CORNELL UNIVERSITY LIBRARIES Mathematics Library White Kail CORNELL UNIVERSITY LIBRARY 3 1924 059 52 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059156152 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. 1991. Cl^arneU University Sibrary BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 MATHEIAATICS A NEW ANALYSIS OF PLANE GEOMETRY FINITE AND DIFFERENTIAL CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager aonllDU: FETTER LANE, E.C. ffillinbutBti: »oo PRINCES STREET Berlin: A. ASHER AND CO. ItijjiB: F. A. BROCKHAUS jjjto Bor*: G- P- PUTNAM'S SONS 6nmbai; anC aalrutta: MACMILLAN AND CO., Ltd. Kotonto: J. M. DENT AND SONS, Ltd. STofeso: THE MARUZEN-KABUSHIKI-KAISHA All nghis fistrvtd A NEW ANALYSIS OF PLANE GEOMETRY FINITE AND DIFFERENTIAL WITH NUMEROUS EXAMPLES BY A. W. H. THOMPSON, B.A. Sometime Scholar of Trinity College, Cambridge Cambridge : at the University Press 1914 CambriSgt : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS PREFACE IT is the purpose of this work to present a new analysis of Plane Geometry. We know that any geometrical theorem may be expressed as a relation in points. We may however look upon Plane Co-ordinate Geometry as having points and lines for its fundamental elements; in relations of which geometrical theorems are going to be expressed. Thus the equation y = mtc + c may be looked upon as a line of co-ordinates {m, c). It is this view that we shall adopt. Now let us denote points by small Latin letters and lines by small Greek letters. Let a,h,c...l be a set of points; a, ^,y ...\ a set of lines. Let us denote the joins and intersections of two points and two lines respectively by drawing a bar over them, thus ab, a/3. Also let us denote the distance of two points a, b by (ah) ; the perpendicular distance from a on /3 by {a^y, the angle between o, ^ by (oj8). Let us use the term ' measure ' to include the three cases. Let us use the notation {""a) Va) to denote the co-ordinates of a and (f., i7«) to denote the co-ordinates of a. Then what Co-ordinate Geometry effects is the reduction of expressions such as (a67S ... efXhfi...) (A) to a function of (xa, j/a), i^^b, Vb) ■•• (?a, »?o) • • • ■ Now let p, a be the Cartesian axes, then a;a = (ap), ya = {atr), and we may put fa=(ap), Va = (P'ra), i.e. the perpendicular from the origin on o. VI PREFACE Hence such an expression as (A) is reduced to a function of measures of the elements occurring taken singly with the reference axes. This idea I have generalized and have reduced the expression to measures of the elements taken in pairs. On p. 109 is an important result which I regret does not appear with Chapter I (it should be read between ^ 20, 21), which states that the square of distances, that perpendicular distances and the sine and cosine of angles are reducible to the quotient of two polynomials in (1) moduli of measures of two points, Ex. | (ab)\, (2) measures of a point and line, Ex. (a^S), (3) sines of measures of two lines, Ex. sin (a/3), (4) cosines of measures of two lines, Ex. cos (o/3), (5) measures of the join of two points and a point, Ex. (a6c), (6) measures of the intersection of two lines and a line, Ex. (aiSy). (7) cosines of the join of two points and a line, Ex. cos (aby). Cases (5) and (7) are reduced in surd form and (6) by means of a point to measures of two elements. Thus we do away with the idea of reference elements. But this brings us to another matter. We know that taking four arbitrary points, there is a relation between the six pairs of measures of two elements. We have also such relations in the case of three points and a line, two points and two lines and in the case of three lines. We have called such relations eliminants, being eliminants of relative position. Suppose we have reduced all our complex measures and noted all our eliminants. The matter of proving a relation between the complex measures reduces to proving the relation between their reductions with the help of the eliminants. Again, a point may be got from another point by a vectorial construction. We have denoted by a„,p the point distant p from a and measured in the direction of a>. We include such derived points in our consideration. To do this formally we take the point of general form a„„^^.^^^...^„^ denoting the point derived by a succession of such constructions. For the corresponding PREFACE vii line we take a»,,p,;»,,p,...»H,p,;„ denoting the line parallel to q> and passing through a„„^;^.p,....^,p,. Now take any complex measure containing such vectorially derived elements. Thus ■ This as we have seen is reducible to a function of measures of pairs of fundamental elements and vectorially derived elements. .In Chapter II we have reduced ia'h)\(a'P), where «' = «„„„;..,,,,...„..,„ (o'6), (a'^), where «' = a„„pj,^,^...^,^,„, to functions of- measures of pairs of fundamental elements and PitPi-.-Pn] including a>i, a)2...(u„ in the fundamental elements. Hence we can reduce any complex measure as above to a function of measures of pairs of fundamental elements and the magnitudes occurring in the vectors. We shall consider yet another class of derived elements. These are elements derived from the fundamental elements by an equation. Thus from the lines Oi, a2...a„ we have the derived line ^ar(xar)+a = 0. Again from the points Oi, aa...a„ and lines /3i, /Sa... /3m we have the derived point SAr (|a,) + XBr cos (1/8,) = 0. where A^, A^ ...An, Bi, B^... Bm are algebraic magnitudes. In Chapter III it is shewn that these can be treated in a similar way to that indicated for vectorially derived elements. Thus if our Geometry comprise only elements derived from fundamental elements by (i) intersections and joins, (ii) by vectorial constructions, (iii) by equational relations, we can reduce any measure of such elements to a function of measures of two elements and algebraic magnitudes. We note the eliminants. We may also have imposed relations stated in the particular problem. With these conditions we must prove relations between certain measures. This is a complete statement of our problem. We have thus stated our problem as a matter of reductional computation. Viu PREFACE The method devised for Differential Geometry is of a similar character. Let a; be a moving point and let a! be its consecutive position. This displacement ofx we shall define by its direction of displace- ment and the amount of its displacement : or in our notation xx and |(xa;')|. We denote asa! by the operative notation rx and \{xa>)\ by &c. Similarly let f be a moving line and ^ a consecutive position. The point of intersection ff we shall denote by the operative notation j)f and the amount of angular displacement (f^') by Sf. We shall first consider Differential Geometry of one displace- ment. Any problem dealing with such Geometry may be reduced to the consideration of such a measure as {•pabcT^ar (aw,) = ...'px^,„4Jc tx^...), and the differential of any complex measure or element. The method of reducing such an expression is developed in Chapters IV — VIII. By means of the principle thus set forth we are enabled to reduce such a measure to measures of the fundamental elements and elements of displacement ra, rb..., pa,p^... and the amount of their displacement. We may look upon ra, rb..., pa,pfi... as fundamental elements. We take note of the eliminants of all our fundamental elements and these elements of displacement. The magnitude of the displacement of each element we must look upon simply as small algebraic quantities. Should there be any imposed conditions we have these and also their first derivatives. Our problem is then com- pletely stated and set forth as a matter of reductional compu- tation. We next come to the matter of elements of displacement of elements of displacement, such as pvx when vx is the line through X perpendicular to rx. We suppose all the fundamental elements that are variable to trace continuous curves and in Chapter VIII we have shewn how to reduce such to elements and differentials of first displacements and the curvature of the curve at the elements. We next consider elements of displacement of elements of displacement of elements of displacement and so on. All PREFACE ix quantities which depend only on the curve traced by a variable point we shall call intrinsic functions of the point. In Chapter VIII it is shewn that such are reducible to elements of a single displacement and intrinsic functions of the curve. Thus any such measure as (pp'^abyd . . . j/"2 a, (xor) = . . . a;^) is reducible to measures of fundamental elements and elements of displacement of these elements and intrinsic functions of these elements. We also need the differentials of such measures and such derived elements as pv^ ahyd ...v"Sar {xa^ = 0. These problems are the most general problems of Differential Geometry. The last chapter is a chapter on Integration adopting these ideas. In the Miscellaneous Examples I have endeavoured to illus- trate the method. As the present work is intended as a presentation of method, I have not tried to make the examples exhaustive of well-known properties. The large modern theory of singularities of curves I have not considered at all. I must apologise for the rather amateurish manner of the state- ment of the axioms, which are ticketed with large Roman numerals. These are the axioms which form the basis of the symbolic procedure of the text. This method must not be worked out by using a figure. The result is generally a hopeless quandary of sign. It must be worked directly from the axioms and deduc- tions after having translated the conditions of the figure into a statement in symbols. As will be seen, no more than these axioms are required so far as the domain defined is concerned. The axioms may be divided into two main classes from a natural standpoint, axioms of actual properties and axioms of convention. The latter have firom time to time been given by successive writers for the purpose of comprehending many cases in one though I believe they appear in a connected form for the first time here. There are two main points in regard to a system of axioms. The first is that they should be sufficient, the second X PREFACE that they should be consistent. The proof that they are con- sistent I have not attempted. The fact that they have not yet yielded a contradiction is a powerful argument of their consist- ence. Axioms (XV), (XVIII) have been proved visually. But no doubt with a few more fundamental axioms of superposition and orientation, these could be deduced. There are two main ways of considering Geometry. One is by sight or figure. The other is by a symbolic representation of the figure. The former method I shall call the Visual method: the latter the Symbolic method. Visual Geometry, as it is known, is of a synthetic or transfor- mational character but there is no reason why it should not be analytic or reductional. A few cases can be cited in which a theorem in Visual Geometry is worked out by a reductional process. The conditions of the problem may be represented by certain magnitudes determined by the problems, such as areas, distances: and what is to be proved is also a relation between the same magnitudes. The working after this is a matter of reductional computation. As an alternative method to the Visual method we have the Symbolic method. As a rule the method of Symbolic Geometry is of a reductional character. The method of the text belongs to the Symbolic method. In some cases, however, the process is easily visualised and coincides with the treatment of Visual Geometry. As regards its accomplishments, the method of Visual Geometry manifests unexpected power, a power which, however, is not sustained. An illustration of this power is afforded by Hart's proof of Steiner's construction of Malfatti's problem. The Symbolic method is characterized by its complete grasp of the problem : compared with the method of Visual Geometry it lacks the power of its transformations. The method of the text has an advantage over Co-ordinate Geometry in the matter of sign. The method of the text gives a more automatic account of sign than does Co-ordinate Geometry, PREFACE XI as the text will I think shew. In a Cartesian system, a line is represented by an equation. Now an equation gives no 'sense' to a line which I think explains this deficiency. Casey has given a convention which applies to Co-oidinate Geometry which states that the perpendicular from a point on a line is positive or negative according as it is on the same