall have the means of defining 
 
 the position of this point by the help of a second equation, — (-^= 1, 
 
 a, "/ 
 where a is changed into a^ and h into 6j. 
 
 As a and h are henceforward to be assumed as variables, we must 
 adopt some appropriate notation to designate their variable cha- 
 racter. It will improve the symmetry of the notation if we put 
 
 0=^, h=~; and thus the preceding equation becomes 
 
 a?| + yw=l; (1) 
 
 and this is the tangential equation of a point. 
 
 As this expression is of constant occurrence, being the link which 
 unites the tangential and projective systems of coordinates, it may 
 with propriety be called the dual equation. 
 
 The position of a straight line will evidently be determined, if we 
 make f= constant, ti= constant. 
 
 It must be borne in mind that, as f , v, f are the reciprocals of 
 straight lines, such quantities as a^, yv, or P? are abstract numbers ; 
 a, y, and P being straight lines. 
 
 As X and y are the projective coordinates of a current point in a 
 plane, so f and v are the tangential coordinates of a line which may 
 be termed the limiting tangent. 
 
THE TANGENTIAL EQUATIONS OF A 
 
 Fig. 1. 
 
 ON THE TRANSrOKMATION OF COORDINATES. 
 
 2.] Assuming the ordinary rectangulax axes of coordinates, 
 we shall imagine them first turned 
 round through an angle 6, retaining 
 the same origin, and afterwards 
 suppose them displaced in parallel 
 directions to another origin. 
 
 Let ^ and - denote the intercepts 
 
 of two rectangular axes, OX, OY, 
 
 hy a fixed straight line; =■ and — 
 
 the intercepts of the axes OX,, OY, 
 
 made by the same straight line, and 
 
 let P, the perpendicular firom the origin on the straight line, make 
 
 the angles \ and X, with the axes OX and OX,, and let 6 be the 
 
 angle XOX,. 
 
 Hence X=\,+^, 
 and cos\=cos X, cos 0— sinX, sin 6 ; 
 
 but Pf = cos X, Pf , = cos X„ Pu = sin X, Pw, = sin X, ; substituting and 
 dividing by P, we find f =cos 6 .^,—sm.a .Vi; and a like expression 
 may be found for v. Hence, when the axes of coordinates are 
 turned round through the angle 6, we may pass from the old system 
 to the new, firom f , v to f „ w„ by the equations 
 
 f = cos d . ^,— sin . Vp 
 v=sia0 .^i+coa0 
 
 .f,.J" 
 
 (2) 
 
 ON THE TRANSLATION OF COORDINATES. 
 
 3.] Let the axes 0,X, and 0,Y, be drawn through the point O, 
 parallel to OX and OY. Let 0,X,=i, 0,Y,=-, OX=i, 
 
 OY=-. Let jB and q be the projective coordinates of the point O 
 
 on the new axes O.Y, and 0,X,; join O 
 and X„ O and Y„ also O and O,. 
 
 Now the whole triangle 0,X,Y, is the 
 sum of the three component tnangles 
 0,0X„ 0,OY„ and OY,X,; or, sub- 
 stituting their equivalent expressions. 
 
 Fig. 2. 
 
 we obtain f 
 
 
POINT AND A STRAIGHT LINE IN A PLANI 
 
 In like manner i;=r 
 
 Fig. 3. 
 
 l-p^-qv,' 
 
 Hence the expressions for the old coordinates^ in terms of the 
 new, translated in parallel directions, are 
 
 when the axes are first turned romid through the angle 6 and then 
 
 translated in parallel directions, 
 
 f._ cos0.^,—wa.0.v, _Bin5.fj + cos0,w, . . 
 
 ^~ \-pi,-qv, ' "" l-p^i-qv 
 
 4.] If we wish to change the inclination of the axes of coordinates 
 
 &om a right angle to an angle to, we 
 
 may easily effect this, the axis of X 
 
 remaining the same, by substituting 
 
 w,— cosw.f ■ 
 
 -^ — = 2- for V. 
 
 smcu 
 
 Let the new axis of Y, make the 
 
 angle m with the former axis of X. 
 
 Then half the area of the triangle 
 
 XOY=t:-: and half the area of the 
 
 same triangle is OZ=— , multiplied 
 
 by the sum of the perpendiculars let fall on it &om X and Y, that is, 
 1 l/sino costB\ /u,— cosa>.f\ ,_. 
 
 5.] The perpendicular from the origin on the straight line whose 
 tangential coordinates are f and v, is manifestly , « . a - 
 
 To determine the length of a perpendicular on the straight line 
 whose tangential coordinates are f and v,from the point whose pro- 
 jective coordinates are p and q. 
 
 Let O, be the point whose projective 
 coordinates are 0,k.^q, Ofisap; let 
 0,Q=P. 
 
 Then the area of the whole triangle 
 OCD = OpC + 0,OD + DO,C, or 
 
 ing by ^v, and reducing, 
 
 t_ ^-p^-qv 
 
 6.] To find an expression far the value of the angle between two 
 given straight lines whose tangential coordinates are given. 
 
 B 2 
 
 P=- 
 
 (6) 
 
4 THE TANGENTIAL EQUATIONS OF A 
 
 Let be the angle between the two given straight lines whose 
 tangential coordinates are f , v and ^„ v,. 
 
 Let these lines make the angles ^ and <^, with the axis of X ; then 
 t e 
 
 tand)=-, tanrf>,= -', and 0=<f>—^,; 
 
 , , -, tanrf)— tan<^, gt;,— g,f ,-x 
 
 hence tang= ., . ,^ ■ . — ^= 1^ . ; . ■ ■ ■ v) 
 
 1 + tan 9 tan <p, ??/ + 1"^/ 
 
 when the lines are parallel, ^v—^,v=0,'\ /g% 
 
 when the lines are at right Sing\es,^^i + vv,—O.S 
 
 7.] To find the tangential coordinates of a right line passing through 
 two fixed points whose pry ective coordinates (a^) and [a^^ are given. 
 
 Let the tangential equation of one point be a^+fiv= 1, and that 
 of the other point be a^f + ^,u=l. 
 
 Eliminating from these equations f and v successively, we get 
 
 ^-a^-afi'^-ufi-a^, ^ ^ 
 
 The line will pass through the origin when - = o-- 
 
 8.] Let us resume the tangential equation of the point {x, y), 
 namely a?f +yu = l. Multiply it hy P, the perpendicular from the 
 origin; bearing in mind that Pf=cosX, Pu=sin\, we obtain 
 
 cos \ . «+ siuX . y— P=0, 
 which is a common form for the equation of a straight line in pro- 
 jective coordinates. 
 
 Now if on this line we let fall a perpendicular from the point 
 (xg/^, the length of this perpendicular will be 
 cos X . a?, + sin \ . y^— P. 
 Thus the /orm of the equation of the straight line in {x, y) becomes 
 the length of the perpendicular let fall upon it from the given point 
 {xgf^ when we substitute x, and y, for x and y. 
 
 Let ¥{x,y)=0, P, {x,y)=0 be the projective equations of two 
 straight lines meeting La a point, then the equation of any other 
 straight line passing through the same point will be F [x,y) 
 ±i¥, (a?, y) =0, i being any constant whatever. It is clear that the 
 values of w and y which at the same time satisfy the equations 
 P (ar, y) =0, F, {x, y)=0 will also satisfy the equation of any other 
 straight line at the point of intersection, and therefore will satisfy 
 the equation F (ar, y) + i F, {x, y) =0. 
 
 To find the value of i in the preceding equation, — 
 If in F(af, y) we write x„ y, instead of x, y, then F {x,, y,) becomes 
 the perpendicular on F (x, y) =0, and F, {x^, y,) becomes the per- 
 pendicular on Fj (ar,y)=0; hence i denotes the ratio of the per- 
 pendiculars let fall from any point on the two lines F (x,y)=Q, 
 P/(«>y)=0, and will denote the anharmonic ratio of the four 
 straight lines F {x, y) =0, F, (a?, y) =0, F (ar, y) ±i F, (a?, y) =0. 
 
POINT AND A STRAIGHT LINE IN A PLANE. 5 
 
 When i= — 1, the anharmonic ratio becomes the harmonic ratio. 
 
 9.] Now let l—x^—yv=0=Y (f, v) be the t^igential equation 
 of the point (xy), and 1— as'jf— y,w=0=V( (f, u) be the tangential 
 equation of the point (jr^y^) ; hence it follows that V (f, v) 
 + jYi (f, w)=0 will be the equation of another point on the line 
 passing through the points {xy) and (a^^y,) , 
 
 On the given straight line whose tangential coordinates are f^ and 
 Vf let faE the perpendicular z from the point (xy) ; the length of this 
 perpendicular e will be P (I —x^,—yv^, where P is the perpendicular 
 from the origin on the given line f, and w^; see (6) . Hence it follows 
 that if in the tangential equation of the point [xy) , namely, V (f , v) = 0, 
 we substitute f , and v^ for ^ and v, and then multiply the expression 
 by the perpendicular from the origin on the Hue ^ju,, we shall 
 obtain the length of the perpendicular upon it, so that if V(f,t;) =0 
 be the tangential equation of the point (xy), P.V (f^u,) will be the 
 lenffth of the perpendicular letfall on the line (^,1;,) from the point (xy). 
 
 In the same way, if the perpendicular let fall from the point 
 {xffi) on the straight line whose tangential coordinates are ^, and v, 
 he z„ then r;=P(l— a?,f,— y;V,) ; hence 
 z^ l-x^,-yv, 
 
 If, now, iu the tangential equation of a third point which lies on 
 the straight line passing through the two points {xy) and {x^,), 
 namely 
 
 (1 -x^-yv) ±j (1 -x,^-yiu)=0, 
 
 we substitute f, and u, for ^ and u, and multiply by P, we get, Zn 
 being the length of this perpendicular, s„=:z±jzr 
 
 It is obvious that the perpendicular from the third point is z +jzi, 
 and from a, fourth point in the same straight line is z—jz,. 
 
 , "When the perpendiculars z and z, are equal, it may easily be 
 shown that the line whose tangential coordinates are f , and v^ is 
 parallel to that whose coordinates are ^ and v; for iu this case 
 
 ^ii+yvt=a!^f-^yf)„ 
 or 
 
 y-Vi^ Si. 
 
 x—x, u/ 
 hence the line which passes through the points {xy) and {x,y^ is 
 parallel to that whose coordinates are f / and v,. 
 
 When this third perpendicular z,, is 0, or when the third point 
 is on the intersection of the lines whose tangential coordinates are 
 
 f and V, ^, and v,, we get 0=z±jz„ oTJ=-, or j is the ratio of 
 
 the two perpendiculars let fall from the points {xy) and {x^y,) on 
 the straight line whose tangential coordinates are ^, and v,. 
 
 10.] We shall more clearly exhibit the duality of the relations 
 
6 EQUATIONS OP A POINT AND A STRAIGHT LINE IN A PLANE. 
 
 between projective and tangential coordinates if we write down the 
 foregoing propositions, side by side, as follows : — 
 
 Fig. 5. Fig. 6. 
 
 Projective Coordinates. 
 
 Let F(x,y)=0, F, (x,y)=0 be the 
 projective equations of two straight 
 lines. 
 
 lietathirdsiraight {tn«ABbe assumed 
 passing through their intersection, and 
 let the prqi'ecttve equation of this line he 
 
 T(x,y)±iF,Cx,y)=0; 
 
 from a point in this straight line, 
 whose coordinates are x, and y,, let 
 two perpendiculars he let fall on the 
 given straight lines ; the ratio of these 
 perpendiciuars will be t^ and the four 
 lines F(a:,y)=0, F.(x,y)=0, F(:r,y) 
 +iF, (x,y)=0 will form an anhax- 
 monic pencil. 
 
 Let F(a;,y,)=z, 
 
 and F,(x,y,)=z„ 
 
 and F (x,y.) ±i F, (x,y,) = 0. 
 
 Seeing that it is the expression for 
 the perpendicular from the point xj/,, 
 which IS on the line, 
 
 F(xy)+iF,(xy-) = 0. 
 Hence z+iz,=0, 
 
 Tangential Coordinates. 
 
 Let Y (I, v) =0, V, (& «) =0 be the 
 tangemticd equations of two points. 
 
 Let a third point be assumed on the 
 line passing through them whose tan- 
 gentud equation shall be 
 
 V(|,v)+jV,«,«)=0; 
 this point will be on the straight line 
 passing throu^ the given points whose 
 equations are v (f , «) =0, v, (f ,«) =0 ; 
 from these two points letperpendiculars 
 be let &11 to a line whose tangential 
 coordinates are ^, and «„ then P. V(£,w,) 
 and P. V, (!,«,) wiU be the length of 
 these perpendiculars. Let them be put 
 z and z,. Let the third point be as- 
 sumed as not only on the line passing 
 through the two given points, but also 
 on the line whose tangential coordinates 
 are (, and i;,. Hence the tangential 
 equation of this point becomesv (|u) 
 i/V, (|w)=0; but as this point is 
 on the line whose coordinates are |, 
 and v„ the perpendicular from it on 
 this line must be 0. Hence 
 
 V(&w,)±iV,tf,«)=0; 
 but it has been shown that 
 
 V(«,v,)=p 
 z+jz,=Q, 
 
 hence 
 
 ■■-J- 
 
 As the principles involved in the preceding theory may appear 
 
THE TANGENTIAL EQUATIONS OF THE CONIC SECTIONS. 7 
 
 somewhat obscure to be^nners, especially how the form of an 
 equation between two variables, when equated to 0, may by a sub- 
 stitution of like quantities become a line of given length, it would 
 seem better to err rather in fiilness of explanation than to assume 
 as easy that which to some minds may at first sight be difficult of 
 comprehension. 
 
 CHAPTER II. 
 
 ON THE TANGENTIAL EQUATIONS OF THE CONIC SECTIONS. 
 
 11.] We shall include the tangential equations of the circle in 
 those of the central conic sections, as nothing is gained in facility 
 of investigation by taking them separately ; and we shall commence 
 with the simplest forms of the equations of those curves, taking the 
 central sections apart &om the parabola, as their tangential equations 
 are essentially distinct. We shall assume the projective equation of 
 the ellipse, referred to its centre and axes, as the basis of investiga- 
 tion, and proceed thence to the more general forms of the equations 
 of these curves. We shall commence with the equation of the 
 ellipse, as we can always pass &om it to that of the hyperbola, by 
 changing b into V — 1^- 
 
 12.] The projective equation of a tangent to an ellipse passing 
 
 through the point (a?^,) on the ellipse, whose equation is -^ + Tf = 1, 
 
 may be written -^■\--^ = i- 
 
 In this equation x and y are the current coordinates, and the 
 limiting tangent meets the axis of x at the point where y=0; at 
 
 g2 1 
 
 this point a? =—; let this distance be x, hence Xf=a^^. In like 
 
 Xf ff 
 
 manner y,=b'^v; 
 
 hence a^f + 6V=1 (10) 
 
 is the tangential equation of the ellipse referred to its centre and 
 axes. 
 
 Let the axes of coordinates now be conceived to revolve, through 
 the angle 6, round the origin, and be then translated, parallel to 
 themselves, to a point whose coordinates are —p and —q. 
 
 The formulae of transformation are, see (4), 
 
 .. cos ^. ^,— sing, v/ ._ Bva.6 .^) + CQ% 6 .V, ,,,. 
 
'}• (14) 
 
 8 THE TANGENTIAL EQUATIONS 
 
 If we substitute these values of f and v in equation (10), omit- 
 ting the traits as no longer necessary, we shall find 
 
 [aScos2^ + 6«sin«5-i>«]P+ [6«co8«5 + a«sin*0-g*]w«l .^^. 
 
 + 2[{a«-6*)sin0cos5-j9g]|w+2p?+2gu=l. J 
 Hence, the tangential equation of a conic section being in its 
 most general form 
 
 aP+a^„2+2y8fu+27f+27^u=l, (13) 
 
 and equating the coefficients of this equation, term by term, with 
 those of the preceding one, we shall have 
 
 {a^—b^)siadcoa6—pq=^;p=y, q="i,- 
 
 In the first place, we may observe that the halves of the linear 
 coefficients of the general equation represent the projective coordi- 
 nates of the centre ; for p and q, the projective coordinates of the 
 centre, are equal to 7 and 7^ which are the halves of the coefficients 
 of the linear terms in f and v. 
 
 Comparing the three remaining coefficients, and introducing the 
 values of ^ and q, we shall have 
 
 O«COs20 + *2siii20 = O + 72,O« sin* 0+62 008^5=0^ + ^2^ 
 
 and (a2-4«) sine cos 5=/3+y7;,- .... (15) 
 
 hence (ai* — b^) cos 20 = (a + rf) — {a, + 7 «),^ 
 
 and (a*-**) sin 20=2(/3+77,) ; 
 
 and also tan 20=^-^1^0^. 
 
 («+7*)-(«i+7/*) J 
 Since 
 
 a2 + 62=(o + 7*) + (a,+7,2), 
 
 (a« + J*)«=(a+7*)«+2(a + 7«) (o,+7«) + (a,+7«)«, 
 and 
 
 (a«-42)'=(a+7«)«-2(a+72) (a,+7;) + («<+7,T408 + 77,)«: 
 subtracting, we obtain the result, 
 
 a2A«=(a+7«)(a,+iy;)-(^ + 7y,)« (18) 
 
 Again, since a* + 6«= (a+7*) + (a,+7/), 
 
 and «'-6^=\/[(o + 7')-(«/+7/*)T+408+77,)«, 
 
 adding these equations together, we obtain the result, 
 
 2a«=(a+7«) + {a, + r^^) +v/L(a+7«) -(a,+7«)/+4(/3 + 77)«, (19) 
 the upper sign being taken for the major axis, the lower when we 
 require the value of the minor axis. 
 
 > • • (16) 
 
 (17) 
 
OF THE CENTRAL CONIC SECTIONS. 9 
 
 13.] When the section is an hyperbola, i* must be negative, or 
 in (18) we must have 
 
 (/3+ry/)">(«+7*) K+7,') (20) 
 
 When the conic section becomes a circle, the two semiaxes in (19) 
 become equal ; hence the quantity under the radical must vanish ; 
 and as this quantity is the sum of two squares, we must have each 
 square separately equal to 0, or 
 
 a+7*=a,+7«, and/3+77,=0 (21) 
 
 When these two relations hold, the conic section becomes a circle. 
 "Hie corresponding property in projective coordinates shows that 
 the origin of projective coordinates must be at a focus. 
 
 Hence the relations between the coefficients of the general tan- 
 gential equation of the conic section which indicate that the curve 
 is a circle, namely 
 
 a+y«=a,+y« and ^+yy,=0, 
 when translated into the projective equations of a conic, namely 
 
 A+C2=A,+C« and B + CC;=0, 
 show that the origin must be at a focus. 
 14.J Let c*=a«— A«, then from (19) 
 
 c4= (fl«-6S)'=[(a+7*) - (a,+y,2)]='+4(/S+yy,)2. 
 Let D be the distance of the origin from the centre, then 
 
 D*=yHyA 
 Now let a=ai and /8=0, then we shall have c*= (y^+ y,*)*; hence 
 c=D, or the orig^ is at a focus of the curve ; and as 
 
 \.aD.20=-^^ or tan ^=5^', 
 
 thus the axis of the curve passes also through the new origin. 
 
 When the two conditions a+y'=o,+y^* and /8+yy,=0 are 
 satisfied, the curve is a circle ; and the origin is at a focus when 
 
 a = ap and /3=0. 
 
 15.] The origin of coordinates is on the curve when 
 
 oo,-/3*=0 (22) 
 
 When the origin of coordinates is on the curve, through this 
 point there can be drawn only one tangent to it, and at this point 
 f = 00, v= 00. Let i'=wf ; then' the general tangential equation 
 of the curve, o|* + o,w* + 2/8f w + Zy^-\- 2y,v= 1, may be changed into 
 (a + K«a;)f + 2/3wf + 2(7+ny,)f =1, or, dividing by |«, 
 
 (a + »'«,+ 2yS«) + 2(5:^/^ = ^, 
 
 or a + n^a, + 2ffn=0, since ^= x. 
 
10 THE TANGENTIAL EQUATIONS 
 
 _A-4- V'iS* cut 
 
 Solving this equation for n, n = — tii — C /. 
 
 a. 
 
 Now, in order that n may have only one value, we must have 
 
 ^=aa,, OTn=\/—; 
 V Uf 
 
 when ^<aa, the origin must be within the curve j when the 
 origin is outside the curve, n must have two values, and therefore 
 
 16.] To find an expression" for the angle which the asymptote to 
 an hyperbola makes with the axis of X. 
 
 Let PP + Qi;« + 2Rfu=l (a) 
 
 be the tangential equation of an hyperbola, referred to its centre and 
 any rectangular axes passing through the centre. As the asymptote 
 may be defined as a tangent to the curve at an infinite distance, 
 
 and which passes through the centre, at which point -z=0, — =0, 
 
 let T be the tangent of the angle which the asymptote makes with 
 
 the axis of X, then t= - or f =tu ; substituting we obtain 
 
 or, solving for t, t= == — p (b) 
 
 Now, the direction of the coordinates continuing unchanged, let 
 them be translated to a new origin, such that the projective co- 
 ordinates of the centre of the curve on the new axes shall be —7 
 and —/,; then 
 
 1=1 F ' "=1 1 • •••(c) 
 
 Substituting these values in (a) and reducing, we find 
 
 (P-7*)r+(Q-7,^)v«+2(R-yy,)?w + 2r£ + 2y,i;=l. 
 Comparing this expression with the general equation of the curve, 
 
 a^ + a,v^ + 2/8fw + 2yS + 27,1; = 1, 
 we shall have P— 7^=0, Q— y^*=Oy, R — 7>y^=^; 
 hence P=a+y^ Q,—ai+y^, R=/3+yy,. 
 
 Substituting in the expression (b) for the angle which the asymptote 
 makes with the axis of X, we obtain 
 
 _-(^ + 77/)± V(^+y7/)"-(a + y^)(a,+y/') 
 
 T = 
 
 ffl + y 
 
 ,2 
 
 (23) 
 
OF THE CENTRAL CONIC SECTIONS. 11 
 
 In'ordCT that this value of t may be real, we must have 
 
 .G8+yy/>(a+7')(a,+ y«) (24) 
 
 17.] It is not difficult to show that when we have the relation 
 03+77/= (a + y^)(«,+y«), 
 the tangential equation breaks up into two linear equations^ the 
 tangential equations of two points. Let the assumed linear equations 
 "^ m^ + nv—l=iO, aaimJS + n,v + l=0. ... (a) 
 
 Multiplying them together, the resulting equation becomes 
 
 mnii^+min^v— mf + mnf v+ 7171/1^— niV+m^ + nv=l. . (b) 
 Comparing this equation with the normal equation of the conic 
 section, of«+a,i;«+2/3£i;+27f+2y,i;=l, 
 
 and equating like coefficients, we get 
 
 mm,=a,nn,= ai,min+mn,=2fi,m—m,=2fy,n—n,=2y,; . (c) 
 
 hence m,+m=2 Va+y', n;+w=2 \/a,+y^, 
 
 and (»!,+»»)*(«,+»)*= 16(a+7*){o^+y«). 
 
 Now m—m,=:2y, n—n,=2Y, ; 
 
 hence (»»—»»,)(«— »()=4yy„ 
 
 and 4^8 = 2 (»»,w + mn,) ; 
 
 hence 4(j8 + 77,) = (m + m,) (ra + n,). 
 Consequently, equating these values, we find 
 
 Since »»+»», =2 V^+y'^ and jm— in;=2y, 
 
 we find jM=y+ V^+yS m,= —y+ Va + yS 
 
 and «=y,+ Va/+y/S »/=—%+ Vo/+yf- 
 
 Hence the equation of the curve is broken up into 
 
 and VaT7f+ s/ a,+y^ -f^-ltV-\=0, \ ..(e) 
 
 18.] When a=0, the axis of Y touches the curve, and when 
 Oy=0 the axis of X touches the curve. 
 
 When ■= is a tangent, the ciirve coincides with the axis of X ; 
 
 then - =0, since this tangent meets the axis of Y at the origin. 
 The general tangential equation of the curve may be written 
 
 «/+ ^{«f + 2yf-l} + ^{2^f + 2y }=0; 
 hence, when - =0, a. must be =0. 
 
12 
 
 THE TAKOENTIAL EQUATIONS 
 
 In like maimer it may be shown that nrhen the axis of Y touches 
 the curve we must have a=0, 
 
 19.] To find the values of ^ and v when they are tangents to the 
 curve. 
 
 When the limiting tangent coincides with the axis of X, then 
 (as in the preceding article) a,=0, and the tangential equation 
 becomes^ when divided by v (wmch in this case is infinite), 
 
 -{a^+2yf— l} + 2/3^ + 2y,=0. Hence, when the axes of co- 
 ordinates are tangents, ve find 
 
 $S + Y,=0, /3t; + y=0 (25) 
 
 20.] Besuming the general tangential equation of the conic 
 section (13), let it be solved for v. 
 
 \ a, J a, 
 
 ---'■-^— "'-^-^ OP 
 
 Now let 
 
 Om- V a, )^ a, ' OP~ \ a, ) 
 
 and 
 
 Fig. 7. 
 
 r 111 
 
 ''^ Oin' OF On "^ "^ anthmetical progression j therefore 
 
OF THK CENTRAI, CONIC SECTIONS. 13 
 
 Om, OP, On are in harmonical progression ; hence QO being = ^, 
 
 QO, Qm, QP, Qra constitute an harmonic pencil ; and as Qm, Qn 
 are tangents to the curve, QO, QP pass respectively through the 
 poles, one of the other ; hence QP passes through the pole of QO, 
 which is the axis of X ; hence 
 
 is the equation of the pole of QO which is the axis of X. In the 
 same way it may be shown that a^+fiv + y=0 is the equation of 
 the pole of the axis of Y, and the two simultaneous equations 
 
 a^ + ^v+y=0, a,v+^^ + Yi=0 .... (27) 
 
 determine the polar of the origin. 
 
 If we solve these equations, we find for J and v 
 
 ^^7,-^ ,=^y=^/ (28) 
 
 These are the coordinates of the polar of the origin *. 
 
 21.] To find the species of the conic section when the reciprocal 
 of the perpendicular on the tangent from the origin shall be a 
 rational function of the tangential variables. Let the equation be 
 
 a|« + a,v« + 2/3fw + 27? + 2y^v=lj (a) 
 
 then ta being the reciprocal of the perpendicular, and \ the angle 
 it makes with the axis of x, we shall have cos \=^, 'srsin\= v, 
 hence 
 
 acos*\ + a(Sin*\+2jysin\ cos\+2{ycos\+7ysin\)— =— 5, 
 
 or 
 
 „ l-Yg-7,» 
 
 ^(0 + 7*) cos*\+(a,+7,') sin*\+2(/8+77j) sin\ cosX 
 
 Now, in order that this may be a rational function of f and v 
 independently of \, we must have 
 
 a + 7'=aj+7' and/9 + 77^=0; 
 hence ^^^-^=^p^, (29) 
 
 * Let Ax'+A,y''+2REy+2Ca;+2C,y=l be the projective equation of a 
 central conic section, "x and y lieing the coordinates of the centre ; then, as is 
 shown in all works on the subject, 
 
 - BC,-AC - BC-AC, 
 
 *= AA -B" ' J'= AA,-B=' 
 which expressions for the ordinates of the centre are analogous to those for the 
 tangential coordinates of the polar of the origin. 
 
14 
 
 THE TANGBNTIAL EQUATIONS 
 
 or the curve must be a cirde, in order that «r may be a rational 
 function of f and v*. 
 
 22.] We shall showforther on that if F(a7, y) =0 andV (f, w) =0 be 
 the projective and tangential equations of the same curve, we may 
 pass from the one system to the other by the help of the following 
 relations, taking the partial differentials as follows : — 
 
 1= 
 
 dP 
 d/r 
 
 dP , dF 
 da? dy* 
 
 37= 
 
 dF 
 dy 
 
 "dF . dF 
 dx dy' 
 
 dV 
 dv 
 
 ^ 
 
 (30) 
 
 ■dV.dV 
 
 dV 
 d|^ 'di;' 
 
 23.] Let Y(x, y)=0 and V(f, w)=0 be the projective and tan- 
 gential equations of the same curve, then 
 
 dV dVj. dF dF 
 
 dV ^ . dV 
 df^'^dj^" 
 
 =0. 
 
 For brevity put 
 
 TT— dV.^dV 
 ^=dr^+di7''' 
 
 dF dF 
 di'^'^dy^ 
 
 w dF ^dF 
 da? dy* 
 
 (31) 
 
 • Let Ait»+A,ys+2Ba:y+2Cx+2C,y=l (a) 
 
 be the proiecfiye equation of a conic section. Let x=r cos 0, ^=r sin a. Sub- 
 stituting these values and reducing, we find 
 
 Acos'tt-f A, nn^ a)-|-2B sin a cos (a-|-2(C cos oi+C, an e>) -=-j, 
 
 or, solving, 
 
 - = C cos a+C, edn a 
 
 }■ 
 
 0>) 
 
 + V[A+(Xl-^-[(A,-^C,») -(A4-C»)]sin» q,-|-2(B+CC,) sin a cos a.^ 
 
 Multiply this equation by r, putting x for r cos 0, and y for r sin a, there finally 
 results 
 
 l-Car-Cy 
 
 V(A+C»)+[(A+C,»)-(A,+C,»)]8in»a.+2(B+CC,)Mn«>co8«' 
 
 (c) 
 
 Now in order that r maj be a rational function of x and y, the coefficients of the 
 trigonometrical quantities under the radical sign must vanish, or we must have 
 A-hC»=A,+C,', and B+CC,=0, and 
 
 „_ l-Oj:-C.y 
 VA-I-C* ■ 
 
 
 
 But when r is a rational function of .t and y, it may easily be shown that the 
 ori^ is at a focus. 
 
OF THE CENTRAL CONIC SECTIONS. 15 
 
 dV dV 
 Then, as shown in (30), x=-fr> V—'TT > 
 
 dV^_dV^ 
 hence aiv—y^= „ .... (32) 
 
 In like manner we may show that 
 
 dF _dF 
 
 ^•^-y?=^Sr^' ^^^ 
 
 hence the truth of the proposition. In fact each expression is the 
 value of tan 0, the angle between the perpendicular and the vector 
 line. 
 
 Of this formula an elementary proof may be easily given. Since 
 
 e=\-a, 
 
 v_y 
 
 n tan \— tana f x xv—yJE 
 
 tan^=:j-—: — ^— =-2 = ^ , '' ; 
 
 l+tan\tana yv x^ + yv 
 
 but x^+yv = l; 
 
 hence taii0=xv—y^ (34) 
 
 24.] To apply these formulse. Let us assume the general pro- 
 jective equation of a central conic section, 
 
 Aa^+Aff^-i-2Ba!y-l=Y{x,y)=0 (a) 
 
 Now 
 
 ^=2A^+2By, |=2Ay+2B., and g-+f ^=2- • (b) 
 
 Hence f=Aa7+By, and v=Ay+Bx (c) 
 
 From these equations, finding the values of ^ and y, and substituting 
 them in the preceding equation, the result becomes 
 
 A^f+Aw«-2Bfu=AA,-B«- (d) 
 
 Comparii^ this formula with the general tangential equation of a 
 conic section, its centre at the origin, namely 
 af« + o,u« + 2/3?i; = l, 
 
 ^efind a=^^^-J_,;« -__-^, ^=_^_^^, . . (35) 
 
 so that we may at once pass &om the projective to the tangential 
 equation of the curve. 
 
 The coefficients of the projective equation appear with some 
 slight alterations : thus A,, the coefficient of y®, becomes the mmie- 
 rator of the coefficient of J*; and + 2B is changed into — 2B, while 
 the absolute term becomes AA'— B*- 
 
16 THE TANGENTIAL EQUATIONS 
 
 On Asymptotes. 
 
 25.J An asymptote may be defined as a tangent to a curve, one 
 of the projective coordinates a? or y of the point of contact being 
 at an infinite distance. Let a? =00; then in the dual equation 
 
 x^+yv = l, if ar be infinite, we shall have -= — ^. 
 dV 
 
 dv 
 
 This is the general equation of an asymptote to a curve. 
 
 To apply this theory. Assume the general tangential equation 
 of a curve of the second order, 
 
 dV dV 
 then ^f+J^w=0=2oP+2a,w*+4j8|w + 27^+27,v; . (a) 
 
 and the equation of the curve, multiplied by 3, gives 
 
 2a^+2aii^+4,^^v+4y^+4ff,v=2 (b) 
 
 Subtracting the preceding expression from this equation, thereresults 
 
 7f + y/''=l, (c) 
 
 the tangential equation of the centre of the curve, since y and y, 
 are the projective coordinates of the centre. From this we may 
 infer that whether the asymptotes be real or imaginary, they must 
 pass through the centre of the curve. To determine the angle 
 which the asymptote makes with the axis of X, let r be the 
 
 e 
 tangent of this angle, then t=-. Substituting this value of f, 
 
 and dividing by i^, we get 
 
 o7^ + »,+ 2/8t+(7t+7,) i=0; 
 
 but (c) gives -=yT+'y,. Introducing this value of -in the pre- 
 ceding expression, we find 
 
 («+7V+203+ry,)T+a,+78=O. 
 Now, as this is a quadratic in t, there must be two asymptotes, each 
 of which passes through the centre ; and if we solve tlus equation 
 for T, we get 
 
 ..- - 0+7Y/) ± ^(^ +77/)^- (a+7^) (a/+7,") 
 a + 7* 
 
 which is real only when (a+7®)(o/+7,*) is less than (fi+yy,)^. 
 
OP THE CENTRAL CONIC SECTIONS. 17 
 
 This expression is identical with that found by a very different 
 method in sec. 16. 
 
 26.] Let UB resume the consideration of the central conic referred 
 to rectangular axes passing through the centre, namely 
 Af2 + A,w* + 2Bfi;=l. 
 
 Retaining the axis of X, let us assume a new axis of Y' passing 
 through the centre and making the angle a> with the axis of X. 
 
 Hence, by (5 ),«=-* — : —; and substituting in the preceding 
 
 equation the value here assigned to v, we get 
 
 (A sin* Q) H- A, cos* w — 2B sin cd cos a>)^+ A,v^ 
 + 2(B sinw— A, cos<a)fi;,=sin*o). 
 Let us assume such a value of a> as will cause the coefficient of the 
 
 A 
 
 rectangle ^v to vanish; then tan a)=r^, and substituting this value 
 
 of tanm in the preceding equation, the equation of the curve 
 referred to oblique axes becomes 
 
 [AA,-B*]f+(A* + B*)u*=A, (37) 
 
 If we draw the limiting tangent parallel to the axis of X, f is=0, 
 
 1 A*+B* 
 
 and -^=— i-j =b^ the semidiameter conjugate to the axis of ^. 
 
 If we draw the limiting tangent parallel to the axis of Y, or make 
 
 AA;-B« g , 2^., AA^-B* A2+B* 
 — 'j^ =af; hence a^+b,^= — ^ + '^ > 
 
 or a,^ + b^=A + A,; but A + A^is the sum of the squares of the 
 semiaxes o* and A*, hence ai^ + b^=a^ + b^, a well-known theorem. 
 
 The same formidae for the parallel translation of axes will hold 
 whether the systems of coordinates be right-angled or oblique, the 
 coordinates of the centre of the first system being drawn parallel 
 to those of the second. 
 
 27.] To determine a general expression in any plane curve for 
 the distance between the point of contact of a tangent and the foot 
 of the perpendicular let fall firom the origin upon it. 
 
 As this line is the projection of the radius vector upon the tan- 
 gent, it may with propriety be called the protangent, and may be 
 written t. 
 
 Let V(f, i;)=0 be the tangential equation of the curve, then 
 ^=a?*-|-y*— />* (a), and write V instead of V (f, v). 
 dV dV 
 
 Now ^=_iL_^, andy = ^y^'^^^ . . . . (b) 
 
18 THE TANGENTIAL EQUATIONS OF THE CENTRAL CONIC SECTIONS. 
 
 Squaring these values of x and y, and subtracting from them hT~9» 
 
 the square of the perpendicular, we obtain this remarkable and 
 useful formula, 
 
 dV dV. 
 
 /= 
 
 (dVj.^dV ) 
 
 Idr^+di;^ 
 
 (r+«*)^ 
 
 (38) 
 
 If we apply this formula to the tangential equation 
 A^ + A,i;^+2Bgi;=l, of a conic section, where t is the distance 
 measured from the point of contact along the tangent to the foot 
 of the perpendicular, we shall find this expression, 
 , (A-A,)gu + B(.,«-P) 
 
 Now,ifthe limiting tangenthe drawn parallel to the axis of X, f=0, 
 
 and t='Rv; but - is the perpendicular on this tangent from the 
 
 centre ; therefore P^ = B . Hence, in the general tan gential equation 
 of the conic section, B denotes the area of the triangle between the 
 axis of Y, the perpendicular tangent to it, and the diameter drawn 
 through the point of contact. 
 
 When the tangent is parallel to the axis of X, the general equa- 
 tion becomes A,u^=l, or P*=Ay; hence 
 
 1" A; P . 
 ^^=^=-=tana,; 
 
 or to, the coordinate argle, is the angle between the conjugate dia- 
 meters of the curve. Hence by the use of oblique tangential coor- 
 dinates we may derive the properties of the conjugate diameters, as 
 we may those of the axes, by the help of rectangular coordinates*. 
 28.] Let \ be the angle which a perpendicular P, let fall on a 
 tangent to an ellipse which touches at a point whose radius vector 
 is r, makes with the axis of X, we shall have 
 
 o*cos*\-f-6'*sin2\=P«r«; (39) 
 
 * We may obtain a similar expression (in projective cooTdinates) for the tan- 
 gent of the angle between the perpendicular on the tangent and the radius vector 
 of the point of contact. 
 
 For tan'fl=p5=-^^; hence -^p5-=p.,-j5- 
 Now ;'- ^^'' .,='= W_ 
 
 Or 
 
 
THE THEORY OF TANGENTIAL COORDINATES. 19 
 
 for Pf =coa \, and Pu=sin \, we have also a^^=x, bh}=y. Hence, 
 substituting, a* cos* \ + J^ sin* \ = P^r*. 
 
 In like manner we may show that if to be the angle which the 
 diameter of an ellipse makes with the axis X, we shall have 
 
 cos* ft) sin* ft) 1 ,.-, 
 
 ~1^+-F-=PV (*°^ 
 
 To express the value of a semidiameter drawn to the point of 
 contact of the limiting tangent in terms of f and v : — 
 
 Since cos\,=Pf, sinA,= Pu, substituting these values in (39), 
 
 r*=a*f« + 6V (41) 
 
 CHAPTER III. 
 
 We may now illustrate this theory by its application to a few 
 examples. 
 
 29.] The prodtict of pairs of perpendiculars let fall from two points 
 on a straight line is constant ; the line envelopes a conic section. 
 
 In the expression given for the perpendicular in (6) , 
 
 where p and q are the projective coordinates of the point, while f 
 and V are the tangential coordinates of the Une, let^ = + c and g' = ; 
 that is, let c be the distance between the points and the middle 
 point of this line, taken as origin ; then 
 
 and let PP,=6*, the resulting equation becomes 
 
 (6* + c*)|* + 6V = l, 
 
 the tangential equation of a conic section, of which (i* + c*) and 6* 
 are the squares of the semiaxes. 
 
 30.] The vertex of a right angle moves along the circumference of 
 
 adding these expressions, and subtracting: . . , and taking the square root, we 
 
 find 
 
 dF _AV 
 
 tanfl=.iLL3L!. 
 df , iF 
 
 di;''+d^y 
 
 If we apply this expression to the equation .\.x'-+ X!/-+2Kvi/=l, we shall 
 find tantf=(A-A,)a:y+B(.r=-J/S). 
 
 c 2 
 
20 THE THEORY OP TANGENTIAL COORDINATES 
 
 a circle; one side passes through a fixed point; the other envelopes a 
 conic section. 
 
 Let the line joining the fixed point with the centre of the circle 
 be taken as the axis of X, let this distance be c, the equation of 
 the circle being a7'+y*=a*; then the tangent of the angle "which 
 the line that passes through the fiscd point makes with the axis of 
 
 X is —S- — , and the tangent of the angle which the limiting tangent 
 makes with the same axis is — -; and these angles are complements 
 one of the other ; hence J' > = 1 • The dual equation (1) gives 
 
 finding the values of y and x, substituting them in the equation of 
 the circle a?'+y*=a', we find 
 
 when c-=a, a^=l, or ^=-, the tangential equation of a point m 
 
 the axis of x, at the distance a &om the origin ; when c:^a the 
 curve becomes an hyperbola. 
 
 31.] Tangents are drawn to an ellip se from any point of a con- 
 centric circle whose radius is sja* + 6* ; the line joining the points of 
 contact envelopes a confocal conic. 
 
 Let t and u be the projective coordinates of the given point on 
 the circumference of the _. 
 
 circle, then fi + u^=a^-\-b^ ■'^^S- 8. 
 
 is the equation of the circle ; 
 and the polar of this point 
 with reference to the ellipse 
 
 is -j + Tg = l; andthis gives 
 
 t=a%u=b^v. Substituting 
 these values of t and u in 
 the equation of the circle, 
 we find 
 
 the tangential equation of an ellipse whose semiaxes a and i are 
 given by the equations 
 
 ««=. «' r» b* 
 
 A«=- 
 
 and these sections are confocal for 
 
 The line drawn from the point (t u) to the point of contact of the 
 
APPLIED TO EXAMPLES. 21 
 
 polar of (t, u) with the interior confocal curve is a normal to the 
 latter. 
 
 The tangent of the angle which the polar of (/ u), the tangent to 
 
 the interior confocal curve, makes with the axis of X is — -: hut 
 
 V 
 
 the tangent of the angle which the line joining the point (t u) with 
 (^; Vi)) the point of contact of the polar of (/ u) with the confocal 
 curve, is 
 
 _ /6* *!_x 
 
 u — y, b^v—b^v I a^-\-b^ !''_" 
 
 ^- a^ + b^ 
 Hence these lines are at right angles, the one to the other. 
 
 32.] The sum of the perpendiculars let fall from n given points 
 in a plane to a straight line in the same plane is constant. The straight 
 line envelopes a circle. 
 
 Let the projective coordinates of the n given points on the axes 
 of coordinates, their origin and direction being arbitrary, be pq, 
 Pill) Pifilw ^^- Then the length of one of the perpendiculars 
 
 on the given line is — , ^ ; for the next point it will be 
 
 VP + K* 
 
 ^£l=£^&c. 
 
 Let the sum of the perpendiculars be nc, then 
 
 n—{p+p,+p„&c.)^—{q-iirq,-\-qn%i,c.)v=ncV^ + v^. . (a) 
 
 Let P and Q be the coordinates of the centre of gravity of all the 
 points ; then 
 
 p +p, +p„ &c. = «P, q + q, + qi, - nO,. 
 Substituting these values in (a), and dividing by n, we get 
 
 l_p^_Q„=c V?+Ii^. 
 Reducing, 
 
 (c«-P*)f«+(c2-Q«)w«-2PQfi; + 2Pf+2Qu = l, . (b) 
 comparing this expression with the normal form, 
 
 of + o,i;2 + 2/3^1; + 2yS + 2y,v=l, 
 which becomes the equation of a circle when 
 
 a + y^=ai+y,% and fi + yyi=0, see (21), 
 relations which hold between the coefficients of the preceding 
 equation. 
 
 When the sum of the perpendiculars is 0, c =0, and the tangen- 
 tial equation of the locus becomes P^+ 011 = 1, the tangential 
 equation of a point of which the projective coordinates are P and Q, 
 the projective coordinates of the centre of gravity of the system of 
 n points. 
 
22 THE THEORY OF TANGENTIAL COORDINATES 
 
 33.] Perpendiculars are drawn to the extremities of the diameters 
 of an ellipse. They envelope a curve ; to determine its tangential 
 equation. 
 
 Let the equation of the ellipse be 
 
 a^ + b^-^' ^^^ 
 
 and as the diameter of the ellipse r is perpendicular to the line 
 whose coordinates are ? and v, we get r'^=x^ + y'^=-=2^^; and as 
 
 1 t 
 
 r is a mean proportional between x and ^, ^=^^?= £237-2* ™ the 
 
 same way y = zg— — 2- Hence, substituting, 
 
 i^+vr=^,+^, (b) 
 
 The rectification of this curve gives one of the best illustrations of 
 the geometrical interpretation of the first elliptic integral. 
 
 34.] Tioo semidiameters of a conic section and the chord joining 
 their extremities contain a given area. The curve enveloped by this 
 chord is a similar conic section. 
 
 Let -9 + ^=1 (a) be the equation of the conic section; x,yi and 
 
 Xfi yii the coordinates of the extremities of the semidiameters ; then 
 
 ,,=!=£!, y„^l^. . . . . (b) 
 
 Substituting these values of y, and y,, successively in the equation 
 (a), we shall have 
 
 [a2|2 + 6V]a?2_2a2fa.^ + a2(l_AV)=0; . . . (c) 
 
 the substitution of y„ would give an equation of precisely the same 
 form. Hence 
 
 '+^"-«2f« + 6V' ' "~ a^^ + bV ' • • W 
 combining these expressions, we get 
 
 V^' ^«) (a^^+AV)a •••(e) 
 
 Let the area of the given triangle be -5- ; it may be shown that 
 it is also equal to 5d£j/. 
 
 Hence _=a^', or »«=i.x_^. 
 
APPLIED TO EXAMPLES. 23 
 
 Substituting the preceding value of {x—Xi^, wo obtain 
 
 Leta2^2^6V = M. Then m2M2=4.(M-1), or 
 
 a'^J2+6V=Jn± Vn^n (f) 
 
 Since the area of a triangle generally is 7r^> ^ being the 
 
 contained angle, and since it has been assumed equal to — -, 
 
 n=sin^; hence 
 
 Z r-, ,- -, 2(l+cosd)) 1 1 
 
 -5[l+ \/\-n^\= ^ • 2^ = 1 or J,' 
 
 ^ sin^ ^ ''o® o 
 
 consequently the equation of the sought curve becomes 
 
 sin'liflS'f + 6V}=1, or cos2|[a2|2 + 6V] = l, . (g) 
 
 accordingly as we take the upper or lower sign. Thus there are two 
 concentric ellipses enveloped by the revolving chord, such that the 
 sum of the squares of the coincident axes will be equal to the 
 squares of the axes of the original ellipse ; for 
 
 o2 sin* I + a* cos^ | = a^, and b^ sin^ | + 6* cos* | = 6*. 
 
 Hence if a polygon of n sides be inscribed in a conic section, the 
 sides being inversely as the perpendiculars let fall upon them from 
 the centre, this polygon will circumscribe a conic section similar to 
 the given one. 
 
 35.] The straight line which joins the points of intersection of 
 two focal vectors, containing a given angle 6, with a conic section, 
 envelopes two conic sections having their foci coincident with the 
 focus of the given section ; and if e and e, be the eccentricities of 
 the loci, e that of the given section, p and p, the parameters of the 
 loci, P that of the given section, we shall have the following rela- 
 tions between the eccentricities and parameters of the three conic 
 sections, 
 
 ^+e^=e\ p^+p,^= Pa- 
 llet the equation of the given section be 
 
 x^ y^ 2ex_b^ -« 
 
 a* 0* a a* 
 the origin being placed at a focus, and the axes drawn parallel to 
 the principal axes of the section. 
 
24 THE THEORY OF TANGENTIAL COORDINATES 
 
 Let (^,x,), {y„ii!,^ be the coordinates of the points in which the 
 sides of the given angle 6 intersect the cnrve : the equation of the 
 line passing through those points is 
 
 y-y,=|^" (^-^,) ; (b) 
 
 or if 
 
 y,=mx„ . . (c) y„=m/c„ .... (c,) 
 
 be the equations of the sides of the angle, we shall find, eliminating 
 y„ y„ between (b), (c), and (c,), 
 
 y— mar,= '"^'~"''^" {x—x^ (d) 
 
 Xf—Xf, 
 
 Let ^ and v denote the reciprocals of the intercepts of the axes 
 of X and Y by the limiting tangent (d) ; then 
 
 -=»ii; + ?, . . (e) — =OT,i/+f. . . . (f) 
 
 Now, eliminating {x„ y^ from the three equations (a), (c), (e), 
 we shall find the quadratic equation 
 
 (a«-fr'u2)m«-2(*«oev+6*?w)m + A9-2oc6«f-i''£«=0. . (g) 
 
 But this is precisely the equation we should have found for m, ; 
 hence m and m, are the roots of (g), or 
 
 hence n.-n.,=^^tmi+l^\^p±:l}l. 
 
 Let the quantity under the radical sign be written M ; then 
 
 l+mmi a'— A'M ' 
 or, solving this quadratic equation, we shall find 
 ^_ a^(l±cosg)^ 
 
 or, replacing for M its value, reducing and taking the lower sign, 
 we find 
 
 b\^+x^) . 26'gc.g ^ ... 
 
 7 37+ 4 =1- . . . (h) 
 
 (A« + a«tan«|j *9+o«tan«| 
 
 Had we taken the upper sign, we should have found for the tan- 
 gential equation of the locus 
 
 ne + v") . ib^ae.S . 
 
 ^+ ^=1 W 
 
 6«+a«cot«| 6« + o«cot«| 
 
APPLIED TO EXAMPLES. 25 
 
 Now in these equations, as the coefficients of f and v are equal, 
 .he foci of these sections are at the origin, or coincide with the 
 focus of the given section. 
 
 To determine the axes &c. of these loci. The tangential equa- 
 tion of a conic section whose semiaxes and eccentricity are A, B, 
 and e, the origin of coordinates being at a focus and parallel to the 
 axes of the section, is 
 
 B«(S2 + y')+2A6.f = l (m) 
 
 Comparing this equation (m) with (h), we get 
 
 b* h^ae 
 
 B*= a, Ae= 
 
 ft2 + fl«tan*|' J^ + o^tan^l 
 
 hence e=ccos-, and -r-=— cos -, or^=PcoB-. 
 Had we taken the upper sign, we should hare found 
 
 hence e2+e,2=c«, p^+pi^=V'': 
 
 when ^ is a right angle, the two loci coincide. 
 
 Had any other point except one of the foci beeu chosen, we 
 should have found for the locus a curve whose tangential equation 
 would be of the fourth degree — ^the curve in this particular case 
 separating into two distinct curves, each of which is a conic section. 
 
 Had the given section been an equilateral hyperbola, and a 
 right angle, a parabola would have been the locus. 
 
 When the given angle revolves round the centre instead of the 
 focus, the tangential equation of the locus is 
 
 36.] A straight line revolves in a conic section, having always a 
 constant ratio to the parallel diameter ; it will envelope a similar 
 conic section. 
 
 Let c be the chord, 2r the parallel diameter, n the ratio. Let the 
 tangential equation of the conic section be 
 
 a%^ + b\^=l (a) 
 
 Let c, the limiting tangent, cut the axes of coordinates at the dis- 
 tances •= and - from the centre. Let 2a, be the diameter conjugate 
 to 2r„ and x the distance between c and r measured along a,. Then 
 
 _2 «.2_/.9 
 
 c .r . . a, a? . o, , or J— „ • 
 
26 
 
 THE THEORY OF TANGENTIAL COORDINATES 
 
 Let the tangent to the curve be parallel to the chord c, then 
 p : I : : a, : a7„ at ^f -.^ :: xf : af, 
 
 In the same way v^ = v^ i — g— 1. Hence, substituting in (a), 
 
 a2S« + 6V==- 
 
 j;but-5=n^; hence 
 
 are 
 
 a^{l-n^)^ + l^{\-n')v^ = \; (b) 
 
 when the line is indefinitely small n^O, and we get the original 
 equation of the curve. When the revolving chord is equal to the 
 parallel diameter, n = \, and the equation becomes . f + . i;= 1. 
 In order that this relation may hold, we must have ?=oo , v=cc , 
 
 or ^=0, -=0, or the chord c must pass through the centre. 
 
 37.] The product of the sides of a right-angled triangle diminished 
 by fixed quantities, is constant ; the hypotenuse will envelope a conic 
 section. 
 
 Let the sides of the triangle be taken as the axes of coordinates, 
 and let the subtracted lines be a and b. Then by the terms of the 
 
 question (t—o)! b\=c^. Since the sides of the triangle 
 
 ^ and -, reducing, 
 
 {c^-ab)liv + a^+bv=\, (a) 
 
 the tangential equation of a conic section. 
 
 Hence |a and \b ai'e the coordinates of the centre. 
 
 Since a and a,, the coeflBcients of the squares of the variables, do 
 not appear in this equation [sec. 18.], the sides of the triangle are 
 tangents to the curve. 
 
 When (?=ab, the equation becomes 
 af+6w=l, the tangential equation of a 
 point. 
 
 38.] Let the sides of the rectangle 
 OPQR be produced, and cut by the trans- 
 versal ABCD ; to find the tangential equa- 
 tion of the curve to which this line is always 
 a tangent, under certain conditions. 
 
 Let 
 
 OP=fl, PQ=6, 0A=^, OB=i; 
 then we shall have 
 
 Fig. 9. 
 
APPLIED TO EXAMPLES. 27 
 
 ? ? w 
 
 RB=l:i*^, PA=l-=^^, PD=1^^, 
 w g V 
 
 AB^=^+^, CD^ = (^.^-^.)(l-«f-H^ . (a) 
 
 , , . ADxBC 
 
 (**) ^^^^""^ ab3^cd="- 
 
 Let ^ + u*=cr^. Now, if we substitute the values of these lines 
 as given above in J and v, we shall find 
 
 (l-a^-bv) -"' 
 
 or rSt) + a£ + bv=l (b) 
 
 n—1 ^ ' 
 
 If we submit this equation to the test in (20) , we shall find that 
 it is an hyperbola, since re+l>m—l, and the curve touches the 
 axes of coordinates, since the coefScients of the squares of the vari- 
 ables are wanting. 
 
 AC X BD 
 
 (/8) Let -pg — p=:=M. Substituting the values of these lines 
 
 above given, we find — ^v + a^+bv=il. It may easily be shown 
 
 that this is the tangential equation of the hyperbola. For, assuming 
 the form of the general equation 
 
 - ab r, a b 
 
 we find a=0, a,=0, ^=^P> 2~'^' 2~'*''' 
 
 and as the curve will be an hyperbola when 
 (/3+yy/>{«+7')(«i+y/'), 
 we shall find on substituting, 2 + w>»' 
 
 AD^ + CB* 
 (y) To find the curve when — ==- — = m*. Substituting the 
 
 AB 
 
 values of these lines, the resulting equation becomes 
 
 a5'|2^.jv-2af-26u=««-2, 
 the tangential equation of an ellipse, parabola or hyperbola, accord- 
 ingly as »2>2, m2 = 2, or w«<2. 
 
28 THE THEORY OP TANGENTIAL COORDINATES 
 
 /S-, T ■ CB' + AD'_^a Substituting the values of these straight 
 lines, we get by reduction 
 
 the equation of an hyperbola, since ^g^ is greater than 1. 
 
 39.] The vertex of an angle of constant magnitude moves along 
 the circumference of a circle ; one side passes through a fixed point ; 
 to determine the curve that will be enveloped by the other. 
 
 Let C B D=^, then while one side of the given angle passes 
 through C, the other side B D touches j,. jq 
 
 the locus. 
 
 Let the given point C, whose distance 
 from the centre O of the circle is c, be 
 taken as the origin ; then the equation 
 of the circle, whose radius is a, will be 
 
 (a?-c)2 + y«=a«- . . (a) 
 
 Let the constant angle be 6, whose 
 
 tangent is m, and let ^, ^, be the angles 
 
 which the moving lines make with the 
 
 aris of X. Then 0=<f>+<f>i. 
 
 V B 
 
 How ta,n0=m, tan^=-, tan<^(=-. 
 
 Hence »»= =, or m=- r^ (b) 
 
 But x^ + yv= 1, hence m= — —y-. Eliminating y and x succes- 
 sively, we get 
 
 m{e+v^)' " m(5«+i;«) ^' 
 
 Substituting these values in the equation of the circle, putting 
 for m its value tan 0, we shall obtain 
 
 (a2-c«) sin25(|«+w«) + 2c8in0(sine.|+cos^.i;) = l. . (d) 
 
 This is the equation of a conic section whose focus is at the 
 origin. 
 
 If we compare this equation term by term with the general tan- 
 gential equation of a conic section, 
 
 a£2 + a^„s + 2/8fw + 2y5 + 2yi/=l, (e) 
 
APPLIED TO EXAMPLES. 29 
 
 we shall have 
 
 o=a,= (fl«-c')8m«^, )8=0, y=csiii*^, yi=camdcoa0. (f) 
 
 To detenuine the semiaxes and eccentricity of this curve. In 
 (19), the general equation of the axes of a conic section, we find 
 
 2A?=a + y^+a, + y,±{[{a + y^)-{a, + Y^)Y + 4>{^ + yy,nK 
 If we substitute the preceding values, we shall have 
 
 ,^, , 2 A^-B* c f- • • (g) 
 
 and therefore e^= — r-s — , or c=-. \ 
 
 Hence, as e is independent of 0, all the enveloped curves will be 
 similar and unifocal. 
 
 The coordinates of the centre are manifestly 
 
 y=csin*0 and y,=csin^cos5. 
 
 Hence the distance of the centre of the curve firom the origin is 
 D=csin^. 
 
 When c=a, or the origin is on the circle, equation (d) becomes 
 
 2a sin* ^. f + 2a sin ^ cos . v= 1, 
 
 the tangential equation of a point on the circumference of the circle. 
 It is manifest that the line joining this point with the origin is the 
 chord of the segment of the circle which contains the angle 0. 
 
 40.] An angle of given magnitude revolves round a fixed point, 
 intersecting by its sides two given straight lines ; the line which Joins 
 the point of intersection envelopes a conic section. 
 
 Let the fixed point be taken as origin of coordinates, the axes 
 of coordinates being rectangular. Let 
 
 \x + liy=\, ... (a) and \fe-\riiy=\ . . . (a,) 
 be the projective equations of the two fixed straight lines. Let 
 
 y=mx, . . . . (b) and y=mfc (b,) 
 
 be the equations of the sides of the moving angle, and let 
 
 a?f + yi»=l ........ (c) 
 
 be the dual equation. Eliminating x and y between (a), (b), and 
 (c), we get 
 
 m=- , and also m,=- ' (d) 
 
 fl — V ' /*/— w 
 
 Let the revolving angle be 0; then tan^=.i '-; or, substituting 
 
 the preceding values of m and »w„ we obtain 
 
 t-.n ff - (/^/-M)g+ (X-\,)i;4- y-X/^; 
 
30 THE THEORY OF TANGENTIAL COOHDINATES 
 
 Should the two revolving lines coincide, 0=0, and the numerator 
 becomes ; or it becomes the tangential equation of a point, and 
 the point is the intersection of the two given lines. Or it becomes 
 when the denominator of (e) is infinite, or f = oo , i;= go , and the 
 limiting tangent passes through the origin. "When the angle is 
 a right angle, the denominator becomes ; but this expression is 
 the tangential equation of a conic section whose focus is at the 
 origin ; and the projective coordinates of the centre, y and 7,, are 
 given by the equations, 
 
 ' Wi + fifii " Wi + fifif 
 
 When the two fixed lines are parallel, and equidistant from 
 the origin, \,= — \, fj.i=—fi, and the denominator of (e) becomes 
 ^ + v^=\^+fi^, or the locus becomes a circle. 
 
 When the lines are at right angles, the constant term in the 
 denominator vanishes and the curve becomes a parabola, as we shall 
 show further on. 
 
 If in the equation (e) we substitute X or X, for J, and fj. or ^, 
 for V, we shall find the equation satisfied independently of ; hence 
 it follows that the fixed lines themselves are tangents to the locus. 
 
 Taking the polar of the above, we get the elementary proposition, 
 that if two lines, each passing through a fixed point, contain a con- 
 stant angle the locus vrill be a circle, since the primitive has its 
 focus at the origin. 
 
 This theorem gives, perhaps, the simplest method of describing 
 a conic section by means of a ruler. Let any point be assumed in 
 a plane, iu which let two straight lines be drawn. If a right angle 
 with sides of indefinite length be made to revolve round this point, 
 cutting the fixed lines always in two points, the line which always 
 joins these points vriU envelope a conic section, of which a focus is 
 at the origin. 
 
 41.] An angle of given magnitude revolves round a point in the 
 plane of a conic section, cutting the curve in two points ; the line 
 joining these points will envelope a curve whose tangential equation 
 is of the fourth order. 
 
 Let the projective equation of the conic section be 
 
 Aa;2^.A^2 + 2BiBy-|-2Car+2C,y=l, .... (a) 
 
 the vertex of the angle being taken as origin. 
 
 Let y,=mx, (b) 
 
 be the equation of one of the sides of the angle ; substituting and 
 dividing by x^, 
 
 A + A,»i2 + 2Bi» + 2(C + C,»w) 1 = ig ; 
 
APPLIED TO EXAMPLES. 31 
 
 but a.s x^+yv=l, —=^ + mv; eliminating x, we shall obtain the 
 
 resulting equation, arranging according to powers of m, 
 
 [A, + 2C,u-w2]m2 + 2[B + Ci;+C,^-fi;]m + A + 2Cf-r=0. (c) 
 
 Now the tangents of the angles which the revolving lines make 
 with the axes of X are the roots of this quadratic equation ; hence 
 
 ^ ,^ _ -2[B + C,g + Cv-gv] 
 
 and 
 
 A + 2 C g-g^ , , 
 
 ""'"""= A, + 2C,v-v^ (") 
 
 Tt is obvious that 
 
 tane=.'"'-'"" 
 
 1 + M^TM,^ 
 
 To obtain the value of m,—m,i, we must square (d) and subtract 
 4 (e) from it. 
 
 Hence, after some reductions, we shall find 
 
 + 2(BC,-CA,)0 + 2(BC-C,A)t;+B*-AAJ^ 
 divided by the coefficient of nu^, and 
 
 1 +m,m„= - [?5' + w2-2C?-2C,u-A-AJ 
 divided by the same coefficient of m^. Hence 
 tan e[r'+u2_2Cf-2C,u-A-A,] = [(C,« + A,)?«+ (C2 + A)i;«) 
 -2(B + CC,)fu + 2(BC^-CA,)f + 2(BC-C,A)u + B2-AAJi.J^ ' 
 
 If we now square both sides of this equation, the resulting for- 
 mula will be of the fourth degree of the tangential coordinates f 
 and V. 
 
 Without proceeding to this expansion, we may make two suppo- 
 sitions, which will lead to remarkable results If we suppose the 
 two sides of the revolving angle to approximate and finally to 
 coalesce, the line which joins their extremities will ultimately 
 become a tangent to the curve itself, and therefore exhibit its tan- 
 gential equation; but if tan 6=0, the second member of the pre- 
 ceding equation (f) becomes 0. 
 
 So that if kJ!'^ + k,y'^ + 2Bxy-\■2Cx + 2C^y = \, be the projective 
 equation of a conic section, the tangential equation of the same 
 section, referred to the same axes, wUl be 
 
 (A,-|-C,^)r + (A+C^)i'''-2(B+CC,)?v+2(BC,-A,C)?| 
 
 -)-2(BC-AC> + B2-AA;=0. j ^^' 
 
 Hence also, as in the general tangential equation of a conic section, 
 
 the halves of the coefficients of the linear terms are the projective 
 
32 THE THEORY OF TANGENTIAL COORDINATES 
 
 J- X .r .u ^ ^X. f BC,-A,C , BC-AC 
 
 coordinates of the centre ; thereiore . ' _'^ and . . _^^ are 
 
 the projective coordinates of the centre of the conic section, whose 
 projective equation is (a) — a result obtained by a very different 
 method in the theory of projective coordinates. 
 
 When 5 is a right angle, tan becomes infinite, the second side 
 of the equation vanishes by division, and the first member becomes 
 
 r , v^ 2Cg 2C,y _ 
 
 A+A, A+A, A + A, A+A,~ ' * * ^ > 
 
 the tangential equation of a conic section whose focus is at the 
 
 Q Q 
 
 origin, the coordinates of whose centre are . . and -j — j-. 
 
 When the angle is a right angle, the two branches of the curve 
 whose tangential equation is of the fourth degree coalesce, so to 
 speak, into one conic section. 
 
 42.] A right-angled triangle has its right angle at a focus of a 
 conic section, while the hypotenuse envelopes the curve ; one acute 
 angle of the triangle moves along a given straight line, the other will 
 describe a conic section. 
 
 Let the origin be at the focus. Then the tangential equation of 
 the given conic section will be 
 
 a[^+i^) + ihS+Zy,v=l (a) 
 
 Let px + gi/=l (b) 
 
 be the projective equation of the given straight line, and let 
 
 y=ma; (c) 
 
 be the equation of one of the sides of the right angle ; and as the 
 hypotenuse meets the straight line in the point {xy), we shall have 
 from the dual equation 
 
 x^+yv=l; (d) 
 
 eliminating w and y between (c), (b), and (d), we shall have 
 
 m=^P^ (e) 
 
 Now as the other side of the right-angled triangle is separated by 
 a right angle firom the former, we shall have 
 
 y,= — -' (f) 
 
 aiid xi£ + y,v==li (g) 
 
 eliminating m between (e) and (f), we obtain the resulting equation, 
 y,^-xiu=py,-gxi; 
 
APPLIED TO EXAMPLES. 33 
 
 combining this with the dual equation, x^ + yf)—!, we obtain 
 
 *~ */"+y," ' ^t+i/!' • • • • w 
 
 Now, if the hypotenuse were a fixed line, ^ and v would be con- 
 stant quantities, and from the last equation we might determine 
 the corresponding values of le, and y,. 
 
 Let us assume that f and v are connected by a linear equation 
 such as Pf+Qu=l. 
 
 Substituting in this equation the preceding values of ^ and u, the 
 resulting projective equation becomes 
 
 'Px + Q.y-i-(Py-Qia;)(py-qa!)=x^+y^, . . . (i) 
 
 the equation of a conic section passing through the origin ; hence, 
 if a right-angled triangle revolve round a given point, and one angle 
 move along a given straight line while the hypotenuse passes 
 through a fixed point, the other angle will describe a conic section 
 passing through the origin. 
 
 Again, let us assume that and v are the tangential coordinates 
 of the limiting tangent to the curve. Substituting in (a) the values 
 off and V given in (h), we shall find, after some reductions, 
 
 [l + 2y,p— aq^] ai^+[l + 2yq - a/>«] y'^ + 2 [apq -yp— y,q'\ xy 
 —2yx—2y!y=a, 
 
 the projective equation of a conic section. 
 
 43.] Assume the tangential equation of a conic section referred 
 to its axes as axes of coordinates, namely 
 
 c202 + JV=l (a) 
 
 Let the axes of coordinates be conceived tobe^r*^ turned round 
 through an angle 6, and then translated, in parallel directions, to 
 a point whose projective coordinates are —p and —q. The formulse 
 by which this double translation is made are given in (4), and are 
 
 1 -/?/-?"/ ' "~ l-K;-?"/ ■ • • • n 
 Substituting these values in the original equation, we get 
 
 [o2 cos2 e + b^ siv? -p^] ^ + [o2 sin* d+b^ cos* 9 - g*] v«) 
 
 + 2[{a^-b^)sm0cos0-pq'\^v + 2pS + 2qv = l. ) ^'^' 
 
 This is the tangential equation of the conic section referred to a 
 new origin and other rectangular axes. Hitherto no relation has 
 been assumed as existing between p, q, and ; but if wc assume 
 
 tan 0=?-, and p^ + q^ = a^-b^, . . . . (d) 
 
34 
 
 THE THEORY OF TANGENTIAL COORDINATES 
 
 (e) 
 
 and substitute these expressions in the preceding equation, we shall 
 obtain 
 
 a* cos^ e + b^ sin^ 6 -p^= 6% a^ sin^ d + b^ cos^ e-q^= b^, 
 and 2[(a2-6«) sin 6 cos O-pg] =0. 
 Heuce the preceding equation is reduced to 
 
 b^^ + v^)+2p^ + 2qv=l (f) 
 
 Now this is exactly the form of the tangential equation of the 
 conic section when the focus is the origin of coordinates. We may 
 hence infer that if a concentric circle be described passing through 
 the foci of the ellipse, and any point on this circle be taken as that 
 round which the right angle revolves, we shall have the same results 
 as if the focus had been selected. Let 6 be the angle through which 
 the first system of rectangular coordinates is turned, the radius of 
 the circle being 
 
 y/a^—b^, then p= tja^—b^ cos 6, and q=: Va^—b* . sin 0. 
 
 There is no difficulty in making this construction. Construct the 
 
 Fig. 11. 
 
 focal circle. Let the coordinates be turned through the angle 0, 
 make the angle O, O Y,=X O Y„ and draw O, X„ parallel with O X, ; 
 0/X„, OiY„ will be the new system of coordinates, and O^T=jb, 
 01 =q. A point taken anywhere in this circle will enjoy the 
 tangential properties of the focus. 
 
 44.] A triangle is inscribed in a conic section; two of its sides 
 always pass through two fixed points, the third side envelopes a conic 
 section. 
 
APPLIED TO EXAMPLES. 35 
 
 Let the line joining the fixed points be taken as the axis of Y, 
 
 Fig. 12. 
 
 and the diameter conjugate to it as the axis of X. Let the distances 
 of the fixed points to the origin be h and h,. 
 
 Let the projective equation of the conic section be 
 
 Aa^ + K,y^ + 2Cx=\; (a) 
 
 and as the line [xy) {x,yi) passes through the point {x, y), we shall 
 have the dual equation 
 
 a;f + yw=l (b) 
 
 Eliminating x and y successively between these two equations, and 
 putting 
 
 M=(A + C*)v«+A^-2CA^-AA„ . . . (c) 
 we shall have 
 
 _A|-Cy^--WM 
 '^~ A^+Aw* ' 
 
 _A^f--Ci^+jM/M 
 "'' A^ + Aw* ' 
 
 y- 
 
 Av + C ug + g VM 
 ' A^ + Aw* ' 
 
 Au + Cug-g V M 
 ^'~ A^ + Au* ■ 
 
 (d) 
 
 The signs of the radical VM must be so assumed as to fulfil the 
 conditions 
 
 {x,-x)i+[y,—y)v=0, and x^-\-yv = l. . . . (e) 
 We have also 
 
 and 
 
 
 ,,„_ ./(g-C)-f V M. ,.^_ ./(g-C)- VM 
 
 y ' — A,r+Au«"""' y^ A,r+Aw« ■ 
 
 (f) 
 
 D 2 
 
36 THE THEORY OF TANGENTIAL COOKDINATES 
 
 Now, as the point [u, t) is on the conic section, we shall have 
 A^*+A,M« + 2C<=1, and also Aa!^ + Aff^ + 2Cx=l. . (g) 
 Subtracting the latter from the former, and dividing by {t—x), we 
 shall have 
 
 A{t+a;) +A,{u+y)(j=^) +2C = 0; 
 
 and as this line passes through the fixed points of which the coor- 
 dinates are y=h, w=0, we shall have 
 
 \ X / t—ai 
 Eliminating u from the preceding equations, we shall have for the 
 value of t, 
 
 x{Ap-\) „> 
 
 l-2Ca?-'2A,%-|-A,&2 v; 
 
 In the same way we shall obtain for the other side of the triangle, 
 passing through the points («, () and (y, a?;), 
 
 a^,(A,V-l) . 
 
 l-2C^,-2A^y< + A,A« ^' 
 
 Comparing these two values of t, we shall have, all necessary reduc- 
 tions made, the following equation : 
 
 A,(A«-V) [x + a:^-2Ca?j?J +(A,*A«A«-l)(a7-a^,)l , .. 
 
 = 2k}l^[kf^^-\)y^x-2Afl{A}lf-Y)yx,. J" ' ^"'' 
 Substituting the values of x, y, x,, and y^ given in the preceding 
 formulae (d), we shall obtain the resulting condition 
 
 A,{h^-h^ (f-C) - {A,^h%'-l)v y/M + A,h{Afi^-l)v{^-C)^ 
 
 -1-A^A(A,V-1) s/M + Afi,{A,h'-l) VM li^) 
 
 -AA(A,A'-l)«(f-C)=0; J 
 
 or reducing, 
 
 A,{h-h,)lA,{h + h,)-{Afih,+ l)v-]{^-C) 
 
 = {Afih,-l)lA,(h + h;)-{A/ik,+ l)v-]^/M, 
 or, eliminating the common factor between the brackets, 
 
 A,{h-hi){^-C) = {Afih,-l) VM. 
 Now, substituting for M its value given in (c), namely 
 
 M = (A -1- C«) i;« -H A,?«-2CA,?- AA;, 
 we shall have 
 
 v« (A^«-1)(AA'-1) .> p.,_, 
 a/(A,M-1)*(A+C«) ^«~^^ -^' • • • (1) 
 the tangential equation of a conic section. 
 Now, if we invert the preceding demonstration step by step, 
 
APPLIED TO EXAMPLES. 37 
 
 substituting tangential coordinates for projective coordinates, and 
 reciprocaUy, we shall establish the reciprocal theorem, that if a 
 triangle be circumscribed to a conic section, two of its angles always 
 resting on fixed straight lines, the third angle will describe a conic 
 section. 
 
 45.] A series of central conic sections having the same centre, and 
 their axes in the same direction, but such that the differences of the 
 reciprocals of the squares of their axes is constant, the tangents 
 drawn to a point on each, their intersection ivith a common diameter, 
 envelope a concentric hyperbola, if the intersected curves be a series 
 of ellipses, and an ellipse if the intersected curves be hyperbolas. 
 
 Let a«f9 + 6V=l (a) 
 
 be the tangential equation of one of the curves ; let y=nx . (b) 
 
 be the equation of one of the diameters; and let ^=-3 + p (c) 
 
 be the relation of the semiaxes. Then 
 
 a^^=x, . . . (d) b%=y (d,) 
 
 Between the five equations (a), (b), (c), (d), and (d^) elimiuating 
 a, b, X, and y, we get for the equation of the envelope 
 
 hHv^-^+h^(l=^^v = l (e) 
 
 When the angle which the common diameter makes with the axis of X 
 is half a right angle, m = 1, and the equation becomes A* (u*— P) = 1, 
 the equation of an equilateral hyperbola referred to its axes. 
 
 In any series of concentric conic sections, in which 70 =-5 + 70. we 
 ■^ 0* o* A* 
 
 shall have h^=-^—r~=-~, or all the conic sections will have the 
 a*— 6' e* 
 
 same Minor Directrices. 
 
 Curves and curved surfaces having the same minor directrices, 
 
 or minor directrix planes possess properties reciprocal to those of 
 
 confocal curves and confocal sui'faces, as shall be shown further on*. 
 
 * It may be instructive to compare, with the brevity and simplicity of the pre- 
 ceding solution, the ordinary method by which questions of this kind are solved. 
 The equation of the tangent through the point x, y, is 
 
 '^i . y, -I 11.1 
 
 -,.^+^,, = 1, oraSj-,=-,+p; 
 
 X fix fix 
 it becomes, since y,=nx,, ^^+"^''J'+ ^a y=l' 
 
 Eliminating x, between this equation and that of the curve, we find 
 
 Vx+ny i 
 
 L a' 
 for the equation of the tangent. 
 
 Eliminating a between the two equations V=0, and jt=0, we obtain, after 
 
 
38 THE THEORY OK TANGENTIAL COORDINATES 
 
 On Polygons inscribed and circumscribed to Conic Sections. 
 
 46.] There is a large class of questions haying reference to poly- 
 gons inscribed and circwOTScribed to curves and also to curved sur- 
 faces, which in many cases may be very simply treated. Questions 
 relating to polygons inscribed in conic sections must be solved by 
 the ordinary procedure of projective coordinates j while those which 
 have reference to circumscribed polygons must be investigated by 
 the help of the methods and formu^ of tangential coordinates^ as 
 explained in the preceding pages. As the properties of space are 
 dual, so must the methods of investigation be dual also. It would 
 be bootless to apply tangential coordinates to the investigation of 
 the properties of inscribed polygons, and equally futile to use pro- 
 jective coordinates in the discussion of circwmscribed polygons. It. 
 is further to be observed that in this class of questions the variables, 
 such as as and ^ or ^ and v, become given constants, while the usual 
 coeflBcients A, A„ &c. or a, «,, &c. become unknown but determi- 
 nate quantities of the first order. Thus if it were required to show 
 that a conic section may be determined by five given points 
 through which it is to pass, or by five given straight lines which it 
 is required to touch, the variables x and y or f and v in the formal 
 equations of the curve A^+Aff^+2Ba!y+2Cx+2Ciy—l, or 
 «l* + a,i'*4-2(8fu + 2yf + 2y,u=l, become given constants, while 
 the coefficients become the unknown quantities ; and as there are 
 five of them, there must be five equations to determine them, and 
 therefore there must be five sets of values of x, y or f, v. And as 
 all the unknown coefficients, such as A, B, C or a, P, y, are linear, 
 the equations by which their values will be ultimately determined 
 are linear also. Hence they are always real, though their values 
 may sometimes be or oc, as the given values of x and y or | and v 
 may turn out. 
 
 We shall apply the method to one or two examples. 
 
 Conic sections are inscribed in the same qvadrilateral, the polar of 
 any point in their plane envelopes a conic section. 
 
 The fixed point being assumed as origin of coordinates, let the 
 tangential equation of one of the sections be 
 
 a|« + o,w«+2/3|i; + 27|:-|-2y,u=l (a) 
 
 The equations of the polar of the origin are, see (27), 
 
 af+j8u-|-7=0 . . (b) and a,v+^^+y,=^0. . . (b,) 
 
 some consideiable reductions, 
 
 /-..^(lz=.>.?(^ (O 
 
 the projective equation of the locus. 
 
 The elimination of a between V= 0, and ^=0, is frequently a matter of great 
 complexity, and is often quite impracticable. 
 
APPLIED TO EXAMPLES. 39 
 
 Now there are four linear equations under the form (a) to deter- 
 mine the five unknown quantities a, a,, /8, y, y,; we may then 
 eliminate any three, and connect the fourth and fifth by a linear 
 equation. Eliminating then a, a^ y, y,, three by three successively, 
 we shaU have 
 
 a = K;8+L, a,=K,0 + l,„ 1 ,. 
 
 7=M/3 + N, y,=M,/8-|-N,, J • ■ ■ ■ ^ > 
 
 where K, L, M, N, K,, li„ M„ N, are known fdnctions of the 
 constant tangential coordinates of the four given straight lines. 
 Substituting these values in (b), (b^), we shall have 
 
 (Kr+« + M)/3+(Lf+N)=0, 1 ,^. 
 
 (K,i; + |+M,)/S+(V + N,)=0;J' ' ■ ' ^' 
 
 eliminating /8 from those equations, we obtain the tangential equa- 
 tion of a conic section . 
 
 When the point chosen is at one of the angles of the quadrilateral, 
 the section becomes a point. 
 
 The origin being placed at this point, as two of the sides of the 
 quadrilateral which are tangents to the curve pass through it, we 
 
 shall have at the origin for one of the lines ^ = 0, - = 0. Let ^s=nv, 
 
 n being the tangent which one of these lines makes with the axis 
 of X. In the general equation, substituting nv for ^, we find 
 
 a»2-l-a,42/8« + 2(yre+y,)-=-2, oras -=0, an^ + a,+2fin=0. 
 
 In like manner for the other tangent, we shall find 
 ani^ + a, + 2^ni=0. 
 Eliminating successively a^ and a between these equations, we 
 shaUfind a = K/3, a,=Kfi. 
 
 Substituting these values of a and a, in (a), the result becomes 
 a=Kl3. a,=Kfi, 
 
 7=M/3-t-N, y,=M,0+-S,; 
 
 hence eliminating ;8, having substituted the preceding values of a 
 and a„ y and y, in the equations of the polar of the origin, (b) and 
 
 the tangential equation of a point. 
 
 47.] A series of conic sections are inscribed in the same quadri- 
 lateral, their centres range on the same straight line. 
 
 From the four tangential equations of the sides of the quadri- 
 lateral we may eliminate three of the five unknown constants a, a,, 
 
40 THEORY OF TANGENTIAL COOKDINATES APPLIED TO EXAMPLES. 
 
 /3, y, and 7^ ; eliminating the three former, we shall obtain a linear 
 resulting equation in 7 and y^ namely 
 
 L7+M7^+N=0, (a) 
 
 the projective equation of a straight line, the coeflBcients 7 and y^ 
 being, as shown in (14), the projective coordinates of the centre ; 
 therefore the centres of all these inscribed curves will be found on 
 the same straight line. 
 
 48.] The method of tangential coordinates supplies a short and 
 simple demonstration of a theorem of some difBculty and much 
 celebrity, due to Newton, that, 
 Tke centres of conic sections in- 
 scribed in the same quadrilateral 
 all range on the straight line which 
 joins the points of bisection of the 
 two diagonals of the quadrilateral. 
 
 Let O A Q. B be the quadrila- 
 teral in which a conic section is 
 inscribed. Let A B and O Q be 
 the diagonals, the line which joins 
 their middle points will pass 
 through the centre of the curve. 
 
 Let the tangential equation of the conic section be 
 
 af«+a,w« + 2)8|i;+2y| + 27,u=l (a) 
 
 Now if O be taken as origin, and O A, O B as oblique axes, then 
 as the curve touches the axis of X, a^=0, and as the curve touches 
 the axis of y, o=0, and thus the tangential equation of the curve 
 is reduced to 
 
 2/3^1/ +2y?+2y,u=l (b) 
 
 Let OA=a, 0B=&, OC=c, OD =d; then, as B C is a tangent 
 
 to the curve, O B=A=-, OC=c=j; hence, substituting in the 
 
 equation (b). 
 
 ^ + V+^-1' °^ 2/3 + 267 + 2c7,=cA, 
 
 cb 
 
 (c) 
 
 in the same way, as AD is a tangent to the curve, rf=-, a=-^; 
 
 hence substituting, 
 
 2$+2dy + 2ay,= ad (d) 
 
 Subtracting (d) from (c), 
 
 2{b—d)y + 2(c—a)y,=cb—ad. . . . (e) 
 
 Now y and y, in the general tangential equation always denote 
 the projective coordinates of the centre. 
 
(f) 
 
 ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 41 
 
 Again, the coordinates of the middle point of O Q axe y,=a, 
 _a 
 
 The equation of the line C B is — h^=l, j 
 
 and the equation of the line AD is - + ^=1, 
 
 a a \ 
 
 and therefore the coordinates of the point Q, their intersection, are 
 
 -_ac{b—d) -_bd(a — c) 
 
 ~ ab—cd' ^~ ab — cd' 
 
 while the coordinates of the middle point of Q are -, ^. 
 
 Now the equation of a straight line passing through two given 
 
 -^', we s 
 
 bd{a — c) 
 
 points heing -f- — ^=^ — ^, we shall have 
 
 or 
 
 or 
 
 b 2{ab — cd) 2 / _a\ 
 y 2~ ac{b-d) a^ V' 
 2{ab-cd) 2 
 
 b d—bf a\ 
 
 2(b—d)x-{-2{c—a)y = cb—ad, 
 which is identically the same as (e). 
 
 The above proof, it will be seen, rests on the simplest elementary 
 principles. 
 
 CHAPTER IV. 
 
 ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 
 
 49.] Let the focus and axis of the parabola be taken as the origin 
 and axis of X. The projective equation of the parabola, referred to 
 its focus, is 
 
 x^ + y^={2k + xf (a) 
 
 Now the tangent to the curve drawn through the point x^, on the 
 
 curve 
 
 being - — — = — , we shall have x=-z when «=0, and v=- 
 
 whena;=0; hence a?, = ^- — , y, — — -y-- 
 
 Making these substitutions in the projective equation (a), we 
 
42 ON THE TANGENTIAL EQUATIONS OF THE FAKABOLA. 
 
 shall have for the tangential equation of the parabola, one fourth 
 of the parameter being equal to k, and the origin at the focus, 
 
 k^ + k^+^=0; (b) 
 
 in this equation there is no absolute term. 
 
 The tangential equation is satisfied by f =0, i;=0 ; for the para- 
 bola admits a tangent at infinity. 
 
 Let the axes of coordinates now be conceived to turn round the 
 origin through the angle 0, and then translated, in parallel direc- 
 tions, to a point of which the coordinates are —p and —q; then 
 using the formulae given in (4) for the transformation of coordinates, 
 namely 
 
 (._ cos ff.fi— sin 0.V, _ain 6.^1+ cos 0.Vi 
 
 ^-Pl-gv, ' " i-p^-qv, 
 
 Substituting these values in (b), the general tangential equation 
 of the parabola becomes 
 
 {k -p cos 0)^+[k-g sin d)v^ \ 
 
 — {psm0+gcos0)^v + cos0.^+sm0.v=O.)' ' 
 Assuming the most general form of the tangential equation of 
 the parabola, let us suppose 
 
 fS^+fiV^+ffSv + hUh,v=0 (d) 
 
 Now, as the absolute term no longer afibrds a guide in comparing 
 the general form of the tangential equation of the parabola with 
 that derived by the transformation of coordinates, and as in this 
 latter form the coefficients of f and v are cos and sin 0, we must 
 reduce the co efficient s of f and v to the same form as in (c) ; hence 
 dividing by >^h^-{-h^, the general form becomes 
 
 s/h^+h^^ s/h^+hy y/h^+k,^^ v/r*+v+ vfhFv^"' ^^^ 
 
 hence 
 
 k-pcoa0= / k-qs\a0= , ^^ 
 
 Bsin ^ + ff cos^= — ■ 3 
 cos0=—-J=, sm0=—^k=. 
 
 or, reducing, 
 
 p_ {f,-f)h-ffh, {f-f,)h,-gh 
 
 (f) 
 
 1 
 
 (g) 
 
ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 43 
 
 Let D be the distance of the focus from the origin. 
 
 Hence, -when f=fi and ^'=0, D=0, or the origin is at the focus. 
 
 When the origin is at the vertex of the parabola, and its axis 
 coincides with the axis of X, the tangential equation becomes 
 
 kv^+^=0 (i) 
 
 We may also express h and A, in terms of ^ and q. Solving (g), 
 
 t,. ^{fi-f)-ff9 f, _ 9{f-f,) -ffP ^•^ 
 
 p^-hq^ ' '" p^ + q^ ■ • ■ ■ ^i> 
 
 In the projective equation of a conic section, when the absolute 
 term is 0, the origin of coordinates is on the curve j while in tan- 
 gential coordinates, when the absolute term is 0, the curve is the 
 parabola. Again,in projective coordinates the condition AAj—B^=0 
 indicates that the curve is a parabola ; so in tangential coordinates 
 the condition aa,—^=0 indicates that the origin is on the curve, 
 as shown in (22). 
 
 50.] In the tangential equation of the parabola /, is =0 when 
 the curve touches the axis of X, and /is =0 when the curve touches 
 the axis of Y. 
 
 To determine the distance of the point of contact on the axis of 
 X from the origin. Let the equation of the curve divided by v be 
 
 ■^+ffS+^+'',^o OT ijr+h^l+ff^+h^o. . (k) 
 
 Now, as -=0, the preceding equation is reduced to g^ + ki=0; 
 
 in the same way, ffv + h=0, gives the distance when the axis Y 
 touches the curve. 
 
 Hence the distances from the origin of the points at which the 
 axes of X and Y touch the parabola are given by the equations 
 
 g^ + h,=0, wad gv+h=0 (1) 
 
 51.] We shall apply this theory to a few examples. 
 The sum of the sides of a right angle is constant. The hypotenuse 
 envelopes a parabola. 
 
 By the terms of the proposition ^-|--=c, or— c.^u + ^+i;=0, 
 
 the equation of a parabola, since the absolute term is =0. 
 
 Comparing this equation with the general tangential equation of 
 the parabola, wc get _ 
 
 f=0,f,=0, g=-c, k=l, //(=1, and^=^^=-j-. 
 
44 ON THE TANGENTIAL EQUATIONS OP THE PARABOLA. 
 
 The locus is a parabola whose parameter is V2c, which touches 
 both sides of the right angle^ and whose axis bisects the right augle. 
 
 52.] An angle of given magnitude moves along a fixed straight 
 line, one side always passes throvgh a fi^ed point, the other side will 
 envelope a parabola. 
 
 Let the fixed line O B and the perpendicular a from the fixed 
 point C be taken as axes of coor- 
 dinates, let the tangent of the *^'S- •'•^• 
 constant angle at B be m ; then, 
 by the conditions of the question, b 
 
 V _m' 
 av 
 or, reducing, mav^ — af u + m^ + u = 0, 
 
 the equation of a parabola, as the absolute term is 0. 
 
 Comparing the terms of this equation with those of the general 
 
 form (c) in [49], we shall have /=0, f,=ma, g= —a, h=m, A,=l; 
 
 hence 
 
 ^ n ^ I '"^o, a n 
 
 tan0=— , k= — -===aco%a, p=a, o=0. 
 
 m' Vl + n*^ 
 
 53.] Parabolas are inscribed in a triangle; the locus of their foci 
 is the circumscribing circle. 
 
 Let the base of the triangle be taken as the axis of X, the origin 
 being placed at an angle of the triangle, n being the tangent of the 
 angle which the second side, passing through the origin, makes with 
 the axis of X. 
 
 Let the base of the triangle be a, and let the two other sides 
 make, with the base, angles whose tangents are n and m, the former 
 line passing through the origin. 
 
 As the base a measured along the axis of X is a tangent to the 
 curve, /,=0; and as ^=nv, the general equation (d) becomes 
 
 {fn^ +gn}v^ + hnv + hf) =0 ; 
 and as at the origin, -=0, fn-\-g=0, or g= —fn. 
 Hence the general equation now becomes 
 
 f^^-fn^v + h^ + h,v=0 (a) 
 
 Again, the tangential coordinates of the third side of the tri- 
 angle are ?=-, v= — . Substituting these values in the last equa- 
 tion, we shall have 
 
 fm—fn + ham + hfl=0 (b) 
 
ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 45 
 
 Substituting in this latter equation the values of h and h, given 
 in (j), first making /,=0, g= —fn, 
 
 ,_ fnq-pf ,_ qf+fnp ,. 
 
 "-p^ + q^' ^'-^^TF" ^' 
 
 Substituting these values in the preceding equation, we get 
 
 f^_j^ + ''«'f^-'^rnpf+aqf+afnp ^^ . . . . (d) 
 
 or dividing by / and {m—n), we obtain the final resultj 
 
 ^2 + g2-flp + fl^(i±^=0 (e) 
 
 ■* ^ ^ ^ m — n ^ ' 
 
 Now, as p and q are the projective coordinates of the focus in the 
 general equation of the parabola, and as they are in this equation 
 the variables of the projective equation of a circle, it follows that 
 the locus of the foci of the inscribed parabolas is the circle circum- 
 scribing the triangle. 
 
 54.] An angle of given magnitude revolves round the focus of a 
 parabola, to determine the curve enveloped by the cord which joins 
 the points in which the parabola is intersected by the sides of the 
 angle. 
 
 Let y*+jj?'=(2o+a?)* be the projective equation of the parabola 
 referred to its focus as origin, and having the axes of coordinates 
 parallel with those of the curve. 
 
 Let y=na! be the equation of one of the lines, then 
 
 v'l + re*=lH ,and -=^+nv; 
 
 eliminating x, 
 
 (4flV— l)w2+4ai;(H-2af)M + 4of(l + af)=0. . . (a) 
 Now the roots of this equation are 
 
 4av(l + 2flf) 4flf(l+«f) „, 
 
 "' + "" 4aV-l ' '''''"= 4aV-l ' ' " ^^> 
 
 hence 
 and 
 Hence 
 and 
 
 64a3|i;g + 64a''|^i;g- 16ag- \Qa^^ 
 '*'"" (4aV-l)« 
 
 ni-n„- 4aV-l ' 
 
 4oV+4a*5«+4af-l 
 1 +«/»«= 4aV-l ' 
 
46 ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 
 
 consequently 
 
 n-n„ _ 4{a«r+«V+agH ,^s 
 
 l+«,M„~4(a2f + aV+a^-l ^> 
 
 Let the expression under the radical be put M, and as ^r — ~ 
 is equal to the tangent of the revolving angle 6, we shall have 
 
 tan^=l^; 
 4M-1 
 
 reducing and solving for M, we shall find, taking the upper sign, 
 
 
 
 M=cot2jr; taking the lower sign, M=tan*5j, 
 
 or, substituting for M its value, 
 
 o2f' + aV + ffle=cot2|, or a«r + flV + a|=tan«|. 
 
 a a 
 
 When is a right angle, tan-=cot h=1j and the two curves 
 
 // fit 
 
 coincide. Its equation then becomes b*|^ + a*!/* + f = 0. 
 
 Since the coefficients of f* and u* are equal, and the rectangle 
 
 disappears, the origin must be at the focus of the curve. Hence 
 
 the loci and the parabola have the same foci. 
 
 We may determine the axis and eccentricity of the two loci as 
 
 follows. While the tangential equation of one section is 
 
 a 
 
 a^^+a^i^ + a^=cot^^, that of the other is 
 
 fit 
 
 a 
 
 Let tan*s=^*- Multiplying the first equation by ^, we get 
 
 Now let A and B be the semiaxes of this curve, and, comparing the 
 coefficients of this equation with the general equation 
 
 a^ + af? + %^%v + 2y| + ^p = 1, 
 we shall have 
 
 a=a't\ a,=a'i^, 7=^, /8=0, y,=0. 
 
 Substituting these values in (19), the formula for finding the axes, 
 we shall have 
 
 2A?=2a^t^+^±\/^ or 2A«=2a«<«+?^, 
 
 or 
 
 4A«=4a2^« + fl2/4 and 4B«=4a«<2; 
 
ON THE TANGKNTIAL EQUATIONS OP THB PARABOLA. 47 
 
 hence — ^ — "IXT*"^'^' ^ *^^ same way we may find, for the 
 eccentricity of the other ellipse, 
 
 or 
 
 cos*^ sin* ^ 
 
 When the two curves coincide, or when 5 is a right angle, the 
 
 eccentricity of the two coinciding loci will be found to be — -= ; 
 
 V 5 
 
 hence, naming the eccentricity c, we shall have 
 
 1_^ sin^ 
 tS /.a "•" /.a ■ 
 
 55.] It is almost needless to observe that, as in the case of pro- 
 jective coordinates, analogous formuUe may be established when 
 the axes of coordinates are oblique, by reasonings precisely similar. 
 A single application of such formtdse may suffice. 
 
 A series of parabolas are inscribed in the same triangle, the line 
 joining the points of contact of each parabola loith the opposite ver- 
 tices of the triangle meet in a point. The locus of this point is the 
 minimum ellipse circumscribing the triangle. 
 
