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TEILINEAR COOEDIMTES. 
 
PRINTED BY C. J. CLAY, M.A. 
 AT THE CNIVEBSITY PEES& 
 
TRILINEAR COORDINATES 
 
 AND OTHER METHODS OF 
 
 MODERN ANALYTICAL GEOMETRY OF 
 TWO DIMENSIONS: 
 
 AN ELEMENTARY TREATISE, 
 
 REV. WILLIAM ALLEN WHITWORTH. 
 
 PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, LIVERPOOL, 
 AND LATE SCHOLAR OF ST JOHN'S COLLEGE, CAMBRIDGE. 
 
 CAMBRIDGE : 
 
 DEIGHTON, BELL, AND CO. 
 
 LONDON BELL AND DALPY. 
 
 1866. 
 
PREFACE. 
 
 Modern Analytical Geometry excels the method of Des Cartes 
 in the precision with which it deals with the Infinite and the 
 Imaginary. So soon, therefore, as the student has become fa- 
 miliar with the meaning of equations and the significance of 
 their combinations, as exemplified in the simplest Cartesian 
 treatment of Conic Sections, it seems advisable that he should 
 at once take up the modem methods rather than apply a less 
 suitable treatment to researches for which these methods are 
 especially adapted. 
 
 By this plan he will best obtain fixed and definite notions of 
 what is signified by the words infinite and imaginary, and much 
 light will be thereby thrown upon his knowledge of Algebra, 
 while at the same time, his facility in that most important sub- 
 ject will be greatly increased by the wonderful variety of expe- 
 dient in the combination of algebraical equations which the 
 methods of modem analytical geometry present, or suggest. 
 
 With this view I have endeavoured, in the following pages, 
 to make my subject intelligible to those whose knowledge of 
 the processes of analysis may be very limited; and I have de- 
 voted especial care to the preparation of the chapters on Infinite 
 and Imaginary space, so as to render them suitable for those 
 whose ideas of geometry have as yet been confined to the region 
 of the Keal and the Finite. 
 
 w. ^ 
 
VI PREFACE. 
 
 I have sought to exhibit methods rather than results, — 
 to furnish the student with the means of establishing properties 
 for himself rather than to present him with a repertory of iso- 
 lated propositions ready proved. Thus I have not hesitated in 
 some cases to give a variety of investigations of the same 
 theorem, when it seemed well so to compare different methods, 
 and on the other hand interesting propositions have sometimes 
 been placed among the exercises rather than inserted in the text, 
 when they have not been required in illustration of any par- 
 ticular process or method of proof 
 
 In compiling the prolegomenon, I have derived considerable 
 assistance from a valuable paper which Professor Tait contri- 
 buted five years ago to the Messenger of Mathematics. My 
 thanks are due to Professor Tait for his kindness in placing 
 that paper at my disposal for the purposes of the present work, 
 as well as to other friends for their trouble in revising proofs 
 and collecting examples illustrative of my subject from Uni- 
 versity and College Examination Papers. 
 
 LiVBEPOOL, 
 
 15 September, 1866. 
 
CONTENTS. 
 
 PAOS 
 
 Prolegomenon. Of Determinants ix 
 
 Chapter I. Of Perpendicular Coordinates referred to two axes ... 1 
 
 Exercises on Chapter 1 7 
 
 Chapter II. Trilinear Coordinates. The Point 9 
 
 Exercises on Chapter II 20 
 
 Chapter III. Trilinear Coordinates. The Straight Line 21 
 
 Exercises on Chapter III 34 
 
 Chapter IV. The Intersection of Straight Lines. Parallelism. 
 
 Infinity 36 
 
 Exercises on Chapter IV 54 
 
 Chapter V. The Straight Line. The Equation in terms of the 
 
 Pei-pendiculars 58 
 
 Exercises on Chapter V 69 
 
 Chapter VI. The Equations of the Straight Line in terms of the 
 
 direction sines 71 
 
 Exercises on Chapter VI 92 
 
 Chapter VII. Modifications of the System of Trilinear Coordinates. 
 
 Areal and Triangular Coordinates 92 
 
 Table of Formulae 96 
 
 Exercises on Chapter VII 101 
 
 Chapter VIII. Abridged Notation of the Straight Line 104 
 
 Exercises on Chapter VIII 115 
 
 Chapter IX. Imaginary Points and Straight Lines 117 
 
 Exercises on Chapter IX 130 
 
 Chapter X. Anharmonic and Harmonic Section 132 
 
 Exercises on Chapter X 145 
 
 Chapter XL Transformation of Coordinates 147 
 
 Exercises on Chapter XI 153 
 
 Chapter XII. Sections of Cones 154 
 
 Exercises on Chapter XII 163 
 
 Chapter XIII. Abridged Notation of the Second Degree 165 
 
 Exercises on Chapter XIII 171 
 
 12 
 
VUl CONTENTS. 
 
 FAOB 
 
 Chapter XIV. Conies referred to a Self-conjugate Triangle 173 
 
 Exercises on Chapter XIV 180 
 
 Chapter XV. Conies referred to an Inscribed Triangle 192 
 
 Exercises on Chapter XV 204 
 
 Chapter XVI. Conies referred to a Circumscribed Triangle 206 
 
 Exercises on Chapter XVI 219 
 
 Introductio:* to Chapter XVII. Notation, &c 221 
 
 Chapter XVII. The General Equation of the Second Degree 226 
 
 Exercises on Chapter XVII 253 
 
 Chapter XVIII. The General Equation of the Second Degree 
 
 continued 255 
 
 Exercises on Chapter XVIII 285 
 
 Ch.\pter XIX. Circles 287 
 
 Exercises on Chapter XIX 30t 
 
 Chapter XX. Quadrilinear Coordinates 307 
 
 Exercises on Chapter XX 321 
 
 Chapter XXI. Certain Conies related to a Quadrilateral 325 
 
 Exercises on Chapter XXI 331 
 
 Chapter XXII. Tangential Coordinates. The Straight Line and 
 
 Point 332 
 
 Exercises on Chapter XXII 343 
 
 Chapter XXIII. Tangential Coordinates. Conic Sections 345 
 
 Exercises on Chapter XXIII 364 
 
 Chapter XXIV. Polar Reciprocals 368 
 
 Exercises on Chapter XXIV 387 
 
 Chapter XXV. Conies determuied by Assigned Conditions 390 
 
 Exercises on Chapter XXV 400 
 
 Chapter XXVI. Equations of the Third Degree 401 
 
 Exercises on Chapter XXVI 421 
 
 Introduction to Chapter XXVII. General properties of Homo- 
 geneous Functions 426 
 
 Chapter XXVII. The General Equation of the n"" Degree 431 
 
 Exercises on Chapter XXVII 451 
 
 Miscellaneous Exercises 455 
 
 Notes on the Exercises. Results and occasional Hints 484 
 
 Index 499 
 
PROLEGOMENON. 
 
 OF DETERMINANTS. 
 
 § 1. Introduction. 
 
 1. If we have m equations involving a lesser number n of un- 
 known quantities, we may determine the unknown quantities from 
 n of the equations, and, substituting these values in the remaining 
 m — n equations, obtain m — n relations amongst the coefficients of the 
 ■m equations. 
 
 In other words, if we eliminate n quantities from m equations, 
 there will remain m — n equations. 
 
 2. If the equations are all simple equations, the solution can 
 always be effected and the vn — n equations practically obtained. 
 
 The notation of Determinants supplies the means of conveniently 
 expressing the results of such elimination, and the study of their 
 properties facilitates the operation of reducing the results to their sim- 
 plest forms. 
 
 3. It must be observed, however, that if the equations be homo- 
 geneous in the unknown quantities, or, in the case of simple equa- 
 tions, if every term of each equation involve one of the unknown 
 quantities, the equations do not then involve the actual values of the 
 unknown quantities at all, but only the ratios which they bear one 
 to another. Thus the equations 
 
 3a; + 4y - 5« = 0, 
 
 5a; -I- 5j/ - 78 = 0, 
 
X OF DETERMINANTS. 
 
 are satisfied if x, y, z are proportional to 3, 4, 5, but they do not 
 involve any statement as to the actual values of x, y, z. 
 
 In this case the number of independent magnitudes, concerning 
 ■which anything is predicated in the equations, is one less than the 
 number of unknown quantities involved in the equations. Thus each 
 of the equations just instanced, involving the three unknown quanti- 
 ties X, y, z, sjjeaks not of the actual magnitudes of those quantities, but 
 of their ratios one to another, which are only two independent mag- 
 nitudes, as is immediately seen by writing the equations in the form 
 
 3^ + 4^ = 5, 
 
 z z 
 
 5^+5^ = 7, 
 z z 
 
 where the equations are exhibited as connecting the two independent 
 
 X It 
 
 ratios - and — . 
 
 §2. Of D^erminants of the second order. 
 
 4. Def. The symbol 
 
 
 is used to express the algebraical quantity afi^ — h^a^, and is called a 
 determinant of the second order. 
 
 The separate quantities a,, 6j, a^, 6, are called the elements of the 
 determinant, and may themselves be algebraically either simple or 
 complex quantities. 
 
 Any hoiizontal line of elements in a determinant is called a row, 
 and a vertical line is called a column. 
 
 Thus the determinant above written has two rows ct,, 6^ and 
 fflj, 6j, and two columns a,, a^ and 6,, 6,. 
 
 = aA-aA = 
 
 5. It follows from the definition that 
 
 Hence a determinant of the second order is not altered by chang- 
 
 
 ing rows into columns and columns into rows. 
 
OP DETERMINANTS. 
 6. It follows similarly from the definition that 
 
 XI 
 
 
 = (a,6, - afi^) = - (».6, - a,6,) = - 
 
 
 Hence in a determinant of the second order the interchange of 
 the tvoo rows changes the sign of the determinant. 
 
 So the intercluxmge of the two colwmns changes the sign of the deter- 
 minant. 
 
 7. IJ ax +by = 0, 
 
 and a'x + b'y = 0, 
 
 be two consistent eqtcations, then will 
 
 a, b 
 a', b' 
 
 = 0. 
 
 Multiplying the first equation by b' and the second by 6, and 
 subtracting, we get 
 
 {ab'-a'b)x=0, 
 
 ab' — a'b = 0, 
 
 therefore 
 or 
 
 8. // 
 and 
 then will 
 
 a, b 
 a, b' 
 
 = 0. 
 
 Q. E. D. 
 
 ase +by + ez =0, 
 a'x + b'y + c'z = 0, 
 
 X 
 
 
 y 
 
 z 
 
 b, 
 
 
 c, a 
 
 a, b 
 
 b', c' 
 
 
 c, a' 
 
 a, b' 
 
 For if we multiply the first equation by c' and the second by c, 
 and subtract, we get ' 
 
 (c'o — ca') X + {be — b'c) y = 0, 
 or (ca - c'o) x = (5c' - b'c) y, 
 
 X y 
 
 °^ be — b'c ca' — da ' 
 
Xll OF DETERMINANTS, 
 
 which may be written 
 
 X 
 
 
 y 
 
 b, c 
 
 
 c, a 
 
 V, d 
 
 
 c', a' 
 
 and therefore, by symmetry, 
 
 X 
 
 
 y 
 
 
 z 
 
 b, c 
 
 ~ 
 
 c, a 
 
 
 a, b 
 
 V, d 
 
 
 c', a' 
 
 
 a', b' 
 
 Q. E. D. 
 
 9. // 
 
 and 
 ilven wiM 
 
 h. 
 
 c 
 
 h', 
 
 d 
 
 a, 
 
 b 
 
 a, 
 
 V 
 
 ax +by = c, 
 
 a'x + b'y = d, 
 
 , and y = - 
 
 c, 
 
 a 
 
 c', 
 
 a 
 
 a. 
 
 b 
 
 a'. 
 
 V 
 
 This follows from the last proposition by writing — 1 for z. 
 
 § 3. Of Determinants of the third order. 
 10. Def. The symbol 
 
 
 a. 
 
 h, 
 
 c 
 
 
 
 
 a', 
 
 V, 
 
 d 
 
 
 
 
 a", 
 
 v. 
 
 c" 
 
 
 
 e expression 
 
 
 
 
 
 V, d 
 
 -b 
 
 a'. 
 
 d 
 
 + c 
 
 a', b' 
 
 b", d' 
 
 
 a", 
 
 c" 
 
 
 a", b" 
 
 and is called a determinant of the third order. 
 
 11. // 
 
 ax + 6y + ce =0, 
 a'x + b'y + dz = 0, 
 d'x+b"y + d'z = % 
 
OF DETERMINANTS. 
 
 xm 
 
 then will 
 
 a, b, c =0. 
 a, b', c 
 a", b", c" 
 
 For the second and tliird equations give, by Art. 8, 
 
 X 
 
 
 b', 
 
 c 
 
 b", 
 
 c" 
 
 y 
 
 
 z 
 
 c, a' 
 
 
 a, b 
 
 c", a" 
 
 
 a", b" 
 
 Substituting these values in the first equation, we get 
 
 = 0, 
 
 = 0, 
 
 a 
 
 b\ c 
 
 + b 
 
 0, 
 
 a' 
 
 + c 
 
 a', b' 
 
 
 b", c" 
 
 
 c", 
 
 a" 
 
 
 a", b" 
 
 a 
 
 6', c' 
 
 -b 
 
 a; 
 
 c 
 
 + c 
 
 a', 6' 
 
 
 b", c" 
 
 
 a", 
 
 c" 
 
 
 o", b" 
 
 a, b, c 
 
 a', V, c 
 
 ft T/' // 
 
 d .^ ^ c 
 
 = 0. 
 
 Q. E. D. 
 
 12. In the foregoing proposition we eliminated the two ratios 
 X : y : z &om the three given equations, and found the result in the 
 form of a determinant. 
 
 We might have proceeded otherwise as follows: 
 
 Multiplying the three equations by A, /i, v (at present undeter- 
 mined multipliers) and adding, we get 
 
 (aX + a'/i + a'V) a; + (6X + 6'/a + 6'V) y + (cX + c /* + d'v) a = 0, 
 
 which must be true for all values of X, ju, v. 
 
 Now by Art. 8 we know that if 
 
 
 t 
 
 
 /* 
 
 
 
 V 
 
 v, 
 
 b" 
 
 
 b", 
 
 b 
 
 
 b, V 
 
 0, 
 
 c" 
 
 
 0", 
 
 c 
 
 
 c, c' 
 
XIV 
 
 OP DETEEMINANTS. 
 
 then the coefficients of y and z in the last equation will vanish, and 
 the equation will reduce to 
 
 (a\ + a'li + a"v) a; = 0, 
 
 so that we must have 
 
 <iX + a'fi + a"v = 0, 
 
 or substituting the values of X : /a : v 
 
 v, 
 
 h" 
 
 + a 
 
 h". 
 
 h 
 
 + a" 
 
 h. 
 
 b' 
 
 c', 
 
 c" 
 
 
 c", 
 
 c 
 
 
 c, 
 
 c 
 
 = 0, 
 
 a, a, a — 0, 
 
 b, V, b" 
 
 c, c, c" 
 which is therefore the result of the elimination. 
 
 But this result must be equivalent to the result obtained by the 
 other method. Hence the two equations 
 
 = 
 
 a, b, c 
 
 = and 
 
 a, 
 
 f 
 
 a, 
 
 a' 
 
 a, b', c 
 
 
 h, 
 
 h', 
 
 b' 
 
 a,b , c 
 
 
 0, 
 
 c, 
 
 c' 
 
 must be identical, and therefore their first members must either be 
 identical or difier only by a constant multiplier. But the coefficient 
 of the term ab'c" in each is seen to be + 1. Hence the two deter- 
 minants are identical, or 
 
 a, a, a" 
 
 b, b', b" 
 
 c, c, c" 
 
 a, 
 
 h, 
 
 c 
 
 = 
 
 a', 
 
 v, 
 
 c' 
 
 
 a", 
 
 h", 
 
 c" 
 
 
 13. Cor. A determinant of the third order is not affected by 
 changing the rows into columns and the columns into rows. 
 
 Care must, however, be taken that the first column becomes 
 the first row, the second column the second row, and so on, and 
 vice versd. 
 
OF DETERMINANTS. 
 
 XV 
 
 14. Since the result of the elimination is the same in whatever 
 order the equations be taken, it follows that the resulting equation 
 
 a, h, c =0 
 
 a, o , c 
 
 a", h", c" 
 
 is not altered in whatever order the rows of the determinant be 
 written. 
 
 Hence the determinants 
 
 a, h, c , a, b, c , a , b , e' 
 
 f 1/ r It Tff tf „ 1 
 
 a, , c a , , c a, o, o 
 
 a,6,c a, 0, c a, 0, c 
 
 -.'' /," >." 
 
 a , , c 
 
 a, b', c 
 
 a, h, c 
 
 &c. 
 
 can only differ by some numerical multipliers, and since the coeflBcient 
 of every term in the expansion of each of them is either + 1 or — 1, 
 they can therefore only differ by the algebraical sign of the whole. 
 
 Since 
 
 a, b, c 
 
 a, o, c 
 
 II -III II 
 a , , c 
 
 and 
 
 = a 
 
 V, 
 
 c 
 
 -6 
 
 a', 
 
 c 
 
 + c 
 
 a.', 
 
 V 
 
 
 v, 
 
 c" 
 
 
 a", 
 
 c" 
 
 
 a", 
 
 b" 
 
 a. 
 
 h, 
 
 c 
 
 = a 
 
 0', 
 
 ¥ 
 
 -b 
 
 c', 
 
 a' 
 
 + c 
 
 V, a! 
 
 a", 
 
 v, 
 
 c" 
 
 
 o", 
 
 b" 
 
 
 c", 
 
 a" 
 
 
 h", a" 
 
 a', 
 
 i', 
 
 d 
 
 
 
 
 
 
 
 
 
 it follows from Art. 6 that 
 a, b, c 
 a, b', c 
 a", b", c" 
 
 and 
 
 a, b, e 
 a , , c 
 a, b, c 
 
 are of opposite algebraical sign. Hence the sign of a determinant of 
 the third order is changed by interchanging its last two rows. 
 
 It will be seen on examination that the effect is the same if we 
 change any other two adjacent rows, or two adjacent columns. 
 
XVI 
 
 OF DETEEMINANTS. 
 
 That is, t/ie sign of tlie determinant is changed wlien any two 
 adjacent rows are iiUercIianged, or when any two adjacent columns are 
 interchanged. 
 
 15. But any derangement whatever of the rows or columns may 
 be made by a series of transpositions of adjacent rows or columns. 
 Such a derangement will or will not aflfect the sign of the deter- 
 minant accoi'ding as it requires an odd or an even number of transpo- 
 sitions of adjacent rows or columns to effect it, thus 
 
 
 a, b, c 
 
 = - 
 
 a', b 
 
 '. c' 
 
 = 
 
 a 
 
 ', I 
 
 ', c 
 
 -■=- 
 
 a , , c 
 
 
 
 a, b', c 
 
 
 a, b, c 
 
 
 „" 7." -" 
 
 a , b , c 
 
 
 a, b', c 
 
 
 
 a", b", c" 
 
 
 a", b", c" 
 
 
 a, b, c 
 
 
 a, b, c 
 
 
 
 a", c", 5" 
 a', c', V 
 a, c, 6 
 
 
 c", a", h" 
 <f, a', b' 
 e, a, b 
 
 
 c", b", a" 
 e\ b', a' 
 c, b, a 
 
 = &c. 
 
 an 
 
 Simil 
 y othe 
 
 arly w 
 r dera 
 
 e may 
 Qgeme 
 
 asceri 
 ntof 1 
 
 lain tl 
 ;he CO 
 
 le sign 
 umns 
 
 of the determinant formed by 
 or rows. 
 
 16. If a row or a column of a determina/nt be mrdtiplied fhrougli- 
 out by any nvumher, the vaiiie of the determinant is m,ultiplied by the 
 same number. 
 
 For the determinant may be deranged till the row or column in 
 question becomes the first row. 
 
 Now 
 
 li.a, lib, IXC 
 
 = l^a 
 
 b', c' 
 
 + /u,6 
 
 &, a' 
 
 + lie 
 
 a', b' 
 
 a', b', c' 
 
 
 b", c" 
 
 
 c", a" 
 
 
 a", b" 
 
 a", b", c" 
 
 
 
 
 
 
 
 = IL 
 
 a, b, c 
 a', b', c 
 
 i a", b", c" 
 
 which proves the proposition. 
 
 17. If two rows qftlie determinant be identical, or if two cdwnms 
 be identical, the determinant vanishes. 
 
OF DETERMINANTS. 
 
 XVU 
 
 For the rows and columns may be deranged until the last two 
 rows are identical, and the determinant takes the form 
 
 a, b, c 
 a , b', c' 
 a , b , c 
 
 = ± a 6', c' =F 6 a', c' ± c a, h' 
 b', c' I a', c' a, b' 
 
 = 0, by Art. 4. 
 Therefore, &c. q. e. d. 
 
 18. Cob. If one row be a multiple of another row or one 
 colum.u of another column, the determinant vanishes. For the mul- 
 tiplier may be divided out by Art. 16. 
 
 19. To shew thai 
 
 a + x, b + y, c + z 
 
 a, 0, 
 
 ^'* 7." V 
 
 a , , c 
 
 a, 
 
 h, 
 
 c 
 
 + 
 
 a. 
 
 h'. 
 
 c 
 
 
 a", 
 
 b", 
 
 c" 
 
 
 a;, y, » 
 
 a , V, c 
 a , , c 
 
 Expanding the first determinant, it takes the form 
 
 {a + x) 
 
 b'. c 
 
 
 
 (6 + 
 
 y) 
 
 a, 
 
 d 
 
 l + ( 
 
 i/ + «) 
 
 
 a 
 
 ', v 
 
 
 
 
 b", c" 
 
 
 
 a", 0" I 
 
 a 
 
 ", b" 
 
 
 a\ b', 
 
 c 
 
 -b 
 
 a', c 
 
 + c d, b' 
 
 
 
 I 4"> 
 
 c" 
 
 
 a", c" 
 
 a", b" 
 
 
 
 
 + 
 
 X 
 
 6', c' -y 
 b", c" \ 
 
 a', c 
 a", c" 
 
 + z 
 
 a, b 
 a", h 
 
 
 a, 
 
 b, c 
 
 + 
 
 X, y, z 
 
 
 
 a, 
 
 b', c' 
 
 
 a, b', c 
 
 
 
 a", 
 
 b", c" 
 
 a", b", c" 
 
 
 Tiiercfor 
 
 B, itc. 
 
 
 li. 
 
 E. D. 
 
 
 
 
 
 
 
 
 
 
 
XVlll 
 
 OF DETEEMINANTS. 
 
 Similarly, 
 
 a +a, 
 
 b, c 
 
 = 
 
 a' +a', 
 
 V, c 
 
 
 a" + a", 
 
 b". c" 
 
 
 a, 
 
 i, 
 
 c 
 
 + 
 
 ^. 5, 
 
 C 
 
 a', 
 
 K 
 
 c 
 
 
 a', b'. 
 
 c 
 
 a". 
 
 h", 
 
 c" 
 
 
 a", b", 
 
 c'' 
 
 And so if each element of any column or row be divided into two 
 parts, the original determinant is equal to the sum of the two deter- 
 minants formed by substituting for each divided element first one 
 of its parts and then the other. 
 
 But note that this operation cannot always be performed at 
 once on more than one column or row. 
 
 Conversely, if a series of determinants are identical except as 
 regards one column or one row in each, their sum is equal to the 
 new determinant formed by retaining in their places the rows or 
 columns that are identical, and adding together the corresponding 
 elements of the row or column which differs. 
 
 20. Jf any row of a determinant be increased by multiples of any 
 other rows, or if any column be increased by multiples of any other 
 columns, the value of the determinant is not altered. 
 
 For, by Art. 19, 
 
 a + mb + nc, b, c 
 
 a' +7nb' +nc', V, c 
 
 tt Iff ff Iff fi 
 a +mb +nc , b , c 
 
 a, b, c + mb, b, c + nc, b, c 
 a', b', c mb', V, c' nc', b', c' 
 
 a , , c mo , , e nc , b , c 
 
 a, b, c , by Art. 18, 
 a', b', c' 
 a", b", c" 
 which proves the proposition. 
 
 This theorem is of the greatest use in reducing determinants. 
 
OF DETERMINANTS. 
 
 XIX 
 
 21. // 
 
 and 
 
 then will 
 
 
 X 
 
 h. 
 
 c, d 
 
 v, 
 
 c', d' 
 
 b", 
 
 c", d" 
 
 ax +hy +CZ + du = 0, 
 a'x + h'y + cz + d'u = 0, 
 a"x + b"y + cz + d"u = 0, 
 
 ~y z 
 
 a, c, d 
 
 
 a', c, d' 
 
 
 a", c", d" 
 
 
 a, b, d 
 a', b', d' 
 a", b", d" 
 
 
 — u 
 
 a, 
 
 b. 
 
 < 
 
 b', c' 
 
 a", 
 
 b", c" 
 
 For if we multiply the first equation by 
 
 c, c" 
 d', dr 
 
 , the second by 
 
 - 1 c, e" 
 
 , the third by 
 
 c, c 
 
 d, d" 
 
 
 d, a 
 
 a, a', a" 
 
 x + 
 
 b, b', b" 
 
 y + 
 
 e, c, c" 
 
 
 c, c', c" 
 
 
 d, a; d' 
 
 
 d, d', d' 
 
 1 
 
 
 , and add, we get 
 
 c, e, c 
 
 r It 
 
 c, c , c 
 
 d, d', d" 
 
 z + 
 
 d, d', d!' 
 
 c, c, c'' 
 
 d, d', d' 
 
 or, in virtue of Art. 17, 
 
 a, 
 
 a, 
 
 a 
 
 c, 
 
 c, 
 
 c 
 
 d, 
 
 d', 
 
 d' 
 
 a, 
 
 c, 
 
 t 
 
 a', 
 
 c', 
 
 I 
 
 a" 
 
 c" 
 
 1 < 
 
 x + 
 
 b, b', b" 
 
 c, c, c" 
 
 d, d', d" 
 
 2/=0, 
 
 x + 
 
 b, c, d 
 b', c, d' 
 , c , a 
 
 -y 
 
 2/ = 0, 
 
 6, c, 
 
 d 
 
 
 b\ c'. 
 
 d 
 
 
 6", c", 
 
 d" 
 
 
 a, c, d 
 
 a', c, d' 
 a", c", d" 
 
 M = 0, 
 
 and similarly the other equations may be established. 
 
XX 
 
 22. If 
 
 and 
 tlien will 
 
 OF DETERMINANTS. 
 
 ax +hy + cz = d, 
 a'x + h'y + cz = d', 
 a"x + h"y + cz = d'. 
 
 b, c, 
 
 d 
 
 b', c. 
 
 d- 
 
 b", c", 
 
 d' 
 
 a, b. 
 
 c 
 
 a, V, 
 
 c' 
 
 a", b", 
 
 c 
 
 c. 
 
 a, 
 
 d 
 
 c', 
 
 a'. 
 
 d' 
 
 c. 
 
 a", 
 
 d" 
 
 a, 
 
 h, 
 
 c 
 
 a, 
 
 b', 
 
 e 
 
 a". 
 
 b", 
 
 c" 
 
 a, 
 
 b. 
 
 d 
 
 a. 
 
 b', 
 
 d' 
 
 a", 
 
 b", 
 
 d' 
 
 a, 
 
 b. 
 
 c 
 
 a, 
 
 b', 
 
 c 
 
 a", 
 
 b", 
 
 c" 
 
 y = 
 
 This follows from the last pFoposition by ■writing — 1 for u. 
 
 [It will be observed that these values of x, y, z obtained by- 
 solving the three simultaneous equations might have been written 
 down by the method of cross multiplication in Algebra.] 
 
 § 4. Of Determina/nts of the fourth order. 
 
 23. Def. The symbol 
 
 Oi, b, c, d 
 
 a', b', c', d' 
 
 a", b", c", d" 
 
 a'", h'", c'", d'" 
 
 is used to denote the expression. 
 
 b', 
 
 c', d' 
 
 -b 
 
 a'. 
 
 c', d' 
 
 + c 
 
 a', 
 
 b', 
 
 d' 
 
 -d 
 
 a', b', c' 
 
 b", 
 
 c", d" 
 
 
 a", 
 
 c", d" 
 
 
 a", 
 
 b", 
 
 d" 
 
 
 a", b", c" 
 
 b'", 
 
 c'", d'" 
 
 
 a'", 
 
 c'", d'" 
 
 
 a'", 
 
 b'". 
 
 d"'\ 
 
 
 a'", b'", c'" 
 
 and is called a determinant of tlie fcntrth order. 
 
OF DETERMINANTS. 
 
 XXI 
 
 24. If 
 
 then will 
 
 ax +bi/ +CZ +du = 0, 
 a'x + h'y + c'z + d'u = 0, 
 d'x + h"y + c"z + d'u = 0, 
 a"'x + V"y + d"z + d:"u = 0, 
 
 a, b, c, d 
 
 a!, b\ c, d' 
 
 a", b", c", d" 
 
 a , O y C , Cl 
 
 = 0. 
 
 For the second, third, and fourth equations give 
 -y z 
 
 
 X 
 
 h'. 
 
 d, d' 
 
 b". 
 
 c", d" 
 
 V", 
 
 d", d'" 
 
 a. 
 
 d. 
 
 d- 
 
 a". 
 
 «", 
 
 d" 
 
 a", 
 
 d". 
 
 d'" 
 
 a', b', d' 
 
 
 a", b", d" 
 
 
 a", V", d" 
 
 
 a!, b', d 
 a , o , c 
 a ,0 ,c 
 
 Substituting these values in the first equation, we get 
 
 b', d, d' 
 b", c", d" 
 V", d", d" 
 
 a. 
 
 d, d' 
 
 + c 
 
 a". 
 
 c", d" 
 
 
 a'". 
 
 d", d'" 
 
 
 a', b', d 
 
 rf lit itf 
 
 a , o , d 
 
 a , , d 
 
 -d 
 
 a', b', d 
 
 a, b", c" 
 
 fti lilt III 
 
 a , o , c 
 
 a, b, c, d = 0. Q. E. D. 
 
 a, b', c , d 
 
 If ifr If Jff 
 
 a , o , c , a 
 
 Iff itif Iff Jiff 
 
 a , , c , a 
 
 Precisely as in the case of the determinant of the third order 
 (Art. 13), we may shew that the value of a determinant of the fourth 
 order is not affected by changing the rows into columns and the 
 columns into rows. 
 
 So the results obtained in Arts. 14—20, will be seen to depend 
 upon general principles, and to hold for determinants of the fouith 
 order. 
 
 W. ^ 
 
XXll 
 
 OF DETERMINANTS. 
 
 25. If 
 
 and 
 then toill 
 
 ax +by +CZ +du + ev =0, 
 ax + h'y + c'« + d'u + dv = 0, 
 a"x + b"y + c"z + d"u + e"v = 0, 
 a"'x + b"'y+c"'z+d"'u+ef"v=0, 
 
 I 6, c', d", e" I " 1 a, c, d", e'" \- \ a, b', d", e" | 
 
 V 
 
 
 \ a, b , c , d I 
 where | a, b', c", d'" \ denotes the determinant 
 a, b, c, d 
 a', V, c', d' 
 
 «" 7." -" J" 
 
 a , , c , ct 
 a , , c , a 
 
 For if we multiply the foiir equations respectively by 
 
 c', 
 
 c'. 
 
 e 
 
 J ~~ 
 
 d", 
 
 d", 
 
 d" 
 
 
 e'", 
 
 e'", 
 
 e'" 
 
 
 c. 
 
 c, c, 
 d", d" 
 
 » 
 
 e", 
 
 
 
 c, 
 
 c, c 
 
 i 
 
 d', 
 
 d', d' 
 
 
 e", 
 
 e", e" 
 
 
 c, 
 
 c, c 
 
 d\ 
 
 d', d' 
 
 e", 
 
 
 and add, we obtain 
 
 a, a, a , a 
 
 c, c, c , c 
 
 d, d', d", d" 
 
 e, e, e", e" 
 
 X + 
 
 whence 
 
 
 X 
 
 
 b, 
 
 c, d, 
 
 e 
 
 V, 
 
 c, d', 
 
 e 
 
 h". 
 
 c", d", 
 
 e 
 
 V", 
 
 c", d'", 
 
 e"' 
 
 b, V, V, b'" 
 
 I It III 
 
 Cj C ) C J c 
 
 d, d', d", d!" 
 
 e, e, e", e'" 
 
 -y 
 
 y = o. 
 
 a. 
 
 c, 
 
 d, e 
 
 a; 
 
 c. 
 
 dl, e 
 
 a", 
 
 c"y 
 
 d", e" 
 
 a-, 
 
 c'", 
 
 d , e 
 
 and similarly the other equations may be established. 
 
OF DETERMINANTS. 
 
 § 5. Of Determinants of the «."" order. 
 
 26. The student ■will have observed that the reasoning of the 
 last two sections is perfectly general. He will have recognised the 
 law by which a determinant of any order is defined with reference to 
 those of the next lower order, and he will have perceived that the 
 proofs of properties of determinants of the 3rd order given at length in 
 § 3 will apply mviaJtis mtUandis to establish corresponding properties 
 for determinants of any order whatever, if they can be assumed to 
 hold for the next lower order. It follows, therefore, by the principle 
 of mathematical induction, that all those properties may be attributed 
 to any determinant whatever. 
 
 27. Def. If we strike out one of the columns and one of the 
 rows of a determinant of the n"" order, we shall obtain a new deter- 
 minant of the (n — 1)"" order, which is called the minor of the original 
 determinant with respect to that element which was common to the 
 column and row. 
 
 Thus the minor of the determinant 
 a, b, c, d 
 a, b', c, d' ... 
 a", b", c", d" ... 
 a , , c , a 
 ■ : : : &c. 
 
 with respect to the element c" is the determinant 
 \ a, b, d .. 
 a, b', d' .. 
 
 fff -I nr T// 
 
 a , o , a 
 
 ■ &c. 
 
 28. Dep. The element which is common to the p"" column 
 and g"" row of a determinant is said to occupy a positive or negative 
 place according as p + q is even or odd. 
 
 c2 
 
OF DETERMINANTS. 
 
 29. The coefficient of any dement of a determinant is tJie minor 
 with respect to that element affected with the sign + or - according as 
 the element occupies a positive or negative place. 
 
 Let the determinant be 
 
 a, b, 
 
 a'l b', 
 
 &c. 
 
 and let the element x occupy the g'* place in the p"" column. 
 
 By making 5' - 1 transpositions of adjacent rows and p-\ trans- 
 positions of adjacent columns, the determinant may be written 
 
 (-1)"+''-'', X 
 
 
 s 
 
 a, b, ... 
 
 
 1 ; a\ b', ... 
 
 
 1 ; ; ; -tc. i 
 
 ^ wc see that the coefficient of a- is 
 
 ± i a, b, ... 
 
 
 a, b', ... 
 