or opposite side of the ongin, but it does not seem to have been developed in conjunction with others. I have been engaged on the present work for the last three years. I claim the method as original. There are some theorems which are original, and most of the general results in the examples I believe are new. The new treatment of the trigono- metric functions is original and necessary for the purpose. We notice here the almost identity in symbol between the method of Grassmann and that of the text, in the case of Geometry of Position. The theorems and proofs on cubic curves in the Miscellaneous Examples have been adapted directly from Grassmann's theorems and proofs as given in Whitehead's Universal Algebra. I was not aware of the method of Grassmann before I discovered the method of the text, though I was aware of a similar method which the Algebra of Invariants afiFords. Not many cases of a general transformation have occurred. One of the best is Example 12, § 28. Most of the Examples given on parallique and orthologique pairs of triangles are readily proved from this. I have laboured to eliminate errors of detail, but no doubt in a new work like this there are some still remaining. The notation I hope will meet with approval. My aim h»s been to make it unambiguous, easily written and as short as possible. The notation of putting r before x for the direction of displacement of a; is ambiguous unless we agree to reserve r for this special purpose. In the Appendix I have given four cases of reduction of pro- ducts of measures. Each is such that, though a component measure is reducible only by radicals, owing to the eliminants the product can be expressed without radicals. Xll PREFACE It is a great pleasure to me to acknowledge my obligations to Mr S. Chapman, Fellow of Trinity College, Cambridge, for reading part of the proofs with me and for suggestions. The terms 'measure' and 'determinate' are due to him. In conclusion I wish to express my gratitude to the readers and officials of the University Press for their close attention and unfailing courtesy. A. W. H. THOMPSON. AprU, 1914. CONTENTS FINITE GEOMETRY INTBODUCTION PAGE Definitions of element, incidence, determinate, measure ... 3 Notation for determinate and measure 3 Classes of derived points and lines considered 4 Sketch of method 5 Fundamentals of differential geometry 6 Method employed for differential geometry 7 CHAPTER I FUNDAMENTALS OF THE GEOMETRY OF TWO, THREE AND FOUR ELEMENTS Axioms 8 Geometry of two points and a line 8 Definition of Sine function ... 9 Greometry of three points 10 Oeometry of three lines 11 Gteometry of two points and two lines 11 Oeometry of three lines and a point 12 Further axioms 13 The value of sin ^ •: 13 z Definition of cosine function 13 Addition formulae for sine and cosine functions .... 14 Further geometry of the triangle 15 Geometry of two lines and a point 16 Geometry of three points and a line 17 Further geometry of two lines and two points 18 Geometry of three lines and a point 18 Eliminants 18 Examples . . 20 XIV CONTENTS CHAPTER II REDUCTION OF MEASURES CONTAINING VECTORIAL ELEMENTS Reduction of (a^...i6)'. Reduction of (a^...^|3) . Reduction of {a^...^b) Reduction of (,aft...4vP) Examples PAGE 24 25 26 26 CHAPTER III REDUCTION OF MEASURES CONTAINING EQUATIONAL ELEMENTS Proof that iar{3!ar) + a=0 represents a line and that any line can be represented by such an equation 30 Reduction of (2a, (a7a,)+o=0 /S) 31 Reduction of (So, (^iv)+a=0 b) 31 Proof that 2 J , (^) + 2 S, cos (^;S,) = represents a point and that any point can be represented by such an equation .... 32 Reduction of (24, (^) + 25, cos (10,) =0 c)* 33 Reduction of (2ii,(|a,)+25,cos(|/3,)=0 y) > 33 Proof that 24,(^,) + 25,co8 (^j3,)=0 represents a direction, when 2i4,=0 .34 Reduction of (2^1, (|a,) + 2£,oo9(^/3,) = y) when 2il,=0 . 34 Examples 34 Method of Analysis 35 DIFFERENTIAL GEOMETRY CHAPTER IV DIFFERENTIATION OF MEASURES OF SIMPLE ELEMENTS Differentiation of \{xa)\ 39 Differentiation of {xa) 40 Differentiation of (^) 40 Differentiation of (|a) 41 Linearity of differentiation of measures of more than one variable element 41 Examples 41 CONTENTS XV CHAPTER V DIFFEBENTIATION OF DETEBHINATE8 OF SIMPLE ELEMENTS PAQB General method of effecting diflFerentiations of determinates . 43 DifTerentiatioD of 3^ 43 Differentiation of |i) . . . . 44 Examples 44 CHAPTER VI DIFFERENTIATION OF VECTORIAL ELEMENTS Evaluation of dxft.„a 46 Evaluation of dx^...^^ 46 Examples 47 CHAPTER VII DIFFERENTIATION OF EQUATIONAL ELEMENTS Beduction of rf {20,(^0,) +o=0} 50 Reduction of rf {2.4, (^,) + 2fi, cos (|^,) = 0} 51 Examples 52 CHAPTER VIII REDUCTION OF MEASURES CONTAINING FUNCTIONAL ELEMENTS Formula for the evaluation of (jxa), {vxa) 54 Formula for the evaluation of (nra), {vxa) 54 Formula for the evaluation of {pta)\ (y^) 54 Formula for the evaluation of {p$a), (v^a) 55 Examples 55 Bule for writing down the differentiation of simply derived elements . 58 Examples 60 CHAPTER IX GENERALIZED DISPLACEMENT OP A POINT Gieneralized displacement given hy t^x, h-ix; t^x, 82^; ■■■'''nX, Sn* • 63 Evaluation ot dx 63 Evaluation of drx 63 Evaluation of {vxa), (rxa) 63 XVI CONTENTS PLANE CURVES CHAPTER X RBDUCTION OF COMPLEX FUNCTIONAL ELEMENTS PAOB Proof for a plane curve that prx=x, rp^^i 66 Definition of px, p^ 65 Reduction of vtx 66 Reduction of v^x 66 Reduction of pvx 66 Reduction of vp( 66 Reduction of y'f 66 Reduction of pv^ 67 Examples 67 CHAPTER XI SUCCESSIVE DIFFERENTIATION OF MEASURES Sucoesaive differentiation of ^(xa)' 69 Successive differentiation of \{xa)\ 70 Successive differentiation of (xa) 70 Successive differentiation of (|a) 71 Successive differentiation of (^a) 71 Examples 71 CHAPTER XII INTEORATION OF HEASCBES Definition of integtation of measures containing point and line variables 73 Examples of integrals 74 Laplace's equation 77 Miscellaneous Examples 78 Appendix 110 Table of Formulae . . 113 Index .119 FINITE GEOMETRY INTRODUCTION § 1. Definitions : (1) The elementary concepts of plane Geometry are points and lines. We shall refer to them as elements. (2) That a line passes through a point we shall also express by saying that the line is incident* in the point, or the point is incident in the line. (3) The determinate of two elements of like kind is the element which is incident in both these elements. Thus the determinate of two points is the line joining them, and the determinate of two lines is their point of intersection. (4) The measure of two elements is a certain quantity deter- mined by these elements, expressing the relation of one in regard to the other. The measure of two points is the distance between them. The measure of a point and a line is the perpendicular distance between the point and line. The measure of two lines is the angle between them. The question of the sign of measures is fundamental. § 2. Notation. Points will be denoted exclusively by small Latin letters a, h, c ... x, y, z: lines by small Greek letters a, ^,7 ... a. The determinate of two elements will be denoted by writing them side by side and drawing a bar oyer the two. Thus the determinate of a point a, and a point h will be denoted by ah. The determinate of a line a and a line yS by a^. The measure of two elements will be denoted by writing them side by side and enclosing them in small brackets. Thus the measure of a point a and a line h is written (a6). The measure of a point a and a line /8 is written {a^), and of two lines (a/3). * See Whitehead's Axiom of Geometry. 1—2 4 INTRODUCTION [2- Vector will be used in the ordinary sense. A vector will be denoted by a small Greek letter with an acute accent. Thus p will denote a vector. The direction of the vector will be denoted by p and the magnitude by p with a circumflex accent, thus p. The letters p, v, t will be used for certain special purposes. § 3. A point may be derived from given elements in the following three ways: (i) by a scheme of determinates only. This is the way points are obtained from other elements in Descriptive Geometry. Thus 067 gives a new point, namely the intersection of the line ah with the line 7. (ii) by a scheme of vectors. Thus if l> denote a vector and a a point, we get another point a,. This denotes the ^-''^^ point distant p from a, measured in the direction of p. (iii) by an equation, as is done in co-ordinate Geometry. Thus if a,, a2...tt„ be n points, and | a variable line, and Ai, A2... An algebraic magnitudes, the equation '^Ar{^ar) = denotes a point; meaning that any line which satisfies the equation passes through a fixed point. There are other ways of getting new points, as by the rolling of one curve upon another; but the above three are the only ones we shall consider. Again, a new line may be derived from a set of points and lines in three corresponding ways : (i) by a scheme of determinates. (ii) by a scheme of vectors and direction. Thus if a is a point and p a direction, a, is the straight line through a, with direction p. (iii) by an equation. If a,, a^.-.a^ be n lines and x a variable point, and Oj, o^ ... an algebraic quantities, the equation Sor {xdi) = denotes a line. § 4. In the present theory a geometrical property depends on an equation in measures. Now the proof that one measure equals one or more other measures, may be effected by reducing 5] SKETCH OF METHOD 5 all the measures to measures of two elements. Also we have, with one exception, for any four unrelated elements, a relation between the measures of the six possible pairs. The proof after this is a matter of algebra. However it is not necessary always to reduce the measures. It would suffice if we could so transform, the measures, without actual reduction, so as to shew their equality. The former method of reduction corresponds to Analytical Geometry ; the latter method of transformation corresponds to Synthetic Geometry. In the text the method of reduction is used uniformly. Our object will be then to classify measures and give a calculus for their reduction. It will be seen that the formulae of Co-ordinate Geometry depend for their use on the fact that they enable one to reduce measures. Instead of these formulae, we have given the actual reduction of such measures as are necessary, and give a definite method for the reduction of more complex ones from these. § 5. Sketch of Method. The measures of two elements are fundamental. They are three in number. The measures containing three elements, we shall call measures of the third order. These are, with one exception, reducible, that is to say, they can be reduced to algebraic or trigonometric functions of the measures of two elements. The measures of two elements are (xy), (xrf), (^ij) where x, y are points and ^, if lines. The order of a function of several measures we define as equal to the greatest number of elements occurring in any component measure of the function: thus \{aiy)\{s^z) is a measure of the third order ; here | (ary) j denotes as usual the modulus of (xy). It is to be remarked that the order of a measure so defined only applies in the case where no two of the elements are identical. A measure of three lines is not reducible. However with the introduction of an arbitrary point, it can be reduced. All measures of the fourth order are reducible. Hence all measures containing three or more elements are reducible. 6 INTRODUCTION [5, So far we have only dealt with simple points and lines. We have next to consider the two other classes of elements stated in § 3. These we shall call (i) the general vectorial point and line, (ii) the general equational point and line. We shall consider first the general vectorial elements. The general vectorial point is «»(»...»• This denotes the point derived from a by a series of vec- tors /{, o- ... y, (o^...*^), (ap*...*ot), (a^...4^0). Having found the reduction of these we can, by the formulae for the reduction of measures of simple elements, reduce the measures of simple and vectorial elements. We proceed in an exactly similar manner in regard to the reduction of measures of equational elements. In other words we have a calculus for the reduction of measures. We have so far considered the geometry of finite concepts only. § 6. We next indicate the ideas upon which differential geometry is built. We proceed as follows : Let a; be a point, of another point near x. We write dx for the small quantity \{xx')\ and rx for omc'. Again, if f be a line, and f another line near f, we write 3f for the small angle (f^'), and p^ for the point of intersection ff . With these definitions we proceed to the differentiation of measures. 