 Let the sides a, b of the triangle be tfdcen as axes of coordinates, 
 then, in the general equation of the parabola 
 
 f^+f,v^+9Sv+h^+h,v=0, (a) 
 
 as the curve touches the axis of X, /,=0 ; and as it touches the axis 
 of Y, /=0. Hence this equation becomes 
 
 gSv + h^ + h,v=Oi (b) 
 
 and as the third side of the triangle is a tangent also to the para- 
 bola, its coordinates being f =-, v=-t, the preceding equation be- 
 
 G O 
 
 comes 
 
 g + hb + h,a=0 (c) 
 
 The parabola touches the axis of X, see (1) in [50] , at a distance 
 
 from the origin =^^, and the axis of Y at the distance -^. The 
 
 projective equations of the straight lines joining these points with 
 
48 ON THE TANGENTIAL EQUATIONS OF THE PARABOLA. 
 
 the opposite vertices, namely y=0, and x=-^, and a;=0, y=6, 
 are 
 
 % — -'^=1, and w=l (d) 
 
 h g a g" 
 
 Taking these as simultaneous equations, the values of w and y 
 
 determined on this supposition are the coordinates of the common 
 
 point. Substituting the values of h and h, derived from these 
 
 equations in the formula g-\-hA + hfl,=0, we get 
 
 aY + b^i«'^ + aba!y-aHy-ab^x=0, . . . . (e) 
 
 the projective equation of an ellipse circumscribing the given tri- 
 angle ; for this equation is satisfied by the three sets of values 
 
 y=0, y=b, y=0, 
 
 x=0, a?=0, x-=a, 
 
 the coordinates of the three vertices of the triangle. 
 
 Let the origin be translated to the centre of gravity of the tri- 
 angle, then iT = a?, + 5, y = y ( -I- K> and the resulting equation becomes 
 o o 
 
 Now, that the ellipse circumscribing the triangle is a minimum 
 when its centre coincides with the centre of gravity of the triangle 
 may be thus simply shown. Let a circle be circumscribed to an 
 eqmlateral triangle, it will have its centre in the centre of gravity 
 of this triangle. On the circle and inscribed equilateral triangle 
 let a right cylinder and right prism be erected, and let them be cut 
 by any inclined plane; this plane will cut the cylinder in an ellipse 
 and the inscribed prism in a triangle, which will be inscribed in the 
 ellipse, and every line that is bisected or dirided in any given ratio 
 in the equilateral triangle or circle will have its projection similarly 
 divided in the triangle and its circumscribing ellipse, and as the circle 
 is the orthogonal projection of the ellipse, while the equilateral 
 triangle is the projection of the triangle inscribed in the ellipse, 
 and which has its centre of gravity in the axis of the cylinder and 
 therefore in the centre of the ellipse, it follows that the minimum 
 ellipse circumscribing a triangle has its centre in the centre of 
 gravity of the triangle. 
 
THE POINT, PLANE, AND STRAIGHT LINE. 49 
 
 CHAPTER V. 
 
 ON THE TANGENTIAL EQUATIONS OP THE POINT, THE PLANE, AND THE 
 STRAIGHT LINE IN SPACE. 
 
 Throttghoitt the preceding Chapters, the point and the straight 
 line have been considered as pole and polar. This is but a partial 
 and inadequate conception, because, in the complete duality of the 
 properties of space, the point, the straight line, and the plane are 
 the polars of the plane, the straight line, and the point. When a 
 curve or a rectilinear figure is given in the same plane with the centre 
 of the polarizing sphere, the reciprocal polar is neither a curve nor a 
 rectilinear figure, but a cylinder or prism standing at right angles 
 to the plane ; and when the centre of the polarizing sphere is not in 
 the plane of the given curve or other figure, the reciprocal polar is a 
 cone or pyramid whose vertex is the pole of the given plane. When 
 discussing the properties of figures in piano or in the geometry of 
 two dimensions as it is called, we leave out of consideration the 
 cylindrical surfaces, the polars of the curves described on the given 
 plane, and deal only with their bases in that plane. 
 
 As a point and a plane may be pole and polar one to the other, 
 so may a straight line be a reciprocal polar to a straight line. Such 
 a line may easily be found ; for, assume any two points in the given 
 straight line, the polar planes of these two points will intersect in 
 a straight line which is the reciprocal polar of the former, and the 
 planes drawn &om the centre of the polarizing sphere through 
 these straight lines will be at right angles, one to the other, and the 
 polar plane of any point assumed on one of the straight lines will 
 pass through the other, as we shall show further on. 
 
 On the Tangential Equations of a Point and Plane in Space. 
 
 56.] Let^, q, r be the projective coordinates of a point on three 
 rectangular planes. The tangential equation of the point is 
 
 p^+qv + r^=l (a) 
 
 The tangential equations of a fixed plane are 
 
 f= constant, ii= constant, f= constant. . . (b) 
 
 On the Transformations of Tangential Coordinates in Space. 
 
 Let three rectangular axes, O X, O Y, O Z, be drawn through a 
 fixed point O meeting a given plane in three points j let the reci- 
 procals of the distances of these points from the origin be denoted 
 by f, V, f J let the reciprocals of the distances of the corresponding 
 points for three other rectangular axes passing through the same 
 
50 
 
 ON THE TANGENTIAL EQUATIONS OF THE POINT, 
 
 Eg. 15. 
 
 origin and meeting the same plane be denoted by f „ v„ g. Let the 
 axis of X, make with the ori- 
 ginal axes the angles X, /it, v; 
 and let the axes of Y, and Z, 
 make with the same axes the 
 angles \„ fi„ v,; X,,, /i«, "«. re- 
 spectively. We are reqmred 
 to express f , v, ? in terms of 
 
 f/'ii' ft- „ . 1 . 
 
 We shall previously give an 
 
 expression for the cosine of the 
 angle contained between two 
 lines drawn from the origin. 
 Let r and r, be the lines, ij) the 
 angle between them, D the 
 distance between their extre- 
 mities, X, fi, V, X„ ft,, V, the 
 angles they make with the axes 
 of coordinates. Then 
 
 and 
 
 or 
 
 but 
 
 D'=r* +r^—2rr, cosf , 
 
 Equating the values of D, 
 
 ^ rr, 
 
 yy ZZ, 
 
 ^^-1 ' =■ cosXcosXj + cos/icos/i, -H cosvcosi',. 
 
 (c) 
 
 When ^ is a right angle, 
 
 cosX cosX^+cos/A cos /i^+cos V cosv^=0; 
 
 and when «^=0, 
 
 cos* X + cos' yu -I- cos' c = 1. 
 
 Let P be the perpendicular let fall from the origin on the given 
 limiting plane, then the angles which the axis of X makes with the 
 three new axes are X, X,, X^,, and the cosines of the angles which the 
 perpendicular P makes with the same three axes are Pf,, Pw^ Pf„ 
 while P^ is the cosine of the angle between P and X ; hence from (c) 
 
 Pf =Pf ; cos X + Pwj cos X^ -I- P^ cos X„, 
 
 or, dividing by the common factor P, 
 
 = f , cos X + u, cos X^ + %^ cos \^. 
 
 By the help of the following formulae we may express the values 
 of the original coordinates in terms of those derived from them ; thus 
 
THE PLANE, AND THE STKAIGHT LINE IN SPACE. 51 
 
 f =coe \.f ( + COS Xf-Vf + COS Xy/.i!"/, ) 
 
 w=cos/*.f( + cos/i,.Vj + cos/i^^.5,>- . . . . (d) 
 
 f=C0S V.f, + COS V,.W( + COS V/y.5).) 
 
 Square these equations and add them, bearing in mind that 
 |' + w* + 5'=0,* + w,* + 5'S, since each is the value of the square of 
 the reciprocal of the same perpendicular P, let fall from the origin 
 on the plane whose tangential coordinates are ^, v, ^, and also £,, t/,, ^f. 
 
 Hence cos* X+cos* /* + cos* v=l, cos* \, + cos* /i,+cos^ v,= l, 
 cos* \„ + cos* fill + cos* v„ = 1, and 
 
 cos X, cos \i + cos fi cos fif + cos V cos v, =0, ) 
 
 cos X cos X„ + cos /I cos n„ + cos V cos v„= 0, I" . . (e) 
 
 cos X, cos \i, + cos ^;Cos fifi + cos V, cos v„=0. ) 
 
 Again, since X, /i, v are the angles which the axis of X, makes 
 with the origiaal axes, and since P makes with the same axes angles 
 whose cosiaes are Pf , Pw, Pf, we shall have for the cosine of the 
 angle between P and X^ the expression Pf , ; hence 
 
 Pfj=Pf cosX + Pwcos/*+Pf cos V ; 
 or, dividing by P, 
 
 fj=cosX.f +cos/t.t; + cosv.f, 1 
 
 i;,=cosX,.f +co8/it,.t; + cos»',.5', >-.... (f) 
 
 f<= cos X„.f + cos Hii-v + cos v„.f. ) 
 
 Here we must also necessarily have cos*X+cos*Xj + cos*Xyj=l, 
 cos* fi + cos* fi, + cos* /*„= 1 , cos* V + cos* v^ 4- cos* v„ = 1 , and 
 
 cos X cos fi + cos X, cos fi, + cos X„ cos fi/i =0, \ 
 
 cos X cos l' + COSX(COS V, + COSX(, cos Vii = 0, > . . (g) 
 
 cos /t cos v+cos/t, cos Vf + cosfiiiCoa v,;=0. ) 
 
 Several demonstrations have been given of the preceding relations 
 between the nine direction cosines ; but nothing can weU be simpler 
 or more elementary than the above. 
 
 57.] To find an expression for the perpendicular from the origin 
 on the plane of which the tangential coordinates are S> v, ^. Let 
 P make the angles X, /«, v with the axes, then 
 
 cos* X + cos* fi + cos* 1'= 1 ; 
 
 but cosX=Pf, C08/a = Pu, cosv=P5'; 
 
 hence P*{|*+w* + ?*) = l, or f + u*+r=p- • • (a) 
 
 To determine the area of the triangle whose vertices are the three 
 points in which a plane is pierced by the axes O X, O Y, O Z. 
 Let S be twice the area of this triangle, P the perpendicular on 
 
 E 2 
 
52 ON THE TANGENTIAL EQUATIONS OF THE POINT, 
 
 it from the origin. Then the solid contents of the pyramid of which 
 
 1111 SP 
 
 the three rectangular edges are -p, -, -^ is ^-pj hut it is also -^ ; 
 
 II6ILC6 
 
 To determine an expression for the perpendicular let fall on a 
 plane from a point of which the projective coordinates are p, q, r. 
 
 Let ^, V, ^ be the tangential coordinates of the given plane ; let 
 a point be assumed of which the projective coordinates are p, q,r; 
 then the volume of the pyramid of which the faces are the three 
 coordinate planes and the limiting tangential plane is manifestly 
 
 = pp-p Let P be the perpendicular on the tangential plane from 
 
 the point O ; then the volume of the pyramid is also equal to the 
 sum of the four pyramids of which the altitudes are p, q, r, and P, 
 while the bases are the triangles made by the axes of coordinates and 
 the limiting plane; and it has been shown that the area of the triangle 
 
 of which the vertices are X, Y, Z is J -7^-^ + pp^ + -^-^ I . Equating 
 
 the volume of the whole pyramid with the sum of the volumes of 
 the four component pyramids. 
 
 Multiply by 6fv5', and the resulting equation becomes 
 
 1 =K + ?!/ + r?+ P{P + w* + S*}*. 
 Hence, finally, 
 
 {r+i^'+n*' 
 
 When the perpendicular is let fall from the origin O„^=0, q=0, 
 r=0, and 
 
 ^"{r+i^+H' ■••(d) 
 
 58.] We may use the preceding formulae obtained for the per- 
 pendiculars let fall from the points O and O. to determine the 
 translation of the coordinate planes in parallel directions. 
 
 Let the coordinates, through O, of the given plane be f, v, f; 
 and let the coordinates of the same plane passing through the 
 point O, be ^„ v,, 5) ; and let the projective coordinates of the point 
 O on these planes be p, q, r. 
 
THE PLANE, AND THE STRAIGHT LINE IN SPACE. 
 Fig. 16. 
 
 53 
 
 ^- 
 
 Now the perpendicular P from the point O on the given limiting 
 plane is 
 
 and the perpendicular P, from the new origin O^ is 
 
 1 
 
 hence 
 
 \i'+v,^+^n^' 
 
 ^ = l-p?,—g,v-r^; 
 
 but it is manifest that these perpendiculars on the limiting plane 
 from the points O and O, are proportional to the segments 
 of the parallel axes of coordinates, through the same points, or 
 
 |-5-j=p or f=p'|^,. Hence 
 ' ^1 
 
 f=i 
 
 ^,?=n 
 
 s; 
 
 (a) 
 
 " 1 -p^-qv,-r^; 1 -p^-qv,-r^; * 1 -p^-qv-r^, 
 
 In like manner, by the help of (d) and (f) in sec. [56] we can 
 always turn the axes of coordinates through certain given angles, 
 and then translate them in parallel directions to another origin. 
 
 59.] On the taru/ential equations of a plane passing through the 
 origin of coordinates. 
 
 When the plane, whose position is to be determined, passes 
 through the origin of coordinates, its tangential coordinates become 
 | = », u=x, §'=00, values which are illusory. Yet the plane 
 
54 ON THE TANGENTIAL EQUATIONS OF 
 
 must have a determinate position in space ; how is this position to 
 be ascertained ? 
 
 Let Pj a perpendicular to the plane^ drawn through the origin, 
 make the angles \, /t, v with the axes of coordinates ; and let the 
 intersections or traces of this plane with two of the coordinate planes 
 (ZX and ZY) suppose make the angles x and -^ with the axes of X 
 and Y. Then the angles which this trace on the plane of ZX makes 
 
 with the axes of coordinates OX, OY, OZ are x, \j and (^— x)- 
 
 But as this trace is also in the plane whose position is to be deter- 
 mined, P and this trace must be at right angles. Hence 
 
 cosX cos;^ + cos/i cos g + cosFCOsf q— xl=0, 
 
 or COS \ COS X + COS f sin x= 0. 
 
 But ^2L- = I . hence | + tan y = 0. In like manner, -c + tan •Jr = 0. 
 COS 1' 6 6 b 
 
 Consequently, when the plane whose position is to be determined 
 
 passes through the origin, the ratios of the tangential coordinates 
 
 f -^f and w-f-f denote the tangents of the angles which the traces 
 
 of the plane make with the axes of X and Y. Hence the position 
 
 of the plane passing through the origin may be determined by the 
 
 equations 
 
 H-tanx.?=0, w + tanV^.?=0 (a) 
 
 CHAPTER VI. 
 
 ON THE TANGENTIAL EQUATIONS OF THE STRAIGHT LINE IN SPACE. 
 
 60.] In defining the position of a straight line given in space, 
 there are two methods we may follow. We may conceive of any 
 two planes out of three passing through the given line and perpen- 
 dicular to two of the coordinate planes ; and the traces on these 
 coordinate planes made by the perpendiciilar planes let fall through 
 the given line enable us, by conceiving planes to be erected on these 
 traces, to determiue the position of the straight line in space. 
 
 But there is another method we may follow. Instead of passing 
 planes through the straight line, perpendicular to the coordinate 
 planes, we may determine the points in which the straight line 
 pierces the coordinate planes ; and if we can ascertain any two of 
 these three points on the coordinate planes, we can fix the position 
 of the straight line. The former is the projective, the latter is the 
 tangential method. Thus the three points which are on the straight 
 line, and also on the coordinate planes, are analogous to the three 
 
THE STRAIGHT LINE IN SPACE. 
 
 55 
 
 pkmei all passing through the straight line and perpendicular to 
 the three coordinate planes. 
 
 61.] Let p, q, p,, r, q,, r, denote the projective coordinates of the 
 three points z, y, w in the three coordinate planes XY, XZ, and YZ 
 through which the straight line passes. Then the tangential equa- 
 tions of these points in the straight line will be 
 
 i>f+gv=l, />;f + rg"=l, and y^w+r,?=l. ... (a) 
 We may deduce any one of these from the other two. Thus, eli- 
 minating P firom the first two, we get for the third ^-^ /!ii_= i . 
 
 P,-P Pi-P 
 This formula may be shown very simply by a diagram. 
 
 Fig. 17. 
 
 It will simplify the notation if, instead of writing the tangential 
 equations in the normal form, we put them under the forms 
 
 ?=/xf+o, v=v^+^ (b) 
 
 If we compare these forms with those given above. 
 
 f=--?+- and «=-^?+l. 
 Pi Pi 9i ii 
 
 (c) 
 
 we shall see that /*= :=tangent of the angle which the line 
 
 . . Pi 
 drawn to the origin from the point in which the straight line 
 
 pierces the plane of XZ makes with the axis of X, while a is the 
 
 reciprocal of the projective coordinate of the same point on the 
 
 axis of X ; and like values may be found for the constants in the 
 
 planes of the tangential equations of the points in YZ and XY. 
 
 It is obvious that to determine the position of the straight line 
 
 two equations are required, namely the tangential equations of two 
 
 of the three points in which the coordinate planes are pierced by 
 
 the given straight line. While one tangential equation determines 
 
 the point, three are necessary to fix the tangential plane. 
 
56 ON THE TANGENTIAL EQUATIONS OP 
 
 62.] To express the cosines of the angles which a straight line 
 makes with t^ awes of coordinates in terms of the constants ft, v, a, /8 
 of the tangential equations of the given straight line, 
 
 ^=liK+a, v=v^+fi (a) 
 
 Assume the general tangential equations of the points in which 
 the straight line pierces the planes of XY, XZ, and ZY, namely 
 
 ^? + ff«=l,i,,0+r?=l, afadv=g?+^, . . (b) 
 
 derived from the two preceding. 
 
 Comparing the coefficients of these equations with those of (a), 
 we shall have 
 
 pI ' pa pa. 
 
 or 
 
 fip,+r=0, ap,= 'l, pfi + qv=0, ap + ^g=:l. . (c) 
 Hence 
 
 ^=^iJ=:^y' P^^Ji^v' ^'=a'J. . . . . (d) 
 
 If we now trn-n to the diagram in page 55^ a short inspection will 
 show (denoting the angles which a parallel line through the origin 
 makes with the axes of coordinates by the letters lOZi, lOY, /OX) 
 that 
 
 cosZOZ= , e.g., Ton, cosZOY= 
 
 {r^+q^ + {p,-p)^i' '-"°' {r*+j«+(/,,-^ni' 
 
 (e) 
 
 cos /0X= ■ - , f' ,P ran- 
 
 HencCj substituting the values of p, q, r, p^ as given in the pre- 
 ceding equations (d), we shall have, putting 
 
 {a« + /3« + (/3/*-ai')«ii= A, 
 
 cos/OZ=^^^^=^, co8/OY=^, cos/OX=|. . (f) 
 A A A ^ ' 
 
 63.] To determine the conditions that two straight lines may meet 
 in space. 
 
 When two straight lines meet, a plane can always be drawn 
 through them. Let 
 
 ^=fj.^+a, u=vf+/3; 0=/t,?+o, and i/=v,?+/8, . . (a) 
 be the tangential equations of the two given straight lines, and let 
 0, V, ^ be the tangential coordinates of the plane which passes 
 through both straight lines. Then, from the four preceding equa- 
 
THE StKAIOHT LINE IN SPACE. 57 
 
 tions, eliminating the three quantities ^, v, ^, we shall have the fol- 
 lowing equation of condition between the eight constants of the 
 two given lines, 
 
 a—ai_fi—ijti 
 
 64.] We may also define a straight line as the locus of the points 
 in which two given planes intersect. 
 
 As the straight line is wholly in one of the planes, this plane will 
 pass through the three points in which the straight line pierces the 
 coordinate planes, and the tangential equations of these points will 
 be satisfied by the tangential coordinates of the two given planes. 
 
 Hence, the equations of two of these points being 
 
 ^=(i^+a and v=v^+fi, (a) 
 
 let the tangential coordinates of the two given planes be ^^ v,, ^^ and 
 
 *«' "/<> sii" 
 
 Now, as the former plane intersects the plane of XZ in the point 
 
 where it is pierced by the common line of intersection of the two 
 
 ' planes, the equation (a) will be satisfied by the coordinates of the 
 
 given planes. Consequently ^/=/iS)+a, and also f/i =/*$)/+ a; 
 
 t _fc 
 hence it=\i — Ir'i but we have also |— ?;=/*(§"— §",). 
 
 f/~ 5// 
 Hence the tangential equation (a) becomes 
 
 |=|/=|Z|«. In Uke manner ^'=^« * . (b) 
 
 65 .J To investigate the conditions that a given line may be found 
 in a given plane. 
 
 Let the tangential equations of the straight line be 
 
 f=K+«, v=v^+^ (a) 
 
 Let ^p v„ $) be the required coordinates to determine the position 
 of the plane. Then, substituting the coordinates of this plane in 
 the equations of the straight line. 
 
 But there are only two equations to determine the three un- 
 known quantities f,, v,, ^,. Hence the problem is indeterminate, 
 as is antecedently manifest. 
 
 66.] To find the tangential coordinates of a plane which shall pass 
 through a given point and a given straight line. 
 
 Let p, q, r be the projective coordinates of the given point ; 
 
 * It is obvious that these equations of a line, the inteisection of ttco givenplanes, 
 are analogous to the projective equatdons of a line joining two given points, namely 
 x-x,_x,-x„ , y-y' _ y,-y„ 
 
 z—z, z,—z„ z—z, z, — : 
 
58 ON THE TANOENTIAI. EQUATIONS OF 
 
 then the tangential eqoation of the given point may be written 
 
 Let ^=fi,^+a, w=yf+/3be the tangential equations of the points 
 in the planes of XZ and YZ where they are pierced by the straight 
 line ; then the plane must of necessity pass through these points, if 
 the given line is to be in the plane ; hence, substituting the values 
 of f and V, we get 
 
 {p/i, + qv+r)^=l—pa—q0: (a) 
 
 this equation will determine the value of 2^; hence those of f and v 
 may be obtained. 
 
 When the point is in the straight line, 2^ must be indeterminate, 
 
 or pii+qy+r=0, l—pa—q^=0 (b) 
 
 67.] To determine the angle between two given planes. 
 
 Let ^,, Vf, ^1 and ^y, v„, and ^i, be the tangential coordinates of 
 the two given planes. Let fall two perpendiculars P and P^ from 
 the origin, making the angles \, fi, v and \„ fi„ v, with the axes. 
 
 Then, tf> being the angle between the planes or between the per- 
 pendiculars to them, 
 
 cos ^=cos X cos X, + cos fi cos fii + cos V cos V,. . (a) 
 
 Now, as Pf,=cos \, VJ^„=coB X,, and making like substitutions 
 for the other angles, we tind 
 
 PP, cos 4"=S£ll+Vfl'u+ U, 
 
 m 
 
 cos A U„-^vfu+iL .y. 
 
 68.] A straight line is perpendicular to a given plane, to determine 
 the relations between the co^cients of the given straight line and the 
 given plane. 
 
 Let £=fi,^+a, and w=v5'+/8, be the equations of the straight 
 line, and let f „ w, {■; be the tangential coordinates of the given 
 plane. Then, as a perpendicular to this plane through the origin 
 is parallel to the straight line, the angle between them is =0. 
 
 Now the cosines of the angles which P makes with the axes of 
 coordinates are f^, v„ ij divided severally by {?* + «*+?*}*; and 
 the cosines of the angles which the straight line makes with the 
 same axes are /8, —a, — y, divided severally by {a*+/8' + 7*}t=A, 
 putting —7 for yS/*— av. Hence the direction cosines may be written 
 
 ^, :ze, and:^, or l=--=^fr:f^lT:^=_. U) 
 A' A' A' Va«+^ + 7«i/f,*+y,«+&*' ^' 
 
 or, reducing this expression, we shall have 
 
THE 8TIUIGHT LINE IN SPACE. 59 
 
 Hence, as this expression is the sum of three squares, we must 
 have each term separately equal to 0, or 
 
 a^,+^v,=0, ai,-yv,=0, ^^,+^^,=0. . . . (c) 
 
 These are the relations that exist between the coefficients in order 
 that a given line may be perpendicular to a given plane. 
 
 "We may obtain a much shorter solution of this problem, but 
 one neither so simple nor so elegant. 
 
 Let ^1, Vi, 5| be the coordinates of the given plane, and ^=fi^+a, 
 v=v^+fi the tangential equations of the given straight line. 
 
 Then as any plane which passes through this straight line will 
 be at right angles to the given plane, we must have f ^, + ww, + S?)= 0, 
 but f=/if+a, v=v^+0} substituting, 
 
 but as the plane through the given straight line is manifestly in- 
 determinate, we must have 
 
 S,^^ + v,v + ^r=0, al+^v,=0, (d) 
 
 or two equations between the four constants of the equations of the 
 given straight line. 
 
 69.] To determine the angles which the straight line, in which two 
 given planes intersect, makes with the axes of coordinates. 
 
 Let the tangential coordinates of the two given planes be ^p v,, ^i 
 and ^11, Vii, ^11 ; then the tangential equations of the straight line in 
 which they intersect are, as shown in (b), sec. [64], 
 
 f-^<=|^'(?-S) and v-v,=^(^-^,). 
 
 Now, if we compare these equations with the general tangential 
 equations of a straight line, 
 
 we shall have 
 
 n — y y > " — y y > "' — y y > ^ — y y' >l 
 
 and -7=(^/t-av)= ^""'~^'> ) 
 
 It is easy to show that perpendiculars to the two intersecting 
 planes are perpendiculars to their intersection ; for as 
 
 cosZOX=^, cosZOY=^, cosZOZ=^. 
 
 Substituting the values of /S, a, 7 given in (a), we shall have 
 
 co8/OX= ''"^'T'''^" , cosIOyMlzM', cos lOZ= ^'^~J'''" . 
 A A A 
 
60 ON THE TANGENTIAL EQUATIONS OF 
 
 Now the perpendicular to one of the planes makes angles with 
 
 the axes of coordinates, whose cosines are ==', =^, ^^ where 11 is 
 
 the reciprocal of P. 
 
 Hence m being the angle between this perpendicular and the 
 straight line in the two planes, we shall have 
 
 cos w - -^ , 
 
 the numerator of which is identically =0; hence o> is a right 
 angle. 
 
 We should have found a like result had we multiplied by f „, v,,, Jjj 
 the tangential coordinates of the second intersecting plane. 
 
 70.] To determine the conditions in order that a given straight 
 line and a given plane may be parallel. 
 
 Let ^„ v„ ^1 be the tangential coordinates of the given plane, 
 ^=fji,^+a, t;=vf+y3 the equations of the given straight line. Now 
 a plane which is perpendicular to this straight line will also be per- 
 pendicular to the given plane. 
 
 Let the coordinates of the second plane which is to be perpen- 
 dicular to the given straight line be |„, w,,, J)/. This plane and the 
 given straight line wiU be perpendicular, see (d), sec. [68], where 
 
 hence 
 
 S„=--v„ and 5/=( '^a'^^ )^ii' 
 
 but as these two planes must be at right angles one to the other, 
 we must have 
 
 ?if« + "/"« + %=0- 
 Substituting the values of f„ and f,„ we get 
 
 fiS-av,-{^fi-av)^,=0 (a) 
 
 71.] A straight line is parallel to a given straight line, to deter- 
 mine the relation between the constants. 
 
 Let f =/if+a, v=v^-\rP be the equations of one of these lines, 
 a,nd ^=/ii^+ai, v=v£+fi, those of the other; then, as these straight 
 lines are perpendicular to the same plane, we must have 
 
 or 
 
 a_a,_ fi—/j,, 
 
 fi-^.-y-y ^^^ 
 
 72.] To investigate an expression for the angle between two given 
 straight lines whose equations are 
 
 i=fi^+a, w=v2:+/3j ^=(i,i;+a„ v=v^+^,. 
 
THE 8TBAI6HT LINE IN SPACE. 61 
 
 Now the angle between two straight lines in space is equal to 
 that between two planes at right angles to them ; and as this angle 
 is equal to that between the perpendiculars let fall from the origin 
 upon these planes^ we shall haye 
 
 „.„,..- a. + "/"« + S5, ... (a) 
 
 We have now to determine the tangential coordinates of these 
 two planes ; and as these planes are given only in direction but not 
 in position, we can only obtain the ratios of the tangential coordi- 
 nates, but not the coordinates themselves. Let ^,^=<f>^„ iit'=4>i^u> 
 Vi=-^^p W/(='^/5|, and the preceding equation becomes 
 
 But it was shown in (d), sec. [68], that when a straight line is per- 
 pendicular to a plane, we must have 
 
 or ^ \ (c) 
 
 Hence 
 
 fifi-av' ^ 0/i-av' ^' fi,fi-a,v; ^' fi^,-a,v, 
 
 Substituting, 
 
 73.] To find an expression for the angle between a given plane 
 and a given straight line. 
 
 Let ^=ti^+a, v=v^+^he the tangential equations of the given 
 straight line, and ^„ v,, §) the coordinates of the given plane. Let 
 a plane be drawn perpendicular to the given line, then the angle 
 between these planes will be the complement of the angle between 
 the given plane and the given line. But since this last drawn plane 
 is perpendicular to the given line, we must have 
 
 /*? + vw+?=0, a^+fiv=0, orf==^u, g== ^^^"""^ v. 
 
 a a 
 
 Substituting these expressions in the general formula for the angle 
 between two planes, we get 
 
 sma)= 
 
 '/a''+^+ {fi(i-avf k/^^ + v^+I^ 
 
62 ON THE TANGENTIAL EQUATIONS OF 
 
 CHAPTER VII. 
 
 ON THE TANGENTIAL EQUATIONS OP SURPACES OP THE SECOND ORDER. 
 
 74.] When the surface is referred to its centre and axes, as origin 
 and axes of coordinates, the transformation of projective into tan- 
 gential coordinates exhibits no difficulty. Thus, let the projective 
 equation of an ellipsoid be 
 
 a^^ b^^ (?■ ' 
 the equation of the tangent plane passing through the point (^ y z) is 
 
 the current coordinates being Xf, y,, and 2,. 
 
 Let yi=0, r,=0, then X(=-z, and — ,^=1. Hence -^— =flf. 
 
 In like manner r=bv, and -^c^, and by substitution 
 
 a«r + 6V + c«?«=l, (a) 
 
 which is the tangential equation of an ellipsoid referred to its axes. 
 
 Let us now refer the surface of the second order to any rectan- 
 gular axes passing through the centre. 
 
 Let its equation, in projective coordinates, be 
 
 A^«+A^« + A,^+2Biry-|-2B>xr+2B,^=l. . . (b) 
 
 Then the equation of the tangent plane passing through the point 
 {x y z) on the surface, is 
 
 [Aa? + B^ + B„y]a:,+ [A;y + B,fr-|-Bir]y,-f- [A,^-l-By-f-B^]z=l,(c) 
 
 in which equation {xy z) is the point of contact on the surface, 
 and x,y,Zi are the current coordinates. 
 Assume 
 
 Ar + B^+B„y=|, A,y + B,^+B^=v, A,^ + By + B^=?; (d) 
 
 solving these linear equations for x, y, and z, putting 
 
 D =AB«+ A,B«+ A„B„«- AA,A„-2BB,Bfl, 
 
 we shall obtain the resulting equations, 
 
 Da?=(B«-A,A„)f + (A„B„-BB>-h (A,B,-BB„)&) 
 
 Dy = (B/-A„A)f+(AB-B,B„)?+(AA-B.B)f, \ . (e) 
 
 Dz=(B,»-A,A)?+ (A,B,-B„B)f+ (AB-B,B„)i;; ) 
 
8VKFACSS OF THE SECOND ORDER. 63 
 
 or dividing by D, and substituting single symbols for the foregoing 
 expressions^ we shall have 
 
 y=a^+^?+/3,,f, C (f) 
 
 Multiply these equations respectively by the three following ex- 
 pressionSj namely 
 
 and adding them, we thus obtain 
 
 These are the relations which exist between the projective and tan- 
 gential equations of the same surface of the second order. 
 
 75.] Let the surface of the second order be now referred to an 
 origin of tangential coordinates other than the centre O. Let the 
 point Q, be so assumed, and through this jmint Q let three rectan- 
 gular planes be drawn, parallel to the original rectangular planes, 
 through the centre O. Let p, q, r be the projective coordinates of 
 the centre O let fall on the new coordinate planes. Let the tan- 
 gential equation of the surface, the origin being at the centre, be 
 as in (g) in the preceding section, 
 
 and, assuming the values of the old tangential coordinates in terms 
 of the new as given in (a), sec. [58], namely 
 
 . f_ ^1 „= Yx y- S 
 
 * \-p}i,-qv,-r^; l-p^-qv-r^i' l-p^-qv,-rZ' 
 
 making these substitutions in the preceding equation, we obtain for 
 the tangential equation of a surface of the second order referred to 
 Q, any point in space, 
 
 (a-p^)^+ {a-q^W+ ia„-r^^ + 2{fi-qr)^v+2{fi,-pr)m ,. 
 + 2{^„-pq)^v + 2pi + 2qv+2ri^l. ) ^ ' 
 
 It is hence obvious that in the general tangential equation of a 
 central surface of the second order, referred to any coordinate 
 planes in space, the coefficients of the linear terms j^, v, ^ will be 
 twice the projective coordinates of the centre on the new coordi- 
 nate planes. 
 
 This is a matter of much importance, as without any calculation 
 at all we can discern the position of the centre. Hence, if the 
 
64 ON THE TANGENTIAL EQUATIONS OF 
 
 tangential eqiiation of a central surface of the second order referred 
 to any system of rectangular coordinates in space be 
 
 L^^+L,i/'+L,^ + 2M^v+ZM,^^+2M„^v + 2^^+2qv+2ri;=l.{h) 
 
 then, comparing this equation with the preceding (a) and equating 
 like terms, we shall have h=a—p^, L,=o,— g', L|,=o„— r', 
 U=p-qr, M,=fi,-pr, M„=^„-pq. 
 
 Hence we may return to the system of coordinates passing 
 through the centre of the surface, if we substitute for the coeffi- 
 cients of that equation, namely a, a„ a,,, /3, jS^, fin, their values just 
 given — ^that is to say, 
 
 o=L+^«, a,=L,+ff9, a„=L„+r*, ) 
 
 /3=M+ffr, ft=M,+iw, S„=M„+i>j.J 
 
 So that, passing from the equation of the surface referred to any 
 system of axes to another parallel system passing through the 
 centre, the general equation (b) will become 
 
 (L+/'')«'+ (L,+ ««)«'+ {L„+r«)?«+2(M+ gr)rw) ,,. 
 
 76.] Let^(^-*,)+g(y-y,)+^(^-r,)=0, ... (a) 
 
 be the equation of a tangent plane to a surface of the second order, 
 f{x, y,z) =0, in which a!,y,z, are the current coordinates and P is 
 written for brevity instead of/(jp, y, z). 
 
 Since x, y, z, are the current coordiuates, let y,=0, z,=0, there- 
 fore x, is the distance from the origin of the point where the axis 
 
 of X is cut by the tangent plane. Hence a?,=^, and we thus get 
 
 dF dF 
 
 ^=dF ^dP dF'*"^^°''=dF W—dF'- ^^ 
 
 d^^+d^J' + dJ^ di*+^y+dJ^ 
 
 and a like expression for f may be found. 
 
 Combining these equations with the dual equation 
 
 x^+yv+z^^l, 
 
 we may obtain the tangential equation of the same surface. 
 
 Thus, let the projective equation of the surface of the second 
 degree/(a?, y, z) =0, referred to three rectangular coordinate planes, 
 be 
 
 Aa?« + A,ys + A,^« + 2Byz + 2B/cz + 23,^^,) 
 
 + 2Ca;-f2C,y + 2C,^=l; f " * ^""^ 
 
SURFACES OF THE SECOND ORDEK. 
 
 65 
 
 taking the partial differential coefficients, we get 
 
 dF 
 
 dF 
 
 d^=2[Aa? + B^ + B„y + C], ^=2[A,y + B,^ + B^ + C,], 
 
 d^ 
 dF 
 
 Ay 
 
 ^=2[A,^ + By + B/B + CJ, and therefore 
 
 dF ^dF . dF „r, r. 
 
 Ay^^Az 
 
 (■(d) 
 
 > 
 
 (e) 
 
 Ax 
 
 hence 
 
 fc_ Aj;+B;g + B„y + C -^ 
 * 1-Ca7-C^-C,^' 
 ,— A,y + B,ja?+Bg + C, 
 1-Ca;-C,y-C,fr' 
 ^_ A,^ + By + B^+CH 
 
 If now from these equations we derive the values of x, y, z in 
 terms of ^, v, and i„ and substitute their values in the dual equation 
 a^+yu+2rf=l, we shall obtain as the result the tangential equa- 
 tion of the same surface — that is to say, «^X?> ^> ?) ~^' 
 
 If we divide, one by the other, the equations (b), we shall obtain 
 
 dF dF 
 
 u dy f_dz 
 
 f~dF' |~dF' 
 da? Ax 
 
 Square these equations, and add, 
 
 /dFY /dFV /dF\« 
 
 
 C 
 
 dFy 
 Ax) 
 
 Now P2(f« + w^ + ?«) = 1, and P2^=cos* \ ; hence 
 
 cos'A,= 
 
 \Ax) 
 
 /dry /dpy /dFy 
 W/ Uy/ \^z) 
 
 (f) 
 
 and like expressions may be found foi cos /i, cos v. 
 
 Hence the angles which the perpendicular on the tangent plane 
 makes with the axes of coordinates derive their expressions from 
 the partial differentiation of the projective equation, and not from 
 that of the tatigential equation of the surface as we might have 
 anticipated. We shall find that like expressions for tlio angles 
 
66 ON THE TANGENTIAL EQUATIONS OP 
 
 which the diameter passing through the point of contact mth the 
 tangent plane makes with the axes of coordinates are derived from 
 the tangential equation of the surface. 
 
 77] . Resuming the general tangential equation of a surface of 
 the second order, 
 
 let us solve this equation for one of the variables, ? suppose, and 
 the resulting equation becomes 
 
 ^^Jl!i±M±li+ VM, (b) 
 
 "■11 
 writing M for the sum of the terms under the radical sign. 5 has 
 evidently two values, because for the same values of f and v there 
 must be two tangent planes to the surface. Now, using the figure 
 in sec. [20], and the reasoning of that article, let 
 
 Om a„ 
 
 1 _ (fa+ftg+y //) V .... (c) 
 
 OP~ a„ ' ' 
 
 Hence ^r— , T=rp, and j=r- are in arithmetical progression, and 
 
 therefore Om, OP, and 0» are in harmonical progression. Conse- 
 quently the plane of XY, the two tangent planes to the surface 
 intersecting in the plane of XY, and the plane which passes through 
 the point P in the axis of Z, and the common intersection of the 
 two tangent planes, in the plane of XY, are four harmonic planes 
 all intersecting in the same straight line in the plane of XY ; there- 
 fore the plane of XY and the secant plane through the point P 
 pass respectively through the pole, one of the other ; hence the 
 plane whose equation is 
 
 passes through the pole of the plane of XY. 
 Consequently the following equations, 
 
 «," +^//f +/3? +y,=oi (d) 
 
 a,^ + /3i; +^^+y„=0,) 
 
 are the tangential equations of the poles of the coordinate planes 
 of ZY, ZX, and XY respectively ; and the combination of the three 
 equations determines the three tangential coordinates of the polai' 
 plane of the origin. 
 
SURFACES OF THE SECOND ORDER. 67 
 
 From the preceding equations, eliminating v and J^, we may find 
 the value of f the tangential coordinate, along the axis of X, of the 
 polar plane of the origin, or 
 
 fc ^ (^"-«/°//)7+(«A-Wy/+(a,^,-<3^«)Y,, M 
 
 Like expressions may be found for v, and ?,. 
 When 
 
 o«,«« + 2/3^//3«-a/3'-«A*-«A'=0> . - • (f) 
 the origin of coordinates is on the surface ; for then the intercepts of 
 the axes of coordinates cut off by the tangent plane to the surface, 
 
 nam.ely ■=, — , y, are each equal to 0. 
 
 When 7=0, 7|=0, 7^=0, it follows that ^,=0, 11,= 0, ?,=0. 
 But when 7=0, y,=0, 7<(=0, the origin is at the centre, and there- 
 fore the coordinates p, — , y of the polar plane of the centre are 
 
 infinite. 
 
 78.] To show that when the surface touches a coordinate plane, 
 the coefficient of the square of the corresponding variable becomes 
 =0. 
 
 When the tangent plane to the surface becomes indefinitely near 
 to the plane, suppose of XY, it cuts the axis of Z indefinitely near 
 to the origin. Hence, when ultimately the plane of XY becomes a 
 
 tangent plane to the surface, -= becomes 0. 
 
 The general equation of the surface may be written 
 
 ««+ («l* + a,w^ + a^i'?+ 2/3,?? + 2/3,,fi; + 2yJ + 27,u + 27„?- 1)^=0. (a) 
 
 But as the second member is =0, since -==0, we must have 
 
 a,f=0. In like manner, when the surface touches the coordinate 
 planes of ZX or ZY, a,=0, or a=0. Hence, when the surface 
 touches the three coordinate planes, the general equation of the 
 surface becomes 
 
 2^i;?-|-2^;S; + 2^,^i; + 2y? + 27," + 27„?=l. • . (b) 
 
 79.] When the surface touches one of the coordinate planes, to 
 determine the equation of the point of contact. 
 
 Let the plane of XY be the tangent plane, thena„=0, as shown 
 in the preceding section, and the general equation may be written. 
 
 (a|« + o,i;2 + 2/3fu + 27?+2y,t;-l)i + 2(/3y + ^,f + 7„)=0. (a) 
 But since 7=0, the first member of this equation is=0. Hence 
 
 F 2 
 
68 ON THE TANGENTIAL EQUATIONS OF 
 
 /3i/ + (3,f +7,;=0 is the tangential equation of the point of contact of 
 the plane of XY. 
 
 Consequently the tangential equations of the points of contact of 
 the surface with the planes of XY, YZ, and XZ are 
 
 ^v+^,S+y,i=0, ^,? + ^„i; + y=0, and ^„f+|3? + 7,=0. (b) 
 
 80.] On the equation of the tangent plane to a curved surface 
 whose tangential equation is <f>{^, v, ^)=0, 
 
 Let the tangential coordinates of the tangent plane be conceived 
 as receiving minute increments, consistently with the permanence 
 of the projective coordinates of the point of contact. Let f, u, f 
 be assumed as having received infinitesimal augments, and thus to 
 be changed into f +df, v + dv, f+df. 
 
 Then, on the ordinary principles of differentiation, 
 
 ^(f. v„ S) =<^(f, V, r) +^J^ df +^^^ 6v+^-^^ d5. 
 
 But <fi{^i, Vp ?,) =0, <^(J, V, f) =0 ; and writing for brevity a capital 
 ^ for ^(j, w, f), we obtain the condition 
 
 If we now, on the same assumed principle, differentiate the dual 
 equation x^ + yv •\- z^=\,^g shall find x&^+yA.v+zdi^=0. 
 
 Equating the coeflBcients of these differentials, first introducing 
 an equalizing factor A, we shall have 
 
 . d* . d^ . d^ 
 A^=-g^, Ay=-^, Ar=-g^. 
 
 Multiplying these expressions severally by f, v, ?, and adding, 
 A /■ »r , . v\ d* ^ d^ d4> . 
 
 But as arf +yw + «?=l, we shall have, finally, 
 A d*e,cl<E> d*^ 
 
 ^=dFf+d^''+-dr^- 
 
 Hence 
 
 d* 
 
 '""d^TTd* "Wji ^^ 
 
 Hl^+diJ-^+dfO 
 Like expressions may be found for y and z. 
 
 These very general and beautiful formula of transition, as they 
 may be called, reduce the passage from the projective to the tan- 
 gential equation of a curve or curved surface, or reciprocally, to a 
 
SURFACES OF THE SECOND OBDER. 
 
 69 
 
 mere mechanical operation as it were ; and the problem is thus 
 reduced in all cases to one of elimination. 
 
 The formulae which exhibit the relations between the projective 
 and tangential coordinates of the same curve or curved surface are 
 simple and symmetrical. 
 
 Let «J>=^(f, V, f)=0 be the 
 tangential equation of a curved 
 surface. The projective coordi- 
 nates x, y, z of the point of 
 contact of the tangent plane may 
 be found from the following ex- 
 pressions : — 
 
 d* 
 dg 
 
 Let F=/(a?, y, z) =0 be the 
 projective equation of a curved 
 surface. The tangential coor- 
 dinates f, V, f of the tangent 
 plane drawn through the point 
 [xyz) may be found from the 
 following expressions : — 
 
 x= 
 
 dF^+di7''+df^ 
 
 y= 
 
 d^ 
 
 dt^ 
 
 d4>^ d* d4> ' 
 df^+diT^+d?^ 
 
 1= 
 
 dF 
 Ax 
 
 dF dF dF 
 
 ■AZ''-^^.y + 
 
 Ase 
 
 Ay 
 dF 
 Ay 
 
 Az' 
 
 ar= 
 
 d^ 
 
 d*^ . d* d4> ■ 
 
 dr^+dir^+dr^ 
 
 ?= 
 
 'dF^ dF , dF^' 
 Ax Ay^ Az 
 dF 
 Az 
 
 > («5) 
 
 dF . dF 
 
 d^^+%2'+^^ 
 
 dF 
 dF^ 
 
 By the help of these groups of equations and the original equa- 
 tions <E>=^(f, V, ?)=0> or/(<^j y> ^)=0> we may eliminate ?, v, f, 
 or X, y, z, and obtain the final eqiiations in x, y, z, or in ^, v, ^. 
 
 CHAPTER VIII. 
 
 ON THE MAGNITUDE AND POSITION OF THE AXES OF A SURFACE OF 
 THE SECOND ORDER. 
 
 81.J If we assume the tangential equation of a central surface of 
 the second order, as given in (a), [74], 
 
 a2f+6V+c«f' = l, (a) 
 
 and refer this surface to another system of rectangular coordinates, 
 also passing through the centre, by the help of the formula of 
 
70 ON THE MAGNITUDE AND POSITION OF THE 
 
 transformation given in sec. [56], the transformed equation will 
 assume the following form, 
 
 TiS^+R,v^+-H.„^+2Kv^+2K^^+2K„^v=l, . . (b) 
 
 H, H(, H„, K, K/, K,( being explicit functions of the nine angles 
 discussed in sec. [56], and of the semiaxes a, b, c. 
 
 Hence the general equation of the surface of the second order, 
 referred to three rectangular axes passing through the centre, being 
 
 a^ + aiu^+a,P + 2^v^+2^;^^+2^„^v=l, . . . (c) 
 
 we shall have to determine the twelve unknown quantities, namely 
 the nine direction cosines, as they are called, and the squares of 
 the three semiaxes of the surface — ^twelve equations, six of which 
 are given by the known relations between the nine angles, and six 
 may be obtained by equating, term by term, the six coeflScients of 
 (c) with the six coeflScients of (b) ; and in this way the problem 
 might be solved. It may, however, be justly surmised that the 
 solution of these twelve equations would lead to very complicated 
 and unmanageable expressions. With the help of other principles, 
 derived irom the following important theorem, we may elude the 
 difficulty. 
 
 82.] To find a general earpresswa for the distance between the 
 point of contact of a tangent plane to a surface, and the foot of the 
 perpendicular let fall from the origin on this tangent plane. 
 
 Let X, y, z be the projective coordinates of the point of contact, 
 and let ^, v, X be the tangential coordinates of the tangent plane to 
 the surface ; and if T be the required distance, we shall evidently 
 have 
 
 T'=^^+y^+z«-;p-pl^, (a) 
 
 We must now eliminate x, y, and z. 
 
 It has been shown in sec. [80] that if ^(f, v, 1^=^=0 be the 
 tangential equation of a surface, 
 
 d* d* 
 dg dg 
 d4>. i^ . d^ ~ A ' ^' 
 
 di^+dir''+d?f 
 
 and like expressions for y and z may be found. 
 
 If we now square these expressions, and substitute in (a), we shall 
 have, after some reductions, 
 
 ^ \M^-^^ +Ldi;^-d?'^J +U^-dF^J ,, 
 
 To apply this expression to the ellipsoid. 
 
AXES OF A SURFACE OF THE SECOND ORDER. 71 
 
 The tangential equation of the ellipsoid is a^S' + b^i^ + c^P=l. 
 
 d4> 
 Hence -T^=2a'J, and A = 2. Like expressions for the other 
 
 variables v and f may be found. Hence 
 
 rr2_ («'-^')"g""'+ {l>^-c^)^v^^+ {a^-c^)^P^ .,. 
 
 By the help of this formula we may determine the magnitude 
 and position of the axes of a surface. 
 
 83.] We may define the axis of a central surface as a line which, 
 drawn from the origin at the centre, to the point of contact of a 
 tangent plane, coincides with the perpendicular let fall from the 
 origin on the same plane. 
 
 In this case T=0 ; but as the numerator of the value of 'P is the 
 sum of three squares, in order that this expression may be =0, they 
 must severally be equal to 0, or 
 
 d* dip d*^ d4> - d<Pf. d*^ . , , 
 
 dF^-dT^^^' d;7^~dr''=^'V-dr^=^- • (^) 
 
 d<E> 
 Now we have shown in [80] that -rp=2a;, and 2a;=2P cos \, 
 
 since in this supposed case P is equal to and coincides with r. But 
 
 d^ 
 cosX=Pf; hence =|-=2F'f; (b) 
 
 and like expressions for the other two variables may be found. 
 
 Let of2+a,«2+a„?2 + 2/St.?+2)8,?? + 2/8„ft;=l . . (c) 
 
 be the tangential equation of a surface of the second order referred 
 to its centre, then 
 
 d<[> d^ 
 
 -^ =2ai + 2^^ + 20,iVi but -^ is also equal to 2P'f . 
 
 Hence a^+/Sy$+/8uW=P'f ; consequently the three resulting equa- 
 tions are 
 
 (a-P»)f+^,?+;S„i;=0,| 
 
 {a,-T^)v + ^„^+fiK=0,\ (d) 
 
 K-P«)?-|-;8y +/8,? = 0.) 
 
 Eliminating f, v, $ from these three equations, which the absence 
 of an absolute term enables us to do, we obtain an equation analogous 
 to the well-known cubic equation for determining the three axes of 
 the surface. 
 
 If we could by any means ascertain that particular position of 
 the tangent plane which would make the perpendicular coincide 
 with the diameter passing through the point of contact, it would 
 follow that f, V, ? would become known quantities, and we could 
 
72 ON THE MAGNITUDE AND POSITION OP THE 
 
 thus calculate the value of P ; but if instead of so doing we elimi- 
 nate f, V, $, we shall ascertain in how many ways P may become a 
 semiaxis of the surface. 
 
 Eliminating £, v, ? from equations (d), we obtain 
 
 -^/(a„-P*)+2^^A=0. J • '' 
 
 a cubic equation which gives us when solved the values of the 
 squares of the three semiaxes. Reducing the preceding equation, 
 and arranging by powers of P, we get 
 
 +/3«a+y3«a,+^>„-aa^„-2/3y3^„=0. J 
 
 Hence we may at once infer that the sum of the squares of the semi- 
 axes is equal to the sum of the coefficients of the squares of the 
 variables. 
 
 We shall recur to this equation, which may be termed the 
 " tangential cubic equation of axes." 
 
 It may have been observed, by those conversant with the subject, 
 that the concentric sphere, which has generally been made use of 
 to determine the magnitude of the axes of the surface, has been dis- 
 pensed with. The principle adopted in the text, that of defining 
 the axes as those perpendiculars on a tangent plane which coincide 
 with the diameters drawn through the points of contact of the 
 tangent planes, possesses the advantage of determining those mag- 
 nitudes from a consideration of the properties of the figure itself. 
 
 81]. To determine the angles which one of the aares of the surface 
 makes with the axes of coordinates. 
 
 The cosine of the angle \ which the axis P makes with the axis 
 of X is Pf ; and a:=P*f . But, as we have shown in [80] , we shall 
 have P^£=/r=ac+/8,^+/8jji/; and finding like expressions for the 
 other variables, 
 
 (a-P«)f+/3^+/8„t;=0,^ 
 
 K-P«)«+/8,^+y3? =0,5> (a) 
 
 (a„-P*)?+/3y +/3^ =0.J 
 
 Let the three roots of the cubic equation resulting from the eli- 
 mination of 0, V, ^ between these equations, and which determines 
 the axes, be t^, t,*, t/, and let 
 
 a— 1^=8, o,— T'=Sp a„— T*=8„; . . . (b) 
 
 also let lf=mi, v=n^; and the preceding equations become 
 
 S;i + )8„+j9ot=0, (■ (c) 
 
AXES OF A SURFACE OF THE SECOND OBDER. 73 
 
 from these equations we get 
 
 squaring these expressions, and adding. 
 
 We may combine this result with the cubic equation (e) in the 
 preceding section, and which may be written in the form 
 
 SB,S,-^S-p,^S-fi/S,+2^^fi,=0. . . (e) 
 Let this expression be successively multiplied by 8, and S^^. The 
 resulting expressions become 
 
 SS,A'-^*SS,-^«V-^/8,S„+2/3^M =0,) (f. 
 
 Adding these two expressions (f ) to (e) and eliminating, we shall 
 obtain a result that will allow of the dividing out of the common 
 factor (j3^— S^„). Hence 
 
 (».H.«4-i)= (^'-^'^"^-^;gqy^^"'-^^'^ . . (g) 
 
 Now, as ?=»»f, v=n^, 
 
 £2 — £4 P^^ 
 
 Consequently 
 
 cos M- (^*_8,s^) + (|3;_8S„) + (^/-8S/ 
 /3„«-88, 
 
 (t) 
 
 '^" "- (^^-8^) + (^-8SJ + (/3/-SS,)- 
 
 We may thus find similar expressions for the other two semiaxes. 
 Hence the problem is completely solved, as we can express the 
 squares of the three semiaxes and the cosines of the nine angles in 
 terms of the coefficients of the given tangential equation. 
 
 On the particular case when the surface is one of revolution. 
 
 85.] In this case two of the axes are equal; and therefore two 
 of the roots of the cubic equation of axes (f ) in [83] become equal. 
 But it is a well known property of algebraical equations that when 
 two of the roots of an equation are equal, one of these wiU be found 
 in the limiting or derived equation. Writing p for P* in the equa- 
 tion of axes, and differentiating, we find 
 
 3p'-2{a + a, + a,;}p- [(fi^-afl,) + (/3,«-ao„) + (^„«-oa,)]; (a) 
 
74 ON THE MAGNITUDE AND POSITION OV THE 
 
 or writing R for a+a^+a,,, and 
 
 Q for i^-afl,,) + {^,^-aa,;j + {^,,^-aa,), 
 the limiting equation becomes 
 
 3p^-2Iip-Q=0, or p=^± VR'+SQ. . . (b) 
 
 o 
 
 One of these values of p is the square of one of the equal semiaxes. 
 
 We may obtain the value of the equal semiaxes from other con- 
 siderations. 
 
 If we refer to the formulae which determine the inclinations of 
 the axes of the surface to the axes of coordinates, as given in (h), 
 sec. [84] , it is evident that, when they become unlimited in numb^r^ 
 the values of the cosines of the angles which these equal axes make 
 with the axes of coordinates must be indeterminate also. If we 
 refer to the expressions for these cosines as given in (h), they will 
 
 become indeterminate^ or of the form t-, if we put 
 
 ^«-8^=0, ^,^-8B„=0, ^/-8S,=0. . . . (c) 
 
 Now reverting to sec. [84] (b), it has been shown that, if t* be 
 one of the roots of the cubic equation of axes, 
 
 a,-T^=S„ a„—T^=S„, 
 or 
 
 a;0„-(o,+a„)7^+T*=S^=/8« . . . . (d) 
 
 on the assumption in (c) . 
 
 Consequently t^ — (o, + a,i)i^ = B?—ap,„. 
 
 Finding similar expressions for the two other axes of coordinates, 
 and adding the resulting equations, we shall have 
 
 3T*-2{a+a, + a„)T«=^«-a/i„+/32-aa„+j3„2-aa,), 
 or 
 
 3t'«-2Rt2-Q=0, (e) 
 
 precisely the same equation as (b) derived from the discussion of 
 the limiting equation. 
 
 It may easily be shown that, when two of the roots of the cubic 
 equation of axes are equal, the three roots are 
 
 R- VR^+SQ R- VR'+3Q ,„. R + 2 \/R«+3Q ,.. 
 3 > 3 , and _ (f ) 
 
 We know that if r, r,, r„ be the three roots of a cubic equation, 
 the sum of the products of the roots, taken two by two, is equal 
 to the coefficient of the third term, 
 
 or rr, + r,r„ + n-„=-Q (g) 
 
AXES or A SURFACE OF THE SECOND ORDER. 75 
 
 Now,ifweinakey=?±?_V^?L±?^, r, and r„ each equal to 
 
 o 
 
 -, , and substitute these values in the preceding ex- 
 
 o 
 
 pressiou, the resulting equation becomes identical*. 
 
 CHAPTER IX. 
 
 ON THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 86.] Let the axis of the surface be taken as the axis of z, the 
 tangents to the vertices of the principal sections as the axes of 
 X and y. Now we may conceive the elliptic paraboloid as gene- 
 rated by the parallel motion of the plane of a variable ellipse whose 
 centre moves along the axis of Z, and whose vertices always rest 
 on two guiding parabolas in the planes of ZX and ZY, having a 
 common axis Z. Let 4k and 4^^ be the parameters of the principal 
 sections passing through the axis of ;;. Let the paraboloid be cut 
 by a plane parallel to the plane of xy at the distance 2. Then, if 
 
 * If we take the cubic equation of axei given in most books on surfaces of 
 the second order, as treated by the method of projective coordinates (see Leroy, 
 ' Analyse appliqu^e & la 04om€trie des trois dimensions/ p. 198), we shall find 
 
 (s-A)(s-A,)(«-A„)-BX*-A)-B,=(«-A)-B„X«-AJ-2BB,B„=0. (a) 
 
 Now, if R be a root of this equation, and we put 
 
 A-R=D, A,-R=D„ A„-R=D,„ 
 
 and if we follow the course indicated in the text, we shall find for the cosines 
 of the angles a, j3, y, which the semiaxis R makes with the three axes, 
 
 (B»-D,D„)+(B,=-D„D)+(B„'-BD,)'" " " ' W 
 
 and like expressions for cos*/3, C0B*y. 
 
 We do not remember to have seen these formulae in any treatise on this subject. 
 
 "When the surface is one of revolution, two of the axes must be equal, hence 
 two of the roots of the cubic equation of axes must be equal, hence the limiting 
 equation to (a) must contain at least one of the equal roots. Differentiating this 
 equation, we shall find the following for the limiting equation 
 [(»-A,)(*-A„)-B"]+[(«-A„)(»-A)-B,=]+[(»-A)(s-A)-B„=]=0,(c) 
 and this derivative, as also the original equation, are satisfied by putting 
 
 D,D„-B^=0, I>„D-B,==0, DD,-B„»=0, 
 and the expressions for the direction cosines become 
 
 „ 
 
 =6' 
 as evidently should be the case 
 
 cosa=Q, cos^=^, C08y=j^, 
 
76 ON THE TANGENTIAL EQUATIONS OF THE PARABOIOIDS. 
 
 X and Y be the semiaxes of the ellipse in which the paraboloid is 
 cut by the parallel plane, we shall have x'=4Az, y'=4A,?; hence 
 
 m+U^=^' ""^ *^' + %'=****^' ■ . • (a) 
 is the projective equation of the elliptic paraboloid. 
 Now ^=2k^, ^=2%, ^ 4M„ 
 
 and 
 Hence 
 
 dF ^dP .dF .,. 
 S^+di^^+d^^=^**'^- 
 
 (b) 
 
 ^ 2kk^ Uz" 2k/ ^ z 
 Substituting these values oijf,y, and 2 in the dual equation 
 
 we shall obtain for the tangential equation of the paraboloid, 
 
 Af+*,w^ + ?=0 (c) 
 
 87.] On the transformation of the axes of coordinates in the case 
 of the paraboloids. 
 
 Assuming the equations given in sec. [56], namely 
 
 ^=^1 cos \ + w, cos \ + ?, cos \„, 
 
 v=^i cos fi + v, cos fj-i + 5( cos fill, 
 
 ?=f, cos v+V/ cos v,+ $y cos Vi,, 
 
 or writing Imn, I, m, n,, Z„ »i^, n,/ for the cosines, 
 
 i=l^i +l,v, +1,^1, 
 v=m^,+ m,v, + m,ff, 
 5=raf^ ^n,v^ +n,X„ 
 hence 
 
 k^=k [P^^ + ;, V + W + 2«M + 2««?A + 2%vA] , 
 V'=*,Kf,* + »w,V + »»„%*+2wM»^,w,+ 2OT»»„?,?,+2»»,»M„wA], 
 
 Arranging according to powers of f , w, and ?, and omitting traits 
 to ^, V, (I' as no longer necessary, and adding, 
 
 [kP+k,m^]^+ \kl^ + A,TO *] u« + [«„* + A,»i/j ?* + 2[Jfc//„ + kfnpi,;\ ?w 
 
 + 2 [M„ + ftjmjw„] f $ + 2 [AZ/, + *„«i»i;] fw + «? + n,i; + w„? = 0. (d) 
 
 When the paraboloid is a surface of revolution and the origin 
 placed at the focus of the surface, its equation becomes 
 
 *(l«+i'* + ?^)+?=0 (e) 
 
ON THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. It 
 
 We may now translate the origin to another point, the direction 
 of the coordinate planes continuing the same. 
 
 Let p, q, r be the projective coordinates of the vertex of the 
 surface on the new coordinate planes ; then, assimiing 
 
 like expressions may be found for v and ?• 
 
 It is plain that the only terms that will be aflfected by the trans- 
 lation of the coordinates wiU be the linear terms j and these will 
 become 
 
 B? —pn^ —qn^v — mf^, 
 
 n,v —qn/u^ —pn^v —mju^, 
 
 Hence the general equation of the paraboloid referred to any 
 rectangular axes in space is 
 
 + \2klln + Zkimm,, + n,r + w„g] ?i» 
 + \%kll„ + 'Hkmm,, + nr + n„p'] f ? 
 + \%kll, + 2kimmi + nq + n,p'] f w + nf + n^w + «„? =0, 
 
 omitting the traits as no longer necessary. 
 
 88.] Given the general equation of the paraboloid to any set of 
 rectangular axes passing through the vertex, to determine the magni- 
 tude and position of the parameters. 
 
 We have shown in the preceding section that the equation of the 
 paraboloid, referred to its vertex as origin, will be 
 
 \kP-{-k,m^-]^+ \kl^ + kim^-]v'+ {kl,f' + k,m,^}^\ 
 
 + 2[kl,1^, + kjm,m,^t,v■^%\kll^, + k^mm^^Jf^ f • • (a) 
 
 Let the general equation of the paraboloid be 
 
 yr +/y +///?" +2^''?+2^/f? + 2i7«?" + Af + V + ^«?=0- ■ (b) 
 Comparing the coefficients of these equations, term by term, we 
 shall have 
 
 kP^k,m^=f, kl,^ + k,m,^=f„ kl„^ + k,m„^=rf„ | 
 
 (f) 
 
 hence 
 
 k\mf->rlfl,f^l,fV^ +k/'[m^m^ + m,^m„^ + m„^m^] » 
 
 + kk,[l,^m'^ + l^m^ + l,i'm,^ + l,^m„^ + l/m^ + PmA . (d) 
 
78 ON THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 and 
 
 ■\-k^\mha^+m^,^+m,^m^] V . (e) 
 + 2ltk, [lliinmi + l^^pijin^f + ll^im,^fn\ . ) 
 
 If we now subtract this equation from the preceding, we shall 
 have 
 
 =kk,[{l,m—lmi)^+(l„m,—l,m,i)^+ {lm,i—l„m)'].) ' 
 Assume 
 
 l/m — Irrii =cos(f>, lumi — /^wi^, = cos j^ and Imn — /„wi = cos ■^. (g) 
 
 Multiply' the first by »j,,, the second by m, and the third by »w^; 
 then we shall have 
 
 litnmii — lmin,i = »m„ cos 0, ) 
 
 l,imim—ljmijm=mcios')(^ > (h) 
 
 /»»„»», — lijmnif = m, cos -^ ; J 
 
 adding, we shall have 0=»»„ cos ^ + »i cos x + ^^i cos y^. 
 
 But as m, m^ m,, are the cosines of the angles which a certain 
 straight line makes with a system of rectangular axes, ^, ^, '^ must 
 be the angles which a straight line at right angles to the former 
 makes with the same axes ; hence 
 
 (Z,JM— /»»,)* + (/,/»«, — /^OT„)'+(/>n,,— /„OT)* = COS*^ + COS*J^ + COS*^ = l, 
 
 and the preceding equation now becomes 
 
 m+f>f„+fJ-(a^+9?+9,?)=i^K • • - (i) 
 
 If we add the expressions in (c), we get 
 
 * + */=/+//+/«. 
 Hence, if p be put for one fourth of one of the principal parameters 
 of the paraboloid, we shall obtain its value from the quadratic 
 equation 
 
 p'- [/+//+/JP+ [ff,+f,f„+f„f-i^+ff!'+ff,n'\ =0. ( j) 
 
 On the Hyperbolic Paraboloid. 
 
 89.] Of all the surfaces of the second order this is the most dif- 
 ficult to form clear conceptions about as to its configuration and 
 limits. Its sheets extending to infinity in different duections, the 
 application of the usual methods of algebraical investigation 
 becomes somewhat vague and indistinct. This surface admits of 
 no circular sections, nor can it be cut by a plane in a closed curve. 
 It is its own reciprocal polar, and is one of those surfaces called 
 gauche by the French mathematicians. 
 
ON THE TANGBNTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 79 
 
 Its genesis and form may best be conceived by the help of the 
 following mode of generation. 
 
 Let three rectangular axes be assumed. In the plane of XZ, 
 suppose^ let a parabola be described, having its vertex at the origin, 
 and its axis of figure coinciding with the axis of Z. Let its equa- 
 tion be a?* = 'ikz. Through the axis of Z let two planes be conceived 
 to pass, equally inclined by the common angle 6 to the plane of XZ, 
 the plane of the above mentioned parabola. These planes may be 
 called the asymptotic planes of the surface. Now let the plane of 
 a variable hyperbola, having its centre always on the axis of Z, be 
 conceived to move parallel to the plane of XY, and let this moving 
 horizontal plane — assuming the plane of XY to be horizontal — cut 
 the vertical asymptotic planes in two straight lines meeting in the 
 axis of Z. Let these lines be the asymptotes of the moving hyper- 
 bola, which shall have its vertex on a point of the guiding parabola. 
 Then the ordinate A of this parabola wiU be half the transverse 
 axis of the hyperbola, and the other semiaxis B will be A tan 0, 20 
 being the angle between the asymptotic planes. 
 
 Let OVV, be the guiding parabola in the plane of ZX. Let 
 OAZA( and OBZB, be the two asymptotic planes cutting the plane 
 
 Fig. 18. 
 
 of XOY in the lines OA, OB. Let HVP, H,V,P, be the moving 
 hyperbola in two successive positions ha^dng its vertex always on 
 the parabola OVV^. 
 
 It is to be observed that while the elliptic paraboloid may be 
 
80 ON THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 generated by the parallel motion of the plane of a variable ellipse 
 guided by two parabolas whose planes are at right angles, and 
 having a common axis, that of Z, the hyperbolic paraboloid is 
 generated by the parallel motion of the plane of a variable hyper- 
 bola guided by one parabola and by a pair of asymptotic planes. 
 
 90.] From the foregoing construction it will be evident that the 
 entire of this surface, while its sheets extend to infinity in opposite 
 directions, will be TrhoUy confined between the asymptotic planes, 
 AOZA, and BOZB; inclined one to the other by the angle 29. As 
 the surface approaches the plane of the guiding parabola or the 
 plane of XZ,it will thin down to an edge; and this edge will be the 
 guiding parabola. 
 
 It will also be shown further on, that while above the plane of 
 XY the surface is confined between the two asymptotic planes 
 whose angle of intersection is 26, below the horizontal plane the 
 surface will be developed between the same asymptotic planes, but 
 in the supplemental angle ir—20. 
 
 91.] The equation of the guiding parabola being a;'=4Axr, let the 
 
 equation of the moving hyperbola be ^—^=1. 
 
 Now, A being an ordinate of the parabola, A'=4A:r; and as 
 B=Atan 0, B'=A^/*=4^A;2r, if tan 6 be put equal to t. 
 Hence the projective equation of the paraboloid becomes 
 
 t^j;'^-y^-4fikz=0 (a) 
 
 To obtain the tangential equation of this surface. 
 
 Putting F for this expression, and taking its partial difiierentials, 
 
 dF_2<2^ dF dF \ 
 
 Ax-'^^^' Ay- ^y' d^- *^*' 
 and L . . . (b) 
 
 dF ^dF ^dF .,., 
 
 dS^ + %y + d^^=*^*^' 
 hence, as in sec. [80], (c). 
 
 Substituting these values of w, y, and z in the projective equation 
 (a), we obtain for the required tangential equation of the surface, 
 
 k}?-kfiv^Jr%=0 (c) 
 
 It will be found that the system of tangential coordinates affords 
 peculiar facilities for the investigation of the properties of this 
 surface. 
 
 92.] To ascertain whether in a tangent plane to this surface there 
 can exist any linear generatrices. 
 
ON THE TANGENTIAL EQUATIONS OP THE PARABOLOIDS. 81 
 
 Let the tangential equation of the straight line which we shall 
 assume as in a tangent plane, and a generatrix of the surface, be 
 
 f=K+a; (a) 
 
 and as this line is to be in a tangent plane, it must satisfy the tan- 
 gential equation of the surface, kP—kt^v + ^=0. Eliminating 5 
 between these equations, we get 
 
 ^'^h-jiT''^ (^) 
 
 Now in order that this equation may break up into factors repre- 
 senting straight lines on the surface, we must have 
 
 :^=;i^°V*=-*" (''^ 
 
 Making these substitutions, and taking the square roots, we shall 
 have as the resulting equation, 
 
 i=z±tv+2a (d) 
 
 This is a very remarkable result. As t the coefficient of v is 
 independent of ft and a the constants in the given tangential equa- 
 tion of the straight line, it will follow that no matter where the 
 point on the plane XZ may be, through which the assumed line 
 passes, the projection of this line on the plane of XY will always 
 be parallel to one of the asymptotic planes ; and as this projecting 
 plane is also a tangent plane to the surface, it will follow that all 
 vertical planes drawn parallel to the asymptotic planes are not only 
 tangent planes to the surface, but they also cut it in linear gene- 
 ratrices. 
 
 93.] To ascertain whether any other tangent planes can be drawn 
 through these linear generatrices besides the vertical planes parallel 
 to the asymptotic planes. 
 . The equation of the paraboloid being 
 
 k^^-k^v'^-{-^=0, (a) 
 
 let £=\i/4-7, (b) 
 
 be the tangential equation of the straight line in the plane of XY. 
 If we eliminate v from these equations, the result becomes 
 
 A{<«-\*)|«-2A7/«f-|-A/V-'^*?=0 (c) 
 
 Now this equation cannot be made to represent the equation of 
 a straight line unless \=^, and the preceding equation (b) thus 
 becomes j= tv -f- 7, 
 
 or, in other words, the line in the plane of XY must be parallel to 
 the asymptotic planes as before. 
 
 On this supposition the equation (c) becomes f^^r^+i, which 
 is identical with (a), sec. [92], if we make 7=3o, for then ^t-=A'- 
 
 6 
 
--^■'i c^) 
 
 8.2 0\ THE TANGENTIAL EQUATIONS OF THE PARABOLOIDS. 
 
 No two successive generatrices of this surface can meet, or be in 
 the same plane. 
 
 They cannot meet, for they lie all in planes which are parallel 
 to one of the asymptotic planes ; nor can they be ever in the same 
 plane, for the projection of these generatrices intersect when they 
 are projected on any plane but that of XY. 
 
 If we do not impose on a rectilinear generatrix the condition that 
 a tangent plane shall pass through it, its equations in the three 
 coordinate planes may easily be found. 
 
 The equation in the plane of XY being S=+tv + 2a, the equa- 
 
 — K 
 tion in the plane of XZ will be 5=Tt^ + ''j while the equation m 
 
 the plane of ZY will become 
 
 4kat 
 
 94.] If we draw a series of tangent planes to the surface, all 
 parallel to the axis of Y, we shall have u=0, and the equation of 
 the surface becomes A;f^ + ?=0, the tangential equation of the 
 guiding parabola, or, in other words, a cylinder whose axis is parallel 
 to the axis of Y circumscribes the surface along the guiding para- 
 bola. 
 
 If we make f=0, the general equation becomes Ai't/*— $=0, 
 which is the equation of a parabola in the plane of ZY and situated 
 below the horizontal plane. 
 
 When ?=0, k^—kt^v'^^0, or f= +tv, or the tangent plane that 
 coincides with the axis of Z cuts the surface in the plane of XY 
 along the sections of the asymptotic planes with the plane of XY. 
 
 Had we assumed the point in the plane of YZ and taken the 
 tangential equation of the right line in that plane i;=v^+/3, and 
 eliminated ^ from this equation and from that of the surface, we 
 should have found 
 
 ^=tv—2^t. 
 
 Or, in other words, the projection of this line on the plane of XY 
 must be parallel to one of the asymptotic planes. 
 
 In the general equation, if we make t=l, or the moving hyper- 
 bola an equilateral hyperbola, and take the asymptotic planes as 
 the coordinate planes of ZX and Z Y, the tangential equation of the 
 hyperbolic paraboloid assumes the very simple form 
 
 2Afi; + ?=0 (e) 
 
APPLICATION OF ALGKBRA TO RECIPROCAL POLAES. 83 
 
 CHAPTER X. 
 
 ON THE APPLICATION OF ALGEBRA TO THE THEORY OF 
 RECIPROCAL POLARS. 
 
 95.] Let a cone, the projective coordinates of whose vertex are 
 p, q, r, envelope a surface of the second order ; the curve of contact 
 is a plane curve. 
 
 The general tangential equation of a central surface of the second 
 
 order, referred to rectangular axes passing through the centre, is 
 
 ar + a,v' + a«?H2^y?+2^,f$ + 2/3,^u-l = 0=<I». . . (a) 
 
 Then, as this is a homogeneous function in f , v, ?, we shall have 
 
 . d*^ , d* d* „ ,, , 
 
 ^=dr^+dl7''+d?=2 (b) 
 
 d* 
 
 df d<I> 
 Now iF=-^ = i-^j hence, as shown in (b), sec. [80], 
 
 y=a,u + /3„?+^?,[ (c) 
 
 X, y, and z being the projective coordinates of the point on the surface 
 touched by the limiting tangent plane. 
 
 Now as these expressions are linear, we may find the values of 
 f, V, ? in terms of x, y, z, and the resulting equations will also be 
 linear. Hence 
 
 |=La? +Mjr +Nr, ) 
 
 w=L;a:+M,y +NfS-,t (d) 
 
 and as the tangent planes to the surface must all pass through the 
 verteK of the circumscribing cone, of which the projective coordi- 
 nates are p, q, r, these tangential coordinates f , v, \ must satisfy 
 the equation pi+qv+r^=.\. Introducing the values of f, v, and 5 
 given above, we obtain the resulting equation, 
 
 (Lp-|-L^ + L„»-)a7+ (MjB + M^ + M„r)y + (Nj9 + N^y + N„r)2'= 1, (e) 
 
 the projective equation of a plane, whose projective coordinates are 
 jr, y, z, and whose coeflBcients L, M, N, &c. are functions of the 
 coefficients of the given tangential equation of the surface (a) . 
 
 Now let fj. Up and ^, be the three tangential coordinates of this 
 plane, we shall have 
 
 f, = Ljo+L,g'-|-V, 1 
 
 i;,= MjB + M,y + M„r,^ (f) 
 
 ?,=NjB-|-N,9-|-N„r. ) 
 
 g2 
 
84 ON THE APPLICATION OP ALGEBRA TO 
 
 These are the remarkable relations that exist between j?, g, r, the 
 projective coordinates of the vertex of the circumscribing cone, and 
 the tangential coordinates J,, v, J^, of its plane of contact ; or, in 
 other words, if p, q, r are the projective coordinates of the pole, 
 fp Vp ^, are the tangential coordinates of the polar plane. 
 
 96.] We may illustrate this theory. 
 
 Let the surface of the second order be an ellipsoid, referred to 
 its axes as axes of coordinates ; its tangential equation is as in 
 sec. [74], 
 
 a*|«+4V + cT=l (a) 
 
 Now referring to the group of formulae (e) in the preceding sec- 
 tion, we shall have 
 
 L=^,. M,=i N„=i and L,, L„, M, M„, N, N, each =0. 
 
 Hence p = a^^„ q=b^v„ r=c'$;, (b) 
 
 Substituting these values of p, q, r, in the dual equation 
 ^f +g'w + r?=l, we shall have 
 
 a%'^+b''v,v + c%K=l (c) 
 
 We may therefore conclude that if an ellipsoid referred to its 
 axes, and whose tangential equation is o^f' + 6'ii* + c'5^=l,'be cut 
 by a plane whose taugeatial coordinates are f,, u,, ?/> the equation 
 
 a^U+b\^ + '^%K=^ (d) 
 
 will be the tangential equation of the pole of this plane. 
 
 97.] Instead of p, q, r, which for the sake of clearness we have 
 hitherto used as the projective coordinates of the vertex of the cone, 
 we may write the more common symbols x, y, and g for p, q, r. 
 
 These equations now become 
 
 v,= '\Lv + Mff + M,^X , or i/,=Y, |- . ... (a) 
 
 From these equations, which on the ground of their importance 
 we shall call The Polar Equations of surfaces of the second order, 
 may the whole theory of reciprocal polars be derived by the appli- 
 cation of the elementary principles of common algebra. 
 
 Thus if the polar plane be fixed, fj, v,, and ?, are constants, hence 
 X, y, and z are constants, or the pole is fixed. When J., Vp and 5, 
 are connected by two linear equations, so also are X, Y, Z ; or if 
 the polar plane pass through a fixed straight line, the pole will 
 also traverse a fixed straight line. 
 
 When 0y, Vp ?, are connected by one linear equation, so also are 
 X, Y, Z ; or when the polar plane passes through a fixed point, the 
 pole traverses a fixed plane. 
 
THE THEORY OF RECIPROCAL POLARS. 85 
 
 When f^, Vf, 5, are connected by an equation of the second degree, 
 so also are X, Y, Z ; or if the polar plane envelopes a surface of the 
 second order, the pole traverses a surface of the second degree. 
 
 Generally, if the tangential equation of a surface be <f>(^, v, ?) =0, 
 the projective equation of its reciprocal polar will be </)(X, Y, Z) =0. 
 
 Should for simplicity the directrix or polaiizing surface be a sphere 
 of radius equal to unity, the tangential equation of a surface being 
 4>{S> "j ?) =0, its reciprocal polar will be <f>{x, y, z) =0, and con- 
 versely. 
 
 By the aid of this very remarkable theorem, and of the properties 
 of tangential equations already discussed, we may reduce the whole 
 theory of reciprocal polars under the dominion of common algebra, 
 with the utmost simplicity. The following are a few of the most 
 obvious subordinate relations that may be derived from this cardi- 
 nal theorem. 
 
 Given the projective equation of a surface /(a;, y, z) =0, or the 
 tangential equation of the same surface ^(f, v, $) =0, we may write 
 down the tangential or the projective equations of its reciprocal 
 polar by simply interchanging the letters x, y, z with f, v, ?• 
 
 Let us conceive a figure composed of points, straight lines, planes, 
 curves of single or double curvature, and curved surfaces, a surface 
 of the second order being taken as the auxiliary or polarizing surface. 
 "We may imagine another figure constructed, whose points, straight 
 lines, and planes shall be the poles, conjugate polars, and polar 
 planes of the planes, straight lines, and points of the original figure. 
 These two figures may be called reciprocal polars*, one of the 
 other. 
 
 From the reciprocal relations between the two equations 
 
 <^(?,v, ?)=0, <i>[x,y,z)=0, 
 
 we may infer the following conclusions : — 
 
 1. 
 A plane is the reciprocal of a point. 
 
 2. 
 A straight line is the reciprocal of a straight line. 
 
 • Let a point be assumed on a surface (S), and the polar plane of this point 
 taken relative to a surface of the second order (C) ; as the assumed point varies 
 on the surface (S), the polar plane envelopes a surface (2), which is called the 
 reciprocal polar of the given surface (S). — Anuales de MathiSmatiques, par 
 Gergonne, torn. viii. p. 201. 
 
 Curt-es and curved surfaces, so related, may with propriety be called reciprocal 
 polars, because it is obviously a matter of indifference -whether the polar plane 
 of the pole on (S) envelopes ihe surface (2), or the pole of the plane enveloping 
 (2) describes (S). [Note to tirst edition.] 
 
86 ON THE APPLICATION OP ALGEBKA TO 
 
 3. 
 
 n planes are the reciprocals of n points. 
 
 4. 
 n planes passing through a Are the reiprocals of n points 
 straight line in a straight Ime. 
 
 5. 
 
 n straight lines in a plane. n straight lines meeting in a 
 
 point. 
 
 6. 
 
 The point in which a plane is The plane which passes 
 pierced by a given straight line, through a point and a given 
 
 straight line. 
 
 7. 
 A cone whose vertex is at a A curve lying in a given plane, 
 given point. 
 
 8. 
 A polygon of n sides in a A pyramid of n sides passing 
 plane. through a given point. 
 
 9. 
 A point on a curved surface. A tangent plane to a curved 
 
 surface. 
 
 10. 
 The point of contact of a tan- The tangent plane drawn 
 gent plane. through a given point. 
 
 11. 
 A cone circumscribing a given A plane section of the reci- 
 surface. procal surface. 
 
 12. 
 A number of surfaces inscribed As many reciprocal surfaces 
 in the same cone. intersecting each other in the 
 
 same plane. 
 
 13. 
 
 A chord joining two points of The intersections of two tan- 
 a surface. gent planes to the reciprocal 
 
 surface. 
 
THE IHEOKY OF RECIPROCAL FOLARS. 87 
 
 14. 
 
 A polyhedron of n faces in- A polyhedron of n solid angles 
 scribed in a surface. circumscribed to the reciprocal 
 
 surface. 
 
 15. 
 A polyhedron of re edges in- A polyhedron of n edges cir- 
 scribed in a surface. cumscribed to the reciprocal 
 
 surface. 
 
 16. 
 A number of surfaces meeting The same number of reci- 
 in a point. procal surfaces touching the 
 
 same tangent plane. 
 
 17. 
 A surface passing through n The reciprocal surface touch- 
 given points. ing n given planes. 
 
 18. 
 A number of parallel straight The same number of straight 
 lines. lines all lying in a plane passing 
 
 through the origin. 
 
 19. 
 A curve in a plane passing A cylinder whose axis is per- 
 through the centre of polarizing pendicular to the plane of the 
 sphere. given curve. 
 
 20. 
 A number of straight lines in As many straight lines per- 
 a plane passing through the pendicular to the plane passing 
 centre of the polarizing sphere. through the centre of the polar- 
 izing sphere. 
 
 21. 
 A polygon of n sides inscribed A polygon of n sides circum- 
 in a curve. scribed to the reciprocal curve. 
 
 22. 
 The sides of a polygon in- The lines which join the cor- 
 scribed in a curve meet two by responding angles of the reci- 
 two on a straight line. procal curve meet two by two 
 
 in a point. 
 
88 ON THE APPLICATION OF ALGEBRA TO 
 
 23. 
 
 The vertices of a triangle move The sides of the reciprocal 
 along three fixed lines. triangle will pass through three 
 
 fixed points. 
 
 24. 
 A conic section. A conic section. 
 
 25. 
 A plane at infinity. The centre of the polarizing 
 
 sphere. 
 
 26. 
 A straight line through the A straight line at infinity, 
 centre of the polarizing sphere. 
 
 27. 
 A plane through the centre of A point at infinity in the per- 
 the polarizing sphere. pendicular to the plane. 
 
 28. 
 n points given on a curve of n tangent planes to the same 
 double curvature. developable surface. 
 
 29. 
 A plane intersects a curve of n tangent planes through a 
 double curvature in n points. point to the same developable 
 
 surface. 
 
 30. 
 n points common to two or n tangent planes common to 
 more curves of double curvatiire. two or more developable sur- 
 faces. 
 
 31. 
 A series of tangent planes As many points of contact of 
 drawn to a curved surface a curved suiface with a tangent 
 through a point on it. plane. 
 
 32. 
 A cusp on the surface of the A curve of contact with a 
 one. tangent plane to the other. 
 
 33. 
 A surface is generated by Its reciprocal polar is gene- 
 straight lines. rated by straight lines. 
 
THE THEORY OF RECIPKOCAL POLARS. 89 
 
 As the reciprocal polar of a surface of the second order is also a 
 surface of the same degree, a great variety of the properties of these 
 surfaces may be derived in this manner, and thus a duality exists 
 between the properties of curves and surfaces of the second degree, 
 which in the general case is found only between curves and surfaces, 
 and their reciprocal polars. 
 
 98.] We shall proceed to illustrate the foregoing principles by 
 their application to a few examples. 
 
 The sum of the perpendiculars let fall from n given points on a 
 plane is constant. To determine the envelope of the plane. 
 
 Let the sum of the perpendiculars be nk ; and let p, q, r be the 
 projective coordinates of one of the points, P the perpendicular 
 from this point on the plane whose tangential coordinates are |, v, 
 and ^. 
 
 Now P=l^=2^, p -k^^^qj?, &c., see (c), sec. 
 
 [57]. _ 
 Taking the sum of all the perpendiculars, we shall have 
 
 n+—{p+p, +p„ + Pu, ka.)i-[q + g j + g»+g„< & c.) v 
 
 -{r-{-r,+r„-^r,„ &c.)?=»A Vr + i'' + ?^ 
 
 Let a, b, c be the coordinates of the centre of gravity of the n 
 points, then 
 
 p+Pi+Pi, kc.=na, q + q, + qiikc.=nb, r+»*, + r„ &c.=nc, 
 and the equation of the locus now becomes 
 
 or, squaring and adding, 
 
 (A« - a«)£2 + (A«-6*)i;«+ (F -c*) f'-2aAf u-2ac|? -2Aci;n 
 
 + 2af + 2Ji; + 2c? = l./ ^^' 
 
 The tangential equation of a sphere whose centre is at the centre 
 of gravity of the n given points, as may be inferred from (d) [75] . 
 
 When the sum of the perpendiculars is = 0, or k=0, the equation 
 becomes af + 6u + c?=l, 
 
 the tangential equation of a point. 
 
 99.] The sum of the squares of the perpendiculars let fall from n 
 given points on a plane whose tangential coordinates are ^, v, ^ is 
 constant, and equal to nk^ ; the plane envelopes a surface of the second 
 order. 
 
 Let the projective coordinates of the given points on the coordi- 
 nate planes be p, q, r, p,, q,, r„ &c., then, as 
 
 l-^f-gi^? _ \-p,^-q,v-r,^kc. ^^ 
 
90 ON THE APPLICATION OF ALGEBRA TO 
 
 we shall hare 
 
 + {r^ + r s + r/ &c.) ?^ + 2 (/»<? +p,q, +p,iq„ &c.) fw 
 
 + 2(;»r+/),r, +;>,/•„ kc.)^t;+2{qr + qir,+g,pr„ &c.)u? |. (a) 
 
 -2{r + 7-, + r„&c.)?+»=n**(r' + ''' + ?')-' 
 Now let the centre of gravity of the n points be taken as the 
 origin of coordinates, and let a, b, c be the radii of gyration round 
 the three principal axes of the system of n points, taken as axes of 
 coordinates, then p^ +pf +p,f + &c. = ««*, j* + qf + q,f + &c. = wi«, 
 r*+r/+r„2+&c.=nc«. 
 
 We shall have also p-\-pf-\-p„ + kc.—0, q-\-q, + q„-\-&c.=0, 
 r+r,-\-r,i+kc.=0,pq+piij,+p,ff„+Sx.=0,pr+p,r,+p,rii+&cc.=0, 
 qr + q,ri+q,f,i+&x;.=0; hence the equation (a) now becomes 
 
 (A«-a')P+{A*-&«)w* + (A*-c2)?*=l, . . . (b) 
 
 the tangential equation of an ellipsoid. 
 
 The distances of the foci of the principal sections of this surface 
 from the centre are independent of k. 
 
 Now k depends on the magnitude of the sum of the squares of 
 the perpendiculars, while a, b, c depend on the relative positions of 
 the n given points. Hence we may infer that if from the same n 
 given points there be let fall different groups of perpendiculars on 
 different planes, the sums of whose squares shall be mA*, nk^, &c., 
 the several tangent planes will envelope as many confocal ellipsoids. 
 
 100.] A series of surfaces of the second order touch seven fixed 
 planes ; the poles of any given plane relative to these surfaces are 
 also on a fixed plane. 
 
 Let the given plane be taken as that of xy, and let the equation 
 of one of the surfaces be 
 
 a^ + a,i^ + a„^ + Ufivi;+5fi^ii+2fi,j£v + 2y^+2y,v+2y,^=li (a) 
 
 the tangential equation of the pole of the plane of xy relative to 
 this surface is 
 
 a„K + Pv + fi^ + y„=0, see (d), sec. [77]. . . (b) 
 
 Now as there are seven linear equations to determine the nine 
 unknown coeflBcients of (a), we may eliminate any six, and connect 
 the three remaining by an equation which also will be linear. Eli- 
 minating then a, a,, ^,,,7, y„ y„, and a, a,, a,,, jS„, y, y,, successively, 
 we shall obtain 
 
 a„=-Lfi +M^, +N, 
 
 7„=L,^ + M,/3,+N, 
 
 :i 
 
 (c) 
 
 L, M, N, L„ M,, N, being determinate functions of the twenty-one 
 constant tangential coordinates of the seven fixed planes. 
 
THE THEORY OF RECIPROCAL FOLARS. 91 
 
 Substituting these values in (b) we get the equation 
 
 (L? + u+L,)/3+(M?+?+M,)/3,+(N;+N,)=0. . (d) 
 
 Now this equation is satisfied, leaving /3 and /3j indeterminate, 
 by putting each of the three factors in (d)=0; solving these equa- 
 tions, we find 
 
 ^=constant, v=coustant, ^=constant; 
 
 the three tangential coordinates of a fixed plane. 
 
 When there are eight fixed planes, it maybe shown, in like manner, 
 that the locus of the poles of any given plane relative to those surfaces 
 is a right line. 
 
 101.] A surface of the second order touches seven givenplanes, to 
 find the locus of its centre. 
 
 Let the tangential equation of the given surface be 
 
 ap + a,v^ + a,}? + 2/8v? + 2/9^? + 2/3„f ? + 27? + 2y,v + 27„? = 1, 
 and let the twenty-one coordinates of the seven given planes be 
 fi. Vp %,; %,„ v,„ $„; ^1,1, y,i„ ^,,1, &c. Substituting these values suc- 
 cessively in the preceding equation, we shall have seven linear 
 equations by which we may eliminate the six quantities a, a,, a^ ; 
 |3, /3p ^1,. The resulting equation will also be linear, and of the 
 form 
 
 Ly-|-M7,-f-Ny„=l, (a) 
 
 which is the projective equation of a plane. Now y, y„ 7^, as has 
 been shown, are the projective coordinates of the centre of the 
 surface. Hence the centre of the surface moves along a plane. 
 When there are eight planes, we may then eliminate y or y,, and 
 the two resulting equations will become 
 
 Ly-fM7,-l = 0, %-hN7«-l=0, 
 
 or the centre will move along a right line. 
 
 102.] If two surfaces of the second order are enveloped by a cone, 
 they may also be enveloped by a second cone. 
 
 Let the vertex of the cone be taken as the origin of coordinates, 
 and let the tangential equations of the surfaces be 
 
 ar-l-a,v«-l-a„?*-|-2;Si;?-|-2yS,f?-|-2/S,^«-|-27?-|-27,v-|-27„?=l,"l ,. 
 a^ + af^ + a,!^ + 2bv^-\-2b^% + 2b,^v-^2t^-^2c,v-\-2c,i;,= \; J 
 
 and as the common tangent planes must pass through the origin, 
 f , V, ?, are the same in the equations of the two surfaces ; but at the 
 
 origin ^=0, -=0, -^=0. At this point let f =^?, v=^^. Sub- 
 stituting these values in the preceding equations and dividing by 
 
 ?=oo, 
 
 a^« + a,f^ + a, + 2fiyjr + •2^,<l> + 20,ff>f=O, 1 
 
 a^^ + a,y^^ + a„+2bylr + 2bi^ + 2biffif=0; j ' ' ^ > 
 
92 ON THE APPLICATION OP ALGEBRA TO 
 
 and as these equations represent the same tangent plane, they 
 must be identical. Hence we shall have, introducing an equalizing 
 factor \, 
 
 a=\a, af=Xa,, aii=\aii, 6=X/3, bf=-\^f, bi,-=.\^fi. 
 
 Making these substitutions in the preceding equations, they become 
 
 Xa£* + Xa,u« + Xxi„?2 + 2X/3?i; + 2X^^? + 2X^„f v + 2cf + 2C;V + 2c„?= 1 . 
 
 Multiplying the former equation by X, and subtracting from it the 
 latter, we get 
 
 2(Xy-c)? + 2(Xy,-c,)i/ + 2(X7„-c„)?=X-l, . . (c) 
 
 the tangential equation of a point which is the vertex of the second 
 enveloping cone. » 
 
 The projective coordinates of this point are 
 
 2(X7-c) 2(Xy-c,) 2(X7,-c„) 
 
 "^ \=V' ^~ X-1 ' ^- X-1 
 
 103.] het a plane cut off from three fixed rectangular axes seg- 
 ments, the sum of which, multiplied by a constant area, shall be equal 
 to the tetrahedron whose faces are the three coordinate planes and the 
 limiting tangent plane ; to determine the surface enveloped by this 
 latter plane. 
 
 Let the three coordinates of the variable plane be f , v, ? ; then 
 
 . . 1 e* 
 
 the volume of the pyramid is = Tr=-i- Let the constant area be rr, 
 
 6gi/? 3 
 
 consequently 
 
 is the equation of the envelope of the tangent plane, the tangential 
 equation of a surface of the second order. 
 
 To find the axes of this surface. Comparing the above with the 
 general tangential equation of a surface of the second order, 
 
 af2 + a,v«+a„?«4 2^i/? + 2^,|?+2a,^u + 2y£ + 2y,v + 27,^=l, 
 
 we shall have 
 
 «=«/=«//=0, 7=7/=yH=0, ^=^,=/3^^a*. . (b) 
 
 Substituting these values in the cubic equation which determines 
 the magnitude of the axes, sec. [83], (f ), we get 
 
 T6_3flV-2a6=0 (c) 
 
 Taking its first derivative, we shall have t*— a*=0. 
 
THE THEORY OF KECIPROCAL POLARS. 93 
 
 Now putting (c) in the form 
 
 T«(T*-a*) -2a*{T^ + a^) =0, 
 
 we find that this equation and its first derivative are satisfied by 
 the root t*=— a*. 
 
 Hence two of the roots are each equal to — o*, and the remaining 
 root is =2a*. This we might antecedently have inferred from the 
 absence of the second term in (c) . 
 
 Consequently as two of the roots are negative and one positive, 
 the surface is a discontinuous hyperboloid or one of two sheets. 
 
 Since the linear terms do not appear in (a), the centre of the 
 surface is at the origin. 
 