 
 ; ; <fcc. 
 
 
 the sign being positive or negative according as p + y is even or odd, 
 which proves the propo.sition. 
 
 § 6. Of Determinants of unequal columTis amd rotes. 
 
 30. Suppose we have to eliminate n unknown quantities from a 
 greater number m of simple equations. As we shewed in Art. 1, 
 we shall obtain m — n equations of relation amongst the coeHicients. 
 
 One method of writing down these m — n equations would be to 
 take the first n of the given equations, and associate with them in 
 order each of the remaining m — n equations. 
 
 From each of the T)i — n groups of w + 1 equations thus form- 
 ed, wo might eliminate the n unknown quantities. Expressing 
 
OF DETERMINANTS. 
 
 XXV 
 
 the resvilt by means of a determinant of the («.+ !)"' order. We 
 should thus have m — n determinants of the (re+l)"' order each 
 equated to zero, constituting the m—n equations sought. 
 
 But it is plain we might have t&ken any m — n combinations 
 whatever of w + 1 equations that could have been formed out of the 
 m equations, provided we took care to introduce all the original 
 equations. Thus there will always be a variety of forms in which the 
 result of such elimination may be expressed. 
 
 31. Suppose, for instance, that we have to eliminate the two 
 ratios x : y : z from the five equations 
 
 a^x + h^y + c,« = 0, 
 a^x + h^y + c^z = 0, 
 
 aiX + 6jy + c^ -- 0, 
 a^x + h^y + Cj« = 0. 
 
 As the result of the elimination we shall have the three equations 
 
 
 = 0, 
 
 6,, ^2, *4. 
 c,, Cj. C4, 
 
 = 0, 
 
 *,' ^9' *5' 
 
 "1' '^z' "5' 
 
 = 0, 
 
 or for any one of these equations we may substitute any other equa- 
 tion obtained by elimination from a different set of three equations, 
 such as 
 
 ttj, ^3, «! =0, or 
 
 
 But it is convenient to express the result of this elimination 
 briefly by the form 
 
 »,i «a. »a' "*> "'!. 
 \, h, K *" *5 
 
 C., Cj, C3, C4, C5 
 
 I'll •'SI 
 
 = 0, 
 
XXVI 
 
 OF DETERMINANTS. 
 
 where the determinant of five columns and three rows indicates that 
 we may select any three of the five columns to form a square determinant 
 and equate it to zero, and tlie triple vertical lines indicate that three 
 such equations may be independently formed. 
 
 32. The example given in the last paragraph wUI sufiSce to sug- 
 gest to the reader the interpretation of any unequal determinant. 
 In most general terms the definition will stand as follows : 
 
 The compound symbol 
 
 6.. K. K - K 
 
 *1) "^2) S ■•• % 
 
 where the number of the quantities a, b, c.k is n (less than m) and 
 the number of vertical lines bounding the determinant is r, is to be 
 understood as eocpressing a system of r independent equations obtained 
 by equating to zero r sevend determinants each form/ed by taking n 
 of the rows of the given unequcd determinant. 
 
 It will be observed, that if the system expresses the result of the 
 elimination of n quantities out of «i independent equations, we must 
 have r = TO — Ji. The notation is, however, found convenient in cases 
 when the original equations, although m in number, are only equiva- 
 lent to some lesser number m' of independent equations. In such a 
 case we shall have r = m —n. 
 
 33. As an example of the case last considered, suppose we have 
 to eliminate the two ratios x : y : z from the four equations 
 
 (a -a )x + (b -b' )y+(c -c )z = Q, 
 {a -a")x + {b' -b")y + {c' ~c")z = 0, 
 (o" -a")x+ (6" - V") y + (c" - c"') « = 0, 
 (a" —a)x +(6"' - 6 )y + (c"' — c ) « = 0. 
 
 These are equivalent to only three independent equations, since 
 any one of them may be obtained from the others by simple addition. 
 
OF DETEEMINANTS. 
 
 XXVll 
 
 There will therefore be only one resulting equation, which may be 
 obtained by eliminating from amy three oi the given equations. If, 
 however, it be desired to have a result recognising symmetrically all 
 the four equations, we may write it 
 
 a — a, a —a , a —a , a —a 
 
 b-b; b'-b", h"-b"', h'"-b 
 
 t I 11 It til m 
 
 e — c, c —e , c — c , c — c 
 
 = 0. 
 
 § 7. Hxamples. 
 Example A. To evaluate the determinarU 
 
 a", a, 1 
 
 ^, /3, 1 
 
 y> 7. 1 I 
 
 Subtracting the third row from each of the others, the determi- 
 nant becomes 
 
 a'-/, a -7, 
 
 P'-y', P-y, 
 
 a'-/, a-y I = (a-y)(/?-y) c + y, 1 
 
 iS'-/. /3-v I P + y, 1 
 
 = (a-v)03-y)(a-)8). 
 
 Example A'. By a similar method we may shew that 
 a\ a. 1 =(a-y)(|3-7)(a-/3)(a + ^ + y). 
 
 ^, A 1 
 /. 7> 1 
 
 Example B. To shew that 
 
 x + a, x + b, x + c 
 
 y + a, y + b, y + c 
 
 z + a, z + b, « + c 
 
 = 0, 
 
XXVIU 
 
 OP DETERMINANTS. 
 
 If o = 6 or 6 = c or c = a, two columns become identical and the 
 determinant vanishes. Therefore if the determinant be not identi- 
 cally zero, a — b, b — c, c—a are factors. Similarly, ifa! = yor y = s> 
 or z = x, two rows become identical, and so y — z, z — x, x—y are 
 factors. 
 
 But the determinant is only of the 3rd order and cannot have 
 more than three factors. Hence it must vanish identically. Q. e. d. 
 
 Example C. To evaltiate the determinant 
 
 tCm X "" Cj tXj ~~ O 
 
 y - c. y, y-a 
 
 z — b,z~a, z 
 Subtracting the first column from each of the others, -we get 
 X, —c, - 6 
 y — c, c, c — a 
 z — b, h — a, b 
 whence we see that the coefl5cient of x is 
 
 c, c-a \, oT ab + ac — a'. 
 
 b — a, b I 
 
 By symmetry, the coefficients of y and z must be respectively 
 be + ah—b' and ca+bc— c'. 
 
 And the terms without x, y, z, are 
 
 0, - c, - 6 I , or - 2dbc. 
 — c, 0, - a 
 -b, -a, 
 Hence the determinant may be written 
 
 ax{b + c-d:) + by{c + a-b) + cz{a + b-c)- 2aAo. 
 
 Example D. To shew that 
 
 P+y, y+a,a + P=2 a, yS, y 
 ^ + /, y + a', a' + P' a', P', y' 
 
 r + y", y'+a", a" + /3" a", /3", / 
 
OF DETEEMINAHTS. 
 
 XXIX 
 
 The first determinant (by Art. 19) ia equal to 
 
 A Y + n, » + /3 
 ^', y + a', a + p 
 P", y + a, a + j8 
 
 or, in virtue of Art. 20, 
 
 P, y + o, o 
 P, -y + a, u. 
 
 y, Y + a, o + /3 
 y', y + a', a' + )3' 
 
 y", y" + a", a" + /3" 
 
 y, a, a + /J 
 
 y, a', a' + P' 
 y", a", a' + P" 
 
 or again, applying the same principle, 
 
 ft y, °- 
 /3', y, a' 
 
 which by Art. 15, is equal to 
 
 y. ". P 
 i, a'. /8' 
 y", a", /3" 
 
 Therefore, &c. Q. K D. 
 Example E. Let 
 L = 
 
 L = 
 
 «. A y 
 a', /3'. y' 
 
 a", /J", y 
 
 «»., 
 
 •^2 
 
 "»3» 
 
 '^a 
 
 ^3. 
 
 '^a 
 
 m„ 
 
 «l 
 
 7W,, 
 
 »». 
 
 »»., 
 
 1, 
 
 Jf.= 
 
 ".. 
 
 s 
 
 . ^.= 
 
 ^„ I». 
 
 
 n,. 
 
 3 
 
 
 ^3> '"'3 
 
 J^« = 
 
 «a. 
 
 'a 
 
 > ^,= 
 
 ^3. ^> 
 
 
 «>. 
 
 1 
 
 
 h> '"\ 
 
 if 3 = 
 
 »».. 
 
 1 
 
 . ^3 = 
 
 l„ m. 
 
 
 w.' 
 
 3 
 
 
 K, m^ 
 
 L = 
 
 then ■<ve have 
 
 Z,Z, + w,if, + n^N^ = IJj^ + m,M^ + m^.V^ = l^L^ + ni3il/3 + »3iV 
 
 ^3. '"a' »»3 
 
 ;,Z, + l^L,_ + ;3£3 = m, J/, + w//, + «i3il/3 = n,N^ + m^iV^ + n.^K^. 
 
So also 
 
 OP DETERMINANTS. 
 
 l^L^ + m^M^ + n^N^ = 
 
 And so Z3Z, + m^M^ + n^N^ = 0, 
 
 and four other like relations. 
 And similarly, 
 
 and five other like relations. 
 Example F. To shew that 
 
 This may either be proved by multiplying out, or as follows. 
 
 Suppose the three equations 
 
 l^x + mj/ + n^z = (1), 
 
 l^ + m^ + ii^z=0 (2), 
 
 l^x + m^i/ + n,^ = (3), 
 
 coexist, 
 
 (3) and (1) by the elimination of x gives us 
 
 J/,y+iygs = (4), 
 
 Similarly (1) and (2) gives us 
 
 i + ]f^=0 
 
 (5). 
 
 The condition that (4) and (5) coexist must be identical 'with the 
 condition of coexistence of (1), (2), (3), that is, the equation 
 
 must be identical with the equation 
 
 l^, »i,, re, =0. 
 
OF DETEEMINANTS. 
 
 TTWI 
 
 And therefore these two determinants can only differ by a con- 
 stant multiplier, which is determined by comparing the coefficient of 
 n^ in each, and found to be Z,, so that 
 
 K,^, =K 
 
 l„ Wl„ TO, 
 
 M,,N^ 
 
 h> »»«. »»3 
 
 
 h' "*3' '"'a 
 
 Q. E.D. 
 
 3)xoiPLE G. To shew thai 
 
 
 
 
 A. 
 
 jjy 
 
 .. ^. 
 
 = { 
 
 «,. 
 
 «»,, «. 
 
 
 
 X„ ^,. i^. 
 
 
 ^». »»«. »2 
 
 
 
 ^., ^s. ^, 
 
 
 ^3. "»3. «3 
 
 We have 
 
 
 I, 
 
 A. 
 
 J/„ A\ 
 
 = 
 
 «.i., ^., ^. 
 
 
 
 A, 
 
 M^,W, 
 
 
 «,-£., ^„ ^. 
 
 
 
 4. 
 
 ^a. 
 
 ^, 
 
 
 hL,, 
 
 ^.. 
 
 •^3 
 
 
 r- 
 
 
 
 1, M^, F, ,byEx. E, 
 0, M^, F, 
 0, M^, F^ 
 
 ^3' ^»' "-3 
 
 ^3.^3 
 
 = iA 
 
 ^3' ™3' »*3 
 
 }', by the last example. 
 
xxxu 
 
 OP DI 
 
 JTEK 
 
 MINANT 
 
 J. 
 
 therefore 
 
 
 
 
 L„ M^, N, 
 
 = { 
 
 ^,, «»,, 
 
 »». 
 
 
 L„ K> ^. 
 
 
 K, m„ 
 
 «. 
 
 
 h, ^3. ^. 
 
 
 h' "^ai 
 
 «a 
 
 Q. E. D. 
 
 Example H. To eliminate xfroni the two eqiMlions 
 
 oaf + bx+ c = (1), 
 
 a'x' + b'x + c'=0 (2). 
 
 Multiplying each equation by x thi'oughout, we get 
 
 aaf+bx' + cx = (3), 
 
 a'af + h'x' + cx=0 (4), 
 
 and eliminating a^, x', x from the four equations (1), (2),, (3), (4), 
 we have 
 
 
 
 0, a, b, c 
 
 = 0, 
 
 
 
 0, a', V, c' 
 
 
 
 
 a, b, c, 
 
 
 
 
 a', b', c', 
 
 
 a, 
 
 c 
 
 '= a, b 
 
 b, 
 
 a'. 
 
 c' 
 
 a', V 
 
 b', c' 
 
 Ex. J. If 
 
 uxx' + vyif + wzz' + M'(y«' + y'z) + D'(«a!' + is'a;) + io'(a^' + a;'y) 
 be zero for all values of x, y, z, t}ten. will 
 
 uvw — uu" — w" — tmo" + 2u'v'w' = 0. 
 
 For since the given expression vanishes for all values of x, y, z, 
 the coefficients of x, y, z must severally vanish. Therefore 
 
 vae' + w'y' + vV = 0, 
 loV + mf + u'z' = 0, 
 t'V + ii'y' + wz' - 0, 
 
OF DETERMINANTS. 
 
 and eliminating x' : 1/ : z', we have 
 
 M, w', v' =0, 
 to', V, u' 
 v', u', w 
 
 ww" + 2u'v'w' = 0. 
 
 XXXHl 
 
 or 
 
 UVW — MM 
 
 ■ w 
 
 Q. E. D. 
 
 Example K. To expand the expression 
 
 ax + ly, dx + w'y, h'x + miy 
 c"x + n"y, bx + my, a'x + l'y 
 h"x + m!'y, a"x + l"y, ex + ny 
 
 according to powers of x and y. 
 
 Putting y = 0, we obtain the term involving a;', viz. 
 
 ' a, c', b' 
 
 i 
 
 c", b, a! 
 \ b", a", c 
 
 So putting aj = 0, we obtain the term 
 I, n', in' y". 
 n", m, V 
 m", I", n 
 
 Suppose the y, y, y in the three columns distinguished by- 
 suffixes, so that the determinant becomes 
 
 ax +ly^, c'x^-n'y^, b'x + m'y^ =0. 
 c"x + n"y^ , bx + my^ , a'x + l'y.^ 
 b"x+'m,"y^, a"x + l"y^, ex +ny^ 
 
 Putting y^ = and y^ = 0, we obtain for the term involving ^y., 
 I, c', V \x\. 
 n", b, a! 
 m", a", c ' 
 
XSXIV 
 
 OF DETEEMINANTS. 
 
 Similarly, we find the terms 
 
 a, n', V afy^ and 
 c",^ m,, a' 
 
 b", I", c 
 
 a, </, m' 
 o", h, V 
 I", a", n 
 
 a^^a- 
 
 Hence the whole coefficient of a?y is 
 
 I, 
 
 if, h' 
 
 + 
 
 n". 
 
 b, a' 
 
 
 m", 
 
 a", c 
 
 
 a, 
 
 n', 
 
 b' 
 
 + 
 
 c", 
 
 m, 
 
 a' 
 
 
 b". 
 
 i", 
 
 c 
 
 
 a, c, m 
 c", b, I' 
 h", a", n 
 
 Similarly the coefficient of xy' is 
 
 a, n', m! + I, c', m' 
 
 c", m, I' n", b, V 
 
 b", I", n m", a", n 
 
 Hence we arrive at the result, that 
 
 ax + ly, c'x + n'y, b'x + m'l 
 c"x + n"y, bx + my, a'x + I'y 
 b"x + m"y, a"x+l"y, ex +ny 
 
 = a, c', b' 
 c", b a' 
 
 I, n', b' 
 n", m, a' 
 m", I", c 
 
 + { 
 
 + { 
 
 I, <f, 
 n", b 
 
 m , a , c 
 
 a, n', Ml' 
 c", m, V 
 b", I", n 
 
 the expansion required. 
 
 
 o , a , c 
 a, n', V 
 o", m, a' 
 
 b", I", 
 
 + 1 I, c', m' 
 n", b, I' 
 m", a", n 
 
 I, n' m' 
 n", m, I' 
 m", I", n 
 
 a, c, m 
 
 c", b, V 
 
 b", a" n 
 
 I, n', b' 
 
 n", m, a' 
 
 m", I", c 
 
 ]x'y 
 
 W 
 
 f 
 
OF DETEBMINASTS. 
 
 XXXV 
 
 Example L. As a pa/rtimUvr case of the last theorem consider the 
 determinant 
 
 U + KO?, It! + Koh, V' + KOC 
 
 w' + Kob, V + Kb', v! + k6c 
 tl + KOC, u' + k6c, to + KC' 
 
 Expanding in powers of k, we obtain 
 M, w', »' 
 
 o', «/, v' + M, 06, tj' + M, m/, ac 
 06. V, M It/, 6', m' to', V, ic 
 
 ae, u', w V, he, w v, u', c' 
 
 + k{ 
 
 the coefiScients of k' and k' vanishing since each determinant therein 
 contains at least two identical columns. 
 
 But further, the coefl&cient of k may be written 
 
 0, a, b, c 
 
 a, u, to', v' 
 
 h, to', V, vl 
 
 e, v', u', to 
 
 Hence the whole determinant becomes 
 
 1, 0, 0, 
 
 a, u, w, 1/ 
 
 0, to', V, u' 
 
 c, t/, u', w 
 
 0, a, b c 
 
 a, u, to', v' 
 
 b, to', V, v! 
 
 c, v', u', w 
 
 — , a, b, c 
 
 K 
 
 a, u, vJ v' 
 
 h, to', V, u' 
 
 c, v', u', w 
 
XXXVl OP DETERMINANTS. 
 
 Example M. To eliminate h and Tcfrom the equations 
 u=ha^+ku'', %u' =h{a'-h'-c') + 2kvw, 
 v=hb' + kv', 2i/ = h (b' -c'-a') + 2kiou, 
 w = he' + kw', 2m/ = a (c" - a' - 6') + Ikuv. 
 
 where 
 
 M = M + w' + I//, V = ly ■VvJ ■¥ u' , w = w + u' + v'. 
 
 The resulting equations will be all the independent equations 
 included in the system 
 
 a', V, c\ a'-b'-c', b'-c'-a", c'-a'-b' =0. 
 u', r', w% 2vw, 2wu, 2uv 
 
 u, V, w, 2u', 2v', 2w' 
 
 But the first column increased by half the sum of the fifth and 
 sixth gives us 
 
 
 
 {u+v + w)u 
 u 
 or, dividing by u throughout, 
 
 
 
 which, being perfectly symmetrical, might have been equally obtained 
 by combining the second, fourth and sixth, or the third, fourth and 
 fifth columns. This shews that the original equations were not 
 independent, but equivalent to only four equations and the result 
 consists therefore of the two equations 
 
 o?, b'. c', 
 
 
 
 u, v', w', u + v + w 
 
 U, V, w, 
 
 = 0. 
 
 [Exercises on Determiaants will be found on pages 455, 456.] 
 
ERRATA. 
 
 r,, line i from bottom, for OY, OX read CY, CX. 
 15, „ 9 „ ioT l^lm read Jc^lmn. 
 
 hi, „ i ., for APB read APC. 
 
 24, „ 12 „ for as, read in. 
 
 48, ,, 3 ,. for 2 J» 003 C read QlnicosC. 
 
 58, „ 8 „ for 2bcrp cos B read 2ca?'^ cos S. 
 
 73, , 7 , /or a rawZ a. 
 
 78, ,, 3 ,, for am A An B am C read 2 sm A &'m B sin i 
 
 95, „ 6 „ for differciice read reference. 
 
 134, Art. 119. QOP is (three times) misprinted for AOQ. 
 0,00, is (twice) misprinted for J OP. 
 
 y 
 
 202, line 7, for — read ^ , and read equations (1), (2), (3) as 
 "Va Ha 
 
 a _|3 
 
 _ _ 7 
 
 '^s.S, 13 A ft7i 
 
 73"2 ^371 7s73 
 
 ° _ P _ 7 
 1301 Oift 7ia3 
 
 So the first determinant ought to be 
 
 follows 
 
 .(i 
 
 7a=ai ^a72j 
 
 7273 
 7ia3 
 
 = 0, 
 
 Page 281, line 10, for ?^ read 
 
 2AII 
 
TRILINEAR COORDINATES 
 
 AND OTHER METHODS OF 
 
 MODERN ANALYTICAL GEOMETRY. 
 
 CHAPTER I. 
 
 OF PERPENDICULAR COORDINATES EEFEUUED TO TWO AXES. 
 
 Before proceeding to explain the method of trilinear coordi- 
 nates, we will introduce a system which may be regarded as a 
 connecting link between the usual Cartesian system, and the 
 trilinear system: — it is the method of perpendicular coordinates 
 referred to two oblique axes or lines of reference. 
 
 1. In the ordinary system of oblique coordinates, the position 
 of any point P with reference to a pair of axes CX, CY, is de- 
 termined by the lengths of two lines PM, PN measured parallel 
 to either axis to meet the other : that is, if x, y be the oblique 
 coordinates of P referred to OX, CY, tlien 
 
 x = PM, and y^PX. 
 
 It is obvious that the position of the point would be equally 
 well determined, if it were agreed to take as coordinates the 
 perpendicular distances of the points from each axis, instead of 
 measuring the distance from each axis parallel to the other. 
 Thus, if we let fall the perpendiculars PM' and PN' on CB and 
 CA, we observe that PM' and PN' might be used, as lawfully 
 
 w. 1 
 
2 OF PERPENDICULAR COORDINATES 
 
 as PM and PN, to detei-mine the position of P, provided it be 
 specified beforehand which system of measurement is intended. 
 We shall speak of PJ/and PN s.s the oblique coordinates, and 
 PJ/ and PN' as the perpendicular coordinates of the point P 
 referred to the same axes CX and GY. 
 
 We shall use a. and /S to denote perpendicular coordinates, 
 reserving x and y to denote, as usual, the oblique coordinates. 
 Thus for the point P 
 
 x = P2I, y = PN: a = PM', /3 = PiV. 
 
 When the axes are rectangular these two systems of mea- 
 surement will indeed coincide, but in the case we have intro- 
 duced of oblique axes they will be distinct. 
 
 Fig. 1. 
 
 The same convention as to the algebraical signs of the coor- 
 dinates will hold equally in the perpendicular as in the oblique 
 coordinates. Thus we shall consider as positive the distances 
 from CX of all points lying on the same side with Y, and con- 
 sequently the distances of all points on the opposite side will be 
 negative. So also the positive side of CF will be that on 
 which X lies, and the negative side the side remote from X 
 
REFERRED TO TWO AXES. 3 
 
 2. If the angle XCYhe denoted by C, then we have 
 
 PM' PN' 
 
 -p^ = sm (7 and -^ = sin C. 
 
 Hence if a, /3 be the perpendicular coordinates of any point 
 whose oblique coordinates are x, y, we shall have 
 
 a = a; sin (7 and ^ = y sin C; 
 
 or x= a cosec C and y = /3 cosec C. 
 
 Consequently, if we have any relation holding good between 
 the oblique coordinates of all points on a locus, we can, by the 
 substitution of 
 
 x = a cosec C, y = /3 cosec C, 
 
 obtain a relation holding good between the perpendicular coor- 
 dinates of the same locus. In other words, we may transform 
 by this substitution the oblique equation of any locus into an 
 equation in perpendicular coordinates representing the same 
 locus. 
 
 For example, the equation in oblique coordinates to the 
 straight line AB cutting off intercepts CA = b and CB= a from 
 the axes is known to be 
 
 X y 
 
 r+ =1- 
 o a 
 
 Hence the equation to the same line in perpendicular coordinates 
 will be 
 
 a cosec G j8 cosec C _ 
 b ■•" a ' 
 
 or aa + b0 = ah sin C; 
 and in a similar way any other equation might be transformed. 
 
 3. Or, instead of taking an equation, we might by the same 
 substitution transform any function whatever of the oblique coor- 
 dinates into an equivalent function of tlie perpendicular coordi- 
 
 1—2 
 
4 OP PEEPENDICULAE COORDINATES 
 
 nates. For example, writing the equation to the same straight 
 line AB in the form 
 
 db — ax — hy = 0, 
 
 we can at once write down the expression for the perpendicular 
 distance of any point [x, y) from it, viz. 
 
 ab — ax — hy . « 
 
 + -j — -^ sin G. 
 
 ~ Ja^ + b' - 2a6 cos C 
 
 Hence if a', /3' be the perpendicular coordinates of the same 
 point, we shall have as the expression for the distance 
 
 ah sin C—aa'— hB 
 
 -I — ^ . • 
 
 ~ 'Ja'+F-2abcosO' 
 
 or if c denote the distance AB, and A the area of the triangle 
 ABO so that 
 
 c' = a' + P-2ab COS C, 
 
 and 2 A = ah sin C, 
 
 then the expression (for the perpendicular distance from AB of 
 the point whose coordinates are a and /S') becomes 
 
 , 2A-aa'-5/3' 
 ~ c 
 
 This is given here merely as an example of transformation 
 of coordinates from the one system to the other, but the result 
 is one which will be seen hereafter to have an important bearing 
 on trilinear coordinates. 
 
 4. The student will do well to examine at this stage of the 
 subject the interpretation of some of the simpler equations con- 
 necting a and yS. 
 
 (1) Consider the equation 
 a = 0. 
 
 It is evidently satisfied by all points on the line CY and by no 
 other : it is therefore the equation to this axis. 
 
REFERRED TO TWO AXES. 5 
 
 (2) Consider the equation 
 
 a = d, 
 
 where <? is a constant. It is equally obvious that this is satis- 
 fied at any point ou a line parallel to CB at a distance d from it 
 on the side towards X. Similarly the equation 
 
 a = —d 
 
 is satisfied at any point on a parallel line at an equal distance 
 on the other side of CY. 
 
 (3) Consider the equation 
 
 a = /3. 
 
 The equation speaks to us of points whose perpendicular 
 distance from CX is equal to the perpendicular distance from 
 CY, and of the same algebraical sign. The locus of such points 
 is the interior bisector of the angle XCY, which is therefore the 
 locus of the equation. 
 
 (4) Consider the equation 
 
 a = -/S. 
 
 This speaks of points whose perpendicular distances from 
 the axes are equal, but of opposite sign. These points will 
 be seen to lie on the exterior bisector of the angle XCY, which 
 will therefore be the locus of the equation. 
 
 (5) Consider the equation 
 
 Suppose P to be a point on this line, join PO and draw PM', 
 PN" perpendiculars on Y, OX, then (by the equation) 
 
 PM'=:m. PN', 
 
 and therefore -dtt = m . ^jp, , 
 
 PM' _ PN' 
 PC~^' PC 
 
 sin PCr=m. sin PCX, 
 
6 OF PERPENDICULAR COORDINATES 
 
 or (in words), Plies on a straight line dividing the angle XOY 
 into two parts POY, PCX such that their sines are in the 
 ratio OT : 1. 
 
 This straight line will therefore be the locus of the equation. 
 
 5. To find the area of a triangle the perpendicular coordi- 
 nates of whose angular points are given with respect to a pair of 
 oblique axes. 
 
 Fig- 2. 
 
 ^ W i* Kf Ax 
 
 Let PQB be the triangle and (a,, /8J, (a„ /3^, (a,, /S,) the 
 perpendicular coordinates of the angular points P, Q, R referred 
 to the oblique axes CX, CY. 
 
 Parallel to GY draw PP', QQ', BR' to meet CX in F 
 
 q; r'. 
 
 Then A P§5 = trapezium P§ ^'i^H- trapezium PPJfJ'P' 
 
 — trapezium QRR'Q'. 
 
 But the trapezium PQQ'P' stands on a base PQ' equal to 
 («i - "s) cosec C and it has a mean altitude equal to J (/?, + ^J. 
 
REFERRED TO TWO AXES. 
 
 Hence its area = - cosec C {^^ + /S^) (a, - a ). 
 
 Similarly, area PBB'F = i cosec C (fi, + y3J (a, - a,), 
 
 and area QBR Q' = - cosec C (/3, + ^^) {o, - a^) ; 
 
 therefore the area of tlie triangle PQB 
 
 = ^cosec (7|(^,+ ^,)(a,-a,) + (^3+/3J (a3-«J-(/S,+;8,) (a^-a, 
 
 = I cosec C |a, (/3, - ^J + a, (^^ - /3J + 83 (/3. - /3,)| , 
 or with determinant notation 
 = - cosec G 
 
 «1. 
 
 «a' 
 
 «» 
 
 /3„ 
 
 ^„ 
 
 /3. 
 
 1, 
 
 1, 
 
 1 
 
 Exercises on Chapter I. 
 
 (1) If (a, ;8) be the perpendicular coordinates of the point 
 whose oblique coordinates (referred to the same axes) are {x, y), 
 and (a, /3') the perpendicular coordinates of the point whose 
 oblique coordinates are {x, y), shew that (a + kcl_, /3 + k/S') will 
 be the perpendicular coordinates of the point whose oblique 
 coordinates are {x + Kx',y + Ky). 
 
 (2) Shew that the points whose perpendicular coordinates 
 are (a, yS), (a, ^'), (a -a', ^-/8') and (a + a',/8 + ^') lie all 
 on one straight line, provided a/8' = a'/8. 
 
8 EXERCISES ON CHAPTER I. 
 
 (3) Find the equation in perpendicular coordinates to the 
 straight line drawn from the origin at right angles to the 
 straight line CX. 
 
 (4) Write down the equation to the other diagonal of the 
 parallelogram two of whose sides are the axes, and one of whose 
 diagonals has the equation 
 
 aa + 5/S = ah sin C. 
 
 "What is the area of this parallelogram? 
 
 (5) The distance between two points in terms of their 
 oblique coordinates (x, y) and {x, y') is given by the formula 
 
 p'={x-xy+(jf-y'Y + 2{x-x'){y-y')cosC; 
 
 hence write down the corresponding formula for perpendicular 
 coordinates. 
 
 (6) Find the area of the triangle one of whose angles is at 
 the origin, and the other two at the points whose perpendicular 
 coordinates are (a, yS) and (a', /S'). 
 
 (7) Find the area of a triangle whose base is of length d in 
 the axis CX, and whose vertex is at the point whose perpen- 
 dicular coordinates are (a, /3). 
 
 (8) Find the perpendicular coordinates of the point bisect- 
 ing the straight line joining the points whose coordinates are 
 
 (a. ^) and (a, ^). 
 
 (9) Find the equation in perpendicular coordinates to the 
 straight line drawn from the origin, at right angles to the 
 straight line whose equation is 
 
 aa + 6y3 = oi sin 0. 
 
CHAPTER II. 
 
 TEILINEAK COOEDINATES. — THE POINT. 
 
 6. We will renew our construction before proceeding to 
 the next step in the development of the system of trilinear 
 coordinates. 
 
 Let BC, GA, AB be any three fixed straight lines forming 
 a triangle, and let P denote some point in the plane of the 
 triangle. 
 
 We have seen that if the perpendicular distances of P from 
 any two fixed straight lines {GB and GA for instance) be given 
 the point P is determinate, and that these distances may be 
 regarded as the coordinates of P. 
 
10 TEILINEAR COOEDINATES. 
 
 Now let a, /S, 7 denote perpendicular distances of P from the 
 three fixed straight lines BG, CA, AB. Then, as we have seen, 
 any two of tliese (e.g. a, j8) may be regarded as the coordinates 
 of P with respect to the corresponding axes [BC, CA), and the 
 remaining perpendicular (7) may he expressed in terms of these 
 two (a and /S), and known constants, by the method of the ex- 
 ample, Art. 3 (or more simply, as we shall shew presently). 
 
 But there are advantages, as the sequel will shew, in regard- 
 ing all the three perpendiculars as coordinates of P, and thus 
 expressing P's position at once with respect to the three fixed 
 axes, or lines of reference (as it is more usual to call them). 
 
 We shall regard as positive the distances from any line of 
 reference to points on the same side with the intersection of the 
 other two lines of reference, and consequently the distances of 
 points on the other side will be negative ; thus 
 
 A lies on the positive side of BG, 
 
 B GA, 
 
 C AB. 
 
 Thus all ]ioint3 within the triangle formed by the lines of refer- 
 ence (which we shall briefly call the triangle of reference) have 
 all their coordinates positive, and points without the triangle of 
 reference have either one or two coordinates negative. 
 
 It will be observed that it is impossible to find a point on 
 the negative side of all the lines of reference, that is, no point 
 has all its three coordinates negative. 
 
 "O" 
 
 7. We have observed that when two of the trilinear coor- 
 dinates of a point are given the third may be calculated. We 
 proceed now to investigate a simple equation, which we shall 
 find connects these three coordinates. 
 
 Let P be any point whose coordinates are a, /S, 7 and join 
 PA, PB, PC, and draw PB, PE, PF perpendiculars on BG, 
 CA, AB respectively. 
 
THE POINT. 
 
 11 
 
 First, suppose P within the triangle ABC as in fig. 3 ; 
 then the triangles 
 
 PBG+PGA + PAB = whole triangle ABC. 
 
 But BO.PD or aa is the double of the area of the triangle 
 PBG: so Jj8 and 07 are the doubles of the triangles PGA and 
 PAB; 
 
 .-. aa + 5/3 + cy = 2A, 
 
 where A denotes the area of the triangle of reference. 
 
 Secondly, suppose P without the triangle. Let it be on the 
 negative side of BG and on the positive sides of GA, AB, as in 
 figure 4. 
 
 Fig. 4. 
 
 In this case the triangles 
 
 PGA + PAB- PBG = whole triangle ABC. 
 
 But in this case a. is negative, so that the lengtli PD = — a ; 
 and therefore — ax represents the double of the area of the tri- 
 angle PBG. 
 
 Hence, as before, 
 
 aa + 5)8 + C7 = 2A. 
 
12 
 
 TEILINEAR COOEDINATES. 
 
 Again, suppose P on the negative side of both BC and AB, 
 as in £g. 5. 
 
 Fig. 5. 
 
 In this case the triangles 
 
 PCA - PAB -PBG= the triangle ABC. 
 
 But in this case both a and 7 are negative, and therefore 
 — aa. and — 07 represent the double areas of the triangles PBC 
 and PAB. 
 
 Hence, as before, 
 
 a(i+b0 + cy = 2A. 
 
 Hence we see that if a, /S, 7 be the coordinates of any point 
 whatever, they are connected by the relation 
 
 oa + 6)8 + C7 = 2 A. 
 
 8. We might have deduced the result just obtained at once 
 from the result in Art. 3, taking care to determine the ambigu- 
 ous sign so that the perpendicular from the origin C should be 
 positive in accordance with the convention of Art. 6. 
 
THE POINT. 13 
 
 Thus we should have written at once 
 
 2A - aa - 6/3 
 7 = 3 , 
 
 or aa + Z>/3 + C7 = 2A, 
 
 but the proof given in Art. 7 is more simple in its character. 
 
 9. Let p be the radius of the circle circumscribing the 
 triangle ABC. 
 
 Then by trigonometry, 
 
 a b c 
 
 sin A sinB sin G 
 
 = 2p; 
 
 hence the equation obtained in Art. 7 may be written 
 a sin ,4 + /3 sin 5 + 7 sin C= —= S suppose. 
 