6] DIFFERENTIAL GEOMETRY Thus we consider the values of d|(a;a) | d(xa) d(^a) d(|o) dx • "dUT' "df' '~djf' r iD8tan( with X, of the expression We define, for instance, , = Limit when a;' tends to identity dx {a/a) — (xa) \{xx')\ ■ It will be found that having differentiated the measures con- taining two elements, all the measures of more elements may be differentiated by a definite method, independent of the method of limits. We next require the differentiation of determinates. It will be found that the differentiation of determinates may be reduced to the differentiation of measures. The definition of the differential of a determinate, say xd, is — ;— = Limit when x' tends to identity with x of -77 — jrf . ax •' \{^x)\ We consider next vectorial elements. First, we require the differentials of a^....^ and a^^,.,^. Having found these we may find the values of the differentials of measures and determinates of vectorial elements. The same holds for equational elements. These are all the formulae we require, and knowing these we may differentiate the most general measure and determinate. At the same time we may reduce measures containing the differ- ential signs p; v, t. CHAPTER I FUNDAMENTALS OF THE GEOMETRY OF TWO, THREE AND FOUR ELEMENTS § 7. The measures containing two elements are (ab), (- , , . , , . •^ ^ ^ :(a6)(cK!)| Now sin(oca6) = — sin(o6ac), .-. (acb) = - (abc), .: (cab) = (abc). Hence (abc) = (bca) = (cab) = - (bac) = — (cba) = — (acb), also (abc)=\ (ca) (ab)\ sin (cac3)) = \(ab) (bc)\ sin (ah be) = I (be) (ca) 1 sin (bccd). Gorollarv «" (ca a^) _ sin ( ab 6^) sin (fee eg) 1 y- \(bc)\ - \(ca)\ -~ \(ab)\ -2R- * (abc) as thus defined is equal to twice the area of the triangle formed by the points. 8-11] FORMULAE OF SEDUCTION 11 We shall call (aic) the standard measure of three points. R ia called the circum-radius of the triangle. § 10. Oeometry of three lines a, /8, 7. We write (0/87) for sin (a/9) . (0^7), .-. iffary) = sin (/So) (^7) = - sin (a/3) (^7) = - (afiy). Now if we put a = fiy, b = ya, c = oyS; we get bc=a, m — p, ab = y, or bc=a, ca = /8, ai = y, or bc=a, ca = 0, ab = y, or bc=ct, cd = l3, ab=y, or one other set tff relations. We shall consider only the first alternative ; the same result follows from any one of them. We have sin (a/3) (a/37) = ^^^ (bcca) (cab) (abcy -\{bc){ca)(al>)\- Hence (0^87) = (/87a) = (70/8) = - (^ay) = - (7^*) = - (ayfi), and (0/87) = (0/87) sin (0^8) = two similar expressions. We shall call (a/87) t^® standard measure of three lines. It is easy to shew that I (a/87) I = 1(7* «/8) sin (7a) sin (a/3) | = two similar expressions. § 11. Geometry of two points and two lines, a, b, y, S. To reduce the measure (abyS). Let 067 = 0. Then dd = ob or oa = ob. Suppose oa = ob, then also ,in(UaB)J-^^l^: siui^bB)J-^^l:^ , \(oa)\ \{ob)\ (ay)_ (oS)-(aS) •■{by) (oS)-(66)' 12 GEOMETRY OF TWO, THREE AND FOUR ELEMENTS [1 , . , . ., ( a^){bS)-(aS)(by) from which (oS) = [ay) -{by) ' Now {abyS) = am{aby)(abyS) (ay){b?,)-{aB){by) \{ab)\ It is easy to see that when od = ob the same result follows. Writing (afryS) for | (ab) | sin ( oi 7) ( aft 78), we have (0678) = (07) (68) - (aS) (by). We shall call (0678) the standard measure of two points and two lines. § 12. Geometry of three lines and a point, a, /3, 7, d. Let a, b, c be three points and d a point in- cident in be. Then (abd) + (adc) = \{bd)\(aFd) + \(dc)\ {adS). Suppose hd = dc = be, then {abd) = {adc) = {abc), and |(6d)|+l(dc)|=|(6c)l, .-. {abd) + {adc) = \{bc)\ {abc) = (aic). The same result follows from the other alternatives to bd = dc = be. Now letd be any point, not necessarily incident in be. Let ad be = e. 4\ Then {dbe) + {dec) = {dbc). //\ {abe)+{aec)={abc), / / \ {abe) = {abd)-¥{dbe), 6 e c {aec) = {adc) + {dec). from which {dhc) + {dca) + {dab) = = {abc). Now let bc = a, ^ = /3, ab = 7- 11-14] DEFINITION OF COSINE FUNCTION 13 Then (dbc) = |(6c)| (da), etc. .-. |(6c)| (da) + |(co)| (dy3) + \(ab)\ (*y) = (abc), .: sin (0y) (da) + sin (7a) (d^) + sin («/8) (dy) = (0/87) by § 9. The same result follows from any of the other alternatives of § 10 ; and is therefore true for any three lines. § 13. With regard to any line a we shall assume that one and only one line yS, passing through a fixed point, can be found, so that (a/9) has any given value, say 6. Further that all such lines through different points are parallel, i.e. the measure of any pair is zero (XVI). Notatiim. We shall write «« for such a direction ; so that (««.) = e. Corollary. o„ „ is parallel to o. 22 Aosiom. We shall assume that i(«i8)| = |(a,„/3a)| (XVII). 2 TT To find the value of sin ^ Let 00, = and let a be a point incident 2 in a, such that oa = a„. 2 2 Then sin (aa,r) = sin (aoa) = — sin (odd) ' _ (««) \(oa)\ ^_J^^^(aa)^ ~|(aa.^«)l IMl 2 = + 1 since the orientation of a in regard to a is counter-clockwise (XVIII). §14. Definition of cosine function. We define the cosine of the measure (oy3) as follows : cos (a;8) = sin (ay3„). Now (a^„) + ()8./8) + (^«)=27r, 2 2 .-. (ayS,)=27r + (a)S) + J. 14 GEOMETRY OF TWO, THREE AND foUR ELEMENTS [l Now sin (a^) = - sin {(o/3) + tt) = sin {- tt - («/3)l = sin {ir - (afi)], • sin {(a^) + f} = sing- (a^)|. Again, (o„ /3) + (M + (««;:) = 2t, 2 2 2 ^ .: sin (a,j8) = sin || + (a/8)| = - sin || - (a^)| = -siu|| + (a^ = -cos(a/3), .-. cos (o|8) = sin (a/3,) = - sin {a^^) 2 2 = sin|j+(«)8)j=sin|^-(a/3)|. Again, cos (|8a) = sin || + (/Sa)l = sin |j - (a/8)j = cos (a^). cos {a,^) = cos || - (a/9)| = sin (o^), 3 (ayS.) = cos || + (o^)| = - sin (a/3), ^ cos {it - (a/3)} = - cos (o/3), cos {it + («/8)} = - cos (a^), cos (o;8) = cos (o/8) = — cos (o/3). cos( § 15. Addition formulae for sine and cosine functions. The three lines ab, ac, 7 where (07) = 0, denote three arbitrary direc- tions. Now {aby) — (acy) = 2Tr + (abac). 14-16] FURTHER GEOMETRY OF THE TRIANGLE 15 Now sin (aby) cos (007) — sin (0C7) cos (067) = sin (aby) sin ( 007^,) — sin (acy) sin (ahy„) 2 2 (by) 2 (07) i ■ / \ n (&C77^) = i"7~iw — Ti I'y formula on p. 12 |(a6)(ac)| -^ ^ ^ |(6c)jsin(77^)(6ca) _ 2 \(ab)(ac)\ _ (abc) = sin(caa6) \{ab){ca)\- = sin (abac) = sin {(aby) — (acy)}. If we put (067) = 6, (007) = ) = sin cos — sin <^ cos 6, and we have the other trigonometric formulae. § 16. FurtJier geometry of the triangle „ ■ . In this notation a = bc, ^=cd, y = ab. We have (^87) + (7a) + (afi) = 27r, .-. - sin (a/S) = sin {(ffy) + (70)] = sin (^y) cos (7a) + sin (7a) cos (3y). Hence - 1 (ah) | = j (6c) | cos (7a) + | (ca) \ cos (0y), similarly - \(ca)\ = |(a6)| cos (0y) + \(bc)\ cos (o^) - \(bc)\ = \(ca)\ cos (a0) + \(ab)\ cos (7a). From which (60)" = (caf + (aby + 2 1 (ca) (ah) | cos ( ca 06 ), and two similar formulae. We may now reduce the standard measure of three points. It may be shewn that 4 (obey = 2 (ca)^ (ahf + 2 (aby (bey + 2 (6c)'' (cay -(bey -(cay -(aby. 1(5 GEOMETRY OF TWO, THREE AND FOUR ELEMENTS [I § 17. Geometry of two lines and a point, a, /S, c. To reduce (a^cy. Let J3 = «c«„, q = Pc^,- 2 2 Now in the triangle pq, y3, a sin (aj99) sm(pq^) sin (/9a) ' sin (pgo) sin (a/8) " Again, we have sin {cp pq) _ sin (^ op) _ sin (/8d) _ sin (a/8) !(c9)r~ !(P9)[ ~ i(P9)l ~ \(pq)\ COS (apq') _ sin (^pqoC) " l(c«/)l_" l(ir/S9)| • .-. tan (pg «) = ^'^^^ = tan (cgc"^), Hence .". sin {pqa) = ± sin (cqcafi). {pqf _ (^qY «i°'(«^) sin= (cgc^) = (a/8cf. Firstly, suppose the sense of a, yS to be counter-clockwise in regard to c, .: (a0cysin={a^) = (pqy = (pcY + (qcf - 2 \(pc) (go) I cos (pcqc) = (pcy + (qcy - 2 \(pc) (qc) \ cos {«^/3,) = (ca)2 + (cySy - 2 (ca) (c/3) cos (ayS), since (ca), (c/8) are both positive. Secondly, suppose the sense of o to be counter-clockwise in regard to c, while that of yS is clockwise. Then the senses of b, yS are both counter-clockwise in regard to c. Hence _(a;8c)' sin= (a^) = (cay + (c^y - 2 (ca) (c^) cos (a^), .■ . (a^cy sin» (afi) = (cay + {c^f - 2 {ca) (c0) cos (a/8). Similarly in the case when both a, yS have senses which are clockwise in regard to c, we obtain the same result. Hence in all cases sin» (ayS) ( ^c^ = (cay + (c^y - 2 (ca) (cy3) cos (a/S). 17, 18] FORMULAE OF REDUCTION 17 Hence sin (a/8) |(0)8c)| = v'(co)» + {c^y - 2 (ca) (c/S) cos (o/3), where the square root has the sign of (a/3). Thequantity (aj8c) = sin(a/8)l(a)8c)( we shall call the standard measure of two lines and a point. § 18. Geometry of three points and a line, a, b, y, d. To find the value of (ahydf. ''^ Let aby = o. In the triangle who.se vertices are a, o, d we have {ady = (aoy + (odf /y + 2 \(ao) (od)\ cos (adod). Similarly {bdf = (bof + (odf + 2 | (bo) (od) \co8 (Food). We shall consider only the case in which ad = bo=ab; in the other cases, the theorem can be proved in a similar manner. Multiplying the first equation by \{bo)\ and the second by \(ao)\ and subtracting, we have {ady\{bo)\ - {bdy \{ao)\ = (aoy \(bo)\ - (boy \(ao)\ + (ody{\(bo)\-\(ao)i}, (a dy\(bo)\-(bdy\(ao)\ - \{ab)\ Now sin (007) = sin ( 067), . («7) _ (h) . " \(oa)\-\(ob)\' \(oa)\-\(ob)\ = \(ah)\, (ay) _ (by) ^ (arf)-(h) \(oa)\ \(ob)\ \(ab)\ ' (ody=' l+|(ao)(6o)|. \(ob)\=, (h) .IWI." (ay)-(byy _ ^ (bdy(arf)-(ady(by) (07X67) (^., T. G. 18 GKOMETBY OF TWO, THKEK AND FOUR ELEMENTS [l The quantity (aiyd) = |(a6)| sin (aby) (ahyd) we shall call the standard measure of the elements a, b, 7, d {ahydf = [{bdy (07) - {adf (67)} {(07) - (67)} + («7) (^7) («&)' = (6d)» {a^y + (ad)' (67)' + (07) (h) {{aby - (adf - (bd)% § 19. Further geometry of two lines and two points, a, fi, c, d. To find the value of {a^cd). (cda/S) |(«/8c)|sin(«^) (ca)(d;8)-(c^)(da) V(ca)» + (ciS)" - 2 (ca) (c/3) cos (a/3) ' where the square root has the sign of (a/3). If (oyScd) = (a)8c) (a^cd), it is clear that (o/3cd) = (cda^S). § 20. Geometry of three lines and a paint, a, y8, c, S. To find the value of sin(a^cS). • /=^*^ (^8)-(c8) sm (o^c8) = -^ _ (a/38)-(c8)sin(tt/3) (aySc) Now (aySS) = (ca) sin (/88) + (c/S) sin (Sa) + (cS) sin {afi). ■ /"=^s^ (ca)sin(/88) + (c/3)sin(Sa) .-. sm (a/8c8) = i — ^^ / a \ ^^ — ^ (ac) sin {I3S) - (fie) sin (g8) (a/8c) We shall call (o/ScS) = (a/3c) sin (a^c8) the standard measure of a, /3, c, 8, so that (a/Sc8) = (ac) sin (/SS) - (/3c) sin (a8). § 21. Eliminants, Let iS be an arbitrary set of elements. Then a relation between the measure of pairs of elements selected from this set we shall call an eliminant of the set. 18-21] ELIMINANTS 19 1 1 1 1 1 (aby (acy (ady 1 (bay {bey (bdy 1 {cay (cby {ody 1 (day idby (dcy We have for three lines (/87) + (7a) + (a/3) = 27r. For four elements there is with one exception an eliminant hetween the six measures of the six pairs of elements we can get from the four elements. (i) Four points a, b, c, d. We have, see Casey's Analytical Geometry, p. 305, formula (756), = 0. If this be expanded and reduced by the relation {bey = {cay + {aby + 2 \{ca) {ab)\ cos {m oft), we get 2 {bey {ady + 22 \{ca) {ai)\ cos {cdab ) {bdy {edy + 22 I (6c) {ca) {ah)\ 2 \{be) | cos {ca^){ady + {bey {cay {aby = 0. (ii) For three points and a line, a, b, e, S. Now if 0+ + ^|r=2v, then 1 — cos* — cos^ ^ — cos* ijr + 2 cos tf cos cos •^ = 0. Now (6c 8,) + (S„m) + (mbc) — 2v, 2 2 .■. 1 — sin' ( 6c S) — sin* {Bed) — cos* {ca be ) — 2cos(ca6c)8in (Soa) sin (6cS) = 0, . {(68)-(c8)}* {(c8)-(aS)}* , {(68)-(c8)}Kc8)-(aS)} ^ {bey ^ {cay __ I (6c) (ca) I ^ ^ = sin* ( 6c ca), .-. {cay {(68) - {cB)}' + {bey {{cS) - (a8)}* + 2 |(6c) (ca)| cos (6cca) {(68) - (c8)} {(c8) - (a8)} = (a6c)*. .-. (a8y (6c)* + (68)* (ca)* + (cS)* (a6)* -2|(co)(a6)|co8(coa6).(6S)(c8)-2|(a6)(6c)|cos(o66c).(cS)(a8) - 2 |(6c) (co)| cos (6cm) . (a8) (68) = (a6c)*. (iii) For two points and two lines, a, 6, 7, 8. We have sin* (78) (a^aa6)* = (6a^)* + (6a«)* - 2 (6a^) {bag) cos (78), where ay denotes the line through a parallel to 7. 2—2 20 GEOMETRY OF TWO, THREE AND FOUR ELEMENTS [I Now (bOy) = (boy) — (aay) = (baOy) = \(ba) \ sin ( baoy) = \(ba)\ sin (bay) = (bay), .: 8in» (78) (abf = {(67) - (ay)]' + {(bS) - (a«)p - 2 {(by) - (ay)} {(bS) - (ah)\ cos (yS). (iv) For three lines and a point, a, 0, y, d. We have no eliminant in this case. § 22. Examples. 1. Shew that if a, j3, y, 6 be four lines sin (j3y) sin (a6) + sin (ya) sin (|38) + sin (aj3) sin (y8) = 0. To prove this, we have (^8) + (Sa) + (a|3) = 2«-, .-. 08) = 2«- + (aa)-(a/3), (yS) + (8a) + (ny) = 2n-, (yd) = Stt + (a8) + (ya), . • . sin (j38) = sin (a8) cos (a/3) - cos (a8) sin (a|3), sin (y8) = sin (a8) cos (ya) + C08 (a8) sin (ya), and hence we have the above formula. 