 To determine the position of the axes. In equations (h), sec. [84] , 
 substituting for 8(8„— /3* its value 3o*, and finding the same values 
 for the other like expressions, we obtain 
 
 cos*\=5, cos'/x=5, cos*v=-j . . . (d) 
 o o o 
 
 hence the positive axis of the surface is equally inclined to the axes 
 of coordinates. 
 
 If we were to make in the same formula the necessary substitu- 
 tions to obtain the position of the two other axes, we should find 
 
 cos\,=^, and so for the other angles also. This evidently should 
 
 be the case, as the two equal axes have no definite position. 
 
 To transform this equation into one that shall contain the squares 
 of the variables. 
 
 Let the cosines of the angles which the new axes of coordinates 
 OX, OY, OZ make with the original axes be 
 
 as shown in (d) sec. [56] . Then 
 
 Substitute these values in (a), and we shall find 
 
 2o*f ^« + 2o* [ml+nl + mti] u,« + 2a* [m/, + «// + »»;'«;] ?/* 
 + 2a* [mil +m^+ nl, + Ufl + mp, + w,to] v^^ 
 
 ^2a*M^f,^, + 2a« (i±^f,„ -1. 
 
 (e) 
 
 As the two systems of axes are rectangular, we must consequently 
 have, as shown in sec. [56], Z+jra+«=0, /,+m( + «,=0, and 
 
 mil + r«/ + «/; + n,l + mp, + w,»n=0. 
 
94 ON THE APPLICATION OF ALQEBRA TO 
 
 But this last expression may be written 
 
 m (l+n) +ni{l+m) +l,{m + n)=0, 
 but l+n^—m, l+m=—n, m-\-n= — l, 
 
 hence W,+»»»i, + «W(=0. 
 
 Now as l+m+n==0, the square of this expression must be =0, or 
 
 2lm + 2ln + 2mn= — (P + m^ + n^)=—l; 
 and as 
 
 li+m, + ni=0, 2l,mi+2lin, + 2m,ni= — (lp + m,^+n^)= —1. 
 The equation (e) from these substitutions, now becomes 
 
 c«[2f«-w«-r«]=i, (f) 
 
 the equation of the discontinuous hyperboloid referred to its axes 
 as axes of coordinates. 
 
 104.] If a series of planes retrench from a cone of the second degree 
 a constant volume, they will envelope a discontinuous hyperboloid, or 
 one of two sheets. 
 
 In the first place, it may be shown, that if the projective equa- 
 tion of a conic section be 
 
 Air«+A,y*-l-2Bary + 2Ca: + 2C^=l, ... (a) 
 its area S will be 
 
 ^[(A+C')(A,+C«)-(B + CC,)«] ^ 
 
 (AA,-B*)t . . . . (b) 
 
 * This proposition may he proved as follows : — 
 
 Let Ajr'+A,y'+2firy=l, (a) 
 
 be the equation of a conic section ; any rectangular axes passing through the 
 centre being taken as axes of coordinates, we may determme the axes of the 
 section by the following method, 
 
 _. , ,, , dF dF . dF dF . ,„„^ ,„„, 
 
 We shall have ^j,-g_^=0, or ^^=j^, win (33), sec. [22]. 
 
 ,.T dF cosX , dF eosX PcosX 1 . x, . ., 
 
 °'^ dx~ 'F~' "°® xAx~ Tx'~ Vx ~W """^ coincides with r. 
 
 But ^=P=2A.+2B|, by differentiation ; or p=A+Bi^. 
 
 We have also p=A,+B£; hence (^^_a)(^- A,)=B», 
 
 or p-(A-)-A,)p-|-AA,-B»=0 (b) 
 
 This equation determines the axes of the section. Hence the product of the 
 squares of the reciprocals of the semiaxes is 
 
 AA.-E', (c) 
 
THE THEOKY OF RECIPROCAL FOLARS. 95 
 
 X' 
 
 « yS ?« 
 
 Ijet ^+ T2"~^=0j ^ t^e equation of the cone, c being its real, 
 
 a and i its imaginary axes, and let a?f +yii + 2$ = l,be the equation 
 of the secant plane. 
 
 Eliminating z between these equations, we get the equation, on 
 the plane of xy, of the projection of the section made by the secant 
 plane cutting the cone. This equation is 
 
 (?~^y + (^-"')^'-2^''*y + 2f''+2.;y=l. . (c) 
 
 Substituting in the preceding expression for the area, the coeffi- 
 cients of this equation, we get for S, the area of the projection of 
 the section on the plane of XY, 
 
 S being the area of this section, then S sec will bs the area of the 
 section of the cone made by the secant plane, and if P be the per- 
 pendicular from the origin on this plane, the volume of the cone 
 
 COS Q ^ 
 
 wiU be equal to Ssec^.P, but P = — — , hence the volume =^. 
 
 and consequently the area 
 
 ^= vri^s w 
 
 Now let 
 
 Aj<»+A,y'+2Ba!y+2Cj:+2C,y=l (e) 
 
 be the general equation of a conic section, and let the origin of coordinates be 
 translated to a point x=x, + a, y=j/,+b, and make the resulting coefficients of 
 X, and y, severally =0; the equation of the curve referred to its centre will be 
 
 A.r2-|-A^»+2aiy= l-Ca-C,J, 
 
 «• * A i^x. ■ 1 BC,-A,C . UC-AC, ,. , , 
 
 or putting for a and b their values, o= . a- - R8 ' "aXTTW' ^I"**'^'"' («) 
 
 0GC0II16S 
 
 A;r»-HA.y»+2B^= (A+C»)CA.+C.°)-(B+CC,)' 
 
 Dividing by the absolute term, and writing A, A, and B for the new coeffi- 
 cients of 3?, y', and xy, in order to reduce the absolute term to 1, 
 
 - A(AA,-B') A,(AA,-B') 
 
 '^-(A+C»)(A,-|-C,»)-(B-|-CC,)" ^'-(A+C=')(A,+C,»; - (B+CC,)"' 
 
 ^ B(AA,-BO 
 
 ^- (A+C^XA-,+C,'')-(B+CC,)»' 
 
 Gonsequentlv as the area is =—7=^====, the area of any conic section in 
 
 terms of its general coefficients is 
 
 ^_ ir[(A+C')(A.+C,') - (B+CC.)'] ,j. 
 
 [(AA,-B»)]* 
 
96 ON THE APPLICATION OF ALQEBRA TO 
 
 Let this volume be also =iTabc, suppose, hence substituting in the 
 preceding expression this value of S, the resulting expression becomes 
 
 c95S-a«f*_fi*,;2=l^ (e) 
 
 the tangential equation of a discontinuous hyperboloid. 
 
 In nearly the same way it may be shown, that if a series of 
 planes retrench from any surface of the second order a constant 
 volume, the enveloped surface is a surface concentric similar and 
 similarly situated. 
 
 105.] On the cubic equation of axes, when the surface is one of 
 revolution, and the origin at a focus. 
 
 In the general tangential equation, of a surface of the second 
 order, 
 
 aP+a,u«+a„^ + 2^i;?+2/3^?+2^„e»' + 2rf + 2r,i;+2y,^=l. (a) 
 
 Let a=a,=aii, and /3=/3y=/3„=0, then the preceding equation 
 Dccomcs * 
 
 a^ + a^ + a};^ + Zy^+2y,v+'ly,/; = l. . . . (b) 
 
 If we now translate the origin of coordinates to the centre of the 
 surface, in parallel directions, using the formula given in sec. [75], 
 (d), the equation, referred to the central axes of coordinates, becomes 
 
 ^, ., . +Hfy^K+2y,ySv=i.j • • 
 
 Now if we write 
 
 {a+y^=a, {a+y^)=a, (a+V)=a, 
 
 
 yiyu=^> y,!Y=^i> fyi=^u" 
 
 the cubic equation of axes, see sec. [83], (f), will become 
 
 (C) 
 (d) 
 
 + ^*a + ^fa,+^,fau-aaflu- 2/3/3^,^=0.) 
 
 (e) 
 
 Or substitutmg for these coefficients their values as given in the 
 preceding equations (d), and putting t for P*, the cubic equation 
 now becomes 
 
 -[a8 + a«(7«+7,'+7„«)]=0./ " ^' ^ 
 Let a + y* + y,«+y„*=A, (g) 
 
 and the preceding equation may now be written 
 
 7«-(2a + A)T*+(a«+2aA)T-a«A=0, . . . (h) 
 of which equation the three roots are manifestly a, a, and *. 
 Let a and b be the semiaxes of this surface, then as 
 
 «'=«+7'+7/'+7«', and 4«=o, a'-*«=7«+7«+7„», • (i) 
 
THE THEORY OF RECIPROCAL FOLARS. 97 
 
 or the eccentric distance is equal to the distance of the origin from 
 the centre. 
 
 If we now turn to the formulae in sec. [84], (h), by which the 
 positions of the axes are determined, bearing in mind that 
 
 (a-T«)=S, (a,-T*)=S„ 
 or m this case 
 
 and putting for t* the value k, the square of the greater semiaxis of 
 the surface, since 
 
 we shall have 8S,-/3„'=7/(7' + r;'+7«'). \ (J) 
 
 and (8S -^„«) + (8A-/3«) + (8«S-/3/) = (y'+y/+7/,*)^) 
 
 Consequently the formulae for the determination of the inclination 
 of the major axis to the axes of coordinates are 
 
 cos \ = — , „ ' „ , cos u = — , „ *' ==,, 
 
 Vy'+yf+y/ '^ V/+y;«+y/ 
 
 cos v= -^ 
 
 (k) 
 
 Vv'+Y'+y/ 
 
 equations which determine the position of the semiaxis V^- But 
 the line drawn from the origin to the centre of the surface makes 
 the same angles with the axes of coordinates, therefore this line 
 coincides with the semiaxis ^k; hence its origin is on this axis, 
 and k—a=<t^+'i^ + yu, therefore the eccentric distance is equal 
 to the distance of the origin from the centre, or the focus is at the 
 origin. 
 
 Had we substituted o instead of k in the formulae, sec. [84], (h), 
 which determine the inclination of the axes of the surface to the 
 axes of coordinates, we should have found 
 
 ,0 
 
 COS\=^, C0S/A = jr, COS »' = g. 
 
 We may therefore infer that when in the general tangential 
 equation of a surface of the second order, the coefficients of the 
 squares of the variables are all equal, and the coefficients of the 
 rectangles are each =0, the equation represents a surface of revo- 
 lution, the origin of coordinates being at one of the foci, while 
 the coefficients of the linear terms are twice the projective coordi- 
 nates of the centre of the surface. 
 
 106.] Three straight lines, constituting a right-angled trihedral 
 angle, revolve round a fixed point in space, meeting a surface of the 
 second order (S) in three points. The plane which passes through 
 these three points envelopes a surface of revolution (S) of the second 
 
 H 
 
98 ON THE APPLICATION OF ALGEBRA TO 
 
 order, whose focus is at the given point, and whose directrix plane 
 relative to this focus is the polar plane of the fixed point relative to 
 the given surface (S). 
 
 Let the fixed point be taken as origin, and let the projective 
 equation of (S) be 
 
 + 2Cx-\-2C,y-\-%Ctfi=l.] 
 Now it may easily be shown that the projective equation of the 
 polar plane of the origin is 
 
 C^ + C,y + C,^=l* (b) 
 
 Let r be the length of one of the revolving lines, and let \, (i, v 
 be the angles it makes with the axes of coordinates, then 
 a7=rcos\, y^r cos fi, and ^^rcosv. 
 Substitute these values of x, <y, and z in the dual equation 
 x^+yv+e^^l, and it becomes 
 
 f cos \ + 1; cos /i + ? cos )'=-, 
 
 or writing I, m, n for co3\, cos/i, cosj' the preceding formula 
 becomes 
 
 li + mv+nZ,=- (c) 
 
 In the general equation (a), writing Ir, mr, nr for x, y, z, it 
 
 nftfiO TTl GS 
 
 AP + Ap« + A„w* + 2B»ira + 2B/« + 2B,/ot ) 
 
 11 L . . (d) 
 
 +2(a+c,m+c„«)i=-,. ) 
 
 Eliminating r between these equations (c) and (d), we obtain the 
 resulting expression 
 
 KP + Kfn^ + A„»* + 2Bmra + 2B> + 2B„/w» 
 + 2(C/2^ + Cin^v + CX5) + %Ql{mv + m?) 
 + 2C,TO(/f + M?) + 2C„M(/f + mv) 
 =/2j2 + ;„ V + n^S* + 2m«i;? + 2Z«f?+ 2/w»fu . 
 
 * Let the projective equation of a surface of the second order be 
 
 Ar'+A,y='+A„z»+2B5ra+2Bas+2B„ay+2Car+2C,y+2C„z=l. 
 Then it may be shown that the locus of the middle points of all the chords of 
 this suifiice passing through the origin of coordinates is the sur&ce whose equa- 
 tion is 
 
 Aj?+Ay+A,^»+2Byz+2B^+2B,^+Cx+C,y-l-Cz =0; 
 
 and if we subtract this latter equation from the former, we shall find the locus of 
 the intersection of these two surfaces. Subtracting the latter from the former, 
 the equation of the locus is 
 
 Ci+C,y+C„z=l, 
 the projective equation of a plane. 
 
 (e) 
 
THE THEORY OF RECIPROCAL POLAR8. 99 
 
 If we now find similar expressions for the other two revolving 
 lines r, and r„, adding all three equations together, and introducing 
 the six relations of the nine direction cosines given in (e), sec. [56] , 
 we shall obtain for the tangential equation of the enveloped 
 surface (S), 
 
 £8 + „9 + 52_2Cf-2C,v-2C„?=A + A,+A„. . . (f) 
 
 If we substitute -forA4-A,+A„ (g) 
 
 the preceding equation will be transformed into 
 
 a(f^ + i^' + ?')-2aCf-2aC,w-2oC,^=l. . . . (h) 
 
 Now as the coeflBcients of the squares of the variables are equal, 
 and the rectangles vanish, (h) is the tangential equation of a surface 
 of revolution whose focus is at the origin, as has been already shown 
 in sec. [105] . 
 
 The projective coordinates of the centre of (2) are aC, aC,, and 
 aCi,. The cosines of the angles which the major axis of (2) makes 
 with the axes of coordinates are 
 
 ^1 and ^" • 
 
 but these are the cosines of the angles which a perpendicular P 
 from the fixed point the origin, on the polar plane of this point 
 relative to (S) makes with the axes of coordinates ; hence the 
 major axis of (2) coincides with this perpendicular. 
 
 This plane is a directrix plane of (2); for if a and b be the semi- 
 axis of this surface, 
 
 b^=a and a^=a + a^C^ + C'' + C,^), see sec. [105], (i) ; 
 
 hence ^^=(^ + C^ + C,^=^^ or P=~ae, or P is the dis- 
 tance between the focus and the directrix plane. 
 
 107.] To show that the Continuous Hyperboloid admits of linear 
 generatrices. 
 
 The tangential equation of the continuous hyperbejloid 
 a^^ + b%^—c^^=l may be written 
 
 {a^+cK){a^-cK) = {l + bv){\-bv); ... (a) 
 and if we assume the equation of a straight line in the plane of XY, 
 
 p^^-qv = \; (b) 
 
 eliminating v between these equations, the resulting expression 
 becomes 
 
 (aY+*V)^-2*'i'f+*'-s'=cV?'-- • • (c) 
 
 H 2 
 
100 ON THE APPICATION OF ALGEBRA TO 
 
 Now this is the tangential equation of the curve in which the 
 tangent plane through the point {p, q) on the plane of XY cuts 
 the plane of ZX ; and in order that this intersection may be a 
 straight line each side of the preceding equation must be a complete 
 square, or 
 
 or aY + l>Y=a^b^ (d) 
 
 or p and q must be ordinates of the principal section of the surface 
 in the plane of XY. 
 
 If we introduce this value of a^q^ + b^p^mto the preceding equa- 
 tion, the result becomes 
 
 or taking the square root. 
 
 It is obvious that if either c^ were negative, or A' negative, the 
 square root would be imaginary. 
 
 Hence no surface, the squares of whose axes are all positive, or 
 one positive and two negative, can admit of rectilinear generatrices. 
 
 The preceding equation may be written in the usual form, 
 
 p — op 
 
 Asp and q are always ordinates of the principal ellipse in the 
 plane of XY, it follows that every rectilinear generatrix to the 
 continuous hyperboloid must always pass through a point on the 
 principal elliptic section of the surface. 
 
 108.] Let a surface of the second order be cut by a given secant 
 plane ; to determine the tangential equation of the section of the sur- 
 face made by this plane. 
 
 Let the tangential equation of the surface be 
 
 a^^+bhJ' + c^^=l (a) 
 
 Let the coordinates of the fixed plane which cuts the surface in 
 the section whose equation is required be ^„ v„ ^„ and let 0=»n?+ o, 
 v^n^+fi be the equations of a straight line in space. Let this 
 straight line be in the plane whose coordinates are ^„ v,, t,., then 
 these variables must satisfy the given equations, and we shall have 
 
 if-f,=»i(?-5), and v-Vf=nlX-Q. 
 Substituting these values of ^ and v in the equation ot the curved 
 surface, we shall have two resulting values of f, which are the reci- 
 procals of the intercepts of the axis of Z made by two tangent 
 planes passing through the straight line in the plane whose coor- 
 dinates are J„ v„ ^,. 
 
THE THHORY OP RECIFROCAL FOLAKS. 101 
 
 When this line becomes a tangent to the section of the surface 
 the two tangent planes coincide ; hence the two values of ? become 
 equal, and they are given by a quadratic equation whose roots 
 must be equal. Now 
 
 ««=v,«+2«v,(?-g+«»(?-S;)S K . . . (b) 
 
 Multiply the first by a^, the second by b^, the last by c*, and add 
 together these expressions. The result becomes 
 
 0=a«f,* + A V + c%^ -1+2 [a*™?/ + b^v, + €%-] (?- Q 
 + Ka« + n^^ + c*) (f- f/. 
 Now as this equation must have two equal roots, we shall have 
 (a«f « + b^v^ + c%2 _ 1)(otV + nH^ + c*)= [a^m^, + b^nv, + c«5;]^ (c) 
 and this may be reduced to the form 
 
 and if we substitute in this equation for m and n their values 
 
 | — gj _ V — Vj 
 
 the preceding expression becomes 
 
 
 } . . (d) 
 
 This expression may still further be reduced to the form 
 
 The projective equation of a cone circumscribing a given surface 
 pf the second order is 
 
 [aV + *'y/' + cV- 1] [«'^* + *V + c*^* - 1] ) «x 
 
 = [a*^^+**M + cV-l]S I • • I J 
 
 a?„ y^, and ar, being the coordinates of the vertex of the cone. The 
 duality of the two equations (e) and (f) is manifest. 
 
 109.] The reciprocal polar of any surface of the second order, the 
 centre of the directrix surface being on the given surface, is a para- 
 boloid. 
 
 The directrix surface being for simplicity a sphere whose radius 
 is unity, at whose centre the origin ojf coordinates is placed, let the 
 projective equation of the given surface be 
 Aa^+Aff^+A,fi^+2Byz+2B^z+2BifVy + 2Civ+2C,y+2C,^=0, 
 the tangential equation of its reciprocal polar is (see {d),sec. [87]) 
 A|« + A,w« + A„?* + 2Bi;? + 2B^? + 2B„?u + 2Cf + 2C(W + 2C„? =0, 
 the tangential equation of a paraboloid. 
 
102 ON CONCYCLIC SURFACES OF THE SECOND OKDEK. 
 
 CHAPTER XI. 
 
 ON CONCYCLIC SURFACES OF THE SECOND ORDER*. 
 
 110.] The properties of confocal surfaces of the second order, or 
 surfaces whose principal sections have the same foci, are discussed 
 at considerable length in various publications, especially on the 
 Continent, devoted to the cultivation of mathematical science; 
 while the dual properties of their reciprocal surfaces have not been 
 at all noticed, so far as I am aware. It is true that M. Chasles 
 and other geometers who followed him have discussed the proper- 
 ties of cyclic cones ; but the theory admits of much wider applica- 
 tion. 
 
 It is known, and is very easily proved, that every umbilical sur- 
 face of the second order may be described by the parallel motion 
 of a variable circle whose centre moves along a fixed line. 
 
 Concyclic surfaces may therefore be defined as concentric sur- 
 faces of the second order, having their axes coincident, and the 
 planes of their circular sections parallel. 
 
 Concyclic surfaces are the reciprocal polars of confocal surfaces. 
 
 Let a, b, c be the semiaxes of the original confocal surfaces in 
 the order of magnitude. Let a* - b^=h^, a® — c^= k^. In confocal 
 surfaces a, b, and c are supposed to vary, while h and k are constant. 
 
 Let c„ b„ a, be the semiaxes, in the order of magnitude, of the 
 derived surface. 
 
 The radius of the circular section of the derived ellipsoid which 
 passes through the centre is b, ; b, will be a semidiameter of the prin- 
 cipal section which contains the greatest and least axis, hence 6, is a 
 
 semidiameter of the principal section whose equation is ^+-s=l. 
 
 Let be the angle which b/ makes with the greatest axis, then 
 
 1 cos2 e sin* e , , 
 
 and hence 
 
 a,^ 
 
 tan^,=kf=£{i£^ (a) 
 
 1 1_ b^(c^-a^) 
 
 ,2 ^2 
 J, C, 
 
 * It is Btiange how the properties of systems of concentric surfaces of the 
 second order having coincident circular sections or, as they may be more briefly 
 termed, concyclic mr faces, have hitherto almost wholly, at least so far as the 
 author is aware, escaped the observation of geometers ; it is the more remarkable 
 as the theorems connected with the subject are numerous, and many of great 
 elegance. 
 
 So far indeed as the properties of such cones are concerned, and of spherical 
 conies thence derived, M. Chasles has discussed them in two memoirs of singular 
 simplicity and beauty, published in the 'Brussels Transactions,' tome v. 1829. 
 [Note to first edition.] 
 
ON CONCYCUC SURFACES OP THE SECOND ORDER. 103 
 
 This is the angle which the plane of the circular section makes 
 with the plane of the greatest and mean axis of the derived surface. 
 Now in confocal surfaces let b'^—c^=h^, a^—c^ssk^; hence 
 
 Jl_JL 
 
 W_}^-c^_ hf cf _ a,{cf-hf) 
 
 Comparing this with the preceding expression, we get tan^=Tj 
 
 but h and k are constant, hence tan 6 is constant, or all reciprocal 
 polars of confocal surfaces have their circular sections parallel. 
 
 In confocal ellipsoids b^ = c^-\- h^, a^=c^ + k^.) 
 In concychc eUipsoids p=^+j2, ^=p+p-) 
 
 111.] We shall give a few examples of the analogies between 
 these surfaces. 
 
 If parallel tangent planes be drawn to a series of confocal 
 surfaces, and perpendiculars from the centre be let fall upon them, 
 the differences of the squares of these perpendiculars will be inde- 
 pendent of their direction. Thus, let one of the perpendiculars be 
 P* = a« cos* \+b cos* /i + c* cos* v; now 6* = c* + A*, a*= c* + A*, hence 
 
 pa = c* + A2 cos V + A:* cos* \ ; 
 for any other perpendicular on a parallel plane 
 
 P 2= c,* + A* cos* /* + A* cos* \ ;| , . 
 
 hence P*-P*=c*-c,*. j 
 
 In like manner, if there be any number of concyclic ellipsoids 
 iaving coincident diameters, the differences of the reciprocals of 
 the squares of these diameters are independent of their position. 
 
 Let 
 
 cos*o cos*)8 cos*7_ 1 
 
 but 
 
 hence 
 
 1_1.A 1=1+1. 
 
 1 1 cos*a cos*/S , T 1 1 , cos*a cos*^ 
 ^=^* + -^ + -7^' -d also _=-,+ _^+-^. 
 
 Consequently 
 
 1_1-1_1 (b) 
 
 p* p*-c* c*- • ^ > 
 
104 ON CONCYCLIC SURFACES OF THE SECOND ORDER. 
 
 These are simple theorems, but they serve to illustrate the com- 
 plete identity that exists in the analytical investigation of the prin- 
 ciples of duality in all their diversified forms. 
 
 112.] Through a given point three central confocal surfaces may 
 be described : an ellipsoid, a continuous and a discontinuous hyper- 
 boloid, or as they are named by French writers, I'hyperboldide a une 
 nappe and I' hyperboloide a deux nappes. 
 
 Let a, /8, y be the projective coordinates of the point. Let 
 fl2=c2 + F, 62=c2^-A^ a^=cl + k^, b,^ = c,^+h'; a„^=c„^ + k^ 
 b/=Ci^ + h^, then the equation of the surface becomes 
 
 ¥+¥^c^+h^^7^-^' ^^' 
 
 or reducing. 
 
 Now as there are two permanencies and but one variation of sign 
 in this equation, we shall have, by the theory of equations, two of 
 the roots negative and one positive. The product of the squares 
 of the three coincident axes is y' A* k^. 
 
 Let c', c^, C/i^ be the three roots of this equation, then 
 
 or as 
 
 a^=c^ + k^, bf=cf^-h^, 
 we find that 
 
 or the sum of the square of the major axis of the first surface + 
 the square of the mean axis of the second surface + the square of 
 the least semiaxis of the third surface is equal to four times the 
 square of the distance of the given point from the origin. 
 
 Had the equation been solved for J', we should have found two 
 of the values of 6* positive and one negative ; while if the equation 
 had been solved for o^, the three values of a' would have been found 
 positive. Thus the three confocal surfaces passing through a point 
 are an ellipsoid, a discontinuous hyperboloid, and a continuous 
 hyperboloid. 
 
 If the two hyperboloids be assumed as constant and the ellipsoid 
 as variable, the successive points in which the variable ellipsoids 
 meet the curve in which the hyperboloids intersect are " corre- 
 sponding points " in the theory of the attraction of ellipsoids j for 
 since c' c^ Ci^=h^ A' 7', while c^ and c/ are constant, c varies as 7. 
 
 Had the equation (a) been solved for a^, we should have 
 
 (fi—a* [a« -I- /3« + y2 + A* + **] 4- a« [a«A« + a«A« + AV + jS*** + h^k^ 
 -a^h^k^=0, 
 
ON CONCYCLIC SXTRFACES OF THE SECOND ORDER. 105 
 
 in which there are three variations and no permanence of sign. 
 Hence the three values of a' are positive. 
 Since a« + a« + V=a2 + /32+y« + AHAS 
 
 «2+{a^S-A«) + (a„*-A*) or a^ + bl' + c,^=a^+^ + y''. 
 
 113.] A series of coney die surfaces of the second order touch a 
 given plane whose coordinates are f , v, f- To determine the equation 
 of the axes of these surfaces. 
 
 Let ««=! 6*=^, c« = i A'=i A« = i 
 
 then ^2=a«-\^ 'f^a^-K^, 
 
 and the tangential equation of a surface touching the fixed plane is 
 
 
 (a) 
 
 Reducing this equation, and arranging by powers of a, 
 a? -a* [w* + «« + x2] .|. flS [XV + (X^ + ««) |« + \2^ + k%^'] ] 
 
 
 Hence, as there are three variations of sign and no permanence 
 in this equation, the roots are all positive. 
 
 It may be shown that as the three confocal surfaces which pass 
 through a given point are each one of the three central surfaces of 
 the second order, so of the three concyclic surfaces which touch a 
 given plane one is an ellipsoid, the second a continuous hyper- 
 boloid, and the third a discontinuous hyperboloid. 
 
 114.] A common tangent plane is dravm to three concyclic surfaces 
 of the second order, the three points of contact two by two subtend 
 right angles at the centre. 
 
 Let a«^+AV + c«?2 = l, and a^^-^by + c^^=\, . (a) 
 
 be the tangential equations of two concyclic surfaces of the second 
 order. 
 
 Subtracting these equations one from the other, there results 
 
 (fl*-fl,*)f'+(J*-i,V+(c«-c;)?«=0. . . . (b) 
 
 But as the surfaces are concyclic, 
 
 1-1-1-1 1-1-1-1 1-1-1.1 M 
 
 a^~a^^k^' b,^~b^'^k^' c^~d'^k^' ■ ' " ^ 
 
 a*a* 
 hence a'— a,' =-73^, and like expressions for the other axes. The 
 
 preceding equation (b) may be transformed into 
 
 aVf*+**V''' + cV?*=0 W 
 
 Now it has been shown (sec. [96]) that if {x, y, z) be a point of 
 contact of a tangent plane, 
 
 x=a^Ji, y=b^v, z=c^^; (e) 
 
106 ON CONCYCLIC STTKFACES OF THE SECOND ORDER. 
 
 and if «^j X' V' *^^ **^® angles which the diameter 2r through the 
 point of contact makes with the axes^ 
 
 cos^=-=-^, cosx=— , cos-^=— ; 
 
 in like manner for the second point of contact^ 
 
 cos<^,= -^^, cosx,=-^. cosV^,=-^. 
 ~j '« '/ 
 
 Making the substitutions suggested by these transformations, the 
 preceding equation (d) becomes 
 
 /r,(cos^cos^, + cos;^cosxj+cos'^cos'^,)=0, . . (f) 
 
 or r and r^ are at right angles. 
 
 115.] Let there be two coney die ellipsoids, and any point on the 
 external one be assumed as the vertex of a cone enveloping the other, 
 the plane of contact will meet the tangent plane to the first surface 
 through the vertex of the cone in a straight line, such that the diametral 
 plane passing through this line loill be at right angles to the diameter 
 which passes through the vertex of the cone. 
 
 Let A, B, C be the semiaxes of the external ellipsoid, a, b, c 
 those of the internal ellipsoid. 
 
 ^^* A«~o* A** B'~62 A*' C*~c2 A*' •••(*) 
 
 and let p, q, r be the projective coordinates of the vertex of the 
 cone. 
 
 Now Bsp, q,r are the projective coordinates of the point on the 
 external surface through which the tangent plane is drawn, we shall 
 
 p=K%, q=-B,\, r=C%, (b) 
 
 f,, v„ and {) being the tangential coordinates of the tangent plane 
 to the external surface through the vertex of the cone. 
 
 Again, as p, q, r are the projective coordinates of the vertex of 
 the cone circumscribing the interior surface, the tangential coor- 
 dinates of ^e polar plane of this point will be given by the equations 
 
 p = a%„ q=l^„, r=c%„ (c) 
 
 f//' "«» ?« being the tangential coordinates of the polar plane of the 
 vertex of the cone with respect to the interior surface. 
 
 Now the equations of the right line which is the intersection of 
 the two planes whose coordinates are f„ v„ ^, and f„, v„, ^,„ are 
 
 f-f/=fef" (?-?/), and v-v,='^' (?-?;) : see sec. [64]. 
 
ON CONCYCLIC SURFACES OP THE SECOND ORDER. 107 
 
 manner ?„-?,= p, and v„-v,-^. 
 
 Consequently the equations of the right line now become 
 
 ^ A2~ »-V^ CV' 
 or 1=1 f+/>(-p-g-,); also v=2f+<y(^—^__^^. . (d) 
 
 The general equations of a straight line in space are as in sec. [61] , 
 
 ^=fi^+a, v=v^+^ (e) 
 
 Equating the coefficients of (d) with those of (e), 
 
 It has been shown in sec. [62], (f), that if a straight line be 
 drawn through the origin parallel to the straight line whose equa- 
 tions are (e), the direction cosines which this line makes are 
 
 cosZOX=|, cos/OY=^, cos/OZ= ~^^^~'"'l . (g) 
 
 If in these equations we substitute for /j,, v, a, ^ their values, we 
 shall have 
 
 cos ZOX = 2 , cos ZOY= —Pixi—^J' 
 
 cos/OZ=^^(i,-^,). 
 
 (t) 
 
 Now the cosines of the angles which D= \'p^.+ q^-{-r^ the dia- 
 meter drawn through the vertex of the cone makes with the axes, 
 
 are £.^S^^^ — - multiplying these expressions two by two, we shall 
 
 have 
 
 p cos /OX + g cos /OY+ r cos ZOZ _^ ... 
 DA ' • • • W 
 
 or the diameter thrmigh the vertex of the cone, on the surface of the 
 exterior ellipsoid, is perpendicular to the plane passing through the 
 origin and the straight line in which the tangent plane to the exterior 
 ellipsoid drawn through the vertex of the cone, and the polar plane of 
 the vertex of the cone with reference to the interior ellipsoid intersect. 
 116.] Let a cone envelope an ellipsoid, so that the plane of contact 
 shall touch a second surface confocal with the former. The line 
 
Consequently 
 
 108 ON CONCTCLIC SURFACES OF THE SECOND ORDER. 
 
 drawn from the vertex of the cone to the point of contact of this 
 tangent plane will be at right angles to it. 
 Let the tangential equations of the surfaces be 
 
 A2^ + BV + C»^=1, and a«f« + AV+c«?«=l. . (a) 
 Let 
 
 h^=a^+f, B«=*'+A (?=c^+f^. . . . (b) 
 
 Let p, q, r be the projective coordinates of the vertex of the cir- 
 cumscribing cone, and let x,, y„ z, be the projective coordinates of 
 the point of contact of the polar plane with the interior ellipsoid, 
 and let ^,, v,, ^, be the tangential coordinates of the polar plane. 
 Hence, as in sec. [115], 
 
 p=k%, q=Wv„ r=C%,) 
 
 p-x,= {A?-a^i, or p-x,=Pi,;^ ,jx 
 
 so also ?—y< =/*"/> r—z,=f^^,. j • • ' K ) 
 
 Now the cosines of the angles which the line drawn from the 
 vertex of the circumscribing cone to the point of contact of the 
 tangent plane being X,, /x,, v,, we shall have 
 
 cos V, r—z, f/ cos V, r—z, ^' ' ' ' ^ ' 
 
 But if we let fall on this tangent plane to the interior surface 
 a perpendicular P from the centre m idling the angles \, fi, v with 
 the axes, we shall have 
 
 COS\_Pf,_f^ cos fJ,_Vi 
 
 cos v"? ~^i' cos v"?/ 
 or 
 
 \=\, n=ii„ v=v„ 
 
 or the line from the vertex of the cone to the point of contact of 
 the interior tangent plane is parallel to the perpendicular from the 
 centre on the same tangent plane, and is therefore perpendicular to 
 this latter. 
 
 To determine the length P, of the line from the vertex of the 
 cone to the point of contact of the tangent plane to the interior 
 surface. 
 
 Since P«= (i»-a?,)«+ {q-y,)^+ {^r-z^\ 
 
 V=m?-^y?Hr) = ^ or ?,?=/*. . . . (f) 
 
 Hence the product of the perpendiculars from the centre and 
 from the vertex of the cone on the interior tangent plane is con- 
 stant. 
 
ON CONCYCLIC SURFACES OF THE SECOND ORDER. 109 
 
 P, is the normal passing through the point of contact of the 
 tangent plane to the interior surface ; to determine the coordinates 
 of the point in which it will meet the plane of XY, suppose. The 
 projective equations of this normal are 
 
 To determine the point where this normal meets the plane of 
 XY, we must put 2=0, and the preceding equations hecome 
 
 Xi=a%, yi=b\, z,=c%. 
 
 but 
 
 Hence 
 
 Now ^1 and v, are the reciprocals of the segments cut off the 
 axes of X and Y by the trace on the plane of XY made by the 
 tangent plane, and x and y are the coordinates of the foot of the 
 normal. Hence this curious but well-known relation, that if we 
 construct the ellipse the squares of whose semiaxes are a*— c* and 
 a^—d^, the foot of the normal and the trace of the tangent plane on 
 the plane of XY will be pole and polar with respect to this section. 
 
 To determine the length of the normal N from the vertex of the 
 cone to the plane of XY, 
 
 W=(p-x)^+{q-y)^ + r''. 
 
 But we have shown that x= («^— c')f, and y=(6*— c*)i/,. 
 
 Hence N« = (A« - a« + c*) %« + (B* - A* + c«) V + C"?,'. 
 
 It may easily be shown that A*— a* + c* = C*, and that B'— 6* -f- c* 
 is also equal to C*. 
 
 Hence N«=C''(?/-l-v/+S;*)=p, or NP=C«- But we have 
 
 shown that V,V=P, subtracting, P(N-P,) =C*-/'=c«. 
 
 117.] Parallel planes are drawn to a series ofeonfocal ellipsoids ; 
 to determine the locus of the points of contact. Let 
 
 S+P+S=i (^) 
 
 be the equation of one of the ellipsoids ; and as they are confocal, let 
 
 a^=(^+k^, *2=c« + A* (b) 
 
 Let f , V, f be the tangential coordinates of one of the parallel 
 
 planes, then 
 
 x=a% y=b%, z = c^^; (c) 
 
 and as these planes must be parallel, let 
 
 ?=»»?, v=w?. (d) 
 
110 ON CONCYCLIC SURFACES OF THE SECOND ORDER. 
 
 From these eight equations we must eliminate a, h, c, f, v, ?, 
 and this elimination will afford ns a double locus for the point 
 
 W V z 
 
 'Now a?=fl«|=(c« + F)OT?, y={c^ + h^)n^, and ^=c*?; from these 
 three equations, eliminating c and f, we get 
 
 nh^x—mk^y + mnfjc^ — h^)z=0, . . . . (e) 
 the equation of a diametral plane of the surface. Now as 
 
 a«=|=— =— and c^=a^-k^, 
 
 we find 
 
 ««= J^, J«=-*^, and <^=a--k^=J^^. 
 x—mz y—nz x—mz 
 
 Substituting these values of a*, 6', c* in (a), we get, after some 
 reductions, 
 
 a?" y* xr^ wyz (\—m^xz _ ... 
 
 k^^h^ F A« "^ »bA« ~ ^ > 
 
 This is the equation of a one sheet or continuous hyperboloid 
 whose centre is at the centre of the confocal ellipsoids. 
 
 Hence the locus of the points of contact of the parallel tangent 
 planes with the confocal ellipsoids is a curve, the intersection of 
 the diametral plane (e) with the discontinuous hyperboloid (f ). 
 
 118.] To a series of concyclic surfaces talent planes are drawn 
 touching the surfaces in the points where they are pierced by a 
 common diameter ; to find the surface enveloped by these tangent 
 planes. 
 
 Let a«p + AV+c«?«=l (a) 
 
 be the tangential equation of one of the surfaces, c being the 
 greatest axis, and as they are concyclic, we shall have 
 
 Let ^,v,^he the tangential coordinates of one of the tangent 
 planes, and let x, y, z be the projective coordinates of the point of 
 contact of one of these planes ; then let x=mz, y=^m; . (c) 
 
 also let x=a^^, y=b''v, z=c^^; . . . . (d) 
 
 now between the eight equations (a), (b), (c), (d), we have to elimi- 
 nate the six quantities x, y, z, a, b, c. 
 
 Since x==a% f=5= J='»^(|+p) = '»c«?(i + i). 
 
 Hence 
 
 
ON CONCYCLIC SURFACES OF THE SECOND ORDER. Ill 
 
 but ^= .g. o. J equating these values of -^, we get 
 
 {k^-h^)mn^+mh^v-nk^^=0, (e) 
 
 the tangential equation of a point at infinity. 
 
 ,.J2iS^, also 6-*!(i^z±«^ c-*!(iz;^). . (f) 
 
 Substituting these values of a^, 6% c' in the tangential equation 
 of the surface, we get 
 
 A«f' + AV-A«?2-«AH+^^-^^f?=l, . . (g) 
 
 the tangential equation of a continuous hyperboloid. Hence, as 
 the plane envelopes a hyperboloid and passes through a point 
 situated at infinity, the locus is an hyperbolic cylinder. 
 
 The reader wDl doubtless have remarked the duality that exists 
 between several of the foregoing problems. In the two latter espe- 
 cially, the diametral plane in the confocal surfaces is the polar 
 plane of the point at infinity, the common direction of the axes of 
 all the cylinders which envelope the concyclic siurfaces ; while these 
 cylinders are themselves the polars of the several curves in which 
 the diametral plane cuts the confocal surfaces. 
 
 It wUl be shown, as we proceed with these investigations, that 
 every surface of the second order that admits of circular sections 
 has four directrix planes parallel to the planes of the circular sec- 
 tions, two by two. In the case of the elliptic paraboloid two of the 
 directrix planes are at infinity. These planes may be called the 
 umbilical directrix planes. It will also be shown that in every such 
 surface there are four foci, the poles of the umbilical directrix planes, 
 and distinct from the foci of the principal sections of the surface. 
 By the help of these umbilical directrix planes and corresponding 
 foci may all the properties of spheres and surfaces of revolution of 
 the second order be transformed and transferred to surfaces with 
 three unequal axes. 
 
112 ON THE SURFACE OF THE CENTHES OF 
 
 CHAPTER XII. 
 
 ON THE SURFACE OF THE CENTRES OF CURVATURE OF AN ELLIPSOID*. 
 
 119.] It is well known to geometers that the lines of greatest 
 and least curvature at any point on the surface of an ellipsoid are 
 at right angles to each other, and that they may be constructed by 
 the intersections of two confocal hyperbolas, one continuous, the 
 other discontinuous, or as they are usually called, a single-sheet 
 one and a double-sheet one. It is also known that these three 
 surfaces are reciprocally orthogonal, or that any two of them cut 
 the third along its lines of curvature where the three intersect in 
 a point. If we fix on the ellipsoid as the surface whose lines of 
 curvature are in question, and normals be drawn to the surface of 
 the ellipsoid along any given line of curvature, the radii of curva- 
 ture will not only lie on these normals at the successive points, but 
 they will all, taken indefinitely near to each other, constitute a 
 developable surface, and the line of centres of curvature will con- 
 stitute its edge of regression. Hence if we draw tangent planes to 
 the two hyperboloids at this point, they will intersect in the normal 
 to the ellipsoid, and will also be tangent planes to the above deve- 
 lopable surface. 
 
 Let the equation of the ellipsoid referred to its axes and passing 
 through the point (a?, y, z,) be 
 
 a« + ^ + ^-l> (a) 
 
 then the equation of the tangent plane passing through the point 
 {x, y, zj) will be 
 
 a* ^ A« ^ c« ' ^°> 
 
 and the tangential coordinates of this tangent plane will be 
 
 P=^ v=^' t—^ r„\ 
 
 Let the equation of the tangent plane to one of the hyperboloids 
 passing through the point {x,y,z^ be 
 
 * The con^deration of this surface, first imagined by Monge, but not discussed 
 by him, will be found to throw some light on the nature of the vmbilici, and of 
 the lines of curvature passing through them, relative to which there has been 
 
CUKVATURE OF AN ELLIPSOID. 113 
 
 or, as the surfaces are confocalj we may put 
 
 Hence b^=a^+b^-a^, cf=a^ + c^-a^, J • • • W 
 
 and the preceding equation may be written 
 
 The tangential equation of the hyperboloid passing through the 
 point {x, y, z^ is therefore 
 
 fl,T+«+A«-a*)w^+K« + c2-a2)^=l. . . (g) 
 Hence we have x,=a^^, yi=b^v, z,=zc^^. (h) 
 
 But as the tangent planes are at right angles^ one to the other, 
 we must have 
 
 S^ + v,v+^=0; (i) 
 
 or, substituting for f „ v„ 5) their values as given in (c), the preceding 
 equation becomes 
 
 3?+|v+S?=0; (j) 
 
 substituting for x,, y,, z, their values given in (h), we obtain 
 
 p+v'+!:'=(«'-««)|g+^+gJ (1) 
 
 If we now refer to the tangential equation of the hyperboloid 
 (g) we shall find 
 
 or [^+i^+}^){a^-a^)-a^^ + b^i^+(^^-l. . . (m) 
 
 Comparing this equation with the preceding, we may eliminate 
 (o*— a *)., and obtain as the tangential equation of the " surface of 
 centres " 
 
 (f8 + „2 + 5*)s=||+^' + |J(a8f' + 6V + c«r-l). • (n) 
 
 This is the tangential equation of the " surface of centres of cur- 
 vature," or, as it may for brevity be called, the surface of centres. 
 
 120.] This surface consists of two sheets — one generated by the 
 successive normals to the surface along one line of curvature, the 
 second by the successive normals along the corresponding line of 
 curvature. Let a perpendicular F on a tangent plane to the surface 
 of centres make the angles X, /it, v with the axes of coordinates, 
 then as Pf s=cos X, Pi;=cos/t, Pf=cosj', the last equation may 
 
 I 
 
114 ON THE SUBFACB OF THE CENTBES OF 
 
 be written 
 [?^+S<e+2^^](„«co8«\+A«cosV+c«co8«v-P«) = 1. (a) 
 
 Now the first member of this equation represents ^, the inverse 
 
 semidiameter squared of the original ellipsoid, making the angles 
 \, fi, V with the axes ; and a^ cos* \ + A' cos* /* + c* cos* v = P * is the 
 square of the perpendicular on a tangent plane to the ellipsoid 
 parallel to the tangent plane to the surface of centres. Hence 
 
 R«=P,*-F, (b) 
 
 whence we have this remarkable property of the surface of 
 centres : — 
 
 Any two parallel tangent planes being drawn to the surface of 
 centres and to the ellipsoid, the difference of the squares of the coin- 
 cident perpendiculars let fall upon them from the centre is always 
 equal to the square of the coinciding semidiameter of the ellipsoid. 
 
 Assume the original equation (n) written in the form 
 
 4*c*£«+a«c*i;*+fl«6*?*= (A«-c*)*a*(;«w*+ (ffl*-i*)*c*Pw*l 
 + (a«-c*)*6»|«§* }■ ^'^' 
 
 Then, by giving to f a constant value, we might determine the 
 tangential equation of the section made in the plane of XY by 
 the cone whose vertex is in the axis of z, and which envelopes the 
 surface of centres. 
 
 But it will be better to determine the sections of the surface made 
 by the principal planes j and this may be effected by putting ?, t/, f 
 successively equal to » and 0. Hence we shall have in the planes 
 of YZ, XZ, XY, the sections whose tangential equations are 
 
 (a*-4*)«A.*+ (a*-c*)*4*(;*=4*c*l • .u i ^^r. ^ 
 
 c*u*+6*?*= (j*-c*)*r«« } '" **•" P^^' °^ ^^' 
 
 (*«-c«)*a«r'+(o*-4*)«c»|*=a*c*l . ,, , ^^„ 
 a*S*+c*P=(fl*-c*)*£*?« / '"^ ^^^ P^°^ °^^^' r^ W 
 
 (a*-c*)*6*|*+(6*-c*)«o*i/*=a«4«l • , , 
 «*£* + «V=(a«-A*)*pu* J "^ *^' P'*°" °^^Y- 
 
 Consequently the sections of the surface of centres in the prin- 
 cipal plMies are two in each ; one an ellipse, the other the evolute 
 of an ellipse. 
 
 It may easily be shown that the tangential equation of the evo- 
 
 a a 
 
 lute of the ellipse whose projective equation is %+^=l will be 
 «*P+iV=(a«-6«)9{«„s (e) 
 
CURVATURE OF AN ELLIPSOID. 115 
 
 The projective equation of the evolute of an ellipse of which the 
 seiniaxe8areaand6is(oy)*+(6jr)*— (a*— i*)*=0; . . . . (f) 
 taking the partial differentials of this expression, we shall have 
 
 dP_2 b dF_2 a dF dP 2 , 
 
 d*-8(4^' ¥~3(^ di* + d^^=3 («'-*')■ 
 
 But, as we have shown in sec. [22], 
 
 m dP 
 
 fr_ da? Ay 
 
 ^~A¥ , dp ' dP , dP • 
 
 di*+d^J' di^+d^y 
 
 Consequently ^^= ^JLiy ' 2"'=i;^(a^- 
 
 Substituting in the dual equation jcf + yi;=l, the values of these 
 expressions, we obtain the tangential equation of the evolute of an 
 ellipse, 
 
 a*f« + iV=(a«-6«)«f2w« (g) 
 
 If in the equations (f ) and (g) we make w=ooory=0, or0=oo, 
 or d? =0, we shall have for the semiaxes of the evolute in the axes of 
 XandY, 
 
 A=f!=^, B=J!=^ ,« 
 
 Hence the semiaxes of the evolute are inversely proportional to 
 the semiaxes of the ellipse. 
 
 It is worthy of notice that, while the axes of the elliptic sections 
 of the surface of centres in the three principal planes are functions 
 of the three axes of the ellipsoid, the axes of the evolutes are fimc- 
 tipns only of the axes of the principal planes in which they happen 
 to be. Thus c does not appear among the constants of the evolute 
 in the plane of XY. 
 
 From the foregoing investigation it follows that the sections of 
 the surface of centres in the principal planes are two in each— one 
 an ellipse, the other the evolute of an ellipse. 
 
 On the umbilical lines of Curvature. 
 121.] Among the French mathematicians there has been much 
 difference of opinion as to the nature of the lines of curvature which 
 pass through an umbilicus of the ellipsoid. Some hold, with Monge 
 and Dupin, that the two lines of curvature which everywhere else 
 on the surface are at right angles to each other, here merge into 
 one. This is such a violation of the law of continuity, that others 
 adhere to the opinion of Poisson and Leroy, to the effect that at 
 an umbilicus the radii of curvature are all equal, and that there is 
 
 i2 
 
116 ON THE SUBFACE OF THE CENTBES OF 
 
 an infinite number of rectangular systems of lines of equal curva- 
 ture all passing through the umbilicus. 
 
 An examination of the surface of centres will demonstratively 
 show that the latter opinion is the correct one. 
 
 For this purpose let a tangent plane to the surface of centres be 
 drawn through the umbilical normal. Now the projective coordi- 
 nates of the umbilicus are 
 
 and the segments of the axes of x and z cut off by the normal are 
 
 -^ V(&^-0(fl'-c^) -_ -/(a'-c«)(g«-y) _ (b) 
 c ' a 
 
 Hence the tangential coordinates of the normal in the plane of xz 
 are 
 
 ?'=(a*-c«)(6*-c')' ^=(o«-c«)(a«-J«)' " ' ''^^ 
 
 Now, substituting these values of 2[ and in the equation (c), sec. 
 [120], of the surface of centres, we shall have for the value of w' 
 the following expression : — 
 
 Or an infinite number of tangent planes may therefore be drawn 
 through the umbilical normal to the surface of centres. 
 
 The principal sections of the surface of centres in the mean plane, 
 or in the plane of XZ, of the greatest and least axes, possess some 
 very curious properties. 
 
 The ellipse and the evolute in the plane of the greatest and least 
 axes of the ellipsoid have four points of contact ; and in these four 
 points they are severally touched by the four umbilical normals ; 
 and these normals are the radii of curvature of the umbilici ; and 
 
 their value is severally = — 
 ' ac 
 
 To show this, let the ellipse and evolute in the plane of xz be 
 
 supposed to have the same tangent. Their equations are 
 
 ( *« - c*) «a«5* -f (o* - i«) VP = aV, 1 
 and a«?2 + c«02=(a9_cS)«^gs, J ^^' 
 
 Let and ^ be the same in both equations. Reducing on this 
 supposition, we shall find 
 
 (a«-c2)(6s-c«)?*=c«, 
 hence 
 
CURVATURE OF AN ELLIPSOID. 
 
 117 
 
 but these are also the tangential coordinates of the umbilical normal 
 found in (c) . Hence the four umbilical normals form an equilateral 
 parallelogram or lozenge, and touch externally the ellipse, and inter- 
 nally the evolute, at the same four common points ; and the common 
 
 distances of these four points from the umbilici are severally =— , 
 
 or they are the centres of curvature of the umbilici. 
 
 Fig. 19. 
 
 ac 
 
 These points may be called the centres of umbilical curvature. 
 122.] We may write the tangential equation of the central ellipse 
 
 in the form 
 
 K:^)%.^+(*!y!L'f.?=i (a) 
 
 Let X and r be the projective coordinates of the centre of umbi- 
 lical curvature. 
 
 Thenj?= ^ ^a ' g, z= ^ ^ S- 
 
 Substituting for f and f their values as given in (f), sec. [121], 
 we shall find 
 
 -..^ («'-^T . -,^^EZ^ (b; 
 
 x" 
 
 ■a^{a^-cy ~cV-c') 
 
118 ON THE SUBFACB OF THE CENTRES OF 
 
 The length of the side of the quadrilateral made by the intersec- 
 tions of the four umbilical normals is — (o*— c*) ; and it is divided 
 at the common point of contact of this lunbilical normal with the 
 
 h h 
 
 ellipse and evolute into the segments — {b^—(F) and — {a^—b^), 
 
 so that the length of this line and its segments made by the point 
 of contact are respectively 
 
 I(a9_c9), A(6s_c«),and-(fl2_A9). , , . (c) 
 
 123. The areas of the umbilical parallelogram, of the ellipse, and 
 of the evolute circumscribed by and inscribed in the four umbilical 
 normals, have certain reciprocal relations which are independent of 
 the axes of the ellipsmd. 
 
 (ffl'-&*)(&«-c') 
 
 The area of the ellipse =ir- 
 
 ac 
 
 The area of the paraUelogram =2^'^'~'^^^ 's/(a^-b^)(b^-t^) 
 
 ac 
 
 The area of the evolute =-^- -■ 
 
 8 ac 
 
 „ area of ellipse x area of evolute _ 3 /ttY / x 
 
 (area of parallelogram)^ ""8x3/ ' 
 
 an abstract number independent of the axes of the ellipsoid. 
 
 When the mean semiaxis b of the ellipsoid is a mean proportional 
 between the semiaxes a and c, or b^=ac, the lunbilical ellipse 
 becomes a circle whose radius is a—c. 
 
 K we examine the ellipse and correlative evolute in either of the 
 other principal planes, we shall find that they have no common 
 point of contact or intersection. 
 
 An inspection of the expressions for the axes of the eUipses 
 and evolutes in the three principal planes will show that these 
 axes are so related that the axes of the central ellipse in any one 
 of the three coordinate planes are the axes of the evolutes in the 
 two other planes, one in each evolute. Thus, if the axes of the 
 ellipses in the three coordinate planes be hk, hjc„ and h,/ci„ the axes 
 of the evolutes in the same coordinate planes will be h/c^, h,^, and 
 Mp so that the twelve axes of the three ellipses and three evolutes 
 are reduced to six. 
 
 Thus, as in figure 20, we pass from Z to /3 along the elliptic 
 quadrant, from to a along the quadrant of the evolute, firom a to 
 7 along the elliptic arc, from y to Y along the quadrant of the 
 evolute, from Y to X along a quadrant of the ellipse, and from X 
 to Z along a quadrant of the evolute. 
 
CUaVATURE OF AN ELLIPSOID. 
 Rg. 20. 
 
 119 
 
 Thus in our course along the 
 from Z, we traverse 
 
 edges of regression," setting out 
 
 the plane of ZY along an ellipse, whose quadrant is Z^, 
 the plane of YX along an evolute, „ „ is /3a, 
 the plane of XZ along an ellipse, „ „ is ay, 
 the plane of ZY along an evolute, „ „ is 7Y, 
 the plane of YX along an ellipse, „ „ is YX, 
 the plane of XZ along an evolute, back again to Z. 
 
 From the previous discussion of the properties of the centre of 
 surfaces it follows that it consists of two distinct sheets which touch 
 only in four points, the centres of umbilical curvature ; for it is only 
 in these four points that the radii of curvature of any point on the 
 surface of an ellipsoid are equal. 
 
 On the projective equation of the surface of centres of an ellipsoid. 
 
120 
 
 ON THE SURFACE OF THE CENTRES OF 
 
 124.] Let 
 
 ?+^+S=I'. aT + *'"'+c«?'-l = G, and P+i;« + ?«=R, (a) 
 a o tr 
 
 then the surface of the centres of corvatnre may be written 
 
 PQ=RS or 4>=PQR-»-l=0 (b) 
 
 Let the partial differentials of this expression be taken succes- 
 sively with respect to f, v, and f; hence 
 
 dP . QR + dQ . PR -2PQ . dB 
 
 d4>=- 
 
 R» 
 
 (c) 
 
 But 
 
 consequently 
 
 ^-2f dQ_ dR_ 
 
 d^ 
 
 -=2[^ + PRa«-2PQ]fR-»; . . 
 
 or, since PQ=R', we obtain 
 
 f =2f[§+P«^-2R]R-. .... 
 
 We have now to find the value of the expression 
 d^ ^ , d^ . d^ ^ . 
 
 dF^+dir*'+d?f=^- 
 
 Making the necessary substitutions. 
 
 (d) 
 
 (e) 
 
 or 
 
 hence 
 
 or 
 
 A=2[QP+PQ+P-2R]-2,or A=^; 
 
 d$ 
 
 df.A" 
 
 ^-|-Pa«-2R^ 
 
 =x. 
 
 x=- 
 
 6' 
 
 ^+?'''+^?'-^«+^+F"'+7.i:'-2?'-2«'-2?' 
 
CtTRVATURE OF AN ELLIPSOID. 
 
 121 
 
 and reducing; 
 
 
 
 y— 
 
 \Q-'i)'^<-t)'^-h\ 
 
 T -_ ^ 
 
 aS + ^2 + c2 
 
 >, 
 
 ^=-'= SS 3 C3 
 
 (f) 
 
 ^ U2 ?2" 
 2-4- — 0.5- 
 
 as-t-ja + ca 
 
 having found like expressions for ^ and z. 
 
 If we introduce the relations established in the second form of 
 the equation of the surface of centres, see (c), sec. [120], we may 
 easUy show, though not at first sight apparent, that the preceding 
 equations satisfy the criterion of duality, 
 
 x^+yv+z^=l. 
 
 By the help of these three equations and the tangential equation 
 of the surface of centres, we may eliminate f, w, 5> and express the 
 projective equation of the surface of centres in terms of x, y, and z. 
 
 The projective equation of the surface of the centres of curvature 
 has been given by Dr. Salmon, Professor of Divinity in the Uni- 
 versity of Dublin, and published in the Quarterly Journal of Pure 
 and Applied Mathematics of Feb. X858. 
 
 Although this surface has been familiarly known to the conti- 
 nental mathematicians since the time of Monge, none of them has 
 ventured to grapple with the enormous difficulties which stand in 
 the way of exhibiting its projective equation, or its equation in aiyz. 
 These difficulties have been surmounted by Dr. Salmon ; and the 
 resulting equation, which is of the twelfth degree, contains no fewer 
 than eighty-three terms. 
 
 125.] To show the power and exemplify the reach of the combi- 
 nation of the methods of projective and tangential coordinates, it 
 will be an apposite illustration to discuss the reciprocal polar of the 
 " surface of centres." This investigation will afford a further instance 
 of the great law which pervades all geometrical truths, that if one 
 method of investigation be more easily applied to the discussion of 
 the properties of curves or curved surfaces, their reciprocal polars 
 will be best investigated by the other. 
 
 The reciprocal theorem to that of the surface of centres is the 
 following : — 
 
122 ON THE 8X7BFACE OF THE CENTRES OF 
 
 Let there be three concyclic surfaces cf the second order touched 
 by a common tangent plane in three points; these points, two by two, 
 will subtend right angles at the centre, and the locus of all the points 
 of contact with the two variable hyperboloids will be a sutface which 
 may be called the " surface of contacts." 
 
 Its projective equation may be found as follows. 
 
 Let the tangential equations of two of the surfaces^ having a 
 common tangent plane, be 
 
 a«f +*2i;S + c«g«=l, and 0;«|*+J,V+c«?»=l. . . (a) 
 
 Subtracting one of these equations from the other, we shall have 
 
 (a«-o;)r'+(6«-J,«)t;« + (c«-c;)r«=0; . . (b) 
 
 and as these sur&ces are concyclic, we shall have 
 
 fl« a*~6« 6«~c« c«~A« ^^' 
 
 Making these substitutions in the preceding equation, there results 
 
 flVP +*'*/*«* +c%T=0 (d) 
 
 Let the projective equation of one of the hyperboloids be 
 
 X? . y? . z? 
 
 ZLj-ei-o-il-— I i^\ 
 
 a^^bf^c*~^' ^^> 
 
 and the equation of the tangent plane to this surface passing through 
 the point x,, y„ z, be 
 
 *i Vi z, 
 
 ^,^+^y+-4^=i; ...... (f) 
 
 and as this must coincide with the second of (a), we shall have 
 afl^=Xp bfv=y„ c«f=ar„ \ 
 
 Substituting these values in (d), we shall obtain 
 
 aV2^ + i^*,V + cVC«=^+%V^'=0,. . (h) 
 J . "i "i f^i 
 
 or, reducing, 
 
 (i-^)(''V+4^,*+c«r,«)=(:r«+y*+z,'). . . (i) 
 
 Equation (e) may be written in the form 
 
 or 
 
CURVATUBE OF AN ELLIPSOID. 123 
 
 Eliminating the quantity (—^ , j between this and the preceding 
 
 expression, we find for the resulting equation 
 
 omitting the traits as no longer necessary. 
 
 This is the equation of the " surface of contacts." 
 This equation may be written 
 
 = (ft«-c«)«flV^'+ (a'-c')'A'*'^'+ (o'-*'')'c«a?V- (m) 
 If, instead of taking the concyclic surfaces with independent 
 axes, and thus investigating the equation of the " surface of con- 
 tacts " directly, we had derived the concyclic surfaces from the 
 confocal surfaces of which they are the reciprocal polars, we should 
 have obtained a projective equation for the surface of contacts more 
 nearly in accord with the tangential equation of the surface of 
 centres than the one above given. 
 
 To show this, let the radius of the polarizing sphere be r, the 
 radius of the polarizing sphere being quite arbitrary ; and let a, b, c, 
 be the semiaxes of the concyclic ellipsoid; then, writing ;r, y, z tor 
 
 h v> K, 
 
 r^ , r^ r« 
 a=-, o=-r, c= — ; 
 a, b, c, 
 
 making these substitutions in (n), sec. [119], and omitting the 
 traits as no longer necessary, we shall find 
 
 (^*+y«+r«)«=[g+g+^(aV+AV 
 
 <^ + c^z^-r*), . (n) 
 
 which is identically the same in form as the tangential equation of 
 the surface of centres. 
 
 This equation may also be reduced to the form 
 
 = (i9_c«)«a«yV+ (a'-c*)**'^'-^^+ (a*- A')*cVy«.J ^°' 
 It should be observed that while the axes in the confocal sur- 
 faces are in the order of magnitude a > 6 > c, in the concyclic surfaces 
 they are in the order of magnitude c>b>a. 
 
 We may show that this surface of contacts has four edges of 
 contact perpendicular to the plane o{xz. 
 
 To show this, let a?* =7-3 — =77-3 — n7» ^—ts eww — «• 
 
 Substitute these values of x and z in (o), and we shall have 
 
 ^=-, or any point taken on the axis of Y wiU be on the surface. 
 
124 THE METHOD OF TANGENTIAL COOBDINATES 
 
 The sections of the surface of contacts in the plane of xz will be 
 the curves 
 
 and 
 or 
 
 (ffl«-A*)«cV+ (A«-cS)«aV=r*a«c«, 
 
 (o*_j.)sf_ + (ft3_c8)»f._^. 
 
 (P) 
 
 of which the former is the reciprocal polar of the evolute of an 
 ellipse^ while the latter is an ellipse. 
 
 We shall find that these two curves have four common points of 
 contact. For if we make the variables x and z in each equal, the 
 resulting value of z will be given by the expression 
 
 {a^-(?)\b^-^fz*-2{a^-(?) (4«-cV'^-2'+c*»*=0 J 
 but as this is a perfect square the two values of z^ merge into one, 
 and the resulting value becomes 
 
 (ffl*-c*)(6«-c*) ' 
 in like manner 
 
 r«=- 
 
 (q) 
 
 ■(a«-c*)(a«-6«)' 
 and these are precisely the values of x^ and z'^, which, substituted 
 in the equation of the surface of contacts, give the value of y under 
 
 the indefinite form ^=7^- 
 
 CHAPTER XIII. 
 
 ON THE APPLICATION OF THE METHOD OF TANGENTIAL COORDINATES 
 TO THE INVESTIGATION OF THE PROPERTIES OF TRANSCENDENTAL 
 AND OTHER CURVES OF A HIGHER ORDER THAN THE SECOND. 
 
 On the tangential equation of the Caustic by reflection of the 
 Circle*. 
 126.] Let the projective equation of the circle be 
 
 a;*+y*s=4a' (a) 
 
 * The general solution of this problem long baffled the skill of the most expert 
 analysts; at length M. Gergonne announced, 'Amudes de Math^matiques,' 
 tom. XV. p. 346, " J'^tais, depuis quelque temps, en possession de I'^quation de Ia 
 caustique par inflexion sur le ceicle, qui n'avait encore 6\i& donn^e par personne ; 
 mais je I'avais obtenu par des calculs trop prolizes, et sous ime forme trop peu 
 ^Iggante pour songer a la publier," &c. 
 
 Some tuie after, the complete solution was given in the seventeenth volume 
 of the same work bv M. de St. liauient, but in a most complicated and un- 
 manageable form. — [Tfote to first edition.] 
 
APPLIED TO TRANSCENDENTAL CURVES. 
 
 125 
 
 Let the axis of X pass through the radiating point, or " radiant," 
 as for shortness it may be called. Let p be the reciprocal of RO, 
 
 Fig. 21. 
 
 >ai 
 
 the distance of the radiant firom the centre ; then, as the lines RF, 
 PC make equal angles with the radius FO, we shall have 
 
 RF:FC::RO:OC; 
 
 but 
 
 OC=^, OD=a7, FD=y, 
 
 hence y'+(*+^) = y'+(^-|) • = ^« =p 
 
 or reducing, and putting 4a* for y*+a?*, we get, dividing by the 
 factor (f— p), 
 
 (b) 
 
 ^-''=2^«- 
 
 If we draw a tangent to the circle at F, meeting the axis X in t, 
 
 then Ot:= — and jt-s^tj— ; hence yr-, P, and— p are in arith- 
 X 2a* Ot Ot '^ 
 
 metical progression, and therefore Or, ^ and are in harmonical 
 
 progression, which should be the case, seeing that FR, FO, FC, FT 
 
 constitute an harmonic pencil. 
 
 1— jrf 
 Since the dual equation gives y^ — — , combining this with (b) 
 
 and the projective equation of the circle a*+y*^4ffl^ we may eli. 
 minate y and x, so that the resulting equation becomes 
 
 4o«(f«+w*)[l-o*(?-p)«]=H-4a«p?, . . . (c) 
 the tangential equation of the caustic of the circle by reflection. 
 
126 THE METHOD OF TANGENTIAL COORDINATES 
 
 This equation may also be written 
 
 4««(f + i;*)[l-o{?-p)][l+a(f-p)]=l+4flVf. • (d) 
 When the radiant is at infinity^ or p=0, the equation becomes 
 
 4ffl«(P + i/«)(l-fl«?«)=l (e) 
 
 We shall show further on that this is the tangential equation of 
 an epicycloid, the radius of the base circle being twice that of the 
 rolling circle*. 
 
 If we solve (c) for v we obtain 
 
 .= l-2a-g(g-p) . ...... (f) 
 
 2aV'l-a*{f-p)« ^ ' 
 
 Hence, if we assign a series of values to ^, we shall obtain corre- 
 sponding values for v, so that the caustic may be defined or set out 
 by the successive positions of the limiting tangent. 
 
 * The learned and accompliahed editor of the Mathematical Papers in the 
 ' Educational Times,' Mr. W. J. Miller of Huddersfield, derives the projective 
 equations of the bicusped hvpocycloid and cardioid from the general tangential 
 equation of the caustic as follows. 
 
 " As an example of the method of finding theprq/ective equation from the tan- 
 gential equation, let us take the catacanstic of the circle for parallel rays ; then, 
 by putting p=0 in equation (c), the tangential equation is found to be 
 
 ♦ = 4o»(l -a»f )(f +1/') - 1 =0. 
 "Now assume dn'^=2<i'({'+u'); then from ♦=0 we have 
 2i»=f=l-cot'^, 2a»«'=cos'^co1?^; 
 
 also ^=8o»{'(l -2o»^-oV) =4fl»f (cos» Vr+cot' V), 
 
 and ^=8a?«(l - a^f ) =4a»« cosec' V ; 
 
 d* . . d* . , , 
 
 •■•d|^+s;''=**^'''' 
 
 hence, applying the formulae in Art. 2, we have 
 
 2(a!'— «?)=3«i'cosec'^— coaec'^, 2y'=a'cosec'^; 
 
 and, eliminating ^, the projective equation is 
 
 4(.r«+jr'-«»)'=27o*y», 
 
 which shows that the catacaastic of tfie cirde, for parallel ratft, is a two-auped 
 epiegeloid whose base is concentric with the reflecting circle and has its radius 
 (a) half the radius of that circle. 
 
 " We may further observe that tiie tangential equation of the Cardioid, or one- 
 cusped epicydoid, referred to the centre 01 the base as origin, and the radius (a) 
 through the cusp as poritive axis of £, is 
 
 27<r'(l-flf)(f+«»)=4. 
 
 This may be obtained from the equation in Question 1492 by putting 
 — y=fi=(3a)-i ; for it can be easily shown (see Parkinson's 'Optics,' Art. 72) 
 that, when the radiant point is in the eireumference of the reflecting circle 
 (y= —p), the caustic is a Cardioid the radius (a) of whose base is one third of 
 tuat of -the reflecting circle." 
 
APPLIED TO TRANSCENDENTAL CTTRYES. 127 
 
 In this expression, trhen p=0, or the radiant point is at infinity, 
 we obtain 
 
 In the general equation (c) of the caustic, if we put f = — p, the 
 resulting equation becomes 
 
 4a«(?« + w«)=l, (h) 
 
 the tangential equation of the circle which generates the caustic. 
 Hence, if the radiant be outside the circle, the tangent drawn from 
 the radiant to the caustic is also a tangent to the circle. 
 
 If we assume, on the opposite side of the origin on the axis of X, 
 a point B., which shall be equidistant &om the origin, so that 
 OR^ORp then ^=p, and the general equation (c) becomes 
 
 4aV = l, (i) 
 
 or the tangent to the caustic drawn from the point B, will cut the 
 vertical diameter of the circle in its circumference. 
 
 If in the equation (f ) we put v=0, or make the limiting tangent 
 vertical, then we shall have l=2fl®f (j— p), or, solving this equation, 
 
 f=p±j:2+E^ (j) 
 
 Hence, when the limiting tangent is vertical, it cuts the horizontal 
 axis in two points at unequal distances from the vertical diameter 
 of the circle ; but when p=0, or the radiant is at infinity, these dis- 
 tances become equal. 
 
 To find when the axis of X is a tangent to the caustic. 
 
 When the limiting tangent coincides with the axis of X, then 
 w = oo ; substituting this value of w iu (f ), we obtain 
 
 a(|-p)=lorf=i+p; (fc) 
 
 when p=0, the cusp is at the middle point of the radius. 
 
 The tangent to the caustic parallel to the axis of X cuts the 
 axis of Y at the distance 
 
 from the origin; for if in the general equation we put f =0, the 
 resulting value of - becomes 
 
 2aVl-ay (1) 
 
 If the radiant be at infinity, p=0, and the horizontal tangent to 
 
 the caustic is also a tangent to the circle. 
 
 127.] There are two cases of the general theorem which may 
 
 repay discussion. 
 
128 APPLICATION TO TRANSCENDENTAL CURVES. 
 
 Let the radiant be at the extremity of the horizontal diameter of 
 X; then /o =„-, and 2ap = 1 . Substituting this value in the general 
 
 Aril 
 
 equation of the caustic, we find 
 
 «*(f'+«^)(3-2af)=l (a) 
 
 Now this, as we shall show further on, is the tangential of the 
 
 cardioid, the radius of the rolling circle, as also that of the base, 
 
 being =fa, or a third of the radius of the reflectiug circle. 
 
 If we put 0=0 in (a), the limiting tangent becomes horizontal, 
 
 and cuts the vertical diameter of the circle at the distance i^Sa 
 
 firom the origin. 
 If we put v=xi , the limiting tangent must pass through the 
 
 3 
 
 centre; and this condition gives f = ^, or the cusp is at the dis- 
 tance -5- from the centre. 
 o 
 
 If in this equation (a) we put t;=0, or make the limiting tangent 
 vertical, the points in which it cuts the axis of X will be given by 
 the cubic equation 
 
 2a8js-3a«f« + l=0; 
 
 or, putting the equation for the moment under the form 
 
 2a?8-3a;2 + l=0, (b) 
 
 the three roots of this equation are +1, —1, +^. Hence the 
 points in which the vertical limiting tangent cuts the axis of X are 
 a, —a, and 2a. 
 
 Since the vertical limiting tangent cuts the axis of X at the dis- 
 tance a from the centre, and the cusp at the distance fa from the 
 centre, it is clear that the distance between the cusp and the point 
 
 where the vertical limiting tangent intersects the axis of X is -. 
 
 3 
 128.] If in the general equation, (d) sec. [126], we put i;=0, or 
 make the Umiting tangent vertical, we sh^ have the following 
 biquadratic equation to determine the points in which the limiting 
 tangent, when vertical, cuts the axis of X — ^that is to say, 
 
 4a*f*-8fl«pf-4o9(l-ay)^+4aVf + l=0. . . (a) 
 Now this expression is the square of the following, 
 2aT-2a»p^-l=0; 
 or, solving this quadratic, 
 
 2i^=ap± V2 + aY (b) 
 
 These are the values of the segments of the axis of x cut off by 
 the vertical limiting tangent. 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 129 
 
 CHAPTER XIV. 
 
 ON EPICYCLOIDS AND BYFOCYCLOIDS. 
 
 129.] The theory of tangential coordinates may be applied with 
 singular facility to the inyestigation of the properties of curves of 
 this clasS; which comprises all those cases in which circles are sup- 
 posed to roll, either on the outside or the inside of other circles 
 assumed to be fixed, carrying along fixed points on their circum- 
 ferences, which like tracing-points describe the curves in question. 
 
 Let r be the radius of the rolling circle, R=2Mr that of the fixed 
 circle. At the beginning of the motion let the fixed or tracing- 
 point coincide with the point of contact of the two circles. Through 
 the centre of the fixed circle let the axes of coordinates be drawn, 
 the axis of X passing through the first point of contact A. Let O 
 be the centre of the fixed circle, Q, the point of contact, B the 
 
 Fig. 22. 
 
 extremity of the common diameter OQ, P the point on the rolling 
 circle which has traced out the arc AP of the epicycloid, and which 
 coincided with A at the beginning of the motion. As the curves 
 are assumed not to slide, one on the other, the arc AQ=PQ. 
 
 Now the principle assumed in the mise en equation is this, that 
 P is always on the circumference of an ever varying circle whose 
 centre is the instantaneous point of contact Q of the two circles. 
 Hence the magnitude and position of the instantaneous radius is 
 always changing, and the limiting tangent to the epicycloid is 
 always at right angles td this radius. QP is the changing radius ; 
 and BPC is the limiting tangent at right angles to PQ. Hence it 
 follows that the limiting tangent CP and the common diameter 
 OQB always meet in the same point B on the rolling circle. 
 
 Let the angle which the common diameter of the two circles 
 
130 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 makes with the axis of X be ^, then the arc described by the 
 tracing-point of the moving circle is 2nrif> ; for as the arcs of the 
 circles which have been in contact are equals putting for the 
 moment this angle as -^j r^=2nr^ or '^=2«^. Hence the angle 
 
 OBC=Mi^, OC=i and the angle BCD = (n+l)^. Conse- 
 quently in the triangle OBC, OB : OC : : sin OCB : sin OBC, or 
 2{n + l)r : ^ : : sin (m-|-1)^ : sinn^, or 
 
 fr_ 8in(w-H)<^ , . 
 
 *^~2r(«+l)sinM^ ^' 
 
 It is manifest that the limiting tangent meets the axis of Y at a 
 
 distance - from the origin, which wiU be equal to OC x tan OCY, 
 
 or -=^tan (m + 1)^ ; or substituting the value of f found above, 
 
 _ cos(w-H)0 „ 
 
 ''~2r(» + l)sinn«^" '' ' 
 
 Eliminating the trigonometrical functions of ^ between these 
 two equations, we shall obtain an equation between ^ and v, the 
 tangential equation of the epicycloid. 
 
 When the circle rolls on the inside of the fixed circle, we must 
 take its radius with the negative sign, so that the formulae 
 
 .._ sin(« + l)^ _ cos(m + l)^ 
 
 ^~2r(» + l) sinra^' " ~2r (re ±1) sin m^ • • ■ W 
 will answer for hypocycloids as well as epicycloids. 
 Squaring these expressions and adding, we find 
 
 4r«(« + l)*(|«-|-i;*)sin*«0=l, . . . . (d) 
 
 which enables us in all cases to find the value of sin n^, and also of 
 cos n<f> in terms of ^ and v. 
 
 130.] We shall now proceed to apply this theory to the following 
 
 selected cases — namely, when r=-^ or »=1, when r=B, or »=i, 
 
 B, 
 whenr=-i- or re=2: we shall also consider the case when the 
 
 radius of the circle rolling inwardly is one half that of the base 
 circle. 
 
 (a) In the general formulae for epicycloids, let n=l, and the 
 
 expressions (a) and (b) in the last section become 2rf= ^^ . ^ . and 
 
 2 sm ^ 
 
 2rv= ^^?-?| also 16r«(5« -|- w«) sin^ ,/,= ]. 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 131 
 
 Now 2rg= "°jJ'°^'P = COS <(,, or sin« ^ = 1 - 4r«^. Eliminating 
 sin* <f>, tte equation of the epicycloid becomes 
 
 16r«(£2 + w*)(l-4r«|«) = l (a) 
 
 But this is the equation that we found for the caustic by reflexion, 
 see sec. [126] (e), when the radiant is at infinity. If we put 
 2r=a,the equation becomes 4a'(g' + w*)(l — a'|*) = l, which is iden- 
 tical with (e) in sec [126] . 
 
 (/3) To determine the epicycloid when n=|. 
 The general formulae become 
 
 sinOj^ 2,,^ cos(l-,j)^ . . . (b) 
 f sin|^ f smi^ 
 
 Let <l>=20. The equations now become 
 
 „ , sin 3d _ cos 3d 
 3rg= ■ „ , 3rv= . „ ■ 
 sm e sm d 
 
 Hence 9r«(P + 1^) sin» 6=1. 
 
 Now 3r^sind=sin30cosd + cos2d8ind. Dividing by sind, 
 
 3>^=2cos«d + (cos«d— sin«d)=3— 4sin«d, or 4sin«d=3{l-rf), 
 
 hence 27r'(f* + w*)(l— rf)=4j or if we put r=|a, this equation 
 becomes 
 
 «*(P + «'*)(3-2flf) = l (b) 
 
 This is the tangential equation of the cardioid, and is identical with 
 the equation for the caustic by reflexion, when the radiant is at the 
 extremity of the horizontal diameter of the reflecting circle, see (a), 
 sec. [127]. 
 
 (y) Let ra=2, or let the base circle receive four undulations of 
 the epicycloid. The general equations become 
 
 - fc sin 36 „ cos3(f>) 
 * sm 2(f> "" ^'*' ' 
 
 or 6V«(f2 + ««)8in' 
 
 sin20' sin2<^C (<^) 
 
 L«.2^=l. ) 
 
 Let sin 20 =c, cos 2^ = V 1 — c*- Then 
 
 sin0 + cos0= ^1+c, sin(/) — cos0= v'l — c, 
 2sin0= Vl+cH- Vl — c, 2cos0= -/l+c— </l—c, 
 
 and sin (p cos 0=o- 
 
 Hence 6rf . c=sin 30=sin 2<^ cos <^ + cos 2(f> sin </>, 
 and Grv . c= cos 3^= cos 20 cos 6 — sin 20 sin 0. 
 
 Multiplying these expressions together, we shall have 
 6V«^w*=sin 20 cos 20 cos* 0— sin 20 cos 20 sin« 
 + cos* 20 sin cos 0— sin* 20 cos sin 0. 
 
 k2 
 
 w 
 
132 ON EPICYCLOIDS AND HYPOCYCLOtDS. 
 
 Hence 6V'Ji;c'=sin2(^cos'2^ + (cos'2^— sin*2^) sin^cos^. 
 
 Substituting the values for these sines and cosines as given in (d), 
 we shall have 6*r*fi/c*=3c— 4c* ; or dividing by t? and squaring, 
 
 6V^t;« 9 24 
 
 —^9 ^-c^ + 16; (e) 
 
 and if we substitute for -5 its value as given in (c), namely 
 6V(^ + t;'), on making the necessary reductions we shall find 
 
 27r2[432»^fw«-27r2(^ + u2)+2](f' + t;9) = l. . . (f) 
 
 This is the tangential equation of the quadrantal epicycloid. 
 
 Another solution of this question may be given. 
 
 131.] Since R=4r, ra=2, and the general equations become 
 Gr^ sin 2^= sin 3<^ and 6rv sin 2^=cos 30, consequently 
 
 36r«(^ + t;«)sin220=l, and sin3<^=-;=L=^. . . (a) 
 
 Let 36r2(f2 + i;«)8in«2(^=a2iir*sin«.2(/.=l. . . . (b) 
 
 hence 
 2^ 
 
 = 2sin*20cos'0 + 2cos220sin*0 + 4sin20cos 2<^8in0cos<^. (c) 
 
 Now 8m'2<^=;^, cos«20=— .-^ =-5-,, 
 
 writing M for a^w*— 1 ; 
 2cos^0=°^+ >^^, 2sin«0=?^:iLlH. 
 
 A'sr 
 
 Making these substitutions in the preceding value for sin 3^, we 
 shall have 
 
 2^«a8«r=a3««+ v'M(4-a2«r9). 
 
 Putting a=f R, we shall obtain for the tangential equation of the 
 quadrantal epicycloid : — 
 
 (3R)«(|2 + i;2)|V=[16-27R2(^ + i;*)]«. . . (e) 
 
 On the Epicycloid whose base is a Semicircle. 
 
 132.] In this case R=2r, and, as generally R=:2Mr, n=\, and 
 the general formulae in sec. [129] become 
 
 f._ sin 20 _ cos 20 
 
 ^~4rsin0' ""iTsm^ ^^' 
 
 Eliminating the trigonometrical functions, which presents no 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 133 
 
 difficulty, we shall finally obtain as the tangential equation of the 
 semicircular epicycloid : — 
 
 16r«{l-r«f)(^+„«)=l (b) 
 
 When the limiting tangent is parallel to the axis of X, f =0, and 
 
 v=-T-, as we might have anticipated. 
 
 When rf = 1 , the equation becomes 16r*(f* + u*) xO = l,ori/=Qo, 
 hence the limiting tangent at the cusp passes through the centre of 
 the base circle. 
 
 On Hypoeyelmds. 
 
 The general formulae, as given in sec. [129], which hitherto have 
 been used to obtain the tangential equations of epicycloids may 
 with the same facility be applied to the investigation of the pro- 
 perties of hypocycloids, by taking the radius of the rolling circle as 
 negative. 
 
 133.] To determine the equation of the hypocycloid, when the 
 radius of the rolling circle is one half that of the base circle. 
 
 In the general formulae, sec. [129] , 
 
 ^ jinjre-l)^ „^.^_ cos(m-l)^ 
 ^ (w— l)sinM^' (w — l)sinn^' 
 
 let «=1, and they become 
 
 2rS='p^^=M, 2n,=52i(lfl)i. . . . (f) 
 * (1 — l)sm<^ (1 — l)sm^ 
 
 How are these expressions to be interpreted ? 
 
 Since (1 — 1)^ is an indefinitely small angle, we may write (1 — 1)^ 
 
 instead of sin (1 — 1)0, and then dividing by (1 — 1), ^=-^-T ^ 
 
 finite quantity ; but since 
 
 cos (1-1)0=1, «'=-(riny^i^=5=*' •••(g) 
 
 hence, as f is finite and v is infinite, the limiting tangent always 
 coincides with the axis of X, as we might have anticipated ; for the 
 locus of the tracing point on the hypocycloid is the diameter of the 
 base circle. 
 
 134.] On the hypocycloid whose radius is one fourth that of the 
 base circle. In this case ra=5, and the general equations (c) in sec. 
 [129] become 
 
134 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 hence 
 
 TEpih-'^h^ ^"^ 
 
 If we put R for the radius of the base circle, R=4r, and the 
 equation of the hypocycloid becomes 
 
 f' + i;«=R«f2„2 (b) 
 
 Since ^ and - denote the intercepts of the axes made by the 
 
 limiting tangent, and since the sum of the squares of these inter- 
 cepts is constant, we may infer that if a line of constant length 
 revolves between the sides of a right angle, it will envelope a qna- 
 drantal hypocycloid. 
 
 135.] To determine the involute of the guadrantal Hypocycloid. 
 
 The projective equation of this curve is 
 
 a?*+yi==a* (a) 
 
 Let / be the length of the string or elastic radius measured from 
 the extremity A of the horizontal axis of the hypocycloid at the com- 
 mencement of the motion at G ; then at any point P on the hypocy- 
 cloid, the varying radius PT of the involute at this point will be the 
 line I, plus s the arc of the hypocycloid AP ; and as this line is per- 
 pendicular to the limiting tangent QT, it is also a tangent to the 
 hypocycloid. Now in sec. [5] we have shown that the length of a 
 perpendicular let fall from a given point P on a limiting tangent is 
 
 = — ^ ; or as /+ s is the length of the perpendicular, and the 
 
 VP + ir 
 
 projective coordinates of P are ar and — y, we get 
 
 /4-,=i5^^ (b) 
 
 but ~ derived from (a) is the tangent of the angle that the varying 
 
 radius (which is a tangent to the hypocycloid) makes with the 
 axis of X, and this line is at right angles to the limiting tangent 
 QTj hence 
 
 t;^dy_yi 
 
 f da: ar* 
 
 or 
 
 consequently 
 
 - x=- * 
 
 (f^+u^)^' (e-^v^)y 
 
 (c) 
 
ON EPICYCLOIDS AND HVPOCYCLOIDS. 
 
 135 
 
 ti G 
 
 Now the are s, measured from A to P, is = f a*y*, or » = ^a -^ ^. 
 
 Substituting these values of s, x, and y in the expression for the 
 perpendicular (b)^ we obtain on reduction 
 
 [(?+«)f^+(/+iy]'=r+»'S • . 
 
 (d) 
 
 the tangential equation of the involute of the quadrantal hypocycloid. 
 It is obvious that when I, the length of the strings the continuation 
 of the elastic radius *, is very large as compared with a, the equa- 
 tion approximates to the tangential equation of a circle ; and when 
 i=0, or when the tracing point begins with A, the vertex of the 
 hypocycloid, the equation becomes 
 
 {a^ + ^-ff=^+v^ (^) 
 
 136.] On the projective equation of the involute of the quadrantal 
 Hypocycloid. 
 
 The projective equation of the quadrantal hypocycloid is 
 
 a?»+y*=a* (a) 
 
 The tangential equation of its involute, as shown in the preceding 
 section, is 
 
 ^ + v^=l[k + a)^+{k-a)v^Y, . . . (b) 
 hence 
 
 ^=[4^(A + a)-2]?, ^=li^(k-a)-2]v, . (c) 
 
136 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 and 
 
 A=Sf+^«'=2«S putting f +i;*=tiT'; 
 Qf av 
 
 consequently 
 
 ^^x=:l2is[k + a)-l'\i; ^^y=[2,s[k-a)-l]v. . (d) 
 
 The equation of the curve (b) may be written 
 
 It may also be put under the forms 
 
 ■er=kv'^-avr^ + 2a^, or ■a=kia^ + a'B^-2av^,) 
 or • . (e) 
 
 2a^=isf-{k-a)is^, 2av^={k+a)vr^—a. ) 
 
 K we square the expressions in (d) we shall have 
 
 Adding these expressions, bearing in mind that 
 
 0« + i;2=CT«, and that i^-v^=' ^^~^'°'\ from (e), 
 
 dividing by u*, and putting 
 
 U for [a;« + jr* + 4(A«-o«)], 
 we shall have 
 
 Ui!r2=8Ai!r-3 (g) 
 
 If we subtract the equations in (f ) one from the other, we shall 
 find 
 
 or putting 
 
 V for a(a?«-y«) ^A^kljc^-a^), 
 
 the resulting equation becomes 
 
 ViBr»=8A«iir«-5AtJ + l (h) 
 
 If we multiply this equation by 3, and add (g) to it, the resulting 
 equation becomes 
 
 3V«r«=(24A«-U)i!r-7A (i) 
 
 Eliminating ir from the two quadratic equations 
 U«r*=8A«r-8, 1 
 
 3Vt!r«=(24A«-U)«r-7*,J ^^ 
 
 we shall obtain the resulting expression in terms of ar*, y*, k, and a, 
 the projective equation of the involute of the quadiantal hypo- 
 cycloid. The elunination of vt gives the final result, 
 
 U iW- 13A«U + 128/t«] =V[18*U + 128A»-27V], . (k) 
 where 
 
 U=a?« + y«4-4{F-o'), and Y=a{x'^-y^) + 4,k{k^~a% 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 137 
 
 137.] On the tangential equation of the trigonal Hypocycloid. 
 In this case R=s3r j hence, as B,=2nr, w=f j and if we substi- 
 tute this value of n in the general equations, 
 
 f._ 8in(w — 1)<^ _ cos(ra— 1)<^ , . 
 
 ^~2r(n—l)smn4>' *'~2r{n-l) sinn^' ' • ' ^^> 
 
 putting <l>=s20, we shall have 
 
 f._ sing _ cosd ... 
 
 ^~rsin3^ ""r sin 30 ^^' 
 
 Hence 
 
 rS({9 + u«)sin«3g=l (c) 
 
 Dividing the expressions in (b) one by the other, we shall have 
 said= — 7===, cos 5= — ,, . sin 20= i 
 
 COS 20=^2-^2. 
 
 But sin 30 = sin 20 cos + cos 20 sing (d) 
 
 Substituting the preceding values of these expressions, 
 
 sin3g=M!d±iH!r:m 
 
 But (c) gives sin 30= />a ■ au - Hence, equating these values 
 
 of sin 30, 
 
 rf[3i;«-f2]=f*+i;« (e) 
 
 We shall arrive at a more symmetrical equation of this curve if 
 we choose the inclination of the axes at such an angle as will enable 
 us to place two successive cusps of the curve on the axes of coordi- 
 nates. Let the coordinate angle be 120°. Then, by the help of 
 the formula given in sec. [4], which enables us to pass from a rec- 
 tangular system to an oblique one, that is to say, 
 
 V, — ^COS(k> 
 
 w=-* — r , 
 
 sinci) 
 
 as m^ 120°, we shall have v = — '-r=- • 
 
 V3 
 
 Substituting this value of v in the preceding equation, we shall have, 
 putting B for the radius of the base circle, 
 
 Rfy(?+w)=r' + w*+ff, (f) 
 
 a symmetrical equation in ^ and v. 
 
138 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 This equation may be written in the form 
 
 Now i and - are the segments of the axes of coordinates cut off 
 
 by the tangent to the hypocycloid. Let I be the length of this 
 tangent between the axes, and s, s, the segments of the axes cut off 
 
 by it ; then, as ^—m+-;^+t^> ^^ ^^^^ ^^^^ 
 
 R(« + «j=Z*, (h) 
 
 or 
 
 S + 8, 
 
 or the sum of the squares of the sides is to the sum of the sides in 
 a constant ratio, while in the quadrantal hypocycloid the sum of the 
 squares is constant. 
 
 Let P be the perpendicular from the centre on the limiting tan- 
 gent to the trigonal hypocycloid. It may be shown that 
 
 138.} On the tangential equation of the hexagonal Hypocycloid. 
 Let the radius of the revolving circle be one sixth of the radius 
 of the base circle, or R=2nr=6r or «=3. 
 Hence the general expressions now become 
 sin 2.^ cos 2^ 
 
 *^* sin3^' * sin 3^ 
 Consequently 
 
 16r«(f*+w«)sin«3^=l (a) 
 
 We shall also have 
 
 sm 2A= — ,J . cos 2d> = — -==r-, 
 2cos«<^= ^* , 28in^^= ^' . 
 
 (b) 
 
 * It is beside the purpose of this work further to develope the numerous and 
 beautiful properties of this curve, which in accordance with analogy I have named 
 the trigonal hypocycloid. The ol^ect of this work is rather to develope new 
 methods of investigation than to discuss at length properties of particular curves 
 or surfaces. There is the less need to do so in the present case, as the properties 
 of the tricusp hypocycloid have been made the subject of profound investigations 
 by MM. Chaales, Cremona, and Steiner on the continent, and amongst ourselves 
 by Messrs. Clifford, Laverty, Townsend, and others. The subject will be found 
 treated with much elegance and research in the mathematical portion of the 
 ' Educational Times ' and other like publications. 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 139 
 
 Now 
 
 2 sin* 3^= 2 sin* 2^ cos* ^ + 2 cos* 2^ sin* </)]^ 
 + 4 sin 2<f> cos 2^ sin ^ cos ^. j 
 
 (c) 
 
 Substituting the preceding values of these expressions, and 
 equating the two values of sin S<f>, we finally obtain 
 
 [l-8r*($* + w*)]*(|* + u2) = 64r*i;2(3p-i;*)*. 
 
 (d) 
 
 the tangential equation of the hexagonal hypocycloid. 
 
 To exhibit the tangential equations of epicycloids and hypo- 
 cycloids of more numerous convolutions than those above inves- 
 tigated, would require the solution of cubic and still higher equa- 
 tions. 
 
 139.] On the tangential equation of the Cycloid. 
 
 The cycloid may be considered as an epicycloid, the radius of the 
 fixed circle being infinite. 
 
 Let A, the initial position of the moving point in contact with 
 the horizontal straight line, be taken at the origin of coordinates, 
 and let the circle be conceived as having rolled forward so as to 
 bring the point A into the position P. 
 
 Then, as in the case of epicycloids, the limiting tangent is at right 
 angles to the momentary radius PQ 
 of the circle whose centre is the point 
 of contact Q of the rolling circle with 
 the horizontal line or axis of X, the 
 other extremity being the tracing- 
 point P. 
 
 Hence the limiting tangent always 
 passes through B, the extremity of the 
 vertical diameter of the rolling circle. 
 
 Let and u be the tangential coor- 
 dinates of the limiting tangent, and let 
 r.2<^ be the arc which the circle has rolled over, ^ being =QBX ; 
 then manifestly, since QX = arc QA, plus the tangential coordinate 
 AX, we shall have 
 
 2rtan^=2r<^-l-g (a) 
 
 Pig. 24. 
 
 (b) 
 
 Now tan^=^, and «^=tan-'(^jj consequently we shall have 
 
 2rf [|-tan-(|)]=l, or 2r [v-ftan-'(p]=l, 
 
 the tangential equation of the cycloid. 
 
 If, instead of taking the origin at the extremity of the curve, we 
 take' it at the centre. of the base, or at the distance nr from the first 
 then using the formulae given in sec. [3] for the trans- 
 
 origin. 
 
140 
 
 ON EFICYCLOIDS AND HYFOCYCLOIDS. 
 
 »>=; 
 
 formation of coordinates^ namely 
 
 . g, „_ 
 
 l-p^^-qv/ " 1-p^^-qv/ 
 
 by which the translation of the origin and axes of coordinates in 
 parallel directions may be effected. In this case we shall have 
 p^nr, and 5=0. Hence the fonnulje now become 
 
 V 
 
 f= ^1 v=- 
 
 " I-tttB; l-nrf/ 
 
 (c) 
 
 Substituting these values in (b), we shall have 
 2^/ + 2r?,[|-tan-0] = l. 
 