 10. The equation of Art. 7, or the equivalent form just 
 obtained plays a very important part in trilinear coordinates. 
 It enables us to make every equation involving a, jS, 7 homoge- 
 neous, for, since 
 
 aa. + 5,S + cy 
 
 2A 
 
 = 1 (Art. 7), 
 
 we are at liberty to multiply any term we please in an equation 
 
 by the fraction — r — , thus raising by unity the order 
 
 of the term. By repeating this operation we can raise every 
 term of an equation up to the same order as the term of highest 
 order, and thus render our equation homogeneous. 
 
 For example, if we have the equation 
 a'+3a7 + 5/3 = l, 
 
14 TRILINEAE COORDINATES. 
 
 we can raise every term to the third order: thus we get the 
 homogeneous equation 
 
 « + 3*7 2A + ^^ [—^A J 
 
 _ f aoL + l^ + cy V 
 ~[ 2A / ' 
 
 which we might proceed to simplify. 
 
 11. If the ratios of the coordinates of any point be given, 
 the point is determinate, and the actual values of the coordi- 
 nates can be found by means of the relation 
 
 aa + 6/3 + C7 = 2A. 
 We may proceed thus : 
 Let the coordinates be proportional toX : fi : v, then 
 
 X fj, 1/ ' 
 and therefore each of these ratios 
 
 _aa + b^ + cy 2A 
 
 Hence 
 
 aX + b/j, + cv aX + hiJL+ cv' 
 
 2XA 
 
 a= — J , 
 
 ax + Dfj, + cv 
 
 oK + hfi + cv' 
 
 2i/A 
 
 'y =^ ■ . 
 
 a\ + &/t + cv 
 
 12. We may however observe that in practice we very 
 rarely require the absolute values of the coordinates. For 
 advantage is almost universally taken of the principle detailed 
 in Art. 10, by means of which our equations in trilinear coordi- 
 
THE POINT. 15 
 
 nates are homogeneous. And it is scarcely necessary to point 
 out that a homogeneous equation in a, /3, 7 will not involve in 
 any way the actual values of the quantities, but will only in- 
 volve their ratios. 
 
 For example, 
 
 a'-3a'/3 + 7'=0 
 may be written 
 
 W/ \.v 'y ' 
 
 Q 
 
 where only the ratios - and — are involved. 
 J 7 7 
 
 Again, if we have to substitute the coordinates of a point in a 
 homogeneous equation it is not necessary to know more than the 
 ratios of the coordinates. 
 
 For suppose the equation were 
 
 a/37 -3a'/3+ 5/3^7 + 7' = 0, 
 
 and suppose the coordinates of the point were known to be pro- 
 portional io I : m : n. 
 
 The actual values of the coordinates may be supposed to be 
 kl, hin, Jen, but it is not necessary generally to know the value 
 of tlie multiplier k ; for if we substitute Jcl, km, kn in the given 
 equation, we get 
 
 kHm - Z¥Fm + 5^-Wn + AV = 0, 
 
 or, dividing by k' throughout, 
 
 Imn - 3Pot + bm'n + w' = 0, 
 
 so k disappears from the final result, and therefore a knowledge 
 of its value was unnecessary. 
 
 We shall conclude this chapter w^ith some examples in 
 which we shall determine the coordinates of several points re- 
 lated to the triangle of reference, leading to results which are 
 continually required in the solution of problems. 
 
16 
 
 TRILINEAE COORDINATES. 
 
 13. To find the coordinates of the angular points of the 
 triangle of reference. 
 
 For the point A it is evident that /3 = and 7=0. Also 
 aa = 2A, hence we can write down the coordinates 
 
 of^, 
 
 
 0, 
 
 0. 
 
 So, of 5, 
 
 0, 
 
 2A 
 
 0, 
 
 and of 0, 
 
 0, 
 
 0, 
 
 2A 
 e 
 
 N.B. The angular points of the triangle of reference are 
 conveniently spoken of as " the points of reference." 
 
 14. To find the coordinates of the middle point o/'BC 
 
 Fig. 6. 
 
 Let P be the middle point, and suppose a, j8, 7 the coordi- 
 nates of P. 
 
 Since P lies on PC we have a = 0. 
 
 Also J/3 = twice the triangle APB 
 = the triangle ABC, 
 
 since APB, APC on equal bases and of the same altitude are 
 equal. 
 

 
 THE POINT. 
 
 Therefore 
 
 
 ^=t 
 
 and so 
 
 
 A 
 
 Hence the coordinates 
 
 are 
 
 
 0, 
 
 A A 
 b' c- 
 
 17 
 
 15. To find the coordinates of the foot of the perpendicular 
 from A upon BC. \ 
 
 Let A A' be the perpendicular and let a, ^, y be the coordinates 
 of A'. Draw A'H, A'K perpendicular to CA, AB respectively ; 
 
 then 
 
 a = 0, = A'ir, y = A'K. 
 
 But 
 
 ~p-T = COS AA'H= cos C ; 
 A A 
 
 
 .: A'R=A'AcoaC; 
 
 
 o 2A „ 
 
 or p = — cos C ; 
 
 
 so 7 = — cos 5. 
 a 
 
 Hence the coordinates are 
 
 
 „ 2A ^ 2A „ 
 0, — cos C, — cos 5. 
 a a 
 
 16. To find the coordinates of the centre of the inscribed 
 circle of the triangle of reference. 
 
 This point is equally distant from the three lines of re- 
 ference ; 
 
 .-. a = ^ = 7; 
 
 a ^ 7 _ flg + &j8 + C7 _ 2A 
 *'l~l~r~ a+b + c ~a + b + c' 
 w. 2 
 
18 
 
 TBILINEAR COORDINATES. 
 
 Hence each of the coordinates is — , where s denotes, as in 
 
 s 
 
 Trigonometry, half the sum of the sides of the triangle. 
 
 17. To fini the coordinates of the centre of the circle cir- 
 cumscribing the triangle of reference. 
 
 Fig. 7. 
 
 Let be the centre, a, /3, 7 its coordinates ; join OB, 0(7 and 
 draw OP perpendicular on BC; then (Euclid, iii. 2), BG is 
 bisected in P. 
 
 Hence the triangles OPB, OPC are equal in all respects. 
 
 Now the angle BOG a.t centre = twice angle BA G at circum- 
 ference. 
 
 Hence ^ BOP= ^BOC = A; 
 
 OP 
 
 ■■ -gp=cot^, or OP =BP cot A, 
 
 So 
 
 i.e. a = 5 cot A. 
 
 /8=|cot5, 7 = |cot(7, 
 
 which give the required coordinates. 
 
THE POINT. 
 
 19 
 
 COE. If p be the radius of the circumscribed circle, these 
 coordinates may be expressed thus : 
 
 a = p cos ^, /3 = p cos 5, 7 = p cos G. 
 
 18. To find the coordinates of the point which divides in a 
 given ratio the straight line joining two points whose coordinates 
 are given. 
 
 Fig. 8. 
 
 Let P„ P, be the given points and (a„ ^„ 7,), (oj, ^^, 7,) 
 their coordinates, m : n the given ratio. Suppose P the point 
 required, and let (a, ^, 7) be its required coordinates ; 
 
 then PJP : PP^ = m : n. 
 
 Draw PD, P^D^, PJ).i perpendiculars on BC, and through 
 Pand P, draw PH and Pj/iT parallel to BC, and meeting P^D^ 
 and PD respectively in H and K. 
 
 Then by similar triangles 
 
 PK : P.B'^ PP, : P^P = m : n, 
 
 i, e. o — «! : ffa — a = »w : «> 
 
 whence na — na, = m<x^ — ma. ; 
 
 or {in + «) a = ma, + na, ; 
 
 ma + na. 
 
 or 
 
 a = - 
 
 VI + n 
 
 2—2 
 
20 TRILINEAR COORDlNATrS. 
 
 m + n 
 
 Similarly we may shew that 
 
 and 7 = —^■^, , 
 
 which give the required coordinates. 
 
 Cor. The coordinates of the middle point between (a, , ^, , 7,) 
 and (of,, /8„ 7,) are 
 
 a, + a, ^, + /3, 7i+7i, 
 
 2 ' 2 ' 2 ■ 
 
 Exercises on Chapter II. 
 
 (10) Find the coordinates of the points of trisection of the 
 sides of the triangle of reference. 
 
 (11) If A' be the middle point in the side BG of the triangle 
 of reference ABC, find the coordinates of a point Pin AA', such 
 that AP= 2A'P. 
 
 (12) If AA' be the perpendicular from the point of reference 
 A upon the opposite side BC, find the coordinates of a point P, 
 so dividing this line that 
 
 AP : PA' = cos ^ : cos B . cos C. 
 
 (13) Find the coordinates of the centres of the circles escribed 
 to the triangle of reference. 
 
 (14) Render the following system of equations in trilinear 
 coordinates homogeneous : 
 
 {la" + mS' + Mc") a" - iMaa + 4A'Z 
 = {la' + mW + tv?) ^ - 4 A9W&/3 + 4A»m 
 = (Za' + mh" + n<?)'f- 4 Anc7 + 4 A*m. 
 
CHAPTER III. 
 
 TRILINEAR COOEDINATES. THE STRAIGHT LINE. 
 
 19. To find the area of a triangle, the, triUnear coordinates 
 of whose angular points are given. 
 
 Let (fflj, ^,, 7j, (Oj, ^j, 7,), (a„ ^3, y^ be the coordinates of 
 the angular points, and let A denote the area of the triangle. 
 
 Now we have alread7 in Art. 5 found an expression for A 
 in terms of the coordinates a and /3 of each angular point. We 
 may however express that result in a form symmetrical with 
 respect to the three coordinates of each point. 
 
 Thus, taking the result of Art. 5, 
 
 A=- cosec 
 
 C [1, ifi, - /SJ + a, (/83 - A) + «3 (A - ^.)} 
 
 = - cosec G 
 
 «1. 
 
 A. 
 
 /8„ A 
 
 cosec G 
 
 - 28 
 
 1, 
 
 1, 1 
 
 
 S, 8, 8 
 
 But diminishing the last row of the determinant by the sum 
 of the first row multiplied by sin A, and the second multiplied 
 by sin S, and remembering that the relation 
 
 asin J. + ^8in.B + 7sin C= 8, 
 
22 
 
 TBILINEAR COOEDINATES. 
 
 is true for the coordinates of each point, we get 
 
 A = 
 
 cosec 
 
 C 
 
 2S 
 
 
 7, sin (7, 7, sin C, 7, sin C 
 «,. /3,, 7i i 
 
 Ks, /Sj, 7=. 
 
 
 1 
 ~2S 
 
 7i> 7s. 73 
 
 
 a perfectly symmetrical expression for the area of the triangle. 
 
 20. To f,nd the condition that three points whose trilinear 
 coordinates are given should lie on one straight line. 
 
 Let (a,, ^,, 7.), (o„ /3„ 7,), (a„ ,83, 7,) be the three 
 points. 
 
 That they should lie on a straight line is the same thing as 
 that the area of the triangle formed by them should be zero. 
 
 Hence, by the last article, the condition is 
 
 = 0. 
 
 «., ^„ 
 
 7i 
 
 «s, /3s, 
 
 7s 
 
 «3, i^a, 
 
 7a 
 
 21. It follows from Art. 20 that the equation 
 
 a, A 7 =0 
 
 «!, ^i> 7, 
 «a, ^s, 7s 
 
 speaks to us of a variable point (a, /8, 7) which lies on one 
 straight line with the points (a^, /S^, 7,), (a„ ^8,, 7,). It is 
 manifest that the equation will be satisfied if (a, ^, 7) denote 
 any point whatever on this straight line : and that it cannot be 
 satisfied if (a, /8, 7) lie elsewhere. 
 
THE STRAIGHT LINE. 
 
 Hence the equation 
 
 o, /S, 7 =0 
 
 IS the equation of the straight line joining the points 
 (an^.,7.), («.,^i, 7J- 
 If L, M, Nha equal to, or proportional to the minors 
 
 23 
 
 /3.. 7, 
 
 /32> 72 
 
 7i, a, 
 
 72 > «2 
 
 
 of the above determinant the equation becomes 
 La. + M^ + iVy = 0. 
 
 Cor. Every straight line is represented by a homogeneous 
 equation of the first order in a, yQ, 7. We proceed to shew 
 that the converse of this proposition is also true. 
 
 22. Every homogeneous equatimi of the first order represents 
 a straight line. 
 
 Let 
 
 la. + »i/3 + «7 = 
 
 be any homogeneous equation of the first order in a, ^, 7. It 
 shall represent a straight line. 
 
 By giving 7 any value (7, suppose) in the system of equa- 
 tions 
 
 la. + m^ + my = 0, 
 
 aa+b^ + cy= 2A, 
 
 we shall get corresponding values {a^, yS, suppose) for a and jS. 
 
 Thus we can find coordinates (a^ , y9, , 7|) representing a point 
 upon the locus of the given equation. 
 
 Similarly by giving 7 another value (7, suppose) we can find 
 the coordinates (a,, j3„ 7,) of another point upon the locus. 
 
24 TEILINEAE COORDINATES. 
 
 But since (a„ /3„ 7,), («,, jS,, 7J represent points lying on 
 the locus of the equation 
 
 la + m^ + ny = (1)) 
 
 we have the relations 
 
 Zo[, + m/9, + M7, = (2), 
 
 and ?», + »iy8, + M7,= (3). 
 
 By means of (2) and (3) we can eliminate the ratios I : m : n 
 from (1) ; thus the equation (1) will take the form 
 
 a, /3, 7 
 a,. A. 7, 
 at, ^j, 7a 
 
 = 0, 
 
 which we know (by Art. 21) to be the equation to a straight 
 line. 
 
 Hence every homogeneous equation of the first order as tri- 
 linear coordinates represents a straight line. 
 
 Note. The only apparent exception is when the two equa- 
 tions la + TOjS + w7 = and aa + Jy8 + 07 = 2 A are inconsistent, 
 that is, when I, m, n are proportional to a, h, c. This case we 
 shall discuss separately in Chapter iv. 
 
 23. By Art. 21 we are able to write down the equation to 
 any strsught line in terms of the coordinates of any two points 
 upon it. 
 
 It is often desirable to express it in terms of any other con- 
 stants which will determine the straight line. For instance, a 
 straight line is determinate when its perpendicular distances 
 from the three points of reference are given; we proceed to 
 determine the equation to a straight line in terms of these three 
 distances. 
 
 Let AP=p, BQ = q, OR = r be the three perpendiculars 
 from the angular points A, B, C upon a straight line PQR: it 
 
THE STRAIGHT LINE. 
 Fig. 9. 
 
 25 
 
 is required to find the equation to the straight line in terms of 
 these quantities p, q, r. 
 
 Let 0, 0' be any two points upon the straight line, and let 
 their coordinates he a, jS, 7 and a', /3', 7', and let p denote the 
 distance between them, then 
 
 pp = twice area A 00" 
 
 = S 
 
 2A, 
 
 a 
 
 0, 
 
 
 
 a, 
 
 /3, 
 
 7 
 
 t 
 
 a, 
 
 /3', 
 
 7' 
 
 by Art. 13, 
 
 therefore multiplying by aa, 
 apap = 2^S 
 
 a, 0, 
 a, 0, 7 
 0.; )8', 7' 
 
 so 
 
 bq^p = 2A5 
 
 0, 
 
 A 
 
 
 
 a, 
 
 A 
 
 7 
 
 a', 
 
 /3', 
 
 7 
 
26 
 
 TRILINEAR COORDINATES. 
 
 and 
 
 c)-7p = 2 AS 
 
 therefore by addition 
 
 {apa + bq^ + cry) p = 2AS 
 
 0, 0, 
 
 7 
 
 
 a, /3, 
 
 7 
 
 
 a', ^', 
 
 y' 
 
 
 ■.2AS 
 
 a, 
 
 0, 7 
 
 
 a, 
 
 /3, 7 
 
 
 t 
 
 ^. y 
 
 and therefore 
 
 opa + hq^ + cry = 0. 
 
 = 0, 
 
 This being a relation among the coordinates a, /3, 7 of any 
 point whatever on the straight line PQB, is the equation to that 
 straight line, and it is expressed in terms of the perpendiculars 
 p, q, r. Therefore it is the equation required. 
 
 24. We shewed in Art, 22, that the equation 
 
 h + m^ + ny = (l), 
 
 must always represent a straight line. 
 
 In Arts. 21 and 23, we have found the equation to a straight 
 line in terms of the coordinates of two points upon it, and in 
 terms of its pei-pendicular distances from the points of reference, 
 but both the equations thus found are particular cases of the 
 general form (1). 
 
 Thus by comparing the various articles we are able to 
 explain the coeflScients in the general equation. We may either 
 interpret I, in, n, as proportional to the determinants 
 
 A, % ' 
 
 %, a. 
 
 
 a.. /3: 
 
 /3., 7, i 
 
 7,. «2 
 
 
 «2. ^« 
 
 where (a,, fi^, y,), (a^, 13^, 7J, denote points upon the line, or 
 we may say that they are proportional to ap, bq, cr, where 
 p, q, r are the perpendicular distances of the line from the points 
 of reference. 
 
THE STRAIGHT LINE. 27 
 
 25. It should be noticed that the equation 
 
 |«, A 7 i = 
 I «.> /3i> 7, : 
 
 '. «„ ^,> 72 i 
 
 will not be altered if we substitute for a^, /3,, and 7,, or for 
 a,, /3,, and 7,, any quantities proportional to them. For this 
 is only equivalent to multiplying the equation throughout by a 
 fixed ratio. 
 
 Hence it is not necessary, in order to form the equation to 
 the straight line joining two points, to know the actual co- 
 ordinates of the points, but it will suffice if the ratios of the 
 coordinates are given. 
 
 Thus if two points be determined by the equations 
 
 \ fi I/' 
 
 A a ^ 7 
 
 and. z-i = —i — ~f , 
 
 \ fj, V 
 
 the equation to the straight line joining them will be 
 
 a, A 7 
 
 \, fl, V 
 
 26. To find the condition that three straight lines whose 
 equations are given should pass through one point. 
 
 Let h + »ny8 + «7 =0, 
 
 ToL + m'^ + n'j = 0, 
 l"a+m"^ + ti"y=0, 
 
 be the three equations. If the three straight lines all pass 
 through one point, all these equations will be satisfied by the 
 coordinates («', /3', 7', suppose), of the point. 
 
 Hence, la +m^' +ny' =0, 
 
 I'a' + m'^ + n 7 = 0, 
 l"a' + J»"/3' + n"y' = 0. 
 
28 TBILINEAR COOBDINATES. 
 
 Therefore eliminating a' : /8' : 7' we get 
 
 = 0, 
 
 I, 
 
 m, 
 
 n 
 
 l\ 
 
 m\ 
 
 n 
 
 r, 
 
 «*", 
 
 It 
 n 
 
 which will be the condition required. 
 
 27. Every straight line passing through the point of intersec- 
 tion of the two straight lines whose equations are 
 
 la +»iy3 +n7 =0 (1), 
 
 l'a + m'^ + n'y = (2), 
 
 wUl have an equation of the form 
 
 1(1L+ m^ + ny + K (l'a + m'^ + n'y) =0 (3), 
 
 where k is an arbitrary constant, and by giving a suitable value to 
 K the equation (3) can be made to represent any particular straight 
 line parsing through the point of intersection of (1) and (2). 
 
 Suppose (a, /8, 7) to be the point of intersection of (1) and 
 (2); therefore these coordinates satisfy the equations (1) and 
 (2): therefore 
 
 la + in/3 +717 =0, 
 
 Z'a + «i'iS + 717 = 0. 
 
 Multiplying the second of these by k and adding, we get 
 
 la + wiyS + W7 + /c (Z'a + wi'/S + n'7) = 0, 
 
 which shews that o, /8, 7, satisfy the equation (3). 
 
 But the equation (3) being of the first order represents a 
 straight line. Hence it represents a straight line passin" 
 through the intersection of (1) and (2). q. e. d. (i). 
 
 Also by giving a suitable value to k the equation (3) will 
 represent any straight line through (a, /3, 7). 
 
 For suppose it be required to make it represent a straight 
 line passing through any point (a', ff, y). 
 
THE STRAIGHT LINE. 29 
 
 The condition that this point should lie on the locos is 
 h! +mff + ny'+ic (fa' + m'^' + n'y) = 0. 
 
 Hence, if we give « the value determined by this equation, 
 i.e., 
 
 _ _ fa' + m ^ ' + ny ' 
 la +mp +ny 
 
 the equation (3) will represent the line joining the points 
 (a, )8, 7), and (a', ^, 7'). Hence we can determine « so as to 
 make the equation (3) represent any straight line through the 
 point of intersection of (1) and (2). Q. E. D. (ii). 
 
 In this case the equation (3) takes the form 
 
 la. + wi/3 + ny I'a + m'^ + n'y _ 
 la' + m^' + ny' I'a' + m'fi' + n'y ~ ' 
 
 which is therefore the equation to the straight line joining 
 («', /8', y), to the point of intersection of the straight lines 
 
 la + m^ + W7 = 0, 
 
 and I'a + wi'/S + n'7 = 0. 
 
 28. If we use u and v to denote the expressions 
 Za + »i;8 + W7 and Ta + m'^ + n'y; 
 
 the third equation of the last article wUl be represented by 
 
 u->r KV = 0. 
 
 Hence we may briefly express our result as follows. 
 
 Jfu and V he any Junctions of the first degree of the coordi- 
 nates, then the equation 
 
 u+ icv=0, 
 
 iciU represent a straight line passing through the intersection of 
 the straight lines represented hy 
 
 M = and V = 0, 
 
 and by giving a suitable value to k, it will represent any such 
 straight line. 
 
30 TEILINEAE COORDINATES. 
 
 29. The following affords a good illustration of the use of 
 the foregoing article. 
 
 B K 
 
 Let the equation 
 
 Za + »n)3 + n7 = (1) 
 
 represent a straight line meeting BC in A, CA in B', AB in C. 
 
 Consider the equation 
 
 m^ + ny = (2). 
 
 From its present form we observe that it is a straight line 
 passing through the intersection of ;8 = and 7 = 0, that is, 
 through A, but if we write it in the form 
 
 {la + 7W/3 + ny) -la=0, 
 we perceive that it passes through the intersection of a = and 
 
 la. + m^ + W7 = 0, 
 that is, through A'. 
 
 Hence it represents the straight line AA'. 
 
 Similarly the equations 
 
 W7 + /a = (3), 
 
 Za + j»/3 = (4), 
 
 will represent BE' and GC respectively. 
 
THE STRAIGHT LINE. 31 
 
 Further let BB', OC meet in a; CO', AA' in J; AA', BB' 
 in c. 
 
 Then the equation 
 
 OTy8-W7 = (5), 
 
 which represents a straight line through A, being equivalent to 
 
 h + j»y3 - {ny + la.) = 0, 
 must pass through a. 
 
 Hence it represents the straight line Aa. 
 
 Similarly the equations 
 
 ny-la = (6), 
 
 h-vi^ = (7), 
 
 will represent Bb and Cc respectively. 
 
 But further the equation (7) may be obtained from the equa- 
 tions (5) and (6) by addition. Hence the straight line (7) passes 
 through the intersection of the straight lines (5) and (6). 
 
 That is, Aa, Bb, Cc meet in a point. 
 
 We conclude this chapter with some Examples of the 
 methods we have been investigating. 
 
 30. To find the equation to the perpendicular from the point 
 of reference A upon the line BC. 
 
 FiEST Method. Let A A' be the line in question. The 
 perpendicular distances of the line from the angular points 
 A, B, C are respectively 
 
 0, c cos B, —b cos C, 
 
 where we give opposite signs to the latter two distances, since 
 they are measured on opposite sides of AA '. 
 
32 
 
 TRILINEAR COORDINATES. 
 
 Fig. 11. 
 
 Hence by Art. 23 the equation is 
 
 ySJc cos B — ycb cos (7=0, 
 or /3cos.B— 7C0S (7=0. 
 
 Second Method. Let A A' =jp, then the coordinates of A 
 are p, 0, 0, and the coordinates of A' are 0, p cos C, p cos B. 
 By Art. 21 the straight line joining these points has the equation 
 
 a, 0, 7 
 
 P, 0, 
 
 0, pcQB C, pcoaB 
 
 = 0, 
 
 or (dividing hyp', and evaluating the determinant), 
 
 /8 cos B- 7 cos (7= 0, 
 which will be the equation required. 
 
 Third Method. Let P be any point in AA', and on AC 
 let fall the perpendicular PE=ifi, and on BA the perpendicular 
 
 Then since the angle PA C is the complement of C, 
 
 PE ^ 
 
 -p2 = coaC. 
 
THE STXJAIGHT LINE. 33 
 
 Similarly, aince the angle PAB is the complement of B, 
 
 FF „ 
 
 jpj=cosB; 
 
 therefore FE : PF= cos G : coaB, 
 
 or /3 : 7 = cos : cos^, 
 
 or yS cos 5 = 7 cos C, 
 
 a relation among the coordinates of any point P in AA', and 
 therefore the equation to AA'. 
 
 31. The perpendiculars from the angular points of a triangle 
 on the opposite sides meet in a point. 
 
 Take the triangle in question as triangle of reference, and 
 call it ABC; then, Art. 30, the three perpendiculars will be given 
 by the equations 
 
 /3cos5 — 7C0S (7=0, 
 
 7 COS — a cos A = Q, 
 
 acoa A — ^ COS B = 0, 
 
 of which we observe that any one can be obtained from the other 
 two by addition; therefore by Art. 27, the three lines pass 
 through the same point. 
 
 32. To construct a straight line whose equation is given. 
 Let loL + wi/8 + W7 = 
 
 be the given equation of a straight line. 
 
 It is required to construct the straight line. 
 
 The given equation will be satisfied if a = 0, and yS and 7 are 
 determined so as to satisfy the equation 
 
 m/3 + «7 = 0. 
 
 But if a = 0, the corresponding values of yS, 7 are subject to 
 the relation 
 
 J/3 + C7 = 2A. 
 TV. 3 
 
34 TEILINEAE COORDINATES. 
 
 From these two equations we obtain 
 
 2nA - 2otA 
 
 "" hn — cm ' " ~ bn — cm' 
 
 which give the coordinates corresponding to a = 0, of a point 
 upon the line. 
 
 Hence we are able to construct the point where the required 
 line meets BC. 
 
 Similarly we can construct the point where it meets CA : 
 and by joining these two points we shall have the straight 
 line required. 
 
 33. It will be understood that when we speak of the 
 straight line Za + »w/3 + «7 = 0, we are using elliptical language, 
 and mean strictly, the straight line whose equattoti is 
 
 la + 9»y3 + 717 = 0. 
 
 So we often speak of a point lying on la + m^ + ny = 0, 
 when we mean that it lies on the locus of that equation. Or we 
 speak of an equation passing through such and such points, 
 when we mean that its locus passes through those points. 
 
 All these modes of expression are of course, speaking 
 strictly, very loose and incorrect ; but as they can hardly lead to 
 any misconception they are not objectionable, and they shorten 
 very much the expression of a mathematical argument. 
 
 It is convenient also to notice that just as the point whose 
 coordinates ars a, /3 and 7 is commonly described as the point 
 (a, /3, 7), so the straight line whose equati n i . la + m^ + ny = 
 may be spoken of as the straight line (Z, m, n). 
 
 Exercises on Chapter III. 
 
 (15) Find the area of the tiiangle whose angular points are 
 the middle points of the sides of the triangle of reference. 
 
EXERCISES ON CHAPTER III. 35 
 
 (16) Find the area of the triangle whose angular points are 
 the feet of the perpendiculars from the points of reference on the 
 opposite sides. 
 
 (17) Find the equations to the sides of the triangle of 
 Ex. 16. 
 
 (18) Find the area of the triangle whose angular points are 
 given by 
 
 a = 
 
 «2/S = Wy . 
 
 /3 = 1 7 = 1 
 
 (19) Shew that the points given by 
 
 a = 
 mP + n7 = . 
 
 /3 = 0} 7 = 
 «7 + Za = j ' Z« + Mi/S = . 
 
 lie all on one straight line. 
 
 (20) Find the coordinates of the points of trisection of the 
 side AB of the triangle of reference. 
 
 (21) Find the equation to a straight line cutting the lines of 
 reference CA, AB in Q, B respectively, where AQ = ^A C and 
 AB = ^AB. 
 
 (22) A straight line cuts the sides BC, OA of a triangle 
 ABC in P, Q and it cuts AB produced in B, shew that if 
 CP : CB = 1 : 3 and 0$ : C4 =2 : 3, then will BA : AB=^l : 3. 
 
 (23) Find the equation to a straight line which cuts off 
 — 1 and (-) respectively from the sides AB, AC oi the tri- 
 angle of reference, and find the coordinates of the point where 
 it meets the side BC. 
 
 3—2 
 
CHAPTER IV. 
 
 THE INTERSECTION OP STRAIGHT LINES. PARALLELISM. 
 INFINITY. 
 
 34. To find the coordinates of the point of intersection of two 
 straight lines whose equations are given. 
 
 Let la, + m^ + ny = 0, 
 
 be the equations to the two straight lines. 
 
 Then the coordinates of the point of intersection must satisfy- 
 both equations, and the ratios of the coordinates will therefore 
 be obtained by solving the two equations together. 
 
 Thus, eliminating 7 we get 
 
 a 
 
 /3 
 
 m, n 
 
 «, I 
 
 1 r 
 
 m, n 
 
 n,V 
 
 and therefore by symmetry, each = 
 
 I, m \^ 
 , m I 
 equations which give the ratios a : y3 : 7. 
 
 But to obtain the actual values of the coordinates we have to 
 introduce the relation 
 
 aa + J/3 + C7 = 2A. 
 
THE INTERSECTION OF STBAIQHT LINES. 
 
 Thus, since 
 
 7 
 
 a. 
 
 /3 
 
 m, n 
 m, n' 
 
 n, I 
 n', I' 
 
 I, m 
 I', to' 
 
 are equal, therefore each of them is equal to 
 
 fflg + 5j8 + cy 
 
 a. 
 
 h. 
 
 c 
 
 I, 
 
 m, 
 
 n 
 
 I', 
 
 m\ 
 
 n! 
 
 or 
 
 
 2A 
 
 
 a, 
 
 b. 
 
 e 
 
 I, 
 
 TO, 
 
 n 
 
 i; 
 
 TO', 
 
 n" 
 
 Hence 
 
 with similar expressions for /8 and 7. 
 
 2A 
 
 TO, n 
 
 = 
 
 
 to', n' 
 
 a, b, c 
 
 
 I, m, n 
 
 
 I' 
 
 , to', n 
 
 37 
 
 35. To find the condition that two straight lines whose 
 equations are given may he parallel. 
 
 Let la. + TO/9 + wy = 0, 
 
 I'a + TO'yS + m'7 = 0, 
 be the two given equations. 
 
 If the two lines are parallel their point of intersection lies at 
 an infinite distance from the triangle of reference. Hence the 
 common denominator in the expressions for the coordinates of 
 the point of intersection, obtained in Art. 34, must be zero. 
 
 That is 
 
 a, h, c 
 I, TO, n 
 
 , TO, n 
 
 = 0. 
 
38 THE INTEESECTION OP STRAIGHT LINES. 
 
 36. To interpret the equation 
 
 oa + 6/3 + C7 = 0. 
 
 We shall shew first that the locus of this equation includes no 
 point other than at infinity; and, secondly, that it includes 
 every point at infinity. 
 
 In order to find the coordinates of points on the locus of any 
 given equation, we have to determine values for a, y3, 7, which 
 will satisfy both the given equation and the perpetual relation 
 
 aa + 5/3 + C7 = 2A. 
 
 In the present case the two equations which have to be com- 
 bined are inconsistent for all finite values of the variables. For, 
 if a, ^, 7 are finite, we get, by subtraction, 
 
 = 2A, 
 
 which is contraiy to our original hypothesis. 
 
 But looking at the equations a little more generally, and 
 remembering that a, /S, 7 may have infinite values, it appears 
 that the result of the subtraction ought strictly to be written 
 
 0.a + 0./3 + 0.7 = 2A, 
 
 an equation which requires that one or more of the variables 
 a, jS, 7 should be infinite. And from considering either of the 
 original equations, we observe that two at least of these variables 
 must be infinite, since if only one were infinite, we should have 
 
 aa + J^ + C7 = 00 . 
 
 But it may be asked, how can the equation aa + bfi + cy = 
 be satisfied by points anywhere situate, since we know by the 
 geometrical construction. Art. 7, that if a, ^, 7 are the coordi- 
 nates of any point whatever, aa + h^ + cy will represent the 
 double of the area of the original triangle? 
 
 True. But when we take any point in the plane of the tri- 
 angle of reference to represent (a, j8, 7), we necessarily take it at 
 some finite distance or other from the triangle. We can make 
 
PARALLELISM. INFINITY. 39 
 
 this distance as great as we please, but we can never actually 
 make it infinite. So when we say that the equation 
 
 aa + Jj8 + C7 = 
 
 represents a locus lying altogether at infinity, we are not contra- 
 dicting, but rather asserting this fact. For to say that every 
 point upon the locus lies at infinity is in fact saying that no" 
 point can he found or drawn which shall satisfy the equation. 
 
 But the statement further implies the following: that al- 
 though no finite point can be found to satisfy the equation 
 
 la. + JM/3 + M7 = 
 
 when I, m, n are proportional to a, h, c, yet when the ratios of 
 I, m, n differ from those of a, b, c by the least possible difference, 
 then such points can be found ; and by making the difference as 
 small as we please, the locus will recede as far as we please from 
 the points of reference. 
 
 This is exactly the meaning which is attached to the term 
 " infinity" in Algebra, where (for instance) the statement 
 
 " + - + - -I- &c. to an infinite number of terms = 1 
 2 4 o 
 
 does not mean that any number of terms which we can actually 
 take will amount to unity, but that by taking as many terms as 
 we please, we can make the sum as near unity as we please. 
 
 But, secondly, any point lying at an infinite distance from the 
 triangle of reference may be regarded as lying upon this locus. 
 
 For, let X be any point at an infinite distance, and let P be 
 any finite point, then we can conceive a straight line joining 
 PX, and by Art. 21, Cor. it will have an equation of the form 
 
 la + m^ + ny = (1). 
 
 Now let Q be another finite point not in the straight line 
 PX, and let the equation to QX be 
 
 Z'a + «t'/3 + «'7 = (2). 
 
40 THE INTEKSECTION OP STRAIGHT LINES. 
 
 Then since FX and QX intersect at infinity they are pa- 
 rallel, and therefore their equations must satisfy the condition 
 investigated in Art. 35, 
 
 i. e. a, h, c = 0. 
 
 Z, m, n 
 V, ml, n' 
 
 But this equation expresses the condition that the three 
 equations 
 
 ax+h^ +cy = 0, 
 
 la + m0 +ny =0, 
 
 I' a + m'^ + n'y = 0, 
 
 should be consistent, or that their loci should have a common 
 point. Therefore the locus of the equation 
 
 aa + 6/3 + C7 = 
 
 passes through the intersection of PX and QX, that is, through 
 X; and so the same locus can be shewn to pass through any 
 point whatever at infinity. 
 