2. Shew also that sin (/3y) sin (86) sin (y8)+sin (yo) sin (y8) sin (08) + sin (a/3) sin (ad) sin (08) +8in (/3y) sin (ya) sin (a/3) = 0. This may be proved in a similar manner. 3. Shew that (^fa)»=/l-2*cos(^)+*2| = {(ija)— jt(|a)} {l + Acosdv)}, * small, = (•?«)+* {(fa) cos (^i;) - (fa)}. Secondly, suppose (^) ts negative : (va)-k(ia) (fy?) (ji^i^^) _ Oy°) (yyV) (/3'a^) (yVg") - (/yyV) (y'y°) Qq'jy) (ygg) sin Oy) sin (ya) sin (o^) sin {8!-/) sin (y'a') sin {alff) = nsinff)n£Vy') t^-'^'^ (<^^-(->>') ('•'^)] =47JA' sin (oa') sin (/3j30 sin (yy') (oo'jS/S'yy'). 9. Shew that (aX)(m5c) + (6X) (mca)+(cX) (OTa5)=(mX) (o6c), (ma) (X/3y) + (m/3)(Xya) + (my)(X we take a line through this point parallel to the given direction. 26 MEASURES CONTAINING VECTORIAL ELEMENTS [H To find the reduction of the measure {a^...4Ji). Let the point a^...# be c; we need (cj)). (c„6) = (6c„) = (6c„) - (cc„)_ = (6c c„) = \(bc)\ sin (6cc„) = |(6c)|sin(6ca)) = (6c«a) = (few) — (co)) = (6a)) -(a^ ...♦&>) = (6(u) — {(aa) — 1p sin (pa>)} = {bato) + 2^ sin (pa>), -■• («(!*... *o6) = (6aa)) + Sj9 sin (/>«»). § 27. To reduce the measure (a^...*«/8). Evidently (a^ ... ♦„ /8) = {tofi). § 28. Examfiet. 1. Shew that (xj3y) = - (^,8) sin (ay) + ( J?y) sin (a/3). We have (i'jSy) = {xxa) sin (/3y) + (a;/3) sin (y.r.) + (jcy ) sin {xa^). 2. Prove that the jperpendiculars of a triangle intersect. Let a y be the points and sides of the triangle. Denote the perpendiculars by Xfiv. (^iu>)={aa^liv) = (flii) cos (av) — (ai<)co8 (aji) 2 = - {abe,) sin (). 9. Reduce ("*i*2...*»*''i»'2">»''»i''8 — 'ii)' +X» (6(4,»'2...<«n_ ,<'''i'2 - 'n-l^") +/««K*j. ..*,_!«*, *j,..*,_,/V) + >'»(<»*i*2...\-l^'»2.->«-l''>') +/l„y« sin (/In »») + WnXn sin {v„K)+k,f^ sin (X„,*„) = («*,*8-*»-i*V2-->-lS'2-»--l) + i. (6cX,) + (in (ca^i,) + if, (a5i. J +jin K sin (m»0+ ^«^» sin (i',X,)+X„pi« sin (X«f»,) +X,S /l,sin(X,/i,)+X, 2 ^,sin(i.,X„) 1 1 n— 1 »— 1 A +fi^ S (/, sin (/%!',)+/*, S Xr8in(X,/i,) 1 1 n— 1 A «— 1 +»„ 2 X,sin(i',X,)+i>, 2 ;ir sin (;*,»„), 1 1 •■• («*i*2-*«K''2">«'"l'2-'n) =(a6c)+ S 2 (6cX,) k.ii.,vr=l + 2 2 2 8inO»,i.,)Ar».. 28 MEASURES CONTAINING VECTORIAL ELEMENTS [ll 10. Shew that (o^y)=(o^/3y)+psm(/3y)8m(po-). We have (op6c)-f(a6c) p2-^2 (a5c) ^ |(po)| cos (^p). a,b, c 2 (o^a)2 (jsftc) = 2 {{oaf - 2p \ {oa) \ cos {dap) + f^} {pbc) = 2 {oa)'^ {pbo)+^ {ahe) - 2^ {{op„) - (ap„)} (;)6c) = 2 {oaf {pbc) + ^ {ahc) - 2^ {op„) {abc) + 2^ (pp,) {abc) = 2 {oaf {pbc) + ^ {abc) + 2^ {abc) \{po)\ cos (pop). 19. Hence shew that 2 {oaf{pbc)^{R^+{pof-{pif}{abc), OtbtC where « is the circum-centre of a, b, c and R the circum-radius. 2 {oa)^{pbc)=2{Su,paf{pbc), where a>=so, p=\{to)\ =2 (»»)' (jb6c) -^p* {abc) + 2p (a6c) | (p») | cos (pia) = {^-fp*-h2p |(p»)| co8(JD«io)}(a6(!) = {R^+{pof-{pi)''}{aic). CHAPTEK III REDUCTION OF MEASURES CONTAINING EQUATION AL ELEMENTS § 29. It is evident that an equation f{{xa,), (xa,)... \{xa,)\, \(xa,)\ ...) = 0, where ai, o^ ... a,, a, ... are fixed, is a locus of a;. We shall consider the following locus a linear function of (a;2,), (aJOs) . . . namely, So, {wor) + a = 0. Let y, zhe two points on the locus, then Sor (yttf) + a = 0, So, {zOr) + a = 0. Hence by subtracting 2a, (yzttr) = 0, i.e. Sa, sin (yzur) = 0, which equation determines the direction of yz. Hence the locus must be a straight line. Conversely, it may be shewn that any line can be expressed in the form of a linear equation. For let ^ be the line, and let a, /3 be any two lines, concurrent with f. Let X be any point on ^, then (xfi) sin (fo) + (aro) sin (/3f ) = 0, and by taking any two lines y, B concurrent with a, we have (xa) sin (yS) + (xy) sin (8a) + (xS) sin (07) = 0. Hence fi, y, S being any three arbitrary lines the equation of f may be expressed as a linear function of (a?/3), (a^y), (xS). 29-31] FORMULAE OF KEDUCTION 31 It is important to notice that the locus given by a linear equation is according to our stipulation two lines ; namely, a line and the line with same position but reversed direction. The signs of the square roots occurring in the following are therefore necessarily indeterminate. § 30. To reduce the measure (Sa, {scBr) + a = 0j8). If y, z be two points on f = 2ar(*«r) + <* = we have seen that Sa, sin (j^5a,) = 0, i.e. So, sin (fa,) =0. .-. 2a,sin{(?^)-(o,/3)}=0. .-. sin (f/8) Sa, coa (a,)8) = cos (f/8) Sorsin (or/S), . to^/'fcm Sa,sin(a,/3) rr • /f=o\ 2arSin(a,^) Hence sin (f/3) = ,^ „^. ==^ , V2o,« + 210,0, cos («,.«,) fta\ Sa,cos(a,/3) COS (f/9) = VSo,' + 22ara« cos (Or^,) Let us give f a certain sense. With this sense we have . ,to\ Sa, sin (a,;9) sin(g;8)= (i), w|v2a," + 22.a,a, cos(a,ag)| where m is either + 1 or — 1. Tu /fcfl\ Sa ,cos(a,;3) ,.., Then cos (f /3) = , . ^ (u), m I V ia,' + 2Sa,a, cos (a,a,)| since the sign of the tangent is independent of the sense of f . Suppose 7 any other line. Then sin (^) = sin {(f/3) + i0y)} = sin (f /S) cos (^87) + cos (I7) sin (,97) Sa, sin (0,7) m I VSa/ + 22re,a8 cos (0,0,) | substituting from (i) and (ii). Hence the sign of the square root depends only on the particular sense of ^ chosen. § 31. To reduce the measure (2a, (««,) + a = 6). Let y be any point on the line, and let a denote its direction. 32 MEASURES CONTAINING EQUATIONAL ELEMENTS [ill Then ( lur {xur) + a = 06) = (y„6) = \(yb)\ sin{a>yb) •Jza/ + zzarUf cos (Oraj) lUr (bt/Or) VSOr' + 220,0, COS (ardg) _ S Or {bOr) — So, (yOr) VSa," + 2^0,0, COS (a,o,) SOr (6«r) + a •. (Sor (war) + a = Ob)- VSOr^H- 2X0,0, COS (a,(»,) Supposing the line 2ar(a;or) + = to have a specified sense, it is important to notice that the square roots occurring in this and the former section have the same sign. § 32. Next we shall shew that where o,, Oj ..., 0i, /Sa... are fixed, is the equation of a point. A given line may be represented by c„,r.^ where c is ao arbitrary point. Let this satisfy the equation, then lAr (C„,,.. «Or) + 25, COS (c^^r-.^-^r) = 0, .-. 2Ar {(arC(f)) + r sin (»^)] + XBr cos (^/8,) = 0, .•. r sin (««/>) tAr = - 2.4, (a^<^) — 25, cos (^^8,). We may change ^ to (^^ and we have r cos (o)^) 2.4, = - 2.4, (a,c^») - 25, sin (<^/8,). Squaring and adding r=(22l,)» = 2^,»(a,c)' + 2 2 ^,4,|(o,c)(a,c)|cos(a;co^) + 22il,5, |(o,c)| sin (^/3,) + 25,' + 225,5, cos ()8,/3,). Since the right-hand side is independent of w, ff> all the lines must pass through the same point. In other words, the equation is that of a point. 31-34!] FOEMULAE OF REDUCTION 33 Conversely, any point can be represeDted by an equation of the form considered. For let o be the point, take two points a, h collinear with o. Let ^ be any line incident in o. Then (af ) (60) = (6?) (ao). Then taking any two lines /3, 7 we have from b, f, /8, 7 m sin (/37) - (6|8) sin (fy) + (67) sin (?/3) = (^^y). Hence (a?) sin (^7) (bo) = (ao) [- (6/8) cos (^^ + (M cos (?^,) + (?/37)]. a, /3, 7 are any three elements, and the equation is of the form considered, proving the theorem. § 33, To reduce the measure (lA, (fa,) + IBr cos (f/Q,) = Oc)^ We have seen that (S^, (fa,) + 25, cos (f/5,) = Oc)» (2^,)» = %Ar''(a,cy + 2 2 .4,.4, |(a,c) (o,c)| cos (a,*a,c) + 22^,5, ia/:^.) + 2fi,» + 225,B, cos (;8,)3.), which is the required reduction. § 34. To reduce (2^, (fa,) + 25, cos (f /S,) = 7). Let c, d be two points on 7, and let a be the point 24, (fa,) + 25, cos (f)8,) = 0. Then ac, ad will satisfy the equation. .•. 2il,(ac«,) + 25,cos(oc/8,) = 0, .-. 2.4, (acur) + 25, (ac/8,„) = 0. Similarly 24,(ada,) + 25, (ad/3„) = 0. .-. subtracting 2.4, |(a,a)| (cda:^) — 25, (cd^^^) = 0, .-. 24, |(a,a)| sin (7^) - 25, cos (7/8,) = 0, i.e. 24,(aa,7)-25,cos(7/S,) = 0, .-. (07) 24, = 24, (0,7) + 25, cos (7/8,). Hence (24, (fa,) + 25, cos (f /8,) = 7) 24, = 24, (7a,) + 25, cos (7/8,). T. 6. 3 34 MEASURES CONTAINING EQUATIONAL ELEMENTS [ill §35. In the case in which 24, = the foregoing results break down. We shall consider this case. From the equation 24, (?«,) + 25, cos (f)3,) = 0, subtract 24, (fc), where c is an arbitrary point. We get 24, |(o,c)| sin (a;c?) + 25, cos (fi^) = 0. Let 7 be any line, and let (I7) = 6. Then 24, |(o,c) j sin {{a,cy) -0\ + 25, cos {(/3,7) - ^} = 0. Hence 24, {(0,07) cos - sin ^ (0,07,)} + 25, {cos (/3,7) cos d + sin (^8,7) sin 6} = 0. Hence {24, (0,7) + 25, cos (^3,7)} cos = {24, («,7,) - 25, sin 0,7)} sin 0, giving independent of the particular line chosen. Hence the equation represents a direction. §36. To reduce (24,(|a,)+25,cos(f/8,)=07>, when 24,= 0. We have from § 35 tan (24, (fo,) + 25, cos (f 5,) = 07) 24,(a,7)+25,cos(/3,7) 24, (a,7,)- 25, sin 0,7)- g 37. Examplet. 1. Shew that SSar&,siii(a,^,) tan(Sa,(^a,)+a=0 s6,(^^,)+6=0) = J|^-^^^^^^^-^ . r 8 Now am(lar{xar)+a=Off)=-r=M^^^^=^, vSa,*+ 2Sa,o, cos (ora,) .-. sin,(2a,(a.-a,)+a=0 26,(.r/S,) + 6=0) _ Sa, sin (2&, (jrj3,) + 6 = n,) 1 , there are ^^ ^ relations between the measures two by two of the n elements. See § 21. Our task is then to prove the vanishing of the function of the "Co measures and m and p quantities by means of these and only these relations between the measures. DIFFEKENTIAL GEOMETRY CHAPTER IV DIFFERENTIATION OF MEASURES OF SIMPLE ELEMENTS § 39. Let X be any point, and ai a consecutive point. Then Kaac')! we denote by &e, xx' by tx. Again let f be any line, and |' a consecutive position, (f |') we denote by 9f , f ^' by jof . § 40. From the preceding it may be shewn that . sind e = k, & constant. The precise value of k is still at our disposal. We shall suppose then that L ?^=1 (XIX). With this stipulation it may be shewn that the sine and cosine functions may now be expressed as the usual infinite series'. § 41. To differentiate \(xa)\. The point a is supposed fixed. We define the differential coefficient or derivative of \{xa)\ as \(xa)\-\(xa)\ , . , dl(a!a)| ' We represent this by — '^V^ • d\(xa)\ _ \(x'a)\-\(a!a) \ Then -d^-J:^ \ixx')\ _ _ ^ \(x'a)\ + \(xa)\ cos (xdax') _ , —\(xx')\coa(ax'x'x) 40 DIFFERENTIATION OF MEASURES OF SIMPLE ELEMENTS [iV by formula on p. 15, = L — cos (x' a axe' ) = — COS (rxxd). Hence —^ — ' = — cos (yxxa). ax The line a;^*, we shall call the normal line at x, and represent It by vx, so that {rxvx) = -^. TT d K'j'ffl)! / \ • / \ Hence —^ — - = — cos{xaTx) = — smyxavx) _ (xavx) _ {vxa) \{xa)\ ~\(xa)\' § 42. To differentiate (xa). {a/xa) We have ^^= L (^''')-(^") ,h,.\{xx')\_ = — L sin {xx'a) = — sin (rara). § 43. To differentiate (|a). The line p^^ we shall call the normal line of f, and represent it by rf . We have from definition dm.- T (r«)-(g«> -(pgar)-(fa) =.i' (IT) "fif (If) _ X - (fg) sin (,>gr) + {v^a) sin (gf) - (?«) by formula on p. 18. . d(ga) . t , 41-47] EXAMPLES 41 § 44. To differentiate (fa). We have ^J^^l.^^ • d? - '■ § 45. Let x,y,e ...,^, Tf, ^ ... he B. number of points and lines. Let /(x, y, z ..., ^,ij, ^ ...) denote a measure depending on the same points and lines. Then the differential oi f{a>, y,z..., f, i), f...) denoted by df{x, y, z ..., ^,T),^ ...)\a defined as the expression f{x\ y, z'..., r, v', r ■••) -/(^. y. z .... ?, f), ? ...). where x' y', z ..., f, if, f ... are near x, y, z ..,, f, 17, %..., small quautities of 9a;, 9f ... being only retained. Then f{a!,y\z!..., f, V. ?'•••)-/(«'. y. ^ -. 1. 'J. ? -) =f{x', y'. z' ..., r, V, ?'...)-/(«', y'. ^' ... r. v, r •••) +/(aj, y', / .... r, V, r -o-zc^'. y, z'..., r, v, r •••)+ - +/(^. y, / .... r, V. r ••■) -/(a^. 2^. ^ - f. v, f •••) + ■- Hence df{x, y, z .... |, j;, f ...) ^ a/(a;. y, ^.•■, f. ^?, ?•••) ^^ ^ df(x,y,z...,l7},^...) dx dy ^ df{x,y,z...,l7,,^...) § 46. We have then the following differentiations : d I {xy) I = — cos (rxxy) dx — cos (ryyx) dy -i(ary)r + |(a^)|'^2'' d (xTi) = — sin (txtj) dx + (jcwj) di], di^) = dv-d^. § 47. Examples. 1. Shew that d{jxyz)={yzTx)dx-r{zxTy)dy+{3yTz)dz. For ^>=|(y^)|^-^)=-|(y^)|8in(ra,-yi)=(y^r^). 2. Shewthat ^(^^0= -8in(Ta:f)rfir+sin(T3rf)rf5>+(a:yi'0<'f- a(^.yC). a{(.rf)-(.yf)} 42 DIFFERENTIATION OF MEASURES OF SIMPLE ELEMENTS [iV 3. Shew that (Jiji) -^^— standard measure of the feet of the perpen- diculars from z on ^, i/f, 1/ respectively. We have \ d {iv^y = id {(^)='+ W - 2 ^z) M cos (I,)} = (|2) (v^) - ("f^) (.vz) cos (I,) - (|«) (t,z) sin (f ,) = measure of the feet of the perpendiculars fro m on g, kI, I) respectively. 4. Shew that rf(fi;f) = (i'f>?f)rf|+(S>")f)) + «')''f)i;X) rfij+sin («X) sin (fij) )) W'^fi+Cp'jf) (l«) rf>; + (fF«) 8in(g,)rfz. 2. Shew that (^0* (d^f )*=M)(yrJ7) Ai; + (3:f) (awy)dy}«+(ary)«(^pf)«df« + 2 {(yO (yr^a;) rf.r + (arf) (arry) dy} (agrpf ) (ay vf) df . 49-51] EXAMPLES 45 Let xy(l = a, then {xya)=0, (fa)=0, .-. {yaTx)dx+{axTy)dy-'r{xyTa)da=0, {avO rff - sin (xaf) da = 0. Now since (raf ) + (faji) + (^ra) = 2w, .