 Now it may easily be shown that the tangent of f^— tan- ■ (^') 1 
 
 - f 
 
 is — . Hence the tangential equation of the cycloid, when the origin 
 
 is translated to the middle of the base, is 
 
 2rrt; + ftan-«(|)]=l (d) 
 
 140.] The vertex V and the hypotenuse AB of a right-angled 
 triangle are given in posi- p. „_ 
 
 tion. The sides are VA **^" ''^^ 
 
 and VB, while VP is the 
 perpendicular on the hypo- 
 tenuse. The line AB is 
 taken always equal to the 
 circular arc PQ. The line 
 drawn through D parallel 
 to VA will envelope a cy- 
 cloid. 
 
 Let VP=2r, and the angle PVQ=<^. Then we shall have 
 
 PQ=2r^=AD, and PA=2rcot^, and cot^=|. 
 ButPD=PA+AD, or 
 
 2r[v+ftan-'(^] = l, 
 
 an equation which exactly coincides with (d), the tangential equation 
 of a cycloid, the origin bemg at the middle point of the base. 
 141.] On the tangential equation of the Logarithmic curve. 
 
 Let the projective equation of the logarithmic curve be y=aJi, 
 and, without detracting from the generality of the expression, we 
 may take a as the base of the system of logarithms whose modulus 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 141 
 
 is m; then ^ = -^; but -/=-£; hence— ^=- A and 
 ax m dx v a v 
 
 y=i—m-; substituting this value of y in the dual equation 
 
 yv=l— «£, we shall find 
 
 £=^1+1) (^) 
 
 and the tangential equation of the logarithmic curve becomes 
 
 r™5^ +e'+'»f=0 (b) 
 
 I) 
 
 \a v/ 
 
 In any curve the subtangent s=lx—^j ; but in the logarithmic 
 
 a?=g+»», as in (a) ; substituting this value of a?, 
 
 8=m, or the subtangent is constant. 
 
 When f =sw, or when the tangent is inclined at half a right angle 
 to the axis of X, y= — m. 
 
 When a?=0, »»0=— 1, and the general equation (b) becomes 
 
 I 1 +l,or-=a. 
 
 \ av J V 
 
 142.] On the tangential eqitation of the Cissoid. 
 
 Let the projective equation of the cissoid be 
 
 a?(a;«+y«)-o/=F=0 (a 
 
 dF dF dP dF 
 
 Then finding the values of -j— , -j— , and of j— ^ + ;i— i we shall 
 
 have 
 
 ^ 3b— 2a? , 4(o-a;)s 
 ax a^sr 
 
 Eliminating x, we shall obtain for the tangential equation of the 
 cissoid, the expression 
 
 2raV=4(flf-l)3 (b) 
 
 143.] On the tangential equation of the Lemniscate. 
 Let the projective equation of the lemniscate be 
 
 (a;« + y«)«-4a«(a?2-yS)=0=F (a) 
 
 Let a7®+y*=r', then we shall have 
 
 r«=4a*(a?«-y*), 8aV=r«(4a«+r*), 8ay=r«{4a2-r2). (b) 
 Hence, using the formulae of transition, and substituting in them 
 the values of x and y in terms of r, we shall obtain 
 
 2a^j^v^ = (4a* - r*) (r^ + 20*) * = 16a« + 12a V 
 
 
 (c) 
 
142 ON EPICYCLOIDS AND HYFOCYCLOIDS. 
 
 Consequently, 
 
 r6(f«+tys)=16a*, and a^{P-v^)=zr*-12a*. . . (d) 
 Eliminating r from these equations, we obtain finally the tangential 
 equation of the lemniscate 
 
 27a*(|« + i;*)'=4[l-a«(02-i;«)]« (e) 
 
 144.] On the tangential equation of the Cardioid. 
 The projective equation of the cardioid is 
 
 (af«+y«-a«)«-4a«{(a?-a)«+y«}s=0=F. . . (a) 
 Hence 
 
 and 
 
 dF dF 
 ^^+^y=l2amx-aY-y-]; 
 
 consequently 
 
 (b) 
 
 ^~ 3a8{(a7-fl)*-jr*} ' ^~^^\{x-a)^-y^]' ' ^^' 
 
 Eliminating x and y between these equations and that of the 
 curve (a), the resulting tangential equation of the cardioid is 
 
 27a«(|* + i;«)(l-af)=4* (d) 
 
 * Mr. W. Spotiiswoode, F.B.S., has ^ven a veiy elegant solution of this 
 
 ?Tohlem in the mathematical papers published in the ' Educational Times ' for 
 865. It is as follows. " To eliminate x and i/ between the three equations 
 
 J(a ^+y'-3a')+2a' y(pfl+y''-Sa' ) (x'+y'-a')' _ 
 "3g{(r-o)Hy'} ~Sv{ix-ay+y'}-4.l(x-a)''+tf^^-"- ' ' W 
 
 "Let ifl+if — a'=r', then the given equations become 
 x(r»-2aS)+2a'=3a»g(r5+2n»-2<M:), (1) y(r'-2a')=Sah,if+2(fi-2ax), (2) 
 r*=4a%r''+2a'-2ax). . . (3) Whence (r'-2a'^x=^$-2a', . (4) 
 (r'-2fl')y=ir«v (5) 
 
 ".-. {f-2ay{3^+f)=(f-2a^\r^+,^) = ^i^{ff+^)-ZaH*^-\-'icfi; (C) 
 also, from (3) and (4), 
 
 ^_ Sr^g-Sa^ _ 8a«+4a'r'-r« 
 
 4(r'-2o>)~ So" "' 
 
 " But multiplying out, and dividing throughout by r*, (6) and (7) become 
 
 9(f+v»)r*-16r'+48fl'(l-a|)=0, (8) 
 
 r«-6o'(l-a|)=0; (9) 
 
 also 
 
 {(8)+8(9)}H-r= g^ves 9i^+v'y~8=0, 
 whence finally 
 
 27«Xf+t^)(l-nf)=4, ^b) 
 
 the tangential equation of the cardioid." 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 143 
 
 On the projective equation of the curve whose tangential equation is 
 
 oV + Z.«P=a«6*(f2 + w*)S (a) 
 
 or the curve touched by one side of a right angle which moves along 
 an ellipse, the other side passing through the centre. 
 
 145.] Let f*+w*=«r*, then the equation of the curve may be 
 
 If we diflferentiate the equation aW{j^ + v'^f—a^v^—b'^^=^=0, 
 we shall have 
 
 ^=4a«A«(f« + v«)5-26*f, and ^=Aa^b^{^ + i^)v-2a'^v,] 
 af dw ( (pj 
 
 andA=^f+^w=2a*6«(r' + v«). ) 
 
 He.« ."_(52^i)l,„a,=^*^^. . . (a, 
 
 If we square these equations separately, and introduce the values 
 of f and V given in (b), we shall have 
 
 a^'nfia^ia^-b^) =4a*w^-4a«w2 + l-4a*6«««+4a«6««*-62OTS (e) 
 
 l^fo^y^a^-b^) =4i*a^'Bfi-4M^^vi* + a^vr^-4!b*'B* + 4b^is^-l ; (e,) 
 
 adding these expressions, and dividing by a^—h^, we shall have 
 
 [a«a?« + JV + 4a'*']«^=4(a* + 6V-3; . . . (f) 
 
 or writing Q* for a'a:* + i*y'+4a*6*, the equation becomes 
 
 Q4«*=4(aH*«)i!r8-3 (g) 
 
 If we multiply (e) by 6' and (e,) by o*, and then add the results, 
 w'e shall have, dividing by (o*— 6'), 
 
 Let a;'+y*=R% and this equation multiplied by 3 becomes 
 
 3fl«*«R«««=3(a*+i«)iir«-3. 
 If we subtract (g) &om this expression, and divide by w', we 
 shall have 
 
 3fl«62R«iii^-Q*w2=-(a«+i«) (i) 
 
 We have now to eliminate «* and ia^ between the equations 
 3ffl«6«R*«^-QV*+ (a« + A«)=0, and Q,'»i!K«-4{a2 + i')w8+3=0. ( j) 
 
 Eliminating tar* and w', we get finally for the projective equation 
 of this curve the following expression, 
 
 Qi2_ (a2 + i2)2Q8-18a5!iS(a2 + 6^)R*Q'') ,j.. 
 
 + 16(a2 + «2)3a262R2 + 27«*6«R'*=0. \ ' ' ' ^ ' 
 
144 ON EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 Substituting for Q* its value [a«jr« + AV +"*«**']» ^^ obtain the 
 projective equation of the curve, which is of the sixth degree. 
 
 When the curve is an equilateral hyperbola, a' + i'^O, and the 
 general equation becomes 
 
 [(a,9_yS)_4o!]8==27a«(a?« + y«)«- 
 
 146.] On the tangential equation of the curve inverse to the central 
 ellipse. 
 
 The projective equation of this curve is 
 
 a«a?« + 6V=(a''+y')* (a) 
 
 Let 
 
 a?9+2,9=r«, (b) 
 
 then the preceding equation becomes a^x^+b^y^=r*. . . (c) 
 
 Hence 
 
 but 
 and 
 
 Consequently 
 
 
 (e) 
 
 s=i^^-, „=(Hr!^; . . . . (f) 
 
 or, substituting the values of x and y given in (d), 
 
 (a*-J«)r«J«=4r«-4aV + flV*-46V+4a«iV-a*Z-2, . (g) 
 
 (a'-4«)r6i;«= -4r«+4aV-6V+46V-4a2A«r« + ffl2Zi* ; (h) 
 
 add together (g) and (h), divide by (a*— 6*), and the resulting ex- 
 pression will become r''{^ + v^)={a^ + b^)r^—aW. ... (i) 
 
 Multiply (g) by b^ and (h) by a*, and the resulting expression, 
 dividing by (a*— i*), becomes 
 
 {6«f*+aV)r6=-4r6+4(o2-|-62)r4_3a«6V; . . (j). 
 
 or if we put U = 6*|*-|-ffl*t;'+4, (k) 
 
 we shall have, dividing by r', Ur* = 4 (a* + i*) r« — 3fl*4* ; . (1) 
 
 and if we write w^ for (J* 4- w*), (i) becomes 
 
 r^ia^={a^ + b'^)r^—aW (m) 
 
 Multiply this expression by 3, subtract (1) from it, and divide by 
 r^ we get 
 
 3r4i!r«-Ur2+(a2 + i*)=0 (n) 
 
ON EPICYCLOIDS AND HYPOCYCLOIDS. 145 
 
 Multiply (1) by (a* + 6«) and (n) by 30*6^ the resulting expres- 
 sions will become 
 
 9»^a262«r2=3Ua*iV-3o«i2(a9 + fi2). | • • ■ ^^ 
 Subtract one from the other and divide by r^ ; consequently 
 
 or 
 
 ,_ 4(a' + &^)«-3Ua^&» , « 
 
 U(o« + 6*)-9o262w« ^'^^ 
 
 (1) may be written 
 
 ._4(oM2*V-3a*6^ 
 U ' 
 and (n) may be written !■ (q) 
 
 ~ 3«* ■ 
 
 Equating these values of r*, we get another expression for r^ — 
 that is to say, 
 
 „a_ U(a« + &«)-9a«6V^ 
 
 U«-12(a« + 6V* 
 
 If we now eliminate r* from this and the preceding equation (p), 
 we shall have the equation 
 
 flSisU'- (a* + 6«)«U« - 18a*6«(a«+ *«) w'U +27a*4*i!i^ 
 + 16{a«-|-*«)Sw2=0j 
 
 and if we substitute for TJ and cr their values, the tangential equa- 
 tion of the inverse curve of the central ellipse becomes 
 
 ^9^2j-i2j2 + aV + 4]8- (a« + *«)«[6*|«-|-aV + 4]2 | 
 
 -18a«A*(a«-|-i')(f* + i^)[*'f* + aV + 4] +16(a2-l-6«)S(f« + i;«) [ (s) 
 
 If in this equation we make 4*= —a', the original curve becomes 
 an equilateral hyperbola, and the preceding equation is reduced to 
 
 27o«{£« + w*)*=4[l -a«(£«-u«)]s, 
 
 which is identical with (e) in sec. [143], writing 2a for a, and is 
 therefore the tange;ntial equation of the lemniscate. 
 
 147.] On the reciprocal polar of the evolute of an ellipse. 
 
 The equation of this curve is 
 
 aV + 6*a?«=a?V, 
 
 and it may be generated as follows. 
 
 Let a tangent to an ellipse be drawn meeting the major and 
 minor axes produced in the points C and D. If ordinates to the 
 
146 
 
 ON EPICYCLOIDS AND HYPOCYCtOIDS. 
 
 axes be erected at these points, the locus of their points of inter- 
 section will be the curve whose equation is given above. 
 
 Fig. 26. 
 
 The tangents to the ellipse at A and B will be asymptotes to the 
 curve. 
 
 Let F=a?V-oV-6'a;2=0; (a) 
 
 then 
 and 
 
 Hence 
 
 
 
 dx 
 
 (b) 
 
 
 consequently ^=-g. In like manner f =-3 
 
 Let 'aa=l, bfi=l, *«=^, y^=^. 
 
 Substituting these values in (a). 
 
 or, as 
 
 b^a* . o«** a*bi i» a» 1 
 
 1 A 1 
 
ON THE INVOLUTE OP THE CYCLOID. 147 
 
 the tangential equation of the reciprocal polar of the evolute be- 
 comes 
 
 (D'Hi)*- («' 
 
 precisely the form of the projective equation of the evolute of the 
 ellipse. 
 
 On the tangential equation of the semicubical parabola. 
 
 14S.1 The projective equation of the semicubical parabola may 
 be written in the form 
 
 27ay«-4rS=0=P (a) 
 
 Taking the partial differentials, 
 
 dF ^^ dF dF dF 
 
 ^=54ay, ^ Ua,^, and D=^j,+-^^=54«y«-12^. 
 
 Consequently a?=p y = . Making these substitutions in (a), 
 
 we shall have 
 
 a^-ws=0, (b) 
 
 the tangential equation of the semicubical parabola. 
 
 On the tangential equation of the OuMcal parabola. 
 
 149.] The projective equation of the cubical parabola may be 
 written under the form 
 
 3a'y-«»=0=F. 
 
 „ dP o c- dF „ , , dF dF T. „ , 
 
 Hence -5-=— 3ar, -5-=3a*, and -3— w + ^-ir=D= — 2a^. 
 da? 'Ay ' Ay" Ax 
 
 3- 1 
 
 Consequently a? =^, y=— — . 
 
 Substituting these values of x and y in the projective equation, 
 we obtain for the tangential equation of the cubical parabola 
 
 9i;+ 40*^=0. 
 
 On the Involute of the Cycloid. 
 
 150.] Let the cycloid be conceived as placed in the position 
 O V C (fig. 27), O being the centre, V the vertex, and C the cusp ; 
 and let the unwinding of the curve be conceived to commence at the 
 point V, and let P Q be the position of the elastic radius. Now P Q 
 is a perpendicular to the limiting tangent LT at P, and 
 
 P Q=the arc Q V = 2Q B=4r cos <^. 
 
 l2 
 
148 
 
 ON THE INVOLUTE OF THE CYCLOID. 
 
 Fig. 27. 
 
 I 
 O A^ DO 
 
 But we have shown in sec. [5] that the expression for a perpen- 
 dicular on a limiting tangent let fall &om a point Q, whose pro- 
 jective coordinates are x and y, is 
 
 1—^—^ 
 
 Vr + w*' 
 
 The angle ABQ=LTO=^, OT= — « and 0U=-. Hence 
 
 Vr + w 
 But in this case as F=4r cos ^, 
 
 4rv=\-\-x^—yv, since |^ is negative. 
 
 (a) 
 
 Now 
 
 y=QLD=2rsm^<f>=^-^ or yv=^^, 
 
 and a7=OC— AC-I-AD, or a;=:2r^(-^— ^j-f-sin^cos^i, since 
 
 the angle which the limiting tangent at P makes with the vertical 
 is the complement of the angle which the elastic radius makes with 
 the same vertical ; hence 
 
 Consequently 
 
 ^-^(i-*)«+^- 
 
 x^-yv=2rsQ-<f>y 
 and therefore 
 
 4rw=H-2rf(|-^). Now tan g-<^j=cot^=^ ; 
 
ON PARM.LEL CVRVES. 149 
 
 consequently' 
 
 4rv-2r^ tan- '0^=1 (b) 
 
 If we now translate in parallel directions the axes of coordinates 
 to the vertex V of the curve, which we may effect by the help of 
 the formulae in sec. [3] , 
 
 2r 
 
 'l+2rv; l + 2rv; 
 the preceding expression becomes 
 
 |«-ftan->(|)| = l (c) 
 
 But this is exactly the expression we have found for the tangen- 
 tial equation of a cycloid when the origin is at a cusp ; see (b), sec. 
 [139] . Hence the involute of a cycloid is a cycloid having its axis 
 parallel to the axis of the former, at the distance 2r from it, and 
 having its cusp at the vertex Y of the original cycloid. 
 
 Since FQ=4rcos<^, and BQ=2rcos^, it is evident that the 
 flexible radius is always bisected by the horizontal tangent to the 
 original cycloid at V, 
 
 When the axes of coordinates are translated in parallel directions 
 to the vertex Y of the cycloid, the tangential equation of the cycloid 
 assumes the very simple form 
 
 2r?tan-'(|) = l (d) 
 
 CHAPTER XY. 
 
 ON PARALLEL CURVES. 
 
 151. J Two straight lines in a plane are said to be parallel when 
 a perpendicular to the one is also perpendicular to the other j whence 
 it follows that these perpendicular distances, wherever taken, are 
 equal. 
 
 So also two concentric circles may be said to be paralle], since 
 the differences of their radii are equd, and they are at right angles 
 to their respective circles. 
 
 We may widen this definition and say that two or more curves 
 are parallel when the differences of their coincident normals are 
 constant. 
 
 In general the parallel curves will be of different orders. Thus 
 while the tangential equation of the ellipse is of the second degree, 
 the tangential equation of its parallel curve will be of the fourth 
 degree. 
 
150 
 
 ON PABAUiEL CUKVES. 
 
 LetNQ, N,Q be two coincident normals to the curves AB 
 and A,B,. Let the constant diflPerence of these normals be h. Let 
 OP=P, aad OP,=P,. 
 
 Fig. 28. 
 
 Now as OX:OX,::P 
 
 V,<^\.l::-^: 
 
 {=¥-^=^> '"4,= V£,'+»,'=-,. 
 
 Hence 
 
 f=l 
 
 i^ ..=. 
 
 and also ■«■=- 
 
 l-Aw/ ""l-Aw/ """ """ 1-A«r/ ■ • (*^ 
 
 Making these substitutions for and v in the original tangential 
 equation of the curve, we shall obtain the tangential equation of 
 the parallel curve. 
 
 We shall now proceed to apply this theory to a few examples. 
 
 152.] To determine the tangential equation of the curve parallel 
 to the ellipse. 
 
 The tangential equation of the ellipse referred to its centre and 
 axis is o*P + b^t^ = 1 . 
 
 Substituting the values above given for ^ and v, we shall have 
 
 [(o«-A«)f«+(A«-A«)w«-l]«=4A9(£«+w'). . . (a) 
 
 Hence it follows that if a series of elastic radii or strings be applied 
 to the evolute of an ellipse, of all the curves that may be thus traced 
 out by a tracing point at the extremity of the radius, there is only 
 one that will be an ellipse; all the rest will be curves of a higher 
 order than the second. 
 
 The length of a quadrant of the evolute of an dlipse is manifestly 
 the difference between the normals to the ellipse at the extremities 
 of the minor and major axes. Let a be the arc of the quadrant of 
 the evolute, then 
 
 _9 1.9 _s in 
 
 (b) 
 
 Let a and /3 be the semiaxes of the evolute. These semiaxes are 
 
ON PARAIXBL CURVES. 151 
 
 evidently the differences between the semiaxes of the ellipse and 
 the coincident normals^ or 
 
 a=a = =ae*, and p=-7 — b= — = — =— i— ; 
 
 a a '^ b b 
 
 consequently 
 
 a b . 
 
 -r~a (''^ 
 
 Now the length of the elastic radius or string that may describe 
 the ellipse by a tracing point at its extremity is the quadrant of the 
 evolute, plus the difference between the coincident semiaxes of the 
 ellipse and evolute ; hence, the length of this radius being I, 
 
 l=s+{a—a), or 1= — r — ha— ae'=-T-. 
 
 This is the length of elastic radius or string that will describe the 
 conjugate ellipse. 
 
 To express the length of this elastic radius in terms of the semi- 
 axes of the evolute. 
 
 Now i=-r, and a=ae^, /8=— r- ; eliminating a and b, we find 
 Hence, if the semiaxes of the evolute of an ellipse be a and /3, all 
 
 Q3 
 
 other lengths of flexible radii or strings greater or less than -5^- — -j 
 
 will describe curves whose tangential equations are of the fourth 
 order. 
 
 Hence also, if we assume the surface of the earth to be an oblate 
 spheroid, and to be covered by a sea or atmosphere of uniform 
 depth, the outer surface of this enveloping shell w^ not be a surface 
 of the second order, but one whose tangential equation is of the 
 fourth order ; and if we imagined successive shells or strata of uni- 
 form thickness to be applied, the equations of the outer surfaces 
 would all be of the same order. 
 
 This is a curious illustration of a breach of the law of continuity. 
 However small the thickness of the shell may be, while the interior 
 surface is that of an ellipsoid the tangential equation of the outer 
 surface will be of the fourth degree. 
 
 153.] The normals of a parabola are increased by the constant 
 line h ; to find the tangential equation of the curve. 
 
 The tangential equation of the parabola referred to its focus as 
 origin, the axis of the parabola being the axis of X, is, as shown in 
 sec. [49] (b). 
 
152 ON PARALLEL CORVES. 
 
 If we now substitute for f and v their values given in the pre- 
 ceding section, we shall have 
 
 the tangential of the curve which is always at the distance A from 
 the parabola, or the c\irve whose normals differ from the coin- 
 cident normals of the parabola by the constant quantity A. 
 
 The parallel curve of the evolute of the ellipse whose tangential 
 equation is 
 
 a«|«-|-JV=(ffl*-6«)«P«'S (^) 
 
 will have for its tangential equation the following expression : — 
 
 [a«|«(l+A'n + *^»^(l+AV) +{A«(a» + 6«)-(a«-i«)*}r«'']'K . 
 
 To find the parallel curve of the quadrantal hypocyclmd. 
 
 Its equation is f-J-i;'=a'j*w'. 
 
 Hence the equation of its parallel curve is 
 
 [l-2aAfw](^« + i;«)=A*(^ + i/')«+a«f«i/'. . , . (d) 
 
 To find the equations of the parallel curves of the involute of the 
 quadrantal hypocyclmd. 
 
 154.] The tangential equation of the involute of the quadrantal 
 hypocycloid is, as in sec. [136], 
 
 [(*+«)?«+ (A-a)i^]*=r + w' (a) 
 
 To obtain the equation of the curve parallel to this curve, we 
 have only to substitute in this equation the values of and v indi- 
 cated in [151], and the equation of the parallel curve, or of one whose 
 normals differ from those of the former by the constant A, is 
 
 [(A + A-t-o)|«+(A + A-o)i;«]«=£«-|-i;«, . . . (b) 
 
 a form exactly the same as the preceding, only having k+h instead 
 of k. Whence it follows that whatever the lengths may be of the 
 successive elastic radii, the difference between the axes of any one 
 of the involutes is constant, and independent of the length of the 
 elastic radius. Hence these successive curves constitute bands of 
 equal width, like concentric circles. 
 
 We may easily find the tangential equation of the curve that is 
 parallel to the semicubical parabola. We have only to substitute 
 
 for £ and v the values -r-^ — and .; — t— in the equation of the 
 1— Aw, 1— Aw, ^ 
 
 curve 0^= v^, as given in sec. [148] . The equation will become on 
 
 reduction 
 
 ««f-AV+(AV-2fl^)fi;« + i;4 = 4. . . . (c) 
 
ON THE TANGENTIAL EQUATIONS OF EVOtUTES. 
 
 153 
 
 In like maimer we may find the equation of the curve parallel 
 to the cycloid, whose tangential equation is 
 
 to be 
 
 2n.[l+|tan-.(r)] = l, 
 
 (d) 
 
 CHAPTER XVI. 
 
 ON THE TANGENTIAL EQUATIONS OF EVOLUTES. 
 
 155.] The system of tangential coordinates supplies a general 
 method of determining the tangential equation of the evolute of any 
 curve whose projective equation is given. To determine the equa- 
 tion of the evolute of a curve by the ordinary methods often requires 
 eliminations that are wholly impracticable. 
 
 Let A B be an arc of a given 
 curve, whose equation is 
 
 F(a7,y)=0. 
 
 Let Qt be a normal and 
 Q,T a tangent to the curve 
 
 AB. Let 0T=|, Ot=J^, 
 
 OD=a:,QD=y; andasTQr 
 is a right angle, Q D*=D T x D t, 
 
 1^ 
 
 Fig. 29. 
 
 
 ■"1 
 
 K-. 
 
 
 . / 
 
 '^>^ 
 
 ^ 
 
 or 
 
 y^=(t-xjU-^j, or [a;-(y«-l-a?«)|]f,= l-*f. 
 
 (a) 
 
 But, as we have shown in sec. [22], f =tt! 
 
 dP 
 dx 
 
 dF 
 
 If we substitute this value of J in the preceding equation, we 
 shall have 
 
 dF _dF 
 
 Ay , , Ax 
 
 r,= 
 
 dF dF 
 
 -, and also i;(=- 
 
 (b) 
 
 dv * da?^ 
 
 dF _dF ■ • • 
 Ay Aae^ 
 
 156.] We may apply this method to a few examples. 
 
 To determine the tangential equation of the evolute of the ellipse 
 
154 ON THE TANGENTIAL EQUATIONS OV EVOLUTES. 
 
 The equation of the ellipse is a*y* + **■*'*—<'***= ^=0; hence 
 Substituting these values in the preceding formuke^ we find 
 
 Introducing these values of x and y in the equation of the ellipse, 
 we obtain, omitting the accents, 
 
 fl2g« + *V=(a«-6«)*|«i/», (a) 
 
 the tangential equation of the evolute of the ellipse. 
 
 157.] To find the tangential equation of the evolute of the pai'a- 
 
 bola. Let the projective equation of the parabola be y' = 4A {2k + a?), 
 
 the origin being taken on the axis at the distance 2k, or the semi- 
 
 dF dF 
 
 parameter, &om the vertex. Hence -j— =2y, -^ =4Ar; making these 
 
 substitutions in the general formulae, we find for the tangential 
 equation of the evolute Ag^=w*, the tangential equation of the 
 semicubical parabola. 
 
 158.] On the evolute of the semicubical parabola. 
 
 Let the projective equation of the semicubical parabola be written 
 in the form 
 
 3ay*-2a^=0=F. 
 
 Then ^=6ay, ^=-6^- 
 
 consequently 
 
 t— "■ 
 
 Hence 
 
 w{a+x)' y{a+x)' 
 ^ ay flw* 
 
 V 
 
 ,s» 
 
 and 
 
 or *=f IT' 
 
 y-va~^~^' 
 
 Substituting these values of x and y in the dual equation, we 
 shall have for the tangential equation of the evolute of the semi- 
 cubical parabola the expression 
 
 4at/«=3J«(3f-2aw«). 
 
 159.] On the evolute of the cubical parabola. 
 The projective equation of the cubical parabola is 
 
 r-a?3=0 (a) 
 
ON THE TANGENTIAL EQUATIONS OP EVOLUTES. 155 
 
 Consequently, f and v being the tangential coordinates of the 
 evolute, 
 
 >_ 3a» 3ar^ 
 
 '^~3a:(a*+a^)' ""S^Ca'+a^) ' • • • • W 
 
 dividing one by the other, a?=flA/^ (c) 
 
 Substituting this value of j? in the equation of the curve (a), we 
 find 
 
 and if we substitute these values of x and y in the dual equation 
 w^-\-yv=\, we shall obtain as a final result the following equation 
 for the evolute of the cubical parabola, 
 
 9?'=a«y(3p + t/')9 (d) 
 
 On the evolute of the quadrantal Hypoeycloid. 
 
 160.] The tangential and projective equations of the quadrantal 
 hypoeycloid are 
 
 J3 + u3=o«Pu2, and ^+y»=a*j ... (a) 
 
 differentiating this latter, and applying the usual formula of trans- 
 ition, we obtain 
 
 dF . _. dF . _. 
 
 d^=^^ *' d^=^^ '' 
 and 
 
 dF dF ,, , . 
 
 Consequently 
 
 tS _g, 
 
 hence ^= , or yj* +xi/*=0 j combining this expression with the 
 
 dual equation x^ + yv=\, we get 
 
 _ g8 _ -ifi 
 
 Substituting these values of x and y in the projective equation 
 of the curve, we obtain 
 
 ^ + v«=o2(J«-w«)2, (b) 
 
 the tangential equation of a quadrantal hypoeycloid, which we 
 may reduce to the usual form by turning the axes of coordinates 
 
156 
 
 ON THE TANGENTIAL EQUATIONS OF EVOLTTTES. 
 
 through an angle of 45°. In sec. [2] it has been shown that this 
 transformation may he effected by putting f=fyCOS^+w, sin^. 
 
 v=^, sin 6—v, COS 6, or in this case. 
 
 ^=4' *^2?=f/+''i' 
 
 i\/2v=^,—v,; hence £*—w*=2^,W|; consequently equation (b) may 
 now be written 
 
 f2 + „9={2a)«Pi;«- . (c) Fig^SO. 
 
 Hence it follows that the 
 evolute of the qnadrantal hy- 
 pocycloid is also a quadrantal 
 hypocycloid whose parameter 
 is twice that of the original 
 hypocycloid, and whose axes 
 make angles of 45° with the 
 original axes. If we take the 
 evolute of this hypocycloid, 
 its parameter will be four times 
 that of the original hypocy- 
 cloid, and its cusps will lie in 
 the same direction. 
 
 On the tangential equation of the evolute of the Lemniscate. 
 161.J Let the projective equation of the lemniscate be 
 
 (a?'+y«)'=4a«(ar«-y«) (a) 
 
 Putting r* for x^+y^, we may write this equation under the forms 
 
 r2= 
 
 »^+4aV« 
 
 4aV«— r* 
 
 8o« 
 
 8as 
 
 (b) 
 
 Hence 
 
 dF dF 
 
 — =4r'a;— 8a'a:, ^^4r*y + 8o*y. 
 
 Substituting these values in the general formulae for determining 
 the tangential equations of evolutes, we shall have 
 
 ?= 
 
 2a« + r» 
 4a** ' 
 
 i/=- 
 
 2o«-r* 
 
 -, or jf=- 
 
 2a^+r^ 
 
 2a^—i^ 
 
 4a^u 
 
 (c) 
 
 Between (b) and (c) we may eliminate x and y. 
 The elimination of x gives 
 
 [1 - 2fl«S2j ^ + 4a8 [1 _ 2a8js-] r* + 4a« =0. 
 The elimination of y gives 
 
 [l+2a«u^r*-4a*[l + 2flV]r«+4a*=0. 
 If we subtract these equations, one from the other, we shall have 
 (?^+w')r2=4-4a2(S«-w*) (d) 
 
ON THE TANGENTIAL EQUATIONS OP EVOLUTES. 157 
 
 Adding these equations together, we shall have 
 
 [l-a«(f-u*)]r«-4a*(f«+wV+4a*=0. . . (e) 
 Eliminating r between these equations, we get 
 
 the tangential equation of the evolute of the lemniscate. 
 
 The preceding equation may be written also under the form 
 8[1 -o*(f«-w«)]2_4[l _a«(52_„2)] (1 +4fl*^V) +a*(f* + w*)«=0.(g) 
 
 We shall now apply this method to a more difficult example. 
 162.] To find the evolute of the curve 
 
 a^a!'' + b^y^={a^+y^f (a) 
 
 This is the equation of the locus of the foot of a perpendicular 
 from the centre to a tangent to an ellipse. Let 
 
 a7*+ys=r', (b) 
 
 and the equation of the curve may now be written in either of the 
 forms 
 
 r*-6V , oV-r* 
 
 "^=1^1^ ""^ y=i^z^ (*') 
 
 If we differentiate (a), and substitute in the general formulae for 
 determining the tangential coordinates of the evolute the values 
 
 thus found for -t- and t-, we shall find, after some simple reduc- 
 tions, 
 
 2r«-4« , o«-2r« ,,, 
 
 If we now eliminate x between the first of (c) and the first of 
 (d), and y between the second equations of the same groups, we shall 
 h^ve two equations by which we may eliminate r. Hence 
 
 [a«^_Asp_4]r«+ [b*^ -a%T +4i«]r«-i'»=0,l 
 [«V-fflV-4]»-*+[flV-fl*6V+4ffl*]r-«-fl*=0.J ' ^^^ 
 If we multiply the first of (e) by a* and the second by b*, we shall 
 have 
 
 [«6fs_a«i9p-4a*]r*+ [a*b*S^-€fib^^+4^b^r^-t^b*=0A 
 llfiv'^-a^b*v^-4A*]r*+[efib*v^-a^^v^+4a^*]r^-M*=0.\ ^ ' 
 
 Subtract one from the other, divide by (a*— 6*)r*, and the result- 
 ing equation will become 
 
 [ffl*f*+ft*w*-4(a«+A«)]r«+ [4a«A«-a*6«J«-fl«i<i;*] =0. (g) 
 Hence 
 
 ,_ a«A«[fl«^+iV-4] 
 ~[a*f«+6V-4(a«+6«)]" , • • • • W 
 If we divide equations (f ) by the coefficients of r*, and then sub- 
 
158 ON REVOLVING ANGLES, PKOJECTIVE AND TANGENTIAL. 
 
 tract them one from the other, we shall obtain another expression 
 for r* — that is to say, 
 
 [{a«-J«)^-4][(c«-A«)i;*+4] ^' 
 
 Between (h) and (i) we may eliminate r ; so that the resulting 
 tangential equation of the evolute of the inverse curve of the central 
 ellipse is a« follows : 
 
 fl«*«[a«£2 + 4V-4] [{a«-J«)P-4] [(o9-&«)i>«+4]'l ...^ 
 
 = [a*P + *V-4 (a« + *«)] *. J • ^' 
 
 If in this equation we make —b^=a^, it will coincide with (g), 
 sec. [161], the tangential equation of the evolute of the lemniscate. 
 
 CHAPTER XVII. 
 
 ON REVOLVING ANGLES, PROJECTIVE AND TANGENTIAL. 
 
 163.] These constitute a large class of problems. It is an obvious 
 question to ask what may be the locus of the point in which a per- 
 pendicular &om a fixed point meets a tangent to a given curve ; and 
 it is equally pertinent to inquire if a right angle move along a given 
 curve, one side passing through a fixed point, what will be the curve 
 enveloped by the other side of the angle. 
 
 To these questions answers have hitherto been obtained by inde- 
 pendent methods, each adapted to the particular case under inquiry. 
 But they may all be solved by one uniform method, as simple as it 
 is universal, and one that requires neither skill nor ingenuity in its 
 application. 
 
 It is proper to give distinct definitions of these curves. When a 
 right angle moves along a curve whose projective equation is given, 
 one side of the right angle passing through the origin, the other 
 side will envelope a curve which maybe called the tangential pedal 
 of the given curve. 
 
 When a right angle moves so that one side shall touch a curve 
 whose tangential equation is given, while the other side passes 
 through the origin, the vertex of the right angle will describe a 
 curve which may be called \h.B projective pedal of the given curve. 
 
 Nothing can be more simple tluin the proof of these important 
 propositions. 
 
 Let XY be a tangent to a curve afi, and OF a peipendicnlar on 
 
 * Frofeasor Price, in his able treatise on the Infiniterimal Calculus, toL j. 
 p. 383, well oheerves, "TheoieticaU}r,the equations to the evolutes of all curves 
 niay be found by means of the g^ren equations ; but the difficulty of elimination 
 is in all cases, save in two or three besides the above, so great as to be beyond 
 the present powera of analysis." 
 
ON REVOLVING ANGLES, PROJECTIVE AND TANGENTIAL. 
 
 159 
 
 this tangent. LetPD=y,OD=ar,OX=i OY=i and let the angle 
 which O P makes with the axis 
 
 of X be ^. Now tan <f»=^ 
 
 Fig. 31. 
 
 and cot <b=-. 
 
 Hence — = 1, 
 
 or y^ — 3nj = 0. Combining 
 this relation with the dual 
 equation a?f + yw = 1, elimi- 
 nating one variable at a time^ 
 we shall have 
 
 y- 
 
 x=-, 
 
 f +««' "'e+v^' "'x^+y'^' "~W+y^' • • ^*^ 
 
 (a) Let a perpendicular be let fall from the origin of coordinates 
 
 on a tangent to the curve a/3, whose tangential equation is 
 
 <j){^, v) =0. The projective equation of the locus of the point in 
 
 which the perpendicular from the centre meets the tangent will be 
 
 s! ■, y 
 
 and 
 
 -s- — 5 for f and v 
 
 ar+y^ 
 
 obtained by the mere substitution of - . . 
 •^ ar+y^ 
 
 in the preceding equation ; so that the projective equation of the 
 
 projective pedal will be ^ ksfy. ^sjp =0- 
 
 (j3) Let AB be a curve (see preceding figure) along which a right 
 angle moves, one side passiag through the origin. The tangential 
 equation of the curve ajS which the other side envelopes, will be 
 
 found by substituting »r^-o and ^ ^ for a? and y in /(a?, y)=0, 
 
 the projective equation of the given curve AB ; so that the tangential 
 
 equation of the envelope will be/ mVIj esT~b \~^- 
 
 These are very remarkable as well as general theorems. They 
 admit of the widest application ; and there are no difficulties of eli- 
 mination to contend with. The solutions are obtained by simple 
 substitution, and they afford another striking illustration of that 
 duality which pervades all mathematical truth. 
 
 164.] We may illustrate this theory by its application to a few 
 examples. 
 
 (a) The projective equation of a straight line is ay + bx^ab ; 
 
 V f 
 
 for V and x substitute their values jz-, — „ and „ „ hence 
 
 " P + w t^ + w* 
 
 av+b^=ab(^+ v^), the tangential equation of a parabola. Hence 
 the tangential pedal of a straight line is a parabola. 
 
 (/3) The tangential equation of a point isa^+bv=l. For and v 
 
160 ON REVOLVING ANGLES, PEOJBCTIVE AND TANGENTIAL. 
 
 SB V ' 
 
 substitute their values g ^ and -a^— 5 ; the resulting equation 
 
 becomes ax+by=x^ + y'^, the projective equation of a circle passing 
 through the origin. Hence the projective pedal of a point is a 
 circle. 
 
 (y) Let the general tangential equation of a conic section be 
 
 Substituting for f and v their values as given above, we shall 
 have 
 
 oa;* + o^« + 2/3a?y + 2(yj7 + 7,y) (ip* + y') = (as* + y«) * ; 
 this is the projective equation of the locus of the foot of a perpen- 
 dicular let fall from any point in its plane on a tangent to the curve. 
 It is manifestly a curve of the fourth degree. 
 
 (S) The tangential equation of a circle when the origin is at the 
 extremity of a diameter is r'u'+Zrf^l ; and if we substitute for 
 
 V and f their values ~J_ g and „ g , we shall have 
 
 the projective equation of a cardioid, the origin being at the cusp 
 of the curve. 
 
 165.] The projective equation of a circle, the origin being any- 
 where in its plane, is 
 
 ^+y*— 2pa:— 2gy=r*— /)'— y' (a) 
 
 Hence, if a right angle move along this circle, one side passing 
 through the origin, the other side will envelope the curve whose 
 tangential equation is 
 
 (r*-i>«-9«)(5« + v«)+2pf-H2gw=l. . . . (b) 
 
 Now, as the coefficients of the squares of the variables are equal, 
 and as the rectangle vanishes, this is the tangential equation of a 
 conic section whose focus is at the origin and whose centre coincides 
 with the centre of the circle, seeing that p and q are the projective 
 coordinates both of the circle and the conic section. 
 
 When the point is within the circle the locus is an ellipse, when 
 on the circle it is an infinitesimal parabola degenerating to a straight 
 line passing through the centre ; and when the point is outside, the 
 locus is an hyperbola. 
 
 Let the tangential equation of the parabola be, see (d) sec. [49] , 
 
 and if we substitute in this equation the values of ^ and v, we shall 
 have 
 
 fa^+fff^+ff^ + ihx+hjy) {a^+y^) =0, 
 
 a curve of the third d^ree which passes through the origin. 
 Let the tangential equation of the ellipse referred to its centre 
 
ON KEVOLVING ANGLES, PKOJECTIVE AND TANGENTIAL. 
 
 161 
 
 aud axes be a*£* + 6*1;* = 1, then the equation of the projective pedal 
 of this curve will be a*ir* + 6'y*= (a;*+y*)*, a well-known curve. 
 
 166.] Let the projective and tangential equations of the semi- 
 cubical parabola be ay* =0?, and u* — a^=Q. If a right angle move 
 along this curve while one side passes through a cusp, the other 
 side will envelope the curve whose equation is ai;*(£^+ii^) =S^; 
 and the locus of the foot of the perpendicular on a tangent to this 
 curve wUl be y*(y*-l-a;*) = aa:^. 
 
 167.] The tangential equation of a cycloid, the middle of the 
 
 base being the origin, is 2r-|u -1-0 tan~'(-H = l (a) 
 
 Consequently the locus of the foot of the perpendicular on a tan- 
 gent to the curve is 
 
 2r J y -J-a; tan-' / - H =i»*-l-y' 
 
 (b) 
 
 An independent geometrical proof of this theorem may be given. 
 Let O C be a perpendicular 
 
 on the tangent PB to the 
 cycloid at P. Let OD=a?, 
 C D = y, and the angle 
 PBQ=^. 
 
 In the adjoining figure, since 
 CD and OD are the coordi- 
 nates of C, the foot of the per- 
 pendicular OC on the tangent 
 PB to the cycloid, we shall 
 have 
 
 0Q=0A-QA=2r 
 
 Fig. 32. 
 
 1-*} 
 
 and 0Q=0D-QD=ir-(2r-y)tan<^. 
 
 Equating these two values of OQ, we shall have the above expres- 
 sion for the locus. 
 
 168.] A right angle moves along a cycloid; one side passes through 
 the centre of the base, the other side will envelope a curve whose tan- 
 gential equation may be thus found. 
 
 When the cycloid is referred to projective coordinates, the origin 
 being at the centre of the base, we know that its equation is 
 
 •cos-M^^ \=x— i^2ry—y^ 
 
 If we make the necessary substitutions, the equation of the en- 
 veloped curve will become 
 
 
 \2r{e 
 
 M 
 
162 ON BEVOLYINa ANGLES^ PROJECTIVE AND TANGENTIAL. 
 
 169.] Let the tangential equation of the cardioid be 
 
 o«(P+w')(3-2af)=l; 
 and if we put a=3c, the equation will become 
 
 27c8(P + w«)(l-2c^)=l. 
 Substituting for f and v their values, we get 
 
 27c' / a;g + yg-2ca; > , 
 
 or 
 
 27c*(a;2+ys-2car) = (a?«+y«)« .... (a) 
 
 is the projective pedal of the cardioid. 
 
 The projective pedal of the evolute of the cubical parabola whose 
 tangential equation is 9 J® = a'u (3f* + u') ^ see sec. [159] , will become, 
 on making the indicated substitutions, 
 
 9a^(^«+jr')*=a'y(ar*+y*)* (b) 
 
 170.] Let the curve be the quadrantal hypocycloid whose tan- 
 gential equation may be written 
 
 SB 1i 
 
 For f and v substitute their values -5 5 and -5-^^ — = i the re- 
 
 suiting equation becomes 
 
 And the polar equation of this curve, putting a;=r cos 0, y =r sin 0, 
 becomes 2r=/sin2&. 
 
 Hence the locus of the foot of the perpendicular on a tangent to 
 
ON KEVOLVING ANGLES, PROJECTIVE AND TANGENTIAI.. 163 
 
 the qaadrantal hypocycloid becomes the looped curve, as shown in 
 figure 33. 
 
 171.] The equation of the logocyclic curve is 
 y^{^a—x)^w{a—xY, 
 as we shall show further on. 
 
 The equation of its tangential pedal is therefore 
 
 2a(f + w«)=f[l + a«(r' + v*)3; 
 or if a right angle move along the logocyclic curve so that one side 
 shall pass through the cusp, the other side will envelope the curve 
 of which the preceding is the tangential equation. 
 169.] The projective equation of the lemniscate is 
 (a;« + y«)«=a2(a?2_j,9). 
 
 The tangential equation of its tangential pedal is therefore 
 a^{]^—v^)=\, the tangential equation of an equilateral hyperbola. 
 
 Hence, if a right angle moves along a lemniscate, one side passing 
 through the centre, the other side will envelope an equilateral hy- 
 perbola. 
 
 172.] The projective equation of the evolute of an ellipse is 
 
 ©•^l)'-. (") 
 
 the equation of the tangential pedal is 
 
 (!)'Hi)'='P+"'*' <"' 
 
 The tangential equation of the evolute of an ellipse is 
 
 62g9 + aV=(«*-6«)9f9u^ (c) 
 
 the- projective equation of the projective pedal is 
 
 The projective equation of the cissoid is 
 
 y^{2r—x) =«* ; (e) 
 
 the tangential equation of its tangential pedal is 2ri^—^=0, the 
 equation of a parabola with its axis in a reverse position. 
 
 Hence, if a right angle move along a cissoid so that one side 
 shall always pass through the cusp, the other side will envelope a 
 parabola. 
 
 The tangential equation of the cissoid is 
 
 3rJi;*=l-3rf; 
 the projective equation of the projective pedal is therefore 
 
 3ri{y^ + xyyi=x^ + y^-2ra! (f) 
 
 173.] The tangential equation of the curve generated bv the cx- 
 
 M 3 
 
164 ON REVOLVING ANGLES, PROJECTIVE AND TANGENTIAli. 
 
 tremity of a line of constant length h added to the normals of an 
 ellipse, as shown in sec. [152], (a), is 
 
 [(a«-A«)fa + (6«-A*)i;«-l]2=4A«(f«+w«). . . (a) 
 If now in this equation we substitute for and v the values 
 
 a^+y« 
 
 and 
 
 ««+y* 
 
 we shall have for the resulting equation of the locus 
 
 a curve of the eighth degree. When A=0, we get the equation of 
 the foot of the perpendicular, 
 
 a«a?«+AV=(ar« + y')*, (c) 
 
 as may be otherwise investigated. 
 
 Hence, if a right angle so move that one of its sides shall pass 
 through the centre of an ellipse while the other side envelopes the 
 cui've parallel to the ellipse, the vertex of the right angle will move 
 along the curve whose projective equation is (b), a curve of the 
 eighth order. 
 
 174. J When we take the parallel curve to the parabola whose 
 equation is given in sec. [153], (a), that is to say, 
 
 [A:(r + i;*) + fl2=A^r(fHu«), (a) 
 
 it becomes 
 
 (a^ + y^)[k+xY=h^a^, ....... (b) 
 
 the projective equation of a con- 
 choid. 
 
 Indeed we might have antici- 
 pated this — because, if we let fall 
 a focal perpendicular to a tangent 
 to a parabola, the locus of the 
 foot of the perpendicular is the 
 straight line perpendicular to the 
 axis touching the parabola at its 
 vertex; and if we produce all these 
 perpendiculars by the constantline 
 h, their extremities will describe 
 a conchoid of which the modulus 
 is h, the rule the vertical tangent 
 to the parabola, and the pole the 
 focus of the parabola. 
 
 Let F A d Q, be the parabola, 
 F the focus, A the vertex, and 
 P Q, P, Q, tangents to it. P 
 moves along the line A P P,. Let 
 PC = P,C,=A, then CT, C^T, 
 parallel to P Q, P, Q., are tangents 
 
ON EIGHT ANGLES REVOLVING BOUND FIXED POINTS. 165 
 
 to the parallel curve, and CC, are evidently points in the conchoid 
 B C C/. 
 
 Hence, if a right angle move along a conchoid, one side passing 
 through the pole, the other side will envelope the curve parallel to 
 the parabola, which has its focus at the pole of the conchoid. When 
 A=0, the conchoid degenerates into the vertical tangent to the 
 parabola, and the curve parallel to the parabola becomes the para- 
 bola itself. 
 
 CHAPTER XVIII. 
 
 ON RIGHT ANGLES REVOLVING ROCTND FIXED POINTS IN THE PLANE 
 
 OF A CURVE. 
 
 175.] Let the tangential equation of the given curve be ^(f , u) = 0, 
 the fixed point being taken as origin. Let a right angle revolve 
 round this point, one side meeting a tangent to the curve in its 
 point of contact ; to determine the projective equation of the locus 
 of the point in which this tangent meets the other side of the right 
 angle. 
 
 The equations of the locus are 
 
 d^ _d* 
 
 ___Jv__ , dg 
 
 "^"d^. d* ' ^-A^ d^ ' 
 
 di7^~df'' dv^ df" 
 
 This theorem may be established as follows : — 
 Let {x„ Pi) be the projective coordinates of the point of contact, 
 and (w, y) the projective coordinates of the required locus. As the 
 tangent to the given curve passes through the points (x,, y,) and 
 (x, y), we shall have the equations 
 
 ^i?-l-yi''=l (a), and a;f+yw=l J . . (a^) 
 
 and as the sides of the revolving angle are at right angles, we shall 
 also have 
 
 yiy+iCfV=0 (b) 
 
 We have also bv the formulae of transition, see sec. [23] , 
 
 d^ 
 df 
 
 j?,= 
 
 '~d^^ d<I> ■ 
 
 (c) 
 
 Eliminating x„ y„ and y from the four equations (a), (a,), (b), and 
 (c), we shall have 
 ^ '' d<D _d^ 
 
 ^=3*7^' ^' ^*'° ^=° dj>,_d0 • • • • (^) 
 
 AtJ^-dS" Av^ dg" 
 
166 ON RIGHT ANGLES EEVOLVING ROUND 
 
 176.] To apply these general formulae to some particular exam- 
 ples. 
 
 The tangential equation of an ellipse, referred to its axes, is 
 
 a2^+iV-l=0=4>. 
 
 d4> d<l> 
 
 Hence -5— =3A^w, -jp=3a*f ; substituting these values in the gene- 
 ral formula (d), the resulting equation becomes 
 
 aW{a^x^-ifbY)=ifl^-b^fxY (a) 
 
 Comparing this equation with the one given in sec. [147], we 
 shall see that the locus is the, reciprocal polar of the evolute of a 
 certain ellipse. 
 
 177.] A right angle revolves round the point of inflexion of a cubical 
 parabola ; one side meets a tangent at its point of contact with the 
 curve, the other side will meet the same tangent in a point, of which 
 the projective equation is required. 
 
 Let the projective equation of the cubical parabola, asshown in 
 sec. [149], be 
 
 3a2y-^=0 (a) 
 
 The tangential equation of the same curve is 
 
 4a2£3_9y (b) 
 
 d4> dO 
 
 Hence g^ = 9' ^=-12aT, 
 
 d4> d4> 
 and A,=^f-^w=3|(3-t-4a«fu). 
 
 Consequently a?= e-„ o^ . and y— ^- 
 
 (c) 
 
 "f(3+4a2|i/) *~(3 + 4a«fv)* 
 
 Therefore |=^, or ^=a^W 
 
 Introducing this value of ^ into the equation of the curve (b), we 
 
 find for the value of v : — ^r— \ / — = v. 
 Qax V X 
 
 Substituting these values of ^ and v in the dual equation, the 
 
 resulting equation becomes 
 
 [y*+ar«]^=12aV, (d) 
 
 the projective equation of the required locus. 
 
 178.] A right angle revolves round the cusp of a sendcubical para- 
 bola ; one of its sides meets a tangent to the curve at its point of 
 contact ; the other side will meet the same tangent in a point, of which 
 the locus is required. 
 
FIXED POINTS IN THE PLANE OF A CURVE. 
 
 167 
 
 Let 3l»«-flP=0=:* 
 
 be the tangential equation of the semicubical parabola, then 
 
 d<I» d^ „ « , ^ d* ^ d* Q*. / V , o\ 
 
 ^^7=6". ^=-3fl£', and A,= g^ ?-^u=3fu(af+2). 
 
 (a) 
 
 Hence 
 
 ai 
 
 and ^ = 7r-- 
 
 «7 2v 
 
 (b) 
 
 (c*) 
 
 Substituting in the equation (a) of the curve this value of t; as 
 derived from (b), 
 
 ^=3i('')^ ^"*(^) gives «=-|^. 
 
 Substituting these values of ^ and v in the dual equation 
 a?f +yt/=l, and reducing, we obtain the projective equation of the 
 locus, 
 
 l2a;V+V=9«^ (d) 
 
 179.] A riffht angle revolves round the vertex of a parabola ; one 
 side of the angle meets a tangent to the curve at the point of contact ; 
 the other side of the angle will meet the tangent in a point whose 
 locus is a curtate cissoid. 
 
 Let ftu*— f=0=^ be the tangential equation of a parabola, the 
 axis being taken in a reverse position. 
 
 Let P O Q be the right angle revolving round the vertex O, one 
 
 Fig. 35. 
 
 T 
 
 ic 
 
 side O P meeting the tangent in P, the other side Q, meeting the 
 tangent in Q. 
 
168 
 
 ON RIGHT ANGLES REVOLVING ROUND FIXED POINTS. 
 
 Now, taking the partial differentials, we shall have 
 
 d4> „, d* , - . d^j. d* oi t L 
 ^=2kv, ^=-1, and ^,= -^-^v=2kv^ + v. 
 
 Hence x= 
 
 d^ 
 dv 
 
 2kv 
 
 A,~2k^v + v' " 2k^v + v' 
 
 Consequently -=2kv or «= 
 
 2%' 
 
 and f= 
 
 2k— X 
 2kx ' 
 
 Substituting these values of f and v in the original equation, we 
 shall find on reduction the equation of the locus 
 
 2y^{2k-x)=x^ (a) 
 
 If now we substitute y^ for 2y*, we shall have y'^{2k—x)=sfi, 
 the projective equation of the cissoid. Hence, the abscissae remain- 
 ing the same, if we shorten the ordinates of the cissoid Q B in the 
 ratio of 1 : sj2, we shall obtain the curve C B C,, the required locus; 
 or if the ordinate of the cissoid be represented by the diagonal of 
 a square, the ordinate of the required locus wiU be its side. 
 
 180.] If the revolving angle, instead of being a right angle, should 
 be one whose tangent is t, we must transform (b), sec. [175] into 
 
 xxi+yy,' 
 
 (a) 
 
 and eliminating x^, y,, and y between this expression and (a), (a^) , 
 with (c) in the same section, we shall have 
 
 «= 
 
 d3> 
 
 d* 
 dv 
 
 and 
 
 [%^-T] 
 
 , ,d4> d<I>„l ' 
 
 [df 
 
 di;' 
 
 d^ d^ 
 dv "^ df ^ 
 
 (b) 
 
 y~ f d4> ^ , d4> I Td^ A^A 
 |df ^+cU7''| + |dF''-d^^r 
 
 We may apply these formulae to one or two examples. An angle 
 whose tangent is t, revolves round the centre ojF an ellipse, one 
 side passes through the point of contact of a tangent, while the 
 other side of this angle meets the tangent to the ellipse in a point 
 of which the locus is required. 
 
LOCI OF REVOLVING ANGLES OTHER THAN RIGHT ANGLES. 169 
 
 Let a'f + 6*w* = 1 be the tangential equation of the ellipse ; then 
 d^ d* 
 
 Substituting these values in the preceding formulae, we shall have 
 
 dividing one by the other, we shall have 
 
 y_ b^v + a^ST v_ a^{y-Tx) ,,. 
 
 a!~a'^-b^vT ^~b^{x+Ty) ^ ' 
 
 v' 1 
 But the equation of the curve gives 0^ + 6^ «=«' and the dual 
 
 equation gives a? +yg=^. From these equations, eliminating ■=, 
 we shall have 
 
 «' + *'^=^' + 2^|+2''p . ■ . . (e) 
 or 
 
 Introducing into this equation the value of ■= as given in (d), the 
 
 resulting equation will be 
 
 {a^-a!^)b*{x+ry)*+{b^-y^)a*{i/-Tx)^=2xya*b^x + Ty){y-rx). (e) 
 
 If we assume the revolving angle to be half a right angle, t=1, 
 and the preceding equation becomes 
 
 r^'+?'-(«'+**)](^+y')=2(«^-6*)^[j;+i5-i]. (f) 
 
 Analogous expressions may be found for the parabola and other 
 curves. The application of the method presents but little difficulty. 
 
 CHAPTER XIX. 
 
 TO INVESTIGATE THE LOCI WHEN THE REVOLVING ANGLES IN THE 
 TWO PRECEDING CHAPTERS ABE OTHER THAN RIGHT ANGLES. 
 
 181 .J We may, with very slight modifications, apply this method 
 to determine the equations of curves generated by the motion of 
 an angle a, whether the angle a move along a given curve, or be 
 the intersection of a tangent with a line drawn from the centre at 
 the angle o>. 
 
170 LOCI OF REVOLVING ANGLES OTHER THAN RIGHT ANGLES. 
 
 We can show that the lines x, y, f, v may be transformed into 
 
 a?= 
 
 ?= 
 
 X V 
 
 (a) 
 
 x^-\-y^-\-Kxy' s^+y^-'trKxy' 
 Let O X and O Y be the axes of coordinates meeting at the angle 
 
 Fig. 36. 
 
 o>. Let X Y be the limiting tangent. Let the revolving angle be 
 O P X. Let P D be drawn parallel to the axis O Y. Let P D=y, 
 O D =a;. Then, as the supplement of the angle YOX = PDO= OPX, 
 the triangles X O P and P O D are similar. 
 
 Hence x=r^i; but r*= «-; — =- — =-, putting k for 2 cos e>. 
 Consequently ^ = p^J^^g„ - 
 
 In like manner we may show that f =— 5- — 5— 
 
 ' ^ x'^+y'^+Kxy 
 
 When we shall have obtained the equation of the locus of the 
 moving angle a> by the help of oblique coordinates, we may pass 
 back again to rectangular coordinates and exhibit the equation of 
 the required locus referred to rectangular coordinates. 
 
 182.] To apply this method to an example or two may suffice. 
 
 To find the projective equation of the point in which a tangent 
 to an ellipse is met by a diameter which makes with it an angle 
 of 60°. 
 
 The tangential equation of an ellipse, referred to its centre and 
 axes, is 
 
 o9^ + 6V=l; (a) 
 
 and if we alter the angle of ordination from a right angle to 60° by 
 the help of the formula in sec. [4], we shall have 
 
 (3o« + 6«)^ + 46V-46«fi/=3; . . . . (b) 
 
LOCI OF REVOLVINQ RIGHT ANGLES. 171 
 
 and if we now substitute for ^ and v their values as given in (a), 
 since 2 cos o)=/ii:= 1, we shall have 
 
 (3a« + 6*)a?* + 4iV-4**a?y=3(a72+y9 + a!y)2, . . (c) 
 
 the projective equation of the curve along which an angle of 60° 
 moves, one side touching the ellipse while the other passes through 
 the centre. 
 
 183.] The tangential equation of the trigonal hjrpocycloid, the 
 axes of coordinates being inclined at an angle of 60°, is 
 
 R(f +i;)?w=P + u* + ^u, see sec. [137]. 
 If we now draw a tangent to this curve and a radius meeting this 
 tangent at an angle of 60°, the projective equation of the locus of 
 this point will be 
 
 R{x+y)ay=[a^+y^+a!y)^- (a) 
 
 Let O X, O Y be the axes of coordinates, inclined at an angle of 
 
 Fig. 37. 
 
 1'20°, so that YOX may be equal to 120°. Let the angle OPQ== 60°; 
 then the above equation will be the locus of the point P, the inter- 
 section at an angle of 60° of the line O P with P Q, a tangent to the 
 trigonal hypocycloid, having two out of three of its cusps at the 
 points X and Y. 
 
 CHAPTER XX. 
 
 ON THE LOCI OF RIGHT ANGLES REVOLVING AROUND GIVEN CURVES. 
 
 184.] A right angle revolves so as to touch with its sides a given 
 curve, to determine the projective equation of the vertex of the right 
 angle. 
 
 Let P and P, be the perpendiculars let fall from the origin on the 
 sides of the revolving right angle ; and let x and y be the projective 
 
172 ON THE LOCI OF RIGHT ANGLES 
 
 coordinates of the vertex of the right angle. Let the tangential 
 equation of the given curve ^(f, i») =0 be transformed into 
 ^[P, sin\, cos\]=0, by Trnting sin\ for Pf and cos\ for Pu. 
 It is manifest that a^+y^^T^ + F^. 
 
 Since the limiting tangent passes through the point {xy), we 
 shall have x^+yv=l or y cos \+a7 sin \=P j and for the perpen- 
 dictdar P, at right angles to the former, y sin \—x cos X=P,. 
 
 Making these substitutions in the given equation of the curve, 
 we shall have 
 
 ^[(y cosX+a?sin\) sin\, cosX]=0; ... (a) 
 and for the perpendicular P, at right angles to the former, 
 
 ^[(y sinX— a?cos\)sin\, cos\]=0. . . . (b) 
 
 Between these equations, eliminating the trigonometrical quantities, 
 we shall obtain the projective equation of the locus in terms of x, y, 
 and the constants of the given tangential equation of the curve. 
 
 To apply this method of investigation to some examples. 
 
 185.] Let the tangential equation of the ellipse referred to its 
 centre and axes be a'P + 6^i;'=l. Multiply by P*, and the result- 
 ing expression becomes 
 
 a* cos^ \ + 4« sin^ X= P« = (y cos \ + a? sin \)«, 
 or 
 
 a*cos*X+ J' sin*X=y' cos' X+2a7y sin X cos X + ^r' sin' X j 
 
 for the perpendicular P,, at right angles to P, we shall have 
 
 a' sin' X + 6' cos' X=y' sin' X— 3a?y sin X cos X + a?' cos' X ; 
 
 adding these equations, 
 
 a' + i«=j?« + y', 
 
 or the locus is a concentric circle whose radius =*/«* + 6', a well- 
 known theorem. 
 
 Let the curve be the parabola. 
 
 186.] The tangential equation of the parabola, see sec. [49], (i), 
 the origin being at the vertex, is 
 
 kv^ + ^=0 (a) 
 
 Mtiltiply by P*, and we shall have k cos*X + P8inX=0, or, sub- 
 stituting for P its value, we shall have 
 
 Acos'X+jrsin'X+y sinXcosX; . . . . (b) 
 
 for the other side at right angles to the former, we shall have 
 
 Asin'X-|-a:cos*X— y sinXcosX; . . . . (c) 
 
 adding these expressions, k+SB=0, or the locus is the directrix. 
 
 To investigate the equation of the vertex of the right angle which 
 envelopes the involute of the quadrantal hypocycloid. 
 
REVOLVING AROUND GIVEN CURVES. 173 
 
 187.] The equation of this involutCj see see. [136], is 
 
 {l+a)e+{l-a)v^= VFh?, (a) 
 
 or, multiplying by P*, 
 
 (f+a) sin'\+(/— a) C08*X=P=y cos\ + a?sin\. . (b) 
 For the other side^ at right angles to the former, we shall 
 have 
 
 (/+o)cos*X+(^— a) sin*X,=y sin\ — ^cos\; . . (c) 
 writing cos 2\ for cos* \— sin* X, squaring these expressions, and 
 adding, 
 
 2P+2a^cos^2\=::x^+y% (d) 
 
 or 
 
 cos2\ 
 
 and 
 
 ~\_ 2a* J' ^^^ 
 
 sin2X=p°^ + ^^;;r+y')J. . . . . (f) 
 
 If we multiply (b) and (c) together, the resulting expression 
 becomes 
 
 2/*-2a*cos*2X={y*-a7*)sin2\-2a^cos2\; . . (g) 
 
 or if we write for brevity M and N for the values of cos 2X and 
 sin 2\ given in (e) and (f ), we shall have 
 
 4i2_(a;2+y*) = (y*_ar9)N-2a;yM, . . . (h) 
 
 the projective equation of the locus of the vertex of a right angle 
 rolling round the involute of the quadrantal hypocycloid. 
 
 A right angle revolves, touching with its sides the curve parallel to 
 the parabola; to determine the locus of the vertex. 
 
 188. J The tangential equation of the curve parallel to the para- 
 bola is 
 
 [A(f^ + i/*)+?]^=A*f(r' + i^*), .... (a) 
 the focus being the origin, see sec. [153] . 
 
 Multiply by P*, first taking the square root, and we shall have 
 
 ^ + P sin \=:A sin \, 
 or 
 
 A+a?sin*\4-y sin\cos\— AsinX=0; . . . (b) 
 
 for the perpendicular at right angles to the former we shall have 
 
 A + a7Cos*\— y sin\cos\+Acos\=0 ; . . . (c) 
 
 adding these expressions, 
 
 2A + a;=A (sinX— cosX) (d) 
 
 Consequently 
 
 sin2X= !^2 '- (e) 
 
174 ON THE LOCI OF EIGHT ANGLES 
 
 Subtracting (c) from (b), 
 
 A(cosX + 8in\)=y sin2X— J^co8 2\; . . . (f) 
 and 
 
 h[am\—cos\)=2k+x, from (d). 
 Consequently 
 
 A* cos 2\= (2k + x)x cos 2\—{2k-\-x)y sin 2X, 
 or 
 
 {2h+w)y\a.n2\={2k-k-w)x-h^; (g) 
 
 and substituting for tan 2\ its value derived from (e), 
 
 \h^-{2k+x)]Y={{2k+x)x-h^Y\2h^-{2kJrx)^, . (h) 
 the projective equation of the required curve. 
 
 When A=0, the equation is satisfied by putting 2k-\-x=0 or 
 x=—2k, and y indefinite, or the locus is the directrix when the 
 curve is a parabola. 
 
 A right angle revolves, touching with its sides a cycloid ; to deter- 
 mine the locus of the vertex. 
 
 189.] Let the tangential equation of the cycloid be 
 
 2rr«+?tan-'(D 
 
 the origin being taken at the centre of the base. 
 
 Multiply this expression by F the perpendicular from the origin 
 on the limiting tangent, the resulting expression is 
 
 2r [cos X + X sin \] =P ; 
 and as 
 
 P=y cos X +a: sin X, 
 we shall have 
 
 [y— 2r]cosX+[a7— 2rX]8inX=0 (b) 
 
 If we take the perpendicular at right angles to the former, we 
 shall have 
 
 I>-2r] sinX-rj:-2»Yx+|)~jcosX=0. . . (c) 
 Eliminating sinX, cosX from these expressions, we obtain 
 
 = 1, (a) 
 
 [2rX-(.-r|)J=(r|;-(2r-y)^ . . 
 But if we eliminate X between (b) and (c), we shall have 
 
 (d) 
 
 8in2X=2^^— 2, (e) 
 
 '2 
 or 2\=siu"' / ^\; substituting this value of 2\ in the pre- 
 
REVOLVING ABOUND GIVEN CURVES. 175 
 
 ceding equation, the resulting expression becomes 
 
 rsin-/?!:^\-(.-,-|)T=(r|y-(2r-,)S . (f) 
 2 J 
 
 the projective equation of the vertex of a right angle revolving 
 round a cycloid. 
 
 A right angle revolves round a fixed point, whose sides meet a curve 
 in certain points, to determine the tangential equation of the curve 
 which is enveloped by the line joining these points. 
 
 190.] Let f{^.y)=0 (a) 
 
 be the equation of the curve, the origin being taken at the fixed 
 point. Let 
 
 y=mx (b) 
 
 be the equation of one of the vectors, and let 
 
 x^+yv=l (c) 
 
 be the equation of the limiting tangent. Now (a) and (c) may be 
 
 written /(a?, m) anda.'=j— — ; eliminating x between these two 
 
 equations, the resulting expression will become 
 
 /(f,v,m)=0 (d) 
 
 The other side of the right angle will give 
 
 f(S,v,m,)=0; (e) 
 
 and as the angle is assumed to be a right angle, we shall have the 
 additional equation 
 
 mm,+ l=0 (f) 
 
 Eliminating m and m, between these three equations, the resulting 
 equation in f and v will be the tangential equation of the sought 
 curve. 
 
 An illustration of this method may suflSce. 
 
 A right angle revolves meeting two given straight lines in the points 
 A and B ; the line which joins the points A and B will envelope a conic 
 section whose focus is at the origin, andvnll also touch the two given 
 lines. 
 
 19L] Let f+|=l, (a), and J + f = 1 (a,) 
 
 be the projective equations of the two given straight lines ; let 
 
 y=nx (b) 
 
 be the projective equation of one of the sides of the right angle, 
 
 and let 
 
 x^-\-yv = \ (c) 
 
176 ON THE LOCI OF RIGHT ANGLES 
 
 be the dual equation ; eliminating x and y between these three eqna- 
 tionsj ve shall have 
 
 «=-.(J^) (-' 
 
 For the other straight line we shall have 
 
 -4;(!^f) w 
 
 But as the revolving angle is a right angle^ w»,+ l=0; eonse- 
 quently, substituting for n and n^ their values, we shall have as the 
 resulting expression 
 
 P + v^-(i + i)?-fi+^V+ — + i=0' • • («) 
 * \a a^l^ \b bj aa, bo, 
 
 the tangential equation of a conic section. 
 
 Since the coefficients of the squares of the variables are equal, 
 and the rectangle f u does not appear, the focus of the conic section 
 is at the origin. The given straight lines are tangents to the curve ; 
 
 for the equation (e) is satisfied by putting |=-, and v=j-, or f=-, 
 
 and i'=T- 
 
 The curve is a parabola when aa,+ i6(=0; for then the absolute 
 term vanishes. But when aa, + 66^=0, the given lines are at right 
 angles. 
 
 192.] A right angle revolves round a point in the plane of a conic 
 section meeting the curve in two points; the linejoining these points will 
 envelope a conic section, one of whose foci is at the given point. 
 
 Let the origin of coordinates be taken at the given point, and 
 let the equation of the curve be 
 
 Aa?«+A,y« + 2Ba?y + 2Ca>+2C,y=l (a) 
 
 Let y=nx {^) and ar^+yv=l (c) ; eliminating x and y between 
 (a), (b), and (c), the resulting equation may be written 
 
 2[B+C^+C..-g..] A+2Cg-p 
 
 If now we take the other side of the right angle, bearing in mind 
 that mm, + 1=0, we shall have 
 
 2[B + C^+ Cv-|u] A, + 2C,v-.,« 
 " A+2Cf-r ''+A + 2Cf-{«=^' • • («) 
 
 Now, as n must have the same roots in each equation, the coeffi- 
 cients must be identical. Equating the absolute term in each we 
 shaU have (A -f-2Cf-S«)2= (A, + 2Ciu-v^)^. ' 
 
 ' And taking the square root on each side, 
 
 (A + 2Cf-J«) = + (A,+ 2C,u-./'). 
 
REVOLVING AROUND GIVEN CURVES. 177 
 
 We cannot, a priori, say which of these signs we should select; 
 but on comparing the second terms, and dividing out the common 
 numerator (B + C^ + Cv— fu), we find 
 
 Consequently the resulting equation becomes 
 
 P + t/«-2Cf-2C,u=A + A, (f) 
 
 Now, as the squares of the variables have the same coefficients, 
 and the term in ^v does not appear, this conic section has its focus 
 at the origin. 
 
 Recapitulation. 
 
 193.J It may be well to recapitulate the propositions that have 
 been established in the preceding Chapters, on the description of 
 curves by the motion of straight lines and angles constrained to 
 move under certain conditions. 
 
 In Chapter XV. it has been shown that a curve may be drawn 
 parallel to another curve at the distance h from it, by substituting 
 
 , and .' for f and v, 
 
 in the tangential equation of the curve «^(0, i»)=0. 
 
 In Chapter XVI. it has been established that if a right angle 
 
 revolves touching at its vertex a curve whose projective equation is 
 
 f(x, y) =0=F, the other side of the right angle will envelope a curve 
 
 (the evolute) whose tangential equation may be found by eliminating 
 
 X and y between the expressions 
 
 dF _dP 
 
 (17/ Q^ 
 
 ^= dF^_dF ' "^ AF^_dF ' ^°d/{x,j,)=0. 
 
 dy da; dy dar^ 
 
 In Chapter XVII. it has been proved that if a right angle 
 revolves so that its vertex shall move along a curve whose equation 
 iaf{x, y) =0, while one side of the angle passes through the origin, 
 the other side will envelope a curve whose tangential equation may 
 be foimd by substituting, for x and y in the preceding equation, 
 
 f 
 
 and 
 
 V 
 
 ^ + w« ^ + v^ 
 
 It has been moreover established in the same Chapter that 
 if a right angle revolves, one side passing through a fixed point, 
 while the other side envelopes a curve whose tangential equation is 
 ^(li v)=0, the projective equation of the vertex of the right angle 
 
 wiU be found by substituting -rr~i ^^^ ~2X~s ^°^ ? ^*^ ^ ^ *^^ 
 
 X -\-y X -\-y 
 
 given tangential equation of the curve. 
 
178 ON PEDAL TANGENTIAL COOEDINATES. 
 
 In Chapter XVIII. it has been shown that if round a fixed point 
 in the plane of a curve a right angle revolves so that one side 
 shall pass through the point of contact of a tangent drawn to the 
 curve, the locus of the point in which the other side of the right 
 angle meets the same tangent will be found by||eliminating f and v 
 between ^(£, v) =0, the tangential equation of the curve, and the 
 equations 
 
 d^ _d^ 
 
 dw , df 
 
 ^=d*— d*-' ^°^ y= dg, d^ • 
 du ^ df " di/ ^ df " 
 
 In Chapter XIX. it is shown how the preceding methods of de- 
 scribing curves by the motion of right angles constrained to move 
 under certain conditions may be extended to the description of 
 curves by the motion of any angle whatever. 
 
 In Chapter XX. it has been established that if a right angle 
 revolves touching with its sides a curve whose tangential equation, 
 ^{£, v) =0, is given, the coordinates of the vertex x and y are found 
 by eliminating f and v between the equations ^(f, t;)=0, a?f + yv=l, 
 xSi+yvi=l, and, as the lines are supposed to be at right angles, 
 
 ^S,+ VV,=:0. 
 
 Finally, if a right angle revolves round a point, meeting in points 
 a curve whose equation is f{x, y), the line joining these points en- 
 velopes a curve whose tangential equation may be found by elimi- 
 nating x, y, and n between the equations 
 
 /(a?,y)=0, xS+yv=l, a;^+y,v=l. 
 
 y=nx, yi=nfe, and w«^-|- 
 
 y,v=l,\ 
 1=0. ] 
 
 CHAPTER XXI. 
 
 ON PEDAL TANGENTIAL COOEDINATES. 
 
 194.J Besides the system of tangential coordinates which has 
 had its principles developed in the foregoing pages, there is another, 
 which for distinction may be called pedal tangential coordinates. 
 The object of the former method is to develope the duality of the 
 properties of space in an analytical form ; while the scope of the 
 latter is to derive new curves from others already known, by the 
 mechanical action, as it were, of a very simple law. 
 
 Let F(a?, y)^0 be the projective equation of any plane curve or 
 other locus. In this equation for x and y let their reciprocals g 
 and V be substituted, x^ and yv being each ^1, and the new equa- 
 tion, Ff|, -j=0, becomes the tangential equation of a curve, in 
 
ON PEDAL TANGENTIAL COORDINATES. 179 
 
 general of a different species, but derived from the projective 
 equation F(a?, y) =0 by a very simple law. 
 
 A few illustrations will render this method of pedal tangential 
 coordinates very obvious and plain. 
 
 Let the projective equation of a straight line be - + |= 1. For 
 X and y substitute ^ and -, and the equation becomes 
 
 ^ + ^=1, or b6?w-o0-6u=O, 
 
 the tangential equation of a parabola, see sec. [49] . 
 
 195.] Let us take the common projective equation of the circle 
 *''+y*=a*. Substitute for x and y their reciprocals, and the equa- 
 tion becomes 
 
 E« + w*=a*r'v«, (a) 
 
 the tangential equation of the quadrantal hypocycloid. 
 
 It is obvious that as p + -j = a^, the portion of the limiting tan- 
 gent which revolves between the axes is of constant length and 
 equal to a. 
 
 Let us assume the central projective equation of the ellipse, 
 
 x^ ifi 11 
 
 -j + Tg=l ; for X and y put their equivalents ^ and -. The result- 
 ing equation becomes 
 
 oT + 6«i;^ = o'6Ti'S (b) 
 
 the tangential equation of the evolute of an ellipse. 
 
 Hence the line joining the feet of the projective coordinates x, y 
 of an ellipse referred to its axes envelopes the evolute of another 
 ellipse. 
 
 Now the semiaxes of this evolute along the axes of X and Y are b 
 
 and a. We may satisfy ourselves that this is so, by putting - and ^ 
 
 successively equal to 0. 
 
 Now if A and B are the semiaxes of the ellipse of whose evolute 
 
 A^c* 
 b and a are the semiaxes, we shall have a=— rjr- and A = Ac', hence 
 
 ^=g aaid A=^,-^„ B=^^-^, (c) 
 
 It is evident that the portion of the limiting tangent which 
 revolves between the axes of coordinates is equal to the semidia- 
 meter of the ellipse which makes an equal angle with the axis of X ; 
 and as the sum of the reciprocals of the squares of any two semi- 
 diameters of the ellipse at right angles to each other is constant, 
 
 n2 
 
180 
 
 ON PEDAL TANGENTIAL COORDINATES. 
 
 SO must the reciprocals of the squares of any two tangents drawn 
 to the evolute at right angles to each other be constant. 
 
 196.] In the base of the evolute AB, let two points M and N be 
 taken so that AN x NB = OM^ ; and if from M and N two tangents 
 be drawn to the curve, of which the intercepted parts are / and ?, 
 we shall have P + l^=a^ + b^. 
 
 Fig. 38. 
 
 It is evident that P+Z*=OM^ + ON' + 0»j* + Ora' 
 
 But OM'+ON*=a2. 
 
 We also have 
 
 ON . On 
 
 h^-. 
 
 + -p-=l, or 0» =A' — jON ; and in like 
 
 .=:-« ,o &'; 
 
 manner Om =6^ — s OM ; hence 
 
 or /'+/,''=a* + 6*. 
 
 197.] Let two tangents at right angles, one to the other, be 
 drawn to two concentric and coaxial evolutes whose axes are con- 
 nected by the relations 
 
 111111 
 
 *« a^~W hf 
 
 "A*" 
 
 Let the segments of the tangents be / and l^, we shall have 
 1+1-1+1 
 
 Let ^ip+38[;=l> or putting -=A, t=B, the result becomes 
 
 A* . B^ 
 
 A* . B* 
 
 js + „s -1> or jaja + ^Sj^j-^jJ 
 
 (a) 
 
ON THE RECTIFICATION OF PLANE CURVES. 181 
 
 but it may be shown that if m be the angle which I makes with the 
 axis of X, ff= , and /i;=-: : consequently 
 
 A* cos* <o + B,* sin* a>=-j^, or, as B* — A*= p, 
 
 substituting. A* + .^ = -s. For /„ we shall have A ,* H — p— = p, 
 consequently 
 
 A* + A* + i,= ^+l; or, as A* + i = B,*, 
 we shall have 
 
 A' + B^=j,+~. or i, + ^,= l+^,. . . . (b) 
 
 On the tangential pedal of the Semicubical Parabola. 
 
 198.] The equation of the semicubical parabola is ay'^=a^, its 
 tangential equation is v^=a^^, and the tangential equation of its 
 pedal is also v^=a^. Hence the pedal curve of the semicubical 
 parabola is the semicubical parabola itself. Consequently if we join 
 the feet of the ordinates of one branch of the curve by a straight 
 line, this straight line will be the envelope or limiting tangent of 
 the other branch. Thus while one of the branches may be traced 
 out by a point, the other may be enveloped by a straight line. 
 
 199.] Let us assume the equation of the fourth degree, 
 
 a*y2 + 6V=a;V (a) 
 
 Substituting for x and y their values j and -, we get 
 
 a«g«+6V=l, 
 
 the tangential equation of an ellipse referred to its centre and axes. 
 Hence the line which joins the feet of the ordinates a? and y in 
 the equation (a) envelopes the ellipse whose semiaxes are a and b ; 
 and if we erect perpendiculars to the axes of coordinates from the 
 points in which they are met by the limiting tangent, these lines 
 will meet on the curve (a), see sec. [147]. 
 
 CHAPTER XXII. 
 
 ON THE RADIUS OF CURVATURE AND THE RECTIFICATION OF PLANE 
 
 CURVES. 
 
 300.] The method of tangential coordinates is peculiarly appli- 
 cable to the rectification of plane curves. 
 
 In the common formulae for rectification derived from the methods 
 
182 
 
 ON THE RADIUS OF CURVATUEE AND 
 
 of projective or polar coordinates, the element ds of the arc is as- 
 sumed to be the hypotenuse of an infinitesimal right-angled tri- 
 angle, of which the sides are the infinitesimals Ax and Ay ; while 
 in polar coordinates, the element As is the hypotenuse of the right- 
 angled triangle of which the sides are Ar and rA0. We must then 
 take the square roots of these expressions to adapt them for inte- 
 gration. Thus 
 
 As= s/Ax^ + Ay^, and As= \/Ar^+r^Ad*. 
 
 Now a little consideration will show that these expressions are 
 arbitrary. They have nothing to do directly with the curvature of 
 the curve itself. They depend on the previous establishment of a 
 particular system of coordinates. Had the system not been in- 
 vented, such a method of rectification would have been any thing, 
 but obvious. 
 
 In the following method, the primary element on which the 
 length and curvature of a given curve depend, is the radius of cur- 
 vature at the extremity of the element ; and it is easy to show that 
 the arc of any given curve may be exhibited as made up of two 
 distinct elements, the one depending on the curvature of a circle 
 whose radius varies from point to point, while the other element, 
 which is arbitrary, is the differential of a straight line. 
 
 As the centres of ciu-vature move 
 along the evolute of the given curve, 
 it would seem that the coordinates of 
 the centre of curvature which isfound 
 on the evolute, must be necessary ele- 
 ments in the rectification of a curve ; 
 but we may elude the difficulty in the 
 following way. 
 
 Let C A, C A^ be the radii of cur- 
 vature at the extremities of the ele- 
 ment of the arc of the curve A G. 
 From an arbitrarily assumed, but 
 fixed point O, let perpendiculars OB, 
 O B, be let fall on the tangents to the 
 curve at the points A and A,. 
 
 A B and A^ B, may be called pro- 
 tangents of the curve, seeing that 
 they are the projections of the radius 
 vector OA on the tangents of the 
 curve. 
 
 Let A B = /, and A,B, = t,. From 
 the point O let fall the perpendiculars 
 O U, O U^ on the radii of curvature 
 A C, A, C. These are equal to A B 
 
 Fig. 39. 
 
 A-ioo 
 
THE KECTIFICATION OF PLANE CURVES. 183 
 
 and A, B,, and therefore equal to t aud t, respectively- Through U 
 draw the line U a parallel to O B^ or to V^ A,. 
 
 Now, though we cannot determine the increment of t, or the 
 value oitf—t from a consideration of the lines A B and A^ B„ seeing 
 that they have no point in common, yet by comparing their equals 
 O U and O U^, the opposite sides of the rectangles A B O U and 
 A, B, OU,, we may see that ^— ^,= UV, ultimately. Let A, be 
 the angle which the perpendicular O B or P makes with the axis 
 O X ; then, putting A C=R the radius of curvature, let 
 A U=0 B=P. Also dX, being the element of the angle between 
 A C and A, C the radii of curvature, and also between the perpen- 
 diculars P and V,, which are parallel to them, we shall evidently 
 have, when the radii of curvature are indefinitely near, 
 
 (R-P)dX=UV,=^-/,=d/ (a) 
 
 Als V,U( is ultimately = P — P^ = dP, and V^U^ = tSX, we shall have 
 ultimately 
 
 i='- M 
 
 or 
 
 d«P_df 
 
 'd^^-~A\ ^°^ 
 
 Introducing this value of -jr- in the preceding equation, we shall 
 
 have 
 
 d^P 
 B=P + ^, ... .(d) 
 
 an elegant and simple expression for the radius of curvature. 
 
 , [I find this formula entered in an old note-book of mine aa having been dis- 
 covered on the 17th August, 1837.] 
 
 201.] Since the elementary arc of the curve As is manifestly 
 equal to Aa + aB and Aa=Pd\, while ah., is equal to UV^, the 
 opposite side of the rectangle aA,V;U, and UV^=d/, ultimately, 
 we shall consequently have 
 
 d«=Pd\+d/ (a) 
 
 Since df=(R— P)dX, the sign of d^ will depend on the sign of 
 (K-P,). _^ 
 
 We have shown in (b), sec. [200], that ^=^; and if we inte- 
 grate the expression (a), we shall haves=JPd\-h^; or putting for 
 
 dP 
 t its value -5^, we shall finally obtain the following simple formula 
 
184 ON THE RADIUS OP CURVATURE AND 
 
 for the rectification of a plane curvCj 
 
 H^^^+w. (^) 
 
 From the preceding demonstration it follows that d/ wiU be posi- 
 tive so long as R>P, that it wiU be =0 when R^P, and that it 
 will be negative when R<P. 
 
 Hence it follows that when at any point of a curve the radius of 
 curvature shall be equal to the perpendicular from the origin on 
 the tangent^ we shall have 
 
 S-o- <«) 
 
 Consequently in the general formula for rectification, 
 
 dP 
 
 d\^ 
 
 the upper sign must be taken when R>P, the lower when R< P. 
 
 When the point O, and C the centre of curvature, are on oppo- 
 site sides of the tangent to the curve, the formula becomes 
 
 dP 
 dX 
 
 *=JPdX±^ (d) 
 
 '=^-1^^'^ («) 
 
 Hence, having obtained the value of P, we may obtain the value 
 of f, the protangent, by simple differentiation; for 
 
 '=!• (0 
 
 202.] These observations may throw some light on a well-known 
 theorem in the rectification of the elliptic quadrant. In Fagnani's 
 theorem there is a critical point at which the quadrant of the ellipse 
 is divided into two sections, whose difference is equal to the differ- 
 ence of the semiaxes a and b; at this point -j-=a— A, and therefore 
 
 d*P 
 
 gr-|=0, or R=P. That is to say, the radius of curvature is equal 
 
 to the perpendicular from the centre on the tangent at the critical 
 point, in Fagnani's theorem. It is easy to show this. 
 
 In the ellipse the radius of curvature at any point is equal to the 
 product of the squares of the semiaxes divided by the cube of the 
 perpendicular from the centre on the tangent at that point, or 
 
 R=p3-- But R=P in this case. Hence P* = a6. 
 
 1 x^ ?/* 1 
 In the ellipse ^^ p=^+^=^. and the equation of the curve 
 
THE BECTIFICATION OP PLANE CURVES. 
 
 185 
 
 or IT .. . 
 
 gives rs+T2=li elimmatmg y and x successively, we shall have 
 
 ^_ a* A* 
 
 \/a + 6' i/a + b 
 
 These are the coordinates of the critical point in Fagnani's theorem. 
 For the angles which this perpendicular makes with the axes, we get 
 
 b .3 a 
 
 a+b a+b 
 
 The tangent of the angle between the perpendicular on the tan- 
 gent and the radius vector of the critical point is 
 
 = — ;=, since t=a—b, and P= i/ab. 
 \ab 
 
 203.] Let the tangential equation of a plane curve he tj) {!£, v) =0, 
 and, assuming the value found for t in sec. [27] , we may transform 
 
 the expression for the radius of curvature, namely R=P + -j:r,into 
 
 dV 
 
 R=- 
 
 + - 
 
 y'|2 + „a-dX 
 
 d* _d*^" 
 dg " di; ^ 
 d^ ^ d4> 
 
 L-dl ^+ diT^J 
 
 V^ + v 
 
 (a) 
 
 While the quadrature of curves may be most easily effected by 
 the use of the integral J yd^ derived from the projective equation 
 of the curves, so their rectification by the aid of formulee derived 
 from their tangential equations may with the greatest facility be 
 obtained. 
 
 Since «= J PdX-l--iT-, we shall have 
 
 *=JPd\+ 
 
 fd* d^- 
 
 dT^-dui 
 
 Id* ^ 
 
 d^ 
 
 vr'+w* 
 
 (b) 
 
 Thus the rectification of a curve, whose tangential equation is 
 given, may in general be very simply effected by the use of these 
 simple and general formulae. 
 
 204.] On the rectification of the circle by the method of tangential 
 coordinates, the origin being taken any where in the plane of the 
 circle. 
 
 Let the general equation of the circle be 
 
 a|'^ + a,uH3/8fu + 2yf-|-27,i; = l; .... (a) 
 
186 ON THE RADIUS OF CCRVATUEE AND 
 
 this equation becomes, writing cos \ for Pf, and sin \ for Pw, 
 P*— Ji(y cos \+ y, sin X)P=a cos* X+ 2/3 cos \ sin X, + a, sin* \. 
 
 Solving this equation for P, we obtain the result, 
 P=7 cos \ + y, sin X 
 
 + V (a + r*) COS* X + (o, + y,*) sin* X + 2 (/3 + 77,) cos X, sin X. 
 
 Now in sec. [14] it has been shown that, when the section is a 
 
 circle, a + 7*=a, + y,*, and fi + yy^ =0. 
 
 Hence P=ycosX+7ySinX+ v'(a + 7*), 
 
 and fPdX=yBinX— 7,cosX+ \/a + y^.X, 
 
 ;, dP , . . 
 
 and A\~ +7/ ^°^ X— 7 sin X. 
 
 Hence j"PdX+^= >^a + y^.\. 
 
 In sec. [ 12], (19), it is shown that the radius of the circle is 
 = ^/a + yK 
 
 205.] On the radius of curvature of the ellipse. 
 
 We have shown in sec. [200] that the radius of curvature of any 
 curve may be expressed in terms of the perpendicular on the tan- 
 gent from the origin, and the angle which tlus perpendicular makes 
 with one of the axes of coordinates ; or 
 
 d*P 
 ^=l' + d^- 
 In the ellipse P*=a*co8*X+4*sin*X, consequently 
 
 p d^ _ g*^* sin-* X- a" cos" X - &* sin'* X + g *6* cos^ X 
 
 dx* e* — ■ 
 
 206.] On the radius of curvature of the parabola. 
 Since P= — -, the origin being at the focus, 
 
 d^_, a±sin*X) 
 dX*~* cos«X • 
 Hence 
 
 T> , d*P 2k 
 
 ^+-Tri= — sT> or asNcosX=2*, 
 dX** cos^ X ' 
 
 eliminating cosX, we obtain B— 
 
 12*)*- 
 
THE KECTI?ICATION OF PLANE CURVES. 187 
 
 0« the rectification of the ellipse. 
 
 207.] The tangential equation of the ellipse referred to its centre 
 and axes is a*f* + 6*1;* = 1 . Multiply by P*, for Pf put cos X, and for 
 Pw put sin X, and then the equation will become P =a v^l — e* siu^ X; 
 _j _ . dP . ae*sinXcos\ 
 
 dX' Vl -e* sins' X 
 
 
 Hence 
 
 
 
 ae* sin X cos X 
 
 «=aJdX v^l— e*sin*X- 
 
 •/l — e^sin^X' 
 
 • • (a) 
 JL — e- siu- A. 
 
 an elliptic function of the second order. 
 
 t is taken with a negative sign as the perpendicular continues to 
 diminish. 
 
 At the critical point, substituting for sin* X and cos* X their 
 
 values — 37? and —7-7, the expression for the curve as far as this 
 point from the extremity of the major axis becomes 
 «=aJdX Vl— e*sin*X— (a— i), 
 
 the limits of integration being X=0, and X=sin~' a / — —. 
 
 V a+b 
 
 On the rectification of the parabola. 
 
 208.] The tangential equation of the parabola is 
 
 ^r + «'') = ?, 
 where k is one fgurth of the parameter. 
 
 Multiply by P*, and the equation becomes P= — and 
 
 COS Ai 
 
 dP 
 
 i=3-j— =Atan sec 6; the positive sign of / to be taken, since the 
 dX 
 
 radius of curvature is always greater than the focal perpendicular. 
 
 Hence s=k\ -+AtanXsecX (a) 
 
 J cos X ^ ' 
 
 This expression for the arc of a parabola is the foundation of 
 parabolic trigonometry. 
 
 On the radius of curvature and the rectification of the cycloid. 
 
 209.] It has been shown in sec. [139] , (d), that if the origin of 
 coordinates be taken at the centre of the cycloid, its tangential 
 equation will be 
 
 l=2r[t; + ftan-(|)] (a) 
 
188 ON THE BADirS OF CURVATCBE AND 
 
 If we multiply this expression by P, it becomes 
 
 P=2r[co8\ + \..sin\]; (b) 
 
 taking its first differential, 
 
 ^=2r\cosX; (c) 
 
 taking the second differential of F, 
 Consequently 
 We have also 
 
 d*P 
 
 ^--^=2rcos\— 2r\sin\ (d) 
 
 d*P 
 R=P+^=4rcos\ (e) 
 
 j'Pd\=4rsin\— 2r\.cos\; . . . . (f) 
 
 and adding the value of -i^=2rX, cos X, as given in (c), we shall 
 have finally 
 
 »=j'PdX + ^=4rsin\ (g) 
 
 If we square the expressions in (e) and (g), we shall have 
 
 R«+«2=4r«j (h) 
 
 or the square of the radius of curvature at any point of a cycloid, 
 together with the square of the arc measured from that point to the 
 vertex, are equal to the square of the diameter of the generating circle. 
 
 This property holds also in the case of epicycloids and hypocy- 
 cloids, with some slight modification. Simple as the theorem is, 
 we do not recollect to have met with it. 
 
 The perpendicular from the centre on a tangent to the cycloid 
 is equal to the radius of curvature at the same point, when 
 XtanX=l. 
 
 For P=2r{cosX+XsinX}=4rcosX=R; 
 
 consequently X tan X=l, or X=cot X (i) 
 
 On the rectification of the evolute of the ellipse. 
 210.] Let the tangential equation of the evolute of the ellipse be 
 
 aV + 6«J«=(o*-i«)*f*t;« (a) 
 
 Then the perpendicular from the origin on the centre will be 
 p _ (a* — A*) sin X cos X 
 
 ~ Vo'sin^x + J^cossx' ^' 
 
 and 
 
 JPdX= Vo^siu*X + Z»*co8^X (c) 
 
THE KECTIPICATION OP PLANE CUKVES. 189 
 
 Now 
 
 dP_ (a'-ftg) (ftg co8*\-fl« sin" \) 
 
 and as 
 
 dP 
 
 dX' 
 we shall have, making these substitutions, 
 
 H^^+^' (e) 
 
 {a*8in*\ + 62cos*X}i 
 ,, andwhenX=90°, s^(= , ..vu^t <.,— oj,-a-j- 
 
 O -TO 
 
 ab 
 
 „<2 i2 9 ig 
 
 When\=0,*,=-^; andwhenX=90°, s^(=— ; hence *,—»„= ^ 
 
 or = — Y— J a result already obtained in a different way. 
 