 But we have already shewn that it passes through no finite 
 point. Hence the equation 
 
 aa + 6/3 + C7 = 
 
 represents a locus lying altogether at infinity, and embracing all 
 points at infinity. 
 
 37. It has already been seen that the equation 
 
 la + »M/8 + W7 = 
 
 when the ratios I : m : n have any values whatever not identical 
 with the ratios a:h:c represents a real and finite straight line. 
 
 Now since the locus is a straight line however closely the 
 ratios l:m:n approximate to the values a : 6 : c, it is a lawful 
 form of expression to describe the limiting locus itself as a 
 straight line. 
 
PARALLELISM. INFINITY. 41 
 
 Thus we are able briefly to express the result we have 
 arrived at as follows : 
 
 Hie equation 
 
 represents the straight line passing through all points at infinity. 
 But it must be remembered that this is but an abbreviated 
 statement of the fact, that a& I : m : n approach the values 
 a : h : c, the locus of the ec[uation 
 
 la + »jjS + W7 = 
 
 will always be a straight line, which can be made to fail by as 
 little as we please from passing through any point whatsoever 
 and every point at an infinite distance from the lines of re- 
 ference; whilst the position to which it approaches will contain 
 no finite point whatever. 
 
 It will be observed that since sin A, sin B, sin G are pro- 
 portional to a, h, c, the equation may be indifferently written in 
 either of the forms 
 
 aa + h^+crf = Q, 
 
 a sin ^ + ;8 sin 5 + 7 sin (7 = 0. 
 
 38. The diflSculty of conceiving such a locus as we have 
 described, may perhaps be lessened by the following con- 
 siderations. 
 
 Let ABC be the triangle of reference, and P any point at a 
 finite distance from it. 
 
 From the centre P, at any finite radius PQ, as large as can 
 be conveniently taken, describe a circle, and suppose that while 
 the centre P remains fixed, the radius of this circle be gradually 
 increased. If this enlargement be carried on indefinitely, the 
 curvature of the circle becomes less and less, and can by snflS- 
 ciently enlarging the radius be made as small as we please. 
 Thus the arc of the circle in the neighbourhood of any point Q 
 upon it can be made as straight as we please: and though 
 
42 THE INTEKSECTION OP STRAIGHT LINES. 
 
 the circle can never become actually a straight line, yet as 
 the radius approaches an infinite length, the circle becomes 
 in eveiy part as nearly straight as we choose, while all its 
 points recede to an indefinitely great distance from all finite 
 points. 
 
 Fig. 12. 
 
 Thus we perceive that as the circle tends to become straight 
 it tends to satisfy the same conditions as the limiting locus of 
 the equation 
 
 la + m^ + wy = 0, 
 
 as I : m : n approach the values a : b : c. 
 
 The consideration of this infinite circle will tend to diminish 
 the diflSculty which would naturally be felt in accepting the 
 following proposition. 
 
 39. Every straight line may he regarded as parallel to the 
 straight line at infinity. 
 
 Let Za+»i^+ny = (1) 
 
PARALLELISM. INPINITT. 
 
 43 
 
 be the eqaation to any straight line. The straight Hue at 
 inJSnity has the equation 
 
 aa + b^ + cy = (2). 
 
 And by Art. 35 the condition that (1) and (2) should repre- 
 sent parallel straight lines is 
 
 a, b, c ; = 0, 
 
 I 
 
 a, h, c j 
 
 I, m, n \ 
 
 which is identically satisfied since two rows of the determinant 
 are the same. 
 
 Therefore every straight line may he regarded as parallel to 
 the straight line at infinity. Q. E. D. 
 
 40. To find the equation to the straight line passing through 
 a given point and parallel to a given straight line. 
 
 Let (a', j8', 7') be the given point, 
 
 and Za+»n;8 + «7 = , 
 
 the equation to the given straight line. 
 
 Let \a + /ij8 + jAy=0 
 
 be the equation required. 
 
 Then since the locus passes through (a', ^', y), 
 
 we have \a' + A'/S' + 1/7' = 
 
 Also since (1) and (2) are parallel, we have 
 
 = 0, 
 
 \, fl, 
 
 V 
 
 I, m, 
 
 n 
 
 a, b, 
 
 c 
 
 or 
 
 m, n 
 h, c 
 
 + /*[«, I 
 c, a 
 
 + v 
 
 I, in 
 a, b 
 
 = 0. 
 
 ■(I) 
 (2) 
 
 (3). 
 
 ,(4). 
 
44 
 
 THE INTERSECTION OF STRAIGHT LINES. 
 
 Eliminating \, fi, v from (2) by means of (3) and (4), we get 
 
 = 0, 
 
 
 
 a, 
 
 ^, 
 
 y 
 
 
 a, 
 
 /9'. 
 
 1 
 
 1 
 
 
 
 m, n 1 
 
 n, I 
 
 I, m 
 
 
 
 b, c 
 
 c, a \ 
 
 a, I 
 
 the equatioi 
 
 1 
 
 required. 
 
 
 
 41. If (a,, /3,, 7,), (a,, ^^, 7,) he the. coordinates of two 
 points, and if L, M, N denote the determinants 
 
 
 ^, 
 
 , -y 
 
 1 
 
 7i, 
 
 a. 
 
 
 «. 
 
 , ^ 
 
 1 
 
 
 ^. 
 
 . 7, 
 
 7j. "a 
 
 
 «j, ;8, i ' 
 
 respectively, then 
 
 will 
 
 
 a, - 
 
 ■«» 
 
 _^,-/8. _ 
 
 7.-7 
 
 , _ 1 
 
 
 b, 
 
 c 
 
 
 c, 
 
 
 a, i 
 
 2A' 
 
 
 M, 
 
 N 
 
 
 N, L 
 
 
 L, ifc 
 
 r 
 
 For 
 
 h, 
 
 c ' = 
 
 0, -c, J 
 
 
 
 M, n\ 
 
 a., A, 7. 
 
 
 
 
 a.. /3.. 7. 
 
 
 
 1 
 
 0, -c, 
 
 
 
 " c 
 
 «,, /3., 2A 
 a., /3., 2A 
 
 
 
 = 2A(a.-a.); 
 
 therefore 
 
 1 
 
 a — a 
 
 
 2A~ 
 
 J, c 
 
 
 
 
 JIf, JT 
 
 
 and similarly 
 
 
 ^.-^ 
 
 !_ _ 
 
 7.-7. 
 
 
 c, a 
 
 
 a, b, 
 
 
 
 
 
 
 
 N, 
 
 L 
 
 
 L, M 
 
PARALLELISM. INFINITY. 
 
 45 
 
 42. By comparing these relations with the result of Article 
 40, it is seen that the equation to the straight line through the 
 point (a, , /3, , 7,) parallel to the straight line joining the points 
 (a.,i8.,7.), and(a„j8„73) 
 
 IS 
 
 a, /3, 7 
 
 «.. ^1. 7, 
 
 = 0. 
 
 43. To find the distance between two points whose trilinear 
 coordinates are given. 
 
 Fig. 13. 
 
 Let P, Q be the two points whose given coordinates are 
 (a,, )8,, 7,), (ffj, Ai 7j)« and let p be the distance between them. 
 On PQ as diameter describe a circle, and in it draw QA', QB' 
 parallel to CB, CA. Join PA, PF , A'B' and through^' 
 draw a diameter A'X. Join XB', 
 
 then A'B" =PA'' + PB" - 2PA'. PB' cos A 'PB' 
 
 = PA" + PB"+2PA'.PB' cos C (1). 
 
46 
 
 THE INTERSECTION OP STRAIGHT LINES. 
 
 But the angles A'XB', A'QB' in the same segment are 
 equal ; 
 
 .-. z.A'XB'=^/.C, 
 
 and .-. A'B' = A'X&m C=PQ sin G 
 
 = psin C; 
 also PA' = or, — ttj, 
 
 and PB' = /3,-iS,. 
 
 Substituting these values of AB', PA', PB' in (1), we get 
 p^ sin' C={a,- a J + (/3, - /3,)' + 2 (a, - a,) {&, - /3,) cos C. 
 
 Similarly we have 
 p' sin' A = {^,- A)' + (7, - 7,)' + 2 (/3, - A) (% " 7») cos ^, 
 p' sin' J5 = (7, - 7 ) + (a, - a,)' + 2 (7. - 7,) (a, - «,) cos 5. 
 
 Thus we have three expressions for the required distance, 
 each of them symmetrical with respect to two of the coordinates 
 of the given points. By combining these expressions in various 
 ways, among themselves or with the identity 
 
 a {a,-a,) + b (/3, - A) + c (7.-7=) =0, 
 
 we might obtain a variety of expressions for the distance, sym- 
 metrical with respect to all the three coordinates of each point. 
 Several such expressions will be found in Chapter vi. 
 
 44. To find the distance "between the two points whose coor- 
 dinates are (a„ /3,, 7,), (a„ /8,, <y^ in a form symmetrical with 
 respect to the determinants 
 
 A. % 
 
 
 7., «i 
 
 
 «.. ^. 
 
 ^., 7. 
 
 ) 
 
 7.. a. 
 
 ) 
 
 «„ ^, 
 
 Let L, M, N denote these determinants, then retaining the 
 notation of the last article, the required distance is given by 
 
 p» sin'^ = (^, - ^,y + (7, - 7.)' + 2 (;3. - /8.) (7, - 7.) cos A. 
 
PARALLELISM. INFINITY. 
 
 47 
 
 But by Art. 41, 
 
 N, L 
 
 7.-7. _ 1 
 a, b 2A' 
 
 L, M 
 
 therefore 
 4AysiiiM = 
 
 N, L 
 
 + 
 
 a, h 
 L, M 
 
 + 2 
 
 c, a 
 N, L 
 
 a, h 
 L, M 
 
 caaA 
 
 = a= {i' + ilf + JV» - 2MN cos A - 2NL cos B - 2LM cos C}, 
 or remembering that 2A sin A = &, (Art. 9) 
 
 p' = |i {i' + IP + JV» - 23fN COS A - 2NL cos 5 - 2ZJy cos C) , 
 p = i V{i'+ 1P+ JV*- 2ilfA"^cos A - 2NL cos J5 - 2ZJ/cos C}. 
 
 45. 2b Jind the perpendicular distance of the point whose 
 coordinates are (o, jS, 7) from the straight line joining the two 
 points whose coordinates are (a,, )8j, 7,) and (a,, )8j, 7,). 
 
 Let^ be the perpendicular distance required, and p the dis- 
 tance between the last two points, then 
 
 pp = twice area of the triangle formed by joining the three 
 points ; 
 
 .•. p = - . (this double area), 
 
 r 
 
 and therefore in virtue of Arts. 19 and 44, 
 
 a, P, 7 
 a. ^.. 7i 
 a , /3.> 7, 
 
 P = ^/[U+M^+ N'- 2MNCOS A - 2NL cosB- 2LM cos 6'} 
 
 Ta + M^ + Ny 
 
 ~ ^{L''+ M »+ N'- 2MNcos A - 2NL cos 5- 2LM cos C] ' 
 
 an expression for the perpendicular required. 
 
48 
 
 THE INTERSECTION OF STRAIGHT LINES. 
 
 46. To find an expression for the perpendicular distance of 
 the point (a', /8', 7') from the straight line whose equation is 
 
 la. 4- «i(8 + 717 = 0. 
 
 Let (oi, /3i, 7,), (cr,, /3j, 7,) be two points on the given line, 
 and let L, M, N denote the determinants 
 
 ^1. % 
 
 
 7i. «, 
 
 
 «„ A 
 
 ^., % 
 
 3 
 
 7„ «2 
 
 S 
 
 ««, /3. 
 
 Then by the last article the required perpendicular is 
 given by 
 
 La' + M^ + Ny 
 
 P ~ a/{L' +M'' + N'- 23IN cos a - 2NL cos B - 2LMcos C } " 
 
 But the equation to the straight line joining (c,, yS,, 7,), 
 («*> /8,, 7,) may be written 
 
 ia+ J//3+iV7 = 0, 
 
 which must therefore be identical with the given equation 
 
 la. + 7H/3 + 727 = 0. 
 
 Hence 
 
 I m n ' 
 
 in virtue of which the expression for the perpendicular becomes 
 
 la + m0 + ny 
 
 Vl^" + »»' + w" — 2mn cos A — 2nl cos B — 2lm cos C} 
 
 Other methods of arriving at this result will be found in 
 Chapters V. and vi. 
 
 Note. The expression 
 
 P+wi°+7i' — 2mn cos^ — 2mZ cos 5— 2ln cos C 
 
 is of such frequent occurrence that it will be convenient to 
 denote it briefly by the symbol {I, m, n}'. 
 
PARALLELISM. INFINITY. 
 
 49 
 
 47. To find the inclination to the lines of reference of the 
 straight line whose equation is 
 
 h + mfi+ny = (1), 
 
 Let 6 be the inclination of the given line to the line of 
 reference BG. 
 
 And let a=k^ (2), 
 
 be the equation to the parallel straight line through C. Then 
 is the inclination of this line to BO, and therefore by Art. 4, (5), 
 
 Bme = h3m{C-e) 
 
 = k (sin C coaO— cos C sin 6), 
 
 (1+k cos C) sm0 = k sin O cos 0, 
 
 k sin G 
 
 tan0=, , ^ 
 
 1 + « cos C7 
 
 But since (1) and (2) are parallel, we have (Art. 35) 
 1, -k, 0=0 
 I, m, n 
 a, b, c 
 
 or {mc — bn) = {na — lc)k; 
 
 therefore substituting in (3), 
 tan = 
 
 .(3). 
 
 (mc — in) sin G 
 
 {na — lc) + {mc — hn) cos G 
 
 _ sin C mc — hn 
 
 c ' m cos G + n cos B — l 
 
 m Bin — n sin B 
 ~ m cos G+n cos B—l' 
 
 Similarly if <f> and ■^ are the inclinations of the same line 
 to CA and AB, we shall have 
 
 nam A — I sin G 
 '^~ n cos A + I COS G — m' 
 
 Z sin i5 — «i sin A 
 
 tan yfr = ^ 5— ;; • 
 
 ^ Z cos £ + «t cos ^ — re 
 
 w. 
 
50 
 
 THE INTERSECTION OF STRAIGHT LINES. 
 
 48. To find the tangent of the angle between the two straight 
 lines represented hy the equations 
 
 la + m$ + ny = 0, 
 
 h' + wy3' + ni = 0. 
 
 Let 6, 6' be their inclinations to the line of reference BC. 
 Then if D denote the required angle between the straight lines, 
 we have 
 
 D = e~ff, 
 
 -- , . -,, tan 6 — tan ff 
 
 UnD = ±Uii {6 -&) = + —- — 3- — -s 
 ^ '1 + tan 6 tan ff 
 
 (main G-n sin5)(m'cos C+w'cos B-l')—{m'am. C—n'am B){m cosC+ n cos B-V) 
 (7neosC+ncosB—l)(in'coa C+n'coa £—l')-(main G—n sin 5)(ni'sin C— n'sin B) 
 
 _ l(m' sitlG ~n' s\nB)+m{n' a\aA—l' SL-aG)+n{l' AnB—m' siuA) 
 ll'+ mm'+ nn!— {ran'+ m'n) coaA — (ni'+ nl) ooaB — {lm'+ I'm) cos G ' 
 
 or (as we may write it), 
 
 I, m, n 
 
 I, m, n 
 
 sin.ll, sin .5, sinC 
 
 ir+ mm'+ nn— {mn'+ m'n) cos A — (nr+ n'l) coa B—(lm'+ I'm) cos C 
 
 49. Cor. 1. The straight lines whose equations are 
 la. + wi/3 + ny = 0, 
 and la + m'^ + w'y = 0, 
 
 are at right angles to one another provided 
 
 W + mm' + nn' — {mn' + m'n) cos A — (jil' + n'l) cos B 
 
 - {Im' + I'm) coa C=0. 
 
 Cor. 2. If the equation 
 
 Ma' + vyS* + w/ + 2u'0y + 2v'fa + 2M>'a/3 = 
 represent two straight lines, they will he at right angles provided 
 u + V + w — 2u' cos A — iv cos B - iw cos 0=0. 
 
PAKALLELISM. INFINITY. 
 
 51 
 
 50. Obs. We shall in the course of the work give several 
 other methods of finding the expression for the angle between 
 two lines whose equations are given. The method in the fore- 
 going article is generally thought to be the .most convenient ; 
 but the student is recommended not to pass over, simply because 
 they lead only to results already obtained, those other methods 
 which we shall give, but to read them as very suggestive exam- 
 ples of the application of trilinear coordinates. 
 
 The methods given in Chapters v. and vi. in particular are 
 offered as very good illustrations of the use which may be made 
 of those forms of equations which it is the special object of 
 those two chapters to develop. 
 
 51 . To determine the sines of the angles of a triangle the 
 trilinear coordinates of whose angular points are given. 
 
 Let P, Q, R be the angular points of the triangle and 
 (a.. /3,, 7.), (a„ /3„ 7,), («„ ^„ 7,) 
 their coordinates. 
 
 Then 
 therefore 
 
 PQ .FEamP=2 area PQR ; 
 2 area. PQR 
 
 Bin P= ■ 
 
 PQ.PR 
 
 
 where 
 
 4= 
 
 /3,. 7, 
 
 , M,^ 
 
 %> «3 
 
 . N,^ 
 
 «,. ^, 
 
 
 /8., 7. 
 
 
 7.. «! 
 
 
 a„ ^1 
 
 4^ 
 
 /3„ 7. 
 
 , M,= 
 
 7.. «! 
 
 , J^3^ 
 
 «,> ^^ 
 
 
 A. 7, 
 
 
 7.. a. 
 
 
 «,. /3, 
 
 4—2 
 
52 
 
 THE INTERSECTION OP STRAIGHT LINES. 
 
 But 
 
 Hence 
 
 
 a, /3. 7, 
 
 «!-«!> ^i-^x 7.-7s 
 ai-«8. A -^8. 7i-73 
 
 2A 
 a 
 
 2 A, A, 7. 
 
 0, A-^.. 7,-7, 
 0, i8.-^3, 7.-7» 
 
 /3,-/33, 7i-7. 
 
 1 
 
 = 2Aa 
 
 c, a 
 
 > 
 
 
 c, a 
 
 > 
 
 a, b 
 a, b 
 
 (Art. 41) 
 
 2A 
 
 a, b, c 
 
 sin A, sin B, sin (7 
 
 Ms, 
 
 L„ 
 
 N, 
 
 sinP=- 
 
 sin A, smB, sin C 
 
 and similar expressions may be written down for sin Q, and sin B. 
 
 52. To fivd the sine of the angle between two straight lines 
 whose equations are given. 
 
 Let the given equations be 
 
 la + m^ + M7 = 0, 
 
PARALLELISM. INFINITY. 
 
 53 
 
 •'-** (<"! > ^1 > 7i) denote the point of intersection of tliese two 
 lines, and let (oj, /3,, 7,) be any other point on the first line, 
 ^id (a„ ^3, 73) any other point on the second. 
 
 Then if D be the angle between the lines we shall have, 
 with the notation of the last article, 
 
 sin^, sin -5, sin G 
 
 sin 2? = + 
 
 ^[L„ M„ N,}{L„ M„.\} 
 
 But 
 
 A_^3_a; 
 
 and 
 
 therefore substituting 
 
 6mJ) = ± 
 the expression required. 
 
 ^^-E-E 
 
 sin^, 
 
 sin 5, sin C 
 
 I, 
 
 m, n 
 
 I', 
 
 m, n 
 
 }Z, m, w} {?, »«', n\ ' 
 
 Other methods of arriving at these results will be given in 
 Chapters V. and vi. 
 
 53. The expression for sin 2) obtained in the last article 
 might have been deduced from the expression for tan J) obtained 
 in Art. 48 ; but the process of squaring and adding the numera- 
 tor and denominator of that expression and resolving the result 
 into its factors would have been tedious, so that it is perhaps 
 more convenient to investigate the sine and tangent indepen- 
 dently. 
 
54 EXEBCI3ES ON CHAPTER IV. 
 
 From a comparison of the results of Arts. 48 aud 52 we can 
 immediately write down the expression for the cosine of the 
 angle, 
 
 viz. cosD 
 
 U'+mm'-\- nn'— (mn'+m'n) coa A- (nl'+n'l) cos B—{lm'+l'm) cos C 
 {I, m, n] {I', m, n'} 
 
 Exercises on Chapter IV. 
 
 (24) Find the coordinates of the point of intersection of the 
 two straight lines whose equations are 
 
 a = 7 cos B, 
 
 /8 = 7 cos A ; 
 
 and find the equation of the straight line joining this point with 
 the point of reference C. 
 
 (25) If the sides QR, RP, PQ of a triangle PQR be repre- 
 sented respectively by the equations 
 
 »Mj8+ M7- 2Za = 0, 
 W7+ la. — 2to/9 = 0, 
 la, + m^ — 2n7 = 0; 
 
 find the equations to all the straight lines joining the points 
 P, Q, B with the poftits of reference. 
 
 (26) Shew that the straight lines 
 
 {a + d)a+{b +rf)/9 + C7 = 0, 
 and {a + d)a+(b-d)fi + cy = 0, 
 
 are at right angles to each other. 
 
EXERCISES ON CHAPTER IV. 55 
 
 (27) Shew that the straight lines 
 asinjB+^sin {B- C) +7sin C cos C= 0, 
 a cos 5 + /3 cos {B- C)+y sin' C =0, 
 
 are parallel, and that each is parallel to the straight line 
 asin(^- C)+^amA+ysm Ccos (7=0. 
 
 (28) Shew that the equations 
 
 a cosec A + ^ cosec B = 0, 
 a cos A+^ cos B — y cos C = 0, 
 represent parallel straight lines. 
 
 (29) Find the condition that the straight line 
 
 la + m/S + ny = 
 may be parallel to the side BC of the triangle of reference. 
 
 (30) Find the condition that the straight line 
 
 h + m/S + ny = 
 
 may be parallel to the bisector of the angle A of the triangle of 
 reference. 
 
 (31) Shew that the straight lines whose equations are 
 
 o + 7 cos 5 = 0, 
 
 /8 + 7 cos ^ = 0, 
 are parallel. 
 
 (32) Find the angle between the straight lines whose equa- 
 tions are 
 
 a — 7 cos ^ = 0, 
 
 /8 — 7 cos .4 = 0. 
 
 (33) The perpendiculars from the middle points of the sides 
 of the triangle of reference are given by the equations 
 
 /8sln5-7sin C+asin {B- G) =0, 
 
 7 sin (7— asin^ + j8sin(C— .4) =0, 
 
 asin.4-;9sin5 + 78in(-4- B) =0. 
 
56 EXERCISES ON CHAPTER IV. 
 
 (34) Straight lines are drawn from the angular points of the 
 triangle of reference so as to pass through the point given by 
 
 la = m^ = W7, 
 
 and so as to meet the opposite sides in the points A', B, C: 
 find the equations to the sides of the triangle ABC. 
 
 (35) Find the equations to the sides of the triangle whose 
 angular points are given by 
 
 (a = 0, andj8+ h =0), 
 
 (/8 = 0, and 7 + »»a = 0), 
 
 (7 = 0, and a + w/3 = 0), 
 respectively. 
 
 (36) If be the centre of the circle circumscribing the 
 triangle of reference, and if AO, BO, GO be produced to meet 
 the opposite sides in A'B'C, shew that three of the four straight 
 lines represented by the equations 
 
 a sec ^ + /3 sec 5 + 7 sec C = 
 
 are the sides of the triangle A'B'C; and construct the fourth 
 straight line. 
 
 (37) Draw the four straight lines represented by the 
 equations 
 
 o cos .4 + /3 cos 5 + 7 cos C = 0. 
 
 (38) Draw the four straight lines represented by the 
 equations 
 
 a + /3 + 7 = 0. 
 
 (39) Interpret the equations 
 
 o sin .4 + /3 sin 5 + 7 sin C = 0. 
 
 (40) Of the four straight lines whose equations are 
 
 la. + m^ + W7 = 
 
EXERCISES ON CHAPTER IV. 57 
 
 two intersect in P, and the other two in P; two intersect in Q, 
 and the other two in Q ; two intersect in R, and the other two 
 in ^'; find the coordinates of the middle points of PP , QQ', 
 RR ; and shew that they lie on one straight line. 
 
 And find the equation to this straight line. 
 
 (41) On the three sides of a triangle ABC triangles PEG, 
 QCA, BAB are described so that the angles QAC, BAB are 
 equal, the angles BBA, PBG are equal, and the angles PCB, 
 QCA are equal; prove that the straight lines, AP, BQ, CB 
 pass through one point. 
 
 (42) Shew that the point determhied hj 
 
 aa. _ b^ _ C7 
 n—l l—m m—n 
 
 and the point determined by 
 
 aa _ J/S _ C7 
 l—m m—n n—l 
 
 both lie at infinity, and shew that the angular distance between 
 them, viewed from any finite point, will be a right angle if 
 
 a^{m-ny+F{n-Tf + c^a- nif = [al, Im, mf. 
 
CHAPTER V. 
 
 THE STEAIQHT LINE. THE EQUATION IN TERMS OF THE 
 PERPENDICULARS. 
 
 54. We have shewn that if p, q, r be the perpendicular 
 distances of the points of reference from any straight line, the 
 equation to this straight line will be 
 
 apa. + hq^ + cr7 = 0. 
 
 We proceed to consider some applications of the equation 
 of a straight line in this form. But it will first be necessary 
 to establish a relation which exists among the perpendiculars 
 
 55. If P> if ''' ^fi *^s perpendicular distances of the angular 
 points of the triangle ABC from, any straight line, then will 
 
 a'j)^ + Pq^ + <?r^ — 2bcqr cos A — 2bcrp cos B 
 
 — 2ahpq cos G = 4 A". 
 
 Let AP, BQ, CB (fig. 14) be the perpendiculars from ABG 
 on the straight line PQR. 
 
 Draw BM, CN perpendiculars upon AF. 
 
 Then AM==p~q, 
 
 and therefore (Euclid, i. 47), 
 
 BM= ± 's/c'-ip-qY. 
 
THE EQUATION IN TERMS OF THE PEKPENDICULARS. 59 
 
 Fig. 14. 
 
 Similarly CN=± Vi" - (r - pY, 
 
 and i?Q = ± Va"- (2 -»•)'. 
 
 But (having regard to algebraical sign in relation to the 
 direction of straight lines) 
 
 BM+NC+BQ = 0. 
 
 Hence 
 
 + V{a' - (2 - r)') + V{i' -(r-py]± ^[c" -(j>- qY) = 0, 
 which when cleared of radicals reduces to 
 
 a'p* + b*q* + cV — 2bcqr cos A — 2carp cos B 
 — 2abpq cos C= 4A°, 
 the relation required to be established. 
 
 N.B. With the notation introduced in Art. 46, Note, this 
 result may be written 
 
 {ap, hq, cr] = 2A. 
 
60 THE STRAIGHT LINE. 
 
 56. To find the perpendicular distances of the points of 
 reference from the straight line whose given equation is 
 
 la. + »i/3 + W7 = 0. 
 
 Let^, q, r be the perpendiculars required. Then the straight 
 line might be represented by the equation 
 
 apa. + iql^ + C9'7 = 0, 
 
 which must therefore be identical with the given equation 
 
 lo. + m/3 + W7 = 0. 
 
 Therefore ^^h^'I , 
 
 I m n 
 
 and since these fractions are equal, each must be equal to 
 
 {ap, hq, cr) 
 {I, m, n] ' 
 
 which by the last article is equal to 
 
 2A 
 {I, ni, n] ' 
 hence 
 
 = ?^ ? _ 2A m _ 2A w 
 
 ^~ a '{l,m,n\' ^~ b ' {I, m, n}' ** ~ ~c ' {I, m, n] ' 
 
 57. The equation to a straight line being given in the general 
 form 
 
 la. + »w/8 + «7 = 0, 
 
 to reduce it to the equation in terms of the perpendiculars. 
 
 "We have only to multiply the equation throughout by 
 
 2A 
 
 {I, m, n] ' 
 
 since by the last article the expression 
 
 ^^ la + m^ + ny 
 {I, m, n} 
 is identical with 
 
 apa. + hq0 + cry. 
 
THE EQUATION IN TEEMS OP THE PERPENDICULARS. 61 
 
 58. To find the perpendicular distance of the point (a', jS", 7') 
 firom a straight line whose equation in terms of the perpendiculars 
 
 is given. 
 
 Let apz + bq^ + cry = 
 
 be tlie given straight line, and let a line be drawn parallel to 
 this through the given point (a', fi', 7'). 
 
 Then if p be the perpendicular distance required, 
 
 p±p, q±P, r + p 
 
 (the upper signs going together and the lower together) will 
 represent the perpendicular distances of the new line from 
 
 A, B, a 
 
 Therefore the equation to the new line is 
 
 aa. {p±p) + JyS {q±p)+ cy {r±p) = 0. 
 
 But, since this straight line passes through (a', ^', 7'), 
 
 aa'ip ±p) + b^'(g ±p)+cy'{r± p) = 0, 
 
 or (oa + J/3' + cy') p = + (apa.' + hq0 + cry') , 
 
 and therefore 
 
 _ apa' + bq/3' + cry 
 
 the expression for the distance required. 
 
 59. To find the perpendicular distance of the point {a, /9', 7') 
 from any straight line whose equation is given in the general 
 form 
 
 loL + »»/8 + W7 = 0. 
 
 Let p, q, r be the perpendicular distances of the straight 
 line from the points of reference. 
 
 Then by the last article the required distance is given by 
 
 apa' + Iq^' + cry 
 P=^ 2A • 
 
62 
 
 But by Art. 56, 
 
 I 
 
 ap 
 
 THE STRAIGHT LINE. 
 
 _h 
 
 cr 
 
 [I, m, n] 2A ' [I, m, n\ 2A ' {I, m, n\ 2^ ' 
 
 Hence the last equation becomes 
 
 _ la.' + m0' + ny 
 ^ ~ [I, m, n\ ' 
 
 the same expression which we obtained by another method in 
 Art. 46. 
 
 60. To find the angle between two straight lines in terms of 
 their perpendicular distances from the angular points of a 
 triangle. 
 
 Fig. 15. 
 
 Let D be the angle between the two straight lines OPQE 
 and OP' Q'R' intersecting in 0. 
 
 And let p, q, r be the perpendicular distances of the former 
 line — and p, q, r those of the latter — from three points ABO 
 forming a triangle. 
 
THE EQUATION IN TEEMS OF THE PEEPENDICULAES. 63 
 
 Then AP'OP=^ABOP-ABOr, 
 
 that is, OF. OP' smD = q.OP-q OF. 
 
 Similarly OP.OP' &mD = r . OP- r OP'. 
 
 Hence eliminating OP, 
 
 (r — q) OP . sin Z) = qr — q'r. 
 
 Similarly {p —f) OQ. sin D = rp — r'p, 
 and {q —p) OR . sin Z) = pq — p'q, 
 
 therefore by addition, 
 
 |(r- - q) 0P+ {p-r)OQ+(q -p) 0R\ sin D ■■ 
 
 2. 
 
 t 
 
 1, 1, 1 
 
 But AABC= AAB Q + CQP- BBP, 
 
 therefore 
 
 2A=p.{OB-OQ)+q.iOQ-OP)-r.{OB-OP) 
 
 = {r-q)OP+{p-r)OQ+{q-p)OR. 
 Hence 2 A sin i> = p, q, r 
 
 P'> 2. »• 
 1, 1, 1 
 
 which gives D in terms of the perpendiculars. 
 
 61. To dediice the expression for the angle between the two 
 straight lines whose equations are 
 
 la + m/3 + W7 = 0, 
 
 and I'oL + »»'/3 + m'y = 0. 
 
 If p, q, r; p', q, r' be the perpendicular distances of these 
 lines from A, B, C, we have by Art. 56, 
 
 ap _iq _cr _ 2A 
 I ~ m n~ [l, m, n] ' 
 
64 
 and 
 
 THE STRAIGHT LINE. 
 
 2A 
 
 ap' _h£ _cr^ 
 
 T~ m~ v! ~ [X, m, n] ' 
 
 But if D be the angle between the lines, we have by the 
 last article 
 
 sin 2) = 
 
 therefore 
 
 sini> = 
 
 2A 
 
 2A 
 
 r t I 
 
 P, 2> »• 
 1, 1, 1 
 
 {l,m,n]{T, m',n'} 
 
 2A 
 aba 
 
 {I, m, n]{l', m, n] 
 
 I m n 
 a' b' c 
 
 I TL '^ 
 a' h' c 
 
 1, 1, 1 
 
 I, m, n 
 I, m, n 
 a, b, c 
 
 k 
 
 m, n 
 
 I', 
 
 t f 
 m , n 
 
 sin^, 
 
 sin B, sin G 
 
 {I, m, n] {I', to', n'} 
 the same expression which we otherwise obtained in Art. 52. 
 
 62. To find the altitude of the triangle whose base is given 
 by the equation 
 
 apa + bq^ + cry = 0, 
 
 and the other two sides by the equations 
 
 ap'a + bgf^ + cr'y = 0, 
 
 and ap"a + bg"^ + cr"y = 0. 
 
THE EQUATION IN TERMS OF THE PERPENDICULARS. 65 
 
 Let 7i. denote the altitude required, and suppose a, /3, 7 the 
 coordinates of the vertex of the triangle, then, by Art. 57, 
 
 , _ ap% + hq^ 4- cr< y 
 
 where a, yS, 7 are to be determined from the equations 
 ap'a + Jj'yS + cr'7 = 0, 
 ap'a. + hq'^ + cr"7 = 0, 
 ai + J/3 + C7 =2A. 
 
 These equations give 
 
 cy _ 
 
 2A 
 
 1, 1, 1 
 /, 2', »•' 
 
 But since the first three of these fractions are equal, there- 
 
 2'. 
 
 r 
 
 
 r, p 
 
 
 P> 
 
 2, 
 
 
 2". 
 
 r" 
 
 
 r", p" 
 
 
 P" 
 
 2" 
 
 
 fore 
 
 each = 2£^^+M+^ 
 
 i^, 2i »• 
 
 P'> 2'. »■' 
 
 ff f/ tt 
 
 p , q , r 
 
 Therefore 
 
 _ apoL + bg^ + cry _ 
 *- 2A 
 
 /'. 
 
 2> 
 
 r 
 
 ;''. 
 
 2'> 
 
 r 
 
 ;"" 
 
 2" 
 
 It 
 r 
 
 1, 
 
 1, 
 
 1 
 
 P' 
 
 2'> 
 
 r 
 
 p 
 
 2" 
 
 r 
 
 63. To find the lengths of the sides of the same triangle. 
 Let p. p, p denote the lengths of the sides whose equations 
 are respectively 
 
 W. "• 
 
66 
 
 THE STEAIGHT LINK. 
 
 apa. + hqfi + cry = 0, 
 op'a + &2'/3 + cr'7 = 0, 
 ap"a + J2"/8 + cr"7 = 0. 
 