-. sin^ (raf) + sin2 {rax^) - 2 sin (raf) sin (ra^) cos {xiiC)=am^ {x^i). ■ • ■ (^^0"= (^)' sin* (raf ) + (a^yTo)« + 2 {xyra) \(xy)\ sin (rof ) cos (^f ), .-. (a;yO''(*»)''=(^y)''(a>'0'W''+{(yaTa:)(jte+((urT3^)rfy}2 - 2 {(j^ora!) f7)rff + ('»2) (|2) (f»!f) dr, CHAPTER VI DIFFERENTIATION OF VECTORIAL ELEMENTS § 52. In this Chapter we shall find the differentials of the vectorial elements, a;^...,i, x^...4„. Having found these we may find the differentials of any measures or determinates containing vectorial elements. §53. To find the value of dx^..,^. Take any fixed line \, then («p*...c6X) = (a;X) — 2p sin {p\), .'. differentiating sin {Ta;^,„^\) dx^,,,^ = sin {jxX) dx + 1dp sin (p\) — S ^ cos (pX) dp ; changing \ to \, we have sin (rxfi ... .jX,) dx^_^ = sin (tjbX^) dx + 'Zdp sin (pX,) - 2^ cos (pX,) dp, i.e. cos (ra;^... J X) . 52-55] EXAMPLES 47 § 55. Examples. 1. Find the value of d \ {pc^...j,y') \ . (^(t... ,iyf = {xyf - 2 1 (.ry) I 2^ cos {xyp) + S^ + 22^o- cos (per), .- . I (^p*...^y) 1 d i (ar^...ii2^) | = | (a^r) | d | (a^^) I - d | (a^^) 1 2 ^ cos (^ p) - I(*y)| 2 {ij) = [sin (rxui) dx + {{zxva) + 'Sp cos (pa)} da + 2 rfp sin (pa) -Spdp cos (po>)]* + (vijz)* dr/^ — 2 (i*!;?) rfij [sin (rxa) dx + {(zxva) + 2 p cos (pa)} da + tdp sin (pa) - Spdp cos (pa)] cos (iju) = [sin (txw) (ij:+{(a;(i*...«B,i;a;i/ii>) +2p cos (pa>)} c{u+2c{^ sin (pa) - 2 prfp cos (piB)P+ (i„,z)2 rfiji! - 2 (vijz) rfi; cos (ijo)) [sin (ra:X) dij + (a;o,iji/X) rfX = 0, and sin (raX) rfz — (zi>X) dX = 0, multiplying second by sin (cm)) and adding sin (i;X) ain {jxa) das + (^Xarm^) dm + (jTmi/ijX) rfij +sin (tzX) sin (^) dia + (aJuTjaa.'ui'i;) di\ — (xainrz) ain (<»ij) d2+{Xai]zy dxi^z=0. .'■ (Xmri!>ydXai)Z = —ain (ra;o)) (v^) sin (ai;) rf.r + (tjz) (iix) da + {pJiXa) (zxa) dr) + (.Tuijtz) rfa. T. G. CHAPTEE VII DIFFERENTIATION OF EQUATIONAL ELEMENTS § 56. As ne discussed the differentials of vectorial elements, so we shall in a similar manner discuss the differentials of equa- tional elements. § 57. To find the value of d {2a, (xa^) + a = 0}. We have from formula of § 30 sin (f/3) 2o,. cos (o^/S) = cos (fyS) Sa, sin (o,/S), where f stands for tarixor) + a = 0. Differentiating, with y8 fixed, — df cos (fyS) lar cos (Or/3) + sin (f/3) (Zdar cos (Or/S) + Sa, sin (a,/8) rfor} = df sin (fyS) Sa^ sin (OriS) + cos (f/8) {2 da, sin (o,/3) — 2 a, cos (of,jS) dot,} .'. d^ {sin (f/8) 2ar sin (oryS) + cos (f/S) 2 a, cos (a,/8)} = 2 da, {sin (f )8) cos (a,/8) — cos (f /8) sin (a,/3)} + 2a,do, {sin (f/3) sin («,/8) + cos (f/3) cos (a,/3)] df 2a, cos (!«,) = 2da, sin (fa,) + 2a,dar cos (fa,), 2 da, 2 a, sin (a,a,) + 2 a,da, 2 a, cos (a,a,) , j^ r a r g 2 a, 2 a, cos («,«,) r * 2 (a,da, — a,da^ sin (o,a,) + 2 a,.'' da, + 2a,a, cos (a, a,) (da, + da,) _ r+£ 2 a," + 2 2 arttg cos (a,o,) .-. d {2 a, (pcttr) + a = 0} 2 (a,da4— a,da,) sin (a,a,) + 2a,^da, + 2 ara, cos (a,a,)(da, + da,) r+£ 2 a,* + 2 2 a, a, cos (a,a,) 56-58] FORMULA OF REDUCTION 51 § 58. To find the value of d {%A , (|o,) + 25, cos {^^r) = 0}. We have from formula of | 34 Dififerentiating, with 7 constant, we have writing a for the equational point 2il,.sin(Ta7)da r = 2 |(aa,)| sin (00^7) dA^ + 2^, sin (Tar7) da, r r - 2djB, cos (7/8,) + 2B, sin (7/3,) d/S,. r r Changing 7 to 7^, XAr . cos (Tti7) da~%\ (aor) \ cos (aai-y) dAr H- S -4^ cos {ra^y) da,. - 2d5, sin (7/8^) - 25^ cos (7^8,) d^,. r r Squaring and adding, {ddf-ilA^)^ r = 2 {aarf (dil,)» + 2^,» {da^f + 2 (d^^)" + 2fi,» (d/S,)'' r r r r + 2 2 |(aa,) (oa,) I dJ^dil* cos (oo^oa^) + 22 .4^ j1« cos (Ta,Ta,) da^da^ + 22 d5,d5, cos (/3, A) + 22 JB,5,d/3,d/8, cos (;8,/8,) + 2 2 dArA,da,(aarvat) - 2 2 di4,d£, (aa,/8,) r, » -2 2 dArB,d^,{aarvP,) r,» -2 2 J.,da,dfi, sin (ra^/S,) - 22 ^,da,5,d^, cos (Ta,y8,) -2 2 dfi,5,d/3,8in(/S,/S.). ,,* 4—2 52 DIFFERENTIATION OF EQUATIONAL ELEMENTS [VII Hence (day.(lArY r = 'S.{dA;f{l.Ak''(arahy + 2 S A^Ail (orah) (arO*)! cos (ara^arat) r h M=ft + 2 2 AHBt(anar^t) + ^B^'' + 2 S 5*^4 cos (/Sft/Sfc)} A, t h A+* + 2 dArdA,{-(ara,y(lAky + l,An'{araky r+» A A + 2 -4 A ^ i |(ar aft) (a, at) I COS (a^aA a,«i) A4* + 2 2 ^ABi(aAa,/3t) + 2 25A»+4 2 BtBtCOs{0„^t) A, A; A A+i: + 2 Aft" (0,01)"+ 2 AHAt\{a,ah)ia,at)\coa(aMa^ A A^:!- + 2 2 AHBt(aHa,fit)] + 2 2 j4a . 2 d4, J,da, {2 A^ (ahOyvas) -tB^ sin (/Sfcva.)} - 2 2 ^ft . 2 d^,dB. {2 A^ (anaS.) + I.Bh cos 0^/8.)} - 2 2 ^A . S dArB.dfi, [IAh (,aHa,vp.) - 2 B* sin (/Sa/S,)} k TfS h h + CZA^y[l(dBry + tBr'{d^ry + 2 2 ArA,cos(rarTa,)darda, A r r r4=» + 2 2 dBrdB, COS (0r^,) + 2 2 B,£,d/3,d/8,cos(/3r/3,) - 22 ^rdard£,8in(Tar/8,)— 2 ArdarB,d^r(ioa(rar$g) -2ldBrB,d$,sin(0rMl § 59. Examples. 1. Reduce rf2a,(a;a,) + a=0 /3. Let 2o,(jPa,) + a=0 /3 = 2. Then (2ar(arar)+a=0 z)=0, Oir)=0. .-. 2a, (za,)+a=0, O)=0, -• . - sin («/3) rf« + (zi//3) d/3 = 0. .-. 2a,sin{(TZ/3)-(a^)}dz=2rfa,(«a,)+2o,(zi«ar)<£(ir+ofot. . • . sin (tzj3) tfe 2 a, cos (0,^) - cos («/3) cfo 2 a, sin (o^/S) = 2 da, {zor) +2 0, (zvo,) da, + da. . • . cos (Tzff) dz 2 o, sin (or/S) = (2I//3) d/3 2 a, cos (o^S) - 2 da, (20,) - 2 a, (zko,) da, - da, .-. (dz)2 {2 a, sin (a,/3)}«={(zv/3) d0}2 {2a, sin (a,/3)}» + {(21-/3) d|3 2 a, cos (a,/3) - 2 da, (zo,) - 2 a, (zwi,) dor - daY = {zv^fdffi {2 a,2+2 2 a,a, cos (a,a,)} + {2 do, (zo,) + 2 a, (zi/Q,) dar + da}* - 2 (zKjS) d/3 2 a, cos (a,/3) {2 do, (za,) + 2 a, {zvor) do, + da} , substituting for z= io^O^oiy+a^O^ we get the value of {dzf. 58, 59] EXAMPLES 53 Reduce d{xa)-k=0 /3. "We have, putting {xa) -k—0 j3 = z, (za)=k, («/3)=0, . • . - sin (na) dz + {zm) da = dk, - sin («^) dz-\-{zvP) rf/3=0. Now sin* (rza) + sin' (t2/3) - 2 sin («a) sin (jz^) cos (o^) =sin2 (a/3), . ■ . sin' (oS) (d«)2 = {(zKn) da - rf*} * + (z..j3)2 d^ — 2 (ai-zS) d0 {{zm) da - dk) cos (o/S) = (««.)« rfa* + {zv^Y d^-2 (zva) {zv?) da dfi cos (a3) - 2 (zva) dadk+2 (zv/3) <;^<2it cos (a)3) +(e;t2. .-. sin*(«ij9)(rfz)'=((a?o)-*=0 /3«.)2da2 + ((a!a)-i=0 ^v^fd^ -2((ji;a)-i=0 /Sko) ((a;a) - * = /SvjS) darf|3cos (a/3) -2rfad*((a;o)-it=0 /3va) + 2d3d/t ((a;o)-*=0 /3k)3)co8 (a/3)+d*^ .-. 8in«( = -S?^4<^«^)- See Ex. 6, p. 42. Formula for the evaluation of (y^a). We have seen ("f") = f^. • Formula for the evaluation of(v^a). We have ("?«) = (?*) — 5 • § 63. Examples. 1. Reduce (rgija). We have from § 51, Ex. 1, (|i,a)S dfia = (pti,) (,a) rf| + (^,1) (|a) di,, (.ptv) (■?«) dj + (y;!) (^") dy 2. Reduce sin (r|i) a). sm(rtna)=— \-- ^ L rf|ij Jaconst. ^ (p|i)) sin (i)a) rfg + (pi)g) sin (gg) di; I V(p|»0''Sj2+(p,|)!rf.;2 + 2 (p|^)Tp^osl|^)(2p^ ' from Ex. 6, p. 42. 3. Reduce (vxya). , — > d{xya) |(ya)|cos(yaarj) (rii:y)oij;+i(j;a)| cos(aa!a;i^) {Tyx)dy {jxy)dx+{Tyx)dy ' from Ex. 5, p. 42 56 MEASURES CONTAINING FUNCTIONAL ELEMENTS [VIII 4. Shew that _ {ayf (yrx)^d3^ + {axf (j-iy)' (fy'+2 1 (ax) (ay)| cos (axa^) (yrx) {.vnf) dxdy ~ {(yrr) dx + {xry) dyf 5. Reduce (p^a). _ . ,_ , — I Qyo) 1 cos (xyya) (ray) dx + 1 {xa) \ cos (yxxa) (r yA) rfy (aya) . (Ta:y)ote+(Tjra;)rfy a = xya {xyayxy„){TXy) dx+{xyaxxi/,) {ryx) dy = -"'"(^"^ • ^ (r^)?f) dl-sin (fij) (gijrt) dz 7. Reduce (Ta;^...i4o). ('".Tpif ...ud) ) _ . . .8m(TXm)cb;+{cucva)cla+ldpam(pm)-SpdpcoB(pa)+da>^pcoa{p) dx + {aXpt ... ^iu.v v) S dp sin (po>) - sin (aa) 'Spdp cos (pa) + sin (aa>) daSp cos (p(i>)]/(&) =8in (raJia) sin fa») -j- + sin (oia) S jH sin (pa) — sin (no)) 2 3 j - cos (pa) CMu £hu da +(ax) — (x^ ...^x) coa(am)+axa (am) 2 ^ cos (pu) Q (aa) S j^ sin (pa) - + (a;a) + 2^ sin (ap). = sin (tj;o>) sin (aa) -r- + sin (aa) 2 j^ sin (pa) — sin (aa) ^p-fi cos (pa) 11. Shew that (/»2a,(a?ar)+a=0 ^) = [2ar'«^— So,a,{rfo,(>'arO,/9)+da,(a,i'ag3)} + 2 (ara,^)(arda,-a,clar)+a2daram(arff)—daSarSin(ar^)] r,8 r ' r -~[Sar^dar+ S ara,(dar + da,) cos(ara,) + S(arda,- a,dar)ain (Tfi») rf|ij = - rf (rfiB|ij) ^ ''8in(«,)^ .-. sin2 (iq) d^. sin (T|ijrfa.)= - [{(Klirrfo)) dl+di-i/rfa) rfij} sin (|ij) __+(^rfii)cos (ti) (dj-di)-\ ^ =8in (Tf<»^[(pf 7) rf| -sin (rf<»|) (/?7f ) rf,], .-. sin^ (I7) rf^ sin* ((a) rffa sin (rl^rfa)) = -8in8(f») [d (^r,) (pil)di-d(Cai) (pni) dr,] =(.Pin) d^ipCa) sin (mij) dC+(paO sin (f^) rfu} - (Pvl) «''> {(K*>) sin (»f ) rff + (Pfflf ) «n (f I) ''»>} • 58 MEASURES CONTAINING FUNCTIONAL ELEMENTS [VIII 14. Find the value of sin {tx^tija). sin {rx^ryt) dxf= - d {ry&x^ = +rf { — {TyijS)-\-p sin {pTy&)} = sin {rxryi) dx+dp sin {pTy&) — pdpcos{prryir), ■ . sin {TX^ryt) dx^dy& = dxd {ytrx) - dpd {yip) — pdpd (ytpi,) = dxd {{yrx) + a- sin (trrx)} — dpd {{yp) + o- sin (55Jai ... an-i<'nXai aj ...au_] a„ir) rf.j:ai ai ... a„_ia„ = {(6xOia, ... a„_ia„ir) — (paroi O] ... a„_, a„.rai Oj ... a„_iO„ir)} X d xai O] ... a,i-i a^ ,-. (a.a,a, ... a„_i a„)2 ^ (Soioi ... a„_i o„ 5) = (6j;a,ai ...o„_ia„»)(o,,a„-i)(a,i-l«B-l)---(''l<»l)("l''^) - (Sola, ... a„_,a„wa„) (a„a„_i)(iin-iaB-i) •■• (aiai)(aiT-ir) = (a„ a„ _ 1) (a„ _ , a„ _ 1) . . . (ni ai) («! tJe) I (6a„) I cos (6o„ 3SI a, . . . a„ _ 1 aj. 8. Reduce T- l(;roia,a2...ana» 6)1- (xajoi ...ana«)*rf|(iFOia,a2-"«n<'ii6)i = — (araioi ...a„a„)''cos(r^Oiaia2...o„a„ ajaioiota ... o„a„fe) X (2 XCI] ai 02 . . ■ 'i) (<«i«i) ("i'''^) f*^- 62 MEASURES CONTAINING FUNCTIONAL ELEMENTS [VIII 9. Reduce -=- {xa-i aj 02 . . . a„ a„ /3). d ^ {xaj^aio.^ ... a^OnY ^{xaiaiUi ... «nO«/3) = —{xa^ai ... a„a„)^ sin (rXOiaiOi... a„u„ j3) d.vaioi ... a„a„/ote =sin Oa„) (a^an) (a„a„_i) ... (ajoi) (aiUi) (o,tx). <; 10. Reduce 37 (|ai ai 02 . . . «« _ 1 a„ |8), (|ai ai 02 ... otn-i Oa)^ rf (^ai Ui a2 ... fl!„_i a„^) =sin(/3a„)(anan-i)(an-iai.-j) ••• (a2«i) (|+...+(_)»p'|+(_)«-l(„|(,). We have (•'"^)=p5 -(!«), differentiating (>''|o)=-(i/|a)+ p'|, and (i.»|a)= -(„i^a)+p"$ = -p«+p"l + (|a), Hence the general formula. Again («^fa)=-(i''fa)+p"'l = -p'i+p"'i+(„ia), giving the other general formula. 2. Reduce (pv^ia). ( pv^$a) = (Z)!/ . via) = (,pv$a) + COS (vga) pv^ = (^|a) + COS (|a) /)| + sin (f i) p'|. 3. Shew that pi'"|=p(»)|, (^"l«)=(l«.)-^. 4. Shew that Hence shew that (piz-la) -(p.."-'|a)=p(— 1)| cos |(« - 1) I - (|a)| . that (pp»|a) = (p|a)+p(»-»)|oos |(» -l)|_(|a;| +p(-==)|cosj(«-2)|-(|a)} +eto. etc. We have (pi;»|a)'=(pi'. i'""l|n)=pl»-')|cos (i'""*|a) + (^i/''->|a) = (?>,'»-l|a) +p(»-»| cos |(« - 1) I - (|a)l . 5—2 68 REDUCTION OF COMPLEX FUNCTIONAL ELEMENTS [X 5. Find the values of (»*•- 'aso), {v*^xa). (v«»i-a)=(i'*»-'.»xa) ~\pxdxj \px/ \pxdxj V/MT/ ~\pxdxj \px) \pxdx) \px) 6. Find the vahie of ( pv'^xa). (pv''xa) = {pv'^"^ vxa)={pvxa) + {-j—\ p»a; COS ] (n — 2) ^ - (x^a)^ +(si)""U)'»'{<"-"i+<'"')*-*- CHAPTER XI SUCCESSIVE DIFFERENTIATION OF MEASURES § 74. We propose to find the successive differentiations of the measures of two elements containing one variable element. § 75. (i) Siuxxssive differentiation of J {xdy. = 1 — (rxa) px, = — {pxf (vxa) - p'x {jxa), \ [^ ^'"'^" ^~k {("^X/^)' + {rxa)p'x\ = — {pxf (v'xa) — ^pxp'x {vxa) — {rxa) p'x = — (pxy + {{pxy — p"x} (rxa) — Zpxp'x {vxa), i ( j-)° ('»»)' = - 2/Mf/o'« + px {{pxy - p'x] {vxa) + {3 {pxYp'x- p"'x} {rxa) + 3pxp'x{l + px{Txa)] - 3 \{p'xy + pxp'x] {vxa) = — opxp'x + \{px)* - ipxp'x — 3 {pxy} {vxa) + {6 {pxy p'x - p"'x} {rxa). And generally, | ^^Y' {xay = -4„ {rxa) + B„ {vxa) + C„ , where A, B,C are polynomials of par, p'x, p"x ..., and C7»+i = Cn + -On- 70 SUCCESSIVE DIFFERENTIATION OF MEASURES [XI (ii) Sttcceasive differentiation o/|(a;o)|. d {vxa) _l—px (txo) (yxa)' f—Xu \i_ p'miTxa) (pa:)' {vxa) 3 {vxa) W '^*"^' \(^\ \{xa)\ \{xa)\* Spx {rxa) {vxa) 3 {vxa)' + K^^)p + \{xa)]' ' W !^*"^!-|(^a)I i(a;a)|»+^^''"q \{xa)r \{xa)\ \{xa)\'>l , .Zpxp'x . , , , 4/3'a; — (i/a;a) .\ \, + (raia) (i»a;a) .. "^ -.. ^ -^ (a;a) ^ "^^ (^") — {rxa ^|(^a)|' + ^''*''Ml(^a)P |(:ra)K - (TxayvxaY i^ - lii?^)* (iii) Successive differentiation of {xa). d{xa) dx = — sin {rxa), (; = p'a; cos (wa) + (pa;)" sin (Ta;a), ^ j (a;a) = /»"a; cos {rxa) + Sp'xpx sin (raw) - (|Ba;)» cos (raxt) = {p"x — {pxY} cos (raja) + Spxp'x sin (raw), ^) («w) = - ^ {(/»«')* - /»"«} cos (raw) - {(/ja;)'' - pa;p"a;} sin {rxa) + 3 T- {pxp'x) sin (raia) - 3 (pa;)' p'a; cos {rxa) = {6 (par)= p'x - p"'x] cos (raw) + sin (Ta;a) {4^a!p"a; + 3 (par^ - (pa;)'}. And generally, (^J* (aw) = il„ sin (rara) + £„ cos (raia), where ^, 5 are polynomials in px, p'x, p"x... and i 76, 76] EXAMPLES 71 (iv) Successive differentiation o/(^a). d(^a) . and generally, (0'(l«) = ("»?«). (A \^ J \2n— 1 ^j (?a) = P"^'» ? - p'^« f + . . . + (-)» p'r + (-)»-' (r?a). (v) Successive differentiation of{^a). Jt) (?««) = 0, »i> 1. Thus we see that we have general expressions for the nth differentiations of the measures of two elements in which the variable element is a line. In some cases this renders the line more useful as a variable than the point. It will be seen in the next chapter that both (^a) and F {(^a)} can also be integrated in known quantities. § 76. Example*. 