 On the rectification of the semiciMcal parabola, and its radius of 
 
 curvature. 
 
 211. J The tangential equation of the semicubical parabola is 
 
 \^=k^, see sec. [148], 
 
 Multiplying this equation by P, the perpendicular from the ori- 
 gin, and referring the angle X to the normal passing through the 
 cusp of the curve as the axis of X, we shall have 
 
 ^ , sin® X 
 
 V=k — 5^ (a) 
 
 cos*X ^ ' 
 
 Hence 
 
 Jk 
 PdX=AcosXH -, (b) 
 cos X ^ I 
 
 and 
 
 dP 2A , ^ * 
 
 jT= s^— ACOSX -: . . . . (c) 
 
 dX cos^X cosX ^' 
 
 consequently 
 
 dP_ 2k 
 dX~cos®X 
 But at the cusp «=0 and X=0, hence 
 
 «=2*(sec»X-l) (e) 
 
 This is a very simple and elegant expression for the arc of the 
 semicubical parabola as compared with that usually given in pro- 
 jective coor^ates — that is to say, 
 
 ,_ (9a?+4a)t-(4a)t 
 27ai 
 See Gregory's examples, p. 416. 
 
 ,=JPdX-H°^=-^+C (d) 
 
190 ON THE KADirS OP CXTKVATrRE AND 
 
 ,ve 
 
 • (f) 
 
 If we differentiate the expression (c), we shall have 
 d»P _ 6Asin\ _ Asin^ 
 dX«~ cos*\ cos'X > ' • ■ 
 sin^X 
 
 butP=* — s^- 
 cos*\ 
 
 „ ^ d^P 6isin\ , . 
 
 Hence ^ = ^ + i\^=-^S^ ' ^S) 
 
 a simple expression for the radius of curvature of the semicubical 
 parabola. 
 
 It may easily be shown that the following relation exists between 
 the arc, radius of curvature, and the perpendicular, on the tangent, 
 fh)m the cusp of the semicubical parabola 
 
 6P 2k ( 2k \i „. 
 
 R~s + 2k \s + 2kj ^ ' 
 
 On the rectification of parallel curves. 
 212] . Let s, be the arc of the parallel curve, and s that of the 
 primitive. 
 
 Applying formula (b), sec. [201], we get 
 
 cr.-, dP, , , T, T> . 1 J dP, dP 
 «/=J^'d^+ dX ' ^^* ^'=^ + *' ^^-^ ^=dX' 
 hence 
 
 «,=JPd\ + A\+^ (a) 
 
 But the arc of the primitive curve is 
 
 H^'^^+^I- (^) 
 
 consequently 
 
 Si—s=h\; (c) 
 
 or the difference of the corresponding arcs of any two parallel curves 
 is equal to a circular arc whose radius is h, the constant difference 
 of the normals of the two curves. 
 
 On the radius of curvature, and the rectification of epicycloids and 
 hypocycloids*. 
 213.] Assuming the general expressions given in sec. [129] for 
 the tangential coordinates of these curves, namely 
 sin (« + 1) ^ _ cos (re + 1) 
 ^~"2r{M+l)sin«^' '''~2r(M + l)sinw<^* 
 
 • In the 49th proposition of the first book of the ' Principia,' Newton shows, by 
 apuiely geometncal method, that all epicycloids and hypocycloids are rectifiable. 
 He does not, however, discuss the relations which exist between the arcs and 
 radii of curvature of these curves. 
 
THE RECTIFICATION OP PLANE CURVES. 191 
 
 squaring these expressions, adding- them, multiplying by F', and 
 taking Qie square root, we shall have 
 
 P=2r(w + 1) sinn<^ (a) 
 
 Let X be the angle which P makes with the axis of X ; then an 
 
 inspection of figure 40 will show that (n + 1)^=-+X, 3^= — -rr. 
 
 For 
 
 BXC=OZX + ZOX or (b + 1)<^=^+\. . . (b) 
 
 Hence 
 and 
 
 JPdX= -2r^^^tiI%osra0 + C, 
 
 consequently 
 
 , = rpdx + g=-2ri?^+licos«,^ + C. 
 J d\ n ^ 
 
 At the cusp, the arc is ^0, consequently 0= —2r- + C. 
 
 Hence, by subtraction, s=2r (I — cosn^) ; or, as this lat- 
 
 ter factor is =2 sin*(-^ j, we shall have finally 
 
 . (2«+l) . ^M\ , . 
 
 s=.^r- ^sin*(-^l (c) 
 
 n 
 
 This is a general formula for all epicycloids, and also for hypo- 
 cycloids when we write 2n— 1 instead of 2n + l. 
 
 d'P 
 Since R, the radius of curvature, is =1'+ j^i* *°^ ^^e 
 
 .^— =2n» cos n^, we shall have 
 
 C1A« 
 
 ■VdX/~d.^ 
 or 
 
 d^ 
 dX« 
 
 and as P=2r(n+ !•) sin n^. 
 
 d /dP\_ d /dP\ d^_ 
 
 cixAdx/-d.^-Vdxjdx-"'f*' 
 
 „ . , d<& %rv? . . 
 
 = — 2r»sinnffl. 37 = — =-sm«A; 
 
 ^ dX n + 1 ^ 
 
 „ „ d'P 2r(2re+l) . , ,,. 
 
192 
 
 ON THE KADIUS OV CtjaVATURE AND 
 
 This is a general form for the radius of curvature of aU epicy- 
 
 '^ °Siiice, when n<j>=0, R=0, we may infer that in this class of 
 curves the radius of curvature is =0 at the cusp. , 
 
 The radius of curvature will be a maximum when sm. n<j) is a maxi- 
 mum, or when if>^^. 
 
 Let us apply these two general formulae — 
 „ „ /2n + l\ . . 
 
 (e) 
 
 We may first give a simple geometrical illustration of these for- 
 mula. Let B P Q be the rolling circle, and A Q the fixed circle ; 
 
 Fig. 40. 
 
 bisect the angle P B Q by the line B D, and join Q, D, then evi- 
 dently 
 
 it)- 
 
 PQ=2rBin^, and DQ^=4r*sin«l 
 
 consequently 
 
 R=(?!^)PQ, and ,=^^±11dq'. . . . (f) 
 \re+l / nr ^ 
 
 Hence, while the radius of curvature of an epicycloid or hypopy- 
 cloid, at any point P of the curve, is proportional to the line PQ 
 (the instantaneons radius, the chord of the arc QP which has rolled 
 
THE RECTIFICATION OF PLANE CURVES. 193 
 
 along the fixed arc AQ), so the arc of the curve up to that point 
 from the cusp is proportional to the square of the chord of half the 
 arc QDP. 
 
 214.] We may now apply these formulae to determine the lengths 
 of the arcs and the radii of curvature of several of these curves. 
 
 (o) In the cardioid whose equation is given in sec. [1 30] , we have 
 E,= 2Mr=r or 2n=l ; substituting this value of n in the general 
 formulae, we shall have 
 
 « = 16rsin«(^), (a), and R=|^ 8in(|), . . (b) 
 when 0=7r, 
 
 s=8r, R=|^ (c) 
 
 (/3) In the semicircular epicycloid whose equation is given iu 
 sec. [130], a, we have ^=2nr=2r, or m=1 ; hence 
 
 s = 12rsin*|, R=3rsin^ (d) 
 
 (y) In the quadrantal epicycloid as R=2»r=4r, «=2, and 
 
 s = 10rsin*^, R=— rsin2(^ (e) 
 
 o 
 
 (S) In the trigonal epicycloid, as the base circle is three times 
 the rolling circle, 2w=3, 
 
 16 
 
 ?= — rsin- 
 o 
 
 '&)■ "-T-'-e*)- ■ ('» 
 
 215.] On the curvature and rectification of hypocycloids. 
 
 The formulae in this case are, s=4r( jsin^f-^j and 
 
 IU=2r ( _, J sin w^. 
 
 (a) Let the rolling circle be one half the base circle, then 2«=2, 
 or« = l. Hence 
 
 as the hypocycloid is in this case the diameter of the base curve, we 
 must evidently have R= 00, and s =4r. 
 
 (/3) Let the curve be the quadrantal hypocycloid whose equation 
 is given in sec. [134] . 
 
 As, in this case, the base circle is four times the rolling circle, we 
 must have «=2, and the resulting expressions become 
 
 s= 6r sin^ 0, R=6r sin 2<^. 
 
194 ON THE RADIUS OF CURVATURE AND 
 
 (y) In the trigonal hypocycloid, as 2»=3, m=|, we shall have 
 
 16r . o/SM _ _ . 3(^ 
 s = — sm^f^j, R = 8rsin-^; 
 
 sin-? is a maximum when d)= 120°. 
 4 
 Hence the length of one of the arcs of the trigonal hypocycloid 
 
 is =—^, and the sum of lengths of the three arcs is ^16r. 
 o 
 
 It is needless to pursue these illustrations further ; what is very 
 remarkable is this, that though we may not be able to give the 
 equation of an epicycloid or hypocycloid either in tangential or 
 projective coordinates, we may notwithstanding find finite and 
 exact expressions for their arcs and radii of curvature. 
 
 Thus, if the base circle be 100 times the rolling circle, we shall 
 have for s, » = (8^)r sin^ (50^) , 
 
 R=2r.^sin(100^). 
 
 216.] If we eliminate the angle between the two equations, 
 sec. [213] , (e) , we shall have the following relation between any arc 
 of an epicycloid and the radius of curvature at its extremity, 
 
 (« + l)m2 + »V=4ra(2»+l)* (a) 
 
 There are several important consequences which may be drawn 
 from this equation. 
 
 (a) When *=0, R=0, or at a cusp the radius of curvature =0. 
 When R=0, s=0, or ws=4r-(2n+l). 
 
 We may write the preceding equation in the form 
 
 (« + l)m2 + w2(4r-s)«=16«*/-«+4«ra. . . (b) 
 
 If we now assume » = ao , or the base circle a straight line, dividing 
 by »*= (»+ 1)^, we shall have 
 
 R2 + (4r-»)2=16r2, (c) 
 
 a property of the cycloid established in sec. [209], (h). 
 
 On the radius of curvature of the cubical parabola. 
 
 217.] If we measure the angle \ from the normal to the curve 
 passing through the point of inflexion at the origin, the equation 
 of the curve ^=a^i^, when multiplied by P*, will become 
 
 P«cos\=o«sin3\; (a) 
 
 difiPerentiating this expression, 
 
 ^U)=^3?x(3''°^'^ + --'^)' ••••(b) 
 
THE RECTIFICATION OF PLANE CURVES. 195 
 
 or 
 
 ^(^)=l-2cos''\+-J^; (c) 
 
 a' \d\/ cos' A, ^ ' 
 
 diflperentiating again, 
 
 4/dP\« 4P/d«P\ „ . , , 4 8m\ 
 F^dx) +^VdA?j=^''"''''°^'^+^^^- • ^^^ 
 
 If now we square (b) and divide by P', we shall have 
 
 4/dP\* _ . ^ ^ , 6siu3X , ain^X , , 
 
 a'XaKJ cos\ cos^X. 
 
 and if we subtract this expression from (d), the result will be 
 
 4P/d'P\ sin\,- , . ,^ ,^, ... 
 
 The equation of the curve gives 
 
 4 P* _ 4 sin^ X cos' X 
 o' ~ cos'*X 
 
 adding these expressions. 
 
 4Pf„ d'P) SsinX , „ „ d«P 
 
 we shall have finally 
 
 R=f a{sin\cos^X)~2 (g) 
 
 Since the radius of curvature is oo when X=0 and when X=90', 
 there must be an intermediate value for R when it becomes a mini- 
 mum. For this minimum value, tan X= — ^. 
 
 On the rectification of the involute of the quadrantal hypocydoid, 
 
 218.] The projective equation of this curve being a* + y*=r*, the 
 tangential equation of its involute, see sec. [136], (b), is 
 
 [(/+a)P+(/-a)u«]= VF+^'. 
 
 Multiplying by P', the preceding expression becomes 
 
 / + a (sin* X — cos' X) = P, 
 
 and -jT-=2asin2X; we have also f PdX=iX— 5 8in2X; consequently 
 dX "^ A 
 
 s=JPdX + ^=^ + ^sin2\ (a) 
 
 It may be remarked that r=4a. 
 
IQQ ON THE EECTIPICATION OF THE 
 
 219.] On the rectification of the curve whose tangential equation is 
 
 (0^ + «T=|+^' ^^) 
 
 This is the equation of the curve enveloped by one side of a right 
 angle, which moves along an ellipse, while the other side passes 
 through the centre. 
 
 Let the angle \ be measured from the minor axis of the elupse, 
 and, multiplying by P*, we shall have 
 
 , „ fsin'X , cos'Xl -r, b ,, . 
 
 and 
 
 dP Ae^ sin \ cos \ 
 
 (c) 
 
 d\ {l-c9sin*\}3 
 consequently 
 
 r-oj^ dP iC dX , Ae'sinXcosX ,,. 
 
 J dX Jvi— c*sm'X [1— c*sm*X]ir 
 
 This curve gives the simplest geometrical illustration of the first 
 elliptic integral. On the surface of the sphere its true geometrical 
 exponent is the spherical parabola, as will be shown in the second 
 volume of this work. 
 
 *0n the rectification of the inverse curve of the central ellipse. 
 
 220.] Ce probleme merite d'etre discute & cause de Inelegance 
 remarquable de sa solution, qui depend de devaluation d'une inte- 
 grale elliptique de troisieme espece h, parametre circulaire. 
 
 On dit que deux courbes sont inverses I'une de I'autre lorsque le 
 produit de leurs rayons vecteurs superposes est constant, c'est-^ 
 dire que : 
 
 Rr=c«, 
 
 Soit: 
 
 I'equation de Pellipse,le centre etant au pdle, et soit 'Rr=kab : on 
 aiu-a, pour la courbe inverse, I'equation : 
 
 k^(aY + 6 V) = (x^ + y2)«. 
 
 On peut simplifier la discussion, sans restreindre la generalite, 
 en prenant A=l. L'equation de la courbe inverse h, I'ellipse, le 
 centre etant au p61e, est alors : 
 
 aV + 6'a;2=(^^+J'*)' (1) 
 
 • This demonstration is transcribed from an article in the ' Annali di Mate- 
 matica piira ed applicata," serie 2, tomo ii. fasc. 1, p. 84. 
 
INVERSE CURVE OF THE CENTRAL ELLIPSE. 197 
 
 Si I'on pose : 
 
 x=rcos^j y=rsin<^, (2) 
 
 cette Equation devient : 
 
 a« sin^ ^ + 62 cos« ^=rS (3) 
 
 d'ou I'on tire, apres quelques reductions simples, 
 
 ds^ _ g^sin^^ + &*co3*^ . . 
 
 d4>^~ a^sas?<f,-\-b^co&^4, 
 
 La substitution : 
 
 a' tan ^=6* tan \, (5) 
 
 change cette formule en la suivante : 
 
 rf^^~ffl«cos«\+42sin«X' ^ ' 
 
 et puisque on en tire aussi : 
 
 d^_b^ cos^<f> J ^ cos*<^_ o^ ,_, 
 
 d\~^ toPIi' "^^ ™®™® I''® ^^^X - 0* cos* \ + 6* sin* \' ^ ' 
 
 substituant et simplifiant on obtient : 
 
 ds^ «W 
 
 dX~{a*cos^\+b*sin^X) Va*co?X+Fsin^' 
 ou: 
 
 *-^ ' . . (8) 
 
 Faisant : 
 
 a»- 62 , , a*-b* 
 
 d\ a* 
 
 „. 0' etr_^=m, (9j 
 
 
 on a: 
 
 m= — 5 — c*, m>c^, 
 
 et par suite, integrant : 
 
 d\ 
 -m siu* \] Vl — c* sin' \' 
 integrale elliptique de troisieme espece h, parametre circulaire, car 
 
 Imaginons le cylindre droit dont la base est Tellipse aux demi- 
 axes a et b, et la sphere decrite du centre avec un rayon= \/ffl* + 6'. 
 Cette sphere coupe le cylindre suivant une ellipse spherique. 
 
 Soient a et /3 les demi-angles principaux de cette ellipse spherique, 
 alors : 
 
 ^'^^"^IF+b'^' ^^^^ ^'"'ifi + b^' ^""^^ " = i^' ta°'^=^2- (11) 
 
198 ON THE RECTIFICATION OP THE 
 
 D'ici on tire : 
 
 a''-^" tan»a-tan^/3 a^-b^ ^ sm^a-sm^ ^ 
 a* ~ tan^a ' a^ sin^a ' 
 
 ffl2 tan a 
 Eflfectuant ces substitutions dans Tequation (8) 11 vient : _ — _ — 
 tan/3 . r* dX .j^. 
 
 Or dans les Philosophical Transactions pour 1852^ Part II. 
 p. 319, j'ai montre que I'expression d'un arc d'ellipse spherique 
 qui resulte de ^intersection d'un c6ne aux demi-angles a et /3 avec 
 une sphere concentrique est donnee par la formule : o-= 
 
 tan^ . „f d^ .,„» 
 
 tana 
 
 Soit e I'excentricite de la base plane du c6ne, savoir e'^ = — , 
 
 2e I'angle des deux lignes focales, et 2?; 1' angle des deux axes cy- 
 cliques, c'est-li-dire des deux droites normales aux sections circu- 
 laires du cone ; d'apres le meme Memoire on aura : 
 
 5 tan'o— tan^/3 • , sin^a— sin^/S ,, .. 
 
 e^= 7—5 -> sin297= ^-5 tL . (14) 
 
 tan^a sin*o ^ ' 
 
 et: 
 
 tan^6=""''^-/'^'^, 
 cos^a 
 
 et la precedente expression de I'arc de 1' ellipse sphferique deviendra : 
 
 .=*-^8in^f ^^ . (15) 
 
 tana J[l-e2sin20] \/l-8in*i;.sin««^ 
 
 Designant par m et m deux parametres conjugues, on a : 
 
 (l-m)(H-M) = l + c2, (16) 
 
 ainsi qu'on pent le voir dans tout ouvrage elementaire sur les inte- 
 grales elliptiques; done si I'on fait m = e* et c«=8in*»i, on a 
 
 sin^a— sin^fl 
 
 n= 5 -. ou 
 
 cos^a 
 
 w=tan^ e. 
 
INVERSE CURVE OF THE CENTRAL ELLIPSE. 199 
 
 Les trois quaatites : e excentricite de la base du cone, 26 angle 
 des ligues focales et 2ij angle des lignes cycliques, sont liees par 
 I'equation simple : 
 
 l—e^=cos^rj.cos^e (17) 
 
 Le coeflacient de I'integrale (12), c'est-li-dire -^^sin/3, est ce 
 
 tana 
 
 qu'on nomme quelquefois le criterium de circularite, car : 
 
 tan^^ / c«\ 
 
 tl,^^^^'^=(l-'»)(l-J' 
 par svtite on pent ecrire : 
 
 <r=^U-m){l-'^L- ■ ,^.f^^ ^ . ^ . (15*) 
 
 * \ »»/J[l— msin*^] v'l— c*sm*^ 
 
 Notre conique spherique a ses arcs principaux supplementaires, 
 
 car puisque tan a = j, tan /3 = -, on a tan a. tan /3=:1, d'ou o+/3=— . 
 
 EUe est egale k sa reciproque tournee d'un angle droit. Les axes 
 focaux de I'une sont les axes cycliques de I'autre : par consequent 
 la rectification de Pune depend de la quadrature de I'autre, ainsi 
 qu'on pent le voir dans le Memoire cite ci-dessus. 
 
 II, De la rectification de la courbe representee par I'equation .- 
 
 aV-iV=(a7« + y2)2 (18) 
 
 Posant x=r siu<^, y = rcos 4>i il vient : 
 
 a*sin2<^— fi'cos^^rrr^, (19) 
 
 d'ou: 
 
 ds^ _ ce* sin^ ^ + b* cos^ ^ 
 d(fi^~a^am^^-b^cQs^<f> 
 Prenant : 
 
 a^tau^<t> = b^sec^\, (21) 
 
 et faisant les substitutions necessaires, on trouvera : 
 
 e?s _ «i V«* + 6* cos* \ -ooN 
 
 d\~ b^ + a^cos^\ ' ^ ' 
 
 ou: 
 
 ds__ a6[flV6'-6*sin'X] 
 
 rf\~[a2 + 62_a2sin*\] VaVfi*-6«sin«\' " ^ ' 
 ou encore : 
 
 b^ 
 
 a,f>_;,2r.r.c2A (^^) 
 
 ab [^-^W^^ 
 
 dX -/a* + 6* 
 
 ly^^w^^l^T^-^^"^ 
 
 (24) 
 
200 ON THE RELATION BETWEEN TANGENTIAL AND 
 
 ou enfin : 
 
 ds _ b^ 
 En faisant : 
 
 
 (35) 
 
 =»». :Ar2=c^ (26) 
 
 flS + j* ' a' + 6* 
 la demiere equation devient : 
 ds b(a^-b^) 1 b^ 
 
 d\ as/a^ + b^ll-maia^X] y/l-c^ain'X a\/a^ + b^' V'l-c'sin'x' 
 
 ou, intrant : 
 ^_b a^-b^ C dX, b^ r d\ 
 
 Si la coTirbe est la lemniscate, on a a=6, et cette expression 
 devient : 
 
 CHAPTER XXIII. 
 
 ON THE RELATION BETWEEN TANGENTIAL EQUATIONS, AND THE 
 SINGULAR SOLUTIONS OP THE DIFFERENTIAL EQUATIONS OF 
 PLANE CURVES. 
 
 221.J The equation ¥{x,y) =0 is the integral known as the sin- 
 gular solution of the equation ^(f, v)==0. For if between the 
 
 three equations *(f, v) =0,a;f +yu=l, and 3^=— 5 we eliminate 
 
 aa; v 
 f and V, we shall find the resulting differential equation 
 
 *('.., |)=o/ 
 
 or using the usual notation ■A=P, we shall have the differential 
 equation of the same curve, *^{x, y, /?) =0. 
 
 Since x^+yv=l, andj9=^, we shall find for f and v, 
 
 ?^ — — — , and 1;= fa) 
 
 (27) 
 
DIFFERENTIAL EQUATIONS OF PLANE CCKVES. 201 
 
 To illustrate this theory. 
 
 (a) Let us assume the tangential equation of the evolute of the 
 ellipse aV+6«fS'=c*^i;*. 
 
 For f and v substitute their values as given in (a), and the 
 resulting equation wiU become 
 
 '^;^''p'-i^^ (^) 
 
 DiflFerentiating this expression (b), we find 
 
 ?!=_y_ or „^ g%*-«V (c) 
 
 Hence 
 
 {y-pxY= ^j 
 
 but the equation of the curve (b) gives 
 
 Comparing these two values, 
 
 substituting the value of ^ derived from (c), we finally obtain 
 
 or, taking the cube root, 
 
 a%* + JM = c*, (d) 
 
 the projective equation of the evolute of an ellipse. 
 
 (/3) As another example let us take the tangential equation of 
 the ellipse 
 
 a«r'+6V=l (a) 
 
 Substituting the values of x and y, 
 
 a^p^ + b^={y —pxy. 
 Differentiating this expression, we get 
 
 Hence 
 
 ^-S w 
 
 ^-^^=^« (') 
 
 Substituting these values in the preceding equation, 
 
 or reducing, 
 
 oV+52^*=ffl«6^ (d) 
 
 the projective equation of an ellipse. 
 
 It is beside the object of this work to pursue the subject further. 
 
202 
 
 ON THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS. 
 
 CHAPTER XXIV. 
 
 ON THE GEOMETRICAL THEORY OF RECIPROCAL POLARS. 
 
 222.] To this simple but very beautiful theory is due the widest 
 development of pure geometry that has been effected in modem 
 times. Nothing can be simpler than the elementary principles on 
 which this theory rests, nothing more beautiful than the results to 
 which it leads. 
 
 Let a point to and a straight line AB be assumed in a plane. 
 
 Fig. 41. 
 (ft) 
 
 <l 
 
 d 
 
 
 
 
 
 / 
 
 
 \ 
 
 TT 
 
 y 
 
 
 -A. 
 
 P 
 
 
 _B 
 
 From this point a> let fall a perpendicular mP on the line AB. 
 With a> as centre, let a circle (H) be described, r being the radius. 
 In the line » P let a point ir be assumed such that airXaV shall 
 be equal to r*. tt is called the pole of the line A B with reference 
 to (II) ; and they are termed reciprocal pole and polar. The circle 
 whose centre is to and radius r may be called the polarizing circle, 
 as being the instrument by which the polars of given points and 
 lines may be exhibited. The polarizing circle or sphere, a dotted 
 line, will be denoted by the symbol (fi). 
 
 It is obvious that any conic section might be used as a polarizing 
 curve instead of a circle ; but no advantage would be gained by com- 
 plicating this instrument of geometrical discovery. 
 
 223.] If we now pass to space of three dimensions, let there be 
 assumed a point and a plane in space. Let a sphere (H) be de- 
 scribed, having its centre at the given point at, and from this point 
 let fall a perpendicular a P upon the plane A B, and let a point nr 
 be assumed on this perpendicular, so that oiirx w P shall be equal 
 to -B?. -IT is the pole of the given /^/ane AB with reference to the 
 sphere (12). 
 
ON THE GEOMETRICAL THEORY OF RECIPROCAL POLARS. 
 
 203 
 
 Through the point a>, and touching the given plane at P, let 
 another sphere (11^) be described. Let w P P, be a great circle of 
 
 Fig. 42. 
 
 this sphere, and through P let a line J J, be drawn at right angles 
 to this plane. Let the common secant plane of the two spheres be 
 drawn, C D being the intersection of this secant plane with the 
 plane of the great circle w P P,. Through the line J J^ let any plane 
 (n,) be drawn cutting the great circle of the sphere (O,) in the 
 straight line P Py. Join w P^ meeting the common cord of the two 
 great circles of the two spheres (fl) and (fl,) in the point tt,. It 
 is manifest that tt^ is the pole of the plane P P^ J Jp since 
 
 O) TTy X tU Pj= O) TT X O) P=:R*. 
 
 Hence it follows that the poles of all the planes drawn through 
 the straight line J P J^ perpendicular to the plane of the great circle 
 » P P( wUl range along the common cord of the great circles of the 
 two spheres (f2) and (fl,). 
 
 It is manifest that the angle between the planes (11) and (ITj) is 
 equal to the angle between the perpendiculars let fall upon them 
 from the centre o of the polarizing sphere (H). 
 
 The common chord of the two great circles along which the poles 
 of the planes range is in a plane at right angles to the line J P J^ 
 through which all the planes pass ; and these two straight lines are 
 called coryugate polars, one of the other, with reference to the polar- 
 izing sphere (H). 
 
204 ON THE GEOMETRICAL THEORY OF RECU'ROCAL FOLARS. 
 
 Indefinite planes and straight lines, or planes and straight lines 
 given only in position, will be denoted by letters enclosed in brackets. 
 Thus (11) and (/) will denote a plane aud a straight line given in 
 position only, and otherwise indeterminate. 
 
 224.] We shall now proceed to develope some of the results which 
 follow from these elementary and obvious principles. 
 
 If any point ir, be assumed in a plane (11), the polar plane (II ,) 
 of this point will pass through ir the pole of (II), 
 
 Fig. 43. 
 
 Join CO and ir,. Let the polar plane (11,) of tt, cut the line a ir, 
 in Q., and the line <a P in Q. Then 
 
 WTT, xa> Q,=«» Qxo>Ps=R'=o)ffXo»P, 
 
 or TT coincides with Q. 
 
 When any number of straight lines are parallel to one another, 
 their conjugate polar s will all lie in the same plane. 
 
 For the conjugate polar of any one of them will be in a plane 
 passing through the polarizing centre w, and at right angles to the 
 given straight line. Hence all these planes will be parallel ; and as 
 they all pass through the point w they will be identical. 
 
 It will much facilitate the study of this theory if these two prin- 
 ciples are firmly grasped, and rendered familiar ; that is to say, the 
 polar plane cf any point assumed in a straight line will pass through 
 the conjugate polar of this straight line, and the polar plane of any 
 point assumed in a plane will pass through the pole of this plane. 
 
 225.] The reciprocal polar of any plane curve is a cone whose 
 vertex is in the pole of the plane of the curve. 
 
 For if we inscribe a polygon in the given curve, the polar plane 
 of every vertex of the polygon will pass through the pole of the 
 plane of the polygon, as we have just now shown, and the conjugate 
 polar of any side of the polygon will be the line in which two con- 
 
ON THE GEOMETRICAL THEORY OF RECIPROCAL POLARS. 
 
 205 
 
 secutive polar planes of two consecutive vertices of the polygon in- 
 tersect. Let the sides of the polygon now be indefinitely multiplied 
 and diminished in magnitude, the polygon will ultimately coincide 
 with the plane curve, its limit ; and the polygonal pyramid will 
 become a cone. 
 
 Throughout the following pages the polarizing circle or sphere 
 will be denoted by the symbol (11) ; the primitive or normal surface, 
 which is to be transformed, by (S) ; and the reciprocal polar of this 
 surface by (S) ; and if {I) or (m) or (n) be any straight lines, their 
 conjugate polars will be denoted by the symbols (X), (fi), (v). 
 
 236.] Fi'om the centre to of the polarizing circle (fl) a perpen- 
 diular mVis let fall on a tangent drawn to the curve (S) at T. 7%e 
 radius eoT produced iviU be perpendicular to the corresponding tangent 
 drawn to the reciprocal polar curve (2) ; and the perpendicular <bP 
 on the tangent to the curve (S) at T will pass through the point of 
 contact T of the corresponding tangent to the reciprocal polar (2). 
 
 Let the vector wT be produced to w, so that <a'Ix<ovr=s.\ 
 Draw the tangent sV, indefi- 
 nitely near to T P, and let the Fig- 44. 
 two perpendiculars wP, wP^ be 
 produced to meet in the points 
 T and Tf, the perpendicular to 
 to-er erected at tsr. Now, as by 
 hypothesis (S) and (2) are reci- 
 procal curves, 
 
 6> T X 6) ■eT=(o P X a> t=r2, 
 
 T «r P T is therefore a quadri- 
 lateral that may be inscribed 
 in a circle. Hence the angle 
 CD T P=a) T w^o) T/ -a, since at 
 the limit the angle t co t, va- 
 nishes. Consequently a circle 
 may also pass through P t t^ P^ 
 and therefore 
 
 «»Px<»T=ft>P^X 0>T,= R*; 
 
 and therefore r^ must also be 
 
 a point on the curve (2), or t t 
 
 an indefinitely small portion of 
 
 the line «r t t, erected perpendicular to the line w «t at the point w 
 
 is a tangent to the reciprocal polar (2) of the original curve. 
 
 Hence when two curves (S) and (2) are reciprocal polars, one of 
 the other, the radius vector through the point of contact of the one 
 will be perpendicular to the corresponding tangent of the other ; or 
 if through a point assumed on (S) we draw a tangent, the polar of 
 this point of contact will be a tangent to (2), and the pole of the 
 
206 ON THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS. 
 
 tangent to (S) will be the corresponding point of contact on (2) ; 
 or in other words, the radius vector drawn to a point of the one (S) 
 will coincide with the perpendicular let fall on the corresponding 
 tangent to the other (2). If r and P, be these quantities, we shall 
 always have rP,=R^, a constant quantity. 
 
 227.] To find the conjugate polar of the normal to a tangent plane 
 applied to the primitive surface (S). 
 
 As the normal passes through the point of contact of the tangent 
 plane to (S), its conjugate polar will lie in the tangent plane to (S) ; 
 and as the normal and tangent plane to (S) are at right angles, one 
 to the other, the vector line from the origin to the point of contact 
 of the tangent plane to (2) will be perpendicular to the vector plane 
 drawn from the origin through the conjugate polar of the normal^ 
 which lies, as we have shown, in the tangent plane to (2) . 
 
 Hence the conjugate polar of the normal to a curved surface (S) 
 is the straight line in the tangent plane to (2), through which and 
 the origin if a plane be drawn, it wUl be perpendicular to the radius 
 vector drawn from the origin to the point of contact of the tangent 
 plane to (2). 
 
 228.] (ffl) A plane (II) passes through a given point ir, and a given 
 straight line (I) . The pole ir of this plane (11) is the point in which 
 the polar plane (IT,) of tt, is pierced by the conjugate polar (X) of 
 the given straight line. See fig. in sec. [224] . 
 
 For as this latter point is in the straight line (\), its polar plane 
 wUl pass through (Z) j and as it is in the plane (II,), its polar plane 
 will pass through tt,. Hence, conversely, the plane passing through 
 TT, and (l) must have for its pole the intersection of (11,) with (\). 
 
 08) A straight line (/) -and a plane (H) are at right angles one to 
 the other. The plane drawn through the pole tt of (11) and (\), the 
 conjugate polar of (/), will be perpendicular to the line drawn from 
 CO, the centre of the polarizing sphere, to «r,, the foot of the perpen- 
 dicular (/). 
 
 Let the given plane (II) be the plane of the paper, suppose ; then 
 (I) will be vertical, and therefore (X) wiU be horizontal. Let the 
 line (1) pierce the plane (H) in sr,. Then, as «•, is a point in (H), 
 the polar plane of vr, will pass through v the pole of (H) j and as tj, 
 is a point in the straight line (I), the polar plane of vr, will pass 
 through (X). Hence the line drawn from to to vr, will be at right 
 angles to the plane which passes through ir and (X) . 
 
 (y) Or this proposition may be proved more simply thus : join the 
 point w„ assumed as the intersection of the line [1) with the plane 
 (O), to to, the centre of the polarizing sphere. Hence (11,), the 
 polar plane of la-,, wiU be at right angles to the line vr, co ; and as is, 
 is in the plane (U) its polar plane (H,) will pass thi-ough ir, the pole 
 of (H) ; and as w, is a point in the straight line (/), its polar plane 
 (O,) will pass through (X), the conjugate polar of (Z). 
 
ON THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS. 207 
 
 If a straight line (/) be perpendicular to a given plane (11), the 
 conjugate polar (\) of this straight line will lie in a plane parallel 
 to (11) passing through w the centre of (fl), which therefore may 
 be called the central plane. It will also lie in the polar plane of 
 the point in which (/) intersects (11), therefore (\) will be the inter- 
 section of these two planes. 
 
 When the line (/) makes an acute angle with the plane (IT), the 
 central plane will cut {I) still at right angles, and (\) will be the 
 intersection of this central plane with the polar plane of the inter- 
 section of [1) with (11). 
 
 (S) A straight line (/) lies in a plane (11), the conjugate polar 
 (\) of (/) will pass through the pole tt of (11), and lie in a plane at 
 right angles to (/) , passing through a the centre of (fl) . For if 
 through (/) we draw two tangent planes to the polarizing sphere 
 (fl), the line (X) which joins the points of contact with the sphere 
 will be the conjugate polar of (/) and will be in a plane at right 
 angles to it, and as the line {I) lies in the plane (11) its conjugate 
 polar (\) will pass through tt the pole of (11). 
 
 229.] When points are assumed along a straight line (/) passing 
 through o) the centre of the polarizing sphere (O), the polar planes 
 of all these points will be parallel, seeing that they must all pass 
 through the conjugate polar (\) of (Z), which is at infinity, since (/) 
 its polar passes through to the centre of (XI) . 
 
 CHAPTER XXV. 
 
 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS APPLIED TO THE 
 DEVELOPMENT OF A NEW METHOD OF DERIVING THE PROPERTIES 
 OF SURFACES OP THE SECOND ORDER WITH THREE UNEQUAL AXES 
 FROM THOSE OF THE SPHERE. 
 
 230.] M. Chasles and other geometers have shown how the pro- 
 perties of surfaces of revolution of the second order may be derived 
 from those of the sphere, by the method of reciprocal polars. But 
 they have not extended their researches so as to include the case of 
 surfaces with three unequal axes. 
 
 In a memoir entitled " Recherches de geometric pure sur les 
 lignes at les surfaces du second degre," published in 1829, M. Chasles 
 has shown how the properties of surfaces of revolution may be de- 
 rived with singular simplicity from those of the sphere. But this 
 great geometer has omitted to apply the method to obtain the cor- 
 responding properties of surfaces with three unequal axes*. This 
 omission may be supplied from the following considerations. 
 
 * " Quand on emploie une sphere pour surface auxiliadre, s'il se trouve une 
 autre sphere dan? la fi<rure qu'on vent transformer, il lui correspondra dans In 
 
208 GEOMETRICAL THEORY OF RECIPROCAL POLARS 
 
 By the help of one reciprocation we may pass from the sphere to 
 surfaces of revolution having two equal axes. By a second reci- 
 procation we may proceed from the properties of these surfaces to 
 those of surfaces with three unequal axes. 
 
 Thus this method of dual transformation enables us to obtain 
 the reciprocal polars of properties which are themselves reciprocal 
 polars. The second reciprocal surface exhibits like properties to 
 those of the sphere. This may need some explanation. Suppose^ 
 for example, that n points lie in the same plane with respect to a 
 sphere ; the reciprocal of this property would be that in a surface 
 of revolution we should have n planes passing through the same 
 point, and we should have for the reciprocal of this, again, « points 
 lying in the same plane, with reference to a surface with three un- 
 equal axes. 
 
 231.] It is a well-known theorem that if a point be taken in the 
 plane of a circle (S), and made the centre o> of the polarizing cir- 
 cle (fl), the reciprocal polar of the circle (S) wiU be a conic section 
 having one of its foci at the centre cd of the polarizing circle (A), 
 and its axis in the line joining the centres of (S) and (Xl) . 
 
 Of the several proofs that may be given of this cardinal theorem, 
 the following is perhaps the most elegant, as it is certainly the most 
 simple. 
 
 Since the product of the segments of a chord of a circle passing 
 through a fixed point is constant, the product of the reciprocals of 
 these segments will be constant also ; and as these reciprocals are 
 coinciding perpendiculars let fall from the polarizing centre to on 
 two tangents to the reciprocal curve (2), these tangents will be 
 parallel, because the extremities of the cord of the circle (S) and 
 the polarizing centre a, are in the same straight line. Conse- 
 quently we obtain this property of the reciprocal curve (2), that 
 the product of two perpendiculars let fall from a point on parallel 
 tangents to the curve is constant. But this we know to be a pro- 
 perty of the conic sections, that the product of perpendiculars let 
 fall horn, a focus on parallel tangents is constant. Hence the truth 
 of the propoeition. 
 
 When the point of intersection of the chords is on the circum- 
 ference of the circle, a chord of the circle passing through this point 
 is =2a cos ^, ^ being the angle which this chord makes with the dia- 
 
 R* 
 
 meter 2a passing through this point; hence =- sec ^ is the length 
 
 of the perpendicular let fall from this point on the corresponding 
 tangent of the reciprocal curve; but this we know to be the expres- 
 sion for a focal perpendicular on a tangent to a parabola. 
 
 nouvelle figure une surface du second degr£ de rfivolution.; on n'aura done point 
 les propri^t^s g^Sn^iales d'un surface du second degr^ qnelconque." — Chaslcp, 
 ' ApeT9u Historique,' p. 233. 
 
APPLIED TO SURFACES OP THE SECOND ORDER. 209 
 
 232.] The polar of the centre O of (S) w D D„ the directrix of the 
 conic section (2) . 
 
 Let 01 be the centre of the polarizing circle (il) , and O the centre 
 of (S) . Let the diameter Q, Q, be drawn and the parallel tangents 
 
 Pig. 45. 
 
 Q P and Q^ P^. Let the polars of the points Q and Q, be taken, 
 vrhich will be tangents «r T, isf, T, to the curve (S) : also let the poles 
 of the tangents P Q and P, Q., to the circle be taken ; these will be 
 the points T and T, of contact of the tangents w T and -a, T, to 
 the reciprocal curve (2) . These points T and T, will be on the 
 same straight line passing through a>, since the tangents P Q, P, Q, 
 to the circle are parallel. Take D D, the polar of the centre O ; and 
 as Tw, T,tT-j, and DD, are the polars of the three poles Q, Q„ 
 and O respectively, which all lie in the same straight line, the dia- 
 meter of the circle, these three lines must meet in a point, or the 
 point Y in which the tangents meet must be on the polar of O the 
 centre of the circle. But it has been shown in [231] that <o is the 
 focus of the conic section (2) ; consequently D D^ is the directrix. 
 233.] If in the transverse aans of a surface of revolution (2) we 
 assume a point V as the common vertex of two cones of revolution 
 circumscribing the sphere (S) and the polarizing sphere (Xl) whose 
 centre is at at the focus of (2), if the base of the former cone pass 
 through the focus of (2),/Ae base of the latter will pass through the 
 centre of (2). 
 
210 GEOMETRICAL THEORY OP RECIPROCAL POLARS 
 
 Let a principal plane be drawn through the centres of (S), (fl), 
 
 Fig. 46. 
 
 
 f 
 
 > 
 
 
 
 
 
 
 
 
 X/^ 
 
 tt. 
 
 V. J--.^ 
 
 
 / / 
 
 
 
 ^ 
 
 
 
 
 CO 
 
 
 
 V,. 
 
 y \ 
 
 
 
 J<^ 
 
 ^ 
 
 XjsN-,., 
 
 Q 
 
 ' \ 
 
 r 
 
 
 and (2). Let parallel tangents be drawn to the principal section 
 of (2) at the points T and T^ ; the diameter T T, will of course 
 pass through C the centre of (2). 
 
 Let the reciprocal polars Qw, Qyor, of the points T and*T, be 
 taken ; they will touch the circle in the points Q, and CL, ; and as 
 by supposition the line Q Q,, passes through ta, the centre of the 
 polarizing circle (H), the two tangents Q^sr, Qf'sr, wiU meet 
 in a point Y. And as the point C the centre of (2) lies in the 
 diameter TT,, the polars of the three points T, T„ and C will 
 meet in V. Hence V o) : R : : R : C ai. But as V t and V t, are 
 by hypothesis tangents to the polarizing circle {Q), we shall have 
 also V(u : R :: R : Cm. 
 
 Consequently the base of the triangle Y t r^ passes through C 
 the centre of (2). 
 
 As the force of the preceding demonstration may not be at once 
 obvious to every reader, it may be observed that the point Y is the 
 pole of the line T C T, by the method of reciprocal polars, and there- 
 fore Y «» : R : : R : C o), R being the radius of the polarizing circle. 
 But in the circle t a> t, whose radius is R^, we have by common 
 geometry Y «» : R, : : Ry : C w ; but when R = R^ the two proportions 
 become identical. 
 
 We may give another demonstration of this important proposi- 
 tion. 
 
 Let a and b be the semiaxes of (2), and c the eccentricity. Then 
 
 Y<aX(ii)C = R2; (a) 
 
 and we have shown in [232] that the polar of O the centre of (S) 
 
AFFLIlLD TO SURFACES OP THE SECOND OKDER. 211 
 
 was the directrix j therefore 
 
 '"0 = a-(&) (^) 
 
 But 
 
 V<»X6)0 = Qa> ; (c) 
 
 V6j:= ^j -': and Qw — v^-; . . . . (d) 
 
 consequently 
 
 Vo,=^a(l-c«) = — (e) 
 
 But Vo) X wC =R* ; hence a>C=ae, the distance between the centre 
 and the focus. 
 
 Therefore, while the base of the cone VQQ^ determines the focus 
 of (2), the base of the convertical cone wUl determine the centre 
 of{S). 
 
 234.] Let a sphere be assumed as the polarizing surface (11) , and a 
 prolate spheroid or ellipsoid as the primitive or normal surface (S) , 
 from which the properties of the reciprocal surface (2) are to be 
 derived. We have therefore three surfaces to consider : — the primi- 
 tive or normal surface (S), which is to supply the properties that 
 are to be transformed ; the auxiliary or polarizing sphere (H), the 
 instrument of transformation ; and the derived or polar surface (2) . 
 
 Let A and C be the semiaxes of the primitive surface (S), the 
 prolate spheroid or elongated ellipsoid of revolution having its prin- 
 cipal circular section whose radius is A in the horizontal plane or 
 plane of XY, and its third semiaxis C, greater than A, vertical. 
 Let D be the distance between O the centre of the prolate spheroid 
 (S) and (o the centre of the polarizing sphere (fl) . Let a, b, c, in 
 the order of magnitude be the semiaxes of the derived surface (2) . 
 
 We shall now establish the following relations between a, b, c 
 and A, C, D, R — R being the radius of the polarizing sphere (fl), 
 which is represented by the circle whose centre is » : — 
 
 R«A . R' J R'A , , 
 
 °=A^:iD^' ^= VA^D' ' ""^"^ C VA^-D' ' • ^*^ 
 
 Thus we shall be enabled to express the semiaxes a, b, c of the 
 reciprocal polar (2) in terms of the semiaxes A, C of the primitive 
 surface (S), and the distance D between its centre O and the centre 
 01 of (12) the polarizing sphere. 
 
 We may observe that when a the centre of (fl) is on the surface 
 
 of (S), D = A, and the above values of a, b, and c become infinite, 
 
 or the surface becomes a paraboloid, of which the semiparameters 
 
 • • , X- ** c* R* , R'A , . , „ .. 
 
 of the principal sections — , — are -r- ana -pg-, which are nnite 
 
 quantities. 
 
 p2 
 
212 GEOMETKICAI, THEORY OF RECIPKOCAL FOLAKS 
 
 When D>A, the values of b and c hecome imaginary, -while a 
 
 Fig. 47. 
 
 continues real. Hence the reciprocal polar (S) is in this case a 
 discontinuous hyperholoid, or one of two sheets. 
 
 235.] It is obvious that there are two surfaces of the second order 
 which cannot be derived from the sphere — ^those whose generatrices 
 are straight lines, the continuous hyperholoid and the hyperbolic 
 paraboloid. Hence surfaces of the second order may be divided into 
 two classes — ^iuto those which have umbilici or points of drcular 
 contact with tangent planes, and those which admit of contact only 
 along straight lines. 
 
 236.] Let AB QQ, (fig. 48) be the circular section of the ellip- 
 soid whose radius is A, and which for clearness we assume to be in 
 the plane of XY. Let a point <o be assumed at the distance D from 
 the centre O of this circle, and let this be the centre of the polar- 
 
APPLIED TO SURTACES OF THE SECOND ORDER. 
 
 213 
 
 izing sphere (O). Through o> let the perpendicular Q Q, be drawn, 
 and at these points Q Q, let tangents drawn to the circle meet in V. 
 From V let two tangents Vt, Vt, be drawn to o>tt„ the great 
 
 
 
 Fig. 48. 
 
 
 
 
 t T 
 
 
 
 
 ^ 
 
 r 
 
 X 
 
 4 
 
 O u) 
 
 / \ 
 
 . sV 
 
 
 \\ 
 
 
 \ * 
 
 ^ 
 
 
 ^ 
 
 -:ii^ 
 
 c^ 
 
 r. 
 
 
 
 
 
 
 circle of the polarizing sphere (fl) in the plane of X Y. Through 
 T, T, let a straight line be drawn meeting the diameter O a> in C j 
 C will be the centre of the surface (2) by [233] . 
 
 Through T and T, let parallel tangents be drawn to the reciprocal 
 polar of the circle j this will be a conic section whose focus will be 
 at (0 by the proposition in sec. [231] . Take the reciprocal polar s of 
 the points A and Bj the extremities of the diameter of the base 
 circle ; let one of them pass through a, it will be a tangent to the 
 conic section at a the extremity of the major axis. Let a and b be 
 the semiaxes of this conic section in the plane of XY, then 
 o)V=CT=&; and as Q, is the pole of the tangent 'it, we shall 
 have Q>Qxo>^=R'; or as at=b, and 0)0= VA' — D', the result 
 becomes 
 
 *=-v^^' ^'^ 
 
 The distances of the polar focus to from the vertices of the trans- 
 verse axis of the principal section of (S) in the plane of XY are 
 manifestly 
 
 aTd^°^a:id ^^ 
 
 Half the sum of these expressions will be equal to a the transverse 
 axis of (2), or 
 
 a= 
 
 A«-D«' 
 
 (c) 
 
214 GEOMETRICAL THEORY OF RECIPROCAL POLARS 
 
 Kg. 49. 
 
 t T 
 
 237.] To determine the third semiaxis c. 
 
 On the base AHH^E let us conceive one half of the prolate sphe- 
 roid to be erected, and an ordinate toz drawn through a> in the prin- 
 cipal plane of the spheroid (S) whose semiaxes axe A and C ; 
 
 then 0)^ : OB : : VA^-D^ : A, or o>z= ^ VA'-D« ^^^ ^^ ^^ 
 
 A. 
 
 tremity r of r is the pole of the tangent plane to (2) parallel to the 
 plane of XY ; consequently c x a)2=B,* ; or, putting for az its value. 
 
APPLIED TO StJaPACES OP THE SECOND ORDEA. 215 
 
 Hence the values of the three semiaxes of the polar surface (2) 
 in terms of the semiaxes A and C of the primitive surface (S), the 
 radius R of the polarizing sphere (XI), and the distance D between 
 the centres of (S and (fl) are, as we have shown, 
 
 „_ R'A R^ , R^^A 
 
 * = T9 — tS9> "= — , . „ ^ . and c= ==. . (b) 
 
 Conversely, we may express the constants of (S) in terms of 
 a, b, c, the semiaxes of (2) — 
 
 ^ b^'^- cb' b* ^*^^ 
 
 Hence 
 
 C b 
 
 A=? ^^) 
 
 or the ratio of the axes of the primitive (S) is equal to the ratio of 
 the axes of the reciprocal polar (2) in the plane at right angles to 
 the transverse axis 2a. 
 Let 
 
 -^=e^-^=eSand-35-=,^ . . 'e) 
 
 then e, e, and ij are the eccentricities of the three principal sections 
 
 of (2) in the planes of XY, XZ, and YZ. 
 
 C*— A^ b^—c^ 
 Since — ^ — = — p— =»?*, it follows that the eccentricity of the 
 
 primitive surface (S) is equal to that of the principal section of (2) 
 in the plane of YZ. 
 
 We find also c = -^ (f) 
 
 • Accordingly, therefore, as D<A, or D=A, or D>A, the prin- 
 cipal section of (2), in the plane of XY, will be an ellipse, parabola, 
 or hyperbola. 
 
 We may express the eccentricities of the three principal sections 
 of the reciprocal polar (2) in terms of the semiaxes A and C of (S) 
 and of D the distance between the centre O of (S) and a> the centre 
 of the polarizing sphere (il), 
 
 , D« „ C«-A« , C^+DSi-A* 
 
 ft* 7j* ^~ — — _. ^* ^ _ __i_^^^^— , 
 
 C — J^i> 'I — QS > ^ — Qi 
 
 We have also the simple relation between the three eccentricities 
 
 l-e«=(l-e«)(l-^») (g) 
 
 238.] We shall now proceed to develope some very beautiful and 
 general properties of umbilical surfaces of the second order having 
 three unequal axes. 
 
216 GBOMETaiCAL THEORY OF EECIPROCAL POLAKS 
 
 The point w, the centre of the polarizing sphere (II), has been 
 shown to be a focus of the principal section of (S) in the plane of 
 XY. 
 
 This point w may be called Xhe polar focus. 
 
 Let us take the polar planes of the two foci F and F, of (S), see 
 fig. 47. The distance of one of the foci of (S) from the polar focus 
 (o is Fo>= VD* + C*— -A.*; and the length of the perpendicular «r 
 let fall from a> on the polar plane of F, will be 
 
 computing the value of this expression from (c) in the preceding 
 section, 
 
 zT=— (a) 
 
 ae 
 
 We should find the same value for ■sr,, the perpendicular on the 
 polar plane of F,, the other focus of (S) . 
 
 239.] Now, as the two foci F, F,, and O the centre of (S) are on the 
 same straight line (the major axis of (S) perpendicular to the plane 
 of XY), the polar planes of these three points will all meet in the 
 same straight line ; and as the polar of O, the centre of (S) , has been 
 shown, see [232] , to be the directrix of the conic section in the 
 plane of XY, whose focus is a, and which is a principal plane of 
 (2), the two polar planes of the foci F and F, of (S) wiU intersect 
 in the directrix of the principal section of (S) in the plane of 
 XY. 
 
 As these planes are of the primest importance in this theory of 
 surfaces of the second order, and as we shall show that they are 
 parallel to the circular sections of the surface (S) in eveiy case, we 
 shall denote them by the symbols (A), (A^), and call them the con- 
 jugate umbilical directrix planes of a surface of the second order. 
 
 It is obvious that as the point ca, the centre of the polarizing 
 sphere {€1), may be taken at the distance D on the other side of O 
 the centre of (S), there will be in general /o«r umbilical planes 
 passing two by two through the directrices of the principal section 
 of (2) in the plane of XY. The lines in which the umbilical direc- 
 trix planes intersect two by two on the plane of XY may be called 
 the polar directrices. 
 
 240.] The inclination i of the umbilical directrix plane may be 
 
 thus found, sini= ~~p"' ^^* "' ^ ^** been shown above, 
 
 — ae 
 e 
 
 be 
 
 is = — 
 
 ae 
 
APPLIED TO SURFACES OF THE SECOND ORDER. 217 
 
 Consequently smt=—, . (a) or tan«i=i^j >--. . (b) 
 
 and cos i=-, a simple expression for the cosine of i. 
 
 Now it is shown in every elementary work on this subject, that 
 the inclination of the planes of the circular sections of an ellip- 
 soid to the trifocal plane is given by the same formula, 
 
 tan«e=y «'^ 
 
 w by 
 
 Since the sum of the squares of any three conjugate diameters 
 is equal to the sum of the squares of the axes, if we put m for the 
 umbilical semidiameter, b and b being the two other conjugate 
 diameters, we shall have 
 
 ««-l-2i«=a« + J2 + c2, or M* = a« + c«-J«. . . (c) 
 
 241 .J The angle which a diameter of (2) passing through an um- 
 bilicus makes with an umbilical tangent plane is thus found. From 
 the centre let fall a perpendicular on the umbilical tangent plane 
 which is parallel to a circular section. Hence, as generally, 
 
 afilP=abc; but in this case a,= bf=b, we shall have P=-r- ; but <^ 
 
 being the angle which the diameter 2u makes with the umbilical 
 tangent plane, 
 
 . , ac , . 
 
 si°9=j^. (^^ 
 
 or 
 
 *^^'^= (aa-&°)(&8_c«) (b) 
 
 There are therefore two cases in which the umbilical diameter is 
 perpendicular to the umbilical directrix plane (A) — ^when a = b, or 
 b=c. 
 
 In the former case the derived surface (2) is an oblate spheroid ; 
 in the latter it is an ellipsoid of revolution round the transverse or 
 major axis. 
 
 It may easily be shown that this umbilical angle ^ is a minimum 
 when the umbilical semidiameter u=6, or when the sphere which 
 passes through the central circular sections of the surface passes 
 also through the umbilicus. 
 
218 GEOMETRICAL THEORY OF RECIPROCAL FOLARS 
 
 ac 
 In this case sin 9 = tj- 
 
 Let M be prolonged to meet the umbilical directrix plane (A) ; 
 
 u 
 its length U measured from the centre C is=- ; for 
 
 - :U : : sin A : sini. Butsini=i-, and sinA=-j— ; 
 e ^ be ^ bu 
 
 therefore 
 
 U=^ (c) 
 
 242.] In every umbilical surface of the second order there are 
 fourfod, independently of the foci of the principal sections. They 
 are, two by two, on the umbilical diameters of the surface, and are 
 also to be found on the vertical which passes through the polar 
 focus CO at right angles to the plane of XY. 
 
 In fig. 47 V and v, on the vertical axis cd Q Q^ and on the umbi- 
 lical semidiameters C U and C U, are the umbilical foci of the 
 surface (2). 
 
 Let 6 be the angle which the umbilical semidiameter u makes 
 with the plane of XY ; then, as 
 
 cos* ^ sin* ^_ 1 
 
 we shall have 
 
 ,03. g_ aV-c*) ^ «*(a*-6*) _a*.« 
 
 Hence 
 
 Cv=ue (a) 
 
 Consequently the point v is the pole of (A) with respect to the 
 surface (S). 
 
 The distance of the umbilical focus v from the plane of XY is 
 thus found. Let this distance be z, then z=ae tan 6. 
 
 But tan^=^j consequently XT =^ (b) 
 
 The segment of this axis z passing through the polar focus, and 
 
 cut oflF by the umbilical directrix plane (A), is r.=— ; 
 
 for cosi=-; consequently tani=:T-, and 
 
 -_/a \^ • be 
 
 .,_^-_«ejtanz=- (c) 
 
APPLIED TO SURFACES OF THE SECOND ORDER. 219 
 
 243.] The ordinate through w in the plane of XY is =— ; but 
 
 be 
 the ordinate through a> in the plane of XZ is = — . 
 
 This follows at once by putting ae for x in the equation -5 + -2= 1- 
 
 26c 
 
 This line — , passing through the polar focus and the two umbi- 
 lical foci, may be called the principal parameter of the polar reci- 
 procal surface (2). 
 
 The segment of the axis of Z cut off by the umbilical plane 
 
 (A) is 
 
 a ^ . ace ac , , 
 
 =- tan*=-^— = 1— (a) 
 
 e ebi) bri ^ ' 
 
 The distance between the points on the axes of X and Z (passing 
 through the centre) in which they are cut by the umbilical plane 
 
 may be thus found: putting H for this distance, CX=-j but 
 
 cos 
 
 i = -; hence 
 
 6 
 
 H=^, (b) 
 
 a simple expression, in which the three eccentricities of the prin- 
 cipal sections are involved. 
 
 The length of the perpendicular from the centre on the umbilical 
 
 directrix plane (A) is P;=- sin i, or 
 
 ^.=E- («) 
 
 The perpendicular P^^ from the polar focus <» on the umbilical 
 dttectrix plane (A) is=-(l— e*) ami; or, putting for sin t its 
 
 value 7-, -^""^ (^) 
 
 In like manner we may find an expression for the perpendicular 
 let fall from the umbilicalfocus v on the umbilical directrix plane (A) . 
 Let this perpendicular be P,j, ; then it may easily be shown that 
 
 ^"'=k (^) 
 
 The perpendicular P„„ let fall from the polar focus a on the tan- 
 
p 
 
 220 GEOMETRICAL THEORY OF RECIPROCAI. POLARS 
 
 gent plane through the polar directrix touching the surface (S) at 
 the extremity of the principal parameter is given by the formula 
 
 The distance of B, the extremity of C, the major axis of (S) from 
 Q> is VC«+D«; but P„„ y/C^+W=W (see fig. 47). Now 
 
 C^=^, and D«=»V^. 
 
 Substituting these values of C and D in the preceding equation, 
 we get 
 
 If we add together these three perpendiculars on the umbilical 
 plane (A), P, from the centre C, P^from the polar focus co, and P„, 
 from V, the umbilical focus, we shall have 
 
 ^'{P,+P„+P„,}=fl«+6«+c* (g) 
 
 244.] Let (D) be the directrix plane of the primitive surface 
 (S) whose major axis is C, and eccentricity 17. Hence the distance 
 of the pole B of (D) (which is parallel to the plane of X Y) from it is 
 
 «8=^. ButC = ^j see (c), [237]: consequently «bS=^ j 
 
 or the extremity B, the pole of (D), coincides with v, the umbilical 
 focus. 
 
 The radius of the circular section of (S) whose plane passes 
 
 through the umbilical focus is equal to the semiparameter — of 
 
 the surface (2). 
 
 For we manifestly have m' : A' : : «*(!— e*) : r*, or 
 
 .-A^(l-e^)=^; (,) 
 
 when e=b, the radius is, as we know, =— in the ellipsoid of revo- 
 lution. 
 
 The radius of the circular section of the surface (2) which passes 
 through the polar focus o> is 
 
 "^ aA~^) (^) 
 
 245 .] In the major axis of (S) there are seven remarkable points— 
 
APPLIED TO SURFACES OP THE SECOND ORDER. 221 
 
 the centre O, the two foci F and F„ the extremities B and B, of the 
 major axis of (S), the points D, D^ in which this major axis meets 
 the directrix planes (D) and (D,). Now these seven points, O, F, 
 F,, B, B;, Dj and D,, all range along the major axis of (S) ; hence the 
 seven reciprocal polar planes will all intersect in the same straight 
 line, the polar directrix of the surface (S), or the directrix of its 
 principal section in the plane of X, Y (see fig. 47) . 
 
 The polar planes of F and F, are the umbilical directrix planes 
 (A) and (A,), passing through the same polar directrix ; the polar 
 planes of the points B and B,, the extremities of the major axis of 
 (S), will touch the reciprocal surface (2) at the extremities Q Q, of 
 its principal parameter, while the polar planes of the points D and 
 D, will pass through the umbilical foci v and v. of the surface 
 
 A tangent plane is drawn to the vertex of (S) cutting the umbi- 
 lical directrix plane (A) in a straight line (t). The plane drawn 
 through this line (/) and the polar focus at will make with the plane 
 
 of XY an angle whose sine is =t- 
 
 The principal parameter of the surface (S) is a mean proportional 
 
 between the parameters of the principal sections in the planes of 
 
 XY and XZ. 
 
 be . . b^ c* 
 
 For — is a mean proportional between — and — 
 a a a 
 
 be 
 246.] Let L = — be the principal semiparameter of the surface 
 
 (S) ; and if we substitute for a, b, c their values as given in terms 
 
 of A, C, D, and R, see [237], (c), we shall find li^-^. This is a 
 
 very remarkable result. The value of L is independent of A and D. 
 Consequently, if we assume the minor axis A of the surface (S) to 
 vary, while the major axis C remains constant, the reciprocal polar 
 surfaces (S), thus generated, will all touch in the same point; and 
 if we further assume A=C, (S) will become a sphere, and (2) will 
 be an ellipsoid of revolution round the transverse axis, so that the 
 series of reciprocal polar surfaces (2) derived from the variation of 
 A in (S) will all touch the ellipsoid of revolution the reciprocal 
 polar of the sphere whose radius is C, and also the tangent plane 
 to all these surfaces which passes through the polar directrix in the 
 plane of XY and touches all the reciprocal surfaces at the extre- 
 mity of their common semiparameter L. This plane may be called 
 the parametral tangent plane. If we assume the polar focus &>, 
 whose distance from the centre of (S) is D, to range along the 
 
222 GEOMETEICIL THEORY OF RECIPROCAL POLARS 
 
 transverse axis, while the semiaxes of (S) continue unchanged, then, 
 as the distance of the vertex of the principal semiparameter from 
 the plane of XY is constant, the vertices of the semiparameters of 
 all the reciprocal polars (S) derived from the variation of D will 
 range along a straight line parallel to the plane of XY, and at the 
 
 distance -^ from it; and the distance of the umbilical focus v 
 
 J. IDS / f^S K 9 
 
 from the plane of XY is — or r<i~ — • ^^ ^^^ ^^ ^ constant 
 
 expression independent of D, the umbilical focus will also range 
 along a straight line parallel to the plane of XY. 
 
 The principal parameter of the elliptic paraboloid is 2 's/kk,, %)c 
 and 2ki being the parameters of the principal sections intersecting 
 in the transverse axis ; and the principal semiparameter is the ordi- 
 nate z through the focus to of the section in the plane of XY. k is 
 greater than k^. 
 
 The coordinates of the umbilicus are 
 
 
 r«-^^*^°d-,=::j-^, (a) 
 
 The length of the umbUical normal from the umbilicus to the 
 
 axis of X is =— • 
 a 
 
 Hence the length of the umbilical normal is equal to that of the 
 principal semiparameter of the surface (S) ; and as the length of the 
 perpendicular from the centre of (S) on the umbilical tangent plane 
 
 is =-T-, consequently the rectangle under the normal and perpen- 
 dicular will be equal to c^, the square of the least semiaxis. 
 
 247.] To find the polar plane of the extremity of the major oasis 
 o/(S). Since this point B is on the major axis of (S), its polar 
 plane will pass through the conjugate polar of the axis of (S) — ^that 
 is, through the polar directrix of the surface (S) ; and as this point 
 is on the tangent plane to (S) at B, its polar plane will meet the 
 surface (S) ; and as this point is on the surface of (S), its polar plane 
 will touch the surface of (2) ; and as the tangent plane at B to (S) 
 is parallel to the plane of XY, which contains the polar focus, the 
 point of contact will be on the perpendicular to the plane of XY 
 passing through the polar focus. (See fig. 47.) 
 
 This point so found on the surface (2) at the extremity of the 
 vertical ordinate passing through the polar focus », is the extremity 
 of L, the principal semiparameter of the surface ; and the tangent 
 plane passing through this point and the polar directrix in the plane 
 
APPLIED TO 8TJRPACES OP THE SECOND ORDER. 333 
 
 of XY is tlie parametral tangent plane to the surface (S), a surface 
 haviiig three unequal axes. 
 
 248.] We may justly infer from the simplicity of the values found 
 for the quantities which determine the umbilical directrix planes, 
 the umbilical foci, as also the lines and points connected with them, 
 that they have a real existence with relation to surfaces of the second 
 order (S) having three unequal axes, as much as the corresponding 
 magnitudes in spheres and surfaces of revolution, especially as the 
 properties of these latter surfaces may be derived from those of (2) 
 by simply changing c into b. 
 
 The reader may desire to see those simple expressions brought 
 together under one view. 
 
 Let a, b, c be the semiaxes of the umbilical surface (S) having 
 
 a*—b^ a^—c^ b^—c^ 
 
 three unequal axes ; and let — ^ — =e*, — j — =e*, and — p — =iy*. 
 
 These may be called the first, second, and third eccentricities of 
 the principal sections of the surface (2) . Let u be the umbilical 
 semidiameter . Then «* = o^ + c* — 6*. 
 
 (a) The perpendictilar from centre on umbilical directrix plane (A) is 
 
 (j3) The perpendicular from centre on umbilical tangent plane is P= -j-. 
 (y) The perpendicular from umbilical focus on umbilical directrix 
 
 plane is P„= ^. 
 
 (y,) The perpendicular from polar focus on umbilical directrix plane 
 
 is P,„ = — • 
 
 (8) Let i be the inclination of umbilical directrix plane to plane of XY, 
 
 . . n 
 
 cost= — 
 
 (e) Inclination fp of umbilical diameter to umbilical plane (A), 
 
 ^^9- {d^-V)(b>-c'y 
 
 (f ) Bistonce of polar focus a from the centre = ae. 
 
 Distance of polar directrix from the centre = 
 
 be 
 (tl) The ordinate pasdng through the polar focus <o is — =L, 
 
 which is called the principal semiparameter of (2). 
 
 (0) The portion of the umbilical diameter « between the centre and 
 
 umbilical directrix plane (A) = -• 
 
224 GEOMETRICAL THEORY OP RECIPROCAL POLARS 
 
 (t) The portion of the ambilical diameter u between the centre and 
 
 the umbilical focus v= ««*• 
 
 (k) Distance of umbilical focus v from the plane of XY, or from a the 
 
 bet/ 
 polar focus = — • 
 
 (X) The portion between the plane of XY and the umbilical plane (A) 
 
 be 
 measured along the semiparameter is = —• 
 
 (ji) Inclination of umbilical diameter u to plane of X Y being ^, tan ^ = ^. 
 
 (i/) Inclination to plane of XY of the tangent plane pasdng through 
 polar directtiz and touching (2) at extremity of principal para- 
 
 ce. 
 
 meter beingj, tany= j- 
 
 _ oe bcn 
 
 (o) The coordinates of the umbilicus are x=-, «= — ■• 
 
 (ir) The length of the umbilical normal = — » 
 
 andia therefore equal to the principal semiparameter L of (S). 
 
 (/>) The radius of circular section of (S) whose plane passes through the 
 
 umbilical focus is = — > 
 
 a 
 
 and is therefore equal to the principal semiparameter L. 
 
 (it) The radius of circular section of (S) whose plane passes 
 
 through the polar focus &> is *"'— ~il — a 1" 
 
 (r) The squared reciprocal of perpendicular from the polar 
 focus u on the tangent plane at the extremity of the 
 
 principal parameter L is jjl ■-^+j- — 5 I. 
 
 (v) Distance between the points on the axes of X and Z in which they 
 
 are cut by the umbilical plane (A) = . ■ — 
 
 eri 
 
 (0) A plane being drawn through the umbilical focus v parallel to 
 
 the plane of XY, the principal axes of this section will be u and — 
 
 a 
 
 (^) The distance of the foot of the umbilical normal fi«m the centre of (2) 
 is aee. 
 
 (a) The distances of the feet of the principal normals to the vertices of the 
 principal sections of (S) passing; through the axis of X are at* and oe* 
 respectively. Therefore the distance of the feet of the three normals 
 from the centre are in geometrical progression, or the distance of the 
 foot of the umbilical normal from the centre is a mean proportion 
 between the distances from the centre of the feet of the normals of the 
 vertices of the principal sections. 
 
APPLIED TO. SUBPACES OP THE SECOND OEDER. 
 
 225 
 
 249.] On the hyfeeboloid op two sheets and its asymptotic 
 
 CONE. 
 
 Fig. 50. 
 
 When to, the centre of the polarizing sphere {€)), is outside the 
 primitiTe surface (S), D is greater than A, and the reciprocal polar 
 (S) becomes a discontinuous hyperboloid, while the reciprocal of 
 the plane curre of contact of the cone whose vertex is a, with 
 
226 GEOMETRICAL THEOKY OP KECI7K0CAL FOLAKS 
 
 the snrface (S) becomes the asymptotic cone of the reciprocal 
 polar (2). 
 
 The cone -whose vertex is at w, and whose base is the common 
 section of (S) with it, is manifestly supplen^ntal to the cone the 
 reciprocal polaa- of this section, whose vertex is at C. The former 
 may be called the ^oZar cone, the latter the asymptotic cone. 
 
 When a cone envelopes a surface of revolution (S), the lines drawn 
 from the vertex of this ctme to the foci of the siuface (S) are called 
 ttie focal lilies* of this cone ; therefore the planes of the circular sec- 
 tions of the supplemental cone are at right angles to these focals ; 
 hence the planes of the circular sections of the asymptotic cone are 
 at right ai^es to these focals : but the umbilical directrix planes 
 (A) and (A,) are also at right angles to these focal lines ; and there- 
 fore the planes of the circular sections of (2) which are parallel to 
 the umbilical directrix planes are also perpendicular to the focal 
 lines of (S) ; consequently the planes of the circular sections of the 
 asymptotic cone are parallel to the circular sections of (S). 
 
 250.] The centre C of the surface (2) may be thus found. Gene- 
 rally, the polar plane of w, the centre of (fl) urith respect to (S), is the 
 polar plane o/C, the centre of'x^) with reject to (il). 
 
 For the plane t r,, fig. 50, is the polar plane of » with respect to 
 (S), while it is the polu: plane of C with respect to (11) . 
 
 The plane of contact of the polar cone with the surface (S) 
 divides it into two segments, the remoter one of which is the reci- 
 procal polar of the sheet of (2), of which to is the focus, while the 
 nearer segment of (S) is the reciprocal polar of the remoter sheet 
 of (2). 
 
 It is worthy also of remark that when to, the polar focus, is within 
 the surface (S), see fig. 48, the point V is the common vertex of 
 two cones circumscribing (S) and (XI). The plane of contact of 
 the former will determine the polar focus to; and the plane of con- 
 tact of the latter with (11) will determine C, the centre of (2). But 
 when to is without the surface (S), the two cones will have a common 
 base, and their vertices will be at the centre and focus of (2). 
 
 251.] The reciprocal polar of the continuous hyperboloid is also a 
 continuous hyperboloid. 
 
 * That the lines drawn from the yeitex of a cone to the foci of a surface of 
 reTolution which it ciicumscrihes are the /ocab of the cone may be thus simply 
 shown. Let/ and/, be the lines drawn from the vertex of the cone to these 
 foci ; let p and p, be the perpendicularg let fall from these foci on the tangent 
 plane to the cone. Let <f> and <f), be the angles which the lines / and / make 
 with the tangent plane. Then we shall evidently have j^, mn ^ din ^, =pp, = V. 
 So also for any other like tangent plane, 
 
 ff^^sai^,=pp,=V; 
 consequently Bin^sin^,=sin^dn^,, or the product of the sines of the angles 
 which/,/ make with any tangent plane to the cone is constant — a well-known 
 property of the /ocafo of a cone. 
 
APPLIED TO SraPACES OF THE SECOND OKDEH. 227 
 
 For if we take the continuous hyperboloid whose principal section 
 in the plane of XY is a circle, toid if we assume in this plane a point 
 «B the centre of the polarizing sphere (il), the reciprocal polar of 
 this circle will be a conic section having its focus at the point <o ; 
 and the conjugate polars of every pair of generatrices of the pri- 
 mitive hyperboloid (U) will be a pair of generatrices on the reci- 
 procal polar (T), a continuous hyperboloid also, whose principal 
 section in the plane of XY will be a conic section. 
 
 When the polar focus to is taken on the circumference of the 
 circle which is the principal section of (U) in the plane of XY, 
 then the reciprocal section of this circle is a parabola, and the polar 
 reciprocal (T) becomes a hyperbolic paraboloid. 
 
 On the oblate spheroid. 
 
 253.] When the polar focus w is assumed to coincide with C the 
 
 Rs R* 
 
 centre of (S), then D=0, and consequently fl=-r-> *~X' ^^^ 
 
 c^>r : and also e=0, e^^rj'= = — 
 
 Since cos i =-= 1, i = Oj in this case therefore the umbilical planes 
 
 are parallel to the plane of XY, the plane of the princijpal circular 
 
 c 
 section, at the distance - from the origin; the polar directrix in 
 
 which the two umbilical directrix planes intersect vanishes to infi- 
 nity, and the two umbilical foci are on the axis of Z, at the distance 
 ce iram. the plane of XY. 
 
 q2 
 
238 
 
 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS 
 
 CHAPTER XXVI. 
 
 ON THE APPLICATION OF THE THEORY DEVELOPED IN THE PRECEDING 
 CHAPTER TO THE DISCUSSION OF SOME THEOREMS AND PROBLEMS. ' 
 
 253.] It will be well to premise the following lemmas before we 
 proceed to apply the principles established in the foregoing pages. 
 
 Lemma I. 
 To express the base A B of the triangle A B C in terms of the 
 perpendiculars on the polars of the points A^ B, and the distance 
 
 ofthe pole ofAB from C, Let the angle AC Bbe =9; letCbethe 
 polarizing centre. Then 
 
 ■ T, CA.CB.sLnd , ,„, R« Rs r« 
 
 AB= j^ ; but CA=— , CB=— , and CD=— ; 
 
 Ui» P P, r 
 
 consequently 
 
 AB= 
 
 R'rsin^ 
 
 m 
 
 Lemma II. 
 
 From a given point R let a perpendicular R P be let fall on the 
 given siraight line CV, and let the point O be assumed as the 
 centre of the polarizing circle whose radius is R. 
 
 Let B Q be the polar of the point R, and S the pole of the given 
 
APPLIED TO THEOREMS AND PROBLEMS. 
 
 229 
 
 straight line C V. From S let fall a perpendicular S T on the polar 
 of B.. It may easily be shown that the ratio of the lines drawn to 
 the poles R, S from the centre O of the polarizing circle will be the 
 same as the ratio of the perpendiculars let fall from these points on 
 the polar straight lines ; for 
 
 OR_R^ . R^OC OV OR-OV RV RP OR_OS 
 OS OB • OC~OB~OQ~OS-OQ~SQ~ST °^ B,T~ ST 
 
 Pig. 52. 
 
 
 The proposition is equally true when C V P and T Q B are planes. 
 For through O R P draw a plane, and through C Y and Q B planes 
 perpendicular to the plane O R P, the above demonstration will 
 still hold. 
 
 ' 254.] If from any point on an umbilical surface of the second order 
 perpendiculars be let fall on two conjugate umbilical directrix planes, 
 the rectangle under these perpendiculars will have to the square of 
 the distance of this point from, the polar focus a constant ratio. 
 
 It has been shown by M. Chasles and other geometers — ^indeed 
 it follows obviously from the corresponding property in piano — ^that 
 if from the foci of an ellipsoid of revolution perpendiculars be let 
 fall on a tangent plane, their product will be equal to the square 
 of the semi- minor axis of the ellipsoid. 
 
 Let the reciprocal polar of this property be taken. The pole of 
 the tangent plane (T) to (S) will be a point Q on (2) ; the polar 
 planes of the two foci F, F, of (S) will be the conjugate umbilical 
 directrix planes (A), (A^) to (S) ; hence the perpendiculars are let 
 fall from one point Q on the surface (S) to its two directrix planes 
 (A) and (A,). 
 
 But the distance of the focus of (S) from the polar centre w is 
 
230 
 
 THE GEOMETRICAL THEORY OF RECIFROCAI. POLAR8 
 
 VC+D*— A', see sec. [238]; and the product of the perpen- 
 diculars is^ as we have assumed, equal tb A^. 
 
 Fig. 53. 
 
 Consequently the ratio of the product of the two perpendiculars 
 from the foci of (S) on a tangent plane to it, to the product of 
 the distances of the two foci of (S) from the polar centre, is 
 _ A« 
 
 C«+ D«-A«" 
 
 But from the preceding Lemma II. it follows that this must also 
 he the ratio of the product of two perpendiculars let fall froia the 
 point Q on the directrix planes (A), (A,) to the square of the radius 
 r drawn from Q, to the polar focus oi. Putting/? and ^, for these 
 perpendiculars, we shall have 
 
 ^_. 
 
 A« 
 
 (a) 
 
 C«+D«-A« 
 
 We may determine the value of this expression in terms of a, 4, c 
 by the help of the formulae in sec. [237], (c) ; hence 
 
 A* 
 
 fl'C' 
 
 ,9-9 
 
 a'c 
 
 Now it has been shown in sec. [243], (c), that the perpendicular 
 from the centre on one of the conjugate umbilical directrix planes is 
 ac 
 
 be 
 
 Let this perpendicular be P. 
 
 Then we shall have, finally, ^=i 
 
 ps 
 
 (b) 
 
AFPLIEB TO THEOKEMS AND PROBLEMS. 231 
 
 (o) Restuning the equation ^= .,°/ ,. , let the surface be 
 
 the elliptic paraboloid; let the semiparameters of the principal 
 sections be 2k and 2k„ then A* =2ak, c^=2ak,. Substituting these 
 values in the preceding expression, we shall have 
 
 since a in this case is infinite. 
 
 Therefore in the elliptic paraboloid ^=-t (c) 
 
 (/8) When b=c the surface (2) becomes an ellipsoid of revolution 
 
 t^c^ a* 1 
 
 round the transverse axis, and rsi-o « becomes -, — ro=-5, a 
 
 o*(o*— c*) a^—b^ c* 
 
 well-known property of the ellipsoid of revolution. 
 
 (y) From the polar focus en let a sphere be described with a 
 
 constant radius r, it will cut the ellipsoid (2) in a curve of double 
 
 curvature ; so that if from any point on this curve perpendiculars 
 
 be let fall on the conjugate umbilical directrix planes, thieir product 
 
 will be constant, since r is constant. 
 
 (S) Ifwe assume the theorem as established, ^'=797-5 — sr, and 
 
 make this latter expression =1, we shall have c= — • 
 
 Hence it follows that if the three semiaxes of an ellipsoid be 
 
 a, b, and — , the product of the pair of perpendiculars let 
 
 fall from any point on the surface to the two umbilical directrix 
 planes will be equal to the square of the distance of this point from 
 the polar focus. 
 
 (e) The above relation is equivalent to -j + yg — j=0. When b is 
 
 also =c, -5=0, or ffl=QO ; or in order that this property may 
 hold in a surface of revolution, it must be a paraboloid*. 
 
 * A rample algebmcal proof may be ^ven of this very general and important 
 theorem. 
 
 It has been shown in eec. [S7], (e), that the length of a perpendicular let fall on 
 ft plane whose tangential coordmates are £, v,- i, from a g^ven point (x, y^), is 
 
 Now the tangential coordinates of one of the conjugate directrix planes (A) are 
 
232 
 
 THE GEOUETBICAL THEOKT OF SXCIFROCAL FOLA.BS 
 
 5,4 
 
 255.] When a=b, D=0, since D* =-75- (a*— 6*) ; therefore the 
 
 polar focus w becomes the centre of (S), which becomes an oblate 
 spheroid having its principal circular sections coincident and in the 
 plane of XY. Consequently the four umbilical directrix planes 
 coalesce, two by two, parallel to the plane of XY, and therefore 
 parallel to one another. These planes may be called the minor 
 directrice planes, seeing that they are perpendicular to the minor 
 axis of the oblate spheroid. The perpendiculars let fall &om any 
 
 Fig. 54. 
 
 6 
 
 
 / 
 
 -\ 
 
 y^ / ]B 
 
 V 
 
 CO 
 
 ^ U> 
 
 point on this surface on the minor directrix planes are evidently in 
 the same straight line. Assuming the genial equation 
 
 j>p,_ a^(? 
 
 pp,_ c« 
 
 or--v= 
 
 D8xD8,_A* 
 
 Substituting these values in the preceding expression, 
 
 In the same way we shall find for the perpendicular let fall on the other con- 
 jugate directrix plane, 
 
 _ Vc-\racex-\-dbifs 
 
 Hence 
 
 The equation of the surface, the origin being at the focus in the plane of XY, 
 or at the polar focus, when multiplied by rfj^c*, is 
 
 i»<r'x'+oVjr'+o»6's?-26>e»a«r-6V=0 ; 
 adding this expression to the preceding, we get 
 
 PP, _ oV 
 x'+y'+z' V(a'-c'y 
 
APFLIEI) TO THEOBEHS AND FBOBLEHS. 
 
 233 
 
 Now - is the distance of the minor directrix plane &om the 
 
 centre =/^ suppose; consequently ^'=-j. 
 
 When c^==a^—c'^, or 2<^=a*, we shall have^,=r'. 
 256.] The sum of the products, taken two by two, of the perpen- 
 diculars let fall from the fow umbilical foci on a tangent plane to 
 
 (S) is equal to 2fi* sin* vH =- cos* v. 
 
 We have shown in sec. [57], (c), that if P be the perpendicular 
 let fall firom a point whose projective coordinates are x, y, z on ti 
 plane whose tangential coordinates are i, v, ^, we shall have 
 
 P= 
 
 _ 1— a?g— yw— .gg 
 
 Now the projective coordinates of the four umbilical foci are 
 
 x=—ae, 
 
 y= 0. 
 bcT) 
 T' 
 
 z= 
 
 x= 
 
 ae, " 
 
 y= 
 
 0, 
 
 Zss- 
 
 bet) 
 
 a'} 
 
 x=—ae, 
 y= 0, 
 
 ben 
 z— — —^. 
 
 x=ae, 
 
 y=o, 
 
 bcq 
 
 z=- 
 
 a I 
 
 Consequently the perpendiculars are 
 
 _ a + a*eg— &c<7 a—cP^e^-Yboi) 
 
 Vr + v« + P' " *" Vf' + w'+f' 
 a + a^e^ + b'^CT) a—a^e^— bctji;, 
 
 "TP- V£* + «* + ?'' ''^'- -v/f* + „* + ?^ • 
 
 (a) 
 
 (b) 
 
234 THE QEOHETBICAI. THEORY OP RECTFBOCAL POLARS 
 
 Therefore a<'¥T?,+a^pp,=: p+J' + ^ ' ' " ^^^ 
 
 and if we add to the numerator of this fraction the tangential 
 equation of the surface multiplied by 2a^, 
 
 2a*P + 2a*A V + 2a V5« - 2a« = 0, 
 we shall finally, on reducing, have 
 
 PP,+i»p,=2[*«Bin«i'+^cos«K] (d) 
 
 In this formula u is the umbilical semidiameter, and v is the 
 angle which the parallel perpendiculars make with the axis of Z. 
 When i=:c, u=a, P=/>, ^^p„ and the formula becomes PP,=d*, 
 a well-known expression for the product of the perpendiculars from 
 the foci of a sur&ce of revolution on a tai^ent plane. It is remark- 
 able that the sum of the products of these perpendiculars varies 
 only with the value of v, and is entirely independent of the values 
 of \ and fi; X, it, and v being the angles which the perpendicular 
 from the centre on the tasogent plane makes with the axis of coor- 
 dinates. Consequently, if round the axis of Z passing through the 
 centre of (S) we describe a right cone whose vertical angle is 2v, 
 and the four perpendiculars from the umbilical foci on a tangent 
 plane be all drawn parallel to a side of this central cone, the sum 
 of their products, two by two, will be constant, since v is constant. 
 See (d). 
 
 257.] The locus of the feet of perpendiculars let fall from the 
 foci of (S), a surface of revolution, on a tangent pluie to this sur- 
 face is a sphere. 
 
 Taking the reciprocal polar of this property, we may infer that if 
 through the polar focus of (S), a surface of the second order, a plane 
 and a straight line be drawn at right angles to each other, the line 
 meeting the surface in the point r, and the umbilical directrix 
 plane (A) in the straight line (S), the plane which passes through 
 the point r and the straight line (S) will envelop a surface of revo- 
 lution (2^) whose focus will coincide with the polar focus of (S), 
 and whose directrix plane will pass through the polar directrix of 
 tiie polar focus a> in the principal section in the plane of XY. 
 
 Let (T) be the tangent plane to (S) ; and let cr, v, be the feet 
 of the perpendiculars let fall from the foci F, F, on the tangent 
 plane (T), and let (11), the polar plane of «r, cut the umbilical 
 directrix plane in the sliraight line (8). 
 
 Now the pole t of (T) the tangent plane to (S), will be a point on 
 the surfiace (2) ; and as «r is a point on the tangent plane (T), its 
 polar plane (11) will pass through t, the pole of (T) ; and as «r is a point 
 on one of the focal perpendioilars, the polar plane (11) of «- will 
 pass through the conjugate polar of this perpendicular, which is 
 
APPLIED TO THEOREMS AND PROBLEMS. 235 
 
 the intersection of the planes (11) and (A), the polar planes of ■sr 
 and F. But as the point vr is always found on the surface of a 
 sphere, its polar plane (S)t must envelop a surface of revolution 
 (S() whose focus coincides with a, the polar focus. Since w^, the 
 foot of the other perpendicular, is also on the tangent plane (T) to 
 (S), its polar plane (11^) will also pass through r and intersect the 
 conjugate umbilical directrix plane (A,) in (S,) ; and as the plane 
 which passes through (8^) and a> is also perpendicular to the line 
 <oT, it will follow that the two planes through at and the lines (5), 
 (8j) are identical. 
 
 When (S) becomes an oblate spheroid, the polar focus a becomes 
 the centre of (2) ; and hence follows this theorem : — 
 
 Through the centre to of an oblate spheroid let a diameter and a 
 diametral plane be drawn at right angles one to the other, the dia- 
 meter meeting the surface (S) in the point r, while the diametral 
 plane meets the directrix plane (A) in the straight line (S), the plane 
 r{S) will envelop a concentric sphere whose radius is c. 
 
 258.] The algebraical proof of this theorem is somewhat com- 
 plicated. To afford a means of comparing it with the geometrical 
 method, it may be thus given. 
 
 The origin being taken at the polar focus, and the axes of coor- 
 dinates parallel to the axes of the figure, let the projective equation 
 of the surface be 
 
 a^ y"^ z^ 2ea?_6*. 
 flS + ^a+cS a ~a^' 
 
 or if {Xf y, z,) be the projective coordinates of the point on the sur- 
 face, and we multiply the preceding equation by a%^<?, we shall 
 have 
 
 i«c«ar«+o«cV + «**W=**'^ + 2**c2aca?,. ... (a) 
 
 • Let (=fi^+a, i;=v?+/3 (b) 
 
 be the tangential equations of the straight line in which the planes 
 (11) and (A) intersect ; then, as this straight line (S) lies in a plane 
 passing through the origin, we shall have 
 
 us=f =-', and v=^=-; see sec. [59]. 
 ? ^/ ? ^/ 
 
 Consequently zJ^=xl^-\-aZp and z^=yfi+fiz,. 
 
 Now, as the line (b) lies in the umbilical directrix plane (A), of 
 which the coordinates are 
 
 ^ ae n y "^ 
 
 substituting these values of ^, v, and ^ in (b) and reducing, we 
 shall find 
 
 xjb{bc^—ari) =j?,c(4'f +ae), 
 and y, {bc^ — aij) = zphv. 
 
 } (c) 
 
236 THE OEOMETBICAL THEORY OF RECIPKOCAL POLABS 
 
 As the plane^ whose tangential equations are required, passes 
 through the point of which the projective coordinates are x^ y, z^, 
 and whose tangential coordinates are 0, v, ^, we must have 
 
 xS+y,vJrz,};=\ (d) 
 
 Now substituting in this equation the values of x, y, z, given in 
 the preceding equations, we shall have, for the values of z,, x,, and y,, 
 
 b{bc);-av) ^ 
 
 ^ c{b^^+ae) 
 
 '~b^c{^+v^ + ^)+ ace^ - abr,\^ 
 
 b^ev 
 
 ^'~ A*c(f + „2 + 5«) + ace^-abv^- ) 
 
 (e) 
 
 If we now substitute these values of Xi y, z, in (a), the projective 
 equation of the surface (S), we shall have, first writing 
 
 V=A«c«(f +«« + ?«) + 2c«flef-aV-c«, . . . (f) 
 and making these substitutions, for the resulting equation 
 
 6V|« + 26Vacf + c^a^e^ + a%*b^v^ + a^l^c^i? - 2^c€^^+ a*b^v^ ^ 
 = 62 V« + 262V (o2»j« + c2 - c'flef - abcr/^) + ffl*6 V + 2o«6 Vi;« + 6 V 
 -262cV7;2c|:-26*c*ac?+o26Ve«f'-2o863«j3?-268cSa^5 
 + 2a26»c8c77|? + a^*c^v^t? + 2Y b^c^ae^ + 2b^c^a?en^ 
 +U^c*aei - 26Vo«e2^ - Z1^<Pa^eri{l;+ 2Y a^c^e^ + 2o*c«c V 
 +2a2c*c«-2a8c*cSf-2a86c»c'i7f. 
 
 Eliminating, combining, and reducing, the preceding equation 
 becomea 
 
 Y{2bci,K-c^e^-a^)=0; (h) 
 
 and this is satisfied hj putting Y==0, or 
 
 6V(P + i/2+^)+2ac2ef=aV+c*. . . . (i) 
 If we refer to sec. [105], (i), and apply the principles there laid 
 down, we shall find, putting A and B for the principal semiaxes of 
 this surface, 
 
 . abc J _o **c« 
 A= - a . — -. and B*=-5-5 s; . . . . fil 
 
 B' be 
 consequently -^=— , or the principal parameters of the enveloped 
 
 surface (S,) and the reciprocal polar (2) of (S) are equal. 
 
 Let e be the eccentricity of this enveloped surface of revolution, 
 we shall find 
 
 c e* €« )i2 
 ^=A"'°'T*=p-^2, (k) 
 
 (g) 
 
APPLIED TO THEOREMS AND PROBLEMS. 
 
 237 
 
 e being the eccentricity of the principal section of {%) in the plane 
 
 When a=A, or the surface is an oblate spheroid, A=B, or the 
 enveloped surface is a sphere. 
 
 259.] Through a given straight line four planes are drawn, two 
 as tangents to a surface of revolution (S) ; the other two are drawn 
 through the foci of this surface ; it is known that the angles between 
 each tangent plane and focal plane are equal. 
 
 The conjugate polar of the straight hne in which the four planes 
 intersect will be a chord of the surface (Si) on which the poles of 
 these four planes will lie ; and as two of the planes pass through 
 the foci of (S), their poles S and B, will lie on the two umbilical 
 directrix plianes (A) and (A^) ; and as the two other planes are tan- 
 gent planes to (S), their poles t and r, will be on the surface of (2). 
 
 Hence, if in any surface (2) having three unequal axes a straight 
 line be drawn meeting the surface in two points r and r, and the 
 umhilical directrix planes (A) and (Ay) in two points also, S and S„ 
 the armies which each pair of points subtend at the polar focus <a will 
 be equal to each other, or the angle toS will be equal to the angle TiwS,. 
 
 Fig. 56. 
 
 When the secant line rr^S, becomes a tangent to the surface (2), 
 
238 THE GEOUETBICAL THEOBT OF BECIFKOCAL lOLAKS 
 
 the two points t and t, coalesce into one t, and we obtain this other 
 remarkable theorem : — 
 
 If toe draw any straight line touching the surface (S) and meeting 
 the umbilical directrix planes (A) and (A,) in two points 8 and S„ the 
 segments of this line between the point of contact t and the umbilical 
 directrix planes will subtend equal angles at the polar focus to. 
 
 When (S) becomes an oblate spheroid, the umbilical directrix 
 planes (A) and (A^) become parallel to the plane of XY, which 
 contains the principal circular section of (S). The polar focus o> 
 coincides with its centre. Therefore, if a straight line be drawn 
 touching an oblate spheroid, the segments of this line between the 
 point of contact and the directrix planes (A) and (A^) will subtend 
 equal angles at the centre of (X). 
 
 260.] Should the secant line be drawn parallel to one of the 
 umbilical directrix planes, (A) suppose, and meeting the other um- 
 bilical directrix plane (A,) in S, and the surface in the points t and t,, 
 the rectangle 
 
 Stx8t,=So»*, (a) 
 
 being the polar focus. See 6g. 57. 
 
 Hence this theorem : — If through any point S assumed on one of 
 the directrix planes (A^) a secant be drawn parallel to the other 
 directrix plane (A), and meeting the surface in the points t and r,, 
 the rectangle under these segments will be equal to the square of the 
 distance of h from, the polar focus <o. 
 
 Fig. 57. 
 
 If the secant be drawn through the umbilicus, it will become a 
 
APPLIED TO THEOREMS AND PROBLEMS. 239 
 
 tangent to the surface. Consequently if through u the umbilicus 
 of the surface (2) a tangent be drawn to meet the umbilical direc- 
 trix plane (A,) in S„ the distances of the point S, &om the umbilicus 
 u and the polar focus to will be equal. 
 
 A direct geometrical proof of this theorem may be given. Let 
 WTT, be a section of (2), a surface of the second order having three 
 unequal axes. Let UA and UA, be the umbilical directrix planes 
 cutting the plane of XY in the polar directrix UV. Let un be a 
 tangent plane through the umbilicus u parallel to (A) and cutting 
 (A,) in the line nS,. Let the secant t t, be drawn parallel to the 
 plane (A), and therefore meeting this plane in 8^,, a point at infinity, 
 and the other umbilical plane (Ay) in the point S. 
 
 Now, as by the preceding part of the proposition the lines aS and 
 mSfi drawn to meet the secant tt^SS, in the umbilical directrix planes 
 (A) and (A,) make equal angles with the lines tor and ar,, we shall 
 have the angle Sq>t= angle St/uj or the triangles Sa>T and Stor, are 
 similar, or St : Sta : : So) : 8t,, or STxSTy=So>* (b) 
 
 When the secant tt^SS^ is drawn in the umbilical tangent plane 
 uSfl, then 8t x Stj evidently becomes nu^ ; but it is also equal to 
 n«i) ; hence nu=na) (c) 
 
 We may also from this theorem conclude that when any point S 
 is taken on one of the umbilical directrix planes of a cone, i. e. the 
 planes drawn through the vertex parallel to the circular sections of 
 the cone, and from this point a line be drawn parallel to the other 
 circular section, and meeting the cone in the points t and t„ the 
 rectangle St x St^ will be equal to SV*, V being the vertex of the 
 cone. 
 