 And let 0, 0', 0" denote the angles opposite to these sides, 
 and Oo, O'o', &o" the perpendiculars from the angles on the 
 opposite sides. Then 
 
 p" sin O = Oo. 
 
 Fiii. Hi. 
 
 But by Art. 60, 
 
 ginO=+-^ 
 -2A 
 
 1. 1 
 q, r 
 
 and by the last article, 
 
 Oo = 
 
 P^ 
 
 ?. 
 
 r 
 
 J>'' 
 
 !Z'> 
 
 r 
 
 p" 
 
 ?"> 
 
 r" 
 
 1, 
 
 1, 
 
 1 
 
 y. 
 
 i, 
 
 r' 
 
 p" 
 
 Q", 
 
 r" 
 
THE EQUATION IN TERMS OP THE PERPENDICULARS. 67 
 
 therefore substitutiiis 
 
 p"=2A 
 
 and similar expressions can be written down for the sides 
 p and p. 
 
 
 
 P' 
 
 1 
 
 , »• 
 
 
 
 
 p\ 
 
 q, r 
 
 
 
 
 p". 
 
 9% r" 
 
 
 1, 
 
 1, 
 
 1 
 
 
 1, ] 
 
 , 1 
 
 P' 
 
 ?' 
 
 r 
 
 
 p', q, '■' 
 
 p 
 
 It 
 
 r" 
 
 
 n II tt 
 
 P > i , '■ 
 
 64. To find the area of the same triangle. 
 
 We have only to express half the rectangle contained by the 
 base and the altitude. 
 
 Therefore by the last article 
 
 area = A 
 
 
 
 
 
 p, q, r 
 
 
 X 
 
 
 
 
 
 /. 2'. '•' 
 
 
 
 
 
 f II II 
 
 
 
 
 />. 
 
 q> »■ 
 
 
 p, q, r 
 
 
 II 
 
 p • 
 
 q", 
 
 r 
 
 t 
 
 q. r 
 
 
 f, 2", r" 
 
 
 p> 
 
 q> 
 
 r 
 
 1, 
 
 1, 1 
 
 
 ] 
 
 , 1, 1 
 
 
 h 
 
 1, 
 
 1 
 
 65. COE. The expression just obtained is homogeneous 
 with respect to j), q, r, and of zero dimensions, hence it will 
 not be altered if we substitute for p, q, r any quantities pro- 
 portional to them. 
 
 Now suppose that the equation to the base, instead of being 
 given in the form 
 
 apx + bq^ + cry = 0, 
 
 is given in a perfectly general form 
 
 la. + »i/8 + W7 = 0, 
 
 r>— 2 
 
68 
 
 THE STRAIGHT LINE. 
 
 then - , ^ , - are proportional to p, q, r, and may be sub- 
 stituted iorp, q, r in the expression for the area. 
 
 And so with respect to the other two sides of the triangle. 
 Hence we obtain the following theorem : 
 If ia + w»/3 + M7 = 0, 
 
 la + 7w'/3 + n'<y = 0, 
 ra + m"/3 + n"7 = 0, 
 
 he the equations to any three straight lines, the area of the triangle 
 ■which they contain is 
 
 
 
 
 
 
 I 
 
 m 
 
 n 
 
 • 
 
 
 
 
 
 
 
 
 a' 
 
 T' 
 
 c 
 
 
 
 
 
 
 
 
 V 
 
 m' 
 
 m' 
 
 
 
 
 
 
 
 
 a' 
 
 b ' 
 
 c 
 
 
 
 
 
 
 
 
 I" 
 
 m" 
 
 n" 
 
 
 
 
 A 
 
 
 
 
 a' 
 
 b 
 
 c 
 
 
 
 
 ^ 
 
 I 
 
 m 
 
 n 
 
 
 r 
 
 m' 
 
 n 
 
 
 I" 
 
 m" 
 
 n". 
 
 
 a' 
 
 b' 
 
 c 
 
 
 a' 
 
 b ' 
 
 c 
 
 
 a' 
 
 T 
 
 1 
 
 
 V 
 
 m' 
 
 n' 
 
 
 r 
 
 m" 
 
 n" 
 
 
 i. 
 
 m 
 
 n 
 
 
 a' 
 
 b ' 
 
 c 
 
 
 a ' 
 
 b ' 
 
 T 
 
 
 a 
 
 T' 
 
 c 
 
 
 1, 
 
 h 
 
 1 
 
 
 1, 
 
 1. 
 
 1 
 
 
 1, 
 
 1, 
 
 1 
 
 or (multiplying numerator and denominator by aVc^, 
 
 ahc^ 
 
 
 
 
 
 I, m, 
 
 n 
 
 
 
 
 
 
 
 
 V, m', 
 
 n' 
 
 
 
 
 
 
 
 r, m". 
 
 n" 
 
 
 
 
 I, 
 
 m. 
 
 n 
 
 
 t, m', 
 
 «' 
 
 
 I", 
 
 m", 
 
 n" 
 
 t, 
 
 m', 
 
 n 
 
 
 7" _" 
 
 I , m , 
 
 ft 
 n 
 
 
 I', 
 
 m', 
 
 n' 
 
 a, 
 
 i, 
 
 c 
 
 
 a, b, 
 
 c 
 
 
 a. 
 
 h, 
 
 c 
 
EXEECISES ON CHAPTER V. 69 
 
 Exercises on Chapter V. 
 
 (43) Find the equations to the straight lines through the 
 angular points of the triangle of reference parallel to the straight 
 line 
 
 opa + hq^ + cry = 0. 
 
 (44) Find the equation to the straight line through the 
 centre of gravity of the triangle of reference and parallel to 
 the straight line 
 
 apa. + bq^ + cry = 0. 
 
 (45) Find the equation to the straight line bisecting AB 
 and cutting AC at right angles. 
 
 (46) Find the equation to the straight line parallel to BC 
 at a distance d from it on the side remote from A. 
 
 (47) Find the area of the triangle whose sides are given by 
 
 the equations 
 
 h^ + cy = 0. 
 
 cy + aa. = 0, 
 
 aa. + ip = 0. 
 
 (48) Find the area of the triangle whose sides are given by 
 
 the equations 
 
 m^ + M7 = 0, 
 
 n7 + k = 0, 
 
 Za+»i/3=0. 
 
 (49) Find the area of the triangle whose sides are given 
 by the equations 
 
 aa — 6j8 + cy = 0, 
 aa + J/3 — C7 = 0. 
 
 (50) Shew that the angle between the straight lines 
 
 an (p + s) + J^ (j- + s) + C7 (»• + s) =0, 
 
 aa (/ + s) + 5/8 {q + «') + <^ ('*' + «') = ». 
 is always the same whatever be the values of s and «'. 
 
70 EXERCISES ON CHAPTER V. 
 
 (51) Througli the points of reference A, B, C straight 
 lines are drawn parallel respectively to the straight lines 
 
 apa + hqP + err/ = 0, 
 
 ap'a + hc[^ + cr''^ = 0, 
 
 ap'a + bq'^ + cr'y = ; 
 
 shew that they will meet in a point, provided 
 1, 1, 1, 1 ' = 0. 
 P, q, r, p 
 
 p', q\ '•'. q 
 
 „ n ., i 
 
 P . a , r , r I 
 
 (52) Shew that if j? — q = c, the straight line 
 
 apOL + bq^ + cry = 
 is at right angles to the line of reference AB. 
 
 (53) Apply the result of Art. 55 to find the equation to the 
 straight line for which p — q = c and r = 0. 
 
 (54) Shew that it p — q = 0, the straight line 
 
 apa + bqP + cny = 
 is parallel to the line of reference AB. 
 
 (55) Apply the result of Art. 55 to find the equation to 
 the straight line for which p — q = and r = 0. 
 
 (56) PQBQ' is a parallelogram of which the diagonal QQ' 
 coincides with the line of reference CA, and the points P, B 
 lie on BG, AB, respectively. If the hase PQ be represented 
 by the equation 
 
 apa + bq^+cry=-0, 
 
 find the altitude of the parallelogram in terms of ^, q, r. 
 
CHAPTER VI. 
 
 THE EQUATIONS OP THE STRAIGHT LINE IN TERMS OP THE 
 DIRECTION SINES. 
 
 66. When we say that the equation 
 la 4 »i/9 + ny=0 
 
 represents a straight line, we do not mean that any values what- 
 ever of o, /8, y which satisfy the equation will be the coordi- 
 nates of a point upon the line : for unless a, j3, 7 also satisfy 
 the relation 
 
 they will not be the coordinates of a point at all, although a 
 point may be found haying its coordinates prcportional to 
 them. 
 
 In other words, if the equation 
 
 la + mfi + ny = 
 
 is to be regarded as a relation among the coordinates of any 
 point upon the line and not merely a relation among their ratios, 
 we must regard the equation 
 
 aa + ijS + C7 = 2A 
 as understood to be simultaneously satisfied. That is, the coor- 
 
72 
 
 THE EQUATIONS OF THE STRAIGHT LINE 
 
 dinates of any point on the straight line must satisfy the 
 simultaneous system 
 
 la + inl3 + n7 = 1 
 
 aa + b/3 + cy = 2AJ " 
 
 We may therefore with the greatest strictness speak of this 
 system of two simultaneous equations, as representing tlie 
 straight line, or defining the coordinates of any point on it. 
 
 Instead of these two equations we may use any equivalent 
 pair of equations obtained by combining them. And if a', y8', y 
 denote known coordinates of any point upon the line, we can 
 express the system of equations in a very convenient form. 
 
 Thus : since (a', y8', 7') lies upon the line, we have 
 la + m^' + wy' = \ 
 and aa! + b^' + 07' = 2Ai ' 
 
 in virtue of which relations the original system can be put 
 into the form 
 
 Z(a-a')+«i(y8-/9')+n(7-7') =0) 
 
 -7')=or 
 
 a{a- 
 
 a') + b 
 
 [iS-/3') + 
 
 c (7 - 7') = 
 
 or 
 
 
 a - a' /3 - /3' 
 
 
 7-7' 
 
 
 m, n 
 
 - n, I 
 
 
 I, m 
 
 
 b, c c, a 
 
 
 c, a 
 
 which we may write 
 
 
 o-a' 
 
 _P-^_ 
 
 = 1- 
 
 -7 
 
 \ ft, V 
 
 where \, /t, v are proportional to the determinants 
 
 m, n 
 
 
 n, I 
 
 
 I, m 
 
 b, c 
 
 7 
 
 c, a 
 
 9 
 
 a, b 
 
 or (which is the same thing), where \, /i, v satisfy the relations 
 
 a\ + bfi + cv = 0, 
 and lK-\-mii + nv=0, 
 
IN TEEMS OF THE DIRECTION SINES. 
 
 73 
 
 67. It follows that if a', 0, y' be the coordinates of any 
 point, the system of equations 
 
 a — a' _ /3 — /S' _ 7 - 7 
 X, /i 1/ 
 
 will represent a straight line provided 
 
 oK + hfi, + cv= 0, 
 
 and if the equation to the same straight line in the ordinary 
 form be 
 
 Is. + m^ + W7 = 0, 
 
 the ratios I : m : n will be determined by the equations 
 
 a7 + /S'm + 7'n = ; 
 that is, the equation in the ordinary form will be 
 
 a. A 7 
 
 \, [I, V 
 
 = 0. 
 
 68. We proceed'to obtain the equations to a straight line 
 in the form 
 
 a — a'_yS — ;8'_7 — 7' 
 \ /u. V ' 
 
 without reference to the ordinary form. 
 
 Let OP be the straight line whose equations are to be found, 
 and let a, /S", 7' be the coordinates of the fixed point 0, and 
 o, (3, 7 those of any point P upon the straight line : and let p 
 be the distance between these two points. 
 
 It will be observed that p like a, ^, 7 is a variable quan- 
 tity dependent upon the position of P. 
 
 Through draw Oa, Oh, Oc parallel to the lines of refer- 
 ence BC, CA, AB respectively, and so that the angles lOc, 
 
74 
 
 THE EQUATIONS OF THE STRAIGHT LINE 
 
 cOa, aOb may be the supplements of the angles A, B, C 
 respectively. 
 
 Fig. ir. 
 
 Let 6, <f), ■^ denote the angles POa, POh, POc all measured 
 in the same direction from the initial line OP. 
 
 then 
 
 Draw the perpendiculars PA', PB\ PC, 
 PA' a -a' 
 
 sin ^ = ■ 
 
 OP p • 
 
 Also, due regard being had to the algebraical signs, 
 sm ^ = " — ^ , 
 
 and 
 
 sm'^ = 
 
 _ 7-7 
 
IN TERMS OP THE DIRECTION SINES. 75 
 
 Hence, if X, /x, v be proportional to the sines of the angles 
 6, <}), -^j we have 
 
 a — a'_^ — /9'_7-7' 
 X /J, V 
 
 relations among the coordinates (a, /3, y) of any point on the 
 given line, and therefore representing the given line. 
 
 And ftirther, if \, fi, v be not only proportional but actually 
 equal to the sines of the angles 0, <f>, yfr, we may write 
 
 « — a'_ y3 — j5' _ 7 — 7' _ 
 X ~ fJL ~ V ~''' 
 
 where p is the distance between the points (a, fi, 7) and 
 (a, fi", 7')- 
 
 69. Def. The sines of the angles which any straight line 
 makes with the three straight lines of reference may conveni- 
 ently be termed the direction sines of the straight line. 
 
 70. To find the relations among the direction sines of any 
 straight line. 
 
 Let X = sin 5, /t = sin <l>, v= sin i/r be the direction sines of 
 any straight line, 
 
 then (Art. 68), <}>-d = 7r-C, 
 
 and ■\lr — (f> = -7r — A. 
 
 Hence we have sin = — sin ( C + ^), 
 and Bmylr = sin{A—<}>). 
 
 Consequently we may write 
 
 \ = -sin(C+^),| 
 fi = sin <f>, c 
 
 V = sin {A — 0), ' 
 
 and these equations will, on the elimination of <^, lead to two 
 
76 THE EQUATIONS OP THE STRAIGHT LINE 
 
 equations among X, /*, v, and the angles of the triangle of 
 reference. 
 
 Performing the elimination between the first and second, and 
 between the second and third equations, we get 
 
 \' + fi^+ 2\fi cos C= sin^C, 
 
 and fjj' + v^ + 2fiv cos A = sin"^. 
 
 By symmetry we must also have 
 
 v' + \^ + 2p\ cos B = sin'B ; 
 
 but this does not express any new or independent relation, 
 being obtainable from the two former by the elimination of fi. 
 
 Also since 
 
 o — a _ ;S — /S' _ 7 — 7' 
 X fi 1/ ' 
 
 and since the simple function of the numerators, 
 
 a(a-a')+6(/3-/9')+c(7-7'), 
 
 is zero, the similar function of the denominators must be also 
 zero, i.e. 
 
 aX + J/i + cv = 0, 
 
 a different relation among X, fi, v, but not an independent one, 
 for this must also be implied in the former equations, since 
 they were shewn to express the necessary and sufficient relations 
 among X, /i, v. 
 
 Hence we arrive at the conclusion that the equations 
 
 g-a '^ ;8-/3' _ 7-7' _ 
 X fi V 
 
 will represent a line passing through the point (a, ^, 7'), 
 /3 being the distance between this, point and the variable point 
 («j A 7)1 provided, and provided only, that X, fi, v satisfy the 
 conditions 
 
IN TERMS OP THE DIRECTION SINES. 77 
 
 a\+hll + CV — 0, 
 
 /i' + v' + 2fiv cos A = sinM, 
 v^ + \' + 2v\ cos P= sin'^, 
 X' + yu.^ + 2\/i cos (7 = sin^ C, , 
 
 ^ 
 
 ■which are equivalent to only two independent equations. 
 
 71. The required conditions of the last article are given hy 
 any two of the four equations just written down, or by any two 
 equations that can be formed by combining them. We pro- 
 ceed to obtain two such which are sometimes more convenient, 
 as involving all the coordinates symmetrically. 
 
 It will be sufficient to start with the first two equations, 
 
 a\ + J/i + 01^ = (I), 
 
 /i" 4- r' + 2/xi' cos J = sin'J. .....'. (2). 
 
 From (1) we get, transposing and squaring, 
 jy-l-cV+ 260/^1; = aV, 
 
 or 2/ti/ = ■ 
 
 he 
 
 Substituting this in (2), we get 
 
 Jc/i' + Icv"^ + (aV - Sy - c=i/') cos A=hc sinM, 
 
 whence 
 
 \V cos A + /x'oi cos B + -^ac cos G=hc sinM, 
 
 and therefore (since sin^, sin 5, sin C are proportional to 
 a, b, c) 
 
 X" sin A COS A + fi' sin B cos B + v' sin C cos C 
 = sin A sin B sin G, 
 
 or X' sin 2 A + /tt' sin 25 + v" sin 2 C 
 
 = 2 sin j1 sini? sin C. (3), 
 
 a result to be remembered. 
 
78 THE EQUATIONS OF THE STRAIGHT LINE 
 
 Again from (1), 
 
 and therefore 
 
 so 
 
 -&M= 
 
 ■■ ciK + cv, 
 
 ll"^ 
 
 a\fi + cfiv 
 
 /i — 
 
 b 
 
 ^? = 
 
 a\v + bfiv 
 
 c 
 
 Substituting these in (2), we get 
 
 acKfi + cV" + o^^v + h'fiv — 2fiv cosA = — be sin" A, 
 or a^fiv + acKfi + abXv + be sin'M = 0, 
 
 or fj-v sin A + p\smB+ X/i sin C 
 
 + sin-4 sin 5 sin C=0 (4), 
 
 another notable result. 
 
 And similarly we may form ad libitum a variety of equations 
 connecting X, fi, v, each one implicitly contained in the system 
 of equations in Art. 70. > 
 
 72. It may well be noticed that each of these equations 
 (except the simple equation a\ + b/ji + cv = 0, which only in- 
 volves the ratios oi \ : fj, : v) furnishes us with a different 
 expression for the distance between two points whose coordinates 
 are given. 
 
 For let (a, yS, 7), (a', /8', 7') be the two points, and let 
 X, fj,, V be the direction sines of the line joining them, then 
 
 0^— a' _ /3— /y _ 7 — 7' _ 
 
 and \, /t, V satisfy the equations of the last article ; 
 
 therefore from equation (3), Art. 71, we get 
 
 , ^ (g - a')' sin 2^ + Q - /g')' sin 2g + (7 - 7')' sin 2 C 
 ^ sin A sin B sin C ' 
 
 and from equation (4), 
 
 ,_ (S-g'K7-7>i"^+(7-7')(a-a')sing+(a-a')(;3-j8')sinC 
 '' ~ sin ^ sin 5 sin C ' 
 
IN TEEMS OF THE DIRECTION SINES. 
 
 79 
 
 two expressions for the distance, perhaps more interesting from 
 their symmetry than useful in practice. 
 
 73. Let p, q, r be the perpendiculars from the points of 
 reference on the straight line whose equation is 
 
 a — a'_jS — yS' 7 — 7' 
 
 ■P- 
 
 Fiff. 18 
 
 Let be the inclination of this straight line to the line of 
 reference BO, then 
 
 \=sin5=+ -i^Il^; 
 a 
 
 therefore q — r=±aX, 
 
 r — p=± bfi, 
 
 J) — g = ± cv, 
 
 the upper signs going together, and the lower together, since 
 we must have by addition 
 
 = a\ + bfi + cv. 
 
80 THE EQUATIONS OF THE STRAIGHT LINE 
 
 Hence, substituting in the equation 
 
 lj.'+v'+ 2/ij' cos A = aiii'A, 
 (Art, 70), we get 
 
 (r-^ + (^' + Hr-p){p-q) ^^^ ^ ^ ^j„,^^ 
 
 or c' (r -j>y + P{p- qf +2{r-J>) ip-q) he cos A = 5V sinM, 
 or o^ + jy + cV — 2bcqr cos ^ — 2carp cos £ 
 
 — 2a6pg' cos C = JV sin" J. = 4 A", 
 
 the same relation among the perpendiculars from the points of 
 reference on any straight line, which we have already obtained 
 in Art. 55. 
 
 74. If instead of substituting in the equation 
 
 /I.' + v" + 2fj,v cos A = sin'-il, 
 we had taken the equation (4) of Art. 71, viz. 
 
 fiv sin A + v\ sin B + X/isin C+ sin A sin B sin C = 0, 
 we should have obtained our result immediately in the form 
 
 a^{p — q) ip-r) +b' {q-r){q -p) + c'{r-p) (r — q) = 4A*, 
 a form in which we shall liereafter find it useful. 
 
 Or if we had substituted in the equation (3) of Art. 71, we 
 should have got 
 
 {q - rf cot A + [r -pf cot B+[p- q)' cot C = 2 A, 
 
 another useful foi-m. 
 
 75. To find the angles between the straight lines 
 o — a' _ /3 — /8' 7—7' 
 
 and 
 
 \ 
 
 H- 
 
 V 
 
 ^h 
 
 a- a' 
 
 ^-0 
 
 t 
 
 _y-y' 
 
 V 
 
 = p. 
 
IN TERMS OP THE DIRECTION SINES. 
 
 81 
 
 Let 0, 0' be the angles which the straight lines make with 
 the line of reference CA. 
 
 Then referring to Art. 70, we have 
 
 sin ^ = fi, 
 
 and ain {A — <j)) = v, 
 
 or sin^cos^ — cos-4 sin^ = i/, 
 
 sin A cos tf) — ficoa A = V. 
 
 Therefore 
 
 sin A cos ^ = v+ fi cos A, 
 
 , v + fiCosA 
 i lb = — . — . — . 
 
 and 
 
 COS) 
 
 sin^ 
 
 Similarly, we have 
 
 sin <f>' = /*', 
 
 ,, v'+u'cos^ 
 
 cos d> = f r . 
 
 ^ am A 
 
 Now if D denote the required angle between the given 
 straight lines 
 
 therefore 
 
 sinJ) = sin(^~f) = ± ^('' + ^''^°^^l-;'^'' + ^°°^^^ 
 
 (1). 
 
 sin^ 
 
 ~ am A 
 
 Or we may write it 
 
 amD = 
 
 1 
 
 M. " 
 
 1 
 
 V, X 
 
 1 
 
 \, fi 
 
 sin^ 
 
 /->' 
 
 ~" sin B 
 
 v', V 
 
 sin G 
 
 X', /*' 
 
 So also 
 
 T^ /J jn (l> + U cos ^) (v' + /*' cos ^) , 
 COSi? = COS (<i- <6) = ^ ^ ^T-3 - + H'f^ 
 
 uix' + vv + [iol + /t'y ) COS A , . 
 
 = SiiM ^^^' 
 
 w. 6 
 
82 THE EQUATIONS OF THE STRAIGHT LINE 
 
 and by symmetry 
 
 vv + \\' + {vhl + y'\) cos B 
 sin'-B 
 
 _ W + tuL + (X/t' + Vm) cos P 
 sin^C? 
 
 76. To _)?n«? iAe angle between the straight lines 
 g— a' _ /3— ;3' _ 7 — 7 ' 
 
 g-a' _ /3-j8' _ 7-7 
 
 \' ~ /i' k' ' 
 
 toAere X, /x, i* and V, /*', v' are «o< cgwoZ Jwt owZy proportional 
 to the direction sines. 
 
 The expressions of the last article were obtained on the sup- 
 position that \, fi, V and V, fi, v were actual direction sines. 
 But if we could reduce them to a form in which they would be 
 of zero dimensions in \, /it, v and also in X', /*', v\ they would 
 still be true when these quantities are only proportional to the 
 direction sines. 
 
 Now we have (Art. 71) 
 
 _ V/t' + k' + 2/^y cos ^ _ Vi^ + X' + 2i/X cos B 
 sin A sin ^ 
 
 _ VX' + /x.' + 2X/A cos (7 
 sin 
 
 and similar expressions connecting X', /i', v', in virtue of whicli 
 the results of the last article may be written 
 
 sm D = - 7 - r- I- / _ j^ 
 
 V Oi' + v' + 2/iv cos A) (/*'* + i;'* + 2/it' i;' cos ^) 
 
 nn« D ^ ^^' + '"^' + ( ^''' + Z^'") COS ^ ,p-r 
 
 V (lit" + i'^ + 2/ii' cos ^) (/Lt" + I/" + 2/4' J/' cos ^) ■' 
 
IN TEEMS OP THK DIRECTION SINES. 
 
 83 
 
 whence 
 
 tan jD = 
 
 {fiv — fjdv) sin A 
 
 Hfi' + vv + (jiv + /jfv) cos A 
 
 = &c. 
 
 and these expressions for the sine, cosine and tangent of D are 
 of zero dimensions in \, fi, v, and also in V, fi, v, and are there- 
 fore still true in the case before us when \, fi, v and \', fj!, v are 
 only proportional to the direction sines. 
 
 77. To deduce expressions for the angle between two lines 
 whose equations are given in the form 
 
 la + m0 + ny =0, 
 
 I'a + m'^ + m'y = 0. 
 
 (Compare Arts. 48, 52, 61). 
 
 If (a', P, </) be the point of intersection of these two lines, 
 the lines may be expressed (Art. 66) by the equations 
 
 a — a 
 
 /3-/3' 
 
 7-7 
 
 and 
 
 m, n 
 
 
 M, I 
 
 
 I, m ' 
 
 b, c 
 
 
 c, a 
 
 
 a, b 
 
 a- a' 
 
 
 ^-/3' 
 
 
 7-7 
 
 m, n 
 
 n', I' 
 
 
 h, c c, a \ a, b 
 
 which are of the form of the given equations of the last article. 
 
 We must consider what the functions 
 fiV — fi'v, 
 fjLfjL + vv + {fiv + li'v) cos A, 
 and fji'+j^ + SfivcosA, 
 
 become when we substitute 
 X = 
 
 m, n 
 
 fj.= 
 
 n, I 
 
 V = 
 
 I, m 
 
 b, c 
 
 
 c, a 
 
 
 a, h 
 
 and similar expressions for \', ii, v. 
 
84 THE EQUATIONS OP THE STRAIGHT LINE 
 
 I. We get 
 fiv — ii!v = 
 
 /4, V 
 
 = 
 
 
 n, I 
 
 
 I, 
 
 m 
 
 
 f^',v 
 
 
 c, a 
 
 ! 
 
 a, 
 
 b 
 
 
 
 
 n',l' 
 
 I', 
 
 1 
 
 
 
 
 
 c, a 
 
 ) 
 
 a, 
 
 h 
 
 
 I, m, n 
 I , m , n 
 a, h, c 
 
 See Prolegomenon, Example F. 
 
 II. fjifjb +w'+ [fiv + fiv) cos A 
 
 = {na - Ic) {n'a - I'c) + {Ic - ml) {I'c - m'h) 
 
 + {{na — Ic) {lie — m'b) + {Ic — mh) {n'a — tc)] cos A 
 
 = a^\lT + mm' + nn' — (win + m'n) cos A 
 
 - {nl'+n'l) cos 5- {lm'+ Tm) cosC\. 
 
 III. 
 
 /*" + »'''+ 2/ii/ cos A 
 
 is the same expression as the last with \', /*', v written for 
 \, II, V respectively. 
 
 Therefore it reduces to 
 
 a' {f + »K° + n' — imn cos A — 2nl cos B — Urn cos (7}, 
 
 or with the notation of a former chapter, 
 
 o°{Z, m, w}°. 
 
 Hence we can write down the following expressions for the 
 trigonometrical ratios of the angle between the two straight lines 
 whose equations are 
 
 la, + m^ + M7 = 0, 
 
IN TEBMS OP THE DIRECTION SINES. 
 
 85 
 
 VIZ. 
 
 8mD = 
 
 sin^ 
 
 I, m, n 
 ?, m', n 
 a, &, c 
 
 -T- {Z, m, w} {7, jm', n'}, 
 
 or 
 
 smi) = 
 
 ?, 
 
 m. 
 
 n 
 
 «', 
 
 m. 
 
 n 
 
 sin^, 
 
 smB, 
 
 sin G 
 
 {I, m, n] {I, m', n) 
 
 COS I) 
 
 _U+ mm'+ nn— {mn'+ m' n) cos A— {rd'+n I) cos B— (lm'+ Tm)cosC 
 
 and tan I) 
 
 {I, m, w}{f , m', n'} 
 
 I, m, n 
 
 r, m', n 
 
 sin A, sin B, sin G 
 
 U'+mm'+nn — {mn'+m'n) cos^— {nl'+n'l) cosB— {lm'+ I'm) cos G' 
 
 78. To find, the direction sines of the straight line whose 
 equation is 
 
 la + m^ + W7 = 0. 
 
 Let \, fi, V be the direction sines required. The angle which 
 this straight line makes with the straight line 
 
 I'a + 9n'/3 + w'7 = 
 was shewn, Art. 77. to be given by 
 
 sin 2) = + 
 
 I, 
 
 m, 
 
 n, 
 
 I', 
 
 m', 
 
 n' 
 
 sin -4, 
 
 sin B, 
 
 sin C 
 
 {I, m, n} {I', vi, n'} 
 
86 
 
 THE EQUATIONS OP THE STRAIGHT LINE 
 
 Hence writing iti = 0, n = 0, and dividing numerator and 
 denominator of the fraction by I, we get 
 
 \ = 
 
 m, n 
 sin 5, sin C 
 
 {?, OT, «} 
 
 So writing T = and n' = 0, we get 
 
 n, Z 
 
 sin C, sin ^ 
 
 M = - 
 
 {Z, m, «} 
 
 Z, m 
 sin^, sin 5 
 
 and similarly v^ ^-^^-^ 
 
 which give the direction sines required. 
 
 79. COE. 1. With the same notation the expression for 
 sin B may be written 
 
 sm 
 
 _ _ Z'\ + m'/ii + nv 
 ~ {r, m', n'} ' 
 
 which therefore gives us an expression for the angle between the 
 two lines 
 
 and 
 
 g — g' _ i8 — yy _ 7 — 7 ' _ 
 Z'g + m'yS + n'7 = 0. 
 
 CoE. 2. The two straight lines expresed in Cor. 1 are at 
 right angles provided 
 
 Z'X+ m'/it + nv — + {Z', to', w'}. 
 
 80. To find ike, perpendicular distance of the point (g', /S", 7') 
 from the straight line whose equation is 
 
 la + m^ + nr/ = 0. 
 
IN TERMS OP THE DIRECTION SINES. 87 
 
 Let ^=^:n^=rzV^^ (,) 
 
 be the equation to the perpendicular from («', /S', 7') on the given 
 line. 
 
 Then the length of the intercept, or the value of p at the 
 point of intersection of the two lines, is obtained by combining 
 their equations ; thus (1) gives us 
 
 a = a.'+\p, ^ = ^' + fip, 7 = 7' + i/p, 
 
 and substituting in the given equation, we get 
 
 la' + j»j8' + ny + (Z\ + mfi + nv) p = 0, 
 
 whence la' + m^ + ny' 
 
 l\ + m/j, + nv 
 
 But since the lines are at right angles, we have by the last 
 Cor. 
 
 l\ + mfi + wi» = + \l, m, n], 
 
 therefore ?«' + >»/3- +^7' 
 
 '^ - {I, m,n} 
 
 as before in Arts. 46, 59. 
 
 Cor. The distance of the point (a', ^, 7') from the straight 
 line apa + bqfi + cry = is given by 
 
 P= ± 2^ (opa + h^ + cry), 
 
 since it was shewn in Art. 55 that 
 
 {op, hq, cr] = 2A. 
 
 81. To find the equations to the perpendicular from (a', /?, 7') 
 on the straight line whose equation is 
 
 la + JM/S + W7 = 0. 
 
 Let \, /*, V be the direction sines of the required line, and 
 \', fi, v those of the given line, then from the expression for 
 cos D in Art. 75, 
 
88 THE EQUATIONS OF THE STRAIGHT LINE 
 
 since the cosine of a right angle is zero, 
 
 = /t/i' + ml + [)iv + yiv) ca%A, 
 or 0=/i(/i' + i;'cosjl) + v(i'' + /*'cos^). 
 
 But it' V 
 
 
 «, ? 
 
 
 Z, m 
 
 •> 
 
 
 c, a 
 
 
 o, 6 
 
 
 tituting 
 
 I, mcos-4 — n 
 
 + !» 
 
 Z, m — mcos^ 
 
 a, J cos u4 — c 
 
 
 a, I 
 
 - c COS -4 
 
 = /i 
 
 But 6 — c cos -4 = a cos (y, and c — 6cos-4 = acos^, there- 
 fore dividing by a, we get 
 
 = /* (m — Z cos 5 — w cos ^) + v (Z cos C + n cos ^ — m), 
 
 or 
 
 m — n cos A — I cos (7 m — Z cos B — tn cos ^ ' 
 and therefore 
 
 I —m cos (7 — m cos B ' 
 
 Hence the equations to the perpendicular will be 
 
 g-a' ^ ^-i8' 
 
 Z — m cos C — n cos 5 m — w cos A — I cos (7 
 
 ^ 7-7' 
 
 M — Z cos B — m cos ^ ' 
 
 82. The equations of the last article 
 
 I— m coa C—n cos B m — ncosA — lcosG 
 
 V 
 
 n- 
 
 I cos B — m cos A ' 
 
 .(1). 
 
IN TEEMS OF THE DIRECTION SINES. 
 
 89 
 
 express the conditions that X, fi, v may be the direction sines of 
 a line at right angles to the line 
 
 h + m/3 + ny = 0. 
 
 But since they imply the relation 
 
 a\+h/i + cv = 0, 
 
 they express only one further condition. 
 
 To find this one condition in a symmetrical form, we have 
 from the first of the equations in (1), 
 
 I {/i-i-X cos (J) — m {\ + fjL COS C) +n {/M cos ^ — \ cos A) = 0, 
 
 whence dividing by C, and remembering that a\ + h/jL + cv = 0, 
 we get 
 
 - (fjbCQsB—v cos 0) + T- (vcos C — \ cos A) 
 a ' 
 
 or 
 
 or 
 
 + -{\coa A — fi cosB) = 0, 
 
 l_ 
 a' 
 
 m 
 b' 
 
 n 
 c 
 
 = 0, 
 
 \ cos A, /J, cos B, V cos C 
 1, 1, 1 
 
 I, m, n 
 
 X sin 2A, /J. sin 2B, « sin 2 (7 
 
 sin .4, sinJ5, sin C 
 
 = 0; 
 
 a result which the student acquainted with the differential cal- 
 culus could have written down at sight from the consideration 
 
 that 
 
 IX + rrifi + nv 
 
 had to be made a minimum subject to the relations 
 
 X''sin2J.+At'sin25+i/'sin2C=2sin^ sin^sin C, 
 
 (equation 3 of Art. 71) 
 
90 EXEECISES ON CHAPTER VI. 
 
 and Xsin^ +/t sinjB+ j/sinC=0, 
 
 whence we must have 
 
 IS\ + mS/jL + nSv = 0, 
 \ sin 2A.h\ + fi sin 2B . S/i + v ain2C . Sv = 0, 
 sin ^ . 8\ + sin ^. B/i + s\nC.Sv= 0, 
 and eliminating the differentials, the result is obtained. 
 