1. Find (jy{xa^),n=\,2,3. d ,_ ., {rata) px {vxa) 2 (rxa) {vxa) (f)\xa^)=-fp^^=.-e^) + \dxj ^ dx {xaf (xaY /dy. ._ p'x(vxa) , px{l-px{Txa)} , 2px{vxt \dk)^'"''^>~ \xaf ^ J^^ +~(^ {xa)* yxa)^ 2px {vxaY 2 (rxa) {1 — p* (rxa)} 8 {rxa) {vxa)* (xa)* (xa)* (xaf (xa)* ^ ' \(xd^ (•«o) J (xa)* 4px (uxa)* _ 2px (rxa)* 8 (rxa) (vxa)* '*■ (xa)* (xa)* "•" ^c(^ • 72 SUCCESSIVE DIFFERENTIATION OF MEASURES [XI 2. Find ^^^^(1^3), „= 1,2,3. We have -=x ((afi) = - ■ ,,t\^ . af^* sin' (fa) -i....(.8).''°''^'-.":g.;'°"'^°' ^ sin' (|a) -«i„/„m sin' (!«)- 2 (p|a) cos (gg) (^y (1^/3) =sin (a/S) . {4 sin (f.) cos (fa) - 2 (pia) sin (fa)} sin (fa) {sin' (f «) - 2 (pfa) COS (f g)} . 3 cos (fa ) sin«(fa) =^^ f^ ^'"' (^') '^**^ (^°) - ^ (P*") ^^ +^ *=°^' (^"^J^- CHAPTER XII INTEGRATION OF MEASURES § 77. We define integratioa as the inverae process of differ- entiation. Thus, for example, ^ (?a) = ("?«)• We have with the usual symbol of integration, /("?«) df = (fa). If (7 be a constant /' § 78. Integration may as usual be defined as the limit of the sum of a series. For the integrand being an algebraic quantity we must have, if /(f) be a function of absolutes containing f, that f. where h = (oa,) = (a,ajj) = . . . = (o,^,a„) = ^-^ ; a, )8 are the limits between which f takes all values. § 79. In the foregoing it is generally necessary to suppose that f envelopes a curve, and so also when we consider the integration of measures of a variable point, we have in general to suppose that x traces a curve. We have as before and f/iic) dx= L {/{a) + f{a,) + ... +f{a„ or b)} k, 74 INTEGRATION OF MEASURES [XII where /i= |(aa,)i = \i(y) dy, where y = (aw). ^^^ j (^ "^ ^^'^^ dx = -j^{y) dy, where y = (m/S). (7) / (^a) cos (raja) da; = trapezium (mna) + seg a;. 79, 80] EXAMPLES OF INTEGRATION 75 This we prove by the summation process. Let m, n be two points on a curve. From points on the curve draw ordinates to a. Let a; be a point on the curve, x' a consecutive position. Let p, p' be the feet of the ordinates. Then it is easy to see that, since dx cos (raw) = \{pp')\ , J m (aw) cos (rara) dx = trapezium (mna) n + area between (mn) and the curve. Hence the indefinite integral is I (xa) cos (rxtt) dx = seg x. (8) From (6) we have I ^ — ^ sin (Sa/S) dx = ~ I sin ydy= cos y = cos (m/8), (rxa) I \(xa)f {(ayS) — («;8)} dx = cos (ira/S). If (a^) = 0, we may put ff — ai, and we have w Xi, rfa; = - (aft) ! cos ixa ah\ J l(a:a)l» (9) Again from (6) we have I 7 — s • / a\ dx=—\ (sin «)-' d« J (a;a)?' sin (aa/3) ./ ^ ^'^ "^ = -logtani2', = — logtan^(Sa/8). /i (rxa) dx \(xa)\{(a^)-(x^)} From which we have (Ta;a) da; 1 fl log tan ^ (aiaa6). \{xa)\ (xah) \(ah)\ (10) We have also from (6) I V / . „ , -. = — / cosec" « dy = cot w = cot (a>d8), J (xay sin^(^/8) j ^ ^ ^ "- '^''' L- 1 /" (rxa)dx , — . from which j ,(^^)_(^^)}. = cot (a;a^), and therefore (rxa) dx _ 1 "(«a6)^ ~(aA)i J (*. cot(««a6). 76 INTEGRATION OF MEASURES [XII (11) ({{Txa)-{rxb)}^(mb)dx = j(y)di/,yiheTey = (xa h For ^ji> (y) dy = "^^ ^fiy)dy= {(rxa) - (ra^)} (xab). § 81. Integrals of measures of elements containing a line variable : (1) I d? = - (|a), where a is a fixed line. (2) J4>i^a)d^=-j{y) dy, where y = (?«). (3) |(?a)d? = linp|-(i'?a). For dififerentiating R.H.S. we have (4) j (i/f a) (fa) df = J" <^ (y) dy, where y = (fa). where y = (f*/8). (6) In(5)put <^(y) = ^,.then f (p^a) sinMfa) ,«,__ 1 f ^= _i_ sin (fa) j sin* (fa) (fa/8)^ "^ sin(a^)J f sm{a^) (fa,8) ••j(fa/3)*"^ sin(a)8)(fa;3) (7) In (5) put (y) = y, then f (ff")(fa/3) rf>:^ 1 (f^/3)' J sin' (fa) ^ 2 sin (a/3)" (8) In (5) put ^(y) = ^, then /•_(£fa)d^ 1_ loB(fafl) j sin (fa) (fa/8) sin (a/3) '^^B U«/5^ (9) /• (Pg^)cos(fa) ^ _(p^^i. g ^ ' j sin' (fa) * sin (fa) 80-83] A GENERAL THEOREM 77 For differentiating r.h.s. we have .. p|sin(ga) (pgg) cos (g«) ^ (jjga) cos (fo) '^^ sin (fa) "^ 8in»(|a) sin''(?a) " (10) fi^ ^ ' J sin" (?«) df = - 2 seg pf - (p^y cot (?a). § 82. Integrals of point elements may be converted into integrals of line elements by the substitution x=p^, and integrals of line elements into integrals of point elements by the substitution f = tx. § 83. A function of measures of a variable point is differentiated along the normal and integrated along the tangent of a curve. If the integral be such that it is independent of the path of integration, the function satisfies Laplace's equation. For let the function of measures be reduced to a function of measures of the variable point x and two fixed lines a, ^ at right angles, thus F{{xa), {x^)]. Then the integral iis • i,3-p — r sin {vxa) + _ j-^ sin (yx^) ■ Since this is to be independent of the path of integration it must be equal to a function of (aro), (x^), say ^ {(aw), (a;/8)}, •■• s(ijr)^^°^''*"> + a(^)''''^''*^> sin (tx^) + aT-pT sin (Tj;a) -/I. ■dx. ■a(a«)^'"^'"'^^"a(a;/S)- = 5-7^ sin (raja) +57^ sin (rxfi). dF 8^ dF 8^ d{xfi) ^d{xay d{xa) d{x0y ■'• ^F d(xay^ dFF d(x^y 0. MISCELLANEOUS EXAMPLES. 1. A curve is given by the general equation f{\{xa)\, 1(^6)1 ... (xa), (a:/3)...}=0, to find the radius of curvature at the point x. Difierentiating once , 3/ d\{xa)\ ^ df djxa) Q_ 8 I {xa) I dx 3 (xa) dx df (yxa) _ 3/ . , X „ r\ 3|(a,Yi)||(ara)| (l{xa) Again, differentiating df i X _px {rxa) _ (vxa)\ '■d[{xa}\\\^\ ~\(M)r WFI 3'/ (vxaf df j_l px (rxa) (vxa)^ ^d\{xa)\^ (xa)^ + 2 -,, ,, sin* (rxa) + S ../ , cos (rXa) px=0, o {xay (Xa) .-. -ax{S -,' . cos(T^a) + 2 -|/ , ,77 — CTf 3y (v:ra)* 3/ f 1 _ (y^.^ .IL sin* Irxa) (ii) ^3|(^)|M^«)'' ^3Ka:a)ll|(a«)| (aw)s/+^3(a!a)2"° ^'^^°^-^^' Eliminating tx, vx from (i) and (ii) we get px. 2. Find the radius of curvature at a point of a curve given by bi-radial co-ordinates. Let a, & be the points of reference. Let :t; be a point of the curve. Let r=\{xa)\, s—\{xh)\. Let \{ab)\<=c. To find px in terms of r, s. dr _ (yxa) di _ (^vxh) dx r dx 8 rdr_{vxa) ■ ■ sdt'l^) W' 1 0^ _ 1 rfs ofo ^Pr_ _ Uv^a) _ {y^xV) \ ' ' r dx i dx drdsdx~ \(vxa) (vxb)) ^ _ f 1 - irxa) px _ 1 — {rxh) px\ ~ \ {vxa) {vxb) J _ (vxb)- (vxa) ((rxb) _ (rxa) ) (vxa) (vxb) \(vxb) (vxa)j ^ (vxb) — (vxa) (rxvxah) (vxa) (vxb) (vxa)(vxb)^ ' , ,. , IS , . (I dr 1 ds ds d^ ) , > , ,, .-. (xab)px=(vxb)-(vxa)-}^- _--_ + _ ^ („^) (.^6). MISCELLANEOUS EXAMPLES 79 From a, h, x, vx we have the eliminant {dbxf = {avxf {hxf + {hvxf {ax^ - {{axY+ (hxf - (flhf) iflvx) {hvx). From (i) we have , {yxa)=rdrldx, {vxVj-tdsjdx. Let {hxxS)=0. Then {dxf (,ahx)^=i^s^dr^+d^+2drd» cos 0), .■ . {dxf = -^^2 ^^^ + tfo" + ^rdi cos 6), 1 dr 1 d> a? dr d» (dr ds ds dt dVl '^~ s da- r da- t^^dada\rda sda drda-d^j' where 4A''=2r»<2 + 2c2(r«+»»)-r*-»«-c*, da^=dr^+d^ + 2drdscoae, ei—jS — gi C08fl = TT . 2r« 3. Rectify the curve, whose tangential equation is We have by integration SArPin pS - {"iar)] 25r/'(l/3r), where ?^?^=/(y), which gives the value of linpf . 4. Rectify the parabola ($g) sin (|8) = a. We have from this J (l«) rff =o J cosec (^8) d$, .-. lin^|-(i'f»)=-alogtani(|8), .-. lin pi = (|«) cot (f 8) - a log tan i (f 8). 5. Find the conic of closest contact of a curve. The equation of a conic touching the tangent tx at :r of a curve and also having the same curvature is px (i/vxy +2h(^vx) i^Tx) + 6 {yrxf = 2 (yrar). Differentiating three times and putting y=x, we have by means of the formulae on p. 70 2h{-Spx) = 2{-p'x},.:h=^. Hence the family of conies having foiu- point contact is px {yvxf + -^ {yvx) (j)Tx) + h {yrxf = 2 (t/rx). 80 MISCELLANEOUS EXAUPLES Again, differentiating four times and putting yx, px { - 8pa.'»} + 2A { - V-f} + * {«P^} = 2 (p.r3 - p"x). Hence 6=_p^.+_ P_ _ . £_. Hence the conic of closest contact is 6. Similarly, in tangentials, shew that the conic of closest contact at £ is - 9 (p£)* sin« (,f ) + dpp (,^£) {3p| cos (,f ) +p'| sin (,|)} + (9p^-3pfp"|+4p'^2) (,p«)2=0. 7. From the equation | (xs) | + 1 (x^) | = constant of a point of an ellipse, deduce directly that (f«) (£«')= constant of a line of an ellipse. From I (,r») I + 1 (xs') \ = constant we have (yxii) (vx£) \{x»)r \xx>^)\ "' from which ) — ( + \ — /, =0, (txi) (txs) .•. integrating log (^») + log (|«') = const. 8. Shew that the join of the intersection of the normals at the extremities of a focal chord of an ellipse with the middle point of the chord is parallel to the major axis. Let the ellipse be \(xs)\=Me(xS), » the focus, d the directrix. Instead of differentiating we may obtain the equation of the tangent at y in the form of an equation. For we have from Exs. 3, 4, p. 20, |(^»)| = |(y»)|-*|(ay)|co8(^y»)) ^ g^^^j, (.xy(i) = (y!t)+i{(y»)-(xi)] ) If the point xy{ is a point on the curve, then ^ will be the tangent at x if r^ = 't is ultimately zero. {xO ^ Employing this method I (y») I - i I (a^)| cos (iyy») = e (y«)+eifc{(ya) - (xi)}. Hence the equation of the tangent is i'^iiw) - « (^8)+e (yd)=0. This equation may also be obtained by differentiating |(y«)| = e(y8) and putting ry = xy. MISCELLANEOUS EXAMPLES 81 If 2 be a point on the normal zj is perpendicular to this. Hence the equation of the normal is cos (^^) -e cos (^a)=0, i.e. («y«)-e(sa)+e(ya)=0 wherea the major axis=«j „. Similarly if y be the other extremity of the curve and z now the intersec- tion of the two normals (zy's) -e(ia) + e (y'a)=0, .-. adding (ya) + (y'a)=2(za). 9. MacCvllagh's Theorem. If a chord pp' of a conic pass through a fixed point o, then tan ^.(^«o) tan ^ (/>'»ld)= constant. Let Ou be the chord pp', and sk be ip. Then o^^=p, so that |(o««a«)! = «(o<»«a8). Reducing (o„ s) = e {(o*a) sin (<»8) — (oS) sin ()cos(»o<»)-sin(l6<»)cos(o«X)}. Put l(>o)\ = i:{o8), .'. -=y sin ((>«X)-oos (S«X), where y depends only on u. Writing x=Un^{pssb), sin(o»X) = - ^, cos(o«X) = r-t_ Hence ii^ (e - it) - 2.i;^ + (e + ifc) = 0, an equation defining X in terms of a>. ' ' e-k The theorem is true when o traces a conic with the same focus .and dii-ee- trix as those of the given conic. 10. o and o' are two fixed points, x any point on the curve _1 1 1 \{xo)\ X^o')\~e' Prove that the distance between x and the consecutive curve obtained by changing c to e+hc is ultimately he 7 3e' a'c* ' where r=\{xo)\, r' = \{xo')\, a = \(oo')\. [Smith's Prize.] T. 6. 6 82 MISCELLANEOUS EXAMPLES Difl'ereutiatiug the equation 1 1 1 \{xo)\ |(W)! c along the normal we have 1 1 , — V J ofc —J cos (vxxo) dp - ^2 cos (vxxo )dp = ^, where dp is an element along the normal. And we have also diflferentiating along the tangent These two equations give the required value for dp. To eliminate to;, vx put {txxo)=6, {rxxo') = ff, and we have the following equations sin 6 siu ff _1 dc "? ^ ~'^dp' cos6 oosd'_- "72 -^ -"■ also 6-ff = , where = {xo'xd) EUminating 0, 6' we have ■Ji* + r'« - 2r'r''' cos _ 1 rfc 7J!/2 ~ c^dp' _dc rV2 ''~"c2Vj-< + r'4-jy(r2+/>'-a2) ■-. , by means ot »• — r = 3c2 a^c* c v/ 11. In a system of curves defined by an equation containing a variable parameter investigate at any point the normal distance between two curves. [Cayley.J Take the general equation f{\{xa)\, 1(^6)1, ... {xa\ (x») ...} = c, where c is a variable parameter. Differentiating along the normal dp is ,, .. sin (rxxa) - S - . . cos Orxa) \=dc, and along the tangent 2 -, I , . , cos (txxo) + 2 - / ■ sin (rxa) = 0, 3|(xa)| ^ ' d{xa) ^ which two equations enable us to eliminate tx. 12. From the theorem that the circumcircle of a triangle circumscribing a parabola passes through the focus shew by differentiation that if an isosceles triangle circumscribe a parabola, the join of the vertex with the point of contact of the base is incident in the focus. MISCELLANEOUS EXAMPLES 83 Let a, /3, y be the sides of any triangle circumscribing a parabola of which the focus is *. Then we have, since the circumcircle passes through «, («j3) (ly) sin (/3y) + (»y) (»a) sin (ya) + («a) («|3) sin (a^) = 0. Differentiate with regard to one of the lines, say y, keeping the other elements fixed. We have W {(«t) sin (/Sy) + (»y) cos (/3y)} + (»a) {(JI17) sin (ya) - (»y) cos (ya)} = 0. . .(A). The condition that s, the vertex a/3, and the point of contact py oi y should be coUinear is (a/3» y5y)=0, .-. (a^y)(«»y)-(a3.T-)(.T,)=0, .- . (»i7) {(»a) sin Oy) + (»/3) sin (ya) + (»y ) sin (o^)} - (»y) {(*a) sin (j3i7) +(»/3) sin (17a) + {svy) sin (a/3)} =0. Hence the condition is {(»a) sin Oy) + (Xw, ^a). Prom Ex. 1, § 55, (va^u.ca) = {(i/a:a)+c cos (raroi)} dx+c {xaa) da, when c is constant. -•. (yXa.eX lJ=c[cos(TXa)dx+dTx{xx^ l^a)], 'px 'px since (i»r,r) = constant, = 0. 