 261.J Tangent planes (T) and (T,) are drawn at the extremities 
 of the major axis of (S). They are cut by a third tangent plane 
 (T„) in two straight lines (/) and (Z,). The planes (V) and (V,) 
 passing through the focus F and the lines (/) , (/,) are at right angles, 
 as is very generally known. We may thus polarize this theorem. 
 
 The poles of the tangent planes (T), (T,) to the extremities of the 
 major axis of (S) are the extremities t, T; of the principal parameter 
 L of (2). The pole of the third tangent plane (TJ is a point T^jOn 
 the surface of (2) ; and S, the pole of (V) , is a point on the umbilical 
 directrix plane (A). But as the three planes (T), {T„), and (V) all 
 pass through the same straight line {I), their poles t, t,,, and 8 will 
 all range on the same straight line t„ t S, the conjugate polar of (Z) . 
 In the same way the poles t,,, t,, and S, of the three planes (T,), 
 (T„), and (Vy) will all range on the same straight line r,,, r,, and o„ 
 the conjugate polar of [l,) . But as the focal planes are at right 
 angles, their poles B and S, on the umbilical plane (A) wiU sub- 
 tend right angles at the polar focus to. Hence we obtain this very 
 general theorem : — 
 
240 
 
 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS 
 
 (a) Let the principal parameter CC, of a surface of the second 
 order (S) having three unequal axes be the base of a triangle whose 
 vertex is the point G anywise on the surface ; and if the sides of this 
 triangle be produced to meet the umbilical directrix plane (A) in two 
 points S and S,, the points S and S, will subtend a right angle at the 
 polar focus a. 
 
 Fig. 58. 
 
 (/3) When the surface (S) becomes an oblate spheroid, the um- 
 bilical directrix planes (A) and (A,) become paridlel to the plane 
 of XY, which contains the principal circular section of the oblate 
 spheroid, and the polar focus coincides with the centre. 
 
 Hence this theorem, see fig. 54 : — 
 
 If the shorter axis of an oblate spheroid be taken as the fixed base 
 of a triangle C CjP inscribed in the spheroid, whose sides are pro- 
 duced to meet one of the directrix planes (A) in two points Q and Q,, 
 these points, Q, and Q,, loiU subtend a right angle at the centre of the 
 oblate spheroid {%). 
 
 (7) When the surface (S) becomes a surface of revolution (S) 
 round the transverse axis, the polar focus becomes ihe focus of (S), 
 the umbilical directrix plane becomes the directrix plane (D), 
 whence we may derive this theorem : — 
 
 If the parameter of a surface of revolution (S) be the beue of a 
 triangle inscribed in it, and the sides be produced to meet the direc- 
 trix plane (D) in d and d,, the points d, d, will subtend a right angle 
 at the focus F of (S). 
 
 (S) We maj extend this theorem to any angle. If two fixed 
 tangent planes (T) and (T,) be drawn to a surface of revolution (S), 
 and a third tangent plane (T,,) movable and which intersects the 
 two fixed tangent planes in the straight lines (Q and (/,), the vector 
 planes (Y) and (Y,) drawn through the focus F and the straight 
 
APPLIED TO THEOREMS AND PROBLEMS. 
 
 241 
 
 lines in which the tangent planes intersect are inclined at a constant 
 angle to each other. This is a known theorem. 
 
 The poles of the vector planes (V) and (V,), since they pass 
 through F, are on the umbilical directrix plane (A); and as the 
 planes (V), (T), and (TJ aU intersect in the same straight line (Q, 
 their poles will be on the conjugate polar of this straight line — that 
 is, on the chord which joins the poles t and t,^ on the surface (S) ; 
 and as the pole of (Y) is on this straight line, it must be the point 
 S where this line tt„ pierces the umbilical directrix plane (A). 
 The same may be said of the other chord t„ t^ S,. Therefore the 
 angle Sq>S,, the angle between the plane (V) and (V,), is constant. 
 We may therefore infer that. 
 
 If a fixed chard be taken in a surface (S), and this fixed chord 
 c c, be made the base of a triangle whose vertex G (see fig. 58,' is any- 
 where on this surface, the sides of this triangle Gc, Gc, being produced 
 to meet the umbilical plane (A) in two points S and S^, the points 8 
 and Si will subtend a constant angle at the polar focus a. 
 
 262.] If we take the reciprocal polar of (7) in the preceding 
 section, we may derive another theorem equally new. 
 
 263.] Since the two points t and t,, the extremities of the para- 
 meter of (S), and the focus F w; kq 
 of (S) in fig. 59 are aU three on *^S- »«• 
 the same straight line, their 
 polar planes (0), (©,), and (A) 
 wiU all meet in the same 
 straight line (S) ; but the two 
 former planes are tangent 
 planes to (2), while the latter 
 is the umbilical plane (A). 
 Since d is in the directrix 
 plane (D) of (S), the polar 
 plane of d will pass through 
 the umbilical focus v, the pole 
 of (D), in (2) ; and as </ is a 
 point on the chord of contact, 
 /, t„ the polar plane of d will 
 pass through the intersection 
 (S^) of the two tangent planes 
 (0) and (©„) to (2). Hence 
 the polar plane of the point d 
 in the directrix plane (D) to 
 (S) is the plane which passes 
 through V, the umbilical focus 
 of (2), and the straight line 
 (d.) the intersection of the 
 tangent planes {&) and (©„) to(2); and therefore the conjugate polar 
 
 d 
 
 
 (D) d 
 
 \ 
 
 
 / ^ ^ 
 
 
 
 
242 THE GEOMETRICAL THEOEY OF KECIPROCAL POLARS 
 
 of the line F</ will be the intersection of this plane with the umbilical 
 directrix plane (A) of (2). In the same way it may be shown that 
 the conjugate polar of Fd, is the line in which the plane through v 
 and the intersection (:&„) of the two tangent planes (0,) and (0„) 
 cuts the umbilical directrix plane (A). But the lines ¥d and Yd, 
 are at right angles ; therefore the planes passing through the polar 
 focus to and their conjugate polars are at right angles also, whence 
 we may derive the following theorem : — 
 
 If through a straight line (■&) in the umbilical directrix plane (A) 
 of a surface (2) of the second order, two tangent planes (0) and (0^) 
 be drawn meeting a third tangent plane (0„) in the straight lines 
 {^i) and (S„), the planes through these two lines and the umbilical 
 focus V will cut the umbilical directrix plane (A) in two straight 
 lines, through which and the polar focus w if two planes be drawn, 
 they will be at right angles to each other. 
 
 264.] The vector plane (V) drawn through the focus F, and the 
 line of intersection (/) of two tangent planes (T) and (T^) to a sur- 
 face of revolution (S), is at right angles to the line drawn from the 
 focus F to the point d in which the chord of contact t, t, of the 
 two tangent planes (T) and (T,) meets the directrix plane (D) of (S) . 
 
 Let us reciprocate this property. 
 
 Since the three planes (T), (T,), and (V) all intersect in the same 
 straight line (/) , their poles t, t,, the points of contact of two tangent 
 planes (0) and (©,) to (2), and the point v, the pole of (V), will 
 all range on the same straight line ; and as this latter plane (V) 
 passes through the focus F, its pole v will be on the umbilical 
 directrix plane (A) . 
 
 Again, as «f is a point on the directrix plane (D) of (S),its polar 
 plane will pass through v, the umbilical focus ; and aa dia a, point 
 on the chord of contact t, t,, its polar plane will pass through the 
 conjugate polar of /, t,, namely (3) the intersection of the tangent 
 planes (0) and (0^) ; and as fif is a point on the focal line Yd, its 
 conjugate polar (8) will lie in the umbilical directrix plane (A) ; 
 consequently the conjugate polar of Yd is the line in which the 
 plane that passes through v and [b) meets the umbilical directrix 
 plane (A). But as Yd is at right angles to the plane (V), the vector 
 line <uv will be at right angles to the plane to (S) . See last figure. 
 
 Hence this theorem : — 
 
 If two tangent planes (0) and (0^) are draum to a surface of the 
 second order (2), having three unequal axes, meeting in the straight 
 line (3), and if r and t^ be the points of contact of these tangent planes, 
 let the chord r t, meet the umbilical directrix plane (A) in v ; and let 
 the plane drawn through v, the umbilical focus, and (d) the intersection 
 of the tangent planes (0) and (0,) mset the umbilical directrix plane 
 (A) in the straight line (8), the line tav will be at right angles to the 
 plane drawn through m, the polar focus and (8). 
 
APPLIED TO THEOREMS AND PROBLEMS. 243 
 
 265.] If the surface become an oblate spheroid, then the polar 
 focus to will coincide with the centre of this surface (2), and its 
 umbilical directrix plane (A) will be parallel to the plane of XY 
 which contains the principal circular section of (2). Therefore 
 the preceding theorem may be thus modified : — 
 
 If two tangent planes (@) and (@,) be drawn to an oblate spheroid, 
 and their chord of contact be produced to meet the directrix plane 
 in V, and if a plane be drawn through the umbilical focus v and the 
 line (d) in which these two tangent planes intersect and meeting the 
 directrix plane in the line {8), the diametral plane C(S) will be at 
 right angles to the diameter Cv. 
 
 266.] Let a tangent plane (T) be drawn to a surface of revolu- 
 tion (S) touching it in the point t, and cutting the directrix plane 
 (D) in {d) . The vector plane (V) drawn through the focus F and 
 the straight line (d) will be perpendicular to the focal line drawn 
 from F to t. Let us take the reciprocal polar of this theorem. 
 
 Since the three planes (D), (T), and (V) all pass through the 
 same straight line {d), their poles wUl be on the straight Une (S), 
 the conjugate polar of {d) ; and as v is the pole of (D), (S) will pass 
 through u. As the pole of (T) will be a point t on (2), (8) will 
 pass through this point j and as (V) passes through F, its pole wOl 
 be the intersection of (S) with the umbilical directrix plane (A) . 
 Again, the conjugate polar of the line joining F and t will be the 
 straight line in which the tangent plane (©) to (2) intersects the 
 directrix plane (A) ; but the line drawn from the polar focus m to 
 the pole S of (V) will be perpendicular to the plane passing through 
 the intersection of the tangent plane (0) with (A). 
 
 Hence this theorem : — 
 
 ijTa tangent plane (0) be drawn to (2), touching it in the point r, 
 an(l cutting the umbilical directrix plane in the line (d), and if through 
 the umbilical focus v and the point t a straight line be drawn meetirig 
 the directrix (A) in the point S, the line «aS drawn from the polar 
 focus o> toB will be at right angles to the plane draionfrom ta through 
 the straight line (d). 
 
 When the surface becomes an oblate spheroid, the resulting 
 theorem is as follows : — 
 
 If a tangent plane be drawn to an oblate spheroid cutting one of 
 the directrix planes (A) in a straight line (d), the diametral plane 
 drawn through (3) will be at right angles to the diameter Or drawn 
 through the point of contact t. 
 
 267.] Two tangent planes, (T) and (T,), are drawn to the primitive 
 surface (S). The focal vectors (/) and (/,) drawn from F to the 
 points of contact t and t, are equally inclined to the vector plane 
 (V) drawn through the focus F and the intersection (/) of the tan- 
 gent planes (T) and (T). 
 
 Now, since the vector plane (V) passes through F, its pole will 
 
 R 2 
 
244 THE GEOMETRICAL THEORY OF RECIPROCAL POLARS 
 
 be on fhe directrix plane (A) ; and as it passes through the straight 
 line {[), its pole will be on the polar of (/)— that is, the chord t t, 
 ■which joins the points of contact of the tangent planes (0) and {&,) 
 to (2), which are the polar planes of the points t and tj — and there- 
 fore is the poiut \ in which the directrix plane (A) is pierced by 
 the line t t,. The conjugate polar of the line F^ is the straight line 
 in which the tangent plane (©) cuts the directrix plane (A) . So 
 also for Ft,. The resulting theorem is therefore as follows : — 
 
 Let two tangent planes, (€)) and {&,), be drawn to a surface (2) of 
 three unequal axes, cutting the umbilical planes in two straight lines 
 (8) and (SJ. Let a straight line be drawn through the points of con- 
 tact meeting the directrix plane (A) in X. !/%e line drawn from the 
 polar focus a to X is equally inclined to the planes a{S) and <»(8j).' 
 
 When the surface becomes an oblate spheroid, the foregoing 
 theorem is thus modified: — 
 
 If two tangent planes are drawn to an oblate spheroid, and cutting 
 the directrix plane (A) in two straight lines (8) and{S,), while the 
 chord through the points of contact meets it in \, the diametral planes 
 0(8) and 0(8,) are equally inclined to the diameter OX. 
 
 268.] Two tangent planes, (T) and (T,), being drawn to the pri- 
 mitive surface (S), the focal vector lines (/) and (/,) drawn to the 
 points of contact t and t, are equally inclined to the focal vector/^, 
 drawn to the point d where the chord of contact 1 1, meets the direc- 
 trix plane (D) of (S) . 
 
 The conjugate polars of the focal vectors (/) and (/,) are the 
 straight lines (j) and (j^, in which two tangent planes (€)) and 
 (@,) to the polar surface (2) intersect the umbilical directrix plane 
 (A) ; and as the point d in which the chord of contact 1 1, pierces the 
 directrix plane (D) is on this plane (D), its polar plane (U) will 
 pass through v ; and as this point d is on the chord 1 1„ its polar plane 
 (U) will pass through (d), the intersection of the tangent planes 
 (ft) and {&,). Hence the conjugate polar of (/„) is the straight 
 line in which this plane (U) intersects (A) . 
 Hence the following theorem : — 
 
 If two tangent planes, (©) and {&,), are drawn to a surface (2) 
 meeting the directrix plane (A) in two straight lines (8) and {S,), and 
 if through the intersection of these tangent planes {d) and the umbi- 
 lical focus V a plane (U) be drawn cutting the directrix plane (A) in 
 a straight line {x)> the planes 0(8) and to{Si) will be equally inclined 
 to the plane a)(x). 
 
 When the surface becomes an oblate spheroid, the theorem is 
 thus modified : — 
 
 When two tangent planes to an oblate spheroid are drawn meeting 
 in the line (A) , and the directrix plane (A) in the straight lines (8) and 
 (8() — and if a plane be drawn through the umbilical foctis v and the 
 intersection (3) of the tangent planes, and cutting the directrix plane 
 
APPLIED TO THEOREMS AND PKOBLEMS. 245 
 
 (A) in the straight line (x), the diametral planes 0(S) and 0(8,) 
 will make equal angles with the diametral plane «i>(x)> 
 
 269.] Two tangent planes, (T) and (T,), are drawn touching the 
 primitive surface (S). From any point p in the line [l) of their 
 intersection two tangents to (S) are drawn touching the surface (S) 
 in the points of contact t and t,. The vector planes (V), (V,) through 
 the focus F and these tangents^ / and p t, will be equally inclined 
 to the vector plane (V,,) which pssses through F and {I), the inter- 
 s3ction of the tangent planes (T) and (T,). Let us take the dual 
 of this property. 
 
 The polar plane (11) of the point/*, since it is in the intersection 
 (/) of the tangent planes (T) and (T,), will pass through its con- 
 jugate polar T and t„ the chord which joins the points of contact of 
 the tangent planes (@) and (©,) drawn to (2) ; and as ^ is a point 
 in the line p t, its conjugate polar will he the intersection of the 
 plane (11) with the tangent plane (@). Hence the pole of the 
 plane (V) will be the point on the umbilical directrix plane (A) 
 where it is pierced by the intersection of the planes (11) and (0). 
 In the same way the pole of the plane (Vy) is the point in which 
 (A) is pierced by the intersection of the planes (11) and (©,), and 
 the pole of the plane (V J is the point in which the directrix plane 
 (A) is pierced by the chord of contact t t, ; but these poles subtend 
 equal angles at the polar focus w. 
 
 Hence the following theorem : — 
 
 If two tangent planes, (0) and (@,), to a surface of the second order 
 be cut by another plane (H) passing through the points of contact 
 T r„ and cutting the tangent planes in two straight lines r ir and t, it, 
 if the three sides of the triangle r t,, t ir, and t, tt be produced to meet 
 the directrix plane in the points \, S, andS/, the angles XwB and XtuBj 
 will be equal. 
 
 270.] If two tangents are drawn from any point j» to two points 
 / and t, on the primitive surface (S), the focal vector planes (V) 
 and (V,) drawn through the tangents pt and pt, are equally inclined 
 to the vector plane (V„) drawn through the chord of contact 1 1,. 
 
 Since the three vector planes (V), (V,), and (V,,) all pass through 
 the focus F of (S), the poles of these three planes will lie on the um- 
 bilical directrix plane (A) . Draw the plane (11)^ the polar plane 
 of p. Now the conjugate polar of the line p t will be the inter- 
 section of the planes (H) and (@), namely (vr), and the conjugate 
 polar of the line p t, will be the intersection (isr,) of the planes (H) 
 and (©,), while the conjugate polar of the line 1 1, will be the line 
 (■5) in which the tangent planes (0) and (©,) intersect; therefore the 
 poles of the three vector planes (V), (V,), and (V,,) will be the three 
 points in which the lines (w), (w,), and (S) meet the directrix plane 
 (A) . From these relations we may obtain the following theorem : — 
 
 Two tangent planes, (0) and (0,), are drawn to a surface having 
 
24!6 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS 
 
 three unequal axes. A third plane (11) is drawn through the chord 
 of contact cutting the tangent plattes in the lines («r) and (wy) . These 
 lines and the line {^), in which the tangent planes intersect, are pro- 
 duced to meet the directrix plane (A) in the three points ir, ir,, andS. 
 The angle ira>S=-7rfii>S. 
 
 When the surface is an oblate spheroid, we obtain the following 
 theorem : — 
 
 Two tangent planes, and a secant plane through the chord of contact, 
 are drawn to an oblate spheroid, the diameter drawn through the point 
 in which the common intersection of the two tangent planes meets the 
 directrix plane (A) is equally inclined to the diameters which pass 
 through the points in which the common intersections of the secant 
 plane with the tangent planes meet the directrix plane. 
 
 271.] Through the same straight line (Z) two tangent planes 
 (T), (T,) and a plane (V) are drawn to the surface and focus F of 
 (S). From the other focus F, perpendiculars P, P; are let fall on 
 the tangent planes (T) and (T,) ; the line joining the feet of these 
 perpenficulars will be at right angles to the plane (V) . 
 
 This may easily be shown, as the tangent planes make equal angles 
 with the focal planes passing through the same straight line. 
 
 Let us now take the polar of this theorem. 
 
 Since the two tangent planes (T), (T,), and the focal plane (V), all 
 three pass through the same straight line {I), and (V) also through 
 the focus F, their poles t, t,, and v will range along the same chord 
 of (2), T and T, being points on the surface, and v the point of inter- 
 section of this chord with the umbilical directrix plane (A) ; and as 
 P, P, are perpendiculars to the tangent planes (T) and (T,), their 
 conjugate polars wiU lie in the planes drawn through to at right 
 angles to on- and wt,, sec. [257] ; and as these perpendiculars P 
 and Pypass through the second focus F^ of (S), the conjugate polars 
 of P and Fj will dso lie in the second umbilical plane (A,). 
 
 Consequently the lines in which the planes through m at right 
 angles to tar and mr^ meet the second umbilical directrix plane (A^) 
 are the conjugate polars («r) and (w^) of the perpendiculars P and 
 Py. Therefore the polar planes of the feet of the perpendiculars P 
 and P, on the tangent planes (T) and (T,) are the planes r{ia) and 
 t/w,) ; therefore the line which joins the feet of these perpen- 
 diculars P and P, must be the line in which the two planes intersect. 
 Let this line be (\) ; hence the plane a){X) wiU be at right angles to 
 the line <ov. 
 
 Hence we may derive the following theorem : — 
 
 If through a surface (2) with three unequal axes a chord be 
 drawn meeting this surface in the points t and t, and the umbi- 
 lical directrix plane (A), and if through «a the polar focus planes be 
 drawn at right angles to on and wt^ cutting the second directrix 
 plane (A,) iu {is) and (isr,), and planes be drawn through T('Br) and 
 
APPLIED TO THEOREMS AND PKOBLEMS. 247 
 
 t/wj) intersecting in the straight line (X,),the plane <o{\) will be at 
 right angles to the straight line wv. 
 
 272.] The vector planes (V) and (V,) passing through the focus 
 F of a surface of revolution (S) and the straight lines (d) and (d,), 
 in which two tangent planes (T) and (T,) cut its directrix plane (D), 
 are equally inclined to the plane (G) drawn from the focus F through 
 the intersection (l) of the tangent planes (T) and (T,). 
 
 Hence, in the reciprocal polar (2) of this surface (S), the poles 
 of the tangent planes (T) and (T,) are two points t and t, on the 
 surface of (2) ; the conjugate polar of the line (I) in which these 
 tangent planes intersect is the chord t, t, ; and as the plane (G) 
 passes through {!) and F, the pole 7 of this plane (G) will be the 
 point in which the chord t t, meets the umbilical directrix plane (A) . 
 Since (V) passes through the focus F^ its pole will be on the umbi- 
 lical directrix plane (A) ; and as the line [d) is the intersection of 
 the three planes (D), (T), and (V), consequently (6), the conjugate 
 polar of {rf),will pass through the poles of (D), (T), and (V) — that 
 is, through the umbilical focus w, the point t on the surface of (2), 
 and S the intersection of the line (8) with the plane (A) . In the 
 sameway we may find the point S,, the intersection of the line(S,) with 
 the directrix plane (A) . Now as the plane (G) is equally inclined 
 to the planes (V) and (V), the poles of these planes will subtend 
 equal angles at the polar focus. 
 
 Hence we derive the following theorem : — 
 
 (a) Let there be a triangle whose vertex is at the umbilical focus, 
 and whose base is a chord of the surface (2). If the three sides 
 
 Fig. 60. 
 
 be produced to meet the umbilical directrix plane (A) in three 
 points V, S, and h,, the line drawn from the polar focus w to r will 
 be equally inclined to the focal lines o»S and <oh,. 
 
 (/3) Should (2) become a surface of revolution (S), we may derive 
 the following theorem : — If in a surface of revolution (S) a chord 
 
248 THE GEOMETRICAL THEOHY OF RECIPROCAL FOLARS 
 
 be drawn meeting the directrix plane in d, and the surface in the 
 points c and c„ the angles c¥d and c,Frf, are equal. 
 
 When the chord c c, passes through the focus P, the angles cFd 
 and CfFd, are right angles. 
 
 (y) When (2) becomes an oblate spheroid, let a chord to the 
 surface T T, be drawn. Let v be the umbilical focus on the minor 
 axis at the distance ce from the centre, measured along the vertical 
 axis OZ ; let the sides v r and v t, of the triangle v t t, be produced 
 to meet the minor directrix plane (A) in the points o and 8,, and 
 let T T, meet the same plane in the point v. The diameter Ov will 
 be equally inclined to the diameters OS and OS,. 
 
 273.] Conjugate polar straight lines polarized by a sphere (fl,) 
 are always at right angles. When they are on the surface of the 
 sphere (that is, on a tangent .plane to the sphere), they are still at 
 right angles. Hence, if we polarize this sphere (fl,) by the polar- 
 izing sphere (il) , we shall obtain a surface of revolution (S), whose 
 focus will be a, the centre of the polarizing sphere (fl) ; and as the 
 conjugate tangents pass through a point on the surface of the sphere 
 (ft,), their conjugate polars will lie on a tangent plane to (S) j and 
 as the conjugate tangents lie on a tangent plane to (ft,), their con- 
 jugate polars will pass through a point on the surface of (S) ; and 
 as the conjugate tangents to the sphere (ft,) are at right angles, 
 the planes passing through the focus of (S), and these conjugate 
 tangents to the surface of (ft,), will contain the conjugate polars 
 which are tangents to (S), and these planes will be at right angles 
 to each other. 
 
 If we now polarize this theorem, the umbilical directrix plane (A) 
 to the surface (S) will be the polar plane of the focus F of (S). 
 Now the poles of these two rectangular planes which pass through 
 the focus F of (S) will be found on the corresponding umbilical direc- 
 trix plane (A); and as these planes pass through two straight lines 
 which are found on a tangent plane to (S), and touching the same, 
 their poles will be found on the reciprocal conjugate polars, see 
 [247], which touch the surface (2), and are in a tangent plane to 
 it. Therefore these poles are the points in which the two conjugate 
 tangents to the surface (X) pierce the umbilical directrix plane (A). 
 Hence the following theorem may be derived : — 
 
 Let any pair of conjugate tangents to the surface (2) be produced 
 to meet the umbilical directrix plane (A) in two points a and j8, the 
 lines drawn from the polar focus toto these points will be at right angles. 
 
 When these conjugate tangents are at right angles, they are 
 tangents to the lines of greatest and least curvature of the surface 
 (2), as we shall now proceed to show. 
 
 274.] Conjugate tangents are parallel to a pair of conjugate dia- 
 meters of the central plane of the surface (2) parallel to the tangent 
 plane containing the conjugate tangents. 
 
APPLIED TO THEOREMS AND PROBLEMS. 249 
 
 A very elegant and at the same time a purely geometrical proof 
 of this proposition may be established by the method of limits. 
 
 Join Q, the point of contact of the tangent plane^ to C the centre 
 of (S). Through the centre C, and parallel to the tangent plane 
 at Q, let a plane section of the surface be drawn ; then the centres 
 of all the sections parallel to the tangent plane at Q, will have their 
 centres on the straight line QC. Let a plane section of the surface 
 be drawn parallel and indefinitely near to the tangent plane at Ct, 
 and let k be the centre of this infinitesimal section. Through the 
 diameter QttC let two planes be drawn cutting the diametral plane 
 parallel to the tangent plane in the conjugate diameters GA and 
 C6^ and the infinitesim^ section in the ^ameters Ka and xfi. Let 
 two tangents be drawn parallel to the diameter ku in the infini- 
 tesimal section, the centre of the line joining the points of contact 
 of these tangents wiU be on the line QC ; and as these parallels lie 
 in the plane of the variable section^ the two tangent planes to the 
 surface (2) through these tangents will intersect in a straight line, 
 which (Euclid, XI. 19) will be also parallel to the plane of the vari- 
 able section, and to the two tangents which it contains, and there- 
 fore parallel to the diameter of the surface CA. 
 
 Let this variable section be conceived to move parallel to itself, 
 it will have its centre always on the diameter QC ; and when at 
 length it comes ultimately to coincide with the fixed tangent plane 
 at Q, the two movable tangent planes to the variable section, or 
 indicatrix* as named by Dupin, will continue always parallel to 
 the diameter CA drawn in the parallel diametral plane, until they 
 at length coincide, and we shall have their ultimate intersection 
 parallel to the diameter CA along the tangent Qa. These planes 
 will therefore be the constituent elements of a developable surface 
 whose edges lie along the lines Qa, Qip.,, and which touch the 
 surface (2) along the line Q^, Q^/S,. 
 
 If we now take the normal radii of curvature along the indicatrix 
 and perpendicular to the tangent plane at Q, these radii of curva- 
 ture, multiplied by the perpendicular distances between the tangent 
 plane at Q and the plane of the indicatrix parallel to it, will be 
 ultimately equal to the squares of the corresponding semidiameters 
 of the indicatrix ; but as these perpendicular distances are all equal, 
 the normal radii of curvature round the point Q will be ultimately 
 as the squares of the corresponding semidiameters of the indicatrix. 
 Now, as the axes are the greatest and the least of all the semidia- 
 meters of the indicatrix, the greatest and the least radii of curva- 
 ture will be in normal planes at right angles to each other, and 
 passing through the point Q. We may also infer that, since 
 the sum of the squares of any pair of conjugate diameters is equal 
 to the sum of the squares of the axes, the sum of the radii of cur- 
 * Dupin, ' D^veloppements de G^om^trie/ p. 48. 
 
250 THE ODOMETRICAL THEORY OF RECIFKOCAL FOLARS 
 
 vature along any pair of conjugate tangents will be equal to the 
 sum of the radii of greatest and least curvature. And since, more- 
 over, the sum of the squares of the reciprocals of the axes of the 
 indicatrix is equal to the sum of the squares of the reciprocals of 
 any other two diameters at right angles to each other, and as the 
 curvature of any curve is inversely proportional to its radius of 
 curvature, it will follow that the sum of the curvatures of any two 
 normal sections passing through Q and at right angles to each 
 other is constant. 
 
 275.] From these principles we may derive a simple method of 
 determining the lines of greatest and least curvature at the point 
 Q. Through this point Q let us conceive two planes to be drawn 
 parallel to the umbilical directrix planes (A) and (A,). These 
 planes will be parallel to the central circular sections of the surface 
 (X) ; and as the central circular sections parallel to them cut any 
 central plane section in two equal straight lines, making equal 
 angles with the axes of this section, it will foUow that the Imes 
 drawn through Q on the tangent plane to (S) bisecting the angles 
 between the traces on this tangent plane of the two planes parallel 
 to the umbilical directrix planes drawn through the same point Q 
 will be tangents to the sections of greatest and least curvature*. 
 
 From the theorem established in [273] , we may derive the follow- 
 ing very elegant geometrical method of determining the position of 
 the lines of greatest and least curvature. 
 
 Let the lines of greatest and least curvature, which are at right 
 angles, be conceived to meet the umbilical directrix plane (A) in 
 the points a and |3, which lie in the straight line in which the tan- 
 gent plane at Q meets the umbilical plane (A) . Now, as the angles 
 aQ/3 and aa>/3 are right angles, a sphere described on a/3 as diameter 
 would pass through the points Q and o>, since the angles in a hemi- 
 sphere are right angles. Hence if we draw a tangent plane to (S) 
 at Q meeting the umbilical plane (A) in the straight line (S), and 
 if we join Q the point of contact, and o> the polar focus, and bisect 
 this line Qa in the point ir, and through ir conceive a plane to be 
 drawn at right angles to the line oQ, and meeting the line (S) in 
 the point 7, the sphere described with this point as centre and 
 through the point a will cut the line (S) in the points a and /3 
 equally remote from y, and wiU also pass through Q by the con- 
 struction. Now as ao>j3 is a right angle, being the angle in a 
 semicircle, a and j3 must be points in which a pair of conjugate 
 tangents to (2) at Q meet the umbilical plane in the line (S) on (A) . 
 Therefore the lines oQ, j3Q are a pair of conjugate tangents to (2) 
 at Q. But as Q is a point on the surface of the sphere, the angle 
 aOfi is a right angle ; therefore the conjugate tangents at Q are at 
 
 • This very elegant construction is due to M. Chasles : see • Recherches de 
 Geometrie pure,' p. 78. 
 
APPLIED TO THEOREMS AND PROBLEMS. 251 
 
 right angles, and are therefore tangents along the lines of greatest 
 and least curvature. 
 
 There is a remarkahle case in which this construction fails. 
 When the point Q is an umhilicus, the tangent plane through Q is 
 parallel to the umhilical directrix plane (A) . The line (8) there- 
 fore in which they intersect recedes to infinity; so also do the 
 points a and /3 in which the conjugate tangents at Q are supposed 
 to meet it. How is this to be interpreted? The solution is as 
 follows. Let a pair of conjugate tangents Qa, Q^ be drawn in the 
 tangent plane at the umbilicus Q tending to meet the lines ma, ci>/3 
 drawn from the polar focus a> in a focal plane parallel to the tangent 
 plane and umbilical directrix plane in the points a and |3 ; but as 
 these points have receded to infinity, the lines Qa and ma wUl be 
 parallel, so also will the lines Q/S and and 6>/3. Hence the angles 
 aQ/3 and aa>/3 will be equal. But the lines Qa and Q^ were assumed 
 as conjugate tangents. Hence the angle a&i/S is a right angle. 
 But aa>/3=aQj3. Hence aQ/3 is a right angle. Consequently any 
 pair of conjugate tangents at the umbilicus will be at right angles. 
 
 276.] The vertex A of a cone circumscribing (S), a surface of 
 revolution, is on the directrix plane (D) of this surface. The plane 
 of contact (K) will pass through the focus F, and the line drawn 
 iirom A to F will be perpendicular to the plaue of contact (K). 
 Consequently, as the plane of contact (K) passes through F, its 
 pole K will be on the umbilical directrix plane (A) ; and as the 
 vertex k of the circumscribing cone to (S) is on the directrix plane 
 (D) of (S), its polar plane (G) will pass through v, the umbilical 
 focus. Therefore the conjugate polar of the line k¥ is the straight 
 line in which (G), the plane of contact of the cone circumscribing 
 (2), meets the umbilical plane (A) . 
 
 This theorem consequently follows : — 
 
 . If a cone whose vertex k is on the umbilical directrix plane 
 (A) be circumscribed to a surface (2), the plane of contact (G) of 
 this cone with (S) will pass through v, the umbilical focus, and will 
 cut (A) in a straight line (7) . The line drawn from the polar focus 
 o> to the vertex k of the cone will be at r^ht angles to the plane drawn 
 from a through the line (T) in which (G), the plane of contact of 
 the cone circumscribing {2), m£ets the umbilical directrix plane (A). 
 
 277.] The lines (/), (/() drawn from the foci of a surface of revo- 
 lution (S) to a point t on its surface make equal angles with the 
 tangent plane (T) at that point. Taking the dual of this theorem. 
 
 The pole t of the tangent plane (T) to (S) is a point on the sur- 
 face (2), and the polar plane of the point t is the tangent plane (0) 
 through T ; and the conjugate polars of the focal lines (/) and (/,) 
 in (S) are the lines in which the tangent plane (©) to (2), the polar 
 plane of the point t, intersects the umbilical planes (A) and (A,). 
 
 We may therefore infer that 
 
252 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS 
 
 If a tangent plane (0) be drawn to (2) touching it in the point t, 
 and cutting the conjugate umbilical directrix planes (A) and (A,) in 
 the lines (8) and (S,), the planes drawn through these straight lines 
 and the polar focus a> will be equally inclined to the line an; drawn 
 from the polar focus m to the point of contact t. 
 
 When the surface becomes an oblate spheroid it will hence follow 
 that, If a tangent plane be drawn to an oblate spheroid cutting the 
 parallel directrix planes (A) and (A^) in the straight lines (S) and 
 {Sf), the diameter through the point of contact will be equally inclined 
 to the diametral planes 0(S) and 0(8,). 
 
 When one of the secant planes is drawn parallel to one of the 
 umbilical directrix planes (A), the line in which it meets this 
 umbilical plane wiU recede to infinity. Hence the tangent plane 
 drawn through the umbilicus will meet the other directrix plane 
 (A() in the straight line (S^), so that any point in this straight line 
 will be equally distant from the umbilicus and the polar focus to — 
 a result already obtained by a different method in sec. [260] . 
 
 When T coincides with the extremity of the principal parameter 
 L of the surface (2), the tangent plane at t will cut the umbilical 
 planes (A), (A^) in their common intersection, the polar directrix 
 which lies in the plane of XY ; or the principal parameter of the 
 surface (S) is at right angles to the plane of XY — ^a result long 
 since obtained. 
 
 278.] It has been shown, in note to sec. [249] , 
 
 That if a cone be circumscribed to (S), a surface of revolution 
 round the major or transverse axis, the lines drawn from the vertex 
 of the circumscribing cone to the foci F, F^ of (S) will be (/), (/,), 
 the focals of the cone. 
 
 Taking the reciprocal polar of this theorem, the polar plane of k, 
 the vertex of the cone circumscribing (S), will be a plane section 
 (K) of (2), and the conjugate umbilical directrix planes (A) and 
 (A,) of (2) are the polar planes of the foci of (S), namely F and F, ; 
 and the straight lines in which the plane (K) cuts the umbilical 
 directrix planes (A) and (A,) will be the conjugate polars of (/) 
 and (/,), the focals of (S) . And if through the polar focus a> and 
 these two lines we draw planes, these planes will be perpendicular 
 to the focals of (S), and therefore parallel to the circular sections 
 of the cone whose vertex is at o>, and whose base is the plane sec- 
 tion (K) of (2). Therefore we may infer that 
 
 If a secant plane be draum cutting the surface (2) in a plane 
 section (K), ami the conjugate umbilical directrix planes (A) and 
 (A,) in the straight lines (S) and (S,), the planes drawn through the 
 polar focus a and these straight lines (S) and {S,) will be parallel to 
 the circular sections of the cone whose vertex is at to and whose base 
 is{K). 
 
 (a) When the surface is one of revolution round the transverse 
 
APPLIED TO THEOREMS AND PROBLEMS. 253 
 
 axis, the two umbilical directrix planes (A) and (Ay) coalesce into 
 (D), and therefore the lines in which the secant plane cuts the 
 directrix planes coalesce into one ; therefore the circular sections of 
 the cone coalesce into one, or the cone is a right cone. Conse- 
 quently it follows that the cone whose vertex is at the focus of a 
 surface of revolution (S), and whose base is any plane section of 
 this surface, is a right cone ; and its circular section is parallel to 
 the plane drawn through the focus F and the straight line in which 
 the base of the cone cuts the directrix plane (D) of (S) . 
 
 (fi) When (S) becomes an oblate spheroid, to coincides with C, 
 the centre of (S); the umbilical directrix planes (A) and (A^) 
 become parallel to the plane of XY. Hence we derive this very 
 elegant theorem : — 
 
 The circular sections of a cone whose vertex is at the centre of an 
 oblate spheroid, and whose base is any plane section of this surface, 
 are parallel to the two diametral planes which pass through the 
 straight lines in which this base intersects the minor directrix planes 
 (A) and {A,) of the oblate spheroid (S) . 
 
 (7) If in a surface (S) having three unequal axes a secant plane 
 be drawn which shall pass through the line in which the umbilical 
 planes (A) and (A|) intersect, i. e. the directrix of the principal 
 section in the plane of XY; then, as the secant plane cuts the 
 directrix planes (A) and (A^) in the same straight line, there can 
 be but one plane drawn from the polar focus o parallel to the cir- 
 cular sections of the cone, or, in other words, this cone is a right 
 cone. Hence we obtain this other remarkable theorem : — 
 
 The cone, whose vertex is at the polar focus a of a surface (S) 
 having three unequal axes, and whose base is a plane section of this 
 surface which passes through the polar directrix, is a right cone whose 
 circular section is parallel to the plane of XY. 
 
 279.] A simple algebraical proof of this theorem may be given. 
 
 The equation of the ellipsoid when the origin is at the focus cu 
 
 and the axes of coordinates are parallel to the axes of the figure, is 
 
 i9c«a:«-|-a*cV-ho«iV-26*c«aeiJ7-6V=0. ... (a) 
 
 The equation of the secant plane passing through the directrix is 
 
 kx-{-^z=\; (b) 
 
 but A=— ^, and B=t, h being the distance measured along 
 
 the axis of z, at which it is cut by the secant plane. Hence the 
 equation of the secant plane becomes 
 
 -r-=6* + aca;; (c) 
 
 A 
 
 or, squaring this equation and multiplying it by c*, 
 
 M„S-S 
 
 AV-«-2iVaea?+aVc*a?«-^-^=0 (d) 
 
254 THE OEOMETBICAL THEORY OF RECIPBOCAL FOLABS 
 
 If we add this expression to the preceding equation, we shall 
 
 The equation of a right cone of which the axis of Z is the axis. 
 
 be 
 WhenA=— , the equation becomes x^+y^=0, the equation of a 
 
 vertical straight line. This we might have anticipated ; for when 
 
 A=— , A=L, the seraiparameter of the surface (S) ; and we have 
 
 shown in sec. [246] that a tangent plane to (S) which passes 
 
 through the polar directrix touches the surface at the extremity of 
 
 the semiparameter. 
 
 be 
 When h>— or greater than L, the secant plane falls outside the 
 
 surface, or the cone becomes imaginary. 
 
 280.] In a surface of revolution (S), the sum of the reciprocals 
 of the segments of any focal chord is constant. This is a well- 
 known theorem. Let C and A be the semiaxes of (S), C being 
 greater than A ; and if/ and f, be the focal chords, we shall have 
 
 i+^=?^ (a) 
 
 Now as /+/( is a line drawn through the focus F of (S), and 
 meeting the surface (S) in the points t and t/, the polar planes of 
 F; t, and t/ will be the umbilical directrix plane (A) and the two 
 tangent planes (S) and (0^) to (Z), all meeting in the same straight 
 line (S) on (A). But by the lemma established in sec. [253], 
 
 RVsing 
 ^ P^— (^) 
 
 Now P is the perpendicular on the umbilical plane (A) let fall 
 from (0, -as, and «r^, the perpendiculars from the same point on the 
 tangent planes (@) and (0^) ; B and B^ are the angles between the 
 umbilical plane (A), and the tangent planes (@) and (€>,) to (S). 
 Through &> and the perpendiculars P, •a, and ts^ on the umbilical 
 and tangent planes let a focal plane (IT) be drawn cutting the line 
 (8) in the point k. From k let a straight line be drawn to a> ; this 
 wUl be r, and the preceding expression, adding to it that for the 
 other segment/,, becomes 
 
 / V,~R«r jsin ^"^^ sin 6^ ~ A«" 
 
 (c) 
 
 From sees. [238] and [237] we find that P=-, and ?-,=J^. 
 ^ ■> ^ ■> ae A* R*oc 
 
APPLIED TO THEOREMS AND PROBLEMS. 255 
 
 Consequently 
 
 rlsm.e^ si.ue,J~ c^ ^ ^> 
 
 Now, if through the polar focus o) we draw a plane parallel to the 
 umbilical directrix, plane (A), this plane will cut (S) in a circular 
 section, and the line in which the secant plane (11) and the tangent 
 plane (€)) intersect will meet the plane of this circular section in a 
 point ; and if h be the distance of this point from to, we shall have 
 h sin 6='B!. Making the same substitution for the other tangent 
 plane, we shall have 
 
 *±*'=??e (e) 
 
 re* ^ ' 
 
 Hence we may derive this theorem. 
 
 If through any straight line on the umbilical directrix plane (A) 
 two tangent planes be drawn to (2), and if through the three per- 
 pendiculars let fall from ea on these three planes a secant plane (11) 
 be drawn cutting the tangent planes in lines which produced meet the 
 plane drawn through w parallel to the umbilical directrix plane (A) 
 in the points r and t,, and if ojT=h, o>T^=hi, and u>K=r in the tri- 
 angle Tier I, the base tcot, will have to the line eoie a constant ratio, or 
 
 h+h, _2b^e 
 
 r c^ (f) 
 
 When the surface (S) becomes a surface of revolution (S) round 
 the transverse axis, c=b and e=e, hence 
 
 ^'=2« (g) 
 
 We may also derive this other theorem in the conic sections. 
 
 If from any point in the directrix a pair of tangents be drawn to 
 the curve, they will cut off equal segments from the ordinate passing 
 through the focus. 
 
 Let the equation of the tangent to the curve in rectangular coor- 
 
 dinates be — r+^'^l* ^^^ ^ *^® tangent passes through a point 
 
 b (CL -J- CtT 1 
 
 whose coordinates are Y and —ae, Y= '-; and as the 
 
 tangent passes also through a point whose coordinates are U and 
 
 a ^_ b^{ae+x,) ^^^ if d i,e the distance from the focus to the 
 e' aey, 
 
 ft* 
 directrix, D=— • 
 
 Squaring, adding, and taking the square root, 
 
 r= ^W+D^=^^^^^; consequently -=e. 
 * aey, r 
 
256 THE GEOMETRICAL THEORY OF RECIPROCAL FOLARS. 
 
 Y Y+Y 
 
 So also for the negative side, —'=e, or -'=2e. 
 
 281. J In a surface of revolution (S) the sum of any two focal 
 vectors /and/; drawn to any point < on the surface is constant and 
 equal to the major axis of (S) ; or 
 
 /+/,=2C. (a) 
 
 Now the polar planes of the two foci F and F, of (S) are the 
 umbilical directrix planes (A) and (A,) of the surface (S) ; and the 
 polar plane of the point / on (S), to which the focal vectors are 
 drawn, is a tangent plane (@) to (S), cutting the umbilical direc- 
 trix planes (A) and (A^) in the straight lines (S) and (£,), which are 
 the conjugate polars of (/) and (/,) . 
 
 Let d and d, be the angles which the tangent plane ((R>) makes 
 with the umbilical planes (A) and (Aj) . Let F and nr be the per- 
 pendiculars let fall firom the polar focus &> on the planes (A) and 
 (@). Let (11) be a secant plane passing through a, the polar focus, 
 and through P and «-, cutting (S) the line of intersection of the 
 planes (A) and (0) in the point r. Join t and o>, and let r be 
 equal to ra. 
 
 Now, assuming the proposition established in lemma I. p. 228, 
 
 , ,1 , , Wr sin 6 
 
 we shall have/= — =5 . 
 
 Pw 
 
 Let fft be the angle which the plane drawn through to and (S), the 
 
 conjugate polar of (/), makes with the vertical ordinate through to 
 
 the polar focus, and let I be the distance from the polar directrix 
 
 to the foot of r; then r : / : : sin t : cos A, or r= r, and /sind= o',. 
 
 cos q> ' 
 
 Therefore r sin 0= ' Now sin »=t-, see (a), sec. [240], and 
 
 P=— , see (a), sec. [238] ; consequently -^=^. 
 Making these substitutions in (a), we shall have 
 /=— Tg -'sec^j and for/, we shall find /,= -r^ —' sec ^;. 
 
 But /-|-/,=2C = -T— , see sec. [237] ; consequently 
 
 5[sec0 + 8ec^J=- (b) 
 
 Let h and h, be the segments into which the tangent plane (0) 
 divides the distance between the polar focus and the polar directrix, 
 
 then -7-= — . 
 
 Consequently -T- (sec ^ -I- sec <^,)= — (c) 
 
 */ C€ * 
 
APPLIES TO THEOREMS AND PROBLEMS. 257 
 
 Therefore the reciprocal polar of the theorem that in a surface 
 of revolution the sum of the focal vectors is constant, is as follows. 
 
 To a surface (2) having three unequal axes let a tangent plane 
 (0) be drawn, cutting the umbilical planes (A) and (A,) in the straight 
 lines {S) and (8^), and the line joining the polar focus and the polar 
 directriw in the segments h and h,. Let the planes that are drawn 
 from, to through the straight lines (S) and {S,) make the angles <f), ij>, 
 with the vertical ordinate passing through the polar focus <o, the 
 resulting expression becomes 
 
 ^(8ec^ + 8ec0,) =— . 
 
 CHAPTER XXVII. 
 
 282.] It may interest the reader to learn the method by which 
 M. Chasles has derived the properties of surfaces of revolution of 
 the second order (S) from those of the sphere, and how these pro- 
 perties maybe extended to surfaces of the second order (S) having 
 three unequal axes. By this method of double polarization the 
 countless theorems that have been established with reference to 
 the sphere (including the whole of spherical trigonometry) may 
 be extended to umbilical surfaces (S) of the second order having 
 three unequal axes. 
 
 The following propositions are extracted from M. Chasles's work, 
 entitled " Becherches de geometric pure sur les lignes et les sur- 
 faces du second degre," Bruxelles, 1829. 
 
 (a) Deux plans tangens k la sphere font des angles egaux avec 
 le plan mene du centre h leur droite d'intersection ; done 
 
 •Les rayons vecteurs menes d'un foyer aux extremites d'une corde 
 d'une surface de revolution (S), sont egalement inclines sur le rayon 
 vecteur mene au point ou cette corde rencontre le plan directeur. 
 
 We may extend this proposition of M. Chasles to a surface with 
 three unequal axes, and conclude that if any two tangent planes 
 (0) and (€),) be drawn to the polar surface (2) having three un- 
 equal axes, they will cut the umbilical directrix plane (A) in two 
 straight lines (S) and (8,), and one another in a third straight line 
 (d), and these three straight lines will be the conjugate polars of 
 the two focal vectors and of the chord of the surface (S) ; and if a 
 plane be drawn through the straight line (•&) in which the tangent 
 planes to (2) intersect, and through the corresponding umbilical 
 focus V, this plane will cut the umbilical plane in a fourth straight 
 line (y), which will be the conjugate polar of the line drawn from 
 the focus of (S) to the point where the chord meets the directrix 
 plane (D), and which make equal angles with its focal vectors; 
 
258 THE GEOMETRICAX THEORY OF BECIPKOCAL FOLARS 
 
 hence the planes drawn through the polar focus and these three 
 lines (8), {7), (S,) in the umbilical directrix plane (A) will make 
 equal angles, the one with the other. Hence we may infer 
 
 That »/ two tangent planes, (0) and (©,), be drawn to a surface (S) 
 having three unequal axes cutting the umbilical plane (A) in two 
 straight lines (o) and (S,), and through the line {Si) in which they 
 intersect, and the corresponding umbilical focus v a plane (U) be 
 drawn cutting the directrix plane (A) in the straight line (7), the 
 planes drawn through the polar focus w and the straight lines (S), 
 {Si) vnll make equal angles with the plane passing through to and 
 the straight line (7) . 
 
 (yS) Deux rayons de la sphere font des angles egaux avec la 
 corde qui joint leur extremites ; done 
 
 Les plans vecteurs menes d'un foyer d'une surface de revolution 
 aux droites suivant lesquelles deux plans tan gens rencontrent le 
 plan directeur sont egalement inclines sur le plan vecteur mene k 
 la droite d'intersection des deux plans tangens. 
 
 Since the tangent plane (T), the vector plane (U), and the direc- 
 trix plane (D], all three meet in the same straight line (d), the 
 conjugate polar of this straight line (d) will pass through the poles 
 of these three planes, that is to say, through v the pole of (D), t 
 the pole of (T) ; and as (U) passes through the focus F, its pole will 
 be in the umbilical directrix plane (A) — that is, in the point S where 
 it is pierced by the line vt. 
 
 In the same way we may show that 8^ is the pole of the plane (U,), 
 and is also on the line vt,; and as the vector plane (Y) passes through 
 F and the intersection {l) of the tangent planes (T) and (T,), its pole 
 will lie on the directrix plane (A) and in the bne which joins the 
 points T, T, on (2). Now as (U) and (U;) make equal angles with 
 (Y), the lines drawn from to to the points S and S, the poles of (U) 
 and (U,) will be equally inclined to the line tov drawn from a> to v 
 the pole of (Y), 
 
 We may hence infer that 
 
 ^ through the umbilical focus v two straight lines be drawn meeting 
 the surface (S) in the points r and t,, and the umbilical directrix 
 plane (A) in the points S and S,, while the line t t, meets the same 
 plane in v, the lines drawn from, to to the two points S, S, on the umbi- 
 lical plane will be equally inclined to the third line tov. 
 
 (7) Le plan mene par le centre d'un sphere et par la droite 
 d'intersection de deux plans tangens est perpendiculaire k la corde 
 qui joint les deux points de contact, et passe par son milieu ; done 
 
 Le plan vecteur mene d'un foyer d'une surface de revolution k 
 la droite d'intersection de deux plans tangens est perpendiculaire 
 au rayon vecteur mene de ce foyer au point ou la droite qui joint 
 les deux points de contact des plans tangens rencontre le plan 
 directeur. 
 
APPLIED TO THEOREMS AND PROBLEMS. 259 
 
 Let US now take the reciprocal polar of this theorem. 
 
 The pole of the vector plane which passes through the focus of 
 (S) and the intersection of the tangent planes (T) and (T,) to (S) 
 is the point in which the umhilical plane (A) is pierced by the chord 
 of contact of two tangent planes (0) and (&,) drawn to (2). 
 
 Let d be the point on the directrix plane (D) to (S) in which the 
 chord of contact of the two tangent planes (T) and (T,) meets this 
 plane; then, as d is a point on (D), its polar plane will pass through 
 V the umbihcal focus ; and as d is a point in the chord of contact of 
 the tangent planes (T) and (T,) to (S), its polar plane will pass 
 through (*) the intersection of (0) and (0,) ; and as d is also a point 
 in the line passing through (F), the conjugate polar of this straight 
 line dF will be the line in which the polar plane of d meets the 
 umbilical plane (A) ; and as the point d and the intersection of (T) 
 and (T,) subtend a right angle at the focus F of (S), their reciprocals 
 will subtend a right angle at the polar focus a. 
 
 Hence this theorem : — 
 
 If two tangent planes (0) and (0^) be drawn to a surface (2) with 
 three unequal axes, and if through the umbilical focus and their line 
 of intersection a plane be drawn meeting the umbilical directrix plan£ 
 (A) in a straight line, the plane drawn through this straight line and 
 the polar focus will be perpendicular to the line drawn from the polar 
 focus to the point in which the chord of contact of the tangent planes 
 (0) and (0|) pierces the corresponding umbilical plane (A). 
 
 (S) La droite qui va du centre d'une sphere au sommet d'un 
 cdne circonscrit, est perpendiculaire au plan du cercle de contact 
 du cdne et de la sphere ; done 
 
 Le rayon vecteur mene d'un foyer d'une surface de revolution au 
 sommet d'un c6ne circonsciit k la surface est perpendiculaire au 
 plan vecteur mene par la droite d'intersection du plan de la courbe 
 de contact et du plan directeur. 
 
 Now taking the reciprocal polars of the preceding lines and 
 surfaces. 
 
 The conjugate polar of the line joining v the vertex of the cone 
 with the focus F of (S), is the line (k) in which the base (K) of the 
 cone circumscribing (2) cuts (A) the umbilical directrix plane; and 
 as the straight line in which the plane of contact (C) of the cone 
 circumscribing (S) cuts its directrix plane (D) is the conjugate 
 polar of the Ime which joins the umbilical focus with the vertex k 
 of the cone circumscribing (2), and as the pole of a plane which 
 pa^sses through a given point and a given straight line is the point 
 in which the polar plane of the given point is pierced by the con- 
 jugate polar of the given straight line ; hence the pole of the vector 
 plane which passes through the focus of (S) and the intersection 
 of the plane of contact with its directrix plane (D) will be the point 
 in the umbilical directrix plane (A) where it is pierced by the line 
 
 s2 
 
260 THE GEOMETEICAL THEORY OF RECIPROCAL POLARS 
 
 passing through the vertex k of the cone circumscribing (2) and the 
 umbilical focus v of the surface. 
 Hence this theorem : — 
 
 If a cone circumscribe a surface (2)j the line drawn through the 
 umbilical focus of the surface and the vertex of the cone will piarce 
 the umbilical directrix plane (A) in a point from which, if a straight 
 line be drawn to the polar focus, this line will be perpendicular to the 
 plane drawn through the polar focus and the intersection of the plane 
 of contact (K) with the corresponding umbilical plane (A). 
 
 (e) Un cylindre circonscrit k une sphere la touche suivant im 
 grand cercle, dont le plan est perpendiculaire aux aretes du cylindre ; 
 done 
 
 Tout cone circonscrit h. une surface de revolution, suivant une 
 courbe dont le plan passe par un foyer, a son sommet sur le plan 
 directeur, et la droite menee du foyer i ce sommet est perpendicu- 
 laire au plan de la courbe. 
 
 The line which joins the vertex of the cone (C) circumscribing 
 (S) with its focus P is the conjugate polar of the line in which the 
 plane of contact of the cone (K) circumscribing (2) meets the um- 
 bilical directrix plane (A). The pole of the plane which passes 
 through the focus of (S), and the line in which the base of (C) Wts the 
 directrix plane (D), is the point in which the line joining the vertex 
 K of the cone (K) and the umbilical focus u of (2) meets the umbilical 
 directrix plane (A). As the plane of contact of the cone (C) circum- 
 scribing (S) passes through its focus, the vertex of the cone (K) cir- 
 cumscribing (2) will be on the umbilical directrix plane (A) of (2) . 
 Hence this theorem : — 
 
 The plane of contact of a cone circumscribing a surface (2), and 
 having its vertex on the umbilical directrix plane (A), will pass 
 through the umbilical focus v, and will cut the umbilical directrix 
 plane in a straight line. The vector plane drawn through this line 
 and the polar focus will be at right angles to the line drawn from the 
 polar focus to the vertex of the cone. 
 
 (^) Tous les plans tangens k un cone circonscrit k une sphere, 
 sont egalement inclines sur le plan du cercle de contact ; done 
 
 Le cone qui a pour sommet un foyer d'un surface de revolution, 
 et pour base une section plane quelconque de la surface, est de 
 revolution, et a pour axe, le rayon vecteur mene au sommet du c6ne 
 circonscrit k la surface suivant sa section plane. 
 
 Let V be the vertex of the circumscribing cone, F that of the 
 inscribed, and Q. a point on the two cones and the surface of (S). 
 
 Since the cones whose vertices are v and F have one common 
 plane of intersection with the surface (S), the polar cone will have 
 one vertex and two bases, one the plane of contact of the cone (K) 
 enveloping (2) , the other the section of this cone made by the umbi- 
 lical directrix plane (A) ; and as F is the focus of (S), its polar plane 
 
APPLIED TO TBEOHEMS AND PROBLEMS. 261 
 
 will be the umbilical plane (A) ; and as Q is a point on the plane of 
 contact of the cone circumscribing (S), its polar plane will pass 
 through the vertex k of the cone circumscribing (S) ; and as Q is 
 also a point on the surface (S), its polar plane will be a tangent j)lane 
 to (2); and as Q is a point on the line FQ, its polar plane will cut 
 the umbilical plane (A) in the line (Z). Hence the conjugate polar 
 of the line FQ will be the line (I) in which the tangent plane (0) 
 to the cone circumscribing (2) cuts the umbilical plane (A), and 
 the plane which passes through this line and the polar focus will be 
 perpendicular to the line FQ : hence this plane will envelope a cone 
 (supplemental to that of which FQ is the side) whose vertex is at 
 the polar focus and whose base is the section in the umbilical plane 
 (A) made by the cone circumscribing (2} ; and as the former is a 
 right cone, so must also be the latter. 
 
 The conjugate polar of the line joining F, the focus of (S) with 
 V the vertex of the cone circumscribing (S) , is the line in which the 
 umbilical plane (A) and the plane of contact of the cone (K) cir- 
 cumscribing (2) intersect ; hence the plane through the polar focus 
 c0 and this straight line is parallel to the circular section of the cone. 
 
 When the section of the cone (K) circumscribing (2) is parallel 
 to the directrix plane (A), the line in which they intersect is at infi- 
 nity, the plane of contact of the cone circumscribing (2) is parallel 
 to the umbilical directrix (A), and is therefore a circular section 
 of the surface ; the section of the cone circumscribing (2) on the 
 umbilical directrix plane is also a circle. 
 
 (ij) Tous les plans tangens au c6ne qui a pour base un cercle 
 trace sur une sphere, et pour sommet le centre de la sphere, sout 
 egalement inclines sur le plan du cercle ; done 
 
 Le cone qui a pour sommet un foyer d'une surface de revolution, 
 et pour base la courbe d'intersection d'un cone circonscrit k la sur- 
 face par le plan directeur est de revolution, et a pour axe la droite 
 menee du foyer au sommet du cone circonscrit. 
 
 Let Q be a point on the directrix plane (D) of (S) through which 
 passes a side (s) of the cone (C) whose vertex is c, and which cir- 
 cumscribes (S). Join Q with F the focus of (S). Now, as Q is a 
 point on the directrix plane (D), its polar plane (IT) will pass 
 through V, the umbilical focus of (2) ; and as Q is a point on the 
 cone (C) which circumscribes (S),its polar plane will pass through 
 a tangent (\) to the curve which lies in the plane (K), the polar 
 plane of c, and is the reciprocal of the cone (C), sec. [225] ; and 
 as this point Q is on (s), a side of the cone which is a tangent to 
 (S), the conjugate polar (\) will touch the plane section of (2) made 
 by (K), therefore the curve touched by (\) is a plane section of (2) ; 
 and again, as Q is in the line FQ, the conjugate polar of FQ will be 
 the line in which (11), the polar plane of Q, cuts the umbilical 
 directrix plane (A) in the straight line (8) . 
 
262 THE GEOMETRICAL THEOBY OF BECIPKOCAL FOLABS 
 
 Hence the polar plane (II) passes throngli v, the umbilical focua 
 of (2), and through (\), the tangent to the plane section of (2) 
 made by the plane (K), and through the straight line (S) in the 
 umbilical directrix plane (A) ; and the cone whose vertex is v and 
 base the plane section (K) of (2), will cut the umbilical directrix 
 plane (A) in a conic section to which (S) is always a tangent; 
 but as (S) is the conjugate polar of FQ, the plane through the polar 
 focus fi> and the straight line (S) will be at right angles to the line 
 FQ ; and as FQ is a side of a right cone, the p^e through to 
 and (8) will envelope the supplemental cone to that of which FQ 
 is a side ; and as the conjugate polar of the line which joins the 
 vertex of the cone (C) with F is the straight line in which the 
 secant plane (K) meets (A), this plane will be perpendicular to the 
 line Fc, the axis of the cone circumscribing (S), and will therefonre 
 be parallel to the circular section of the cone whose vertex is at the 
 polar focus a, and whose base is the conic section on the umbilical 
 directrix plane (A). 
 
 This is equivalent to the following theorem : — 
 
 Let a plane section (K) of a surface of the second order (2) having 
 three unequal axes cut the umbilical directrix plane (A) in a straight 
 line {k) . T%e cone whose vertex is at v, and whose base is the section 
 (K) of (2), toill cut the umbilical plane (A) in another conic section, 
 which will be the base of a right cone having its vertex at the polar focus 
 Q), and its circular section parallel to the plane drawn through to, and 
 through the straight line in which the plane section (K) cuts the umbi- 
 lical directrix plane (A). 
 
 {0) Tons les plans tangens au c6ne qui a pour base un cercle de 
 la sphere et pour sommet un point du diametre perpendiculaire au 
 plan de ce cercle sont egalement inclin& sur ce plan ; done 
 
 Un cone etant circonscrit ^ une surface de revolution, tons les 
 plans menes par la droite d'intersection du plan de la courbe de 
 contact et du plan directeur de la surface, couperont ce c6ne 
 suivant des coniques qui, etant vues du foyer correspondant au 
 plan directeur, sembleront £tre des cercles concentriqnes ; le centre 
 commun de ces cercles sera sur le rayon visuel mene au sommet du 
 cone circonscrit. 
 
 Let {d) be the line in which the base of the cone (C) drcom- 
 scribing (S) cuts the directrix plane (D). Let c be the vertex of 
 this cone. The conjugate polar of this line (d), since it lies on the 
 plane (D), will pass through the umbilical focus v; and as this line 
 is also in the plane section (C) of the cone enveloping (S), it will 
 pass through k, the vertex of the cone (K) enveloping (2). Hence 
 VK is the conjugate polar of the line (<!) ; and as the secant planes 
 of the cone (C) circumscribing (S) all pass through the line {d), 
 their poles k,, k,,, &c. will be on the line kv ; and as all the partial 
 cones enveloping (S) have the same vertex but different bases, their 
 
APPLIED TO THEOREMS AND PROBLEMS. 263 
 
 reciprocals will have diflFerent vertices, «, «,, «„, &c., and the same 
 ba8e(K). > i u> 
 
 Again, let Q be a point on one of the plane sections of the cone 
 (C) which passes through {d). Then as Q is a point in this plane, 
 its polar plane (11) will pass through k; and as Q is a point in the 
 side of the cone (C), its polar plane (11) will pass through the tan- 
 gent to the section of (2) in the base (K) whose vertex is « ; and 
 as Q is a point on the tangent to the plane section of (C,), its polar 
 plane will pass through a side of the cone whose base is (K). 
 
 Hence (11), the polar plane of Q, is a tangent to the cone whose 
 vertex is k, or «„, and whose base is (K) . Consequently the conju- 
 gate polar of FQ, the side of the cone whose vertex is at F, and 
 whose side is FQ, is the straight line (S) in which the tangent plane 
 (II) to the cone whose vertex is k, and plane section (K) meets the 
 umbilical directrix plane (A) ; and as the former is a right cone, so 
 is the latter ; and as the axis of the cone whose vertex is F and 
 base (C) is the line Fc, so the plane passing through <o and the line 
 in which (K) and (A) intersect is parallel to the circular section of 
 the cone. 
 
 We may therefore conclude that if through v a straight line of 
 indefinite length be drawn, and if in this line a point k be assumed 
 as the vertex of a cone, enveloping (2) along the plane of contact 
 (K), and if any number of cones be described whose vertices range 
 along the line vk having a common base (E), the sections of these 
 cones by the umbilical directrix plane (A) wiU be the bases of right 
 cones whose common vertex will be at the polar focus a. 
 
 (X) Let two cones (K) and (K^) circumscribe (2), a surface with 
 three unequal axes. Let their vertices k and k, be joined to v, one 
 of the foci of the surface, and let these lines be produced to meet the 
 corresponding umbilical directrix plane (A) in two points t and t^. 
 The lines joining these points with the polar focus co are the focak 
 of a cone whose vertex is at to, and whose base is the conic section 
 in the umbilical directrix plane (A), which is the intersection with 
 it of another cone whose vertex is at the focus of the surface, and 
 whose base is the common plane of intersection of the two circum- 
 scribing cones. 
 
 (fi) When the surface is one of revolution, these two cones become 
 identical, and we obtain the known theorem, that the focals of a 
 cone whose vertex is the focus of a surface of revolution (S) and 
 base the common intersection of two circumscribing cones, are the 
 lines drawn from the focus to the vertices of the circumscribing 
 cones. 
 
 When one of the cones becomes a tangent plane (©), we have the 
 theorem that if a cone (K) circumscribes a surface (2), and is cut by 
 a tangent plane (0) in a conic section, the cone whose vertex is a 
 focus of the surface and base this section, is cut by the corresponding 
 
264 ON THE KECIPSOCAL FOLASS 07 CONFOCAL SURFACES. 
 
 umbilical directrix plane (A) in a conic section (C), and the straight 
 lines drawn from this focus to the point of contact of {&) and the 
 vertex of (K) are met by the same directrix plane in two points 
 T and r,. The straight lines joining the polar focus to and the 
 points T, T, are the focals of the cone whose vertex is w and base (C) . 
 
 CHAPTER XXVIII. 
 
 ON THE KECIFBOCAL FOLABS OF CONFOCAL SURFACES. 
 
 283.] The reciprocal polars of confocal surfaces are concyclic 
 surfaces, or surfaces the planes of whose circular sections arp 
 parallel We may apply the method developed in the foregoing 
 pages to the derivation of the properties of one class of surfaces 
 from those of the other, and proceed to show that if we take a series 
 of confocal surfaces as the primitives (S), we shall make no change 
 either in the position or inclination of the umbilical directrix planes 
 (A) and (A^) to the plane of XY, and the distance of the polar focus 
 w from the polar directrix UY will continue the same. 
 
 Let A' and C% the squares of the semiaxes of the primitive sui- 
 face (S), be changed, A^into A,''=A« + F, and C* into C,*=C* + A*. 
 This will introduce no change into the focal distance of (S,) from 
 the centre; for this distance in (S,) is VC,*— A^*= VC*— A*, the 
 same as in (S) . 
 
 We shall assume the distance of to the polar focus from O the 
 centre of (S) to be constant, so also the polarizing sphere (£2). 
 Therefore D and B will be constant. 
 
 284.} The inclination of the umbilical directrix plane (A) to the 
 plane of XY continues unchanged. 
 
 Let i be the angle which denotes this inclination. In sec. [2401 
 
 it has been shown that coa^i=j^j-^ ^. Substituting in this 
 
 expression the values of a, b, and c as given in (c), [237], we shaU 
 find 
 
 "^'^^ C'+D'-A* ' (''J 
 
 and the value of this expression is not altered by changing C^ and 
 A' into C' + A* and A*+*'. The distance between ta, the polar 
 
 6* A* B.* 
 
 focus, and UY, the polar directrix, is manifestly — . Now —=-^ 
 
 andc=^; consequently —=-=:y, a constant quantity independent 
 
 of C and A. 
 The length of the perpendicular let fall irom the polar focus <o 
 
ON THE RECIPROCAL FOLARS OF CONPOCAL SURFACES. 265 
 
 ■f>3 
 
 on the umbilical directrix plane (A) is «r= — The 
 
 value of this expression is not altered by chaneing C® and A* into 
 C«±A«, and A«±A«. 
 
 Consequently, if we have a series of canfocal primitive surfaces 
 (S), (S,), (SJ, &c., their reciprocal surfaces (2), (2,), (2„), &c. 
 will be concyclic. They will have a common polar focus : they 
 will have the same umbilical directrix planes ; and the graphical 
 properties of one set of surfaces may easily be transformed into 
 their reciprocals on the polarized surfaces. 
 
 The portion of the vertical axis passing through the polar focus 
 (o and cut off by the umbilical directrix plaue (A) continues un- 
 changed, when C' and A* are augmented by a constant quantity ; 
 for if (oz be this distance, 
 
 <oz= — , but — = 7^ and ij=_i_r ^. 
 
 R* 
 Hence <oz=—-— -, which is not affected by the constant 
 
 added to C* and A^ 
 
 285.] Every property of confocal surfaces of revolution (S), (S^), 
 &o. implies a corresponding theorem of concyclic surfaces (S), (2,), 
 &c., having the same polar focus oi, and the same umbilical direc- 
 trix planes (A) and (Ay). 
 
 Thus, for example, it' from a point k a secant plane be drawn to 
 a surface (S) cutting it in a section (C) and passing through the 
 two foci F and Fy of (S) , and if from these foci focal chords / and/, 
 are drawn to k, and tangents if) and [t^ from the point k to the 
 section (C) , the angles between / and (^j, / and (f,) will be equal. 
 
 'Let us take the polar of this theorem. 
 
 The polar planes of the points F, F,, and k will be the umbilical 
 directrix planes (A) and (AJ, and also a secant plane (K) cutting 
 (2), and the base of a cone circumscribing (2), whose vertex is «, the 
 pole of (C). Now, as / is a line joining F and the point k, its con- 
 jugate polar will be the line (8) in which the planes (A) and (K) 
 intersect. In the same way the planes (A,) and (K) will intersect 
 
 Again, as c, the point of contact of {/) with (C), is on the surface 
 (S), its polar plane will be a tangent plane (0) to (2) ; and as it is 
 a point on (C), its polar plane will pass through «, the vertex of the 
 cone circumscribing (2) along (K) . Consequently the conjugate 
 polar of the line FA will be the line (t) in which the planes (K) and 
 (®) intersect ; therefore the planes drawn through a> the polar focus 
 and the straight lines (S) and (r), are inclined at the same angle as 
 the planes <»(8,) and a>(T,). 
 
266 ON METRICAL METHODS AS APPLIED TO 
 
 "We may consequently infer 
 
 That if a cone circumscribe (2), a surface having three unequal 
 axes, and if the plane of contact be produced to meet the umbilical 
 planes in two straight lines and cut the tangent planes to the circum- 
 scribing cone, the planes through the polar focus and these straight 
 lines in the directrix planes will be equally inclined to the planes 
 through the polar focus and the straight lines, the intersections of the 
 tangent planes with the secant plane. 
 
 Now, if we take any niunber of surfaces (S), (S,) whose semiaxes 
 squared are C.A?; C^ + k^, A?+k^ ; C^H-**, A« + *«, &c., their 
 reciprocal polars (2), (2,), &c. will all have the same polar focus tt> 
 and the same umbilical planes (A) and (A|) ; and if we cut all these 
 surfaces by a common secant plane, this plane will cut the umbilical 
 planes in two lines (5) and (S|), and tangent planes to the circum- 
 scribing cones in two sets of tangents (t), (t;), (t„) and (t'), (t^), (t"); 
 the angles between one set of planes (S)o>(T|), [h)m{T^, (S)o)(t^), 
 &c. wiU be equal to the angles {8,)o»(t'), (S>{t"), {S,)«(t"'), &c. 
 between the corresponding set of planes. 
 
 CHAPTER XXIX. 
 
 ON METKICAL METHODS AS APPLIED TO THE THEOBY OF 
 RECIPROCAL POLABS. 
 
 286.] Throughout the investigations of the methods developed 
 in the foregoing pages, lines and planes, surfaces and curves have 
 been treated graphically, so to speak. Abstract numbers or their 
 representatives have been rarely admitted. The distinction between 
 graphical and metrical properties is sufficiently obvious. That the 
 opposite sides of a hexagon inscribed in a conic section will meet 
 iu three points which range on a straight line is a graphical pro- 
 perty ; while the theorem that in any conic section the sum of the 
 squares of any pair of conjugate diameters is constant, is evidently 
 a metrical relation. 
 
 We shall find that, by the application of these methods, entirely 
 new classes of properties of curves have been brought to light, the 
 existence of which had hitherto been imsuspected and unknown. 
 We give a simple illustration of this method. 
 
 In section [32] it has been shown that if perpendiculars be let 
 fall bam a series of fixed points A, A^ &c. in a plane on a straight 
 line in the same plane, and if the sum of the perpendiculars be 
 constant, the line will envelop a circle. 
 
 We may take the reciprocal polar of this theorem and say, That 
 if a fixed point O be taken, and a number of fixed straight lines BT, 
 B,T^, Ifc. be drawn, and another current point S be assumed from 
 
THE THEORY OF RECIFBOCAL POLARS. 
 
 267 
 
 which and from the point O perpendiculars are let fall in pairs on 
 these fixed lines, if the reciprocal of the vector OS, multiplied by the 
 sum of the ratios of each pair of perpendiculars on the fixed lines, be 
 constant, the point S will describe a conic section having its focus 
 at O. For the ratios of each pair of perpendiculars we may sub- 
 stitute the ratio of the segments OQ, QS of the line OS. 
 
 Let O be the fixed 
 point, and let A be s Fig. 61. 
 
 one of the fixed points 
 referred to in the pre- 
 ceding theorem, from 
 which perpendiculars 
 AP=P are let fall on 
 the line CP, In the 
 preceding theorem, of 
 which we require the 
 polar, 
 
 P + P, + P„ -I- &c. equal 
 
 to a constant, let R be 
 the radius of the polar- 
 izing circle, and let the 
 line SQ be the polar 
 of the point A, and 
 let S be the pole of CP. Let fall the perpendicular ST =11 on 
 this line. Now, as the point A and the line £Q are pole and 
 polar, and also the point S and the line CP, we shall have 
 R«=OAx OB=OS X OC, or 
 
 OA_OC_OV_OA-OV AV P 
 
 or 
 
 OS~OB OQ~OS-OQ~SQ' 
 
 OAP OA„_ OAxOB 
 
 OS ~n' °^ OS OS X OB 
 
 R«n 
 
 'OS. OB" 
 
 p n 
 
 Let OS=:r, then ^5-,= — ;^^. 
 R* r.OB 
 
 Now, the projective coordinates of the point S being x and y, 
 and the tangential coordinates of the straight line BQ being 
 
 f and V, we shall have 11= — " ; and the length of OB being 
 
 VP- 
 
 V^ + v 
 
 =, the resulting equation will become 
 
 n 
 
 = l—x^—yv or 
 
 P _ 1— J?f— yi; 
 
 (a) 
 
 0B~' ■** ^^ "' R«- r 
 
 Now, as the sum of the perpendiculars, divided by the square of 
 
268 ON METRICAL METHODS AS APPLIED TO 
 
 the constant radius of the polarizing circle, is constant by supposi- 
 
 P + P +P P 
 tion, we shall have ' Rg" " ' " ="^» °^ substituting for 
 
 P, P,, P,,, &c. their values, since OS=r= sjx'^ + i^, the resulting 
 expression will become 
 
 Let the tangential coordinates f, u,'f,, w,, f,„ Wj/Of the fixed lines 
 BQ, B,Q,, B„Q,,&c. be added together, and let |+f^ + ^„&c. = MV, 
 t; + u,+v^^&c.=»U, and the preceding equation will become 
 
 l+Va; + Uy=C V^^+F, (b) 
 
 dividing out by re. Squaring this equation, and reducing, 
 
 (C2-V*)a?2 + (C'-U«)y2-2VUa:y-2Var-2Uj/=l. . (c) 
 
 But it has been shown that if in the general equation of a conic 
 section Aa;« +A<y« + 2Biiy + 2Ca^ + 2C,y=l, A + C«=A,+C,% and 
 B + CC^=0, the origin is at a focus j and these conditions are 
 satisfied in the preceding equation. 
 
 287.] Instead of defining a conic section as the curve of inter- 
 section of a right cone by a plane, the definition adopted by 
 ApoUonius, De la Hire, Hugh Hamilton, and others, we may define 
 a conic section as a plane section of a surface of revolution of the 
 second order, just as a circle may be defined as the curve of inter- 
 section of a sphere by a plane ; and if, moreover, this plane be 
 drawn through one of the foci of this surface, this point wiU. also 
 be one of the foci of the section. The directrix of this section will 
 be the straight line in which its plane cuts the directrix plane of 
 the surface. 
 
 It is needless to dwell longer on these trite and well-known pro- 
 perties of the focus in the major axis, and the directrix at right 
 angles to it. 
 
 Of the theorem that if a plane be drawn through a focus of a 
 surface of revolution, this point will also be a focus of the section, 
 the following simple geometrical proof may be given. 
 
 Let F and P^ be the foci of the surface of revolution (S), and let 
 FQCQ| be a plane section of it passing through the focus P. Let F, be 
 the vertex of the cone whose base is the section FQCQ,. This will 
 be a right cone, as we shall now proceed to show. Through the foci 
 of the surface let any plane be drawn cutting the surface (S) in the 
 conic section AQBQ/ and the cone in the straight lines P.Q and 
 FjQ(. Let a perpendicular PD to the plane QCQjF be erected at 
 F, meeting in D the tangent plane to the surface at Q. Now, as 
 the focal lines FQ and F,Q make equal angles with the tangent 
 plane at Q, the angle DQP«=DQF; hence in the right-angled 
 
THE THEORY OF RECIPROCAL POLARS. 269 
 
 triangles DQP and DQP, DP=DF and F,P=AB. In the same 
 way DP, may be shown to be equal to DF. Hence a sphere de- 
 Fig. 62. 
 
 scribed with D as centre and DF as radius will be touched by the 
 two tangent planes to the cone through QP and Q,P, ; but we know 
 that when a sphere inscribed in a right cone touches a plane section 
 of it, the point of contact will be a focus of this section. Hence F 
 is a focus of the plane section. 
 
 It is easy to show that the centre of the sphere inscribed in the 
 cone is on the directrix plane of the surface (S). 
 
 288.] If, now, we consider a conic section as a plane section of an 
 oblate spheroid, and passing through the centre, this plane will cut 
 the surface in a plane section, and the umbilical directrix planes 
 (A) and (A,) in two parallel straight lines, since the surface is an 
 •oblate spheroid, which has its directrix planes parallel to the plane 
 of XY, and therefore parallel to one another. 
 
 The sections of this oblate spheroid will have, with respect to 
 these lines, properties not identical with, but analogous to, those 
 which belong to the focus and directrix of the major axis. We 
 may determine the position of these parallel directrices, which, as 
 they are perpendicular to the minor axis of the figure, may be called 
 the minor directrices, as those belonging to the major axis may be 
 termed the major directrices. 
 
 We have shown in sec. [243], (d), that the length of a perpen- 
 dicular P let fall from the polar focus q> on the corresponding 
 
 he . . 
 
 directrix plane is — ; but as in this case i = a, the expression becomes 
 
 ae 
 
 - or — ; and as i=a, e=0, or the polar focus a coincides 
 
 with the centre C of the surface (2) . 
 
270 ON METRICAL METHODS AS APPLIED TO 
 
 We may vrrite h for the distance between the minor directrix 
 and the axis of X. 
 
 As the distance of the umbilical focus v from the plane of XY is 
 
 — -, see sec. [242], (b), and as i=aj and 17'= — p — = — g — =e*, 
 
 we shall have for the distance of v from the plane of XY ce; or as 
 
 h=-, we shall have for that distance Ae*. 
 
 289.] In the hyperbola, as the minor axis is imaginary, the new 
 directrices must be drawn in a somewhat different manner. Let 
 ^ be the angle between the asymptotes of the hyperbola, and on 
 the transverse axis let two points be assumed at the distance a sin ^ 
 from the centre ; through these points let perpendiculars to the 
 transverse axis be drawn, these lines are the minor directrices of 
 the hyperbola ; and if two other points be assumed at the distance 
 
 -: — from the centre, these points are the minor foci. It is almost 
 smv 
 
 needless to mention that the distances of the common directrices 
 
 and foci from the centre of the hyperbola are a cos v and 
 
 ""^ '^ cos X 
 
 respectively; hence, in the equilateral hyperbola, where y=-t, the 
 
 4 
 ordinary and minor directrices coincide, as do also the common 
 and minor foci ; whence we may deduce the very general and re- 
 markable conclusion that. 
 
 The common directrices and fod of the equilateral hyperbola 
 possess two distinct classes of properties — those which belong to them 
 as being the common or ordinary directrices and fod, as also that 
 other new and equally extensive class, to which they are in like 
 manner related, as being the minor directrices and foci of a central 
 conic section. 
 
 In the circle the minor directrices are infinitely distant, and the 
 minor foci coincide with the centre, as is also the case with the 
 ordinary directrices and foci. 
 
 290.] We may, however, derive the properties of the minor 
 directrices and their corresponding foci in a much simpler and less 
 arbitrary way by the help of the method of reciprocal polars, from 
 the well-known properties of the common focus and directrix. 
 
 Thus, let A and B be the semiaxes of the primitive ellipse or 
 hyperbola. On 2A let a circle be described, and let this circle, 
 whose radius is A, be taken as the polarizing circle (II) in the appli- 
 cation of the method of reciprocsd polars. Let (2), a new ellipse 
 or hyperbola, be derived, the reciprocal polar of the former. Let 
 a and b be the semiaxes of this derived section (2) . Then mani- 
 
 A* 
 festly i=A, and «=^> since A is the radius of the polarizing 
 
THE THEORY OF RECIPBOCAX FOLARS. 271 
 
 circle (il). It is clear that the reciprocal polar (S) will have its 
 minor semiaxis A or A coincident with the major axis of (S), while 
 its major semiaxis will be coincident with the minor axis of (S). 
 
 The primitive and reciprocal sections (S) and (2) will be similar ; 
 for 
 
 A* F a« ~^' '-^' 
 
 or the eccentricity e is the same for (S) and (S) . B is the semi- 
 parameter of (S), since B = — (b) 
 
 When the given section is an hyperbola, the reciprocal polar is 
 also an hyperbola, having the same centre and transverse axis, and 
 the angle between the asymptotes of the one equal to the supplement 
 of the angle between the asymptotes of the other. 
 
 In the hyperbola the focal distance to the centre is s/a^ + b^, or 
 a sec "x^ while the distance of the directrix is a cos x. The distance 
 of the minor focus from the centre is a cosec ;^, while the distance 
 of the minor directrix is a sin ^. 
 
 When the hyperbola is equilateral, ai ')(^=-, the major and minor 
 
 foci coincide, as do also the major and minor directrices ; so also 
 do the asymptotes. 
 
 The parabola has neither minor focus nor minor directrix. 
 291.] Let us assume the theorem established in lemma I., p. 228, 
 
 R*rsin0 
 
 PP ' ^*^ 
 
 a simple but most important formula due to Poncelet. 
 ' We shall now proceed to make some applications of this theorem. 
 In any central conic section the sum or difference of the distances 
 of any point on the curve from the foci is constant, or 
 
 FC + F,C=2A=24, 
 since the major axis of the primitive (S) becomes the minor axis of 
 (2) in this transformation. 
 
 Now, by the preceding lemma, rC= pp — ; but R=A=i, 
 
 r=OM, OD=P=-, and P,=OP. Making these substitutions, 
 
 FC=Jesin^.|=rp. In like manner F,C=Jesin^. jyp'; adding 
 these two expressions, 
 
 FC + F,C = 2i=»^M^6esin^. 
 
272 
 
 ON METRICAL METHODS AS APPLIED TO 
 
 Now OP=OX sin and 20X = DM + DM^; consequently 
 
 (b) 
 
 TTierefore, if a tangent be drawn to a conic section meeting the 
 minor directrices in p. -„ 
 
 two points M and M„ ^^' 
 
 thesum of the distances 
 of these points from the 
 minor axis is to the 
 sum. of their distances 
 from the centre in a 
 constant ratio e, the 
 eccentricity of the sec- 
 tion. In the hyper- 
 bola the differences 
 must be taken. It is 
 to be observed that 
 while F and ¥, are the 
 major foci of the pri- 
 mitive conic (S),they 
 are the minor foci of 
 the reciprocal conic 
 (2) . The same is true 
 also of the directrices; 
 the major directrices 
 of the one are the minor directrices of the other. 
 
 292.] The product of the focal perpendicvlars let fall from the 
 minor foci P and P, on a tangent to the curve is to the square of the 
 perpendicular from the centre on the same tangent as the square of 
 the semidiameter a, passing through the point of contact Q, is to the 
 square of the semi- major axis a. 
 
 Let IS and ct^ be these perpendiculars. Then «-=P— iecos^, 
 vtf=V-\-becosd (see fig. 63). Therefore 
 
 ori!r,= P«- JV cos^ 0=a^ sin* + A* cos* S— *V cos* 0, 
 
 a*sin*d + 6*co8*^ . „ . j x- xi. • 
 
 or «rsr^= J , smce o is measured from the mmor 
 
 axis. But it has been shown in sec. [28] that 
 
 a* sin* 5 + A* cos* 6>=P*(af* -i- y*) = Fa «, 
 consequently 
 
 ■sroTy a* 
 
 pT-;^- 
 
 293.] Prom any point Q (see fig. 63) on the curve, perpendiculars 
 QT, QT( are let fall on the minor directrices ; the product of these 
 
* — „2«2— „2' (^) 
 
 THE THEOKT OF KECIFROCAL POLARS. 273 
 
 perpendiculars is to the square of OQ, the semidiameterj in a con- 
 stant ratio, or 
 
 QT X QT, _ b^ _h^ 
 
 2A being the distance between the minor directrices. 
 
 This follows at once from the analogous theorem established in 
 sec. [254] for surfaces of the second order. 
 
 The proof by the ordinary methods is very simple : 
 
 QT=^-y,, QT,=^ + y,or QTxQT,=^-^ 
 
 3 
 
 
 (b) 
 
 ¥c 
 29-l.j III the ellipse or hyperbola the ratio of ix-^^i ■ • (a) 
 
 see fig. 63. Now FC, as we have shown in sec. [290], is equal to 
 
 ^sin^.OM; andDT=*-^cos6>=-^^(P-6ccos6l), or T)t=^. 
 
 Consequently, by substitution in {a.),vr=e sin 6 OM. In like man- 
 ner tr, = e sin 6 . OM, ; consequently 
 
 « OM ,, , 
 
 ^roM/ ("^^ 
 
 or, the ratio of the focal perpendiculars on the tangent is the same 
 as that of the distances from the centre of the points in which the 
 tangent cuts the minor directrices. 
 
 295.] Since, as we have shown in sec. [294] , 
 i!r = c sin d . OM, and i!r,=c sin 5. 0M„ w«j,=e2 sin^ 9 OM . OM,; 
 
 bat «^,= -^, and sm0=^j;j=Q^, or ^^'^'&=q^^q^i- 
 Now putting for QT x QT, its value as given in section [293] , 
 , AV 1 11 V. OMxOM, V* 
 
 namely ^, we shall have q^^qm ,= Z^; 
 
 or, if a tangent be drawn to a central conic section meeting the 
 curve in the point Q, and the minor directrices in the points M and 
 M,, the rectangle under the distances of these points from the centre 
 will be to the rectangle under the segments of this tangent as the 
 sqtiare of the perpendicular from the centre upon it is to the square 
 of the semi- minor axis. 
 
 , „. w OM QM QT , . 
 
 296.] Smce ^ = OM,= QM,=-QT,' ^"^ 
 
 it will follow that tlie ratios of the focal perpendiculars on a tangent, 
 of the distances to the centre O of the points M and M, in which 
 
274 UN METRICAL METHODS AS APPLIED TO 
 
 this tangent meets the minor directrices, of the segments into which 
 this tangent is divided between the point of contact Ql and the minor 
 directrices, as also of the perpendiculars from this point Q, vpon 
 them, are the same for all. 
 
 It will also follow that since OM : OM^ : : QM : QM„ the angle 
 MOM, is bisected by the line OQ. 
 
 297.] The sum of the reciprocals of the segments of any focal 
 chord in a conic section is constant and equal to twice the reci- 
 procal of the semiparameter, or 
 
 ±+±—^ U) 
 
 FC'*'P,C~B« ^ ' 
 
 be OM ■ 
 Now, if we refer to sec. [290], we shall find FC= q„ ; in 
 
 like manner FC,= ^J ; and as A=b and B=— , we shall have, 
 
 by substitution, 
 
 0X + 0X, _2a%_ 2ae . , 
 
 OM ~> ~/6*\ ^ ^ 
 
 Consequently we may enunciate the following theorem : — 
 
 If from any point M in the minor directrix of a conic section we 
 draw two tangents to it meeting the transverse axis in the points X 
 and Xj, the distance between these points iviU be to the distance of 
 the point 'M. from the centre as the distance between the foci is to the 
 semiparameter. 
 
 298.] It is a characteristic property of the focal distances of any 
 point on a conic section that they may be expressed in rational 
 functions of the projective coordinates of that point. A like pro- 
 perty will be found to hold with respect to the distances from the 
 centre of the points in which the minor directrices are cut by a 
 tangent to the curve. 
 
 Let the equation of the tangent to the curve be 
 
 t?yi^-\-b'^xfe=c?b^, (a) 
 
 X, and y, being the projective coordinates of the point of contact. 
 Let the coordinates of the point in which this tangent meets the 
 
 minor directrix be - and ^. Substituting these values in the equa- 
 
 tion (a) of the tangent, we shall have x or DM = —^ — ^ . 
 In like manner D,M,=?^^^^'^. 
 
THE THEORY OF RECIPROCAL FOLARS. 275 
 
 Now OM'=Do' + DM*=^+?^^^^; reducing, we shall find 
 
 In like manner 0M;= — ^ — -^, or OM and OM, may be ex- 
 pressed as rational functions otxi and y,. 
 
 299.] The following investigation is deserving of attention, as it 
 supplies an example of the method of deriving new theorems by 
 means of a double reciprocation. 
 
 Prom any point G in the line DD,let two tangents, GC andGC,, 
 
 Fig. 64. 
 
 be drawn to a circle. The product of the tangents of half the angles 
 which the lines GC and GC, make with the line DD, is constant. 
 
 Let Q be the pole of DD,. Then as the angle D,GC=COD, 
 since the angles at D and C are right angles, the angle D,GC =2CAB, 
 and the angle DGC, is equal to 2C,A.B ; but the tangent of the angle 
 
 CAB= yt" ^^ ^^^ tangent of the angle C,AB = ^'. 
 
 ,' BC X BC, 
 Consequently the product of the tangents is equal to a q v A C 
 
 But ?r^ ^rr' is equal to the ratio of the perpendiculars let fall 
 AC X ACi 
 
 r—k 
 from B and A on the chord CC, — that is, as -—7, putting r for the 
 
 radius of the circle and k for OQ. 
 
276 
 
 ON MKTRICAL METHODS AS APPLIED TO 
 
 300.] Now, if we assume any point F in the plane of this circle, 
 and take its reciprocal polar, which will be a conic section having 
 its focus at F, the pole of the line 
 DDy will be a fixed point S within 
 the curve, and the poles of the 
 tangents GC and GC, will be 
 points T T, on the reciprocal polar; 
 and as the three straight lines 
 DD„ GC, and GC, meet in the 
 
 point G, the three poles S, r, and 
 
 T, will range along the straight 
 
 line tSti ; and the angle between 
 
 the lines DD, and GC will be 
 
 equal to the angle between the 
 
 lines FS and Ft. Hence we may 
 
 infer that if a chord be drawn 
 
 through a fixed point 8 in the plane of a conic section meeting the 
 
 curve in the points t and t„ the product of the tangents of half the 
 
 angles 8Ft and £Ft, will be constant. 
 
 301.] If, now, we take the centre of this conic section as the 
 centre of the polarizing circle, the polar of the focus F will be the 
 minor directrix (A) ; the polars of the two points of contact t, t, 
 wiU be tangents {t), {t,) to the reciprocal polar (S) ; and as the 
 three points r, t„ and S range on the same straight line, their 
 polars {t), (/,), and PQ will all meet on the same straight line. 
 Hence we may derive this other theorem by a second polarization. 
 
 If a straight line PQ be drawn in the plane of a conic section, and 
 from any point P in it two tangents be drawn to the curve meeting 
 
 Fig. 66. 
 _T (A) \a_ 
 
 the minor directrix in two points T and 1,, while the given straight 
 line meets it in Q, the product of the tangents of half the angles TOQ 
 and TpQ will be constant. 
 
THE THEORY OF EECIPROCAL POLABS. 
 
 277 
 
 We may diversify this theorem by taking other centres of polari- 
 zation. Thus we may show that if the diameter of an ellipse be the 
 base of a triangle whose vertex is at a focus, the product of the 
 taiigents of half the angles which the sides of the triangle make 
 with the major axis is constant. 
 
 302.] If from any point E in the plane of a conic section two 
 tangents be drawn, and the chord of contact produced to meet the 
 major directrix in the point y, the lines /E and/7 drawn through 
 the focus /are at right angles. 
 
 Fig. 07. 
 
 Fig. 68. 
 
 Hence, in the reciprocal polar, if a chord ab he drawn meeting the 
 minor directrix in G, and a line be drawn from the minor focus F 
 to meet in Q the intersection of the tangents drawn at a and 6, and 
 cutting the minor directrix in C, the lines OC and OG are at right 
 angles. 
 
 For as Q, fig. 68, is the pole of a/3, and F is the pole of 7M, QF 
 is the polar of the point y ; and as / is the pole of GN, therefore 
 C is the pole oify. Again, as ab is the polar of E and GN the 
 polar of/, G is the pole of/E ; but/E and/y are at right angles ; 
 consequently GOC is a right angle. 
 
 303.] The locus of the feet of focal perpendiculars on a tangent 
 to a conic section is the circle described on the majjor axis. 
 
 From any point C of the circle described on 20A, the major 
 axis of (S), draw a tangent CP to the curve (S) and the chord 
 CF through the focus of (S) ; these lines we know are at right 
 angles. 
 
 Taking the polar reciprocal of this theorem, the pole of the tan- 
 gent CP will be a point Q, on (2) ; and as the polar of F is the minor 
 directrix DG, while the tangent to the circle through C is the polar 
 of the point C, as the circle whose radius is OA is the polarizing 
 circle, the point G where this tangent to C intersects the minor 
 directrix will be the pole of FC ; hence the points G and Q subtend 
 a right angle at the centre ; and as Q, G, C are the poles of the 
 
278 
 
 ON METRICAL METHODS AS APPLIED TO 
 
 three lines CP, CF, and CG which all meet in the point C, these 
 three points Q, C, and G- will all range on the same straight line, 
 the polar of C — that is, the tangent at C to the polarizing circle. 
 