 Exercises on Chapter VI. 
 
 (57) The straight line whose direction sines are X, ft, v meets 
 the line at infinity in the point given by the equations 
 
 a 
 
 
 V 
 
 (58) Find the coordinates of the point at which the sides of 
 the triangle of reference subtend equal angles. 
 
 If through this point three straight lines be drawn each 
 parallel to a side and terminated by the other two sides, the rect- 
 angles contained by their segments are equal. 
 
 (59) From the point (a', /3', 7') the straight line is drawn 
 whose direction sines are \, yit, v : find the length intercepted 
 upon this line, between the straight lines whose equations are 
 
 la + m^ + ra7 = 0, 
 
 and Ta + m'/3 + w'7 = 0. 
 
 (60) Shew that if from any fixed point there be drawn 
 three straight lines OP, OF, OF', whose lengths are p, p, p", 
 and whose direction sines are (X, ft,, v), (\', /i', v), (\", ^", v") 
 respectively, then the area of the triangle PP'P"will be 
 
 2A 
 
 1> W. 
 
 vp 
 
 a 
 
 1 ' f 
 
 1. /*P, 
 
 v'p' 
 
 
 1 r, It 
 
 v'-p' 
 
EXERCISES ON CHAPTER VI. 91 
 
 (61) From the middle points of the sides of the triangle 
 of reference perpendiculars are drawn proportional in length 
 to the sides ; and their extremities are joined to the opposite 
 angular points of the triangle. Shew that the three joining 
 lines will meet in a point whose coordinates (a, /8, 7) are con- 
 nected by the equation 
 
 sin (.5 - C) sm{C-A) s,m{A-B) 
 a ^ 7 
 
 (62) From the point 0, {a, P, 7') a straight line is drawn 
 in any direction to meet the straight lines 
 
 la + 9Bj8 + 727 = 0, 
 
 fa + »w'/3 + v'7 = 0, 
 
 (i + f) a + (jw + m) /3 + (» + «') 7= 0, 
 
 in points P, Q, R. Shew that the ratio OP. QB : OQ.PB is 
 equal to 
 
 _ la + m^ + W7' 
 
 "iV + m'^ + nV' 
 
 whatever be the direction of the transversal. 
 
CHAPTER VII. 
 
 MODIFICATIONS OP THE SYSTEM OF TKILINEAR COORDINATES. 
 AEEAL AND TRIANGULAR COORDINATES. 
 
 83. The great principle which distinguishes the modem 
 methods of analytical geometry from the old Cartesian methods 
 is, as we have seen, the adoption of three coordinates instead of 
 two to represent the position of a point, and the recognition of 
 the power thus gained of rendering all our equations homoge- 
 neous. 
 
 This homogeneity of equations will be always attainable 
 whatever quantities x, y, z we may use as coordinates of a point, 
 provided the third, z, be connected with the other two by a 
 linear equation, 
 
 Ax-\-By^-Cz = D, 
 
 Ax + By+Cs , 
 
 ^ 1; 
 
 for (exactly as in the case of Art. 10, page 13) any term in an 
 equation which is of a lower order than another may be raised 
 by multiplying it by 
 
 Ax + Bi/+ Cz 
 B • 
 
 (since this is equal to unity) and we may repeat the operation 
 
AEEAL AND TRIANGULAK COORDINATES. 
 
 93 
 
 till every term is raised to the order of the highest term, and the 
 equation is thus homogeneous. 
 
 84. We have hitherto used the perpendicular distances of 
 the point P from the lines of reference as the coordinates of P, 
 and we have established the relation 
 
 •(1), 
 
 aa + &;8+C7=2A 
 
 connecting the coordinates of any point. 
 
 The position of the point would be equally determinate if 
 we used any constant multiples of these perpendicular distances 
 as coordinates. For instance, we might call the coordinates of 
 P, a', 0, y', where 
 
 a' = \a, ^ = fi^, i = v^, 
 
 and the relation (1) connecting the coordinates of any point 
 would then become 
 
 ai 
 
 
 .(2). 
 
 The particular case in which 
 
 \ = a, /* = J, v = c 
 
 will present the advantage of a very simple relation among the 
 coordinates, for the equation (2) reduces in this case to 
 
 a' + ;8' + 7'=2A (3). 
 
 Fig. 19. 
 A 
 
94 MODIFICATIONS OF TKILINEAR COORDINATES. 
 
 And the quantities a, /S', 7', which in this case will be the 
 coordinates of the point P, are capable of a simple geometrical 
 interpretation. For if PD be the perpendicular from P on BC 
 (fig. 19), we have 
 
 a=aa. = BC.PD 
 
 = 2APBC, 
 
 so ^' = 2^PCA, 
 
 and 7' = 2AP45. 
 
 The coordinates a, /3', 7' of the point P are therefore the 
 double areas of the triangles having P as vertex, and the sides 
 of the triangle of reference as bases. 
 
 85. If a", ^', 7" denote the halves of a', /3', 7', the equa- 
 tion (3) of Art. 84 gives us 
 
 a" + (8" + 7" = A (4), 
 
 as the relation connecting a", /3", 7" if they be taken as the 
 coordinates of P. These coordinates represent the areas of the 
 triangles BPC, POA, PAB, and used often to be called indiffer- 
 ently the areal or triangular coordinates of P with respect to the 
 triangle ABC. These terms areal and triangular have however 
 more recently been applied to the system of coordinates described 
 in the next article, and authors are not uniform in their use of the 
 expressions. It seems convenient to describe these coordinates 
 a", /Q", 7" which represent the actual areas of the triangles 
 PBC, PGA, PAB as areal coordinates, observing that as they 
 represent areas they are of two dimensions in linear magnitude. 
 We can thus reserve the term triangular for the system now 
 about to be described, although it would certainly be preferable 
 to invent a name for them which should indicate the fact (which 
 will immediately appear) that they are of zero dimensions in 
 linear magnitude, expressing not lines nor areas but simply 
 ratios. 
 
VIZ. 
 
 AREAL AND TKIANGULAE COORDINATES. 95 
 
 86. The relation among the trilinear coordinates a, /3, 7, 
 oa + J/3 + C7 = 2A 
 
 may be written 
 
 2A "^ 2A "^ 2A 
 
 If therefore x, y, z denote the ratios 
 
 aa 6/3 07 
 2A' 2A' 2A- 
 
 they will be subject to the very simple relation 
 
 x-\-y + z = \ (5). 
 
 But since x, y, z bear constant ratios to a, /3, 7 they may be 
 used as the coordinates of P (Art. 84) : and on account of the 
 simplicity of the relation (5) just obtained, very great advan- 
 tages attend their use. 
 
 It will be observed that these coordinates {x, y, s) represent 
 the ratios of the triangles FBC, PC A, PAB severally to the 
 triangle of reference ABC. They are (not very appropriately) 
 often spoken of as the areal or triangular coordinates of P, but 
 as we said in the last article, we shall call them triangular coor- 
 dinates, reserving the term areal for tlie system described in that 
 article. 
 
 In speaking of the areas of the triangles PBC, PC A, PAB, 
 the same convention with respect to algebraical sign will have 
 to be adopted as in the case of the perpendicular distances of P 
 from the lines of difference. Thus (as in Art. 6, page 10) the 
 triangle PBC will be considered positive when it lies on the 
 same side of the base BC as does the triangle of reference, and 
 so for the other triangles. 
 
 87. It is important to observe that if the triangle of refer- 
 ence be the same, the triangular coordinates {x, y, z) and the 
 
96 TABLE OF FORMULJE. 
 
 trilinear coordinates (a, /8, 7) of any point P, are connected by 
 the relations 
 
 X _ y _ B 1_ 
 
 ^ ~ P "" C7 - 2A ' 
 
 so that we can at once transform any equation or expression 
 from the one system to the other. 
 
 To exemplify this, and for convenience of reference, we 
 append a table of the principal results which we have already 
 obtained in trilinear coordinates, together with the correspond- 
 ing results for triangular coordinates. 
 
 TABLE 
 
 OF FORMULAE AND OTHER RESULTS. 
 
 In trUinea/r coordinates. \ In briamgvlwr coordinates. 
 
 (i) The coordinates of any point are connected by the rela- 
 tion (Art. 7), 
 
 (ii) The coordinates of the middle point of BO are 
 (Art. 14), 
 
 0,^^. I 0,1,1. 
 
 ' b ' c ' 2 ' 2 
 
 (iii) The coordinates of the foot of the perpendicular from 
 A upon BC are (Art. 15), 
 
 „ 2A ^ 2A 
 0, — cos C, — cos B. 
 a a 
 
 boos C c cos B 
 V, - , - . 
 
TABLE OP FORMULA. 97 
 
 In trilinear coordinates. j In triangular coordinates. 
 
 (iv) The centre of the inscribed circle is given (Art. 16) by 
 
 2A 
 
 1 
 
 a + b +c 
 
 X y _2 _ 
 
 a b c a + b + c' 
 
 (v) The middle point between the two points 
 («,> /Si. 7.) ancl («,. /8„ 7,), is \ (a;., y„ «,) and (a;,, y„ z,), is 
 
 V 2 ' 2 ' 
 
 + % 
 
 ■*i + a'. .Vi + y^ « 
 
 -T^')' 
 
 (vi) The area of a triangle whose angular points are given 
 (Art. 19), is 
 
 A = 
 
 2>S' 
 
 «„ A. 7i 
 
 \ 
 
 Ha. ^8) 7s 
 
 
 a,> f^s> % 
 
 \ 
 
 ^=A 
 
 35,, 
 
 ^1. 
 
 ^1 
 
 a;^, 
 
 ^2. 
 
 2, 
 
 a'a- 
 
 ^3. 
 
 «3 
 
 (vii) The equation to a straight line joining two points 
 whose coordinates are given (Art. 21), is 
 
 a, /3, 7 
 «i, A. 7, 
 «j, ^2, 7i, 
 
 = 0. 
 
 «, y, 3 
 
 (viii) The equation to the straight line whose distances 
 from the points of reference are jj, g, r (Art. 23), is 
 
 qpa + bql? + f^ri = 0. 
 
 ■px + qy + rz = 0. 
 
 (ix) The condition that the three straight lines 
 
 Za + 7M/3 + W7 =0, 
 
 Va. + m'yS + w'7 = 0, 
 
 ra+«i")S+ m"7=0, 
 w. 
 
 ha + my + nz =0, 
 Z'o; + niy + w'2 = 0, 
 X'x+ ■m'y+ «"a = 0. 
 
98 
 
 TABLE OF FORMULA. 
 
 In trilinear coordinates. | In triangular coordinates. 
 
 should meet in a point (Art. 26), is 
 
 I, m, n = 0. 
 
 I, m , n 
 
 111 II If 
 I , m , n 
 
 (x) The equation to the perpendicular from A on BC 
 (Art. 30), is 
 
 /8cos5-7COS (7 = 0. I ycotB-zcotC = 0. 
 
 (xi) The condition of parallelism (Art. 35) of the two 
 straight lines whose equations are 
 
 la + myS + wy = 0, 
 l'a + m'^ + n'y=0, 
 
 is I, m, n ,= 0. 
 
 ?t If 
 , m, n 
 
 a, h, c I 
 
 Ix + my +nz =0, 
 I'x + my + n'z = 0, 
 
 IS 
 
 I, m, n 
 r, m, n' 
 1, 1, 1 
 
 = 0. 
 
 (xii) The equation to the straight line at infinity (Art. 36), 
 aa + &/3 + C7 = 0. I x + y + z = 0. 
 
 (xiii) The perpendicular distance (Art. 46) of the point 
 («', /3', 7) S {x', y, z) 
 
 from the straight line whose equation is 
 
 IS 
 
 la. + }M/S + ny = 0, 
 Id + m^ + w^' 
 
 ha + my + ns = 0, 
 
 b' + my + nz 
 \al, Im, en] ' 
 
 is .^^Ix' + my' + nz' 
 
 [l, m, «} 
 where 
 
 {I, m, nY = l" + m^ -\- n^ — 2mn cos A — 2nl cos B—2lm cos C, 
 and therefore 
 {a?, J»n, en}' 
 
 = a^{l-m) (/ - n) + 6'(m-n) (m- Z)+c'(ra - Z) (re - m). 
 
TABLE OP FORMULAE. 99 
 
 In trilinear coordirmtea. | In tricmgular coordinates. 
 
 (xiv) The perpendicular distance of the same point from 
 the straight line whose equation is (Art 58) 
 
 apa + hq& + err/ = o, 
 
 apa! + hqP + erf' 
 2A 
 
 IS 
 
 px-\-qy -{-rz^ 0, 
 px + qy' + rz . 
 
 (xv) The sine of the angle D between the two straight 
 lines whose equations are 
 
 px + qy +rz =0, 
 p'x + qy + rz = 0, 
 
 apa. + Iq^ + cry = 0, 
 
 
 
 apa. + Iq^ + cr'7 = 0, 
 
 
 rt. 60) 
 
 1 
 2A 
 
 p', 2', »• 
 
 
 1, 
 
 1, 1 
 
 (xvi) The sine of the angle Z> between the two straight 
 lines whose equations are 
 
 h. + OTyS + W7 = 0, 
 T!a. + »w')S + re'7 = 0, 
 
 is (Ai-t. 61) 
 
 I, 
 
 m, 
 
 n 
 
 i; 
 
 m; 
 
 n' 
 
 sin -4, 
 
 sinjB, 
 
 sin G 
 
 {I, m, n} {I', m, n'} 
 
 2A 
 
 Ix + my +nz =0, 
 Tx + m'y + n'z = 0, 
 
 I, m, n 
 I , m, n 
 1, 1, 1 
 
 {al, hm, en] [al , bm, en'} 
 7—2 
 
100 TABLE OF FORMULA. 
 
 In trUinea/r coordinates. 1 In triangular coordinates. 
 
 (xvii) The area of the triangle whose sides are represented 
 by the equations 
 
 apa. +hq^ + cry =0, 
 ap'a. + bq'^ + cry = 0, 
 ap"a+bq"^+cr"y=0, 
 
 is (Art. 64) 
 
 px +qy +rs = 0, 
 p'x + qy + rz = 0, 
 fx^q'y-\-r"z = % 
 
 F, ?, »• 
 
 2 
 
 
 
 i>', i, 1-' 
 
 
 
 P", q, r" 
 
 
 
 P, 4, r' 
 
 
 P . 
 
 q', r' 
 
 p", s", r" 
 
 
 P' 
 
 q, r 
 
 1, 1, 1 
 
 
 1, 
 
 1, 1 
 
 p, q, »• 
 
 r f I 
 
 p, i, f 
 1, 1, 1 
 
 (xviii) The condition that the two straight lines 
 
 Za + 7n/3 + «7 = 0, | fe + my + wa = 0, 
 
 Va. + m'/3 + riy = 0, | fa: + niy + m'a = 0, 
 
 should be at right angles is (Art. 49) 
 
 Zf + »»»»'+ nil \ Ud^+ mniW + nn<? 
 
 — (wm'+ »»'«) cos A <, — {mn' + m'n) he cos A 
 
 — {nl'+n't) cos B I — (nf + n'Z) ca cos B 
 
 — {hn + I'm) cos (7 = 0. 5 — (Zm' + Im) ah cos (7=0. 
 
 (xix) Any straight line drawn through the given point 
 
 («', /S', 7'), I («=', y, «'), 
 
 may be represented by the equations (Art. 68) 
 
 
 X — X _y — y _s — z 
 
EXEBCISES ON CHAPTER VII. 101 
 
 In trUinear coordinates. i In triangular coordinates. 
 
 where \, fi, v are proportional to the coordinates of the point 
 where the straight line meets infinity and are subject to the 
 relation 
 
 aX + b/j, + cv = 0. I \ + /i + j; = 0. 
 
 (xx) Each member of the equations in (xix) will be equal 
 to the distance (p) from the given point to the point 
 
 («. ^' 7), \ {x, y, z), 
 
 provided \, /i, v satisfy a further condition which is expressed 
 by any one of the equations (Arts, 70, 71) 
 
 i^ + V + 2i/\ cos ^ = sin''^, \»Jc cos A + fj-'ca cos B 
 
 \''+/i' + 2Xfi cos G= Bin'C. + T?ah cos C= 1. 
 
 Exercises on Chapter VII. 
 
 (63) Shew that the straight lines 
 
 (m + n) a; + (» + Z) y + (Z + jw) s = 0, 
 and Ix + my + wa = 0, 
 
 are parallel, and find their inclination to the straight line 
 
 (»i — n) a; + (« — Z) y + (Z — «i) 3 = 0. 
 
 (64) If a series of different values be given to y, the equa<- 
 tion 
 
 X (cos'a +p sin'a) + y (cos'/3 ■V'p sin*/8) + z (cos^ +p sinV) = 
 will represent a series of parallel straight lines. 
 
102 ' EXERCISES ON CHAPTER VII. 
 
 (65) Shew that the straigm line bisecting at right angles 
 the aide BG of the triangle of reference is given by the equa- 
 tion 
 
 {x — y-'rz)cotB={x-{-y — z) cot G. 
 
 Hence shew that the three straight lines, bisecting at right 
 angles the three sides of the triangle of reference, meet in one 
 point. 
 
 (66) Draw the four straight lines represented by the equa- 
 tions in triangular coordinates, 
 
 a; cot-4 + y cot B + s cot C = 0. 
 
 (67) Draw the four straight lines represented by the equa- 
 tions 
 
 X cosec A±y cosec B±z cosec (7=0. 
 
 (68) Interpret the equations 
 
 x±y ±e = 0. 
 
 (69) Find the area of the triangle whose sides are given by 
 the equations 
 
 Ix + my + M3 = 0, 
 
 mx + ly — m = 0, 
 
 s = 0. 
 
 (70) Find the area of the triangle whose angular points are 
 given respectively by 
 
 x = and y = fe, 
 
 y = and a = mx, 
 
 a = and x = ny. 
 
 If mn = 1, the result is independent of I. Interpret this cir- 
 cumstance geometrically. 
 
EXERCISES ON CHAPTER VII, 103 
 
 (71) Of the four straight lines whose equations are 
 
 Ix ± my ±nz = 0, 
 
 two intersect in P and the other two in F', two intersect in Q 
 and the other two in Q', two intersect in M and the other two in 
 -R'. Find the triangular coordinates of the middle points of 
 PJ", QQ', RR', and shew that they lie on one straight line. 
 
 Find the equation to this straight line, and compare the 
 result with that of Ex, (40), page 56, 
 
 (72) Prove that the straight line represented in trilinear 
 coordinates by the equation 
 
 is parallel to the straight line represented in triangular coordi- 
 nates by the equation 
 
 a;cot^+^cotP + « cot (7=0. 
 
 (73) ABC is a triangle: through the angular points 
 A, B, G, straight lines he, ca, ah are drawn forming a second 
 triangle dbc : and through a, b, c straight lines are drawn paral- 
 lel to BC, CA, AB respectively, so as to form a third triangle, 
 and so on. Shew that the areas of the triangles thus formed are 
 in geometrical progression. 
 
CHAPTER VIII. 
 
 ABRIDGED NOTATION OF THE STRAIGHT LINE. 
 
 88. In Chapter ill. Art. 27, we used m = and u = to 
 represent the equations to two straight lines, regarding u and v 
 as abridgements or symbols of the expressions la. + m^ + ny 
 and I' a, + »n'/3 + n'7. 
 
 But it will have been observed that the reasoning of that 
 article would have been equally valid if m = 0, v = had re- 
 presented the equations to two straight lines expressed in tri- 
 angular coordinates, or in the ordinary Cartesian system. 
 
 We may therefore write the result of that article in the 
 following more general form. 
 
 If u = 0, V = he the equations to two straight lines express- 
 ed in any system in which a point is determined hy coordinates, 
 then will the equation u-\- kv = represent a straight line pass- 
 ing through the intersection of the former lines. 
 
 And by giving a suitable value to k this equation can be 
 made to represent any straight line whatever passing through 
 that point of intersection. 
 
 89. The following principle will often be assumed. 
 
 If u = (i, D = 0, to = he the equations to three straight lines 
 forming a triangle expressed in any system in which a point 
 
ABRIDGED NOTATION OF THE STEAIGHT LINE. 
 
 105 
 
 ■is determined hy coordinates, then any straight line whatever 
 will he represented hy an equation of the form 
 \u + fjiV + vw = 0. 
 
 Fig. 20. 
 
 For let QB, RP, PQ be the straight lines whose equations 
 are m = 0, v = 0, w = respectively, and let FG be any straight 
 line whatever whose equation it is required to find. 
 
 Since QR, RP, PQ are not parallel {hypoth.), one of them 
 can be found which being produced will meet FG. Let it 
 be QR. 
 
 Let QR produced meet FG in F. Join FP. 
 
 Then, by Art. 88, since FP passes through the intersection 
 oi v=Q, ic = 0, it will have an equation of the form 
 
 V + KW = 0, 
 
 or, as we may write it, 
 
 fiv + VW =^ 0. 
 
 But since FG passes through the intersection of the straight 
 line whose equations are m = and 
 
 fji,v + vw= 0, 
 it will by the same article (88) have an equation of the form 
 
 \u + iiv + vw = 0, 
 which was to be proved. 
 
106 ABRIDGED NOTATION OF THE STRAIGHT LINE. 
 
 We have supposed tliat the three original straight lines form 
 a triangle. All that is necessary however for the validity of 
 the proof is that they should not le all parallel, and that they 
 should not meet in a point. 
 
 If they were all parallel, one could not necessarily be found 
 to intersect PG, and if they met in a point, FP and FQ would 
 coincide, and the equation to FG could not then be determined 
 as passing through the intersection of FP and FQ. 
 
 But the theorem is perfectly true if two of the original 
 straight lines be parallel and intersect the third. 
 
 90. If M + V + w = represent the fourth side of a quadri- 
 lateral whose other three sides are m = 0, v=0, w=Q, it is 
 required to interpret the equations 
 
 v + w; = 0, w ±u = Qi, M+v = 0. 
 
 Let the first three sides be BC, CA, AB (fig. 21), and let the 
 fourth side cut them (when produced) in A', B', C respectively. 
 
 The equation v+w = Q represents a straight line passing 
 through the intersection of m = and u+v+w=Q: also through 
 the intersection of u = and w = Q. 
 
 It is therefore the straight line AA'. 
 
 Similarly, w + m = represents BB, 
 
 and u + v = represents CC. 
 
 Again, let BB', CC, intersect in a; CC, AA' in b; 
 AA', BB in c. 
 
 Then the equation v — w = 0, since it results from the sub- 
 traction of 10 + M = and m + « = 0, represents a straight line 
 passing through the intersection of BB' and CC, or a. 
 
 It also passes through the intersection of u = and w=0, 
 or A. 
 
 Hence, v — w = (i represents the line Aa, 
 
 so w — M=0 Bh, 
 
 and u — v=(i Cc. 
 
ABRIDGED NOTATION OP THE STRAIGHT LINE. 107 
 
 Fig. 21. 
 
 C a c' IJ 
 
 91. The following proposition is very important. 
 
 If W + « + M) = (1), 
 
 -M + « + w = (2), 
 
 M — w + w = (3), 
 
 M + u — w = (4), 
 
 he the equations of four straight lines forming a quadrilateral, 
 then 
 
 M = 0, 
 
 v = 0, 
 
 w = 0, 
 
 will be the three diagonals of the quadrilateral. 
 
 For M = evidently represents a straight line passing through 
 the intersection of (1) and (2), and also through the intersection 
 of (3) and (4), therefore it is a diagonal of the quadrilateral. 
 
 Similarly v = joins the intersection of (1), (3) with that 
 of (2), (4), and so w = joins the intersection of (1), (4) with 
 that of (2), (3) ; 
 
 .'. &C. Q.E.D. 
 
108 ABRIDGED NOTATION OF THE STRAIGHT LINE. 
 
 92. If the equations 
 
 u + v = (1)> 
 
 M + w=0 (2), 
 
 u-v = (3), 
 
 u-w = (4), 
 
 represent the four sides of a quadrilateral in order, then will 
 
 v + w = (5), 
 
 and v—w = (6), 
 
 represent its interior diagonals, and 
 
 u = (7), 
 
 will represent its exterior diagonal, 
 
 and v = (8), 
 
 and «;=0 (9), 
 
 will represent the straight lines joining the point of intersection 
 of the two interior diagonals to the points of intersection of 
 opposite sides. 
 
 Let AA', BB' be the interior, and CC the exterior diago- 
 nal, so that the equations (1), (2), (3), (4) represent AB', B'A', 
 A'B, BA respectively, and let AA', BB' intersect in 0. 
 
 Fig. 22. 
 
 B' 
 
ABRIDGED NOTATION OF THE STRAIGHT LINE. 109 
 
 Then the equation (5) may be obtained either by subtracting 
 (1) and (4), or by subtracting (2) and (3). Therefore it repre- 
 sents the line joining the intersection of (1) and (4) with that 
 of (2) and (3); i.e. the line AA'. 
 
 Similarly (6) denotes the line BB'. 
 
 But M = passes through the intersection of (1) and (3) as 
 well as through that of (2) and (4) ; therefore it represents the 
 line CC. 
 
 Also v = passes through the intersection of (1) and (3) 
 as well as through that of (5) and (6). Hence it denotes the 
 line CO. 
 
 And similarly the equation w = must denote the line 
 
 CO. Q.E.D. 
 
 The student is recommended to examine for himself the 
 modifications which these theorems undergo when one of the 
 straight lines is at infinity. 
 
 93. We now introduce some geometrical terms which will 
 be found convenient. 
 
 Definitions. 
 
 I. Three or more straight lines which pass through the 
 same point are said to be concurrent, 
 
 II. Three or more points which lie upon the same straight 
 line are said to be collinear. 
 
 III. Two triangles ABC, AEG' are said to be co-polar if 
 AA', BB', CC meet in a point, and this point is called the 
 ■pole of the triangles, or the pole of either triangle with respect 
 to the other. 
 
 IV. Two triangles ABC, A'B'C are said to be co-axial 
 if the points of intersection of BC, B'C, of CA, CA', and of 
 AB, A'B' lie in one straight line, and this straight line is 
 called the axis of the two triangles, or the axis of either triangle 
 with respect to the other. 
 
110 
 
 ABRIDGED NOTATION OF THE STRAIGHT LINE. 
 
 94. If two triangles he co-axial they will also he co-polar. 
 
 Let ABC,A'B'C' be two co-axial triangles, and \&i PQR 
 be their axis; P being the point of intersection of BG, B'C, 
 Q that of CA, GA', and R that of AB, A'B'. 
 
 Fig. 23. 
 
 Let M=0, « = 0, tc = be the equations to BG, GA, AB 
 respectively, and a; = the equation to PQR. 
 
 Then, since B' C' passes through the intersection of PQR 
 and BC {x = and m = 0), its equation may be written 
 
 x+lu = (i (i), 
 
 So the equation to G'A' may be written 
 
 .'B + TOU=0 (ii), 
 
 and the equation to A]^ 
 
 x-\-nw=Qi (iii). 
 
 From (ii) and (iii) by subtraction we obtain 
 
 mv — nw=Q (iv), 
 
 which therefore represents a straight line passing through the 
 point of intersection of G'A! and A'B'; i.e. through A'. 
 
ABRIDGED NOTATION OF THE STEAIGHT LINE. Ill 
 
 But this equation, from its form, must represent a straight 
 line passing through the intersection of the straight lines v = 
 and w = 0, i.e. through A. 
 
 Hence (iv) is the equation to AA'. 
 
 Similarly, nw — lu = (v), 
 
 and lu — mv = (vi), 
 
 are the equations to BB' and CC respectively. But (iv), (v), 
 (vi) are all satisfied at the point determined by 
 
 lu = mv = nw. 
 
 Therefore AA', BB', CC all pass through this point, and 
 therefore the triangles are co-polar. 
 
 Therefore any two co-axial triangles are also co-polar, q.e.d. 
 
 95. If two triangles he co-polar, they will also he co-axial. 
 
 Let M = 0, V = 0, M) = be the equations to the sides of 
 the triangle ABC, and let AB C be a co-polar triangle, the 
 point being the pole. 
 
 Let BC, BC intersect in P, CA, C'A in Q, and AB, A'B' 
 in B. We have to shew that F, Q, B are coUinear. 
 
 Let a; = be the equation to PQ. 
 
 Then B C passing through the intersection of BC and PQ 
 has an equation which may be written 
 
 x + lu = (i). 
 
 So C'A' passing through the intersection of CA and PQ 
 may be represented by the equation 
 
 x + mv = (ii)- 
 
112 ABRIDGED NOTATION OF THE STEAIGHT LINE. 
 
 Fig. 24. 
 
 ' P 
 
 Then the equation 
 
 lu—mv = (iii) 
 
 must represent a straight line passing through the intersection 
 of the straight lines (i) and (ii), as well as through the intersec- 
 tion of the straight lines u and v. Therefore it represents the 
 straight line CC. 
 
 Therefore the point is given by the equations 
 lu = mv 
 
 = nw suppose. 
 And therefore OA and OB will have the equations 
 
 mv — nw = (iv) 
 
 and nw— lu =0 (v). 
 
 Now consider the equation 
 
 x + 7iw = (vi). 
 
 It must represent a straight line passing through the intersec- 
 tion of the straight lines (i) and (v), that is, through B. 
 
 Similarly its locus must pass through the intersection of the 
 straight lines (ii) and (iv), that is, through A'. 
 
ABRIDGED NOTATION OF THE STRAIGHT LINE. 113 
 
 Therefore it represents the straight line AB ; but by its 
 form its locus must pass through the intersection of the straight 
 lines w and x, i. e. AB and PQ. 
 
 Hence the three straight lines AA , BE, PQ meet in a 
 point, or (in other words) the point of intersection R of the sides 
 AA', BE is collinear with P and Q. And therefore the tri- 
 angles ABG, A'B'C are co-axial. 
 
 Hence any two co-polar triangles are co-axial. Q. E. D. 
 
 96. The three straight lines which are represented in abridged 
 notation hy the equations 
 
 III +mv + nw =0 (1), 
 
 I'u + m'v +n'w = (2), 
 
 l"u + m"v + n"w = 0' (3), 
 
 will be concurrent, provided 
 
 I, m, n = 0. 
 
 r, m , 11 ; 
 in 'I ti I 
 
 I , m , n ! 
 
 For (Art. 88) if the straight line (3) pass through the inter- 
 section of (1) and (2), its equation must be obtained by adding 
 some multiples of the first two equations. 
 
 Suppose h, k' the respective multipliers, then we must 
 have 
 
 kl + k7 =1", 
 
 km + Tcm=in', 
 
 kn + Icn = «", 
 
 whence, eliminating k, k', we obtain 
 
 w. 8 
 
114 AURIDGED NOTATION OF THE STRAIGHT LINE. 
 
 I, m, 11 = 0, 
 
 7' ' ' 
 
 I , m, n 
 
 , m , n 
 
 the required condition for the concurrence of the three straight 
 lines. 
 
 97. It will be observed that the result of the last Article 
 is precisely the same as that of Art. 26. Indeed Art. 26 is but 
 a particular case of Art. 96. 
 
 For if we regard 
 
 M = 0, v = 0, w = 0, 
 
 as denoting the equations to three straight lines in trilinear 
 coordinates, then (Art, 46) the expressions u, v, w, themselves 
 denote the perpendicular distances of the point (a, ^, 7) from 
 these straight lines. 'And therefore if these straight lines be 
 taken as lines of reference, u, v, w will be proportional to the 
 new trilinear coordinates of the point (a, /3, 7), and may them- 
 selves be regarded (Art. 84) as coordinates of this point, re- 
 ferred to the new triangle. 
 
 The equations 
 
 lu + mv + nw — 0, 
 
 Tu + mv + n'w =■ 0, 
 
 Z'm+ m"r+ n"M) = 0, 
 
 need therefore be no longer regarded as abbreviated expressions, 
 but they may be read as relations among the coordinates m, v, w, 
 and as such may be subjected to the reasoning which in Art. 26 
 is applied to the relations among the coordinates a, /3, 7. 
 
 98. It is interesting to observe that the method of trilinear 
 coordinates originally grew out of the method of abridged nota- 
 tion applied to Cartesian coordinates exactly by the process of 
 thouglit indicated in the last article. In the works of Mr Tod- 
 
EXERCISES ON CHAPTER VIII. 115 
 
 hunter and Dr Salmon, the subject will be found treated from 
 this point of view. As far as we know, Mr Ferrers (in 1861) 
 was the first to publish a work establishing trilinear coordinates 
 upon an independent basis. 
 
 Exercises on Chapter VIII. 
 
 (74) If M = 0, r> = 0, w = be the equations to three 
 straight lines, find the equation to the straight line passing 
 through the two points 
 
 U V 10 , u V w 
 
 7 = - = — ana t, = — = — . 
 
 (75) Find the equation to the straight line passing through 
 the intersections of the pairs of lines 
 
 2au + bv + cw = 0, bv — cw = 0; 
 
 and 2bu + av + cw = 0, av — cxo = 0. 
 
 (76) If s = be the equation to the straight line at infinity, 
 the equations 
 
 M + i; + s = 0, — M + v + s = 0, 
 M + « — s = 0, M — ■y + s = 0, 
 
 represent the sides of a parallelogram whose diagonals are 
 M = and » = 0. 
 
 (77) Let the three diagonals of a quadrilateral be produced 
 to meet each other in three points, and let each of these points 
 be joined with the opposite comers of the quadrilateral : the 
 six lines so drawn will intersect three and three in four points. 
 
 (78) If 5 = be the equation to the straight line at infinity, 
 then the triangle whose sides are 
 
 M = 0, v = 0, w = 0, 
 is co-polar with the triangle whose sides are 
 
 M + & = 0, v + ms = 0, w + ns = 0, 
 whatever be the values of I, m, n. 
 
 8—2 
 
116 EXEECISES ON CHAPTER VIII. 
 
 (79) If ABC, A'B'C be the two triangles in the last ques- 
 tion, and if AA', BG intersect in D ; BB', CAmE; CC, AB 
 in F; shew that the intersections of BU and AB, EF and BO, 
 FD and CA will be coUinear. 
 
 (80) The three points determined by 
 
 
 u 
 
 
 
 V 
 
 
 
 w 
 
 p 
 
 — 
 
 1 
 
 ■ P 
 
 - 
 
 i 
 
 p 
 
 '-2'" 
 
 
 u 
 
 
 
 V 
 
 7 
 
 
 10 
 
 3 
 
 — 
 
 r 
 
 ~4 
 
 — 
 
 It 
 
 1 
 
 -r"' 
 
 
 u 
 
 
 
 V 
 
 
 
 10 
 
 are coUinear. 
 
 It II ? 
 r — p r —p r — p 
 
 (81) If M = 0, v=0, w = 0, x = denote the equations to 
 four straight lines, and if the sum of the expressions u, v, w, x 
 be identically zero, the three diagonals of the quadrilateral 
 foi-med by the four straight lines will be represented by the 
 equations 
 
 u-\-v =Q, or w; + a: = 0, 
 u + w = 0, or x+ v = 0, 
 u + x = (i, or V + w = 0. 
 