14. If li, fj, ... $n be a set of parallel lines fixed in regard to the tangent and normal at a variable point j? of a curve, shew that fj, Is, ... fn envelope a set of parallel curves. 15. To find the polars of a point in regard to an algebraic curve. Let P {(xa)2, {xbf ... (xa), (xfi) ...} =0 be the curve, where P is a polynomial. Let If be the point. Let yi meet the curve in the point yi|. We have /'{(p?o)« ... i^a) ...}=0, . „ f (za)'-^ {(ya)'+(^a)«-(y0)«}+it'(ya)' (ifa)-i(za) \ ■ \ (!-*)« ■■■ 1-* •••/-"> where g|=*. 6—2 84 MISCELLANEOUS EXAMPLES Hence by putting the successive coefficients of *=:0, we get equations in y, z. Looking upon zaaa, variable, these are the equations of the successive {>oIars of y. 16. Find the poles of a line in regard to an algebraic curve. yotatioH. We shall use the noUtion a . bkjk, to denote the jwint aby, where (-^)=-' 17. Shew that (a . bic,ik,^) = - - ■■.-- ir^- • 18. Shew that (aX)+2(a» l+k, 1+S*r 1+2 ir 1 Put the J..H.S. = Pn- Pn-i{l+"skr) + K{aM Then P^ = 1+2*, I . . i',(l+2ii;,)-/'„_,(l + "i'i,)=X:„(a,X). 1 1 n.— 1 n— 2 Similarly /'„.,(1+ 2 *,)- i'„.2 (1+ 2 *,) = *..-! K-i^), 1 1 Pj (1 +2 *r) - A (1 +il) ='f-2 («sX), . adding Pn{l+^k,)-P,{l + i:,)=it,{ar\), 1 2 which gives P„. Notation. We shall use the notation a . ^kjk, to denote the line a/3c, where ,-^^ = ^'. It is evident that this does not completely define the line, as it does not specify any sense. 19. Shew that (a . |3t./^, c) = , - M-^ ~A^ ^^L . , */V+V-2*,*2COs(a/3) where the sign of the square root is arbitrary. 20. 1{ d he the isotomic conjugate of e in regaid to a, 6 ; a,b,c being coincident : find (y)l(o|)' g(yl) ~t(oy)l(o«r For point reciprocation, we shall denote by R„ the reciprocal of any element in regard to o. 86 MISCELLANEOUS EXAMPLES The preceding formulae become 24. Similarly shew that (^°-^'^°^^'"'^ = '*^'-|(o^-)S')WI' 25. Shew generally that ... ._f^ ( Ho^i RqXj Rpji R„X3 Bo^2 ••• Ro^n^o'gM + l) (^V''2€l^3g2 ... ^,x,,,)- A . ^^^^^^^ ^^^^^^^ \(oR„i,)\ ... {oRo^n^,) ' . , . t \-k' (■ft.^1 RcXj Roil RqXj ... .g„j;,i^„g«_i o) (^i^-2€i^3g.....^»§»-i;-A . ^^^^_^^j ^^^^^^^ \(.oR„ti)\ ... (oR^x^^i) • It t ^ t ,. t t \—k' (R'il RoiiRpXi ... fipg, RoSn+l) (€.f=^,f3^2 ... €„f„.,)-A . |^^^__^^^ (o7i.«2)l(o/f<.:^,) ... l(oAol»+.)l ' -*>„>„ .„ s_rr (flpgi Rpji RqXi RoiiRpXj... R„inRo'^ti-\0) 26. Roulette*. One curve rolls on another fixed curve, to find the displacement of the point of contact and the tangent at the point of contact ou the rolling curve. Suppose the curve to roll counter clockwise. Let the senses of description of the curves be counter clockwise. Let X, y, y be three contiguous points on one curve, i/, y', y" three points on the moving curve which take up positions x, a/, if' in its rolling. Let x=y, 3d=i/, cd X r" \s In the rolling jf remains at x', but yy becomes x'li' . I.e. if y be the point of contact on the rolling ourve^ then dy=0, dTy=—drjy+dTX, drx>dTiy, where dr-^y is the displacement of ry when the curve is .fixed. MISCELLANEOUS EXAMPLES 87 Knowing these two displacements, we may find the displacement of any element derived vectorially or equationally from them, by the method of the text. The case of a point on a rolling curve does not come under the classes of derived points considered. The above investigation from first principles is therefore necessary. 27. Find the displacement of a point fixed in regard to the rolling curve. Let z be a point fixed in regard to the rolling curve and y the point of contact on the rolling curve. Let ^=y„,jj' .•. di=Rdio = Rdiry=R(^dTX-dTiy), and TZ=yi^. 28. Find the displacement of a carried line. Let f be the line. Then d^=dTy=dTX-dTiy, 29. Find the radius of curvature of a carried point. We may now no longer concern ourselves with the fixed curve. Let z be the point. Then dTZ=d^ by Ex. 27 I(y2)l " I(.y2)l Here the displacement of y has a different significance from what it has in Ex. 26. In Ex. 26 the considerations of its displacement were due to the rolling. The displacement now is due to the point taking^ up, as we suppose, successive positions on the curve. _ sin (rya) dy d/rx — dr^y _ sin (rya) dy 1 ~/e(drx-dTty)'*'R iPi^x-py^R' By similarly differentiating P' {j,if{px-py)^\i2,z)V find -r- : and so on. az 88 MISCELLANEOUS EXAMPLES 30. Find the radius of curvatxire of a carried line. Let f be the line. Then ^^^^i,' d(^. a) d{ayC) divja) . % I Consider the case of a curve rolling on another curve which is rolhng on another curve. 31. Shew that the pedal triangles of a triangle of points inverse in regai-d to the circuracircle are similar. Let the points be »^ , » nt. p Now if X, y, z be the summits of the pedal triangle of s , (y2)2=(oa)2sin20y) = {^ + p2 - iRp cos (Ja a)) sin» Oy). If y, y\ z are the siunmits for s b«, p -j + «2 - 2 — cos (iS ») Uin« (/3y) 32. If we represent by I^x the inverse point of x in i«gard to a circle centre o, shew that '^^^1 \{ohx){oI,y)\- I„x=^o_ j{i , where R is the radius of the circle. "' |(• |(0i))| ""> Kos)| Ri = 2 r; — r-T— r, sin (ovoz) 2 2 (oa7)« (oyz) {oxf{oyf{ozy R* by theorem on p. 29 which proves the result. Anharmonie or Cross-ratio. Let a, b,c,dhe four points incident in a line X. Then the ratio {ad}/ (bd) is called the anharmonie ratio or cross-ratio of the range of points, and is represented by {ab, cd). If a, j3, y, 8 be four lines incident in a point I, then the ratio ?!5 i'^y} I ^!° ^^y} is called the anhai-monic ratio or cross-ratio sm {ab) I sm (/38) of the pencil of lines and is represented by {o^, yb}. In projective geometry, of the trigonometric functions the sine function only occurs ; hence, for brevity, we shall represent sin (a)3) by (ajS). Thus the cross-ratio of four lines incident in a point is (aS)/ OS)- The cross-ratio of a pair of points a, b and a pair of lines y, 8 we shall define as («8)/ (68) ' and this is written [ab, yb} . 34. Reduce {aoW, yy88^}- {aa'bb', ri «« ) = /-=^ / 1WW\ (aabo) I (Ob So) (ay)(a'y)-(ay')(ffl'y) / (by)(b'y')-{byr)(b-y) ~ {ab) {a'y) - {ab') {a'b) / (68) (6'd') - (68') (6'8) ' Particular cases. When (oy')=0. («'«)-0; (6y')=0, (6'8)=0. Then {W66', ^'W) = {ab, yb} . [a'b', y'8'i. Similarly if (oy')=0, (a'8)=0 ; (6y)=0, (6'8')=0, then {i^^'b^, W^} ={»*'. y8) • («'*> >'*')• * This ratio and its usage are new, as far as I know 90 MISCELLANEOUS EXAMPLES 35 If " * '^l , ", t "!! be two triangles ; and W, 66', c? be represented 037) a p y ) by X, ^, r and ra', 30', yV by I, m, n ; shew that {/*!/, aa'}. {»i», oa'} = l. {^i.,, aa'}= {66' w', 3^, 3V} -(6y)(6'|3) / _WWyl ~ mi<^y) I -(«y')(cW = !6'c,3v'};16o',/3'y}. Sirailariy {»m, aa'} = {/3'y, 6c'}/{3y', Vc). 36. Shew that ^^=^, (gY)(6a)-(aa)(6y) / (a'Y')(6'8)-(a'8)(6'y) ( afty a 6 y , 88-) - („.y) (jg-) _ (aS-) (6y) / (a'y') (6'8') - (a'a-) {Vy') ' 37. In Ex. 35 shew that {mn, aa'} . l='^v, an'} . X, where {ah, cd\.o denotes {oaob, ocod). {mn, aa']. l = {mn, lala'}-={^ff yy', la I'a} W(i3'a) (yg') {y'l) (/3a')(/3'i) ■ (yO(y'«) = {/3y,te'}/{/3'y',H Now (3i)=(e,„„.)=_i_2i_J_, (3'0 = (?^W)=i'S?l^„ l(co)l(oo) (y0=(a6aa) = |^^^yy^^, / •i\ i-:?T'—\ ("'°')(fr'° ) Hence ;«.«, ««'} . ^=^.^lM)f_f^«) .mm, ' {da) (y'a) . (6a') (/8 where f=aaifi =0, since (/6i*)=0. Hence the six points a, 5, c ; d, e,/lie on the curve, where d=a'^a, e=a/3=/3y='ya, f=7m{^. Hence a = de, ^=ef. Also rf=UiCa = tticefe, .•. (a,ec?)=0. and f=ayafi — aiaef, ••■ (aio/)=0, .•. aj = afcd- Hence the cubic (xade afed xbefkyby xc)=0 passes through a, b, c ; d, e, f. As regards k, y, by , we have (ye)=0, (/6,X)=0 Take three other points g, h, i. 94 MISCELLANEOUS EXAMPLES Now let gi=gaaai gc=gade af cd gc. hi = haaai he = hade af cd he. ii—iaaai ic =iade af cd ic. gi=ghft =gb ef. ht=hb^ =hb ef, i^=ibfi =ib ef. Thus the six points gi, A,, t] ; gt, Ag, i^ cau be obtained by linear con- struction from the nine points a,b,c; d, e,f; g, h, i. We proceed to choose t, y, 6^ so that the following equations hold : (giiy^i9i) = 0, {hatybihi)=0, (is^6iii)=0, which are the conditions that g, h, i should lie on the curve. If possible, determine y and i from (iibiykii)=0, without conditioning bj. For this we must suppose t'l^iy ^t'l, and (ijii^)=0. Hence (yj,)=0. Hence since ('ye) = as well _ y= M- 7= i,e and (Hj^) — account for the first equation. The remaining two equations can be written (%27^i6i)=0. (*AayA,6,)=0. Hence k is such that {.tgiygi kh^yhi */)=0, also (/•Mij) = 0. Hence k is one of the points in which tit'a intersects the curve {xf xg^yg^ xh^yhi)—0. We consider this curve {xf xg^ ygi jtAjj y A,) = 0. Put x=y^, where $ is any line. Then (y|/ y| S's y fir, y^ Aj y A,) = (y|/ yf y, yf Ai) =0. Again, put x= /3f . Now g^, Aj, /lie on /3, .-. xf=fi, xgiygi=^ygi, JFAjyAi=/3yA,. Hence /3, y are parts of the curve. Hence the remainder of the locus is another line. MISCELLANEOUS EXAMPLES 95 To find this line we have {xf xg^ygi xkiyhi)=0, .-. {_xf xgt ygihiy liix)=0. This is satisfied if i.e. {xfki)=0, {xgiygihi)=0, i.e. (iP/A,)=0 and {higiygix)=0. i.e. x=fhi higiyg^. Similarly another value is given by therefore the third line is Denote this, for brevity, by X. Then k is incident in iii^ and in /3 or y, or X. If we assume that k lies in /3, the equation of the cubic becomes (xaaUi Pybi xe)=0, i.e. a conic and a line. Similarly if k lies in y, the cubic becomes (xaaUi khi xc)=0, i.e. a couic and a line. The only possibility then is k lies in X. It will be shown that this assump- tion allows the cubic to be of the genei-al type. We shall prove this by showing that the cubic passes through the nine arbitrarily assumed points. Hence let it be assumed that k=iiii\. Accordingly with these assumptions the equations {g\hyl'9i)=^y {hh^ykhi)=(i, {iibycki2) = are satisfied and therefore g, h, i lie on the curve. &i is the point of intersection of */ *^2 7S'i. kh2yhu ••• l>t=kgiyg,kf. Finally therefore it has been proved that the cubic curve {xaaai xbfikybi xe)=0 96 MISCELLANEOUS EXAMPLES passes through the nine arbitrarily chosen points a, b, c d, e, / g, h, i pro- vided that a, A y, a,, 6i, i are determined by the linear constructions a=de, j3=e/, y=eii, a,=af cd, ji;=i,ijX, bi = kg2ygi if. where gi = ffade afcd gc, ki = hade afcd he, g.i=gbef, lii=libef. ii=iade afcd ic. This gives «s the analogue of Pascal's theorem for a cubic. This theorem and analysis are due to Grassmann. 47. In bi-i-adial co-ordinates, shew that Laplace's equation is \oi^ o^ aros r or i oij where r=\{xa)\, s-\{xb)\, 6={xa,xb), a, b being the points of reference. Use the theorem on p. 77. 48. Find the condition that y is a double point of the curve f{\{xa)\, \ixb)\, ■Xxc)\...{xa),(a;?),(^)...}=0, and that i; is a double tangent of the curve /{(?«), (fi), (ic) ... (ga), m, (^) ...}=o. 49. Let ' „' \ ; ' ' ,]■ he two triangles and s any point. Shew that a,P,yi o,p, yj (saa ibff iCy) {loa') (»6/3') (»cy') _ R (a, b, e) («^ W^ Idy) {X^a) (»6' j3) {»ga)+2 (»|a) (pga)' {£f log (|a)=[(p"f + 5p?) (^i)'- Vf (^.)» (-fa) + 12p5(v«a)»(f»)-3 0.|)»(fo)». -2 (/>««)* {(f»)*+3(vfa)»}]/(fa)«. 66. Two points a, b move on two lines c., c^ in such a manner that \{ah)\ is constant. Shew that, if j he the intersection of ab with its consecutive position, q and the foot of the perpendicular from c on ab are isotomic con- jugates in regard to a, b. [The Frincipia.'] We have to shew that (v o6o) = (to t6 634,), when I {ab) \ = constant. ^ ,-^ . \(ah)\(Tah)da |(a6)|(ra6)(i>6a) '{rah){»ba)-{Tba)(vab)' (raft) sin (r6a6») a sin (rorft) =(TaT665ii„). 67. Shew that the conic which has three-point contact with a curve at the point a and has a focus at « has the equation 2 |(^*) (a,)\-(as)*-(sxy+{''^f=2pa^^(xTa). 68. By means of £z. 67 or otherwise find an equation of the locus of the foci of conies having four-point contact at a point of a curve. 69. Shew that i(ia){&>) dS=i (of) {(^) (|5)-Kvfa) (.|6)} -i{(^)(''ib)+(ib)(,^)} + {(ia) + {&>)} {-i(ai) (pf -p"£ + plTf _...) + (l+i)p'|-(2+i)p"'f-l-(34-i)p»f-...} + {(•'^»)+(»f6)} { - i (of) (/.'f -p"'|+p'| - ...) -Jpl + (1 +l)p"|-(2-H)p"|-l-...}, where a is an arbitrary line. since (pob) da+(vba) db = 0, HISCELLANEOXTS EXAMPLES 101 The infinite series occurring are supposed conveigent and differentiable. Assume J(^) (ff,) dS=I^+Ii (|o) (|6)+/jj (via) (..f 6) +^3{(f»)(»f6)+«6)(^fa)} +^4 {(f»)+ (^)}+/6 {(•'|a)+(-|6)}. where the Fb are intrinsic functions of the curve and differentiate. 70. Prove generally ;/• {(£«), (m ... (-I"), wft) ... sin (fa), sin (m ... r, n). Mi) ...> di =^{(?«), (ib)...(via), (vi6)...sin(fa), sin«/3) ... Atf), /j (©...}, where P denotes a polynomial, and the Pa denote intrinsic functions. 