 Hence the points Q, G are on the tangent which touches the 
 circle at C; and as OQ is perpendicular to CP, Q, is the pole of CP ; 
 but CP is at right angles to FC j therefore OQ is at right angles to 
 OG. 
 
 Hence, if two diameters are drawn at right angles in a conic section, 
 one meeting the curve in Q, the other meeting the minor directrix in 
 G, the line QG envelops the circle whose diameter is the minor axis. 
 
 304.] (a) The lines drawn from the foci of a conic section to the 
 point V, in which two tangents to the curve intersect, make equal 
 angles with them. 
 
 Let the angle FVF^=-^, FVQ=(?, F,YQ.,=0,; let/and/^bethe 
 
 Fig. 70. 
 
 focal chords, P and p the perpendiculars from F on the tangents. 
 
THE THEORY OF RECIPROCAL POLARS. 279 
 
 Pj and^, the perpendiculars from F, on the same tangents. Then 
 P =/sin 0, T,=f, sin (i^ + ^), ;>=/sin (^ + 0^) , p, =f, sin 0, ; but 
 
 PP,=j(/;8in08in{^ + ^)=JS 1 
 and |. .... (a) 
 
 i'i'/=^i sin ^, sin {yjr + 0,)=b^,j 
 or 
 
 sin^sin(i^+0)=sin0^sin(i/r + 0,). . . . (1)) 
 But 
 
 2 sin sin {^lr + 0) =co8 1^— cos (1^ + 2^) 1 
 
 and . . . (c) 
 
 2 sin 0, sin (yjr + d,) =cos i^— cos {■^ + 20,) . j 
 
 Consequently cos (■\|r+20)=cos (■^ + 2^,), or 
 
 0=0, (d) 
 
 (/3) The line drawn from a focus F to the point V, the intersec- 
 tion of two tangents which touch the conic section in the points Q 
 and Q„ makes equal angles with the lines FQ, FQ,. 
 
 This is evidently the reciprocal polar of the simple property of 
 the circle, that any pair of tangents make equal angles with the 
 chord of contact. 
 
 (7) Twice the angle between the focal lines drawn to the point 
 V in which two tangents meet, and touch the conic section at the 
 points Q and Q,, is equal to the sum of the angles which the foci 
 subtend at the points Q and Q,. 
 
 Let the normals to the curve at the points Q, and Q^be QN and 
 QN,. Let the angle FQN=F,QN = a, and Fp,N,=FQ,N,= a,. 
 Let QFV=VFQ^=^, and Q,F,V=Vr,Q=iS,. 
 
 But the angle FXF^=2a-|- 2/3, andFXF,=2o, + 2^(; consequently 
 
 But FXF,= FVF, + /3-|-y3(; consequently 2FVr,=2a-|-2a,, or 
 
 2FVF,=FQF,-|-FQ,F, (e) 
 
 305.J If we now take the reciprocal polars of these three theorems 
 (o), (/3), and (y) , we shall find, for (a), that as the two focal vectors 
 PV and F,V, and the two tangents QV and Q^V (fig. 70), all four 
 meet in the same point V, the polar of V (namely QQj, fig. 71) will 
 contain the four poles of these four lines ; and as two of them pass 
 through the foci (fig. 70), their poles Q, Q, will be on the minor 
 directrices (fig. 71) ; and as two of them are tangents to the given 
 curve, their poles P, P, will be on the polar curve ; and as the 
 angles FVQ and F^VQ^ are equal (fig. 70), the segments PQ and 
 VQ, will subtend equal angles at the centre O (fig. 71) ; and as the 
 segments of the tangent which passes through P, namely PM and 
 PMp also subtend equal angles at the centre, the angle POM will 
 be equal to the angle POMy, and therefore the angle QOM will be 
 equal to the angle Q^OM,. And as NN, is a tangent at P,, the 
 
280 
 
 ON METRICAL METHODS AS APPLIED TO 
 
 angle PpN will be equal to the angle PpN,, and the angle QON: 
 the angle Q,ON,. 
 
 Fig. 71. 
 
 11 M _Q 
 
 306.] It has been shown in (e), in the preceding section, that 
 
 2FVr,=rQF,+FQ,P', (fig. 70). 
 
 If we now take the reciprocal polar of this property, the poles of 
 the lines FV and F^V in fig. 70 will be the points Q and Q, in 
 fig. 71, and the poles of the lines FQ, and F,Q in fig. 70 wiU be the 
 points M and M, on the minor directrices, and the poles of the lines 
 FQ, and F^, will be the points N and N, on the minor directrices. 
 Hence the following theorem : — 
 
 If a secant be drawn to a central conic section cutting the carve in 
 the points P and P,, and the minor directrices in the points Q. and Q,, 
 and if through the points P and P, tangents be drawn meeting the 
 minor directrices in the points MM^ and NN^, the sum of the angks 
 which the straight lines MM^ and NN. subtend at the centre will be 
 equal to twice the angle which QQ, subtends at the same centre. 
 
 307.] If from any point Y in the plane of a central conic section, 
 of which the projective coordinates are ^ and q, lines/,/, be drawn 
 to the foci F, F„ and tangents to the curve touching it in the points 
 Q and Q,, the perpendiculars let fall from the point V on the four 
 focal chords FQ, F,Q, ¥0,,, and F,Q, are all equal. 
 
 This may briefly be shown by the method of tangential coordi- 
 nates. 
 
 Let ^, and v, be the tangential coordinates of the focal chord FQ, 
 then the length of the perpendicular VP let fall upon it will be 
 
THE THEORY OF RECIPROCAL POLAKS. 281 
 
 "/f-fr^' ' ^^ sec. [5]. Now l=- and t/, = ^fc^, f and v 
 
 Fig. 72. 
 
 being the tangential coordinates of the tangent YQ which passes 
 through Q. Consequently 
 
 , act— 1 , . 
 
 We must bear in mind that ^, and v, are the tangential coordinates 
 of the focal chord, while f and u are the tangential coordinates of 
 the tangent to the curve passing through the point Q. 
 
 If now we substitute these vjdues of ^, and v, in the value of the 
 perpendicular, we shall have 
 
 „ VP^ {ae-p)b^v+a^gS-age 
 
 1— aef > ■ • ■ • \ > 
 
 from this equation we must eliminate ^ and v. 
 
 The equation of the curve and the dual equation give 
 
 a^^ + b^i^=l, a,ni pS + qv= I (c) 
 
 Eliminating v, 
 
 ^~ flY+*V 
 
 writing M for a^q^+b'^p^—a^^, we shall have also 
 
 aV + 4V 
 
 Substituting these values of f and t; in (b), we finally obtain 
 
282 ON METRICAL METHODS AS APPLIED TO 
 
 Now, this expression for the perpendicular being a function of the 
 projective coordinates p and q of the point V only and the constants 
 of the tangential equation of the curve, it will hold for any one of 
 the four focal chords. Consequently the four perpendiculars will 
 be equal. 
 
 Hence we may obtain the following theorem (fig. 71) : — 
 If through any two points P and P, on a central conic section a 
 secant be drawn, and two tangents through the points P and "P, 
 meeting the minor directrices in the four points M, M,, N, and N,, 
 and perpendiculars be let fall from, these four points on the secant, 
 namely MP, M,P,, NP„, N,P,„, the ratios 
 
 MP M^, NP,, N^, 
 
 MO' up' NO' Np' ^«'' 
 
 will all be equal. 
 
 308.] A simple algebraical proof of this very elegant theorem 
 may be given. 
 
 Let X, yi and a?,^ y,, be the coordinates of the points P and P,. 
 Then the equation of the straight line passing through these points 
 will be 
 
 (y-yi)(^/-a^//)-(y,-y//)(a?-a?/)=0; ... (a) 
 
 or as Xf y, and a;„ y,, are points on the curve, 
 
 a^yf-^b*xf=a^b^; . . (b) a^y,^ + b^x,?=a%^, . . (c) 
 or 
 
 «'(y<-yi/)(y;+yH)+**(^/-'»«)(-»^i+a?„)=o. . . (d) 
 
 Consequently the equation of the secant becomes, by substitution, 
 
 o'(y-y/)(s'/+yfl)+*V-'^/)(«/+^«)=o. . . (e) 
 
 But this line meets the minor directrix in a point of which the 
 
 coordinates are x and -• 
 c 
 
 Substituting these values in the preceding equation, we shall have 
 
 ^ a%^e + e{ahf,y „ + &'a?,y J - a%{x, + x,) 
 
 A*c(a;,+a?„) • • • U) 
 
 Now the tangent through P meets the minor directrix in a point 
 of which the abscissa 2° is given by the equation 
 
 ^^ a\be-y;i 
 
 hex, ^^' 
 
 Subtracting aP from Is, we shall have 
 
 b'ex,{x,+x„) ^°J 
 
 Here i—a!" denotes the distance between the points in which the 
 
THE THEORY OP RECIPROCAL POLARS. 
 
 283 
 
 secant and tangent cut the minor directrix, or a!—x°=QlM, see 
 fig. 71 ; but 
 
 OM= — ." - , see sec. [298] ; consequently 
 
 QM_ (y,fg,-y/g J ... 
 
 0M~ b{x,+x,ii ^' 
 
 But this is a symmetrical expression^ independent of the particular 
 position of any one of the four points M, M^ N, N, on the minor 
 directrices. 
 
 Let MP be the perpendicular from the point M on the secant 
 QQ, ; let <f) be the angle which this secant makes with the minor 
 directrix. Then MPsQM sin ^. Now 
 
 tan^ 
 
 y-y,! _ ^' (^i+a^w) 
 
 consequently 
 Finally, 
 
 
 sin^=-| 
 
 M0-[a*(y, + y„)« + 6V,+^„)*]^' • " " (J^ 
 a symmetrical expression in which x, and y,, x,, and y,, are similarly 
 involved. 
 
 Fig. 73. 
 
 309.] Let VQ, VQ, be two fixed tangents drawn to a central 
 
284 
 
 ON METRICAL METHODS AS APPLIED TO 
 
 conic section, and let t t, be a variable tangent touching the curve 
 in the varying point T ; the tangent t t, will subtend fixed angles 
 at the foci, and the sum of these angles will be equal to the sup- 
 plement of the angle QVQ,. 
 
 Let the normals QN, Q,N, be drawn to the curve at the points 
 Q and Q,, and let the angle FXF, be bisected by the line XNN,, 
 meeting the normals in the points N and N^. Now let the angle 
 
 QFt=TPt=o, Q,Ft,=TFt,=/3, 
 
 also 
 
 let 
 
 Now 
 
 but 
 
 Q,F,T,=TF,T,=a,, QF,T=TF,T=^,; 
 FQN=NQX=7, and P,Q,N,=N,QX=5. 
 FXF,=2y + 2a + 2^, or iFXF,=a + |S + 7; 
 
 iFXF, or F,XN=y +N, therefore a + ^ = N, or tFt,= QNX. 
 
 In like manner a, + /3^=N, or tF,t,=Q,N,X; consequently 
 
 a + fi + a,+ Pi= N + N, = QIQ = supplement of V, 
 
 since YQQ^I is a quadrilateral that may be inscribed in a circle. 
 
 Now applying to this theorem the method of reciprocal polars, 
 we may infer, since the poles of the fixed tangents are two fixed 
 points on the reciprocal (S), and the pole of the movable tangent 
 a movable point on the same curve, and the polars of the three 
 vertices V, t, t, are three chords inscribed in (2), one fixed, the 
 
 Fig. 74, 
 
 others variable, and as the poles of the lines Ft, Ft, are the points 
 
THE THEORY OK RECIPROCAL POLARS. 
 
 285 
 
 in which these chords meet the minor directrix (A), and as the lines 
 Ft and Ft, contain a constant angle, and also F,t and F,t„ we may 
 infer the corresponding reciprocal theorem : — 
 
 If two fixed points A and B be assumed on a conic section, and a 
 third point C variable in position, and if the chords CA, CB be 
 produced to meet the minor directrices (A) and (A,) in the points 
 M, M, and N, N„ the line MM, will subtend a constant angle at the 
 centre O ; so also will the line NN, on the other directrix (A,) ; and the 
 sum of these angles will be constant and equal to the angle AOB. 
 
 That the sum of the angles MOM, and NON, is equal to the 
 angle AOB follows from the theorem established in sec. [305]; for 
 the angle AOM=CON, and the angle BOM, is equal to CON,; 
 consequently the angle AOB = MOM,+NON,. 
 
 310.] When the tangents are drawn to the curve at the extre- 
 mity of the major axis, the supplement of the angle between them 
 is two right angles, since they are parallel. Hence the angle which 
 the moving tangent subtends at one of the foci becomes a right 
 angle, the poles of the two fixed tangents to (S) become the extre- 
 mities of the minor axis of (2), and the preceding theorem becomes 
 thus modified. 
 
 If a triangle be inscribed in a conic section whose base is the minor 
 axis, and whose vertex is variable along the curve, the sides being 
 produced will meet the minor directrix in two points MM,, which 
 will subtend a right angle at the centre. 
 
 When the curve becomes a circle the minor directrix recedes to 
 infinity, the lines OM, OM, coincide with the sides of the triangle ; 
 hence the angle in a semicircle is a right angle. 
 
 311.] Let a quadrilateral be circumscribed to a central conic 
 section, and from one of 
 
 the foci F let vectors FA, J^ Fig. 75. 
 
 FB, FC, FD be drawn to 
 the angles of the figure, 
 and lines Fa?, Fy, Fr, Fm 
 to the points of contact ; 
 and as any two of these 
 adjacent lines make 
 equal angles with the 
 vector line between 
 them, let these angles be 
 a, a; ^, /3; y,i; h, h; 
 then a-|-/3 + 7+8=7r. 
 
 Now the angle between 
 the vectors FA and FB 
 is a-\-P, and the angle 
 between the vectors FC and FD is y + S. 
 
 Let us now take the reciprocal polar of this theorem. The reci- 
 
286 ON METRICAL METHODS AS APPLIED TO 
 
 procal polar of the quadrilateral circumscribed about the conic sec- 
 tion will be a quadrilateral inscribed in a conic section ; and the 
 sides a, b, c, d of this inscribed conic will be the polars of the 
 angles A, B^ C, D of the circumscribed conic, and the pole of the 
 vector FA will be the point Ag on the minor directrix, in which the 
 side a of the inscribed quadrilateral intersects it ; the same may be 
 said for the point Bq. Hence the line Ag B,, on the minor direc- 
 trix subtends at the centre the angle a-{-/3. In the same manner 
 it may be shown that Cq and D,, subtend at the centre the angle 
 7-1-8. Consequently we shall have the following theorem : — 
 
 Let the sides of a quadrilateral inscribed in a conic section meet 
 one of the minor directrices in the points Ag, Bg, Cg, Dg ; the sum of 
 the angles which the segments AgBg and CgDo subtend at the centre- 
 will be equal to two right angles. 
 
 When the curve is a circle the minor directrix recedes to infinity, 
 and the lines OAg, OBg, OCq, ODq become parallel to a, b, c, d, the 
 sides of the inscribed figure. Hence we may infer that the sum of 
 the opposite angles of a quadrilateral inscribed in a circle is equal 
 to two right angles. 
 
 Fig. 76. 
 
 A» Co X)o Bo 
 
 312.] Let ^ be the angle between a pair of tangents to a conic 
 section, ff the focal radii through the point V (the intersection of 
 the tangents), and P the perpendicular let fall from V on one of the 
 focal chords (see fig. 72) ; we shall have 
 
 //,sin^=2aP (a) 
 
 Let the perpendicular from the point V on the major axis divide 
 the angle ^ into the two angles a and tOp or let ^=a + ci>,; hence 
 sin ^ =sin (u cos », -|- sin w, cos at. Let the tangential coordinates of 
 the tangent passing through V be f , v, and f „ v,„ then 
 
 smQ)= — ; ' cos&> = ■ ' , 
 
 SlU «», = — -=^t=r, COS &>,= " • 
 
THE THEORY OF RECIPROCAL F0LAR8. 287 
 
 consequently 
 
 8inrf>= , ^fii-^^ii"! (b) 
 
 We have now to calculate the value of this expression for sin <f>. 
 In sec. [307] it has been shown that 
 
 fc—J^ + gV'M „_a'ff— pVM t _6'u— o VM 
 5, ^ > «'/- £ * f/, j^ ' 
 
 and v„=^9±Pj{^, writing L for 0*5" + A V- 
 L 
 
 Consequently ^i^ii'^^u^i~ — t-—> i^) 
 
 Tj 
 
 and as 
 
 we shall have, by substitution, 
 
 Now, as the focal lines /and/, are drawn from the point V, 
 
 /2=g«+(ae_^)*, and/2=y«+(ae+p)«; . . . (e) 
 or 
 
 fVi' = Up + ?)' + 2(o«-6=')?«-2(«^-6V + {a^-bT] ■ (0 
 
 Consequently (?,* + v,^) (f, « + V) =-^ ; and therefore 
 • . Iy"« + ?/;"/ _ 2^/M _2aP 
 
 Hence, finally, j5f,sin^=2aP, (g) 
 
 where P is the perpendicular from V, the intersection of thejan- 
 gents, and whose value we found in sec. [307] to be flP= VM. 
 
 313.] Hence we may readily obtain a very simple method of 
 finding the curvature of a conic section. 
 
 Let AB be an arc of a conic section, on which two points, P, Q, 
 indefinitely near to each other, are assumed. At these points let 
 tangents to the curve be drawn meeting in s, and intersecting in 
 the angle ^. Let P* = Q» = c ; and let R, R, the radii of curvature 
 at the points P and Q, meet in O. 
 
 Hence R sin <^=PT= 2c ultimately; for ultimately the two sides 
 
 of the triangle P»Q coincide with the side PQ, which is ultimately 
 
 2c 
 equal to PT. Hence sin </> = ^• 
 
 . , ■ 1 .1. X .1 2aP 
 
 But in the preceding article it was shown that sin 9= -sr-- 
 
288 ON METRICAL METHODS AS APPLIED TO 
 
 Hence, eliminating sin ^, 
 
 But when ultimately the point s coincides with the conic section, 
 ■'•" ' V b Fig. 77. 
 
 Hence R=-t, a well-known expression for the radius of curva- 
 ture of a conic section. 
 
 On groups of conic sections having the same minor directrices. 
 
 314.] We shall find that peculiar relations exist between conic 
 sections having the same minor directrices. Let h be the distance 
 between the common centre and oue of the directrices ; then the 
 axes of the curve are connected by the relation 
 1 1.1 
 
 6«~a2'^A*" 
 
 (a) 
 
 A few examples of these properties are given. 
 
 Let a series of concentric conic sections, all having the same 
 minor directrices, be cut by a transversal, the portions of this line 
 intercepted by any pair of these curves will subtend equal angles at 
 the centre ; and if, through every pair of points in which this trans- 
 versal intersects the sections, tangents are drawn intercepted both 
 ways by the directrices, the sum of the angles which any pair of these 
 tangents subtend at the centre is constant, being equal to twice the 
 angle which the common transversal intercepted both ways by the 
 directrices subtends at the centre. 
 
 Let 
 
 "3 + ^2 = 1 and x^ + yv=\ 
 
 b^ 
 
 (b) 
 
 be the projective equation of the curve and the dual equation of the 
 point [x, y). Let y = mx be the equation of the diameter passing 
 
THE THEORY OP RECIPEOCAL POLARS. 289 
 
 through the point {x, y) . Eliminating x and y between these three 
 equations, we shall have 
 
 a'(l-6V)w«-2aa62fu.M + 62(l-a2f2)=0. . . (e) 
 Let n, and n„ be the roots of this quadratic equation, we shall 
 
 Let ^ be the sum of the angles whose tangents are m, and n„. Then 
 
 Now, as the sections are assumed to have the same minor direc- 
 trices, p i~A«' consequently 
 
 a value independent of a and b. 
 
 Let n=tana>, then »,=tan {27r— «b,)= — tan w, ; therefore 
 
 ^==o» + 27r— «B,=a)— Wy ; 
 
 or^is the geometrical difference of the angles a> and «b,; consequently 
 
 tan0=tan(Q)-ia,)=^-j-p^^^. . . . (g) 
 
 Now, when the diflTerence between two variable quantitiesis constant, 
 these variable quantities must receive equal increments ; but these 
 increments are the angles between each successive pair of dia- 
 meters. 
 
 315.] If two diameters at right angles revolve round the centre 
 of two conic sections having the same minor directrices, each dia- 
 meter meeting one of the curves, the line joining these points will 
 envelop a circle. 
 
 Assuming the equation (c) established in the preceding section 
 we shall have 
 
 But when the curves have the same minor directrices 
 
 the equation of a circle. 
 
 When the connected points are on different curves, let r and r, 
 be the two semidiameters ; then 
 
 cos*<^ sin*0_l 1.1 1 _1 
 
 V 
 
290 ON METRICAL METHODS AS AFFILED TO 
 
 consequently 
 
 1 sin^<6 1 ,,, 
 
 a* A* r* 
 
 For the diameter at right angles 
 adding. 
 
 1 cos^^ _ 1 , . 
 
 or the sum of the squares of the reciprocals of the semidiameters, . 
 drawn one to each curve, is constant ; and as 
 
 P the perpendicular on the line joining the feet of r and r, is con- 
 stant, and therefore the locus is a circle. 
 
 The difference of the squares of the reciprocals of any two coin- 
 cident semidiameters of two conic sections having the same minor 
 directrices is constant. 
 
 „ 1 cos* sin* e 1 sin* d , 1 1 . sin* , 
 For;:,=^^ + -p-=^4^^,and^=-,-h-p-> hence 
 
 316.] Let a series of concerdric conic sections, having the same 
 minor directrices, he cut by a common diameter, the tangents dravm 
 through the points where this diameter intersects the curves enve- 
 lop a concentric conic section. 
 
 The solution of this question is simply ohtained hy the method 
 of tangential coordinates. Let 
 
 6V + a*f*=l (a) 
 
 be the tangential equation of one of the series of ellipses or hyper- 
 bolas, and, as they all have the same minor directrices, 
 
 a^'^k^~b^' ^ ' 
 
 let y=nx be the projective equation of the diameter, then 
 
 a^ 
 
 Eliminating o and b between the equations (a), (b), and (c), we 
 find 
 
 the equation of a concentric equilateral hyperbola or ellipse. 
 
 se=» (°) 
 
THE THKORY OF RECIPROCAL POLARS. 291 
 
 On certain properties of the Equilateral Hyperbola. 
 
 317.] A8 the equilateral hyperbola is its own reciprocal, the 
 centre of the curve being the centre of the polarizing circle, the 
 major and minor foci will coincide ; so also will the major and 
 mmor directrices, as shown in sec. [289]. Hence all the focal 
 properties of this curve possess analogous central properties also. 
 For example, we may instance the following : — 
 
 (a) If a tangent to an equilateral hyperbola meet the directrices 
 in the points M and M„ the difference of the distances of these points 
 from the transverse axis will be to the difference of the distances of 
 the same points from the centre in the ratio o/ V2 : 1. 
 
 _ (^) V <i tangent be drawn to an equilateral hyperbola meeting the 
 directrices in M and M„ the ratio of the distances of these points 
 from the centre is equal to the ratio of the focal perpendiculars on 
 the tangent. 
 
 (y) If tv)o diameters at right angles be drawn, one meeting the 
 equilateral hyperbola in the point M, while the other meets the direc- 
 trix in N, the line MN envelops a circle described on the transverse 
 axis as diameter. 
 
 (8) If from any point two tangents are drawn to an equxlatm-al 
 hyperbola, the line joining this point with the focus, and meeting its 
 directrix in M, while the chord of contact meets the same directrix 
 in N, the portion of the directrix MN subtends a right angle at the 
 centre. 
 
 (e) If from a point in the directrix of an equilateral hyperbola two 
 tangents be drawn to the curve, this point, and the point G, where the 
 chord of contact meets the directrix, subtend a right angle at the 
 centre. 
 
 (?) ^ff^om any point in the plane of an equilateral hyperbola two 
 tangents be drawn meeting the directrix in two points M and N, the 
 chord of contact meeting the directrix in Gr, the diameter OG bisects 
 the angle MON. 
 
 (ff) Two chords are dravm from the extremities of the transverse 
 axis of an equilateral hyperbola meeting the curve in any point C, and 
 produced to meet the directrix in the points M andl^. These points 
 will svbtend a right angle at the centre. 
 
 [6) If two fixed points be assumed on an equilateral hyperbola, 
 and a third point variable, the chords drawn through this point and 
 the two fixed points will meet the directrix in two points M and N, 
 which will subtend at the centre a constant angle. 
 
 (k) From any point V let two tangents be drawn to an equilateral 
 hyperbola. Join V with the centre O, through O draw a perpendicular 
 to O V meeting the chord of contact in T. The perpendiculars to the 
 transverse axis through T and V with the two directrices are four 
 harmonicals. 
 
 u 2 
 
292 ON THE LOGOCYCLIC CURVE. 
 
 (\) If two fixed points be assumed on an equilateral hyperbola, 
 and a third point variable, the chords drawn through this point and 
 the two fixed points will meet the directrices in four points M M, arae? 
 N Ny ,■ the sum of the angles at the centre subtended by MM; and NN^ 
 is equal to the angle subtended at the centre by the two fixed points. 
 
 (n) If a straight line be drawn meeting the directrix of an equi- 
 lateral hyperbola in V, and if from any point in this straight line two 
 tangents be drawn meeting the directrix in T and T^, the product of 
 the tangents of the semiangles TOP and T,OP is constant. 
 
 There would be no difficulty in adding to these theorems others of 
 a like kind ; but enough perhaps has been given to show the ex- 
 haustless fertility of the method. The judicious student may easily 
 augment the number; primo avuho non deficit alter. 
 
 CHAPTER )CXX. 
 
 ON THE LOGOCYCLIC CtTRVE, THE TRIGONOMETRY OP THE PARABOLA, 
 AND THE GEOMETRICAL ORIGIN OP LOGARITHMS'^. 
 
 318.] It must doubtless have often appeared strange to geometers 
 that no direct connexion could be traced between those arithmetical 
 quantities called logarithms and certain other corresponding geo- 
 metrical magnitudes. It is true that an imaginary relation between 
 logarithms and circular arcs may be exhibited by means of the 
 theorems of Cotes and De Moivre; but this imaginary relation 
 would seem rather to indicate a real corresponding relation of 
 duality with some other geometrical magnitude. As the principle 
 of duality is of universal application in geometry, and as every 
 property of circumscribed space — whatever be the nature of the 
 bounding lines or surfaces — ^has its correlative or dual, it is a 
 natural anticipation that this duality must exist somewhere for the 
 trigonometry of the circle. 
 
 In the following pages an attempt is made to develop this prin- 
 ciple of duality, and to show the connexion that exists between 
 logarithms and parabolic trigonometry. 
 
 The obscurities which hitherto have hung over the geometrical 
 theory of logarithms will be, it is hoped, now removed. It is pos- 
 sible to represent logarithms, as elliptic integrals usually have been 
 represented, by curves devised to exhibit some special property only; 
 and accordingly such curves, while they place before us the pro- 
 
 • The Bubstance of the following Chapters was embodied in a paper read before 
 the Mathematical Section of the British Association at Cheltenham in 1866, 
 and printed among the Reports of the Association for that year. 
 
ON THE LOGOCYCLIC CURVE. 293 
 
 perties they have been devised to represent, fail generally to carry 
 us any further. The close analogies which connect the theory of 
 logarithms with the properties of the circle will no longer appear 
 inexplicable. 
 
 To devise a curve that shall represent one condition of a theory, 
 or one tnith of many, is easy enough. Thus, if we had first ob- 
 tained by pure analysis all the properties of the circle without any 
 previous conception of its form, and then proceeded to find a geo- 
 metrical figure which should satisfy the conditions developed in the 
 theory, we might hit upon several geometrical curves that would 
 satisfy some of the established conditions, though not all. That all 
 lines passing through a fixed point and terminated both ways by 
 the curve shall be bisected in that point, would be satisfied as well 
 by an ellipse or an hyperbola as by a circle. That all the lines 
 passing through this point and terminated both ways by the curve 
 shall be equal, would be satisfied as well by the cusp of a cardioid 
 as by the centre of a circle ; but no curve except the circle would 
 fulfil all the analytical conditions of the theory of the circle. 
 
 In the same way, no curve but the parabola will satisfy all the 
 conditions of the arithmetical theory of logarithms. 
 
 We shall now proceed to show that if the natural numbers 
 ranging from zero to infinity be represented by the radii of a curve, 
 called the Logocyclic curve^ the logarithms of the numbers repre- 
 sented by these straight lines will be expressed by the residual arcs* 
 of a corresponding parabola described so as to have certain rela- 
 tions with the given Logocyclic curve. 
 
 This theory will decide a controversy long carried on between 
 Leibnitz and J. Bernoulli ou the subject of the logarithms of nega- 
 tive numbers. Leibnitz insisted they were imaginary, while Ber- 
 noulli argued they were real and the same as the logarithms of 
 equal positive numbers. Euler espoused the side of the former, 
 while D'Alembert coincided with the views of Bernoulli. Indeed, 
 if we derive the theory of logarithms from the properties of the 
 hyperbola (as geometers always have done), it will not be easy 
 satisfactorily to answer the argument of Bernoulli — that as a 
 hyperbolic area represents the logarithm of a positive number, 
 denoted by the positive abscissa -)-a?, so a negative number, accord- 
 ing to conventional usage, being represented by the negative ab- 
 scissa — X, the corresponding hyperbolic area should denote its 
 logarithm also. And this is the more remarkable, because by Van 
 Huraet's method the quadrature of the hyperbola itself depends on 
 the rectification of the parabola, as is generally known. All this 
 
 * By the residwd arc of a curve is meant the difference between an arc of a 
 curve, and that portion of the rectilinear tangent drawn at its extremity which 
 is the pri^ection of the radius vector upon it; or, in other words, the residual arc 
 is the difference between this arc and itc protangent. 
 
294 ON THE LOGOCYCHC CURVE. 
 
 obscurity will be cleared up by the theory developed in the text, 
 which completely establishes the correctness of the views of Leib- 
 nitz and Euler. 
 
 It is somewhat remarkable in the history of mathematical science, 
 that although the arithmetical properties of lojjarithms have been 
 familiarly known to every geometer since the time of Napier, their 
 inventor, or rather discoverer, no mathematician has hitherto 
 divined their true geometrical origin. And this is the more sin- 
 gular, because the properties of the logarithms of imaginary num- 
 bers are intimately connected with those of the circle. No satis- 
 factory reason has been shown why this should be so. The loga- 
 rithmic curve, which has been devised to represent one well-known 
 property of logarithms, is a transcendental curve, and has no con- 
 nexion with the circle. Neither has any attempt been made to show 
 how the Napierian base e, an abstract isolated incommensurable 
 number, may be connected with our known geometrical knowledge. 
 Had the circle never been made a geometrical conception, the same 
 obscurity might probably have hung over the signification of tt 
 which has hitherto concealed from us the real interpretation of the 
 Napierian base e. 
 
 This affords another instance, were any needed, to show how thin 
 the veil may be which suflBces to conceal from us the knowledge of 
 apparently the simplest truths, the clue to whose discovery is even 
 already in our hands. The geometrical origin of logarithms and 
 the trigonometry of the parabola ought, in logical sequence, to 
 have been developed by Napier, or by one of his immediate suc- 
 cessors. They had many indications to direct them aright in their 
 investigations. So true it is that men, in the contemplation of 
 remote truths, often overlook those that are lying before their feet ! 
 
 It will be shown in the following pages that the theory of loga- 
 rithms is a result of the solution of the geometrical problem to find 
 and compare the lengths of arcs of a parabola, just as plane trigo- 
 nometry is nothing more than the development of the same problem 
 for the circle. It has been shown, too, elsewhere*, that elliptic 
 integrals of the three orders do in all cases represent the lengths of 
 curves which are the symmetrical intersections of the surfaces of a 
 sphere or a paraboloid by ruled surfaces. These fiinctions divide 
 themselves into two distinct groups, representing spherical and 
 paraboloidal curves ; and by no rational transformation can we pass 
 from the one group to the other. The transition is always made 
 by the help of imaginary transformations, as when we pass from the 
 real logarithms of the parabola to the imaginary logarithms of the 
 circle. When we take plane sections of those surfaces (that is to 
 say, a circle and a parabola), the theory of elliptic integrals becomes 
 
 ♦ " Researches on the Geometrical Properties of Elliptic Integrals," Philo- 
 sophical Transactions for 1862, p. 316. 
 
ON THE LOGOCYCLIC CURVK. 295 
 
 simply common trigonometry, or parabolic trigonometry with the 
 theory of logarithms. 
 
 These views will suggest the reflection, how very narrow is 
 the field of that vast region, the Integral Calculus, which has 
 hitherto been cultivated or even explored ! When we find that 
 the highest and most abstruse of known functions, not only circular 
 functions and logarithms, but also elliptic integrals of the three 
 orders, are exhausted, " used up," in representing the symmetrical 
 intersections of surfaces of the second order, who shall exhibit and 
 tabulate the integrals of those functions which represent the un- 
 symmetrical sections of surfaces of the second order, or generally 
 those curves of double curvature in which surfaces of the third and 
 higher orders intersect? Considerations such as these but add 
 fresh evidence to the truth, how small even in mathematics is the 
 proportion which the known bears to the unknown ! 
 
 319.] The properties of the Logocyclic curve (which we have so 
 named from the similarity of many of its properties to those of the 
 circle, and from its use in representing numbers with their loga- 
 rithms] are very numerous, some are remarkable, and almost all of 
 them may be easily investigated. We shall now proceed to develop 
 a few of the leading properties of this curve. 
 
 On the Equations of the Logocyclic Curve. 
 
 Let r and 6 be the polar coordinates of the curve, its equation is, 
 a being a constant, 
 
 r=o{sec^ + tan5) (a) 
 
 Since sec0=-, tau^ = -, and »-*=ir'+«*, 
 
 X x 
 
 we get for the equation of the curve in rectangular coordinates 
 
 y^{2a-x)=x[x-afA y. 
 
 or {j» + y^){2a-x)=a^x.) 
 
 320.] Let a pai-abola be described whose focus is at the origin, 
 and let its vertical focal distance be put =a. Assume the axis of 
 this curve and the perpendicular to it through the focus as the axes 
 of X and y. Let the vertical tangent OT and the directrix DS of 
 the parabola be drawn, then we shall have the following properties 
 of the Logocyclic curve. 
 
 The directrix DS (fig. 78) of the parabola is an asymptote to 
 the curve; for the perpendicular distance of the point R from 
 this line is a(l— sinfl), since RS = o(sec^— tan^), and this value 
 
 approximates to as ^ approaches to -r. 
 
296 
 
 ON THE LOGOCYCLIC CURVE. 
 
 Let ■^ be the angle which the tangent to the curve makes with 
 the radius vector (or vector, as we may more briefly call it), then 
 
 tan vr = -y- = cos 0. 
 ^ dr 
 
 (a) 
 
 When the vector coincides with the axis a, 6=0, hence tan •^= 1, 
 or the curve intersects at right angles in the point O, making angles 
 of 45° with the axis a. 
 
 Fig. 78. 
 
 The vector »•=« (sec ^ + tan ^) drawn from the focus F of the 
 parabola cuts the loop and one of the infinite branches in the 
 points R and R„ so that 
 
 FR=a(8ec^ + tane), FR, = B(sece— tand) ; . . (b) 
 
ON THE LOGOCYCLIC CURVE. 297 
 
 the product of the segments of this secantj as in the circle, is con- 
 stant and equal to a^. 
 
 These points may be called reciprocal points. 
 
 Since TO=TR=TR,=atane, 
 
 a circle described with T as centre touching the axis in O will pass 
 through the points R and R^ on the vector FS. Hence the curve 
 may be described from point to point with the use only of the rule 
 and compass. Therefore FR^ is always equal to SR, and the angle 
 ROR, is always a right angle. 
 
 321.] Through T draw the straight line TQ at right angles to 
 the line RR^, and touching the parabola in Q ; join the points QR, 
 QR., and draw tangents to the logocyclic curve at the reciprocal 
 
 QT 
 
 points RRj and let them meet in V. Now, as tan QRjT=nv5- ^^^ 
 
 QT = a sec d tan 6, while TR,=a tan 6, .-. tan QR,T=sec 6 ; but 
 tanTRjV or tan -^=008 6 ; .•- ■^ and the angle QR^T are comple- 
 mentary, or QR;V is a right angle. Hence, if normals to the logo- 
 cyclic curve be drawn through any two reciprocal points, they will 
 meet in the parabola conjugate to the logocyclic in a point Q, which 
 point may be called the logarithmic point corresponding to the 
 radius vector passing through RR,. 
 
 Since the tangents of the angles VRT, VR,T are each equal 
 to cos 6, the lines VR and VR^ are equal and equally inclined to 
 the chord RR,. This is also a property of the circle. 
 
 322.] The locus of V, the intersection of tangents to the logocyclic 
 
 at any two reciprocal points R and R^, is a cissoid whose axis is a, 
 
 whose cusp is at O, and whose asymptote is the directrix of the 
 
 parabola. Join VO ; then, since VT =TR( tan i/r= c sin 6, and as 
 
 a perpendicular from O on the vector RR^ is also = a sin 6, VO is 
 
 Ui sin^ 
 parallel to FR, and VO= a sec ^ — a cos = — — g-. But since VO 
 
 is parallel tO TF, the angle VOD=TFO = 0; therefore VO and 6 
 are the polar coordinates of the locus of V. Let this radius be p, 
 then 
 
 a sin* 6 , , 
 
 P=-^^d-' (*) 
 
 which is the polar equation of the cissoid. 
 In rectangular coordinates, since 
 
 apW^ and p« = .^ + ,S .-' + 2/^=^', 
 •^ p cos a '^ " X 
 
 or y«(a-a7)=ir3 (b) 
 
 Hence a straight line VQ. at right angles to a vector VO 
 drawn to the cusp of the curve through a point V on the cissoid 
 
298 ON THE LOGOCYCLIC CURVE. 
 
 envelops a parabola whose vertex is at the cusp, and whose vertical 
 focal distance is equal to a the axis of the cissoid ; or, in other words, 
 the locus of the foot of a perpendicular let fall from the vertex of 
 a parabola on a tangent to it is a cissoid. 
 
 This is easily shown by the method of tangential coordinates. 
 
 Since the line VQ is at right angles to the line VO, VO =/o = 3-. 
 
 Hence, a.s pf=cos0 and pv^&xad, f and v being the tangential 
 coordinates of the line VQ, 
 
 ^=av\ (c) 
 
 which is the tangential equation of a parabola. 
 This may be shown by the ordinary method thus : 
 The equation of the right line VQ at right angles to the line 
 
 VOis 
 
 y-y;+-'(^-a?,)=u=o (d) 
 
 Eliminating y, by the equation of the curve, and making ^—=0, 
 
 we should find the equation of the locus. The work is tedious, but 
 involves no principle of any difficulty, it is therefore omitted. 
 
 323.] The sum of the ordinates of the reciprocal points is equal 
 to the ordinate of the corresponding logarithmic point on the 
 parabola ; 
 
 for y,=r sin6=:a (l+sinfl) tan ^, 
 
 and yii~^i ®i° 0=a (1 — sin 0] tan d ; 
 
 hence y^+y,^=:2a tan 0= the ordinate of Q. ... (a) 
 
 In the same way it may be shown that 
 
 x,+a!ii=2a (b) 
 
 We have also yyfi-\-xfOi,=a^ : . . (c) 
 
 324] . The distances of any point Q on a parabola from its focus 
 and directrix are equal. We may generalize this well-known 
 theorem, and say the distances are equal of any point on a parabola 
 from its logocyclic curve measured along the two normals to this 
 curve drawn through the point Q. This is evident, for the normals 
 QR and QR, make equal angles with the tangent QT. 
 
 Hence, if we conceive the sui-faces of revolution and the para- 
 boloid generated by the revolution of the logocyclic curve and the 
 parabola round the axis DA, a luminous object of the form OR^F 
 between the focus and vertex or the paraboloid would be reflected 
 by the surface of the paraboloid in diverging rays from the con- 
 jugate surface OR, along the diverging lines RQ. 
 
 Through the two reciprocal points R, R^, on the logocyclic curve. 
 
ON THE LOGOCYCLIC CUKVE. 299 
 
 the logarithmic point Q, on the parabola and the intersection on 
 the cissoid at V of the tangents drawn to the logocyclic through 
 the reciprocal points R and R, a circle may be described. This is 
 evident from an inspection of the figure. 
 
 325] . The sum of the polar subtangents FC and FC, belonging 
 to the points R and Rj is constant and equal to 2a ; 
 
 for FC = R,F . tan CR,F =r, tan ■\fr= a (sec 6 — tan 0) tan yfr 
 
 = a (sec^— tan^) cos^, .... (a) since tau'^^cos^; 
 
 hence FC = a (1 — sin^). In like manner FC,=a (1 + sin 0) ; 
 
 therefore FC + FC, = 2a (b) 
 
 The sum of the reciprocals of the polar subnormals is constant 
 
 2 
 and equal to - ; for if v be the subnormal to the point R, 
 
 l_tani/r_ cos _ cos*0 
 
 v~ r ~a(sec^ + tan^) ~«(H-sinfl)' 
 Similarly for the point R,, 
 
 — = ,, , • /), ; consequently - + -=-. . . . (c) 
 
 Hence the locus of the extremities of the subtangents FC and 
 FC( belonging to the reciprocal points R and R, is a cardioid whose 
 diameter is Za, whose cusp is at F, and whose axis is the axis of the 
 figure ; while the locus of the extremities of the polar subnormals 
 is a parabola confocal with the original parabola, and having its 
 
 vertical focal distance equal to -• 
 
 326.] The lengths of the tangents to the curve between any two 
 reciprocal points and the asymptote are equal to one another, or 
 Rf=R;,. 
 
 For in the triangle SR^^ 
 
 n/t, : R^S : : sin R;S^; : sin S^,R„ 
 
 or 
 
 t 
 
 : r : : cos : sin ( - — 
 
 ■fl+t); 
 
 
 
 hence 
 
 t: 
 
 r cos 
 
 
 
 
 ~ cos COS ■^ + sin ^ 
 
 sin'^' 
 
 
 but as r cos 
 
 0=a(l+sin^), and 
 
 cos cos 
 
 Vr= 
 
 isiu'^^, 
 
 
 
 /- " . 
 
 
 
 
 
 smy 
 
 
 
 
 (a) 
 We should find the same expression for the value of R^ 
 
 Hence R/=Rj/, or tsst,= — 
 
 sin^^ 
 
300 ON THE LOGOCYCLIC CURVE. 
 
 Hence also the perpendiculars t'V let fall from the points t and 
 t, on the vector FS are equal to one another and to a. 
 Since St, cos0=a, St,=a sec 0. 
 
 \t,=Ii,C, and Y/=~RC,. 
 As the projection of V/, on FS is equal to 
 ST-SU = a(sec^-tan^), 
 \7, cos 1^ = a (sec — tan ^) = FR, = R,C cos i/r ; 
 therefore V/,=R,C; hence VC = R/,; 
 consequently VC(=R^ (b) 
 
 Therefore the point V is equidistant from the reciprocal points 
 R and R,, and also from C and C,. 
 
 327.] Let p and p, be the perpendiculars let fall from the focus 
 
 F on the tangents drawn through the reciprocal points R and R,, 
 
 then 
 
 . . , , COS0 , o(l + sin^) 
 
 P=rsmir; butsint=-:7(i:p^^^,andr=— ^^^— . 
 
 a(H-sin0) , . 
 
 ^^""^^ ^=v/(l+cos^0) (*) 
 
 T VI a(l— sing) 
 
 In like manner ^'= ^ (l + cos^g) ' ^^^ 
 
 hence ppi = u^sm^ifr. 
 
 In [326] it has been shown that if t, t, denote the lengths of the 
 tangents between the two reciprocal points and the asymptote, 
 
 "'^^sin^t' ^^°*^^ PPfh=<^* (c) 
 
 328.] Let u and m, denote the portions of the tangents inter- 
 cepted betwen the points R, R,, and the feet of the perpendiculars 
 let fall from the origin upon them. 
 
 Then u=r cos ■^, Vj=r, cos •^ ; 
 
 therefore uui=^a^ vos^ yfr. 
 
 Hence also uu,+ppi=a^ (a) 
 
 329.] Let X and \, be the angles which the perpendiculars p and 
 Pi make with the axis FO. Then an inspection of the figure will 
 
 show that \^d + -^— , and \,=fl— i^-l-— . Hence 
 
 \+\=20, (a) 
 
 or the sum of the angles which the perpendiculars on the tangents 
 
ON THE LOGOCYCLIC CURVE. 301 
 
 through the reciprocal points make with the axis are equal to twice 
 the angle which the vector makes with the same axis. 
 
 We may express the perpendiculars on the tangents through the 
 reciprocal points in terms of ■<fr, the angle which these tangents 
 make with the vector, since 
 
 cos0 = ta,U'\jr, sm^=\/{l—t^,ll^^p•), 
 \/(l + cos'' ^)= sec ilr. 
 Making these substitutions in (a) and (b), sec. [327], 
 jB =o{cos ^ + v' (cos* i/r— sin* 1^) },) 
 /Jj^ajcos'v^— V'(cos*-<^— sin* •^)}./ 
 Let V'2sin-^=sin<^, 2 cos*i/r=2 — sin*^, 
 
 or cosi/r= v/(l — isiu'^). 
 
 Hence jd =«{ ^/{l — ^ siu*^)+cos^},J 
 
 ;j, = a{^(l-^sin*<i.)-cos(^}.j ^°' 
 
 These formulae will be found useful in the rectification of the curve. 
 
 On the quadrature of the Logocyclic Curve. 
 330.] Sinco from sec. [319], (b), y= ^"~f'^^^'^ \ 
 
 ax x^ 
 
 " V{2ax-x'^) V{2ax-x^) ^*^ 
 
 Therefore ^yAx=a\~7-T^ k — I— tts — - — Tx- •••(b) 
 
 But al-TTH or=— «V(2ffla;— ar*) +a*versin-'-, . . (c) 
 
 J v[2ax—x^) ' a ^ ' 
 
 *°^ -J V{2ax-x-r '^^^'^-^%^^r-2 ^^^'^-'a- ^ 
 Hence, adding these results together, 
 
 ^ydx=(——)\^{2ax—x)^—^ versin-'-; . . (e) 
 
 no constant is added, for the area begins from x=0. Hence the 
 area of the half loop is found by taking the integral between the 
 limits x=0 and x=a. 
 
 Therefore j'J'yda?=a*-^ (f) 
 
 When we require the area between the infinite branch of the 
 curve and the asymptote, as x>a, we shall have 
 
 Jydar=2- versin-' (~2~) ^(^'J^— «*) +C 
 
302 ON THE LOGOCYCLIC CURVE. 
 
 As the area of the infinite branch of the curve begins with x = a, 
 we shall have 
 
 = — a^ + C, or C = a* r- ; 
 
 4 4 
 
 hence ^^yAx = -^ + C = a^-\-— ; (g) 
 
 consequently the whole area of the logocyclic curve, i. e. the area 
 of the loop and of the curve between the ir finite branches and the 
 asymptote, is equal to 4 a*, or to the square of the distance between 
 the focus and the asymptote, while the difference between these 
 areas is equal to aV. Thus, while the sum of the two areas is equal 
 to the square of 2a, the difference of the two areas is equal to th^ 
 area of the circle inscribed in the same square. 
 
 331 .] The area may be found very easily by a simple transforma- 
 tion. For the loop assume a— ir=a cos 0. Then, substituting and 
 reducing, 
 
 Jyda:=a«cos ^(1 -cos^)d^. 
 
 Hence area of the loopsso*/ sin^ + sin^ cos^— ^1, 
 
 taking the integral between the limits B={) and 6=—. 
 
 The area of loop = a* — —• 
 
 When the area between the infinite branch and the asymptote 
 is required, we must assume 
 
 x—a=as\\\ 6, 
 and Jydx= — a' f sin 5(1 +sin 6)di6 ; 
 
 hence jyda7=— a*cos5 + -^ -^ sin^cos^ + C. 
 
 Since the area begins when 6-^0, 
 
 0=-a2 + C, or C = a«, 
 
 or Jyda:=a*(l —cos &) + — — sin 5 cos 5; 
 
 therefore f2«da!=a'+— :-. 
 
 Jo '^ 4 
 
 332.] If we take the cissoid whose cusp is at F, and whose 
 asymptote is the line DS (see figure 78), its equation may be 
 
 written y^^x/vo^ZT^)'' and if we take the curve (known as the 
 witch or the curve of Agnesi) whose vertex is at F, and whose 
 
ON THE LOGOCYCLIC CURVE. 303 
 
 asymptote is also the lineDS, its equation maybe written, F being tlie 
 origin, y = 2a a/I—-—-)- and the equation of the logocyclic curve 
 
 referred to the same axes being y=(a— a?) a /(n-^ — ), we shall 
 
 have, putting Y^, ¥„, and Yx for any coincident ordinates of the 
 cissoid, the curve of Agnesi, and the logocyclic 
 
 iY„-Y,= +Y, (a) 
 
 Hence the respective areas of these curves must be in the same 
 ratio as these coincident ordinates; or if -we draw a pair of ordi- 
 nates common to the three curves, the area of the logocyclic curve 
 between these parallel ordinates will be equal to the difference be- 
 tween the corresponding areas of the cissoid and half the area of 
 the curve of Agnesi. 
 
 Hence, taking the whole of the areas between these three curves 
 and their common asymptote DS, as the area of the cissoid is three 
 times that of the base-circle, and half the area of the curve of Agnesi 
 equal to twice that of the base-circle, the area of the logocyclic, 
 which is the difference between the areas of these curves, must be 
 equal to the area of the base-circle OAT, as is otherwise shown in 
 sec. [319]. 
 
 333] . Through any two reciprocal points of the logocyclic curve 
 R and R^, let ordinates be drawn to the logocyclic, the cissoid, and 
 the curve which bisects all the ordinates of the curve of Agnesi, 
 and let these ordinates be Y^, Ya', Yc, Y^', and Ya, Yo', and as 
 Xi + a!ii=2a, Xi and Xn being the ordinates of the reciprocal points 
 as shown in [10], we shall have, omitting the traits. 
 
 Hence obviously 
 
 Y,YJ=Y,YJ + Y Y,', (a) 
 
 a curious relation between the six ordinates of the three curves 
 
 drawn three by three through any two reciprocal points of the 
 
 logocyclic curve. 
 
 334] . The logocyclic curve is " inverse to itself." 
 
 A curve is said to be "inverse to itself" when the product of 
 
 any two coincident vectors drawn from the pole to two points, one 
 
 on each branch, is constant. 
 
304 ON THE LOGOCYCLIC CURVE. 
 
 In the case of the logocyclic this is evident ; for, its vectors being 
 the roots of the quadratic equation 
 
 r^-2asec0.r + a^=O, (a)* 
 
 we shall have 
 
 E.=a (see + tan 0) and r=a (sec ^— tan 0), or Rr^a*. 
 This may be shown also in rectangular coordinates. 
 
 Let w, y and x,, y, be the coordinates of any two reciprocal points. 
 
 Then as — ^, ^=r^/= r-^'=^^^.- I° tb« ^^^^^ '"^^"e"' 
 y= a "' ^ . Now, if we substitute these values of a? and y in the 
 equation of the logocyclic curve given in sec. [319], namely 
 
 (a?* + y*) (2rt — ar) = a^x, 
 we shall have 
 
 (57 2 + J, 2) {2a-xi = a^x,; (b) 
 
 that is, we shall reproduce the original equation of the curve by 
 this substitution. This property is peculiar to all curves that are 
 inverse to themselves. 
 
 On the rectification of the Logocyclic Curve. 
 
 335.] It may be shown, if 2 and <r are the arcs of any two 
 " inverse curves " terminating in a common vector which makes 
 the angle with the axis, P and jo being the perpendiculars let fall 
 from the pole on the tangents drawn through the reciprocal points, 
 that 
 
 2 = A^J^, and a=Fjf , .... (a) f 
 
 k being a constant. This is a new formula for rectification and 
 one of remarkable simplicity. It is easily established. 
 Let Rr=A^ and put ru=\. Then R=A*m, and 
 
 ^=M'''-^K*-M-+^>^- 
 
 (b) 
 
 * The polar equation of the circle is 
 
 »-=-2ocosfl.r+o'=*», 
 k bein? the radius of the circle. 
 
 t This expression for the arc of an inverse curve suggests a very beautiful 
 geometrical representation for the velocity of a body in any part of its orbit, 
 subject to any law of central force whatever. For as the velocily in the orbit 
 is inversely as the perpendicular from the centre of force on the tangent, or 
 
 jn •' lis jfc2 " ' 
 
 v^— , we shall have -t^^t^: «• Or the element of the curve inverse to the orbit 
 
 between the vectors drawn through any two consecutive positions of the body 
 will be a direct measure of the velocity with which the body describes that 
 element of its orbit. 
 
ON THE LOGOCYCLIC CURVE. 305 
 
 Now it is easily shown that a//m* + — ) is the reciprocal of the 
 
 perpendicular jB let fall from the pole on the tangent to the inverse 
 curve drawn through the extremity of the vector, and which makes 
 the angle with the axis. Hence 
 
 2= A' I — ; in like manner <r= A* 1 -^ . 
 
 336.] When the curve is the logoeyclic, k=a, and 
 p_ a (1+ sin g) _ a ( 1— sing) 
 4/{l+C0S«g)' ^~*/(l + cos«g)' 
 
 as shown in (a) and (b), sec. [327] . 
 Hence 
 
 l_ (l-sing)(l+cosgg) l_ (l + sing)(l+cos«g) 
 
 P a.cos2gv'(H-oos*g)' p a.cos«g.»/{l + cos*g)' ' ^^' 
 
 therefore 2=-^ ri4^^^Hi+^ . e^g, . . (b) 
 
 v(a)J cos*PV(l — ism^g) ' ^ ' 
 
 i/(2)J co8«gV(l-4sin2g) ^''> 
 
 Adding (b) and (c), we shall have 
 
 2 + ^= ^(2) •« Jeos«g^(f-isin«g) +^(^) -"jTlT^t'-^T 
 
 C no 
 
 Now ^ (2) . a I — 3f)^n _i. • 3/a\ '^ *^® expression for an arc of 
 
 an equilateral hyperbola whose semi- transverse axis is 2a, and whose 
 central vector, drawn to the extremity of this arc, makes an angle 
 o> with the axis, such that sin 6= V'(2) . sin a>. 
 
 The second term is the expression for an arc of the lemniscate 
 whose semi- transverse axis is 2a, and whose vector is inclined by 
 the angle a> to the axis. 
 
 337.] The arc of the logocyclic curve may be exhibited as a 
 function of a hyperbolic arc, an elliptic arc, and a right line, as 
 follows. In a paper published in the Philosophical Transactions * 
 the two following expressions for the arc of an hyperbola have been 
 given, which become when the hyperbola is equilateral (T being the 
 
 arc) 
 
 r Ad 
 
 IcosS'gV'll-isinSg) 
 
 (d) 
 
 y'(2)J{;ir,=2fl*/n —l,^ir,9ft\ .... (a) 
 
 * "Researcheson tlie Geometrical properties of Elliptic Integrals," Philosophical 
 Transactions for 1852, p. 373, (292) (c) and (k). 
 
 X 
 
306 ON THE LOGOCYCLIC CURVE. 
 
 and also 
 
 T = a i^ (2) tan e v/' (1 - i sin^ e) - a ^ (2) Jv/ll - i sin« 5)d5 
 
 '^V{2)}VJX^^J^^'' ^^ 
 
 resuming (d), sec. [336], 
 
 S + o- _ a r d^ a r d0 
 
 2 ~ ♦/ (2) J cos2 dV[l-i sin* 0) "^ v'(2) J */ (1 - i sin* 5) ' 
 and subtracting the two former equations &om this latter, 
 
 ^±^-2T=a*/(2)J' v^(l-4 sin*^)d^ 
 
 -aV'(2)tan^i^(l-isin«5) fe) 
 
 ?tow the integral represents an arc of an ellipse whose semiaxes 
 are \^{2).a and a; and 2T is the arc of an equilateral hyperbola 
 whose semi- transverse axis is 2a. Hence the sum of two such arcs 
 of a logocyclic curve may be represented as the sum of the arcs of 
 an equilateral hyperbola and of an ellipse together with a right line. 
 338.] If we take the diflference of the arcs, we shall find, sub- 
 tracting (c) from (b), sec. [336], that 
 
 2-o- _ C sin ddd r sin 0dd 
 
 2 ~°Jcos20*/(H-cos2d)''"*'j V'(H-cos8^)" • ^^^ 
 
 Let cos ^=tan yfr, tfr being, see (a), sec. [320], the angle between 
 the vector and the tangent to the curve at one of the reciprocal points. 
 Then, introducing the necessary transformations. 
 
 2 — a_ a C i-^lr 
 
 2 sin '^ J cos •^' 
 
 Hence, integrating, 
 
 —2^=^:^-''^°S{Bec-ilr + ta.n^fr) + C: . . (c) 
 
 o 
 —. — -—t is the distance between the extremity of the arc and the 
 smy- ' 
 
 asymptote, and measured along the tangent to the arc at this point, 
 
 as has been already shown in (a), se<;. [326] . 
 
 To determine the constant; when ^=0, ■^=^ir, S=0, <r=0. 
 
 Consequently 
 
 0=a*/(2)-alog{^(2)-H}-|-C, 
 
 or C= alog{v/(2)-».U-aA/(2). . . . (d) 
 
 Hence 
 
 ?:=^=.-«^(2)+«log/-^^|±l I. . . . (e) 
 2 X / o i^secy-Htanyj ^ ' 
 
ON THE LOGOCYCtIC CURVE. 307 
 
 This may be put in another form, since 
 •/(2)+l 
 sec ^ + tan ^^^ (^^^ iTr + tan ifl-)(sec^-tan-f ) 
 
 =seci7rsec^— tani^tani^ + seci/rtanj7r-sec^7rtani/r, 
 or 
 
 {sec jTrsec^— tan iw tau'^} — {seciir tan'^— sec-^tan Jtt} ; 
 and as this may be written 
 
 sec(i7r-T-Vr)-tan(i7r-rVr), (f) 
 
 we shall have 
 
 S-a 
 
 — ^=<_ov'(2)^.alog[sec(i9r-r^)— tan(i7r-r'^)]. . (g) 
 
 Now this last expression represents the residual arc of the para- 
 bola, or the arc diminished by its protanffent, which corresponds to 
 the vector drawn to the loop of the logocyclic, and which is repre- 
 sented by the expression 
 
 r=a[sec (iTT-r'^)— tan (i-TT-rV^)]. . . . (h) 
 (See the Trigonometry of the Parabola, next Chapter.) 
 
 339.] We shall find a remarkable relation to exist between the 
 arc of the logocyclic curve and the arc of the cissoid which is the 
 locus of y, the point of intersection of two tangents touching the 
 logocyclic curve in any two reciprocal points such as R and R, 
 (see figure 78). 
 
 Let S be an arc of the cissoid whose equation is 
 
 p=asm0ta.u0 (a) 
 
 We shall find by the common methods, 
 
 fsing^(l+3cos'g)^^ 
 J cos*« 
 
 (b) 
 
 Make 3 cos* = tan' ^, 
 
 and we shall have 
 
 S 
 
 -af-^ + C (c) 
 
 \^(3) sin ^ J cos <l> 
 
 This is identically the expression that was found in (b), sec. [338], 
 for the diflference of two logocyclic arcs terminating in two reci- 
 procal points on the same vector, or coradial, if one might use such 
 a term. 
 
 The constant may be found from the consideration that the arc 
 S=0, when ^=0, or ^=60; hence 
 
 x2 
 
308 ON THE LOGOCYCLIC CURVE. 
 
 and therefore 
 
 S f 1 2 1 , , r 2 +i/(3) \ ... 
 
 In (e)jsec. [338], a similar expression has heen given for the logo- 
 cyclic residual — ^namely. 
 
 Take the lengths of these curves between the limits 
 
 ^=0 and cos~'ff= ,,„. , 
 V(3) 
 
 or <f> between the limits 60 and 45, and ■^ between the limits 45 
 and 30, then 
 
 and lz^=«{2-V(2)} + alog,/(3)-alog||±^|. 
 Hence, adding these equations together. 
 
 This is the relation which subsists between the arc of the cissoid 
 and the residual arc of the logocyclic between the limits 0=0 and 
 
 340.] The logocyclic curve is the envelope of all the circles whose 
 centres range along the parabola, and whose radii are successively 
 equal to v'{/'— a*), /being the distance of the centre Q of the 
 circle from the focus of the parabola. This follows from an inspec- 
 tion of the figure ; but it may easily be proved as an independent 
 theorem as follows : — 
 
 Let the equation of one of the circles be 
 
 Now o=ffl— atan'5, j8=2atan<^,° /=OBec'^. 
 
 Making these substitutions in the preceding equation, and re- 
 ducing, 
 
 w^+y^ + 2a{ta.n^0~l)a;-Mta.n0y + a^=O=Y; . . (a) 
 taking the differential of this expression with respect to 0, and 
 
reducing. 
 
 ON THE LOGOCYCLIC CURVE. 309 
 
 dV_4fla?tanfl 4oy 
 dd ~ 008*0 "coi^^^ 
 
 or tan0=i. 
 
 Introducing this value of tan in (a), we shall find 
 
 the equation of the logocyclic curve. 
 
 The vertical tangent OT bisects all the chords of the logocyclic 
 curve passing through F. 
 
 The angles VRT or VIl,T='^ and d or OFR are so connected 
 that cos 0=ta,n-^. Hence the maximum ordinate of the loop is 
 found by making ■^=0, or tan ^= cos d, or 
 
 sin»= — ■^ 
 
 If any point Q be taken on the parabola as centre, and through 
 the two reciprocal points on the logocyclic curve a circle be drawn, 
 it will always cut at right angles the fixed circle whose centre is F 
 and radius =a. 
 
 This is evident; for the radius of this circle is equal to 
 (FQ)S-aa=(QT)«, QT being the tangent drawn from Q to the 
 fixed circle. 
 
 341.] It is not difiicult to show that if we put A and A^ for the 
 diameters of curvature of the logocyclic curve at any two reciprocal 
 points of which the vectors are r and 11, we shall have 
 
 ^+^=smir (a) 
 
 If we put C and C, for the chords of curvature of the two reciprocal 
 points and passing through the pole, we shall have 
 
 c+cr' ^^^ 
 
 These simple and remarkable expressions for the curvature of the 
 logocylic curve at any two reciprocal points, are true of all inverse 
 curves whatsoever, at any two reciprocal points. Thus, in the 
 simple case of a circle, let a and b be the segments of any chord 
 
 a + b, then evidently j H 7= 1 . 
 
 a+o a+o 
 
 This very general theorem may be proved as follows : — 
 
 Let iA be the radius of curvature of a curve at any point, r the 
 
 vector from the pole, and p the perpendicular from the pole on the 
 
 tangent to the curve through this point. Then in most elementary 
 
310 
 
 ON THE LOOOCYCLIC CURVE. 
 
 works it is shown that A=2r-T- : and if C be the chord of curva- 
 
 ap 
 
 dp 
 ture through the pole, C = A-=2/>^j and therefore ^=~^and, 
 
 dP 
 for any other curve, 7=r="5p j hence 
 
 /dp dP^ 
 
 dr^dR 
 
 (c) 
 
 Now, in all " inverse curves," as the tangents at any two reciprocal 
 points are equally inclined to the common radius vector, if P, R, 
 and C, are the corresponding quantities for the inverse curve, 
 
 rP 
 ;> : P : : r : R, or p=-^, (d) 
 
 and Rr=A2 ^^^ 
 
 Diflferentiating (d) and (e), 
 
 di»=:dP^+pJ-rP^, andRdr+rdR=Oj 
 
 R^ R 
 
 R" 
 
 hetice 
 
 d/'^.JldP + ^^dr-^R. 
 p Vip tip Wp 
 
 Now¥=P, ^=r, g = l; therefore ^^=¥H-^-f; 
 
 r ' P 
 and therefore as 
 
 Ap dP 
 
 Pr 
 
 R 
 
 (f) 
 
 dR 
 
 j» P , R , dr dR 
 
 ■3-=-j-+l— T— : and as — = — =-, 
 dr dr or ' r R 
 
 we shall have finally 
 
 Ap dP 
 
 dr+5K-^' 
 
 r R 
 
 ^+5^=1 
 
 (g) 
 
 (10 
 
ON THE LOGOCYCLIC CURVE. 311 
 
 342] . The preceding formula will enable usj by a simple trans- 
 formation, to express the relation between the central forces, by 
 which any curve and its inverse may be described, having the same 
 centre of force. For the formula 
 
 ^p dr + *P dR~^' 
 may be written 
 
 and we have also 
 
 p=rsin-s^, and P=Rsin'^; 
 
 ^-- ^^hH^)^^ (^) 
 
 But in every elementary treatise on central forces it is shown that 
 the expression for the centripetal force in any given orbit is as 
 
 f n 3J )■ Putting F for this force, and <t> for the force at the reci- 
 procal point of the inverse curve, we shall have this general expres- 
 sion for the relation which connects the laws of the central forces 
 in the two orbits, 
 
 [r»F-fR'*4>]sin2 4,= l (b) 
 
 Thus, let one of the curves be the focal parabola, the inverse 
 
 curve will be the cardioid, and sin*-Jr=-, while F = :: j. Hence, 
 
 making these substitutions in (b) and putting ^r — k^, we shall 
 have 
 
 ^=i^^ ('=) 
 
 or in the cardioid the force is inversely as the fourth power of the 
 distance, the cusp being the centre of force. 
 
 The polar equation of the curve which is inverse to the focal 
 ellipse is 
 
 R = «{l+ecose), if Rr = 6«- 
 
 a b^ 
 
 Now, in the focal ellipse, F = HlTa' ^^^ ^^^^ "^ — j.m _ \ • Hence 
 
 ^=t|.-£, (d) 
 
 or such an orbit might be described by a body attracted by a force 
 varying inversely as the fourth power of the distance, and repelled 
 by a force varying inversely as the fifth power of the distance. 
 When the centre of force is the centre of an equilateral hyperbola, 
 
312 ON THE LOOOCYCLIC CCEVE. 
 
 the inverse curve is the lemniscate, and 
 
 F=^ sint=^; 
 
 hence R'*=^, or 4»=^x ^; (e) 
 
 or the law of force in the lemniscate is inversely as the seventh 
 power of the distance, the force being attractive to the centre. 
 
 In the same way we might show that as the spiral of Archimedes 
 is the inverse curve to the hyperbolic spiral, and as the law of cen- 
 tral force in this latter is inversely as the cube of the vector, and 
 (o being the vector angle in the former spiral whose equation is 
 
 R =00, we shall have tan -^so), 
 
 or san*'«fr=i =. 
 
 ^ 1+ca' 
 
 Now, in the hyperbolic spiral whose equation is r<o = a, r^F=^; 
 hence, reducing, 
 
 *=*''13+^* (^^ 
 
 or the force in the spiral of Archimedes is inversely as the third 
 and fifth powers of the distance. 
 
 In the same way we might proceed to discover the law of force 
 in other inverse curves. 
 
 A direct expression for the radius of curvatiu-e of the inverse 
 curve, in terms of u, the reciprocal of the radius vector of the ori- 
 ginal curve, may easily be found by putting Il=A*«, and therefore 
 
 lA— L_jlfl_ (g) 
 
 " +'^dP~"d^ 
 
 Let p and p, be the reciprocals of the radii of curvature of the 
 logocyclic curve at the points R and R,j and let p„ be the reci- 
 procal of the radius of curvature of the parabola at a point through 
 which the normal makes the angle '^ with the axis, then 
 
 p + p,=8p„; (h) 
 
 •^ and are connected by the condition tan ^= cos 0*. 
 
 * So far as we have been able to learn, the first notice of " Inverse Curves " 
 will be found in a short paper published by Mr. Stubbs, of Trinity College, 
 Dublin, in the Philosophical Magazine for 1843. Mr. Ferrers has also given a 
 paper on the " Inversion of Curves," in which he combines the principle of 
 " inversion " with that of " reciprocal polars." It will be found at p. 32, voL i. 
 of the Quarterly Journal of Pure and Applied Mathematics. Other notices of 
 this method wiU doubtless have appeared elsewhere ; but the properties given in 
 the test would nnt seem to have been published before. 
 
ON THE TaiGONOMETRY OF THE PARABOLA. 313 
 
 CHAPTER XXXI. 
 
 ON THE TKIGONOMETRY OF THE PARABOLA. 
 
 343.] Let the angles to, ^, and ;^, which we shall call conjiigate 
 amplitudes, be connected by the equation 
 
 tan<u = tan^8ecx+tan;^sec0 (a) 
 
 Hence a> is such a function of <f> and ^ as will render 
 tan [<l>, x\ =tan «f> sec j^ + tan ;^ sec ^. 
 We must adopt some appropriate notation to represent this func- 
 tion. Let the function [^, x] ^® written <f>-'-Xt ^o tl^** 
 
 tan (,'}>-^Xi =tan^ sec^ + tan^^sec^. 
 This equation must be taken as the definition of the function ^-^x- 
 In like manner we may represent by tan (^-rx) the expression 
 
 tan sec x— tan x sec ^. 
 From (a) we obtain 
 
 sec(u = sec(^-^j^)=sec^secj^ + tan<^ tan^. . . (b) 
 If we now diflferentiate the equation 
 
 tan a = tan ^ sec ^ + tan x sec ^, 
 we shall have 
 
 . secc»= — ^.-.secd secvH 2L tanAtanvl 
 
 cos a cos ip ^ '^ cos x I 
 
 H ^ tan <p tany H ^^ sec d> sec y. 
 
 C0S(|> ^ '^ cosx - 
 
 Adding these expressions together, and introducing the relation 
 established in (b), we shall find 
 
 do) _ d<^ I ^X /jN 
 
 cos o) cos ^ cos X 
 
 This is the difiFerential equation which connects the amplitudes 
 to, ^, and X- 
 
 As to, <ft, and x ^^ supposed to vanish together, we shall have by 
 integration 
 
 "'• ^^ (e) 
 
 f do, ^fd.^ Cdx 
 J cos to J cos <f> J cos X 
 
 I cos X 
 
 or, in the more compact notation, 
 
 fseco)d«a=Jsec^d^+Jsecxd;^. . . . (f)* 
 
 * The relation between the conjugate amplitudes a, <p, and x was originally 
 obtained in this way. In the theory of eUiptic integrals, any three conjugate 
 amplitudes are connected by the equation 
 
 cosa)=cos</)Cosx— 8in(/>sinx Vl — t'ein'o) ; 
 
314 ON THE TKIOONOMETKY OF THE PARABOLA. 
 
 Hence, if m, <f>, and x are connected by the relation assumed in (a) , 
 we shall have the simple relation between the integrals expressed 
 in (e). 
 
 If in (a) we make the following imag mary substitutions — 
 that is to say, put V — Isina for tan^, V — Isin^S for tan^, 
 V — 1 sin y for tan w, cos a for sec (}>, cos /8 for sec Xi cos 7 for sec oi, 
 and change ->- into + and -r into — , we shall have 
 
 sin y=sin (a+ ^) =sin a cos /3 + sin /3 cos a, 
 the well-known expression for the sine of the sum of two arcs of a 
 circle. 
 
 We shall show presently that an arc of a parabola measured from 
 the vertex may be expressed by the integral J sec ^d^, being the 
 angle which the normal to the arc at its other extremity makes with 
 the axis, or the angle between the normals drawn to the arc at its 
 extremities. 
 
 ->- and -r may be called logarithmic plus and minus. As exam- 
 ples of the analogy which exists between the trigonometry of the 
 parabola and that of the circle, the following expressions in parallel 
 columns are given — premising that the formulae marked by corre- 
 sponding letters may be derived singly, one from the other, by the 
 help of the preceding imaginary transformations. 
 
 In applying the imaginary transformations, or while tan <f> is 
 changed into V — 1 sin^, sec ^ into cos^, cot ip into — V — 1 cosec <f>, 
 ->- must be changed into +, -r into — , and fsec^d^ into 
 
 The reader who has not proceeded beyond the elements of trigo- 
 nometry may assume the fundamental formula as proved. He will 
 find little else that requires more than a knowledge of plane trigo- 
 nometry. 
 
 t is called the modulus. When we laake t=0, we get 
 
 co8(i)=cos^ cos^ — sini^ sinx or <o=0+x 
 in the trigonometry of the ciicle. When we take the complement of 0, or make 
 »:=1, we get 
 
 8ecia=8ec(^8ecx+tan^tanx or a>=0-^x 
 in the trigonometiy of the parabola. Whence, as above, 
 tan a>=tan (^ sec x+tan x sec if). 
 
 * The advanced reader hardly needs to be reminded that this is the imaginary 
 transformation by which we are enabled, in elliptic functions of the third order, 
 to pass from the circular form to the logariihrmc form, or to pass from the pro- 
 perties of a curve described on the sur&ce of a sphere to its analogue described 
 on the sur£Eu;e of a paraboloid of revolution. See the author's paper " On the 
 Geometrical Properties of Elliptic Integrals," in the Philosophical Transactions 
 for 1852, pp. 362, 308, and for 1864, p. 53. 
 
ON THE TRIGONOMETRY OF THE PARABOLA. 315 
 
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316 ON THE TRIGONOMETRY OF THE PARABOLA. 
 
 Since 
 
 sec (^ -1- 4>) =860* ^ + tan* <^, and tan (^ J- <^) = 2 tan ^ sec ^, 
 
 sec (^ -•- ^) + tan (<^ ->- ^) = (sec ^ + tan <f>) *. 
 
 Again, as 
 
 sec {<f>-^^-*- ^)=sec {(^ -^ <^) sec ^ + tan {<f> -^ ^) tan <f>, 
 and 
 
 tan (^ -L (^ J- <^) =tan (^ -^ <^) sec </> + sec {«^ -^ 0) tan «^, 
 
 it follows that 
 
 sec (<^ -L -J- ^) 4- tan (<^ -L <^ -1- <^)= (sec ^ + tan ^ )^, 
 and so on to any number of angles. Hence 
 
 8ec(<^-^<^-L^...tow^)+tan(<^J-<^-^<^...to«<^)=(sec^+tan<^)'. (|) 
 Introduce into the last expression the imaginary transformation 
 
 tan ^= V — 1 sin <f>, 
 and we get Demoivre's imaginary theorem for the circle, 
 
 cosm^+ V"! sin«<^ = {cos<^+ x/ — 1 sin <^}-". 
 This is a particular case of the more general theorem 
 sec(a-L/3-L7J-S-i-&c.)+tan(aJ-/3J-7-i-S-i-&c.) ) ,„> 
 
 = (seca + tano) (sec^ + tan/3) (8ec7 + tany) (secS + tan S) &c.*/ 
 
 In the circle, 
 
 l±tan|^ /l+sin2. ^ ^^^j 
 
 l-tan<^ V 1— sin2^ ^ ' 
 
 accordingly, in the parabola, 
 
 1+ V^sin<^ _. / l+ V^tan(^-L^) ,. 
 
 1- V^lsin</)~^ 1- i/-ltan(0-'-^)' 
 In the circle, 
 
 tan*./.= ;^!°;t"''"t? ; (bb) 
 
 ^ 2sm2<^ + 8m4<^' ^ ' 
 
 hence, in the parabola, 
 
 sin* 6 - 2 tan (.^-^.^)-tan (,^ -J- ^ -i. j, -J- «f,) 
 
 '^~2tan(<;!)-i-0)+tan(0-i-^-i-^-i-^)- • " ^l^' 
 In the circle, 
 
 cos2<^=co8*^— sin*0; (cc) 
 
 hence, in the parabola, 
 
 sec (<^-'-^)=8ec*^— tan^<^ (yy) 
 
 * Hence 
 
 cos(a+/S+y+8+&e.)+ V^sin (a+^+y+«+&c.)=(coBa+ VITIsina) 
 (cos j3+ V^ sin /3)(cos y+ V ^ sin y)(cos 8+ V^ sin ») &c. 
 
• 
 
 sin^ .. , 
 
 ON THE TRIGONOMETRY OP THE PARABOLA. 317 
 
 In the circle, 
 
 '*' cos*<^cos*;^ ^ ' 
 
 therefore, in the parabola, 
 
 sin«.^-sin«X= *'°^'^"'ff'^i^"^^ - . . . (8S) 
 ^ sec* ^ sec* ;^ ^ ' 
 
 In the circle, 
 
 tan^ = x/IE££iM feel 
 
 ^ V l+co82<^ ^^^> 
 
 Accordingly, in the trigonometry of the parabola, 
 
 ^ Vsec(./,-^^) + l ^"^ 
 
 sin</> ^ 8in(</,-x) 
 
 sin^ sin{x— ^)' ^ ' 
 
 it is easily shown that tan ^, tan ^ and tan ■^ are in harmonic pro- 
 gression. 
 
 Hence it follows in parabolic trigonometry, that if 
 
 tan^ ^ tan(«^-rX) , v 
 
 tan^ tan(x-r^)' ^**' 
 
 sin ^, sin ■)^, and sin yjr are in harmonic progression. 
 
 Let <u be conjugate to •^ and <a, while «a, as before, is conjugate 
 
 to ^ and ^. Then we shall have 
 
 tan w =tan (<^ -^ X "*" V^)* 
 or 
 
 tan (<^ -'-;^-'-'^)=tan ^ sec ;^ sec ■^ + tan ^ sec ■^ sec <}> 
 
 -f-tan'^sec^sec;;^ + tan^tanxtan'^, ....(■») 
 
 sec (^ -"- X -•- ■^) = sec ^ sec ;^ sec •^ + sec ^ tan j^ tan -^ 
 
 + secj^tan'^tan^+sec'^tan^tanx, . . . . {p) 
 and 
 
 sin (A ■ V ■ Tfr)^ sin^ + sinx+sinVr + sin^sinxsin-^ 
 ^r A. T^ l + sin^siu'^ + sin'^sin^ + sin^sinx' 
 whence, in the trigonometry of the circle, 
 
 sin (^+x+'^) =8in ^ cos ^ cos -^-1- sin x cos-^ cos ^ 
 
 + sin ^ cos (f> cos X— sin ^ sin ^ sin '^, . . . . (p) 
 
 cos (^-f-x + V^) = cos ^ cos^ cos'^— cos^ sinxsiu'^ 
 
 — cos^sin'^sin^— cos'^sin^sinxj . . . . (r) 
 tan (A 1 V I ^w *an«^ + tanx+tan^-tan^tanxtanVr 
 vr /^ T/ 1— tan^tan'^— tan'^tan^— tan^tanx 
 
318 
 
 ON THE TRIGONOMETRY OF THE PARABOLA. 
 
 We have here a remarkable illustration of that fertile principle of 
 duality which may be developed to such aa extent in every depart- 
 ment of pure mathematical science. 
 
 The angle ^ -■-</) may be called the duplicate of the angle «^, the 
 angle tf>-^<f)-^^ the triplicate, and the angle (^ J-^to ra terms) the 
 n-plicate of the angle 0. 
 
 The reader will observe that the signs J- and -r connect the 
 angular magnitudes only of the parabola, while numerical quantities 
 are connected by + and — . Thus, in the circle, we have </>+% 
 and a + b indifferently, while in the parabola we must use the nota- 
 tion ^ j-j^ or ^-r^^for angles, but a+b or a— 6 for lines, as in the 
 circle. 
 
 345.] An expression for the length of a curve in terms of a per- 
 pendicular p let fall from a fixed point on a tangent to it, and 
 making the angle 6 with a line passing through the given point or 
 pole, uaxaelj s=^pd0 + t, has been established in sec. [201]. In 
 the following figure. 
 
 p = ST, ^=VST, t=VT. 
 
 Fig. 79. 
 
 Let TI{m.0) denote the length 
 of the arc of a parabola, whose 
 parameter is 4m, measured from 
 the vertex to a point at which the 
 tangent to the arc is inclined to 
 the ordinate of that point by the 
 angle 0. When m = l, the symbol 
 becomes Tl{0}. 
 
 In the parabola whose equation 
 is y'^=^mx, the focus S is taken as 
 the pole, and therefore ^=m sec 0, 
 while PT or /=»» sec^ tan 5. 
 
 The arc of a parabola, measured 
 from the vertex, may therefore be 
 expressed by the formula 
 
 n(m. d) =m sec d tan -frnj* seed d^. ... (a) 
 
 The difference between the arc and its subtangent t has been 
 named the residual arc. 
 
 For brevity, and for a reason which will presently be shown, the 
 distance between the focus and the vertex of a parabola will be called 
 its modulus. Hence the parameter of a parabola is equal to four 
 times its modulus. 
 
 Let H {tn . io),'n.{m . <j>),'n(m . x) denote three parabolic arcsVD, 
 VB, VC, measured firom the vertex V of the parabola. Let, more- 
 over, a, (f>, and x ^ conjugate amplitudes. Then 
 
ON THE TRIGONOMETRY OP THE PARABOLA. 319 
 
 n(m. &>)=?» tan 6) sec<B + »M fseco) d<a,] 
 
 n(m.^)=mtaii<^sec(^ + w Jsec^d(^, >. . . (b) 
 
 n(»». j^)=jM tan x^^^ X"^"^ J '^^^ X ^X'l 
 
 Whence, since J sec to dm =Jsec <j) d^+J sec j^ d;^, because to, <f>, and 
 "X, are conjugate amplitudes, we get, after some reductions, 
 
 Tl{m.w) — n(ni.^)— n(»».^)=2ni taniBtan <f> tan;^. (c) 
 It is not difficult to show that 
 
 tan o> sec m — tan <^ sec <^— tan x sec x—'^ tan to tan <f> tan x- (d) 
 
 Put for tan to, sec to, their values given in (a) and (b), sec. [343], 
 Write (sec* <^— tan* ^) and (sec* j^— tan^;^) for 1, the coefficient of 
 tan <f> sec tf) and tan x sec x ^^ the preceding expression, and we shall 
 obtain the foregoing result. 
 
 346.] Let y, y,, y„ be the ordinates, to the axis of the parabola, 
 of the extremities of the arcs n(»».<B), 11 (m.^), and n(7».;^). 
 Then y = 2m tan to, y^^= 2m tan <f>, y„ = 2m tan x- Therefore 
 
 2ni tan o) tan ^ tan j^; =^-'^' (a) 
 
 We have therefore the following theorem : — 
 
 The algebraic sum of the three conjugate arcs of a parabola, mea- 
 sured from the vertex, is equal to the product of the ordinates of 
 their extremities divided by the square of the semiparameter. 
 
 To exemplify the preceding theorem. Let 
 
 1 v'5 
 
 tan ft) = 2, tan ^ = ^, tan x= — ^> 
 
 then _ 
 
 /= .i V5 3 
 
 secft)=v5, sec 9 = — q-, secx=n> 
 
 and these values satisfy the fundamental equation of condition, 
 
 tan tB= tan tp sec j^+ tan x sec «^. 
 
 Now _ _ 
 
 n(m.w) = »w2 V'5 + mlog{2+ */5), 
 
 n(m..^)=m^ + mlog(l±^), 
 Uim.x)=m^-^ + mlog{^} 
 Hence, since log (3+ V5)=log(?-t_V^)+log(^-±^),weshall 
 
320 
 
 ON THE TRIGONOMETRY OF THE PARABOLA. 
 
 have n(»».i»)— n(fn.^)— n(»».x)=»» V5; • • • (b) 
 
 and m V5 = 2m tan w tan <f> tan ^. 
 
 347.] If we call an arc measured from .the vertex of a parabola an 
 aspidal arc, to distinguish it from an arc taken anywhere along the 
 parabola, the preceding theorem will enable us to express an arc of 
 a parabola, taken anywhere along the curve, as the sum or difference 
 of an apsidal arc and a right line. 
 
 Thus, let VCD be a parabola, S its focus, and V its vertex (fig. 
 80). Let VB = n{»i . ^), VC=n(»i . x), VD=n(m . o>), and let 
 
 ^^'= A. Then(c),sec. [345] showsthat the parabolic arc (VC + VB) 
 
 =arc VD— A, and the parabolic arc VD— VB=BD = VC + A, 
 When the arcs n(m . ^) and 11 (m . ;^) together constitute a focal 
 
 arc, or an arc whose chord passes through the focus, ^ + y=-5, and 
 
 A is the ordinate of the arc VD. Accordingly we derive the fol- 
 lowing theorem : — 
 
 Any focal arc of a parabola is equal to the difference between the 
 conjugate apsidal arc and its ordinate. 
 
 Kg. 80. 
 
 The relation between the amplitudes ^= ( «— X ) *^d «* ^^ tbis 
 
 2 cos( 
 
 case is given by the equation sin2^s= J" t^jam Thus, when the 
 
 1 ^COS €0 
 
 focal chord makes an angle of 30° with the axis, we get cos to=\, 
 or y=10iR. Here, therefore, the ordinate of the conjugate arc is 
 ten times the modulus. 
 
ON THE TRIQONOMEXB.Y OV THE PARABOLA. 321 
 
 When ^=x> (*')' ^^^- P^S], is changed into 
 
 n(»i.o)) — 2n(»».<^)=2ff»tan<atan«<^; ... (a) 
 or as tan o>=2 tan (^ sec <}>, see (ij) of [344], 
 
 n(»i.(o)— 2II(m.^)=4mtan^^sec<^. . . . (b) 
 
 Let ^=45, then 11 fm.^ j is the arc of the parabola intercepted 
 
 between the vertex and the focal ordinate ; and as 
 sec « = sec (^ J- ^) = sec* ^ + tan* <^, 
 we shall have, since tan ^=1 and sec ^ = ^2, sec o> =3 ; therefore 
 
 n(7».8ec-'.3)-2n(»».j)=4m ^2. . . . (c) 
 
 Now, as sec 10=3, tanik>=2 V^, and the ordinate Y=4m ^2, we 
 may therefore conclude that the parabolic arc whose ordinate is 
 4m V2, diminished by this ordinate, is equal to the sum of the arcs 
 of the parabola between the focal ordinate produced both ways, and 
 the vertex. 
 
 It is easy to give an independent proof of this particular case 
 without the help of the preceding theory. 
 
 The length of the parabolic arc whose amplitude is 45° will be 
 foimd by the usual formula to be 
 
 n(m.l)=, 
 
 :»rai/2 + mlog(H- \/2) ; 
 
 and twice this arc is 
 
 2n(m.'^=m2 '»/2 + »»log(3 + 2 '/2), since (1+ V'2)*=3 + 2 ^2. 
 
 The parabolic arc whose amplitude is sec~' 3, is found in like 
 manner to be 
 
 n(m.sec-'3)=»»3.2^2+OTlog(3 + 2 \/2). 
 Subtracting the former equation from the latter, 
 
 n(»i . sec-' 3) -2U (»»-|) =4»M v'2. 
 
 Now the ordinate Y of the parabolic arc whose amplitude is sec"' 3 
 is equal to _ _ 
 
 2b».2 V'2=4»» v/2; 
 therefore 
 
 n(m.8ec-'3)-2n(jre.|)=Y. . . . (d) 
 
 It is easily shown that 4w V2 is the radius of curvature of the 
 extremity of the arc whose amplitude is 45°. 
 
 Y 
 
322 ON THE TBIGONOMETEY OF THE PARABOLA. 
 
 To find a parabolic arc which shall diflfer from twice another 
 parabolic arc by an algebraical quantity, may be thus exemplified. 
 
 Let tan ^=2, tana>=4v'5, sec^= VS, 8eco»=9; 
 substituting these values in (a), sec. [345], we shall have 
 n{»» . sec-' 9)=»n36 '/5+m log (9 + 4 \/5), 
 and 2n(»» tan-' 2) =2m 2 \/5 + «» log (2+ V5)^. 
 
 Consequently, since (2+ V'5)*=9+4'/5, 
 
 n(m . sec-' 9)— 2n(»M . tan-' 2) =»i32 v 5 =2»» tan a tan««^. (e) 
 
 348.] We may in all cases represent by a simple geometrical 
 construction the ordinates of the conjugate parabolic arcs whose 
 amplitudes are <f>, Xi ^^^d en. 
 
 Let BC be a parabola whose focus is S and whose vertex is V. 
 
 Fig. 81. 
 
 Let YS=m; moreover let VB be the arc whose amplitude is ^, 
 and VC the arc whose amplitude is j^. At the points V, B, C 
 draw tangents to the parabola ; they will form a triangle circum- 
 scribing the parabola, whose sides represent half the ordinates of 
 the conjugate arcs VB, VC, VD. 
 
 We know that the circle circumscribing this triangle passes 
 through the focus of the parabola. Now 
 
 VT=mtan<^,VT,=»Mtanx, T,A=mtan^ sec^, TA=»»tanxsec^j 
 hence 
 
 TA + TA=m(tan <f> sec x + ^-ai X ^^^ «^) =»» tan at ; 
 
ON THE TBIGONOMETRY OF THE FABABOLA. 
 
 323 
 
 therefore VT, VT', and TA + AT' represent half the ordinates of the 
 arcs whose amplitudes are ^, x> ^^^ '">• 
 
 When VB, VC together constitute a focal arc, the angle TAT, is 
 a right angle. 
 
 The diameter of this circle is m sec ^ sec x- 
 
 The demonstration of these properties follows obviously from the 
 figure. 
 
 349.] It may be convenient, by a simple geometrical illustration, 
 to show the magnitude of the frmctions sec (^-^x) and tan (<^-'-x). 
 
 Let SV=»», ASV=x, BSV=<^, the line AB being at right 
 angles to SV. Through the three points ABS describe a circle. 
 Draw the diameter SC, and join the point C with A and B. Let 
 fall the perpendicular CT. 
 
 Fig. 82. 
 
 Then m sec (<^ J-x)=SC+CT, and mtan (<^-'-x)=AC+CB. 
 Moreover also it follows, since 
 
 sec (<^ -^ x) + **^ (•/• -^ X) = i^'^ ^ + **^ </>)(sec ^ + tan x), 
 as has been established in (?) of sec. [344] , that 
 
 m(SC + CT+AC+CB)=(SB+BV)(AS+AV). . (a) 
 Of this theorem it is easy to give an independent geometrical 
 demonstration. 
 
 We have manifestly also 
 
 CT(SC + m+SA+SB) = (AC+AT){BC+BT). . . (b) 
 350.] Let m be the conjugate amplitude of a> and y^, while oi is 
 the conjugate amplitude, as before, of (f> and x- Then, as 
 j" sec w d«!>= Jsec w dta + J sec ■^ d'^, 
 
 Y 2 
 
324 ON THE TBIGONOMETRT OF THE PARABOLA. 
 
 and 
 
 f sec 0) do> = J sec d> d^ + J sec x ^X> 
 we shall have 
 
 j'sec»d5=jBec^d^+j'sec%d%+j8ec'^d'^; . (a) 
 and if !!(»».«), II(»».<^), 11 (m.^), and n(»».'^) denote four conju- 
 gate parabolic arcs, 
 
 U(m.l^-U{m.<f,)-U{m.x)-Ti{m.ir) = 
 
 2mta.ii{<j>-^X)tsa{<f>-^ylr}tsa{x->-'^), . • (b) 
 which gives a simple relation between four conjugate parabolic 
 arcs*. 
 
 To exemplify the foregoing formula. Let us assume the follow- 
 ing arithmetical values for the angles 5, ^, %, ■^ : — 
 
 tan5 = 12±^, tan0=l tanx=-:^, tan,^=| 
 
 _ _ (c) 
 
 sec5=i±l^, sec<^=-^, 8ecx=l, sec^=| 
 
 Hence 
 
 n (m.tan-' [l^±^^])=ro{20-|-9 Vl) +»»-^-l-mlog(6 + 3 y/5), 
 
 n (»t.tan-l) = ». J^ + m log {^-^^)> 
 
 n («».tan- J^)=«»iV5 + m log (3W5), 
 
 (4\ 20 
 »i.tan-'^j=»»-n-+mlog3. 
 
 * This latter theorem may he proved as follows : — Since a is conjugate to a 
 and ^, we shall have, by (c), sec. [345], 
 
 n(m . a) — Il(m .a)—Jl(m, ^)=2 m tan a tan d> tan ^ ; 
 and since a> is conjugate to and %> 
 
 n(m . <»)— n(>» . ^)— n(»n . x)=2 mtan » tan ^ tan x. 
 Hence, adding these equations, n(m . o>) will disappear, and 
 
 n(m . o) — n(m . f) — n(m . x)— n(ni . ^)=2>n tan a> [tana tan'^-|- tan ^ tan x]- 
 Now tan a=tan (n-'-^). 
 
 Therefore tan ms=tana> sec iff+tam^ seem. 
 
 But tano>=tan0sec x+tanxsec^. 
 
 Substituting this value in the preceding equation, and multiplying by tan ^, 
 tan « tan iff = tan ^ sec X sec ^ tan ^ + tan X sec sec 4' tan ^ 
 +8ec sec X tan' 4i+i«a ^ tan x tan' ^i, 
 and tan0tanx=sec'^tan0tanx— tan'^tan^tanx- 
 
 Consequently 
 
 tan a tan iff+tan tan x=Csec i^ tan 0+sec tan ^)(8ec x tan ^+eee if/ tan x) 
 =tan (0 -'-«'') tan (x -■- >!'), and «= ^■'- %• 
 
 (d) 
 
ON THE TRIGONOMETRY OF THE PARABOLA. 325 
 
 Adding the latter three equations together, and subtracting the 
 sum from the former, the logarithms will disappear ; for 
 
 =log(6+3v^5); (e) 
 
 consequently 
 
 n(»ra.m)— n(m.^)— n(OT.x)— n(»i.i/r) 
 
 ^^^1604-73 V5) = 2m.2.( 5 + 4 v^5 ^(l2±5^^ . (f) 
 
 since 
 
 tan(«^-X)=2,tan(<^-i-^) =^±1J^, and tan (v-'-^)=li±Li:^. 
 
 6 6 
 
 351.] Let, in the preceding formula (b), ^=;^='^, and we shall 
 have 
 
 n (»» . 5) — 311 (m . «^) = 2»i tan« (^ J- X) = 16ot tanS ^ sec^ <^. 
 
 We are thus enabled to assign the difference between an arc of a 
 parabola whose amplitude is ^ = (^ -■- (^ -^ ^) and three times another 
 arc. 
 
 If in (tr), [344], we make ^=;^='^, 
 
 tan0=4tan^^ + 3tan^ (a) 
 
 Introduce into this expression the imaginary transformation 
 
 tan^= v^ — Isinfl, change J- into +, 
 
 and we shall get sin3^ = — 48in*^+3 8in5, which is the known 
 formula for the trisection of a circular arc. (a) may therefore be 
 taken as the formula which gives the trisection of an arc of a 
 parabola. 
 
 The following illustration of the triplication of the arc of a para- 
 bola may be given : — 
 
 Take the arcs whose ordinates Y and y are 4m and m respectively. 
 Let a and (]> be the amplitudes which correspond to these ordinates ; 
 then, as _ 
 
 Y=:2mtan0=:4m, tan^=2, secci>=V'5; 
 and as 
 
 1 /K 
 
 y=s2»ttan^=m, tan^=^> 8ec^= • 
 
 Now these values of tan m and tan <f> satisfy the equation of condi- 
 tion (a), namely 
 
 4 tan^0+ 3 tan ^=tan w. 
 