 (82) In a given triangle let three triangles be inscribed, by 
 joining the points of contact of the inscribed circle, the points 
 where the bisectors of the angles meet the sides, and the points 
 where the perpendiculars meet the sides ; then will the corre- 
 sponding sides of these three triangles pass through the same 
 point; also the triangle formed by the three points of intersec- 
 tion will be a circumscribed co-polar to the original triangle, and 
 the pole will be on the straight line in which the sides of the 
 given triangle meet the bisectors of its exterior angles. 
 
CHAPTER IX. 
 
 IMAGINARY POINTS AND STRAIGHT LINES. 
 
 99. Let f+f''^^,g+gW-l,h+h'V^ be Irrational 
 values of a, yS, 7 which satisfy the relation 
 
 aa + i/3 + C7 = 2A. 
 
 If instead of being irrational the values had been rational 
 they would have been (Art. 7) the trilinear coordinates of some 
 real point. But being irrational tJiey are said to he the coordi- 
 nates of an imaginary point. 
 
 This must be taken as the definition of the term " an 
 imaginary point." 
 
 Such a point has no geometrical existence, it exists only in 
 respect to its coordinates. In other words, when we speak of 
 an imaginary point we are using a phrase wliich has no strict 
 geometrical application, but is convenient as giving expression 
 to an analytical result, and is very useful in enabling us often 
 to use much more general language than we could without 
 such a convention. 
 
 For instance, suppose we are finding the coordinates of a 
 point of intersection of the loci of two equations (two curves). 
 And suppose we arrive at the result 
 
 Oi=a-\-'JU'-<?. 
 If we use only geometrical language wc shall have to observe 
 
118 IMAGINARY POINTS AND STRAIGHT LINES. 
 
 that this value of a. is only real when l>c. Hence in stating 
 our result we must say that there will be a point of intersec- 
 tion only when J > c. In the language of analysis which we 
 have just introduced we shall be able to state the result more 
 generally: we shall be able to speak of the point of intersection 
 as always existing, but we shall observe that it is real or 
 imaginary according as S > c or b <c. 
 
 100. So, again, if coordinates are to be determined by the 
 solution of a quadratic equation, we shall say that there will 
 always be two solutions, but they will sometimes denote real 
 and sometimes imaginary points. 
 
 We shall shew hereafter (Chap, xii.) that every conic section 
 has an equation in trilinear coordinates which may be written 
 
 Ma'' + v/3"+wy'+2M')87 + 2u'7a + 2M''a/3 = (1). 
 
 Now suppose 
 
 Za + JnjS+ 717 = (2; 
 
 is the equation to a straight line, and that it is required to find 
 the coordinates of the points of intersection of the straight line 
 and the conic, i. e. to find coordinates which will satisfy both 
 the equations, we shall have to solve the equations (1) and (2) 
 simultaneously with the relation 
 
 aa + J;S+C7 = 2A (3). 
 
 By means of the simple equations (2) and (3) we can express 
 /3 and 7 as functions of a of the first order. 
 
 Substituting these values in the quadratic (1) we shall get 
 a quadratic equation to determine ot. 
 
 This quadratic will have two roots (real or imaginary). 
 Hence there will be two points of intersection (real or 
 imaginary). 
 
 Hence every straight line meets every conic section in two 
 and only two points (real or imaginary). 
 
IMAGINARY POINTS AND STRAIGHT LINES. 119 
 
 101. Iff+f-J^, g+gW^, h+hW^ be the coordi- 
 nates of an imaginary point; then f, g, h will he the coordinates 
 of a real point, and f, g, h' will be proportional to the coordi- 
 nates of a point at infinity. 
 
 For if /+/' V^, g + g V^, h + h' V^ be the coordinates 
 of a point they must satisfy the relation 
 
 oa + J/3 + C7 = 2A. 
 
 Hence 
 
 (o/+ bg + ch) + {af + bg + cK) V=T = 2A, 
 
 and therefore equating real and imaginary parts 
 
 af +bg +ch =2A (1) 
 
 and af' + bg' + ch' = (2). 
 
 But the equation (1) shews that (/, g, h) are the coordinates 
 of a point, and (2) shews that/', g', h' satisfy the equation 
 
 aa + iyS + c7=0 
 
 to the line at infinity, and therefore represent a point at 
 infinity. Q. E. D. 
 
 COK. The coordinates of an imaginary point at infinity 
 will be 
 
 /+/'V^, g+g'^^, h + h'^~l, 
 
 where af +bg +ch = 0, 
 
 as well as af + bg' + ch' = 0. 
 
 102. Def. An equation of the form 
 
 (Z+ Z' V^) a+ («i + wi' V^) /3 + (w + n V^) 7 = 
 is said to be the equation of an imaginary straight line. 
 
120 IMAGINARY POINTS AND STRAIGHT LINES. 
 
 We here suppose that 
 
 have no common factor, observing that such an equation as 
 
 {p +p' V^) (Xa + 11^ + vy)=0 
 is not imaginary; it represents a real straight line. 
 
 103. Every imaginary straight line passes through one and 
 only one real point. 
 
 Let the real and imaginary parts of the equation to the 
 straight line be collected so that the equation takes the fonn 
 
 u 
 
 + i,V-l = (1), 
 
 where u and v are rational functions of the coordinates of the 
 first degree, and therefore 
 
 ^ = ^ (2), 
 
 « = (3), 
 
 are the equations to two straight lines (which cannot be coinci- 
 dent, else the imaginary factor would divide out of the original 
 equation and the equation would become real, which is contraiy 
 to hypothesis). 
 
 The equation (1) can only be satisfied by real values of the 
 coordinates when u and v are each zero. (Todhunter's Algebra 
 Chap. XXV.) Hence any real ponit whose coordinates satisfy 
 (1) must also satisfy (2) and (3), and must therefore lie upon 
 both the straight lines (2) and (3). 
 
 Therefore the point of intersection of these two straight lines 
 is the only real point which satisfies the given equation. 
 
 Hence every imaginary straight Hue 2)asses through one and 
 only one real jJoint- 
 
IMAGINARY POINTS AND STRAIGHT LINES. 
 
 121 
 
 104. Every imaginary point lies upon one and only one real 
 straight line. 
 
 For let 
 
 /+/'V^, g+gW^, h + hW^ 
 
 be the coordinates of an imaginary point. 
 And if possible, let 
 
 la. + 9«/8 + ny = 
 
 be the equation to a straight line passing through it. 
 
 Then the coordinates of the point must satisfy this equation, 
 and therefore 
 
 lf+ mg + nh + {If + mg 
 
 + nK) V-1 
 
 = 0, 
 
 and therefore we must have 
 
 lf+mg + nh = 0, 
 
 lf + mg' + nh'==0, 
 whence 
 
 
 I 
 
 
 m 
 
 
 n 
 
 
 
 g, h 
 
 
 h,f 
 
 
 f> 9 
 
 > 
 
 
 ff',h' 
 
 
 h',f' 
 
 
 f\9' 
 
 
 equations which determine l:m.:n, and shew that only one 
 solution is possible. 
 
 Therefore every imaginary point lies on one and only one 
 real straight line. Q. e. d. 
 
 Cor. 1. The real straight line passing through the imaginary 
 point whose coordinates are 
 
 f+fW^, g + g''/^!, h^K4^ 
 
 is represented hy the equation 
 
 f, g\ ^' 
 
 = 0. 
 
122 liMAGINAUY POINTS AND STUAIQHT LINES. 
 
 CoE. 2. The imaginary point whose coordinates are 
 
 /+/V=1, g + g'J^, h + hW^, 
 
 lies on the straight line Joining the point {/, g, h) with the point 
 at infinity [f' : g' : h'), 
 
 CoK. 3. The same real line passes through the two imagi- 
 nary points 
 
 f + f^~\, g+gW^, h + hW^, 
 
 and 
 
 f-fW-1. g-g^I^, h-hW^. 
 
 Cor. 4. ^ two curves intersect in a series of imaginary 
 joints, they will lie two and two upon real straight lines. 
 
 This follows immediately from Cor. 3, when we remember 
 that imaginary roots can only enter into an equation by pairs, 
 the two members of every pair differing only in the sign of the 
 imaginary part. 
 
 105. We have said (Cor. to Art. 101) that the coordinates 
 of an imaginary point at infinity will be 
 
 f + f^ZTi^ g+g'^J^, h + hW^, 
 
 where f, g, h, f, g, h' satisfy the relations 
 af+ bg + ch = 0, 
 af + bg'+ck' = 0. 
 This statement requires a little consideration. 
 
 Let us retm-n for a moment to real points, and suppose 
 \, fi, V are numbers satisfying the relation 
 
 a\ + l(i + cv = (i)j 
 
 then we are accustomed to say that the equations 
 
 «=^ = 7 ^^^ 
 
IMAGINARY POINTS AND STRAIGHT LINES. 123 
 
 represent a point lying at infinity, for if we suppose a!, /3', 7' to 
 be the coordinates of the point determined hy (2), then, since in 
 virtue of (2), a', jS', 7' are proportional to \, (jl, v, it follows from 
 (1) that 
 
 oa' + &/3' + C7' = 0, 
 
 which shews that a', /3', 7' satisfy the equation to the straight 
 line at infinity. 
 
 But if we consider what are the actual values of these coor- 
 dinates a', ^, 7', we perceive (Art. 36, page 38) that two at least 
 and generally all three of them are infinite. But no difficulty 
 practically ensues from this, because we never want the actual 
 coordinates of such a point, it being sufficient to know that the 
 finite quantities X, fi, v are proportional to them : and it is very 
 convenient to speak of the point at infinity whose coordinates 
 are thus proportional to X, /t, v, as the point (X, fi, v), since the 
 quantities X, /i, v, or any quantities proportional to them, satisfy 
 all practical conditions of the coordinates of the point. For, as 
 we have already seen, so long as we have to do with homoge- 
 neous equations we never require the actual values of the coor- 
 dinates of any point, but only the ratios of those actual values, 
 except in theorems connected with the distance of the point from 
 other points ; consequently we shall not expect ever to require 
 the actual coordinates of a point at infinity, since its distances 
 from all finite points are infinite and cannot therefore generally 
 be introduced into problems. 
 
 (It should be noticed that if only two of the actual coordi- 
 nates of a point be infinite and the third be finite, then two only 
 of the quantities X, /t, v will have a finite magnitude, and the 
 third will be zero). 
 
 But, to return to the imaginary points, it follows from what 
 we have said, that whether 
 
 be the actual coordinates of a point at infinity or only propoi-- 
 tional to them, we must still have 
 
124 IMAGIXAKY POINTS AND STRAIGHT LINES. 
 
 af-\-lg+ch = 0, 
 a/'+ J9' + cA' = 0. 
 
 And we shall find it very convenient to speak of such a point 
 as the point 
 
 (/ + /'V^, g+g'-J^X, h + h'-J^l), 
 
 whether the quantities 
 
 f+fj~l, g+g'J^l, h + h'J-l 
 
 be the coordinates, or be only proportional to the coordinates of 
 the point. And indeed, since it is more convenient to deal with 
 finite than with infinite quantities, we shall always suppose that 
 the expressions 
 
 f+fJ-Ti, g+g'J^, h + JiJ^ 
 
 do represent quantities only proportional to the actual coordi- 
 nates. 
 
 In other words, if we speak of the point (m, v, iv) as a point 
 at infinity, we mean the point at infinity determined by the 
 equations 
 
 - = - = 2 
 
 u V w' 
 
 106. Two imaginary straight lines are said to be parallel 
 when they intersect in a point on the straight line at infinity. 
 Hence the condition investigated in Art. 35 may be applied to 
 imaginary straight lines. 
 
 107. Any equation of the fonn 
 
 la." + tna^ + n^ = (i), 
 
 represents two straight lines intersecting in the point C of the 
 triangle of reference. For if fi,, fi^ be the roots of the quad- 
 ratic 
 
 l/j,' + nifi + n = 0, 
 
IMAGINARY POINTS AND STRAIGHT LINES. 125 
 
 the equation (i) can be written 
 
 which shews that it represents the two straight lines whose 
 separate equations are 
 
 The two lines will be real if /i, and /a, are real, and they 
 will be imaginary if /i, and /i^ are imaginary. Or, assuming 
 the condition investigated in Algebra, the straight lines will 
 be imaginary or real according as iln, is or is not greater 
 than jw'. 
 
 For example, consider the equation 
 
 a'' + /3'+2a/3cosC=0. 
 
 Since 1 > cos^C, it follows that this will represent two ima- 
 ginary straight lines. 
 
 The following proposition is interesting. 
 
 108. The two straight lines represented by the equation 
 
 y8' + 7'+2;S7COs^ = (1), 
 
 are parallel, each to each, to the two straight lines represented ly 
 ,y»+a''+27acos-B=0 (2), 
 
 and also parallel, eacK to each, to the two straight lines repre- 
 sented hy 
 
 o^+^+2a.^coa (7=0 (3). 
 
 For the straight lines (1) meet the line at infinity in the two 
 points determined by 
 
 /9»+7'+2;S7COS^ = 0| 
 aa.- 
 
 i^'t cos ^ = 01 
 ; + J/3 + C7 = ol ' 
 
126 IMAGINARY POINTS AND STRAIGHT LINES, 
 
 the latter of which gives us by the transposition, 
 
 or 2jS7 = 
 
 Ic 
 
 Substituting this value of ^7 in the fiist equation, we get 
 
 Ic [^ + 7=) + [V^ + cV - aV) cos A = 0, 
 
 or a'rt cos A + ^hcosB + 'fc cos (7=0. 
 
 Hence the two straight lines (1) meet the line at infinity in 
 the two points given by the symmetrical equations 
 
 a'a cos A + ^h cos B + 7*0 cos C = 
 
 aa + 6/3 + 07 
 
 = 0) 
 
 By symmetry the straight lines (2) will meet the line at 
 infinity in the two points given by the same two equations. 
 
 Hence the straight lines (1) and the straight lines (2) pass 
 through the same two points at infinity, and therefore are 
 parallel. 
 
 And similarly, each pair is parallel to the two straight 
 lines (3). Q.E.D. 
 
 109. Def. We observe that the six straight lines repre- 
 sented by the equations (1), (2), (3), pass three and three through 
 two imaginaiy points on the straight lina at infinity. These two 
 imaginary points will be found hereafter to have some very 
 important and curious properties. We have indeed to refer to 
 them so often that it is convenient to have a special name by 
 which to distinguish them : and on account of properties which 
 we shall shew hereafter to belong to them it is deemed appro- 
 priate to term them the two " circular" points at infinity. 
 
 By this name we shall continually refer to them. 
 
IMAGINARY POINTS AND STRAIGHT LINES. 127 
 
 110. To find the ratios of the coordinates of the circular 
 points at infinity. 
 
 The circular points are given by the equations 
 
 /9' + 7*+2/37COs^ = (1), 
 
 and aa+&/3+C7 = (2). 
 
 The equation (1) determines the ratio of /3 to 7, thus 
 ^ + 2/S7 cos ^ + 7* cos'^ = - 7' sinM, 
 /3 + 7 cos ^ = + 7 J— 1 sin ^, 
 0= — fY (cos A ± J^ sin A], 
 or in virtue of the equation (2), 
 
 |8 __7_ a 
 
 cos A ± J— 1 sin ^ — 1 cos ^ + J^ sin B ' 
 
 the upper signs going together and the lower together. 
 
 Hence the ratios are known. 
 
 111. From considerations of symmetry we at once per- 
 ceive that the result of the last article, giving the ratios of the 
 coordinates of the two circular points as infinity may be ex- 
 pressed in any one of the following forms, 
 
 g ^ 7 
 
 — 1 cos C + >/— 3 sin C cosB ± J— 1 sin 5 ' 
 
 a ^ 7 
 
 cos 
 
 C ± J— 1 sin C - 1 cosA + J— 1 sin -4 ' 
 
 /3 
 
 _ 7 
 
 cos £ + J—l sin B cos A ± J— 1 sin ^ — 1 ' 
 
 By simple addition we can express these ratios in a form 
 symmetrical with respect to the three axes : but such form is 
 more complicated, and one of the forms already written down 
 will generally be more useful. 
 
128 IMAGINARY POINTS AND STRAIGHT LINES. 
 
 By multiplication we obtain 
 
 cos {B-C)±J-1 sin {B- C) cos {G-A) ± J-l sin {C-A) 
 
 7' 
 
 cos{A-B) ± J-y sin {A-B) ' 
 
 a result symmetrical as far as it goes, but when we come to ex- 
 tract the cube roots by Demoivre's Theorem we lose the symme- 
 try, as it is found upon examination that we cannot take similar 
 cube roots and write 
 
 ^ 
 
 cos^'±V-i3in^~cos ^ ± JZr, ,^0-A 
 
 A-B /— - . A-B' 
 cos — - — + V- 1 sm — —^ 
 o 3 
 
 but we must take dissimilar roots as 
 
 B-a , ,— - . B-G 
 cos — - — + V- 1 sm — -— ^ 
 o 3 
 
 . ^ 
 
 cos + V — 1 sm — 
 
 cos IZ^i^il^ ± VZi ,:- 4^^ + ^ - ^ ' 
 
 sm- 
 
 3 - — ^" 3 
 
 a much less convenient form than those already obtained. 
 
 112. Throughout the present chapter we have spoken of 
 trilinear coordinates, and proved the properties of imaginary 
 points and straight lines by the aid of them. 
 
 But all that we have said applies mutatis mutandis to tri- 
 angular coordinates : and in order to adapt our arguments to 
 
IMAGINARY POINTS AND STRAIGHT LINES. 129 
 
 this system it is in most cases only necessary to read unity for 
 a, b, and c severally, as well as for 2 A. 
 
 Thus our results will take the following form. 
 
 In Friangular Coordinates. 
 
 (i) K f+g + h = l, 
 
 and f'+ff'+^' = 0, 
 
 then f + fW^, g + ffW^, h + hW-^, 
 
 are said to be the coordinates of an imaginary point lying within 
 a finite distance of the lines of reference. 
 
 (ii) If f+g + h=o, 
 
 and f' + ff' + h' = 0, 
 
 then f+f'^^, 9+9"^^, h + hW^, 
 
 are said to be the coordinates of an imaginary point lying at an 
 infinite distance from the triangle of reference. 
 
 (iii) The results of Arts. 101—104, 106, 107 will remain 
 unchanged. 
 
 (iv) The two circular points at infinity are represented 
 by the equations 
 
 OS _ y 2 
 
 — a 5 cos C ± V— 1 6 sin C - c cos £ + V— 1 c sin J5 
 
 X y _ f 
 
 or 
 
 or 
 
 a cos C + V— lasin C — * c cos.4 + V— 1 c sin-4 
 
 X y ^_£_ 
 
 a cos .B + V— 1 a sin £ J cos -4 + V— 1 & sin -4 - c 
 
 w. 
 
130 
 
 EXERCISES ON CHAPTER IX. 
 
 Exercises on Chapter IX. 
 
 (83) If the imaginary straight lines 
 
 m4.vV^ = and u +v'V— 1 = 
 
 have a real point of intersection, then the four real straight 
 lines 
 
 u = 0, v=0, m'=0, v' = 
 
 are concurrent. 
 
 (84) The straight line joining the real point (a', ^, y) with 
 the imaginary point 
 
 if+f'^^), 9 + 9^^, h + hW~l 
 is represented by the equation 
 
 Virr=o, 
 
 «, 13, 7 
 
 + 
 
 a, A 7 
 
 «'. 0, i 
 
 
 a', /8', 7 
 
 / 9, A 
 
 
 /'. ^', A' 
 
 and will be real, provided 
 
 a', /?, 7 =0. 
 
 J^ 9^ A 
 /, ^', /*' 
 
 (85) If the value of the multiplier Tc vary, the locus of 
 the imaginary point 
 
 (J+hfJ^\, g + hg'J^, h+kh-J^) 
 
 is a real straight line. 
 
 (86) Shew that the equation in trilinear coordinates 
 
EXERCISES ON CHAPTER IX. 131 
 
 represents two imaginary straight lines parallel to each other 
 and to the real straight line 
 
 a cos A +^ COB B + ycoa G=0. 
 
 (87) Given m = 0, v = 0, w = the equations to three i:eal 
 straight lines in any system of coordinates, and 0, 0, ■yfr three 
 angles together equal to three right angles, shew that the equa- 
 tion 
 
 m' + v" + to" + 2vw sin + 2wu sin <f> + 2uv sin ■\jr = 
 
 represents two imaginary straight lines, and find their separate 
 equations. 
 
 (88) Shew that the equation in triangular coordinates 
 
 (x'-i-^+zyix + y + zf+Axi/z ix^+f+z") {x + y+z)+8a?fz'=0, 
 
 represents six imaginary straight lines parallel two and two to 
 the three lines of reference. 
 
 9—2 
 
CHAPTER X. 
 
 ANHARMONIC AND HARMONIC SECTION. 
 
 113. Definitions. 
 
 (1) Let a straight line AB be divided at P into two parts 
 in tlie ratio m : 1, and be divided at Q in the ratio n : 1, 
 
 then the ratio m : n \s called the anharmonic ratio of the 
 section of AB in P and Q. 
 
 (2) Let an angle AOB be divided by OP into two parts 
 whose sines are in the ratio m : 1, and be divided \ij OQ into 
 two parts whose sines are in the ratio n: \; 
 
 then the ratio to : n is called the anharmonic ratio of the 
 section of the angle A OB by OP and OQ. 
 
 114. It will be observed that if hoth the sections be external 
 or ioth internal m and n will be of the same sign, and therefore 
 the anharmonic ratio of the section will be positive. If one 
 section be external and the other internal, m and n will be of 
 opposite signs and the anharmonic ratio of the section will be 
 negative. 
 
 115. For the sake of brevity the anharmonic ratio of the 
 section of AB in P and Q is often spoken of as the anharmonic 
 ratio of the range of points APBQ, and it is expressed by 
 the symbol [APBQ]. 
 
ANHAEMONIC AND HARMONIC SECTION. 133 
 
 So the anharmonic ratio of the section of the angle AOB by 
 OP and OQ'is spoken of as the anharmonic ratio of the pencil 
 of straight lines OA, OP, OB, Q, and it is expressed by the 
 &ym\io\{O.APBQ]. 
 
 116. When we speak of the range of points APBQ it must 
 not be inferred that the points necessarily occur in the order in 
 which we read them : it must be understood that AB (terminated 
 by the points mentioned first and third) is the line which we 
 suppose divided, and P and Q (mentioned second Ani. fourth) are 
 the points of section. The sections may either of them be inter- 
 nal or external, but we read the letters in the order in which 
 they would come if the first section were internal and the second 
 external. It is found most convenient to adopt this system 
 because in a particular case of most frequent occurrence (the 
 case of harmonic section described below) one section is always 
 internal and the other external. 
 
 117. In expressing the ratios of lines it must be understood 
 that AB and BA denote lengths equal in magnitude but opposite 
 in sign. 
 
 ihus, ^p- p2- PB~ BP' 
 
 118. The anharmonic ratio of the range APBQ 
 -UPBO]-^-^ = ^^^^ 
 
 and the anharmonic ratio of the pencil OA, OP, OB, Q 
 
 \n A nnn^ _- si'^ AOP sin AOQ 
 - I ^' ^^^ V} - gin BOP ' sin BO Q 
 
 amAOP.smBOQ 
 "" ain AOQ.ainBOP' 
 
134 ANHAEMONIC AND HARMONIC SECTION. 
 
 119. If the -pencil OA, OP, OB, OQ cut any transverse 
 straight line in the range APBQ, the anharmonic ratio of the 
 pencil is the same as that of the range. 
 
 Fig. 23. 
 
 smQOP _AP , smAOQ _AQ 
 ^°^ sin PAO'PO sin PAO~QO'' 
 
 therefore 
 Similarly, 
 
 sin QOP ^AP QO 
 sinQOQ AQ'PO' 
 
 sin BOP BP QO 
 
 Bin BOQ BQ' PO' 
 
 therefore by division, 
 
 sin QOP. sin BOQ ^ AP. BQ 
 sin QOQ. s\a.BOP~ AQ. BP ' 
 
 or [O.APBQ =[APBQ]. q. e. d. 
 
 120. To shew that 
 
 we have only to write the values of the several anharmonic 
 ratios as in Art. 118, 
 
ANHAKMONIC AND HAEMONIC SECTION. 135 
 
 Thu3 {APBQ] =^|7||= [BQAP] ; 
 
 and {AQBP] = ^^=.[BFAQ}; 
 
 which prove the proposition. 
 
 121. Similarly we may prove that 
 
 {APBQ] = [PAQB] = [QBPA] = [BQAP]. 
 
 122. If the angle G of the triangle of reference he divided 
 hy two straight lines GP, GQ whose equations are respectively 
 
 /3 = ma. and /S = no., 
 
 then the anharmonic ratio of the pencil GA, GP, CB, GQ is 
 m : n. 
 
 For by Art. 4, 
 
 sin ^CP J smAGQ 
 
 BiuBGP Bin BGQ 
 
 therefore by division 
 
 {APBQ} =m:n. Q. E. D. 
 
 123. It follows that the anharmonic ratio of the section of 
 tlie same angle by the two straight lines 
 
 M + 2ma/3 + n^ = 0, 
 
 {m ± 'Jm^ - Inf 
 hi ' 
 
 which is real when m' > In, i. e. when the straight lines them- 
 selves are real. 
 
 If m? < In the straight lines are imaginary, and unless m — O, 
 the anharmonic ratio is also imaginary. 
 
 But if »« = and I and n have the same sign the two straight 
 lines become imaginary, but the anharmonic ratio of their sec- 
 tion of the angle C is real and equal to negative unity. 
 
136 ANHABMONIC AND HARMONIC SECTION. 
 
 Cor. The two sh-aight lines, whether real or imaginary, 
 which are represented by the equation 
 
 divide the angle of the triangle of reference so that the an- 
 hannonic ratio of the section is negative unity. 
 
 124. If (he angle between the straight lines w = and v = 
 be divided by the straight lines u + kv = Q and u + kv = 0, the 
 anharmonic ratio of the section is k : k. 
 
 Let OA, OB he the two straight lines represented by m = 
 and v=0, and OP, OQ the two straight lines represented by 
 M + «« = and u-\- k'v = 0. 
 
 Through any point 8 whose coordinates are (a', y3', 7') draw 
 a transversal SAPBQ cutting OA, OB in A and B, and OP, 
 O^inPand Q. 
 
 And let u, v be what u, v become when a', /S', 7' are written 
 for a, P, 7, 
 
 Also let \, jjL, V be the direction-cosines of the straight line 
 SAB, and let m, n be what u, v become when \, fi, v are written 
 for a, 0, 7. 
 
 Then we have (as in Art. 80) 
 
 SA=^-^, SP = -^L±Jfl^ 
 m m + Kn' 
 
 SB = -'^, 8Q = -^L^; 
 m m + Kn 
 
 therefore 
 
 < Ap-p>Q]_ -^P-BQ . W m+Kn)'\n m + Kn) 
 
 l^^^V] J_Q^BP- fi^ u' + k'v \ (V _±^\ 
 
 \m m + KnJ' \n m + Kti) 
 
 _ K {nu' — mv') (mv — nu) 
 K [nu' — mv) [mv' — nu) ' 
 
 or {APBQ] = !^,. q.e.d. 
 
ANHARMONIC AND HAKMONIC SECTION. 
 
 137 
 
 125. To find the anharmonic ratio of the pencil formed hy 
 the four straight lines, 
 
 M + KU = 0, M + \v = 0, M + /it) = 0, u + w = Q. 
 
 Put ^ = M + /cu and ■>^=u + fj,v, 
 
 then 
 
 K — fl K — fi 
 
 So the four given equations become 
 </> = 0, 
 
 lofr — /j,^ + \ [tj) — ■\fr) = 
 
 t = o. 
 
 h or < 
 
 •^ = 0, 
 
 K — \ , 
 
 K — V , 
 
 U + -^ = 0. 
 
 Hence by the last article the anharmonic ratio is 
 
 K —X K— V 
 
 X — fi ' V — /m' 
 
 or 
 
 (k - X) ill - v) ^ 
 (ic-v) ifi-X)' 
 
 126. Dep. If the anharmonic ratio of any pencil or range 
 be equal to — 1 the pencil or range is called harmonic. 
 
 An example of this is obtained if the angle C of the triangle 
 A GB be bisected internally and externally by straight lines CP, 
 CQ meeting the base AB in P and Q. 
 
 Fig. 26. 
 
138 ANHARMONIC AND HAEMONIC SECTION. 
 
 For by Euclid vi. 3, 
 
 AP:PB = AC:BC, 
 
 AQ:BQ = AC:BC: 
 
 AP AQ 
 whence PB'^'BQ' 
 
 AP.BQ _^ 
 '"' PB.AQ~ ' 
 
 A.^. f AP.BQ . 
 
 and therefore „„ . ^ = — 1 ; 
 
 i.e. the range APBQ is harmonic, and therefore by Art. 119 the 
 pencil CA, CP, CB, GQ is harmonic. 
 
 127. By reference to Art. 114, it will be seen that if a line 
 or angle be divided harmonically one of the sections must be 
 internal and the other external. 
 
 Thus if APBQ be a harmonic range one of the points P, Q 
 will lie between A and B and the other beyond them : the four 
 points will in fact occur either in the order in which they are 
 read in the numerator of the fraction 
 
 AP.BQ 
 AQ.BP' 
 
 (which expresses the ratio) or else in the order in which they are 
 read in the denominator. 
 
 128. From Arts. 120, 121, it appears that if APBQ be a 
 harmonic range then 
 
 {APBQ} = {PBQA} = {BQAP} = { QAPB] 
 = { QBPA) = {A QBP] = {PA QB) = {BPA Q} : 
 
 in other words, we may read the four letters in any order in 
 which neither A and B are contiguous, nor P and Q. 
 
ANHARMONIC AND HARMONIC SECTION. 139 
 
 129. It follows immediately from Art. 122 that the straight 
 lines whose equations are 
 
 /3 — ma = 0, and y8 + wia = 0, 
 divide the angle C harmonically. 
 
 Or more generally, from Art. 124, the four straight lines 
 
 tt = 0, M — KV = 0, 
 
 t! = 0, M + KV = 0, 
 
 form a harmonic pencil. 
 
 130. To find the equation to a straight line which shall form 
 with the three straight lines 
 
 u-\- KV = Q, M + \w = 0, M + /it; = 0, 
 
 a harmonic pencil. 
 
 Let u-\-w = Q, he the equation required. 
 
 Then by Art. 125, 
 
 {K-v){p.-\) 
 
 or (« - X.) (/* - v) + (« - v) {ii — X) = 0, 
 
 or (/c — 2\ + /i) v + X/t — 2/x« + «\ = 0, 
 
 X./X. — 2/x/c + kK 
 
 or 
 
 /c — 2\ + /li 
 
 Hence the straight line required will be represented by the 
 equation 
 
 M (« — 2\ + /ii) — •« (A./* — 2fiK + /c\) = 0. 
 
 131. To establish relations among the different anharmonic 
 ratios obtained hy taking a range of four points, or a pencil of 
 four straight lines in various orders. 
 
140 ANHAEMONIC AND HAEMONIC SECTION. 
 
 Four letters K, L, M, N can be written in 24 diflferent 
 orders. We hare seen however, in Art. 121, that there are four 
 different orders in which any range of points can be taken with- 
 out affecting their anharmonic ratio. Hence we cannot obtain 
 more than six different anharmonic ratios by taking the points 
 in different orders. We observe also from Art. 120, that the 
 reciprocal of any anharmonic ratio can be obtained by taking 
 the points in a different order. Hence we cannot expect more 
 than three different anharmonic ratios and their reciprocals. 
 
 We may shew this more formally as follows : 
 
 Let 
 
 M + «W = 0, M + Xu = 0, M + /iU = 0, M + ru = 0, 
 
 be the equations to four straight lines OK, OL, 021, ON inter- 
 secting in the point which is given by m = t) = 0. 
 
 And let 
 
 r= k\ + fiv, 
 
 m = K/1 + v\, 
 
 n = KV + \fi. 
 
 Then by Art. 125, 
 
 {KLMN} = (;-'^)y-^) = ^^LUII 
 {k — v)(jj, — \) m — l 
 
 m — n 
 
 or 
 
 ■m 
 
 {KLMN} = -f-^ (1)> 
 
 {KMNL} = -^^ (2), 
 
 and {KNLM}^-^-^ (3), 
 
 and the ratios {KNML}, {KLNM}, {KMLN} are (Art. 120) the 
 reciprocals of these three, therefore 
 
 ^^^^^J = -ll^ W. 
 
ANHAEMONIC AND HARMONIC SECTION. 141 
 
 {KLNM} = -*!^ (5), 
 
 {KMLN} = -'^— (&). 
 
 l — m 
 
 It will be seen (by Art. 120 or independently) that if the 
 letters K, L, M, ^be taken in any other order besides these six, 
 they will still give one of these same six ratios. 
 
 Hence we observe, 
 
 (i) that hy talchig a range of four points in different orders 
 we can only get six different anharmonic ratios. 
 
 (ii) that of these six ratios, three are ike reciprocals of the 
 other three. 
 
 (iii) that the ratio compounded of the first three is negative 
 unity, and so is the ratio compounded of the other three, 
 
 132. Cor. If {KLMN} be any harmonic ratio, any an- 
 harmonic ratio- obtained by talcing the points K, L, M, N in 
 
 different order will be equal either to +2 or to +-. 
 
 For since {KLMN} is harmonic, therefore by equation (1) 
 of the last article, 
 
 7n —n _ 
 
 or m — n — l — m. 
 
 And since m — n and Z — m are equal, each is equal to half their 
 sum ; that is, 
 
 m — n 
 
 l — m n — l 
 
 1 1 2 ■ 
 
 Hence the equations (2), (3), (5), (6) of the last article 
 
 give us 
 
 {KMNL} = {KMLN} = 2, 
 
 and 
 
 {KNLM] = {KLmi] = l, 
 
 and the equation (4) shews that {KN3fL} is harmonic. 
 
142 
 
 ANHAEMONIC AND HAEMONIC SECTION. 
 
 Conversely. If the anharmonic ratio of a pencil or range be 
 either 2 or i , we may obtain a harmonic pencil or range by 
 taking the lines or points in a different order. 
 
 133. We proceed to establish some important harmonic pro- 
 perties of a quadrilateral. 
 
 Let the straight line A'E C meet the three straight lines 
 BC, CA, AB in the points A', B', C respectively, so as to form 
 a quadrilateral. Let the diagonals A A, BB', CC be drawn 
 and produced so as to form a triangle ahc. 
 
 Fig. 27. 
 
 Let 
 
 M = 0, t) = 0, w=0, 
 
 be the equations to BC, CA, AB respectively, where «, v, w in- 
 clude such constant multipliers that the equation to A'B'C 
 may be (Art. 89) 
 
 u + v + w — d. 
 