71. In the cubic {sca^bxii Ci ^1 Oi j;) = 0, find where 8, hi ; ca, CiOj cut the curve and shew that cuts the curve in the points where it cuts the conic {xciiC\fiiaix)=0. To find where S cuts the curve, put x= iC, then {(da^S(diiCiffiai6{)=0. Hence f8 a/3c8=f8...(i) or (d8,c,j3iaf8)=0...{ii). From(i) ({8a^CS)=0, .: {i=M or ^. From (ii) f8 = d8iO,3,a8. Hence the three points of intersection are /38, ca8, 88,c,^ia8. To find where ca cuts the curve, put x=ca(. And we find in a similar manner that the points of intersection are a, cad, oa8jCj/3i[8in (ya) (c6a)(a3)+(aea)(/8y6a)]=0. 75. If in Ex. 74 (/3y6c) = (yaco)=(a^a6)=0, shew that one solution for ^, y, z is bca, ca^, ahy and find the other solution. Also consider the case in which (a6c)=(a^y)-0. 76. If a{Agi,hi)=h(figfih()=c(Jtgnhy) = {f(g^hi), shew that {/(grihg) satisfies a cubic in measures off, g, h, a, p,y and a, b, c. 77. Shew that for a cubic curve if ' ' \ , ,' ' , j- be two triangles o. ftyJ a,pr,y) such that their vertices and the intersections of corresponding sides lie on the cubic, then (,a'^y')=l {affy')=m {a'^)=n(a'ffy), and hence shew that the cubic is not restricted by such a condition. Hence shew that the cubic (xaa xbfi xSy)=0 is a general cubic*. * Dae to OraBsmann. MISCELLANEOUS EXAMPLES 103 78. Find the equation of a circle, the mea8ures of which in regard to the three points a, b, c are *„, ti,, *». Let x,yhe points on the circle. Let ry be the tangent at y. Then (,xy)'=2p(xri/), where p is the radius. It is easy to shew that ta^={ayY-ip{aTy), t,?={hyY-iplJtny), t,^Ho}ff-ip{cTy). If we eliminate ry we find the equation of the circle. Now from o, b, c, y, ry we have 2 (rya) (bey)=0, - .-. i{t.*-{ayy]{bcy)=0, .-. (o6c)<«+S(6cy)«„s=0, where t is the measure oty in r^ard to the circunicircle of a, b, c. If we denote by cir a, b, c the circumcircle of a, b, e the equation of a circle r may be written in the form (arcrr a, b, c)« (abe)+ S (aTy(bav)=0. a, o,c 79. Find the radius of a circle r when (Fa), (r/S), (Ty) are giyen. We have (oa) = r cos (Fa), and two other equations. . • . J-2 sin (j3y ) cos (Fa) = (ajSy). 80. Transform oir l,\;k io the form cir o, r. cir l,\;k'\a equivalent to cir h„ _»> s'lfi-^3Je{tK). 81. If we denote by ra TiTj the radical asis'of the circles Tj, Tj, shew that Kracir o,, r, cir Oj, rj a?)| = | ^— i^ — ^^^^^^^^ ^| . 82. If a circle touch two circles, shew that the perpendicular from its centre on the radical axis of the two circles is proportional to its radius. Let the touching circle be cir o, r and the other circles be cir oi, r^ and cir 02, ra. We have I (ra cir o, , n cir oj, r^ o)i/r= 2(piOi)r I 2(Oi02)»- since (»--ri)i' = (oo,)«, (r-rj)«=(ooj)«, ■'~ ,^ a constant. (0,02) 104 MISCELLANEOUS EXAMPLES 83. Find the condition for a double point at the point h of the curve /{|(awi)l, \{xai)\...{a;ai), (aroj) ...}=0. 84. Find the condition for a double tangent at the line X of the curve /{(^•i), (!«.)- (««i), (fis) ..■}=o. 85. Find the tangent and radius of curvature of the cun-e given by 0{|(«ai)l. K^KOs)! — (a^ai), (jjoj) ... <} =0, V'{|(^i)l. \{xa^)\ ... {xai), {xai)...t)=0, where < is a variable parameter. 86. Defining (ai Oja, 0302 . . . a„ _ i on) as I(«i«2)|8in(ai0,a)((0i02aia3)|sin(a,ajaia3a2)...(n(ia2aia3a2...a„_,n„), shew that (aioi) (0,02) (a|n3)...(aia„_i) (aio„) (0201) (a202) (ajoa)... (a2an_i) (ojon) (0302) (0303) — («30«-l) («3»n) (flil - 2<»h) ("l«l) (ai«2) faiOs)... (a,a„_2) sin (a,a„) (aeffli) (0202) (0JO3) ... (a2an-2) sin (020,,) (os«2) (os^s) ... (asan-2) 8in(a3a„) ...(a,_ia„_2)sin(a„_,a,) 88. Let ' V be a triangle. A circle cuts the sides a, /3, y in the points «>. dJx pXu = 22 cos (t, Jr») -p- —^PrX\ -J- ] cos (TrXa) + 2 sin (TrXa) —s . y. ao> r \ wo J r Ota 91. pq is the chord of a continuous curve cutting off an arc of constant length ; the tangents at p, q meet in t, the bisector of the lines pt, qt meets pq in r; if / be the isotomic conjugate of r in regard to p, q, prove that / is the intersection of pq with its consecutive position. 92. If X and y be points such that xy=TX, then ^ I (■*^) \ = dx— cos {tXt}/) dy. lixi, X2 be the points of contact of two tangents from ^ to a curve, then d+ 2 2 5, (a!^,) + C __J r 2Q ' where Q2=2 4,» (a,cf + 2 £,» + 2 2 4,il, | (o,«) (o.c) | cos ( ^ ^ ) r r*=« + 2 2 BrB.eOB(firff,) +2 2 .4,5, 1 (a,c) I sin (/Sjojic) = 2il,«(a,c)2+S5,»+2 2 ArA.KflrC) {a^)\ cos (a^a^) +2 2 BrB.eosifirfi.) -2 2 J,5,(o,cfl,) = -2 4,il,(a,a.)*+25,«+2 2 iB,5.co8(/3,/3,) «+• r r*f -224,£.(o,A). MISCELLANEOUS EXAMPLES 107 Hence (2 J, (jw,)» +225, («/3,) + 0=0 (t) r V 2 Ar (rfOr)»+ 2 2 B, (d^,) + C r r ~2V- 2 J,iil,(o,a,)>+25,''+2 2 B,B,co8 (/3,/9,)-2 2 J,5.(o,/3,) ' 98. Shew that, when 2 ^,=0 r sin (^Ar {xOrY + 2 2 B, {xfir) + C=0 «) r r - 2 il , (o,«,) + 2 S, Bin (8j3,) _ g T ^-2^ril,(a,o,)i'+2Br»+2 2 5,B,oo8((3,/3,)-2 2 J,B,(oWS.)' r, a r r^s r, g 99. Shew that the nonnal at .a point of a Cartesian oval passes through the synuuedian point of the triangle whose vertices are the point and the focL 100. Find the area of a segment of the curve whose curvature varies as the cube of the sine of the gradient of the tangent. We have p^=a cosec' (|a), .•.pfO'5«)«sin(^)=a4^^. Integrating a I • a?t ■ ■<^^= / (jco)' sin (ra;a; da;, ar=p|, .-. - 2 seg pf- (p{a)» cot iia) ^ (*a)'= - ^ (j>$af. .: aegpi=^(piaf-^(piaycot(ia). 101. Shew that / (xa) dx=]iQX (xa) + il sin {rxa) + fi cos {rxa), where A, B are intrinsic functions. 102. Shew that J (jca)' di!=UTix(xdf+A (rxa) + B (yxa) + C, where A, B, C are intrinsic functions. 103. Shew that /(ara) (jr/3)diF=lina; {xa) {x0) + (xa) {A sin (ra;;3) + Bcos (rx/S)} + {xfi) {(7 sin (rxa) +Dcoa (rXa)] +.F8in (rxa) sin (tx$)+F sin (rxa) cos {rxfi) + Ocoa (rxa) sin (rxff) + 5^ cos (rxa) cos (tx$), where A ... S are intrinsic functions. 104. Indicate the general form of the integral of a polynomial function of (xaf, {xb)K.. {xa), (x^).... 105. Indicate the general form of the nth differential of a polynomial function of (i) (xay, ixby...(xa), {xff)..., (ii) (f«), («6)...8in(|a), 8in(«/3).... 106. Shew how to find the family of rhumb lines of the family of curves (i) f{\{xa)\, |(j«*)|,...(»a), (a;^)...}=variable parameter, (ii) /{(fi), (|6), ... (|a), (l/S), ...}=variable parameter. ADDITION TO CHAPTER I To reduce cos {bead). We have |(6c) (cmOI cos(6c^) = |(ad)|(5c^w) = |(ad)|{(6^)-(c53,)} = \{ad)\ {(6a^)- (caoS,)} = |(6o) (orf)| cos {baad)-\(.ea) (orf)! cos (eaad). Hence 2|(5c)(arf)|cos(6^a5)=(oY) may be taken as irredncible. APPENDIX REDUC3TION OF PRODUCTS OF MEASURES We give four exampleB of reductions of products of measures which possess the property of being reducible without radicals, notwithstanding that the reduction of one or more of the com- ponent measures contains a radical. This is due to the elimin- ants existing between the elements. The examples are (1) (a6c) sin (ay8). (2) (a6c)cos(^r)l(a^)|. (3) |(a;y)|co8(^5)sin(\M), (4) {abc)(a^y). (1) Though the reduction of (aic) contains the radical, the product (abc) sin (ayS) is expressible without radicals. The reduction may be effected as follows : We have (a0bc) = (aa) sin (;86c) + {aa) sin (bed) + (aic) sin (a/3), .-. (o6c)6in (a;8) = (a^bc) + (aa) (bc^) - (afi) (bca) (aa) (a^) 1 (6a) (6;8) 1 (c«) (c^) 1 (2) Here the reductions of both the component measures contain radicals. The product may be reduced as follows : (a6c) cos (^?)i(*y)l = (abc)8m(^ayy^)\(xy)\ 5 (bK) (axy^) (bxy,) (cxy,) \(^y)\ by (I; APPENDIX 111 = 2 (af) \(bc) (xy)\ cob {bcs^) = i2 (a?) {(6y)' + {cxy - {hyf - {cxf} (a?) (ay)>-(aa;)» 1 (6?) (byy-(bxy 1 (c?) (cyy-(cxy 1 (3) Though the redaction of coa(^5') involves radicals, cos (^ f ) sin (\/t) may be reduced without radicals. From the four lines \, ft, xy, f we have sin (fixy) cos (\f ) + sin (xyX) cos (/uf) + cos (xy^ sin (\/*) = 0, ••• j(«y)l cos (iy?) sin (V) = {(a;/t) - (y/it)} cos (\?) -{(a!\)-(y\)}cos(/tf). (4) The product (abc)(a^y) may be reduced without radicals as follows : (a6c)(a^7)= S sin(a/8)(a;87)(a6c) = 2 sin (a/3) 2 -(cL$bc){ay) = 2 {ai){afibc) a,b,e = 2 (a7){(a6)(/8c)-(ac)(yS6)}. Hence {(dtc)(a^y)= (cm) (afi) (ay) (6a) (6y8) (67) (ca) (c;8) (cy) Referring to the result on p. 109 and using (1) and (3), it is easy to see that the square of any measure of the Hrst class, any measure of the second class, the sine and cosine of any measure of the third class is reducible to the quotient of two polynomials in (1) the moduli of measures of two points, (2) measures of a point and a line, (3) sines of measures of two lines, (4) cosines of measures of two lines. 112 APPENDIX Examplet: 1. Shew that sin (head) is reducible rationally by multiplying by sin (Xfi). % Reduce {Obyd)^ by means of the reduction of (i^Se)'. We have {ahyd)*={ahf (ahyd)* = {ahf {(^oO» + (>d)«-2 (yrf)'-2 (y«0 l(«6)l () = costf cosflb — sin 5 sin *... ♦1.6)= -(o6<») + 2 psin(/jl=5 (85)- vp^—v^ (86). (v^l«)=P«-(l«) (87). (p,.^) = (/>|a) + C08(|a)p| (88). CHAPTER XI. \^{xafH^xa) .(89). 1(0^'"'^'"^"^'''"^'"' ^^^' \{j^{xaf=-{pxf{vxa)-p'x{rxa) (91). \ (S'^"^^'" -(p^)»+{(pa:)3-p":r}(r^a)-3pV^ (-*•«) (92)- 9 (rf~ ) (*")*= ~ bpxp'x-\-{(j>xf - ipxp"x - 3 (p'a.-)*} (vxa) + {6 (j,xfp'x-p"'x} (Txa) (93). And generally, s I t- ) (*«)^ = -^ « (^in*) + Sn {vxa) + C„ , where 4, B, C are polynomials of px, p'x, p"x ..., and ^\ \{xa) .(95). (d\*, ,. {fixf 3 .(96). + (T.ra ,)| \{xa)\^ |(^a)I+(^^)(''^")r(Sayp (p.r)3 6px )C\{xa)\ |(:ra)P } . . Zpxp'x , . . , 4p'.r - (''■'^") TZ;^ + ('■^) (''^«) ^^ d(xa) dx ("™) |(.ra)P + (''^"^ ||(^a)|3 |(.ra)|4 (r.m)(-.ra) |^^^^|^ |^^^^|^ (9,). = — sin(T.j;a). .(98). I J- ) (.rn) = p.r C(1N (r.rn) (99). J- I (j;a)=^'.rcos(Ta;«) + (p.r)2sin(T.i;ii) (100). ^ j {xa) = {p".r - (px)'} cos {rxa) + Spxp'.r sin (rXa) (101 ). (^ J^^ ixn) = {6 (pa;)2 p'a: - p"'x} cos (r^a) + siii(T:ra){4pa;p"a:+3 (p'.37)2-(p.i;)<} (102). And generally, irr) (■*'«) = ■An sin ('"■^'a) + B„ cos (r.ra), where A, B are polynomials in px, p'x, p"x ... and Ani.i = A„' + B„px, Bn + i = Bn'-A„px. (df) (^)=P""""f-P"""')l+-+(-)""'pl+(-)"(^0 -(103). (^)""''(f>f) = P<'^-"|-P<''-'>l + ... + (-)"p'f + (-)"-'(«|«)...(104). APPENDIX. (a6c)sin(a/3)= .(105). (aa) (ad) 1 (6a) (J/3) 1 (ca) (c/3) 1 I (a*) I cos ( aby) sin (X,i) = {(ap.) - (bp)} cos (Xy) - {(aX) - (6X)} cos (/ly). . . (106). (af) {ayf-{axf 1 I (107). (if) {byf-{bxf 1 I (cf) (cyf-{,cxf 1 I 2(a6c)cos(33(f)|(j^)| = (a6c)(a^y)= (fla) (afi) (ay) (6a) (60) (6y) (ca) (c/3) (cy) .(108). . INDEX Algebraic curve, method of tinding the polars of a point 83, poles of a line 84 Analysis, method of 5, 35 Anharmonic ratio, of a pencil of lines 89, of a pair of points and a pair of lines 89, of a range of points 89 Area oi a triangle 10 Axioms 8, 9, 10, 13, 39 Bertrand 83 Cartesian oval 107 Casey 19 Cayley 82 Circle, as a third element 96, 97, 98, 103 Circles, radical axis of two 103 Conic of closest contact 79, 80 Cosine function 11, addition formula for 12 Determinate 3, general method of differentiation of 43, notation of 3, quadratic form of differentiation of 43, rule for differentiation of determinates of simple elements 59 Differentiation of |(xo)| 39, {xa) 40, (Ja) 40, (fa) 41, (xyz) 41, (xi/f) 41, (f,2) 42, of xji 43, {)) 44, Jt/z 44, xi/f 44 of Sa,(a;o,) + o=0 50, S.J,(fttr) + Si(,cos (|/3r)=0 51 Sa, (a;oj + a=0^ 52, SJ, (l»r) + 2iJr cos (|j3,.)=0c 53 Directly similar figures 99 Displacement of a line 39 Displacement of a point 39, generalized — 63 Elements 3, derived — 4 Eliminants 18, 19, 20 Evaluation of {rxa), (vxa) 54; (raa), {vxa) 55; (2>f«)"> ("I") 54; (pfa), {via) 55 otprx, VTX, v^x, pvx 65, 66 ; rpf , vp^, v^, pv^, 66, 67 ; i'"f , pi>"^ 67 ; v'^x, pv^'x, 68 Foot of perpendicular from a point on a line 24 Geometry of three points 10, of two points and a line 9, of two lines and a point 16, of three lines 11, of four points 19, of three points and a line 17, 19; of two points and two lines 11, 18, 19; of three lines and a point 13, 18 ; of four lines 108 Grassmann 92, 102 Grave 105 120 INDEX Incident 3 Integration 73, general results in 101, 107 Inversely similar figures 99 Laplace's equation 77 Length of a curve 74, 79 Line, notation of 3, veotoriaUy derived — 1, equationally derived — 4 MacCullagh 81 M'Cay 98 Measure 3, notation of 3, differentiation 39, linearity of differential 41, suocessive differentiation of — 69 Pascal 92 Point, notation of 3, vectorially derived — 4, equationally derived — 4 Principia, The 100 Kadius of curvature 63, 65 ; in bi-radial co-ordinates 78 Beciprocation 8S, 91 Beduction of (afcc) 15, ^'° {^y) 9, {a^f 16, (0JS7) 13, '*'" (6^5d)108, {abyd)' 17, {ahyd) 11, (ydab) 18, ^'"(^cJ) 18, (0^7* (^ 108 (V ...-''>' ^*' (V...-^^^^'