326 ON THE TRIGONOMETRY OF THE PARABOLA. 
 
 But 
 
 and 
 
 n(»i.tan-'2)=»»2v'5+mlog(2+ \^5), 
 
 and three times this arc is 
 
 3n(»».tan-'|)=TO|i/5+»nlog(2+ ^^5), 
 
 since 
 
 Subtracting tbis latter equation from the former, the logarithms 
 disappear, and we get 
 
 n(wi.tan-'2)-3n (m.tan-'|)=^5_^=16»»tanS./.8ec8^. (b) 
 
 Now, as the radius of curvature B is equal to the cube of the 
 
 normal divided by the square of the semiparameter, B=: Z— , 
 
 4 
 since N=2»» sec S. We have therefore the following theorem : — 
 
 The arc of the parabola whose ordinate is equal to 4m, or to the 
 abscissa, diminished by the radius of curvature of its extremity, is 
 equal to three times the arc whose ordinate is m, or one fourth that 
 of the former arc. 
 
 It is evident that the chord of the greater arc is inclined by an 
 angle of 45° to the axis, or the ordinate is equal to the abscissa, 
 while in the lesser arc the ordinate is four times the abscissa. 
 
 This is the point on the parabola up to which the ordinate is 
 gi-eater than the abscissa; beyond this point it is less than the 
 
 Another example of the triplication of the arc of a parabola, 
 or of finding an arc which, diminished by an algebraical quantity, 
 shall be equal to three times another arc, may be given. 
 
 Let 
 
 3 — 
 
 tan^=Q, tan&>=18, 
 
 sec«i>=.X^^ secw=5Vi3. 
 
 These values satisfy the equation of condition, 
 4 tan^^ +3 tan ^=tan 0. 
 

 
 ON THE TRIOONOHETRY OF THE PARABOLA. 327 
 
 Hence 
 
 n{OT.tan-'.18)=m90. Vr3+mlog(18+5 Vl3), 
 
 and three times this arc is 
 
 an (»».tan-'|) =^i|Q2 +^log(18 + 5 i/ 13), 
 since 
 
 ^3+J^y=184-5^13. 
 
 Therefore, subtracting the latter equation from the former, 
 
 n(«».tan-. 18) -sn (,».tan-|)=m351|^ = 16^(|)''( V^y. (c) 
 
 When there are five parabolic arcs, whose amplitudes 
 ^, x> V'j "> ^ "'^ related as above, namely 
 
 we may proceed to obtain in like manner a formula which will 
 connect fire parabolic arcs whose amplitudes are connected by the 
 given law. 
 
 352.] To find the arc of a parabola which shall differ from n times 
 a given arc by an algebraical quantity, may be thus investigated. 
 
 Let <f> be the amplitude of the given arc, then 
 
 n(m.^) =m sec ^ tan ^ + m log (sec <^ + tan <p), 
 and n times this arc is 
 
 »n (m . ^) = nm sec ^ tan ^ + >» log (sec ^ + tan ^) ". 
 Let ^-i-(^-i-^-»-^ to n terms=^, then 
 n(»».4>)=msec*tan4>+j»log(sec<l>+tanO), sec. [344]. 
 Now sec 4>+tan^= (sec </) + tan ^)", as shown in (f). 
 Hence n(»».^) — Mn(wi.«^)=»»[sec «I> tan ^—n sec (}> tan <^]. 
 Let seC(^+tan^=:X, then sec^ + tan<l>=:X", and 
 
 . x+x-» . , \-x-> 
 
 sec 9:= — n — > tan<p^ — „ — • 
 
 We have also sec4>= — 5 , tan<E>= 5 — . Hence 
 
 iTf if.^ TT, ^^ r(\2«_\-!!»)-n(X«_X-i')-i 
 n(»».<E>)— ran(m.<^)=TOl i '-g — i -' . 
 
338 ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 
 
 3 5 
 
 Let n=3, tan ^=3;, sec ^=t> ^•=3. Then 
 
 n(m.4.)-3n{m..^)=5(5j5y. 
 When »=:4j 
 
 U{m.^)-4,U{m.^)=m-^-^ — ; 
 
 and so may n be taken as any other integral number. 
 
 353.] The equation (a), sec. [351],afirords a very simple mode of 
 expressing the real root of a cubic equation. 
 
 Let the cubic equation under the ordinary form be a^+px=:q. . 
 
 Let the parabolic equation tan^o) + 7 tan m= — ■^— be written 
 
 tan^oi + — T- tan»=-r tanfl (a) 
 
 4 4 ^ ' 
 
 by introducing the modulus m ; hence 
 
 Now, since the value of x, found by the ordinary methods, is 
 
 V2^V27^4^V2 V27+4' 
 we shall have 
 
 2x=m *^8ecI2+tann— OT'v'secil— tanO, . . (b) 
 
 while 
 
 m 
 
 =^vi. '«'»=i?\/i- 
 
 When the sign of p is negative, the solution must be sought in 
 the trigonometry of the circle. 
 
 CHAPTER XXXII. 
 
 ON THE OEOHETRICAL OKIQIN OE LOGARITHMS. 
 
 354.] In the trigonometry of the circle we find the formula 
 
 <v— tor. «v tan»a . tan^d tan'd . 
 
 d=tand —^—^ ^+&c. ... (a) 
 
 And if we develop by common division the expression 
 1 cosd 
 
ON THE OEOMBTBICAL ORIGIN OF LOGARITHMS. 329 
 
 and integrate, 
 
 Jd^ f OAO ■ /,.8in»0 sins^ sin?^ , „ „, 
 
 If we now inquire what, in the circle, is the magnitude of the 
 trigonometrical tangent of the arc which differs from its subtangent 
 by the distance between the vertex and its focus ; or, as the sub- 
 tangent is in the circle, and the focus is the centre, the question 
 may be changed into this other, what is the trigonometrical tangent 
 of the arc of a circle which is equal in length to the radius ? This 
 question would be answered by putting 1 for <& in (a), and reverting 
 the series 
 
 - ^ ,-, tan»(l)^tan5(l) tan7(l) . .^ 
 
 l=tan(l) gi^ + — ^ yM + &c. . . . (c) 
 
 By this process we should get, in functions of the numbers of Ber- 
 noulli, the value of tan(l), as is shown in most treatises on trigo- 
 nometry. 
 
 Let us now make a like inquiry in the case of the parabola, and 
 ask what is the value of the subtangent of the amplitude which will 
 give the difference between the arc of the parabola and this sub- 
 tangent equal to the distance between the focus and the vertex of 
 the parabola. Now, if 6 be this angle, we must have 
 
 ll{m.0)—m sec 6 tan 6=m. 
 But in general, as shown in sec. [345], 
 
 ll{m.0)—maec tan 0=m^aec0di0. 
 
 We must therefore have, in this case, J sec^d^=:l. If we now 
 
 revert the series (b), putting 1 for Jsec 0d0, we shall get £rom this 
 
 particular value of the series, namely 
 
 . . - sin^tf sin^fl , sin^d „ 
 
 l=sm^+— 5— +— g— + -y-+&c., . . . (d) 
 
 an arithmetical value for sin 0*. This we shall find to be 
 
 sin 0=-i Tij e being the number called the base of the Napierian 
 
 e T^ 
 logarithms. Hence sec 5-|-tan 0=e ; or if we write e for this par- 
 ticular value of to distinguish it from every other, 
 
 sec E+ tan e=e =2-718281828, &c (e) 
 
 let c^mn 0, then 
 
 , /l+sinflx _/. J, , Bm'6 ,Bio.'6 , Bin's ,em'6 . \ _ 
 
 or 
 
 /1+rinA^ or(l±^ = e. Hence sec 0+tan e=e. 
 \l-smW ' V costf / 
 
S30 ON THE OEOMETRICAI. ORIGIN OF LOGABITHMS. 
 
 We are thus (for the first time^ it is believed) put in possession 
 
 of the geometrical origin of that quantity so familiarly known to 
 
 mathematicians — the Napierian base. From the above equations 
 
 we may derive 
 
 e* + c-* , ci— e-' ... 
 
 sece= — 2 — , tan€= — ^ — , (f) 
 
 or tan 6=1175201195, whence €=-8657695, . . (g) 
 
 or c=49° 36' 18"*. 
 
 The corresponding arc of the parabola will be given by the follow- 
 ing series : — 
 
 n{««.e)=2m [l+^+j^ + j2^ &c.], 
 
 * Mr. Merrifield, F.R.S., has been good enough to calculate the 
 exact magnitude of the angle e, the Napierian base angle. No 
 relation has hitherto been discovered between this angle e and the 
 angle which subtends the arc of a circle equal to unity. 
 
 To find 
 c from sece+tan6=e, or from logj^ tan (^ir+ie)=10'43429 .... 
 The best working formula is : 
 
 if log tan {x + h) =log tan x+r, 
 log A=log (4 Mr sin 2a?) + \r cos %x nearly. 
 
 log tan (a? + A) = 10-43429 44819 0325 
 
 log tan ir =1043423 67350 3220=logtan69°48' 
 
 r = 5 77468 7105 
 
 r is positive, but cos 2a: is negative = — cos 40° 24' ; the correction 
 of log h is therefore subtr active, and the work is as follows : — 
 
 Correction of log A. logr=5-76152 84574 
 
 log r =5-7615285 log4=9-69897 00043 
 
 log i =9-6989700 log M =0-36221 56887 
 
 logBin2a?=9-81165 53569 
 
 ar. CO. log 1" =5-31442 51332 
 
 log cos 2a?= 9-8816918 
 
 5-3421903 
 Correction=2,19882 
 
 0-94879 46405 
 Corrections —2 19882 
 
 A=8-88735 75414 logA=0-94877 26523 
 
 .-. 45°+ie=69° 48' 8"-88735 75414 
 6=49° 36' 17"-77471 5083 
 The last two figures are of doubtAil accuracy. 
 
ON THE GEOMETRICAL ORIGIN OF L0QARITHM8. 331 
 
 since the subtangent in this case is equal to 
 
 m8ec6tan£=-7 (e®— e~*). 
 4 ^ 
 
 355.] If we now extend this inquiry, and ask what is the magni- 
 tude of the amplitude of the arc of the parahola which shall render 
 the difference between this parabolic arc and its subtangent equal 
 to n times the distance between the focus and the vertex, we shall 
 have, as before, by the terms of the question, 
 
 Tl{m.ff) —maecB tan 6=nm. 
 But, in general, 
 
 II(m.0) —m sec 6 tan 5=jm Jsec Q AB\ 
 hence we must have 
 
 »=j8ec6d5=log(8ec^ + tan^), or sec^+tan^=c». 
 
 Now we may solve this equation in two ways : — either by making n 
 a given number, and then determine the value of sec^+tanS, which 
 may be called the base ; or we may assign an arbitrary value to 
 secd+tand, and then derive the value of n. Taking the latter 
 course, let, for example, 
 
 sec ^ +tan 6= 10, then n=log 10 j 
 
 or putting B for this angle, 
 
 secS+tanS=10. 
 
 To represent Numbers and their Logarithms by the Logocyclic 
 Curve and its Conjugate Parabola. 
 
 356.] A parabola whose focal vertical distance is a or 1 being 
 drawn, and also its logocyclic curve, let a vector be drawn to the 
 latter equal to the given number n. Then (see fig. 83) 
 
 N=sec^+tan& (a) 
 
 Let this line meet the vertical tangent in T, the parabolic arc 
 OQ— QT is the logarithm of n. 
 
 It is clear that the infinite branch of the curve from + oo to 
 will give vectors of every magnitude £rom oo to 1, and parabolic 
 arcs from k> to j hence, whUe the vectors or numbers range from 
 « to 1, the parabolic arcs or logarithms range from oo to 0. When 
 the number lies between 1 and 0, the vector representing it is 
 drawn belmo the axis ; its extremity wiU be found on the loop, and 
 the corresponding arc of the parabola will be negative : hence the 
 logarithm of a positive number is equal to the logarithm of its reci- 
 ^ocal, with the sign changed ; for the magnitude of the parabolic 
 arc depends on 6, and 6 is the same in sec d+tan 9, as in its reci- 
 procal sec d— tan 6. 
 
 Hence, while the infinite branch of the logocyclic curve from 
 
332 ON TH£ GEOMETRICAL OBIOIN OF LOGARITHMS. 
 
 + 00 through H, O, pia F, may hy its vectors represent all positive 
 numhers &om + co to +0, the two infinite branches of the parabola 
 will be used up in representing the logarithms of positive numbers 
 firom + oo to + 0; that is, the upper or positive branch of the parabola 
 will be expended in representing the logarithms of positive numbers 
 from + <x> to +1, and the lower or negative branch of the parabola 
 in representing the logarithms of positive fractional numbers from 
 + 1 to +0. There is, therefore, no construction by which we can 
 represent negative numbers or their logarithms ; consequently such 
 numbers can have no logarithms. 
 
 Let vectors be drawn from F to the logocyclic curve equal to 
 e, c*, e*, c*. . . c", e being the Napierian base ; then these lines will 
 meet the tangent to the vertex of the parabola in the points T, T,, 
 Tg . . . T„; and tangents being drawn from these points, touching 
 the parabola in Q, Q^, Q,,„ Q,„, On, the logarithms of these numbers 
 will be 
 
 0Q-QT=1, OQ;-Q,T,=2, OQ„-Q„T„=3, . . . 
 
 0Q„-Q,T„ = (« + 1); (b) 
 
 hence the logarithms of e, e^, e^, e" are 1, 2, 3, . . . n. 
 
 In like manner we should find the logs of e, e^, e^, e^ . . . e« to be 
 
 357.] Let a series of vectors be drawn from the point F to the 
 logocyclic curve in geometrical progression, and let them be 
 
 (sec^+tan^), (sectf+tan^)«, (sec^+tan^)* . . . (see 6+ tan. 0); 
 
 meeting the vertical tangent to the parabola in the points Tp T^„ 
 T„; . . . T«, and let the tangents drawn from the points T,, T„, &c. 
 touch the parabola in the points Q,, Q„, Q^^ . . . Ct,; let the d^er- 
 ence between the first parabolic arc and its protangent be S, or let 
 S be the residual arc, then we shall have 
 
 OQ,-Q,T,=S, OQ„-Q„T„=28, 0Q„,-Q,„T„,=3S, 
 0Q"-Q,T„=n8. 
 
 Or while numbers increase in geometrical progression, their loga- 
 rithms increase in arithmetical progression. 
 
 Assuming the common methods of logarithmic differentiation as 
 known, it becomes evident that the residual arc of the parabola is 
 the logarithm of the corresponding number ; for if the given num- 
 ber r be put under the form 
 
 r=a (secd+tan^, — = a, 
 
 ^ ' r cosp 
 
 integrating 
 
 '^°S^=J^- 
 
ON THB GEOMETRICAL ORIGIN OF LOGARITHMS. 333 
 
 Now the parabolic residual arcs=OQ— QT=al- 
 
 COB e' 
 
 Hence logr= ^=^ (a) 
 
 As every number whose logarithm is to be exhibited must be put 
 under the form sec 5 + tan 0, which is of the form \-^x, since the 
 limiting value of sec ^ is 1, we discover the reason why in deve- 
 loping the logarithm of a number^ the number itself must be put 
 under the form 1+a;, or some derivative &om it, and not simply 
 under that of x. 
 
 If we equate sec ^ + tan B with 1 -f-jr, we shall find 
 
 ^= l-tanig - I'et«=taiii«, . . . . (b) 
 
 1 +M 
 
 then N=sec 5+tan 6 = \ +a;=-r , which is another familiar form 
 
 v—v 
 
 under which a number is put whose logarithm is to be developed 
 in a series. In such a form, u represents the tangent of half the 
 angle that the vector of the logocyclic (which represents the num- 
 ber) makes with the axis. 
 
 358.] Let us assume the relations established in sec. [344] : 
 
 tan(a-i-/3)=tanasec/3-f tan/3 seca,| ,, 
 
 sec (a -"t ^) = sec o sec ^ -J- tan a tan ^. j 
 Therefore 
 
 sec (a-'-|3)-f-tan{a-'-fl)=(seca-t-tana) (secjS+tan/S). 
 In like manner it may be shown that 
 sec (a-i-/3-Ly) -J- tan (o-i-/3-«-y) 
 
 = (seco-|-tano)(secj3-Htan j3)(sec7-|-tan7), . . (b) 
 
 and so on for any number of angles. 
 
 Hence, if we draw vectors from F to the logocyclic, making the 
 angles a, /3, y, S, &c. with the axis respectively, and another making 
 the angle (a-i-/3-'-7-^S-'-&c.) with the same axis, this latter vector 
 will be equal to the continued product of the former, or 
 
 ra.rp.rY.r4=E(a4.pj.fco.)j (c) 
 
 and the sum of the residual arcs of the parabola corresponding to 
 the former will be equal to the residual arc of the parabola corre- 
 sponding to the latter. 
 
 If in (b) we put a^^^yszS, 
 
 sec (a -•-a-'- a -"-to n places) -(-tan (o-'-o-'-a-'-to « places) 
 
 = {seco-|-tana)" (d) 
 
 Change sec into cos, tan into ^{ — 1) sin, and -*- into +, then 
 {cos na+ ^(—1) sin na} = {cos o -I- ^( — 1) sin a}". 
 
331 ON THE OBOHETBICAL ORIGIN OF LOOABITHHS. 
 
 The values of the expressions sec {a-'-a-'- Sec.) and tan (a ->- a -■- &c.) 
 are very easily found. 
 
 Let sec a + tan a= n, 
 
 ., N + N-' . N — N-* 
 
 then seca= — „ — , tana= — - — , 
 8ec(o-'-o)= — - — , tan(oJ-a)= ^ — > 
 
 £i At 
 
 , , ^ , n8+n-» + ^ , , X N^-N-s Ke) 
 sec(a-^a-^a)= 5 — , tan (a -1-0-^0)= 5 — 
 
 sec(a-'-o-'-a-'-to»terms) = — ^ — , tan(fl.-^oJ-a-i-toiterms) = — ^ — . 
 
 This gives a very beautiful and simple law for the magnitudes of 
 the secants and tangents of multiple logocyclic or parabolic angles. 
 But this theorem may be extended ; indeed it follows from (b) . 
 Let 
 
 sec a 4- tan a= n, sec |3 + tan /3=»i, sec 7 + tan y =p, 
 sec 8 + tan 8=5', &c. 
 
 (f) 
 
 Then sec(a-^g-Lyu.8)^'^"^g + '"-'»"'^"'g"V 
 
 and tan(a-L^-.-y^8)='""^g-'"^'""'^''g"'. 
 
 Now, in circular trigonometry, let 
 
 cosa+ V'(— 1) sino=m, co8/3+ V'( — 1) sin^ = «, 
 co8y+i/(— l)siny=p, cos S+ ■/( — 1) 8in8=5, &c., 
 
 then cos(a+j3+7 + e)= — ^-^^^ — 5 — *-— ^J 
 
 and sm(a+^+y + 8)= "^ 2^(-i) -J 
 
 These relations place in a striking point of view the analogies 
 between circular and parabolic trigonometry. It is not a little 
 curious that these imaginary expressions for the cosines and sines 
 of multiple arcs in the circle should have been discovered so long 
 before their real analogues were found for the secants and tangents 
 of multiple angles in parabolic trigonometry. 
 
 359.] Since we have shown that negative numbers have no loga« 
 rithms, at least no real ones, and imaginary ones can only be educed 
 by the transformation so often referred to, this leads us to seek them 
 among the properties of the circle. For as d always lies between 
 
ON THE GEOMBTRICAL ORIGIN OP LOGARITHMS. 335 
 
 and a right angle, or between and the half of ±ir, sec ^4; tan 6 
 is always positive ; therefore negative numbers can have no real or 
 parabolic logarithms, but they may have imaginary or circular 
 logarithms ; for if in the expression 
 
 log{cos * + */-^^sin ■&} =S v'^^, 
 
 wemaked=(2ra + l)7r, we shall get log( — l)=(2ra + l)9r ■>/ — 1. 
 
 Hence also, as the length of the parabolic arc TP, without refer- 
 ence to the sign, depends solely on the amplitude 0, it follows that 
 the logarithm of sec 6 — tan ^ is equal to the logarithm of sec ^ + tan ^. 
 We may accordingly infer that the logarithm of any number is equal 
 to the logarithm of its reciprocal, with the sign changed, since 
 (sec d+tan 0) (sec ^— tan 0)=\. 
 
 When 6 is very large, sec tf +tan 6=2 tan 6 nearly. It follows, 
 therefore, if we represent a large number by an ordinate of a para- 
 bola whose focal distance to the vertex is 1, the difference between 
 the corresponding arc and its subtangentwill represent its logarithm. 
 
 360.] Let f sec ^ A<f>=p, ("sec ^ d^ =g ; then as 
 
 f sec o) A<0 = f sec ^ d^ + J sec x ^X' 
 J sec«ado)=/> + y, and to=^-^x- 
 
 Hence if <j> be the amplitude which gives the residual arc=;>, 
 and X t^6 amplitude which gives the residual arc=g', ^-J-;^ is the 
 amplitude which will give the residual aic=p + q. In the same 
 way we might show that, if ■^ be the angle which gives this residual 
 arc=r, (^-'-X-^V') ^^ *^^ angle which will give this residual arc 
 =p + q+r. 
 
 Let a be the amplitude of the number A, and p its logarithm ; 
 /8 the amplitude of the number B, and q its logarithm ; y the am- 
 plitude of the number C, and r its logarithm. Then 
 
 A=seca+tano, B=sec^+tan/3, C=sec7 + tan7, 
 and log A=^, logB=5', logC=r, or 
 
 /»+? +r=log A-l-log B +log C. 
 We have also 
 
 ABC= (sec a + tan a) (sec j3 + tan /3) (sec 7 + tan 7) 
 =sec(a--/3-i-7) +tan(a-L/3 J-7). 
 
 Now, as ^ is the logarithm of sec a + tan a, q the logarithm of 
 sec ^ + tan /3, r the logarithm of sec y + tan 7, 
 
 p+q+r is the log of sec(a->-/3-i-y) + tan(o-L/3J-7), or of ABC, 
 as shown above. We may therefore conclude that 
 log (ABC) =log A +log B +log C. 
 
 361. J If e be the angle which gives the difference between the 
 
336 ON THE OEOHETBICAL ORIQIN OF LOOARITHHS. 
 
 parabolic arc and its Bubtangent equal to m, (e -^ e) is the angle 
 vhich wiU give this difference equal to 2»», (e-i-e-^e) is the angle 
 which will give this difference equal to 3m, and so on to any num- 
 ber of angles. Hence, in the circle, if d be the angle which gives 
 the circular arc equal to the radius, 2d is the angle which will give 
 an arc equal to twice the radius, and so on for any number of angles. 
 This is of course self-evident in the case of the circle ; but it is 
 instructive to point out the complete analogy which holds in the 
 trigonometries of the circle and of the parabola. 
 
 Hence the amplitude which gives the difference between the 
 parabolic arc and its subtangent equal to the semiparameter is 
 given by the simple equation 
 
 sece,+tan6,=e' (a) 
 
 And more generally, if 6" be the amplitude which gives the difference 
 between the parabolic arc and its subtangent equal to n times the 
 modulus, we shall have 
 
 sece' + tane''=c» (b) 
 
 In the same way it may be shown that if e^ be the angle which 
 gives the difference between the parabolic arc and its subtangent 
 
 equal to -th part the modulus, we shall have 
 
 1 
 sece,+tan€„=c" (c) 
 
 Let the difference be equal to one half the modulus, then n =2, and 
 sec e, -1- tan e, = e*. 
 
 This is easily shown. 
 
 Let €,-^e,= e. Then sec (e, J- e,) = sec e = sec* e, + tan* e,, and 
 
 tan {e, -^ e,) = tan e = 2 sec e, tan e,. 
 Therefore sec (e,-J-ej)-f-tan(e,-Le,) = sece-|-tan€^e 
 
 ^sec* e^+tan' e,-t-2 sec e, tan e,= (sec e,+tan e,)*. 
 Hence sece,4-tane^= Vc (d) 
 
 Since tane= — ^ — , sece = — = — ; 
 
 tan(eJ-e)= — = — , sec (6-"-e)=- 
 
 2 ' 
 
 gS g-3 gS 1 g-8 
 
 tan(6-Le-i-e) = — ^ — , aec{e-^e-^e)= — ^— j 
 tan (eJ-eJ-to n terms) = — = — , sec(6-L€ to n tennB)=: — 5 — 
 
 2 
 
 Therefore 2 sec e tan e =■ tan (« + e) 
 
 28ec (e-'-e) tan (€-'-e)=tan (e-'-c-'-e-'-e). 
 
ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 337 
 
 and generally 
 
 2 sec (e-^e-Lto « terms) tan (e J-e-J-to w terms) = 
 tan (e-J-e-'-e-i-e-^to 2» terms). 
 
 Now 2 sec (e -L e -J- to w terms) tan (e J- e J- to m terms) is the portion 
 of the tangent to the curve intercepted between the axis of the para- 
 bola and the point of contact whose amplitude^ or the angle it makes 
 with the ordinate^ is {e-^e-^to n terms), whUe tan [e-i-e-^e-^e-^to 
 2» terms) is half the ordinate of that point of the curve whose am- 
 plitude is (e-i-e-Le-^e to 2w terms). Hence we derive this very 
 general theorem : — 
 
 Thai if two points be taken on a parabola such that the intercept 
 of the tangent to the one between the point of contact and the aids 
 shall be equal to one half the ordinate to the other, the amplitudes of 
 the two points will be 
 
 {e-^e-^ton terms) and (e-^e-Le-J-e to 2» terms) respectively. 
 This theorem suggests a simple method of graphically finding a 
 parabolic arc whose amplitude shall be the duplicate of the ampli- 
 tude of a given arc. Let P be the point on the parabola whose 
 amplitude is given. Draw the tangent PQ meeting the axis in Q. 
 Erect VT at the vertex=PCl. Through T draw the tangent TP,; 
 the amplitude of the arc VP^ will be the duplicate of the amplitude 
 of the arc VP, or [6-^6-^Xon terms) and {^-^^J-to 2w terms) will 
 be the amplitudes of VP and VP^ respectively. We may therefore 
 conclude that in the circle 
 
 2 cos (^ + 0+to w terms) sin {6-t6 + ion terms) = 
 sin [O + e + e + e to 2m terms). 
 
 362.] To represent the decimal or any other system of logarithms 
 by a corresponding parabola. 
 
 The parabola which is to give the Napierian system of logarithms 
 being cLrawn, whose vertical focal distance m is assumed as the arith- 
 metical unit, let another confocal parabola be described having its 
 axis coincident with that of the former, and such that its vertical focal 
 distance shall be m,. The numbers being represented as before by 
 the vectors of the logarithmic curve whose asymptote coincides with 
 the directrix of the parabola whose parameter is 4m, the differences 
 between the similar parabolic arcs and their subtangents in the two 
 parabolas will give the logarithms, in the two systems, of the same 
 number represented by the vector of the logocyclic cm-ve ; for as 
 all parabolas, like circles, are similar figures, and these are confocal 
 and similarly placed, any line drawn through their common focus 
 will cut the curves in the same angle, and cut off proportional seg- 
 ments. Hence the two triangles FPT and Fwt are similar, and 
 the tangential differences PV— PT and isv—tst are proportional to 
 Im and 4m„ the parameters of the parabolas. 
 
338 ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 
 
 Fig. 83. 
 
 Let log denote the Napierian logarithm, and Log the decimal 
 logarithm of the same number. 
 
 Draw the line FT, making the angle e with the axis such 
 that 8ec€+tan e=e. Then as PV— PT : vrv—ar :: m: m^ and 
 PV— PT=»»=1, since e is the base of the Napierian system, and 
 ov— crT=Log e on the decimal parabola, therefore 
 m I Loge :: m : m,, or m,='Loge. 
 We may therefore conclude that the modulus of the decimal 
 system is the decimal logarithm of the Napierian base e. 
 
 Draw the line FT, making with the axis an angle S, such that 
 sec S+tanS^lO. Now 
 
 P,V— PT, : «r,v— w,T, :: m : nii; 
 but 
 
 P,V— PT,=»»log 10; hence «-,w— w,T,=»»,log 10. 
 
ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 839 
 
 Now in order that 10 may be a base, or, in other words, in order that 
 its logarithm may be unity, we must have «■,»— Br^T,=OT^ log 10=»t ; 
 
 or if m=l, we must have »»,log 10=1, or »»;=; — tj;; that is, the 
 
 parameter of the Decimal parabola must be reduced compared with 
 that of the Napierian parabola in the ratio of log 10 : 1 . Hence, as is 
 well known, the modulus m, of the decimal system is the reciprocal 
 of the Napierian logarithm of 10. 
 
 It is therefore obvious that, as any number of systems of loga- 
 rithms may be represented by the differences between the similar 
 arcs and their subtangents of as many confocal parabolas, the loga- 
 rithms of the Aame number in these different systems will be to one 
 another simply as the magnitudes of the parabolas whose arcs repre- 
 sent them, that is, as the parameters of these parabolas. Accord- 
 ingly the moduli of these several systems are represented by the 
 halves of the semiparameters of the several parabolas. 
 
 The Napierian parabola differs from the decimal and other para- 
 bolas in this, that the focal distance of its vertex is taken as the 
 arithmetical unit, and that the logocyclic curve, whose vectors repre- 
 sent the numbers, has the directrix of the Napierian parabola as its 
 asymptote. 
 
 Hence, if m, the vertical focal distance of the Napierian parabola, 
 be taken as 1, the vertical focal distance m, of the decimal parabola 
 is -4342 &c., or, if jw = l, »m,=-4342 &c. 
 
 363.] Since VT -J- TP> arc VP, therefore 
 
 VT >aic VP-TP >log ¥t. 
 Hence VT or tan 5 is always greater than the logarithm of 
 (sec + tan d) in the Napierian system of logarithms. This may 
 be shown on other principles : thus 
 
 sec ^+ tan ^= 
 
 1+sing «in«2-^cos»2 + 2«^°-2C°«2 l+tan^ 
 
 ' cos^ 7d ~ZS ' ' 5" 
 
 cos*5— sm*n 1— tanjr 
 
 a 
 
 Lettang=M. Then 
 
 log(secd-t-tan^)=log(^)=2(«-h^ + jV^'&c.). 
 g 
 2tanH 
 
 and tan5= ^=2{u+t?+i^+u' + Scc.) 
 
 l-tan«2 
 
 Hence tan > log (sec + tan 0) , 
 
 or ^~^ is always greater than the logarithm of n. 
 2 
 
 z2 
 
340 ON THE OEOMETHICAL ORIGIN OF LOGAHITHMS. 
 
 364.J In every system of logarithms whatever, the logarithm of 
 lisO. 
 
 For when the point T coincides with V, the corresponding point 
 T will coincide with v, whatever be the magnitude of its modulus 
 m,. It is obvious that the circle whose radius is unity is analogous 
 to the parabola whose vertical focal distance is unity, and that the 
 Napierian logarithms have the same analogy to trigonometrical 
 lines computed firom a radius equal to unity, which any other system 
 of logarithms has to trigonometrical lines computed from a radius r. 
 As we may represent difiFerent systems of trigonometry by a series 
 of concentric circles whose radii are 1, r, r,, &c., so we may in like 
 manner exhibit as many systems of logarithms by a series of con- 
 focal parabolas whose focal distances or moduli are 1, m„ ran, &fc. 
 The modulus in the trigonometry of the parabola corresponds with 
 the radius in the trigonometry of the circle. But while in the 
 trigonometry of the parabola the base is real, in the circle it is 
 imaginary. In the parabola, the angle of the base is given by the 
 equation sec + tan 6=e. In the circle, cos 6+ V — l8in^=c*V~i; 
 and making 6=\, we get 
 
 cos(l)+ V^sin(l)=e^^ (a) 
 
 Hence, while c' is the parabolic base, c^~' is the circular base. 
 Or as [sec e+tan e] is the Napierian base, [cos(l) + V — 1 8in(l)] 
 is the circular or imaginary base. Thus 
 
 [cos(l)+ v'^sin(l)]*=cosA+ V^sinS. . . (b) 
 We may therefore infer, speaking more precisely, that imaginary 
 numbers have real logarithms, but an imaginary base. We may 
 always pass from the real logarithms of the parabola to the imaginary 
 logarithms of the circle by changing 
 
 tan into V — 1 sin d, sec into cos S, and e* into e ^-'. 
 
 As in the parabola the angle ^is non-periodic, its limit being ^ir, 
 while in the circle 5 has no limit, it follows that while a number 
 can have only one real or parabolic logarithm, it may have innu- 
 merable imaginary or circular logarithms. 
 
 From P, the focus of the parabola, draw a series of vectors to the 
 logocyclic curve in geometrical progression such as 
 
 m{aec0 + ta,n0), m(8ec0 + ta.n0)^, .... TO(sec5 + tan^", 
 meeting the tangent to the vertex of the parabola in the points 
 T, T„ T„, T„. The line FT will be =m sec 0, the line FT, 
 =»»sec {0-^0), the line FJ!,i=m see {0-^0-^0), &c.; and we shall 
 likewise have 
 
 VT=»»tane, YT,=mtaji{0->-0), \T:„=mt8Xi{0-^0J-0), &c. 
 This follows immediately from (f ) of sec. [344] ; for any integral 
 
ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 
 
 3 J. I 
 
 power of (sec^ + tan^) may be exhibited as a linear function of 
 sec + tan e, writing ® for 6-^6-^0 .. . &c., since 
 
 sec{0-i-e-^e-^0&c.tonO)+ta.n{d-^e-i-e-^0kc.ton0) = {sece+ta.n6)'' 
 Hence the parabola enables us to give a graphical construction for 
 the angle {0 -^ -^ Sec.) as the circle does for the angle {0+0 + &c.) . 
 365.] The analogous theorem in the circle may be developed as 
 follows : — In the circle FBA take the arcs 
 
 Pig. 84. 
 
 AB^BB ^B^By^B((B^(j . 
 Let the diameter be D ; then 
 
 . . &c.=2d. 
 
 and 
 
 FB^Dcosd, FB,=Dcos2d, 
 
 FB„=Dcos33 
 
 ,&c.. 
 
 AB=D sind, AB,=D sin 2*, AB„=D sinSd . . . &c. 
 
 Now, as the lines in the second group are always at right angles 
 to those in the first, and as such a change is denoted by the symbol 
 V— 1, we shall have 
 FB + BA=D{cosS+ V^sind}, 
 
 FB,+B(A=D{cos2d+ V^sin2d} = D{cos3+ V^sinS}'; 
 FB„ + B„A=D{co833+ v'^sin33} = D{cosS+ V"-^sin3}3&c. 
 
 FB„ + B„A=D[co8mS+ v'-lsiuwd]=D[cosd+ V-lsinS]". 
 
 When the points B^, B„ fall below the line FA, the angle becomes 
 negative, and we get 
 
342 ON THE GGOHETBICAL ORIGIN OF LOGARITHMS. 
 
 FB,-B,A=cos3- V^sind, 
 
 FB„-B„A.=cos2d- V^ sin 25= [cos d- V^sinS]'. 
 
 Therefore 
 
 log (FB + BA) =log (cosd+ V^ Bind) =d V^- 
 Let i=sl, then 
 
 log [cos (1) + y/^i sin (1)] = V^. 
 
 Hence generally d v' — 1 is the logarithm of the bent line whose 
 extremities are at F and A, and which meets the circle in the point B. 
 
 It is singular that the imaginary formulae in trigonometry have 
 long been discovered, while the corresponding real expressions have 
 escaped notice. Indeed it was long ago obsei-ved, by Bernoulli, 
 Lambert, and by others (the remark has been repeated in almost 
 every treatise on the subject since), that the ordinates of an equi- 
 lateral hyperbola might be expressed by real exponentials whose 
 exponents are sectors of the hyperbola; but the analogy being 
 illusory, never led to any useful results. And the analogy was 
 illusory from this — that it so happens the length and area of a 
 circle are expressed by the same function, while the area of an 
 equilateral hyperbola is a function of an arc of a parabola, as will 
 be shown further on. The true analogue of the circle is the 
 parabola. 
 
 There are some curious analogies between the parabola and the 
 circle, considered under this point of view. 
 
 In the parabola, the points T, T,, T,,, which divide the lines 
 m (sec 5+tan d), m[8ee {0-^0) + taa (0-^0)] 
 into their component parts, are upon tangents to the parabola. 
 The corresponding points B, B„ B„ in the circle are on the circum- 
 ference of the circle. 
 
 In the parabola, the extremities of the lines m (sec ^-}-tan 0) are 
 on a logocyclic curve; in the circle, the extremities of the bent lines 
 are all in the point A. 
 
 The analogy between the expressions for parabolic and circular 
 arcs will be seen by putting the expressions under the following 
 forms : — 
 
 Parabolic arc — log (sec + tan 0) — subtangent = 0, 
 
 Circular arc + log (cos 0+ V— Isin^)^-'— subtangent = 0. 
 
 The locus of the point T, the intersections of the tangents to the 
 parabola with the perpendiculars from the focus, is a right line ■ 
 or in other words, while one end of a subtangent rests on the para- 
 bola, the other end rests on a right line. So in the circle ; while 
 one end of the subtangent rests on the circle, the other end rests 
 
ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 
 
 343 
 
 on a cardioid whose diameter is equal to that of the circle, and 
 whose cusp is at F. FPA is the cardioid. 
 
 366.] The quadrature of the hyperbola depends on the rectifica- 
 tion of the parabola. 
 
 Through a point P on the parabola draw a line PQ parallel to 
 the axis and terminated in the vertical tangent to the parabola at R. 
 Take the line B,Q always equal to the normal at P, the locus of Q 
 is an equilateral hyperbola. For x=2m sec ^, since PM is equal 
 to RQ, and as before y^2m tan <^; therefore 
 
 a?*— y*^4i»', (a) 
 
 the equation of an equilateral hyperbola whose centre is at V, the 
 vertex of the parabola, and whose transverse axis is the parameter 
 of the parabola. 
 
 The area of this curve, the elements being taken parallel to the 
 axis, or the area between the curve and the vertical axis passing 
 through V, is found by integrating the value of xdy. 
 
 Now 
 
 d7=2msec^, and ^=2ffitan^; 
 therefore 
 
 Jjrdy =4in' Jsec^ <l)i<f>=2m\jn sec (f> tan ^ + m Jsec <^d</>J- 
 But it has been shown in (a), sec. [345], that 
 
 n {m.^)=m sec <j) tan<^ + »i f sec«^d<^. 
 Hence the hyperbolic area 
 
 YAQLR=2m.n(m.^) (b) 
 
344 ON THE OEOxMETRICAL ORIGIN OF LOGARITHMS. 
 
 Therefore, as the hyperbolic area is equal to a constant multiplied 
 into the corresponding arc of the parabola, the evaluation of the 
 hyperbolic area depends on the properties of logarithms. 
 
 It also follows, from what has been established in the preceding 
 part of this Chapter, that hyperbolic areas may be multiplied and 
 compared according to the laws which regulate parabolic arcs. 
 
 Let <f> and be the angles in which the normals to the corre- 
 sponding points of the parabola and the hyperbola cut the axis ; 
 then it is easily shown, since VQ= normal at Q, that 
 
 tand=sin^ (c) 
 
 This relation will enable us to express the hyperbolic area in 
 terms of the angle which the normal to the hyperbola makes with 
 the axis instead of the parabolic amplitude ; for as the parabolic 
 amplitude is related to the normal angle of the hyperbola d by 
 the equation tan d=sin 0, 
 
 2tan</)sec0==— -T — g^=tan2^, . . . . (d) 
 
 and 
 
 sec0 + tan0= v'sec2^+tan25 (e) 
 
 Now 
 
 n(»i . <^) =m sec tan <^ + ot log (sec <^ + tan 0), 
 or, substituting for the preceding values of «^, 
 
 211 (to . 0) =OT tan '16 + m log (sec 2^ + tan 2d) ; 
 but taking the amplitude 20, 
 
 U{m.2ff) =m sec 2^ tan 20 + mlog (sec 2^ + tan 26). 
 Hence, subtracting the former from the latter, 
 
 n [m . 26) — 2n(m . 0) =m tan 26 (sec 26-1). 
 Accordingly 
 
 the hyperbolic area=OTn {m . 26) —m? tan 26 (sec 26— 1) . (f ) 
 Since 
 
 2 tan 20 = 2 tan ^ sec 0, 
 we shall have 
 
 2^=^-^* (g) 
 
 Hence the normal angles and of the corresponding points of 
 the parabola and hyperbola are so related that 
 
 25=<^-"-<^, 
 
 whence we might at once have inferred the relation established in 
 (e), namely 
 
 (sec + tan 0)* = sec 20 + tan 20. 
 
 The corresponding points P and Q on the parabola and hyper- 
 bola are always found in a line parallel to the axis. 
 
ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 345 
 
 If through the points P and Q on the parabola and hyperbola we 
 draw diameters to these curves, they mil make angles with the nor- 
 mals to them at these points, ojie of which is the duplicate of the 
 other. 
 
 For these angles are 20 and 4> respectively, 
 
 but 26=4>-^<l> (h) 
 
 367.] Let Po, P„ Pj, Pg, P^ . . . P„_„ P„ be perpendiculars let 
 fall from the focus on the n sides of a polygon circumscribing a 
 parabola, and making with the axis the angles 
 
 0, 0, 0-^0, 0-1-0-1-0, 0.^0.^0^6, . . . 
 to « terms respectively. 
 Let 
 
 sec^+tan^=a, 
 then 
 
 sec {0-^0) +tan {0-^0)=u^, ) 
 
 sec[0-^0-^0)-\-taji(0-^0-i-0)=i^\ ' ••■(*) 
 
 sec {0-^0-^ . . . to » terms)+tan (d-L^J-to M terms) = a". 
 Hence, as 
 
 2Po=»»(m» + m-<») "I 
 2P, = ot(m1+m->) 
 2P2=ot(m«+m-*) 
 
 2P„ = »i(m» + m-"),; 
 we shall have 
 
 2 . 2 . P„ . P, =»w2(M» + M-») (m> + M->) 
 
 = m* [(m»+^ + M-(»+i)) + (w-i + M-(»-i))] , 
 or 
 
 2P„.P,=»M(P„+i + P„_i). 
 
 But 
 
 P, =m8ec^; 
 therefore 
 
 <,<.»/iX> I*i»+1 + I*n-I 
 
 secair„= jr , ..... 
 
 (b) 
 
 2 ' ('') 
 
 or any perpendicular multiplied by the secant of the first amplitude 
 is an arithmetical mean between the perpendiculars immediately pre- 
 ceding and following it. Thus, for example, Pq=»», P, =m sec 0, 
 'Ps=m see (0-^0), OT 
 
 n n /w + msec(^-«-^ 
 secpjnseco= ^ '-. 
 
 But 
 
 sec {0->-0) = sec^ + tan^ 0; 
 
 hence the proposition is. manifest. 
 
346 ON THE GEOMETRICAL ORIGIN OF LOGARITHMS. 
 
 368.] Again, as hence 
 
 2Po=m(a°+a»), 2.2. PoPi=»»*(«" +«"' +m' + m->). 
 
 2V,=m[u+u-'), 2.2.PiP2=OT«(tt8 + M-a+M'+M-»). 
 
 2P2=m(M« + M-*)» 2.2. P,P3=»»2(m5+m-5+„i + M-1). 
 
 2P3=ot{«?+«-3), 2.2. PsP4=»m8{m7 + W-7 + m1 + m-1). 
 
 (a) 
 
 2P„=m («« + «-»), 2.2.P„_,P„=»»V*""' + «~'*""''+«'+«~')- 
 We shall have, therefore, adding the preceding expressions, 
 
 2[PoP,+P,P, + P,P3 + P3P4 .... Pn-.P»J=) .. 
 
 mC + Pi+Pg + Ps+Py . . . P.„_.+ {«-l)P,], j ■ V'' 
 
 or tivice the sum of all the products of the perpendiojdars taken two 
 by two up to the nth, is equal to the sum of all the odd perpendiculars 
 up to the (2n—l)th-\-(n — l) times the first perpendicular. 
 Thus, taking the first three perpendiculars, 
 
 To=m, P,=msec^, Ti=mHec{d-^0)=m{8ec^0+ta,n^0), 
 Vs=m sec {6-^ 9--- 0)=m (4sec8 0-3 sec 0). 
 The truth of the proposition may be shown in this particular case ; 
 for 
 
 2[PoP, + PiPs]=4m«sec8e=m(P, + Pg + 2P,). . . (c) 
 Again, since 
 
 2Ps„=ot(m«» + m-«»), 
 and 
 
 4P^=ot2(m2" + 2 + m-2"), 
 
 we shall have 
 
 2I^-m8=»»P2» (d) 
 
 Thus, for example, twice the square of the perpendicular on the 
 fifth aide of the polygon, diminished by the square of the modulus, is 
 equal to the tenth perpendicular multiplied by the modulus. 
 
 In the same way we may show that 
 
 4P^-3ff»«P„=»»«P,„. 
 
 fi n sn 
 
 Let »=5 and m=\; then /our times the cube of the fifth perpen- 
 dicular, diminished by three times the same perpendicular, is equal 
 to the fifteenth perpendicular, or to the perpendicular on the fifteenth 
 side of the polygon. 
 
 369.] Since 
 
 logM=M— M-' — i(M« — M-*) +i(M8 — M-3) — J(tt« — M-4), &c,, 
 
 and as 
 
 «— tt-'=2tane, M«— M-«=2tan(^-L^), 
 M«_M-»=2tau (5 J- ^ J- 0-1- ton terms), 
 while 
 
 u = sec5-|-tan0, 
 
ON TBS 8EOHETRICAL ORIGIN Of LOOABITHMS. 
 
 347 
 
 we shall have therefore 
 
 PV — PT 
 
 logu= 2 =tan5-itaii(0J-fl) + itaii(0J-^-re, &c.). (a) 
 
 We may convert this into an exp ression for the arc of a circle by 
 changing -i-into + , tan into V— Isin, and the parabolic arc into 
 the circular arc multiplied by ^ — 1. 
 
 Hence, since PT in the circle is equal to 0, 
 
 Q 
 
 ^=sin^-isin2^ + ^sin30-isin4^, . . . (b) 
 
 a formula given in Lacroix, ' Traite du Calcul DiflFerentiel et du 
 Calcnl Integral,'' torn. i. p. 94. 
 
 370.] In the trigonometry of the circle, the sines and cosines of 
 multiple ai'cs may be expressed in terms of powers of the sines and 
 cosines of the simple arcs. Thus 
 
 cos 2d = 2cos*d-l, 
 
 cos 30= 4cos^0— 3costf, 
 
 cos4d= 8cos*rf- 8cos*0 + l, 
 
 cos 55= 16 cos* e—%0 cos3 5 + 5 cos 0, 
 
 cos 65=32 cos6 5-48 cos* + 18 cos^ 0- 1, 
 
 (a) 
 
 sin25=sin5(2cos5), 
 sin35=sin 5(4cos2 5-l), 
 sin 45 = sin 5 (8 cos^* 5 - 4 cos 5) , 
 sin 55=siu 5(16 cos* 5-12 cos« 5 + 1), 
 sin 65 = sin 5 (32 cos* 5 - 32 cos^ 5 + 6 cos 5) . 
 Hence, in the trigonometry of the parabola, 
 sec(5-^5)=2sec«5-l, 
 sec(5-^5-L5)=4sec»5-3sec5, 
 «ec(5-i-5-L5-i-5) = 8sec*5-8 8ec«5 + l, 
 sec(5-'-5-»-5J-5J-5)=16sec*5— 20sec'5+5sec5, 
 sec(5 -r 5 J- 5 J-5-L5-L 5) =32 sec«5-48 sec* 5 + 18 sec« 5-1, 
 
 tan {e-^ff)= tan 5(2 sec 6) , 
 tan(5J-5J-5)=tan5(4sec*5-l), 
 tan(5-»-5-i-5-^5)=tan5(8sec35— 4sec5), 
 tan(5J-5-J-5-L5+5)=tan5(16sec*5-12sec«5+l), 
 tan (5 -L 5 + 5 -L 5 -^ 5 -L 5) = tan 5(32 sec* 5 - 32 sec8 5 + 6 sec 5) . 
 
 .(b) 
 
348 
 
 ON THE OEOHETKICAL ORIGIN OF LOGARITHMS. 
 
 The preceding formulae may easily be verified. 
 If we add in the above series any two corresponding secants and 
 tangents, the sum will be an integral power of sec ^+ tan d. 
 Thus sec(^--5) +tan{^-^^)={sec ^+tan Of. 
 
 Again, since in the circle 
 cos^=cos^ 
 2cos«0=cos25 + l 
 4 cos*5 =cos 35 + 3 cos ^ 
 
 8 co8*5=co8 4e+4cos 26+ 1 
 and y . . (c) 
 
 sm6=sva.6 
 2 sin* 5= -cos 25+1 
 4 sinS 5= — sin 35+3 sin 6 
 8 sin* 5 = cos 45 —4 cos 25 + 3, 
 
 hence in parabolic trigonometry 
 8ec5=Bec5 
 2sec*5=sec(5-«-5)+l 
 48ec8 5=sec(5-'-5-^5) +sec35 
 8sec*5=sec(5-L5-i-5--5)+4sec(5J-5)+l 
 
 (d) 
 
 taii5=tan5 
 2tan«5=sec(5-A-5)-l 
 4tan35=tan(5 J-5 J-5) — 3 tan 5 
 8tan*5=sec(5 J-5-i-5-»-5) -4sec(5-L5) +3. 
 371.] The roots of the expression 
 
 r2»— 2aif" + l=0 (a) 
 
 may be represented under the form cos A + V — 1 sin A, when a is 
 less than 1. This has long been known. It is not difficult to show 
 that, when a is greater than 1, the roots may be exhibited under the 
 form 
 
 secA+tanA (b) 
 
 Since a is greater than 1, let a=8ec 0, and let 5 be divided into n 
 angles ^, connected by the relation 
 
 ,^ j.<^ j.^ A.^ &c.=5 (c) 
 
 It has been shown in (f ), sec. [344] , that 
 
 sec(^-i-^-'-^J-^ton^) +tan(^-i-^-'-^-'-^ton^) = (8ec«^+tan^)". 
 
 Let sec (^+ tan ^su; then 2 8ec«^=M'+M~', 
 and therefore 2 sec 5= 28ec(0-'-0-'-i^-'-to n^) = «" + u-". 
 
ON THE QGOMETRICAL ORIGIN OF LOGARITHMS. 349 
 
 Substitute this value of 2 sec d in (a), and we shall have 
 r*"— («"+«-»)«» + 1=0, 
 or, resolving into factors, 
 
 (S-0('^""~^)^° ^^^ 
 
 Hence, finding the roots of these binomial factors by the ordinary 
 methods, we shaU have, since M=sec <^ + tan ^, 
 
 z = (sec0 + tan^) (multiplied successively into the n roots of unity) i 
 and I (e) 
 
 (sec^ — tan^) (multiplied successively into the n roots of unity) . J 
 We are thus enabled to exhibit the 2« roots when a>l. 
 Thus, let M=3, then the equation becomes 
 
 2^-2 sec 62^ + 1=0, (f) 
 
 and 
 
 ^->-<j,^<l> = d; (g) 
 
 consequently the six roots are 
 
 (sec.^+taii./,)(l,:zi±^^^), I 
 and y ■ ■ ■ (h) 
 
 (8ec.^-tan^)^l, ~^±^^~^ ). J 
 
 By the same method we may exhibit the roots when a is less 
 than 1, or a=cos 0. 
 
 We might pursue this subject very much fiirther; but enough 
 has been done to show the analogy which exists between the tri- 
 gonometry of the circle and that of the parabola. As the calculus 
 of angular magnitude has always been referred to the circle as its 
 type, so the calculus of logarithms may in precisely the same way 
 be referred to the parabola as its type. 
 
 On parabolic trigonometry as applied to the investigation of the 
 properties of the Catenary and the Tractrix. 
 
 372.] The application of the principles of parabolic trigonometry 
 to the discussion of the properties of those kindred curves the 
 catenary and the tractrix, elicits some singular properties and rela- 
 tions of these curves, as also between the catenary and the para- 
 bola, and affords a fiirther illustration how the invention of new 
 methods enlarges the boundaries of science. 
 
 Let us for brevity assume as known the equation of the catenary 
 referred to the rectangular axes of coordinates OX, O Y ; and let 
 O A=o, the modulus; then 
 
 '=s(«" + «'')^ and«=^(c«— c"). 
 
 (a) 
 
350 PROPERTIES or THE CATENARY AND TKACTRIX INVESTIGATED 
 
 By addition^ 
 
 X —X 
 
 i/+s=ae'; by subtraction, y—»^ae " . . . (b) 
 Multiplying these expressions, 
 
 y«_«2=a2, (c) 
 
 or, the difference between the squares of an ordinate of a catenary 
 and its corresponding arc is constant and equal to the square of the 
 modulus. 
 
 373.] Assume the following relations. 
 
 then 
 
 2sec^=e" + c", and 2 tan ^=c»—c'' 
 
 (a) 
 (b) 
 
 y=asec0, and s=a ta,n0 ; . . . 
 adding these expressions, we shall have 
 
 y + »=o(8ec5+tan^), 
 the form which so often occurs in parabolic trigonometry. 
 
 Now, if we make x„ x,,, x„„ &c. successively equal to a, 2a, 8a, 
 &c. in (a), we shall have 
 
 y +s =a(aecd + ta.n0), 
 
 yi +s, =a{8ec (^J-5) +tan (0-i- 0) }, 
 
 y,,+s„=a{aec{e-^e-^e)+ia.n{e->-e->-d)}, " ^'^^ 
 
 ytf/ + »*H=«{8ec (^J-^-i^^-J-d) +tan (5-«-5-i-ej-^ &c.) }. 
 But it has been shown in sec. [344] that 
 (sec0 + tan^)» = 8ec(dJ-^-i-0 ton terms) +tan(eJ-^-i-e tow terras); 
 
BY THE AID OP THE TRIGONOMETRY OP THE PARABOLA. 351 
 
 consequently 
 
 (y+#)» = a— (jr„+«„)j (d) 
 
 or, if two points be assumed on a catenary, the abscissa of the one 
 being n times that of the other, the nth power of the mm of the first 
 ordinate and its corresponding arc will be equal to the sum of the nth 
 ordinate and its corresponding arc multiplied by a"~'. 
 
 It will also follow froin the preceding expression that if a series of 
 equidistant ordiimtes to the catenary be taken, and the corresponding 
 arc be added to each ordinate, their sums will be in geometrical pro- 
 gression. For 
 
 y+s=y + s, {y+sY=a{y, + s)„ | 
 
 (y + »)3=o«(y„+«„), (y+*)"=o»-'(y„-,+«,_,).j 
 
 374.J The catenary will enable us to represent graphically, with 
 great simplicity, the sum of a series of angles, added together by 
 the parabolic plus -^ . 
 
 Let a set of equidistant ordinates whose intervals are a, 2a, 3a, 
 4a, &c. meet the catenary (fig. 86) in the points b, c, d, k, I; and 
 then let the catenary be conceived as stretched along the horizontal 
 tangent passing through the vertex A. Let the points b, c, d, k, I 
 on the catenary in its free position be conceived to coincide with 
 the points ^, y, i, k, \, when it is stretched along the horizontal 
 
 tangent; we shall then have, since - is successively equal to 
 
 1, 2, 3, 4, &c., 
 
 2y=ei + c-', 2s=c'— e->, 
 
 2y,=e* + e-^, 2s,= e^-e-'^, 
 
 2y„=e> + e-^ 2s„=e>-e-^ ; 
 
 consequently the angle AO/3 or e is such that secc+tane=c, 
 AOy such that sec A07+tan A07=c*, or A07=e-'-6, AOS such 
 that sec AOS+tan AOB=^ or AOS=€-^e-i-e &c. 
 
 Consequently, if we draw lines from the pole O to the points 
 /3, y, S, &c., the angles AOjS, AO7, AOS, &c. will represent the 
 angles 
 
 6, €■>-€, e-^e, e-'-e-'-e, &c. 
 
 Hence, as successive multiples of an arc of a circle give successive 
 arithmetical multiples of the corresponding angle at the centre, so 
 successive multiples of a given abscissa give successive arcs of the 
 catenary which extended along the vertical tangent subtend at the 
 pole O successive parabolic multiples of the original angle. 
 
 375.] Since ^=i{e'—e~'}, we shall have ^=twa.6; but ^ 
 
 is the trigonometrical tangent of the angle which the linear tangent 
 to a curve at the point Jxy) makes with the axis of X. Hence 
 
352 FBOPERTIES OF THE CATENARY AND TBACTRIX INVESTIGATED 
 
 this other theorem : — Let a set of equidistant ordinates meet the 
 catenary in the points b, c, d, k, I, jfc, and at these points let tan- 
 gents to the curve be drawn, they will be inclined to the axis of^by 
 the angles d, 6-^-6, 0-t-0-^0, 0-i-6-^6-^0, &c., which is even a yet 
 simpler geometrical representation than the preceding. 
 
 Hence also it evidently follows that as the limit of the angle 
 which a tangent to the catenary makes with the axis of X is a right 
 angle, the limit of the angle 0->-0-^0 &c. ad infinitum must also be 
 a right angle. 
 
 On the Tractrix. 
 
 376.] Let the length of the constant tangent PD be a. Let 
 
 OU^- and 0D=5 ; then by similar triangles 
 
 y = PT=--^^, and^=OT=^ 7§=j,- ■ (a) 
 
 since 
 
 x^ + yv=i. 
 
 " ^^~~^» 
 
 Fig. 87. 
 y 
 
 c 
 
 
 ^ 
 
 7 
 
 ^ 
 
 1 
 
 Assuming the projective eqnation of the tractrix, as given in the 
 ordinary text-books— that is to say, 
 
 a+ ija^—y^=ye^ 
 
 ?) 
 
 (b) 
 
BY THE AID OF THE TRIGONOMETRY OF THE PARABOLA. 353 
 
 we shall have 
 
 x+ v^o*— w* 1 ^ ,- av 
 
 ——-rt' ^^^ a+ va*— w*=o+— 7==== J 
 
 substituting and reducing, we shall have 
 
 {|.^^f= 
 
 e (c) 
 
 Let d be the angle which the tangent to the tractrix makes with 
 the axis of Y ; then 
 
 tand=^, sec-^= — ^-. > 
 
 or the preceding equation becomes 
 
 (sec Sh- tan :&)«*=« (d) 
 
 If we now take -^^a, 2a, 3a successively, we shall have 
 
 secS + tanS=e, (8ec3 + tan3)*=c*, (8ecd + tand)3=e3. 
 But 
 
 (secd + tan*)*=sec(d-i-d) + tan(S-u*), 
 
 (8ec3 + tand)*=sec (S-Ld-«-S)4.tan (3-»-3-i-d). 
 
 Consequently S is the base-angle in the system of the Napierian 
 logarithms. 
 
 In figure 87 let us take OD=AC=J?=^,and make these substi- 
 tutions in the equations of the catenary and the tractrix, namely 
 (sec^ + tan^)=c» and (secd+tanS)'*=e; or if we put x for j, we 
 
 X 
 
 shall have (sec ■& + tan •&)=£", consequently 6=^; but it has been 
 
 dy dv 
 
 shown in sec. [375] that 3^:=tan ^, therefore ^=tan^, or the 
 
 da? da? 
 
 angle QGC is equal to the angle DUO. Consequently tand= 
 
 tand, or 0=d; and as QD=asecd, and DR=asecd^asecd, in 
 
 the two triangles DCR and DPQ we shall have QD=DR and the 
 
 angle QGC = the angle QDP; consequently the quadrilateral 6PDC 
 
 may be inscribed in a circle^ and therefore the angles DPG and GCO 
 
 are equal to two right angles. Sut the angle GCD is a right angle ; 
 
 hence QPD is a right angle, and PD=:CD=a, and the tangent QG 
 
 to the catenary meets the tangent to the tractrix at right angles to 
 
 the tangent of the latter and at the point of contact P, since PD = a ; 
 
 consequently the tangent PQ to the catenary, since it is at right 
 
 angles to the tangent to the tractrix at P, its point of contact, will 
 
 therefore envelop the catenary, or the catenary is the evolute of the 
 
 tractrix. 
 
 2 A 
 
334 PROPERTIES OF THE CATENARY AND TRACTRIX INVESTIGATED 
 
 Since QP^a tan d, and the arc of the catenary is equal to a tan B, 
 and »a0=^, QP is equa] to the arc of the catenary AQ; and as CB 
 is equal to QP, CB is also equal to the arc of the catenary, and 
 therefore AB is the difiference between the arc of the catenary AQ 
 and its projection AC. 
 
 We may also enunciate this other theorem (see fig. 86) : — 
 
 If with the point O as focus and A a« vertex we describe a para- 
 bola, and from the points /S, y, S, k, \ we draw tangents to the para- 
 bola /9B, 7G, 8D, «cK, XL, the differences between these tangents and 
 the corresponding parabolic arcs will be a, 2a, Sa, 4!a, jfc. 
 
 AG-/8B=o, AG-7G=2a, AD-8D=3a, AK— «K=4a, &c. 
 This is evident from the principles of Parabolic Trigonometry ; for 
 the angles A0^ = 6, AOy=e-J-e, AOS=e-Le-i-e, A0«=e-«-6-»-e-i-c. 
 
 377.] We may further extend these properties of the catenary. 
 To simplify the expressions, let Y^ denote the ordinate of a point 
 on the catenary at which the tangent makes the angle <f> with the 
 axis of X. Let S^ denote the corresponding arc measured from 
 the lowest point, and let H^ signify the corresponding abscissa. 
 
 Then S<^ = a tan <f>, Y^= a sec ^. 
 
 Now let;r, x,, x,j be the abscissae of the three arcs whose tangents 
 make the angles ^, x, &> with the axis of X, and let the equation of 
 condition be simply 
 
 then we shall have the following relations between the corre- 
 sponding arcs and ordinates of the catenary : — 
 
 aS«a=S<^Yx + SxY^, 
 aYto = Y^ Yx + S^Sx ; 
 when x^=x, 
 
 3S'«^ = (Ya)-a)o, 
 2Y«<^=(Ya> + a)a, 
 
 o8Y«=Y>-s4, 
 since Y,«-S,«=a«. 
 
 Let there be four arcs of the catenary whose abscissse x, Xf, «.., x.^, 
 shall be connected by the following relations, 
 
 Xii — X + x^, 
 
 ^///=^« + *« O'' X,ii=X + Xi + X„. 
 
 Let a>t ^, x> '^ ^ ^^^ corresponding angles made by the tangents 
 to the extremities of the arcs Sw, S^, 8%, S^. 
 
 Then we shall have the following relations between the arcs and 
 the ordinates, 
 
 a«Si»= S^ Y^Yi^ + SxY^Y^ + S^Yi^Y^ + S^S^S^^, 
 «» Yw= Y^YxY-f + Y^Sx^yfr + Y^S^Si^ + Yx^S^S^. 
 
BY THE AID OF THE TRIGONOMETRY OF THE PARABOLA. 355 
 
 Hence also 
 
 Y<l>Yx^yt^~\.Y^J \YxJ \Y'>p'r\Y<l>YxYylrJ ' 
 
 or the ratio of the fourth arc nmltiplied by the square of the 
 modulus a to the product of the ordinates of the three preceding 
 arcs is equal to the sum of the ratios of each preceding arc to its 
 ordinate 4- ratio of the product of the three arcs to the product of 
 the three ordinates. 
 We have also 
 
 Let x-=x,=Xii, and a?„(=3a?; 
 
 then we shall have 
 
 an equation which gives the relation between two arcs of the cate- 
 nary, the abscissa of the one being equal to three times that of the 
 other. 
 
 When one abscissa is double of the other, the arcs are related 
 by the equation 2YS=aS,. 
 
 o- . 8 . sec (<AJ-^)— 1 
 
 Since sin* ^ = txTjlTT i ' 
 
 ^ sec (9-»-9) + 1 
 
 and sin^=Yj ^e shall have 
 
 S'_ Y,-a 
 Y*-Y,+a' 
 an equation which enables us to calculate Y, when we know Y, 
 since S*=Y'— a*. Thus the catenary may be constructed by 
 p6ints. 
 
 Let 8, y, sffi, Siffii, SuHm be four arcs and corresponding ordinates 
 of a catenary, whose abscissae are connected by the equation 
 
 Xm ^ X^ -^ X i-y X *, 
 
 then we shall have 
 
 sju y y, Vii yy&u 
 
 yui 2 1 **i [ */*« j '^H 
 
 yyi y&u yyu 
 
 378.] The curvature of the catenary at its lowest point approxi- 
 mates to that of a parabola at its vertex whose semiparameter is 2a. 
 
 Let S be the deflection of the catenai-y from the horizontal tan- 
 gent, then Z=y—a = a (seed— 1). 
 
 s the arc of the catenary near its lowest point may be taken for 
 the horizontal ordinate, but s=a tan^ or *'=a'tan'^=«*(sec'd— 1) . 
 
 2 a2 
 
356 ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 -9 
 
 Therefore j = « (sec d + l); and as sec is nearly equal to 1, we shall 
 
 have ^=^2aB, the equation of a parabola. 
 
 Since the original interyal was assumed equal to a, and as the 
 arc of a catenary is always longer than its projection on the axis 
 of the abscissa, or as A/3 is greater than a, the angle AOj3 or 6 is 
 greater than 45°. 
 
 CHAPTER XXXIII. 
 
 ON SOME PROPERTIES OF CONFOCAL SDRPACES*- 
 
 In Chapter XI. sec. [112] we have shown that through any point 
 of space three confocal suriaces of the second order may be assumed 
 as described, and that these surfaces must be an ellipsoid, a con- 
 tinuous hyperboloid, and a discontinuous hyperboloid. 
 
 This remarkable theorem, first enunciated by Monge, and further 
 developed by Dupin, is one of great importance. The latter geo- 
 meter has shown that the tangent planes drawn to these surfaces at 
 this common point, are at right angles, any one to the other two, 
 and that the tangent planes to any two of them at this point cut 
 the third surface along its lines of curvature. 
 
 We shall now proceed to establish a very remarkable theorem, 
 involving most important results, especially with regard to those 
 curve lines in the principal planes which have been named by 
 M. Chasles excentric conies. The demonstration may appear 
 tedious ; but the results obtained are very elegant and curious. 
 
 379.] ITiree confocal surfaces of the second order intersect in a 
 common point Q, the vertex of a cone which envelops a fourth confocal 
 surface ; to determine the equation of this cone referred to the nor- 
 mals of the three surfaces, at the common point Q, as axes of coor- 
 dinates. 
 
 It is very generally known that confocal surfaces mutually inter- 
 sect at right angles t; the normals of the three surfaces will there- 
 fore constitute a system of rectangular coordinates. 
 
 Let the equation of the fourth surface, an ellipsoid suppose, be 
 
 f|»» •/* S^ 
 
 ^H 35+^=1; (a) 
 
 and let a, )9, 7 be the coordinates of the point Q, the vertex of the 
 enveloping cone. 
 
 * The gulntance of this Chapter appeared in the Camlwidge and DubMn 
 Mathematical Journal, May 1864. 
 t See Dupin'a ' D^veloppements de G^om^trie.' 
 
ON CONFOCAL SURFACES OP THE SECOND ORDER. 357 
 
 Let x-a^m{z-y), y-fi=n{z-y) . . . . (b) 
 
 be the equations of a right line passing through the point [a^y), 
 and piercing the surface (a) in two points. If we substitute in (a) 
 the values of yp and y derived from (b), we shall have a resulting 
 equation of the form 
 
 U2rS + 2Vr + W=0, (c) 
 
 of which the two roots will express the values of the vertical ordi- 
 nates of the two points in which the line pierces the surface. 
 When the line becomes a side of the enveloping cone, the two 
 roots become equal, and we get the weU-known condition 
 
 UW-V«=0 (d) 
 
 If we substitute the values of x and y derived from (b) in (a), we 
 shall have 
 
 and by substitution, the equation of condition (d) will become 
 
 or, putting for m and n their values 
 
 x—a y—& , V 
 
 z—y z—y' ^°' 
 
 the equation of the enveloping cone becomes 
 
 ^ r a(^-g) Pjy-fi) y{z-y) y 
 La* 6* c* J ■ 
 
 (h) 
 
 We shall leave the equation of the enveloping cone in this form, to 
 facilitate its future transformation. 
 
 380.] Let the tangent planes at the common intersection Q of 
 the three confocal surfaces be taken for the coordinate planes to 
 which the equation of the enveloping cone is to be referred, and 
 let afi/C,, Oifiifiii' ''u/'iifiiii ^^ ^^^ semiaxes of the three confocal sur- 
 faces. Moreover, l.etp,Pi,Pn be the perpendiculars from the common 
 centre of the surfaces on the three tangent planes at the point Q. 
 Now the normals, at the point Q, to the three surfaces, being parallel 
 to the perpendiculars p, pp p^,, make the same angles with the ori- 
 ginal axes of coordinates. 
 
 Let X/tK, X'/tV'j ^'V'"^' ^ ^^^ angles which the perpendiculars 
 P> Pi> Pii respectively make with the axes of coordinates. Then, as 
 
ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 each abscissa is equal to the projections of the three new ones 
 upon it, 
 
 X— a=af cos \ + y* cos X' + 2* cos X", -^ 
 
 y—^=af cosfi+y' cosfi'+s' cos/i", >. . . . (a) 
 
 z—'f=a!' cos v+^ cos i^ + z' cos v",J 
 
 but 
 
 C03X=^, cosX'=^, 
 
 cos 
 
 
 (b) 
 
 Finding similar values for /i, fj, |t" and v, v', i^', then substituting 
 the resulting values in (a), we shall have 
 
 La,* fl„« a„^l' 
 
 (o) 
 
 Substituting these values in (h), sec. [379], the equation of the 
 enveloping cone, and eliminating the equivalent terms, we shall 
 have, after some reductions. 
 
 \ca) U« + c„*+c„«JLv + V+^«J 
 \ab) La«^V^VJU« + V *//.'-!' 
 
 + 2 
 + 3 
 
 (d) 
 
 omitting the traits over the xyz as no longer necessary. 
 Now, as the surfaces are confocal, let 
 
 
 
 C/i = c + A:, , 
 - s_-.a 1 I 3. 
 
 (e) 
 
 and as o^ is a point on each of the three confocal surfaces, we 
 
ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 359 
 
 shall have 
 
 
 
 1; 
 
 whence 
 
 ~ {a«-c«)(a«-6«) 
 P - (A«-c*)(6«-a2) 
 ^~ (c«-a«)(c«-62) 
 
 (f) 
 
 >. 
 
 (g) 
 
 If we now perform the operations of multiplication indicated in 
 the equation (d) of the enveloping cone, we shall find that the 
 coefficient of the term xy will be as follows : — 
 
 2«» r/ /9y(^'-c^) y J ya{c'-a') Y ,/ a)8(a^-&«) \'1 
 ' L\ be bfii bifii, } \ca cfl, c,flj \ab ap^ a,ftj J 
 
 -^^-p'L-^ 
 
 + 
 
 ^ 
 
 b%-b,^ 
 
 +- 
 
 e*c, 
 
 — 1 
 
 (h) 
 
 Now, if we eliminate firom this expression the quantities a, /3, 7, 
 by the help of (g), the first term of the preceding expression will 
 become 
 
 r tf(6'-c')(y+V)(c'+A,,')+y(c'-a')(o'+fe,,')(c'+A:.,')+c'(B''-y)(a'+A,.')(6'+A;.,') - 
 L o«6»(r'(6«-<r')((r'-«r')(o»-6=) 
 
 and this expression may be reduced to 
 
 rfl'(ft^-c«) + i«(c<-ffl*) +c^{<^-b*)- \ a 
 
 L a«A«c«(6«-c«)(c«-ffl«)(a«-A*) J*"" " .' 
 
 (i) 
 
 But the numerator of this fraction may be written in the form 
 (i*— c')(c*— a')(o*— 4*). Hence the first member of the coefficient 
 
 of xy in (h) may be reduced to ^S^'^ • 
 
 If in the same manner we calculate the second member of (h), 
 
 we shall find it to be — aft' J' • Hence the coefficient of xu is 0. 
 
 In the same way it may be shown that the coefficients of xs and ys 
 are each =0. 
 
p' 
 
 360 ON CONFOCAL SURFACES OF THE SECOND OBDEB. 
 
 We have now to determine the coefficient of oi^ in the equation 
 (d) of the cone. 
 
 Collecting the terms by vhich a^ is multiplied in that equation, 
 ve shall find their sum to he 
 
 li, in the first member of this expression, we substitute for 
 a, fi, y their values derived from (g), the transformed result will 
 become 
 
 + 62(c2-««) (c2 + *,a)(c'+V) («' + *;)(«' + *«')(*' + *') ■ W 
 + c^a^-b^) {a^ + h?) {a^+k„^ (&* + V) (**+*«')(c' + **) J 
 divided by the denominator 
 
 Making like substitutions in the second member of (j), the 
 resulting expression will become 
 
 4«c«(*«-c*) («« + *,*)(««+ V)(**+*')(c* + *') ' 
 + c«a*(c9-aS)(*2 + it2){A*+A^,9)(B2 + A9)(c»+A«) . (1) 
 
 divided by the same denominator as the preceding expression. 
 
 If we add together the terms in (k) that are multiplied by ifii?, 
 we shall find the result to be 
 
 -a«c«(a«-c2)(6« +*«) {*« + *««) [aV+ (a« +««)*' + *'(*<'+*«*) -*,'*«*], 
 
 and the corresponding term in (1) is 
 
 a V (ffl* - c«) (A* + *,«) (*« + V) [a«c« + A« (a« + c«) + A*] ; 
 
 adding these expressions together, the resulting expression will 
 become 
 
 c«c«(a«-c«)(i«+*,«)(i«+V)(A»-*,') (*«-*,«), . . (m) 
 or, developing this expression, 
 
 [o«4V+ fl*A«c«(* «+*„«) +a*c«*9*„«] (*«-*«) (*«-*„«) 
 - [fl Vi* + o«6 V ( A,« + V) + a V* ,V ] (*' - *') (** - *«*) • 
 
 Hence, if in like manner we develop and combine the remaining 
 terms in (k) and (1), the whole coefficient of 3^ will be the sum of 
 the following twenty-four terms, 
 
 -p^ 
 
p*{)c^-kf){k^-k^), (n) 
 
 ON CONFOCAL SUBVACBB OF THE SECOND OBDEB. 361 
 
 + A Va« + 6 Va* (* « + *,«) + A*a W 
 
 -b^<*a*-bH^a*{k? + V) -b^e^k^k,^ 
 
 + c«a*A« + c*o«A« (*,« + V) + c«4«*/y 
 
 -c*a*b^-€^a^b*{k,^ + k„^) -c«A«VV J 
 
 divided by the denominator 
 
 -a%^c^a^bfc^{1^-c^ {(^-a^ (o«-*«). 
 
 Nov, if we add the terms of this coefficient vertically, the sum of 
 the first column =0, the sum of the second is also =0, and the 
 sum of the terms in the third column may be reduced to 
 
 (6«-c«) (c«-fl«) (o«- &*)*/*„«. 
 
 Hence the coefficient of x^ becomes 
 
 ^m?k,?{k^-k?)[k^-k,^) 
 
 k*a^b?c*aW<^ ^°' 
 
 k*k^k ' 
 The symmetrical factor of this expression — '^ \ will appear in 
 
 the coefficients of y^ and 2% and may therefore be eliminated by 
 division firom the equation of the cone. Hence the coefficient of 
 x^ may thus be reduced to 
 
 p\k^-kr){k^-k,?) 
 
 k^a^b^b^ ^P' 
 
 Let A, B denote the semiaxes of the section of the ellipsoid {a/>f^ 
 conjugate to the diameter passing through the vertex of the cone, 
 and we shall have, by a well-known relation, 
 
 p.A.'B=afifii (q) 
 
 Hence (p) may now be reduced to 
 
 (*^-*,')(*'-V) ,,^ 
 
 ft«A«B* ^ ^ 
 
 From (g) it follows that 
 
 ««+/3«+7«=ffl« + i* + c« + A*+*« + V; • • • W 
 and by a common theorem 
 
 a'+i3«+7«-|-A*+B*=o«+V+c,«=a' + 6« + c«+3*«. . (t) 
 Hence, combining (t) and (u), 
 
 A«-|-B«=(A»-V) + (A«-V). . . . . (u) 
 Now the confocal surfaces {afifii) and ifliPuC,^ intersect in a line 
 of curvature, and for the whole of this line A is constant, or when 
 k and A, are constant A is constant ; hence 
 
 A«=(A«-V), B«=(ft«-V), . . . . (v) 
 
362 ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 and the coefficient of x^ in the equation of the cone, namdjr 
 
 becomes simplj 
 
 F . . . (w) 
 
 By a similar process we shall find the coefficients of y^ and z'^ to be 
 T^ and r-j. Hence we derive the following very simple equation 
 for the enveloping cone, 
 
 «»9 AaS «2 
 
 If we turn to (g) we shall easily perceive that one of the 
 factors in the numerator of the value of ^* must be negative. Let 
 k^ therefore be taken with a negative sign, and let it be greater 
 than Zi*. In order that the value of 7* may be real, since one of 
 the factors of the numerator is negative, two must be negative; 
 hence A/ must be taken with a negative sign ; and that there may 
 not be two negative factors in the value of /8^, k,^ must be less than 
 b^. Now, if a >6 >c, in the order of magnitude we shall have 
 
 b,r=b^-k^, c„^=f-k,K 
 Hence b,^ and c„* must be taken with negative signs. Since 
 bf,f=b^—k^f, Cnf=c^—ki^, b,^ must be taken with a positive sign 
 and c^f^ with a negative sign. Therefore the surface {a/tf^ is an 
 ellipsoid, the surface {aifinc,!) is an hyperboloid of two sheets, and 
 {"ii^iifiiii) '^ ^° hyperboloid of one sheet, (x) may now be written 
 
 hi~ If 2 hi — 
 
 i it 
 
 It is remarkable that the constants in the equation of this cone 
 are independent of a, b, c, the semiaxes of the sur&ce enveloped. 
 Hence, so long as the enveloped surface remains confocal to the 
 three others, it may change in any manner the ratio or magnitude 
 of its axes without changing the species of the enveloping cone. 
 
 When c=0, we get the equation of the cone which stands on the 
 ellipse (a, b) as base, and whose vertex is at the point Q,. 
 
 ^1.] If now with reference to the new system of coordinates as 
 axes, and with the point Ql as centre, we conceive a second group 
 of three surfaces confocal to each other, and a fourth confocal sur- 
 face whose semiaxes shall be k, k„ k/,, the squares of the semiaxes 
 of the intersecting group being 
 
 k^+a% k,^+a% k,?-\-d'; 
 
 kUb% k? + b\ V + A«,[ (a) 
 
 k^ + c^, kj' + c^, V + c^'J 
 
ON CONFOCAI. StTRFACES OF THE SECOND ORDER. 363 
 
 respectively, these surfaces will intersect in the origin O of the first 
 system of coordinates, and the fourth confocal surface (M,*,,) may 
 be enveloped by a cone whose vertex is at O and whose equation 
 referred to the original axes of coordinates is 
 
 ^+S+5=« ^'') 
 
 The original axes of coordinates Ox, Oy, Oz are normals to the 
 second group of confocal surfaces, as Qa/, Qy*, Q^ are normals to 
 the first, and the sums of the squares of the nine semiaxes in each 
 group will obviously be equal to each other, as also an axis in each 
 pair of corresponding surfaces. It is also obvious, from an inspec- 
 tion of (g), sec. [380J, that o, yS, y, the coordinates of the point Q 
 in the first system, become the perpendiculars from the point Q, 
 the origin of the second system, on the tangent planes to the 
 second group of surfaces having their common point of inter- 
 section at O. 
 
 382.] Let two cones having their common vertex on a surface of 
 the second order, an ellipsoid suppose, [ajbfii) envelop two confocal 
 surfaces. The diametral plane of the surface conjugate to the dia- 
 meter passing through the common vertex of the two cones will cut 
 off from their common side a constant length, independent of the 
 position of the common vertex of the two cones on the surface 
 {afifii) . 
 
 Let a, b, c; a, ^, <y be the semiaxes of the confocal surfaces. 
 And, as in the preceding theorem, let {fljbfi^, {Oifii/Ci^, {aiifiinCiii) 
 be the axes of three confocal surfaces passing through the vertex 
 of the cones. Hence we shall have, as in (e), sec. [380], 
 
 a,^=a^ + k^ a,^=a^ + kf, a„?=a^ + k,?,-\ 
 
 al'==a^ + h% a,^=a? + h^, a„f=a^ + h,^.\ ' ' ' ^ > 
 
 The equations of the cones will be, as in (x), sec. [380], 
 
 ■r* «s z^ 3? «' z^ 
 
 J+|. + ^=0; 5+|. + ,-;.=0, . . . . (b) 
 
 or, as these equations may manifestly be written. 
 
 ^^ , y' I ^' -0 1 
 
 af-a^^ a^f-a^^ a^^f-a^ ' ' 
 d^ y^ z^ 
 
 (c) 
 
 ««*-«* ««'-«' 
 
 =0. 
 
 Now the distance between the vertex and the diametral plane is p, 
 and as p coincides with the new axis of x, we shall have 
 
 x=p (d) 
 
 Let D be the length of the common side of the cones ; then 
 
364 ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 And if we find the values of x, y, z from the equations (c) and (d), 
 we shall have 
 
 In (p) and (w), sec. [380] ^ it was shown that 
 
 P - {k^-k?W-k,ry 
 
 
 or, as k^—k^=-a,—a^, k^—k^=a^—a„, , 
 
 («'-«/*)(«'-«/«*)■ 
 
 Now the numerator of (e) may be resolved into the product .of 
 the three factors 
 
 -K'-V)K'-«i')«-0' and «,«-a«=A«, a«-a»=A*. 
 Hence, making the substitutions indicated, 
 
 "-^ V) 
 
 Hence, as the value of D is independent of kp k,,, and of A,, h,/, it 
 will therefore not depend on the two auxiliary confocal surfaces 
 introduced, but the value will continue unchanged wherever the 
 point be taken on the surface of the ellipsoid. Hence D' varies 
 inversely as the product of the squares of the coincident semiaxes, 
 for 
 
 A«=a«-a«, A«=o«-o«. 
 
 When the enveloped surfaces become plane sections, c=:0, /3=0, 
 but J,*=/3* + A*, c,*=c' + A'; hence in this case ft,*=A*, c,*=A*, or 
 'D=a,. 
 
 388.] A cone whose vertex is on a surface of the second order 
 envelopes a confocal surface. To determine the length of the axis 
 of the cone between the vertex and the plane of contact. 
 
 Let the equation of the locus of the vertex of the cone be 
 
 a«+^9+^-i> (a) 
 
 {xff^i) being the vertex of the cone. 
 
 1^ y^ z^ 
 
 ^* 7^+F'^W+F'^'^+k'-'' <**) 
 
 be the equation of the confocal surface. 
 
 The equation of the polar plane of {xff^,} with reference to this 
 last surface is 
 
 ^^/ I yp, I ^^« _i (. 
 
ON CONFOCAL SURFACES QF THE SECOND ORDER. 365 
 
 The equations to the normal at the point {xg//:}) in (a) are 
 cos\ , , cosu , V 
 
 V . cos\ _ c'a?) co8/t_c*y^ 
 
 cos V €?zl cos v~ b^zl 
 Now A*=(j?-a?)«+(y-y/+(ir-2,)«, 
 
 or, substituting in this expression the values derived from the pre- 
 ceding equations^ 
 
 -•=K-'#-a^' <=) 
 
 We must now determine the value of z for the point in which 
 the axis of the cone meets the polar plane. For this purpose, from 
 the equation of the polar plane 
 
 a« + A« ■*" A« + **■*■ c* + A« ~ ^' 
 
 subtract the identity 
 
 J^, + ^=-^, + ^ + ^ 
 
 replacing 1 by its value, 
 
 as + i« + c«' 
 the result will be found 
 
 ^i(j^-^i) ■ yi(y-y<) , g|(g-.g i)_^» r ^,* , y,' , ^i' T u 
 
 o« + A« ^ *« + *« ^ c«+A« ~ Lfl«(a* + /t«)"^A«(i' + A*) c'(c'l-*')J' ^ 
 
 or putting for (x—x^ and (y— y,) their values derived from the 
 equations of the normal (d), we shall find 
 
 ..(£Z£.).*.. 
 
 Whence, cranbining this expression with (e), we shall find 
 
 For any other confocal surface, the vertex of the cone remaining 
 unchanged, 
 
 A'=^, or A: A' ::*«:*« 
 
 P 
 
 384.] To transform the equation of a surface of the second order, 
 
 so that the axes of coordinates shall be the normal to the surface at 
 
 a given point, and the two right lines in the tangent plane at this 
 
 point which are tangents to the lines of greatest and least curvature. 
 
366 ON CONFOCAL SURFACES OF THE SECOND ORDER. 
 
 Let the normal be the axis of x, then the axes of coordinates are 
 the normals to three confocal surfaces passing through this point. 
 Now if a, jS, 7 are the coordinates of this new origin on the surface, 
 substituting the values of x, y, z in the equation of the surface 
 
 a?' y* «•' - 
 
 derived &om (c), sec. [380] , we shall find the following resulting 
 expression : — 
 
 I* ^ 
 
 \ =1. (b) 
 
 af af Vaf a,^ a^f] a^La^ «„* a„,*J 
 ^ 2^ [>£, pgj pj^n ^« fpx, pg, pj£j\ ' 
 o, o, 1.0, Ofi a, II J Oi Lo, Oil Oiiij 
 
 Ci c, Lc^ Cii Cm J Cj LC| Cy Cm J / 
 
 Adding these terms vertically, the sum of the first column is mani- 
 
 2ir 
 festly =1. The sum of the terms in the second column is — '. 
 
 The sum of the terms in the third column is 
 
 Now the cosines of the angles X, fi, v, which the axes of coordi- 
 nates make with the perpen£cular p (let fall on the tangent plane 
 
 through the point (oySy) on the surface {afifilj), are ^> t^*"^; 
 
 and the cosines of the angles X', fi', v' which p, makes with the perpen- 
 dicular on the tangent plane through the point {a/Sy) on the surface 
 
 {aifiifiii) are^*^, ^^> a J and as these planes are at right angles, 
 "// "ii ^11 
 
 cos \ cos Xi+coa/i cos fii + cos v cos v,=0. 
 Hence the third column in the coefiScient of ^^^=0. In like man- 
 ner the fourth column in the coefficient of Zi=0. The fifth column 
 is 
 
 Now, as 
 
 cos«X=^, cosV=TT, cos* 7=^4, 
 n "i '^1 
 
 the coefficient of Xi^ may be written 
 
 cos* X cos* fl 
 
 .* + ** +- 
 
ON CONPOCAL SURFACES OF THE SECOND ORDER. 367 
 
 This expression is =-g, if we denote by r the semidiameter of the 
 
 surface parallel to p. 
 
 In like manner the coefficients of y^ and z,^ are - and 
 
 respectively, r, and r„ being parallel top, and p„. 
 The coefficient of jp^y, is 
 
 'I '■// 
 
 „ r «* /3* V* "1 
 
 multiply the terms of this expression by the equivalent factors 
 V-«i*=V-*/=c«*-C/'=V-A', dividing by this latter, and 
 the expression will be transformed into 
 
 2^, ra« ^ 7« / a« ^' y« \-| 
 
 Now the first of these groups is equal to -j ; and the second, as we 
 have already shown, is = ; hence the coefficient of xff, is 
 
 In the same manner it may be shown that the coefficient of x^, 
 ia—^ii— 
 
 Let r^ and r^i be the axes of the section parallel to the tangent 
 plane at the point (a^y) ; then, as we have found in (u), sec. [380], 
 
 rS J-2 1.2 «2 M 19 
 
 ' U — "■ "■!! ' 'I — " '^t ' 
 
 Introducing into the equation (b) the resulting expressions thus 
 found, the equation of the surface will at length become 
 
 ^^+y!+4_EP/^y_2?k,,+2f=0. . . . (c) 
 
 In this equation the coefficients are the perpendiculars p, pp p,, 
 from the centre on the coordinate planes, and the three diameters 
 of the surface which coincide with these perpendiculars. 
 
 Let x=0; then the equation becomes 
 
 y=V{-\)z^, (d) 
 
 which can be real only when one of the semiaxes r, or r„ is imagi- 
 nary, or, in other words, when the surface is a continuous hyper- 
 boloid. Since 
 
 y=/(-l).^, y=.^(^,).... 
 These are the generatrices of the hyperboloid j and it may easily be 
 
368 ON CON FOCAL SVBTACES OF THE SECOND ORDER. 
 
 shown that they are also the focal lines of the cone whose equation 
 is 
 
 385.] Along a line of curvature tangent planes are drawn to a 
 twrface of the second order. The perpendiculars from the centre on 
 these planes generate a cone of the second order, whose focal lines 
 coincide with the optic axes of the surface or with the perpendiculars 
 to its circular sections. 
 
 Since a line of curvature is generated hy the intersection of twQ 
 confocal surfaces, let the equations of these surfaces be 
 
 Let \, fi, V be the angles which p makes with the axes ; then 
 cos X= w, cos iL=. ?^', cos V = ^'. 
 
 From these five equations, eliminating x„ y,, Zi,a.nip, and putting for 
 
 , — — their values -, -, the resulting equation becomes 
 
 cos y cos V IT z ° ^ 
 
 Now the angles which the focals of this cone make with the 
 internal axis are given by the equation 
 
 a result independent of A; hence all the cones so generated are 
 confocal. Hence the supplemental cones to these have their cir- 
 cular sections parallel to these of the given surface, and therefore 
 these supplemental cones are the intersections of spheres with the 
 given surface. 
 
 The angles which the optic axes make with the vertical axes are 
 given by the same equation, 
 
 ^ , c«(a«-A«) 
 
 THE END OF THE FIRST VOLUME. 
 
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 INDEX. 
 
 Aii^ &' Overton's English Church History 14 
 
 Adney's Photography 10 
 
 Acton's Modem Cookery 20 
 
 Alpine Club Map of Switzerland 17 
 
 Guide (The) 17 
 
 Amos's Turisprudence 5 
 
 Primer of the Constitution 5 
 
 — — 50 Years of English Constitution 5 
 
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 Miscellaneous Works 6 
 
 Sermons 15 
 
 (T.) English Literature 6 
 
 Poetry and Prose ... 6 
 
 Amotts Elements of Physics 9 
 
 Atelier (The) du Lys 18 
 
 Atlierstone Priory 18 
 
 Autumn Holidays of a Country Parson ... 7 
 
 Ayre's Treasury of Bible Knowledge aa 
 
 Bacon's Essays, by Whately 5 
 
 Life and Letters, by Spedding ... 5 
 
 Works s 
 
 Bagehot's Biographical Studies 4 
 
 Economic Studies 21 
 
 Literary Studies 6 
 
 Bailey's Festus, a Poem 18 
 
 Bain's^ames Mill and J. S. Mill 4 
 
 ^■— Mental and Moral Science S 
 
 on the Senses and Intellect s 
 
22 
 
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 Bain's Emotions and Will 5 
 
 Baker's Two Works on Ceylon 17 
 
 Baits Alpine Guides 17 
 
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 ' & Bramwell on Railways, &c 13 
 
 Bauermnn sTAmeraXogy 10 
 
 Beaconsfield's (Lord) Novels and Tales 17 & 18 
 
 ^— — ^^— — ^— Speeches i 
 
 Wit and Wisdom 6 
 
 Becker's Charicles and Gallus 7 
 
 Beesly's Gracchi, Marius, and Sulla 3 
 
 Bingham's Bonaparte Marriages 4 
 
 Black's Treatise on Brewing 20 
 
 Blackky's German-English Dictionary 7 
 
 Bloxam's Metals 10 
 
 Bolland and Lan^s Aristotle's Politics S 
 
 Boultbu on 39 Articles IS 
 
 -s History of the English Church... 14 
 
 Bourne's Works on the Steam Engine 14 
 
 Bavidkr's Family Shakespeare 19 
 
 Brahourne's Fairy-Land 18 
 
 ——— Higgledy-piggledy 18 
 
 Bramley-Moore's Six Sisters of the Valleys . 1 8 
 
 Brande's Diet, of Science, Literature, & Art 1 1 
 
 Brassey's British Navy 13 
 
 Sunshine and Storm in the East . 17 
 
 Voyage in the ' Sunbeam ' 17 
 
 Bray's Elements of Morality 16 
 
 Brmime's Exposition of the 39 Articles IJ 
 
 ^nmiBjs/j Modern England 3 
 
 Buckle's History of Civilisation 2 
 
 Buckton's Food and Home Cookery 31 
 
 Health in the House 12&21 
 
 Bults Hints to Mothers 21 
 
 Maternal Management of Children. 21 
 
 Burgomaster's Family (The) 18 
 
 Cabinet La'wyer 20 
 
 Calvert'sVliii-i Manual 16 
 
 Capes' s Age of the Antonines 3 
 
 Early Roman Empire 3 
 
 Carlyh's Reminiscences 4 
 
 Crtto 'j Biographical Dictionary 4 
 
 Cay&yj Iliad of Homer 19 
 
 Changed Aspects of Unchanged Truths ... 7 
 
 Chesney's Waterloo Campaign 2 
 
 Christ our Ideal 16 
 
 Church's Beginning of the Middle Ages ... 3 
 
 Colenso's Pentateuch and Book of Joshua . 16 
 
 Commonplace Philosopher 7 
 
 Comic's Positive Pohty 4 
 
 Conder's Handbook to the Bible 15 
 
 Conington's Translation of Virgil's iEneid 19 
 — ^— ^ Prose Translation of Virgil's 
 
 Poems 18 
 
 Contanseau's Two French Dictionaries ... 7 
 
 Conybeare 3Xii Nowson' s Si. Paul 15 
 
 Cotta on Rocks, by Lawrence 11 
 
 Counsel and Comfort from a City Pulpit... 7 
 
 Cox's (G. W.) Athenian Empire 3 
 
 >^-^— ^— Crusades 3 
 
 m Greeks and Persians 3 
 
 Creigkton's Age of Elizabeth 3 
 
 England a Continental Power 3 
 
 Papacy during the Reformation 14 
 
 Shilling History of England ... 3 
 
 Tudors and the Reformation 3 
 
 Critical Essays of a Country Parson 7 
 
 Culle/s Handbook of Telegraphy 13 
 
 Curteis's Macedonian Empire 3 
 
 Davidson's New Testament 14 
 
 Dead Shot (The) 19 
 
 DeCaisne3.nALeMa0utsBota.Tiy II 
 
 De TocquevilW s Democracy in Ainerica... 4 
 
 Dewes's Life and I-etters of St. Paul 15 
 
 Dixon's Rural Bird Life , 11&19 
 
 £>an'j American Farming and Food 2i 
 
 Irish Land Tenure 21 
 
 Eastlake's Hints on Household Taste 13 
 
 Edmonds's Elementary Botany 11 
 
 Ellicotts Scripture Commentaries 15 
 
 Lectures on Life of Christ is 
 
 Elsa and her Vulture 18 
 
 Epochs of Ancient History 3 
 
 English History 3 
 
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