 Then, as we shewed in Art. 90, the equation 
 
 V + w = represents the line AA' , 
 
 and v — w=-0 Aa; 
 
ANHAKMONIC AND HARMOKIC SECTION. 143 
 
 therefore by Art. 129, A A! and Aa divide harmonically the 
 angle contained by the straight lines 
 
 v=^...{AG), 
 
 to = 0... [AB). 
 
 Hence the pencil {A . BA' Ca} is harmonic, and therefore 
 (Art. 119) the range in which this pencil is cut by BB' will be 
 harmonic, that is, 
 
 {BcB'a} = -l. 
 
 And similarly, {CaC'b} and {AbA'c} are harmonic ranges. 
 Again, since the range {BcB'a} is harmonic ; 
 therefore the pencil {A . BcB'a] is harmonic, 
 
 and therefore the range in which this pencil is cut by the 
 straight line A'B" C is harmonic, i. e. if Aa meet A'B' C in 
 X, then 
 
 {C'A'B'X} is harmonic. 
 
 These properties may be extended and multiplied almost 
 without limit. 
 
 134. The following geometrical constructions are sometimes 
 iiseiul. 
 
 I. Given three points in a straigJd line to find a fourth 
 point completing the harmonic range. 
 
 Let A, P, B be the given points, and through AB describe 
 any circle, ARBO. Bisect the arc ABB in R, and join RP and 
 produce it to meet the circumference again in ; and through 
 draw OQ at right angles to RO meeting AB in Q. Then Q 
 shall be the point required. 
 
 Join A 0, BO, then by Eucl. III. 27, the straight line OR 
 bisects the interior angle A OB ; 
 
 therefore OQ sA right angles to it bisects the exterior angle ; 
 
 therefore by Art. 126, {APBQ} is harmonic, and Q is the 
 point required. 
 
144 ANHAEMONIC AND HARMONIC SECTION. 
 
 Tig. 28. 
 
 II. Given three concurrent straight lines to find a fourth 
 line completing the harmonic pencil. 
 
 Let OA, OP, OB be the three given straight lines, and let 
 them be cut by any transversal in the points A, P, B. Find a 
 point Q completing this harmonic range and join OQ, then (Art. 
 119) the straight line OQ forms a harmonic pencil with OA, 
 OP, OB, and is therefore the line required. 
 
 135. The constructions of the last article can be made by 
 the ruler alone, without the introduction of the circle, by apply- 
 ing the properties proved in Art. 134. Thus : 
 
 Fig. 29. 
 
EXERCISES ON CHAPTEE X. 145 
 
 In OA take any point P', and let OP, BF intersect in ^'; 
 then let AA', OB intersect in c, and let cF, AB intersect in Q. 
 Join OQ. 
 
 Then, applying to the quadrilateral ^P^'F the prop3rties 
 proved in Art. 133, 
 
 {APBQ} is a harmonic range, 
 
 and therefore 
 
 {O.APBQ) is a harmonic pencil. 
 
 Hence Q is the point, and Q the line, required. 
 
 Exercises on Chapter X. 
 
 (89) If two straight lines OK and OK' intersect a system 
 of parallel straight lines KK, LL\ MM', NN' in K, L, M, N 
 and K', U , M, W respectively, then will 
 
 {KLMN} = {K'L'MN'}. 
 
 (90) A point is taken within a triangle ABC, and OA, 
 OB, 00 are drawn; and through A, B, C straight lines B'C, 
 C'A', AB' are so drawn that each of the angles of the original 
 triangle is cut harmonically. Shew that the points of intersec- 
 tion of BG and B' G' , GA and G'A, AB and AB' are col- 
 linear. 
 
 (91) If through the vertex of a triangle two straight lines 
 be drawn, one bisecting the base and the other parallel to it, 
 they will divide the vertical angle harmonically. 
 
 (92) Any two straight lines at right angles to one another 
 form a harmonic pencil with the straight lines joining their 
 point of intersection with the circular points at infinity. 
 
 w. 10 
 
146 EXEKCISES ON CHAPTER X. 
 
 (93) If K, \, fi be in arithmetical progression the straight 
 line V = will form a harmonic pencil with the three straight 
 lines 
 
 U+ KV = 0, u + \v = 0, U + flV = 0. 
 
 (94) If K,\ ft he in barmonical progression the straight 
 line M = will form a harmonic pencil with the three straight 
 lines 
 
 U + KV = 0, M + Xt) = 0, u + /iv = 0. 
 
 (95) The four straight lines represented by the equations 
 
 M = 0, v = 0, 
 lu^ + 2muv + nv' = 0, 
 will form a harmonic pencil if 8n»' = 9ln. 
 
 (96) The angle between the two straight lines 
 
 lu' + 2muv + nv' = 
 is divided harmonically by the two straight lines 
 
 Tu' + im'uv + n V = 0, 
 provided In', mm', nl are in arithmetical progression. 
 
 (97) The anharmonic ratio of the pencil which the two 
 straight lines 
 
 Zm* + 2«iMv + wv* = 0, 
 
 form with the two straight lines 
 
 Tm' + 2m' uv + 71 V = 0, 
 is equal to 
 
 In' - 2mm' + wf + 2 V (m" - In) (»t" - ZV) 
 In' - 2mm' + nT + 2 '^ (m' - In) {m" - I'n') ' 
 
CHAPTER XI. 
 
 TKANSPOEMATION OF COORDINATES. 
 
 136. Suppose we have the equation to any locus referred 
 to a triangle AB C, and suppose we wish to find the equation to 
 the same locus referred to a new triangle A'B'G'. 
 
 The method of transformation will depend upon how the 
 new triangle is given, and two cases immediately present them- 
 selves ; first, the case in which the new triangle is given by the 
 coordinates of its angular points being assigned, and secondly, 
 the case in which the equations of its sides are given. 
 
 We proceed to discuss these two cases separately. 
 
 137. Case I. When the coordinates of the new points of 
 reference are given. 
 
 Let the coordinates of A', B', C referred to the original tri- 
 angle be (flf„ ^1, 7,), (a„ /3„ 7,), (a,, ^3, 7,) ; and let a', b', c 
 denote the sides of the new triangle A'B" C, and A' its area. 
 
 Also let a , yS', 7' denote the new coordinates of any point 
 whose old coordinates were a, y3, 7. 
 
 Then «!,«,, a, are the distances of BC from the three points 
 A,B, (7' respectively; therefore the equation to the straight line 
 BG referred to the new triangle A:B' G' is (by Art. 23) 
 
 a.a'a' + a.J'/3' + ff/7' = (i). 
 
 10—2 
 
148 
 
 TEANSFOKMATION OF COORDINATES. 
 Fig. 30. 
 A 
 
 And therefore (Art. 58) the perpendicular distance of from 
 the line BC\a 
 
 But this perpendicular distance is the coordinate a ; there- 
 
 fore 
 
 and so 
 and 
 
 « = ^a7 ("i*'*' + "»*'^' + "ac'v'). 
 
 2A' 
 
 
 .(ii) 
 
 equations which express the old coordinates a, j8, 7 of any point 
 explicitly in terms of the new coordinates a', /S", 7' of the same 
 point. 
 
 If therefore an equation is given connecting the old coordi- 
 nates a, /3, 7 of any point on some locus, by writing the three 
 expressions given by (ii) instead of a, P, 7, we at once obtain a 
 new equation connecting the new coordinates a', ^, 7' of any 
 point on the same locus : that is, we obtain the equation to the 
 locus referred to the new triangle A'B' C. 
 
TEANSFOEMATION OF COOEDINATES. 149 
 
 Thus if the equation to any locus referred to the triansle 
 ABC be 
 
 /(a,/3,7)=0, 
 
 the equation to the same locus referred to the triangle A'B'C is 
 
 •^ \ 2A' ' 2A^ ' 
 
 7,g'a' + yjb'^ + 7/7 \ ^ 
 
 But if, as is nearly always the case, the given equations be 
 homogeneous, then 2A' will divide out, and therefore 
 
 If any locus referred to the triangle ABG he represented hy 
 the homogeneous equation 
 
 /(«,/3,7)=0, 
 
 the same locus referred to the triangle AB'C will have the 
 equation 
 
 f[p.,dd + a//3' + a/7', ^/a' + ^^^ + y33c'7', 
 
 7io'a' + 7/^ + 7307') = 0. 
 
 138. CoE. It will be observed that the equation just ob- 
 tained is necessarily of the same degree as the original equation. 
 Hence the degree of an equation is not altered hy transformation 
 of coordinates. 
 
 This is a very important result. 
 
 139. This case of transformation of coordinates becomes 
 very much simpler when triangular coordinates are used. 
 
 For if {x, y, z) be the triangular coordinates of any point 
 referred to the old triangle of reference, and («', y', z) the tri- 
 angular coordinates referred to the new triangle, the angular 
 points of the latter being 
 
 K.yi. «i). (a'.>yi, «»), K. 2^3' «»)> 
 
150 
 
 TEANSFOEMATION OF COOKDINATES. 
 
 then the equations (ii) of the last article become 
 
 so = x^al + x^' + x/\ 
 y = y,x' + y^y'+y^A • 
 z = z^x' + z^' + z^'t 
 
 And the equation f{x, y, z)—0 therefore transforms into 
 /(«/ + x^' + x/, y^af + y^' + y/, «,«' + z^' + z^') = 0. 
 
 140. Case II. When the, equations of the new lines of re- 
 ference are given. 
 
 If the sides of the new triangle be represented by the equa- 
 tions in terms of the perpendiculars from A, £, C; viz. by the 
 equations 
 
 p,aa + qfi^ + r^y = 0, 
 
 p^an + qfiP + r^ = 0, 
 
 p^aa. + qfi^ + r^crf==0, 
 
 then the coordinates of A, B, G referred to the new triangle are 
 respectively : 
 
 new coordinates of .4 , {pi,j>„pg), 
 
 of -S, C?,,?., q,), 
 
 of C, (r,, r,, r,). 
 
 Hence, if (as before) any point have the old coordinates 
 (a, /8, 7) and the new coordinates {a, /S', 7'), then a represents 
 the perpendicular from (a, yS', 7) on the line joining {q„ q^, q^ 
 and (r„ r,, rj. 
 
 And aa. is the double area of the triangle whose angular 
 points in the new coordinates are 
 
 («'. /3', 7'), {q„q^,q,), {r„ r„ r,). 
 
 Hence we have 
 
 aa = 
 
 28' 
 
 
TEANSFOBMATION OF COORDINATES. 
 
 151 
 
 a = 
 
 2aS' 
 
 
 ' ^ 265' 
 
 «'» ^. 7 
 
 i'..i'..i'3 
 
 7 = 
 
 _ 1 I «', ys', 7' 
 
 2ci8' 
 
 
 Therefore, if the hens of the homogeneous equation 
 f{a,^,y)=0 
 
 be referred to the new triangle whose sides are given hy 
 pfla + qfiP + r,c7 = 0, 
 pjaa. + qj>^ + r,C7 = 0, 
 i'aoa + gs'^/S + r,C7 = 0, 
 
 tto e^zMctibn in terms of the new coordinates will be 
 
 f{l 
 
 «', /3', y' 
 
 ?1> ?«> ?3 
 
 1 
 
 ' b 
 
 a', ye-, 7 
 
 
 ) 
 
 = 0. 
 
 141. The cases which occur in practice are generally very 
 simple. The following is an example. 
 
 Let an equation connecting the coordinates a, 0, y involve as 
 terms or factors the three expressions 
 
 Za + wi/S + ny, I'a + m'^ + n'y, T'a + ml'^ + ri'y, 
 
 and suppose we have to transform it to the new triangle of 
 reference whose sides are represented by the equations 
 
 Za + wi/9-+ 717=0, Ta + m'^ + n'y=0, ra + m"^+n"y = 0. 
 
152 .TEANSFOKMATION OP COOKDINATES. 
 
 Then if a', /3', 7' denote the new coordinates of the'point whose 
 old coordinates are a, /3, 7, we have (Art. 46) 
 
 , la + m^ + ny ^ _ I'oi + m^ + n'y 
 , Va + m"^ + «"7 
 
 Hence, in effecting our transformation, wherever the expres- 
 sions 
 
 ln + m^ + wi, ta + m'^ + n'y, l"a + m"^ + n"'Y 
 
 occur, we have only to substitute for them 
 Ka, kP, k 7, 
 
 where «, «', k" are constants and represent the expressions 
 {l,m,n}, {r,m',n'}, {r,m",n"}. 
 
 It follows that if the original equation be made up entirely of 
 the expressions 
 
 Za+»i/9 + W7, loL + m'^ + n'y, roi + m"fi+n"y, 
 
 the transformation will generally simplify it very much. 
 
 It should also be borne in mind, if 
 
 aa+b^ + cy 
 
 occur as a factor or a term in the original equation, that since it 
 is known to represent a constant quantity it cannot be trans- 
 formed into an expression which would denote a variable quan- 
 tity. It can therefore take no other form than 
 
 K{a'a:+b'^ + c'y'), 
 
 where a, V, c' are the sides of the new triangle of reference, 
 and K is a constant expressing the ratio of the areas of the old 
 and new triangle. 
 
exeecises on chapter xi. 153 
 
 142. Example. 
 The equation 
 {la + OT/3 + ny) (aa. + h^ + cyY 
 
 = K (Z'a + m'^ + n'i) {l"a + m"/3 + n'^y, 
 may be transformed into the much simpler form 
 
 a' (aV + J'/3' + c'yT = a:'/9'7", 
 by taking the straight lines 
 
 la. + 7w/3 + ny = 0, 
 la + ?n'/3 + n'y = 0, 
 ra+7?i"j8+n"7=0, 
 as lines of reference. 
 
 We shall very often have recourse to such a transformation 
 as this in order to simplify the equations of curves. 
 
 ExEECiSES ON Chapter XI. 
 
 (98) Transform the equation 
 
 a'' + /3' + 7* + 2^y cos <^ sin i/r + 27a cos 1^ sin 9 
 
 + 2ayS cos ^ sin <^ = 
 
 to the new triangle of reference formed by the straight lines 
 
 ^ cos ^ + 7 sin •^ = 0, 
 
 7 cos ^ + a sin ^ =0, 
 
 a cos ^ + /3 sin ^ = 0. 
 
 (99) Transform the equation 
 
 wa' + uyS" + wV + 2m'/37 + 2t>'7a + 2w'a^ = 
 
 to the new triangle of reference formed by the straight lines 
 ua + w'^ + v'y = 0, 
 w'yS + v'y — 0, 
 (w' — m'io') /S — (ww — mV) 7=0. 
 
CHAPTER XII. 
 
 SECTIONS OF CONES. 
 
 143. Having explained the principles of Trilinear Coordi- 
 nates and exhibited the application of the method to the inves- 
 tigation of the properties of straight lines, we now pass on to 
 apply the same method to carved lines, commencing with the 
 conic sections. 
 
 We shall endeavovir to make our investigation of these curves 
 as independent as possible of the knowledge which we have 
 acquired of their properties by purely geometrical and other 
 methods. At the same time the student must not expect to 
 find in these brief chapters anything like a complete treatise on 
 the properties of the curves, as it is our object rather to set 
 before him such properties as can be advantageously treated of 
 by trilinear coordinates, and to leave for treatment by other 
 methods those properties to which other methods are specially 
 applicable. Success in the solution of a problem generally 
 depends in a great measure on the selection of the most appro- 
 priate method of approaching it ; many properties of conic sec- 
 tions (for instance) being demonstrable by a few steps of pure 
 geometry which would involve the most laborious operations with 
 trilinear coordinates, while other properties are almost self-evi- 
 dent under the method of trilinear coordinates, which it would 
 perhaps be actually impossible to prove by the old geometry. 
 We shall strive to set before the student such a series of propo- 
 sitions as shall best illustrate the use of trilinear coordinates, and 
 
SECTIONS OP CONES. 
 
 155 
 
 at tlie same time put him into possession of such properties and 
 results as are most often called for in the solution of problems. 
 
 144. Any plane section of a right circular cone when referred 
 to suitable lines of reference may be expressed by an equation of 
 the form 
 
 la' + m^^ + ny" = 0, 
 
 where I, m, n are not all of the same sign. 
 
 Fig. 31. 
 
 Let he the vertex, OZ the axis of any right circular 
 cone, and XYZ any plane cutting the cone in a curve, one of 
 whose points is P. 
 
 Through draw two straight lines OX, OF at right angles 
 to OZ and to one another, meeting the plane of the section in 
 X, Y, and let 6, <f>, ■^ be the angles which OX, OY, OZ make 
 with the perpendicular upon XYZ. 
 
156 SECTIONS OF CONES. 
 
 Take XYZ as triangle of reference, and let 
 
 Fx=:a, Py = p, Pz = 'i, 
 
 be the trilinear coordinates of any point P on the curve. 
 
 Let PL, PM, PN be the perpendicular distances of P from 
 the planes OYZ, OZX, OXY. 
 
 Then we hare 
 
 PL . . PM . . PN . , ,.. 
 
 j^=sm0, ■p^=sin^, ■^ = sin'f (i). 
 
 But if to be the semivertioal angle of the cone 
 PX= ON. tan 0), 
 and since ON" = PL' + Pif , (Euclid, l. 47) 
 
 PN* = {PL' + PM') tan= a, ; 
 and therefore, in virtue of (i), 
 
 7" sin'i/r = (a' sin*5 + ^ sin'^) tan'w, 
 or a' sin'^ tan'w + yS" sin'i^ tan'<a — 7" sin'^ = 0, 
 
 which may be written 
 
 Za"+iwy3' + «7' = 0, 
 
 it being observed that n is of the opposite algebraical sign to 
 I and m ; but the absolute ratios of I, m, n are any whatever, 
 since angles can be found with their sines in any assigned ratio. 
 
 145. Any section of a right circular cone, to whatever lines 
 it he referred, will he represented hy a homogeneous equation of the 
 second degree in trilinear coordinates. 
 
 For we have shewn in the last article that when suitable lines 
 of reference are chosen, the conic section will be represented by 
 an equation of the form 
 
 la* + m^ + m7» = 0, 
 
 a homogeneous equation of the second degree. 
 
SECTIONS OF CONES. 157 
 
 Now if we transform our coordinates to any other lines of 
 reference (Art. 138, page 149), the degree of the equation will 
 not be altered. Hence, whatever be the lines of reference, the 
 conic will be represented by an equation of the second degree. 
 Q. £. D. 
 
 146. Conversely. 
 
 Every equation of the second degree in trilinear coordinates 
 represents some conic section. 
 
 For any equation of the second degree (being rendered homo- 
 geneous, Art. 10, page 13) may be written in the form 
 
 wa' + v^+wf+ 2it'(S7 + 2«'7a + 2to'a/3 = 0. 
 Multiplying by u and re-arranging, we get 
 
 (wa + w'^ + v'i)' 
 
 = (m)" - uv) /3' - 2 {v'w - uu) ^y + («'= - mw)y'. 
 
 Now multiplying by w'" — uv and rearranging, we get 
 (w'* — uv) (wa + w'yS + vyY = {(w'" — uv) /3 — {v'w — uu) 7]" 
 4- u {uvw + 2u'v'w' — uu' — vv'^ — ww'^} 7'. 
 
 And if we take the lines 
 
 wa + tOj/3 + v'<y = 0, 
 and (w" — uv) /3 — {v'w — uu) 7 = 0, 
 
 as lines of reference instead of 
 
 a = 0, 
 
 and y3 = 0, 
 
 the equation takes the form 
 
 Ij? + m^ + «7' = 0, 
 
 and therefore represents a conic section by Art. 144. 
 
 Hence, every equation of the second degree in trilinear coor- 
 dinates represents a section of a right circular cone. 
 
158 SECTIONS OP CONES. 
 
 147. A conic section can generally he found to satisfy five 
 simple conditions. 
 
 For the general equation to a conic section has been seen to 
 consist of six terms. It therefore involves six different coeflSci- 
 ents, and by altering the ratios of the coefficients one to another 
 we shall obtain different equations representing different conic 
 sections. 
 
 But since an equation is not altered by altering all its coeffi- 
 cients in a constant ratio, we shall obtain all possible variations 
 of the equation by giving any one of the coefficients an assigned 
 value and varying the values of the remaining five. 
 
 In order to determine the equation to a conic section we 
 have therefore five unknown coefficients to assign. And since 
 each simple condition which the conic satisfies (such as the con- 
 dition of passing through a particular point or touching a parti- 
 cular straight line) will in general give us one equation among 
 the coefficients, it follows that we can generally so assign the 
 coefficients as \haXfive such conditions may be satisfied. 
 
 The following article illustrates this. 
 
 148. To find the equation to the conic section which passes 
 through five given points. 
 
 Let 
 
 («i, ^,. 7.), (a,. ^a> 7»). (cs' /S„ Ts), (a*. ^«, 7,), («., ^^, %) 
 be the coordinates of the five given points. 
 
 And suppose 
 
 Ma' + v^+ wy' + 2w'/37 + 2«'7a + 2w'a^ = 
 to be the equation to the conic. 
 
 Then, since the five points lie upon it, we have 
 «a,' + w/3,' + toy,' + 2!t'/3,7, + 2«'7,o. + 2w'a,/3, = 0, 
 ««/ + f /3/ + W7/ + 2m'/3,7, + 2w'7,a, + 2w'a^^ = 0, 
 
 ««,' + «^.' + W7/ + 2m'A73 + 2«'73«3 + 2m''«A = 0, 
 Mx/ + v/3,' + «07/ + 2m'jS,7, + 2u'7,a. + 2w'a^^ = 0, 
 ««/ + v^,' + Mvy/ + 2m'/3,7, + 2t)'7,a. + 2w'a^, = 0. 
 
SECTIONS OF CONES. 
 
 159 
 
 Hence, eliminating u: v : w. u : v : w',we get 
 a», ^, 7», fiy, rya, a/3 = 
 
 a,'. /8,'> %'. ^i7,. 7i«.. "i/S. 
 «,*, /S.', 7,'. ^7!,. %ai, a A 
 
 «.'. ^3*' 73'. ^373. 73«3. «A 
 
 «/• /3/. 7/, /3474. 74a4. aA 
 a.'. ^.'. 7*'. ^575. 76a«r aA 
 which will be the equation required. 
 
 149. To find the condition that six points whose coordinates 
 are given should lie upon one conic. 
 
 Let the given coordinates he (a,, j8j, 7,), (o^, ^8,, 7,), 
 (a.. /Sj, 73), (a„ /S^, 7j, (a^, ^j, 7J, (a,, ^., 7J. 
 
 The points will lie upon one conic if the coordinates of one 
 of them satisfy the equation to the conic through the other 
 five. 
 
 Hence by the last article the condition is 
 
 «.', A', 7.', A7., 7A, a A =0. 
 a,'. /3,', 7,', ^,7,, 7,a„ afi, 
 
 "', ^a, %\ ^7,. 7s«3. «A 
 a*^ ^A 7*'. /3«74, 74«4. aA 
 
 ««'. /^s', 7b'. ^676. 76''5. <^A 
 
 «a'. ^6*. 76*. /3a7., 7««,. a A 
 
 150. Every straight line meets every conic section in two real 
 or imaginary jooints, distinct or coincident. 
 
 Let Ma' + «/3'+to7'+ 2M'^7 + 2i;'7a + 2w'a/3 = (1) 
 
 be the equation to any conic section, and 
 
 la. + m^ + iVY = (2) 
 
 the equation to any straight line. 
 
160 SECTIONS OF CONES. 
 
 The coordinates of their points of intersection must satisfj 
 both the equations as well as the relation 
 
 aa + 6/3 + c7 = 2A (3). 
 
 Hence, to find the coordinates we may proceed theoretically 
 thus: 
 
 From (2) and (3), which are simple equations, we may 
 express fi and 7 as functions of a of the first degree. We may 
 then substitute these values in (1), which thus becomes a quadra- 
 tic equation in a. Being a quadratic it will give two values for 
 a, real or imaginary, equal or unequal, and the simple equations 
 (2) and (3) will then give a value for /3 and a value for 7 corre- 
 sponding to each value of a. Thus there will be determined two 
 and only two points of intersection, which may however be real 
 or imaginary, coincident or distinct. 
 
 151. If in the argument of the last article the straight line be 
 at infinity the reasoning still applies. Hence every conic cuts the 
 straight line at infinity in two real or imaginary points, either 
 coincident or distinct. 
 
 If these two points be real and coincident the conic section 
 is called a Parabola. 
 
 If they be real and distinct it is called a Hyperbola. 
 
 If they be imaginary it is called an Ellipse. 
 
 152. Def. Tangents which do not lie altogether at infinity 
 but have their contact at infinity are called Asymptotes. 
 
 It follows that an ellipse has two imaginary asymptotes and 
 a hyperbola two real ones. 
 
 In the parabola, since the straight line at infinity meets it 
 in two coincident points, that line is a tangent, and there can be 
 no other tangent touching at infinity. Hence the parabola has 
 no asymptote. 
 
 153. Students who have not commenced Analytical Geo- 
 metry of Three Dimensions may omit the remainder of this 
 chapter. 
 
SECTIONS OF CONES. 161 
 
 But those who have made any progress in that subject will 
 observe that the first article in this chapter is but a particular 
 case of the following more general theorem. 
 
 i/ f{x,y,z) = Q (1) 
 
 he the equation to any surface in rectangular coordinates of three 
 dimensions, and if 
 
 a; cos 6 -V y cos<j) + z cos yjr^p (2) 
 
 be the equation to any plane, then the equation to the section of 
 the surface (1) hy the plane (2) loill he 
 
 f{a.smd, /3sin0, ys'my}r) = () (3), 
 
 the coordinates being irilinear, and the lines of reference the 
 traces of the coordinate planes upon the plane of the section. 
 
 For if P be any point upon the section, x, y, z its coordi- 
 nates regarding it as a point upon the surface (1), and a, /3, 7 its 
 coordinates referred to the traces of the original planes upon the 
 plane of section, then since 6, (j>, -yjr are the angles which this 
 plane makes with the original planes, we have 
 
 x = a sin 0, y = /3 sin <fi, 2=7 sin -yjr, 
 
 therefore substituting in (1), we get 
 
 f {a sin 6, /8sin^, 7sin-\|r) = (3), 
 
 a relation among the trilinear coordinates of any point in the 
 section, and therefore the equation to the section. 
 
 154. It should be noticed that if the surface represented 
 by the equation (1) happen to be a conical surface having its ver- 
 tex at the origin, the equation (1) will be homogeneous in x, y, z, 
 and therefore the equation (3) will be homogeneous in a, /3, 7. 
 
 In all other cases these equations will be heterogeneous, and 
 we shall generally have to make the trilinear equation (3) homo- 
 geneous by means of the identical relation 
 
 aoL + h^ + cy = 2A, 
 as explained in Art. 10. 
 
 AV. 1 1 
 
362 SECTIONS OF CONES. 
 
 155. COK. If/(a;,y, 2)beofthen"'oi-der, then 
 /(a sin 0, /S sin <f>, 7 sin ■<fr) = Q 
 is of the Ji"" order : that is, 
 
 Uvery plane section of a surface of the «* order is a curve 
 of the w**" 01-der. 
 
 The following particiilar case is important. 
 
 Every plane section of a surface of the second order is a conic 
 
 section. 
 
 156. The identical relation among the coordinates a, ^, 7 
 may be obtained directly from the equation to the plane, thus 
 
 a; cos +y cos <^ + z cos i/r =p, 
 but a;=asin0, y=^Bm<f), z^ysinylr; 
 
 therefore substituting 
 
 a. cos 6 sin 6-^ 13 cos ^ sin </> 4 7 cos yfr sin 1^ =p, 
 or a sin 2^ + /3 sin 20 + 7 sin 2i|r = 2p. 
 
 Cor. The equation to the straight line at infinity in this 
 plane loill he 
 
 a sin 26 + P sin 2<^ + 7 sin 2a^ = 0. 
 
 157. To find the section of the el^ysoid whose equation is 
 x^ y^ 3' , 
 
 hi the plane whose equation is 
 
 xcos6 +y cos <f) + z cos ■>lr=p. 
 
 The equation to the section in trilinear coordinates is (Art. 
 
 153) 
 
 a' sin'g /ysin't/) 7^ sin'^r _ , 
 „' + J? ^ /." ~^' 
 
SECTIONS OF CONES. 163 
 
 which has to be rendered homogeneous by means of the relation 
 a cos sin ^ + y cos <j> sin<ji + s cos •^ sin yfr =p. 
 
 Hence the equation becomes 
 
 g' sin'ff ^ sin'<^ 7' sin'>/r 
 
 (« sin 2^ + /3 sin 2^ + 7 sin 2i|r)'' ^ 
 CoE. If the plane of section were 
 
 -+f + -=1, 
 a c 
 
 we should have 
 
 cos = ^ , cos <f) =f , cos 1^ = ^- , 
 
 a ^ b ^ c 
 
 and the equation to the section would be 
 ^ sin'0 + ^ sin'^ + % sW'f 
 
 /8 
 
 /a • n /8 • , 7 ■ ^' 
 ( - sin » + ^ sm ^ + - sm -^ j = 0, 
 
 or a^7 sin ^ sin i/r + Jyx sin i/r sin + ca/3 sin 6 sm^ = 0, 
 
 or ^^ + ^"y" + '^"^ = 
 
 sin sin ^ sim^ 
 
 Exercises on Chapter XII. 
 
 (100) Determine three straight lines which being taken as 
 lines of reference, the equation 
 
 \/87 + fiya + va/S = 
 shall be transformed to the form 
 
 11—2 
 
164 EXERCISES ON CHAPTER XII. 
 
 (101) Shew that the six points given by 
 
 a_^_7 2l=^=1 - = ^ = 1 
 l'~m,~n' m n V n I m' 
 
 "■-§.-1 2 = ^ = 1 - = ^ = ^, 
 n~ m~ I ' I n m' in I «' 
 
 lie all on one conic section. 
 
 (102) More generally, shew that if 
 
 M = 0, V = 0, w = 
 
 be the equations to three straight lines, then the six points 
 given by the equations 
 
 u V w u V _w "_'*'_ '^ 
 I m n ' m n Z ' n I vi' 
 
 n V w u V 10 ii _v w 
 n~ m I ' I n m' in I n ' 
 
 lie all on one conic section. 
 
 (103) Find the equation to the conic section circumscribing 
 tlie pentagon, whose sides taken in order are represented by the 
 equations in triangular coordinates, 
 
 x = i/+z, y = z + x, 
 £C=0, s = 3(a;+,(y), y = Q. 
 
 (104) Sliew that the six points in which the straight lines 
 
 jS cos 5 = 7 cos G, /3 sin 5 = 7 sin C, 
 7 cos C = a cos ^, 7sin C = a sin-^, 
 a cos ^ = ^ cos B, a sin ^ = /8 sin B, 
 
 (drawn from the vertices of the triangle of reference) intersect 
 the opposite sides, lie all on one conic section. 
 
CHAPTER XIII. 
 
 ABRIDGED NOTATION OF THE SECOND DEGREE. 
 
 158. We have already in Chapter viir. considered the in- 
 terpretation of a variety of equations obtained by combining the 
 symbols u, v, w, &c. which we have used to denote expressions 
 of the first degree in trilinear, triangular, or other coordinates. 
 
 But the equations of that chapter were all formed by the 
 addition or subtraction of constant multiples of u, v, w, &c. and 
 were therefore all of them equations of the first degree, and con- 
 sequently represented straight lines. 
 
 We now go on to interpret equations of the second order 
 obtained by combining u, v, w, &c. where these symbols them- 
 selves still represent expressions of the first degree in the coor- 
 dinates, and therefore when equated severally to zero form 
 equations to straight lines. 
 
 159. To interpret the equation 
 
 UV — KWX = 0, 
 
 where w = 0, » = 0, w = 0, a; = are the equations to four 
 straight lines and k is any constant. 
 
 Since the equation is of the second order, it represents a 
 conic section (Art. 146). 
 
 Moreover it is obviously satisfied if m = and w = are 
 simultaneously satisfied, therefore the conic passes through the 
 intersection of these two straight lines. 
 
166 
 
 ABKlDGEi) NOTATION 
 
 Similarly it passes through the point determined by 
 M = and x = 0; 
 also by v=0 and w=0, 
 
 and V = and x = 0; 
 
 therefore the equation represents a conic circumscribing the 
 quadrilateral in which u and v are opposite sides, and w and x 
 opposite sides. 
 
 Fig. 3-2. 
 
 160. We may notice the particular case when x = repre- 
 sents the straight line at infinity. The equation then represents 
 a conic passing through the points in which w is cut by u and 
 V, and having its asymptotes parallel to these last two straight 
 lines. This will be more clearly seen after reading Chapter 
 xvir. 
 
 Fig. 33. 
 
OF THE SECOND DEGREE. 167 
 
 161. To interpret the equation 
 
 UV — KW^ = 0. 
 
 We may regard this as a particular case of the last equation, 
 ■where the opposite sides w = 0, a; = of the quadrilateral have 
 become coincident. 
 
 Each of the lines ?« = 0, u = therefore meets the conic in two 
 coincident points on tlie coincident straight lines w = and 
 a:=0. 
 
 Hence m = and v = represent tangents, and w = repre- 
 sents the chord of contact. 
 
 Cor. Consider the particular case 
 
 UV — KS^ = 0, 
 
 where s = is the straight line at infinity. The lines v, = 0, 
 « = are now tangents touching the conic at infinity, and are 
 therefore asymptotes. Hence the equation uv — ks' = repre- 
 sents a hyperbola whose asymptotes are m = 0, « = 0. 
 
 162. Any two conies may be regarded as intersecting in four 
 real or imaginary points. 
 
 For the two conies will be represented by two equations, 
 each of the second degree, solving these together with the 
 relation 
 
 na + J/3-|-C7=2A, 
 
168 ABltlDGED NOTATION 
 
 if the coordinates be trllinear, or 
 
 x + y-f z= \, 
 
 if they be triangular, we obtain four different sets of real or 
 imaginary values for the coordinates, satisfying both the equa- 
 tions. These will determine four real or imaginary points lying 
 on the two conies. But since two or more of the solutions of the 
 equations may be identical, it follows that two or more of the 
 points of intersection may be coincident. 
 
 163. Cor. Two conic sections have in general six common 
 chords, since four points can be joined two and two in six 
 ways. 
 
 We say in general, because if some of the four points of 
 intersection be coincident, some of the common chords will 
 become also coincident, while others will become common tan- 
 gents. 
 
 Def. The chord joining ant/ two of the points of intersec- 
 tion of two conies, together with the chord joining tlie other two 
 points, constitute a pair of common chords of the two conies. 
 
 Thus if P, Q, R, S be the four points of intersection, the six 
 common chords can be arranged in the three pairs QB, P8; 
 RP, QS; PQ, RS. 
 
 164. Definitions. If two of the four points of intersection 
 of two conies are coincident, the conies are said to touch one 
 another. 
 
 If the other two are also coincident, but not coincident with 
 the first two, the conies are said to have double contact with 
 each other, and the common chord is then called the chord 
 of contact. 
 
 If three of the points coincide, the points are said to have 
 tiiree-pointic contact, or to have the same curvature at the point 
 of contact. 
 
OF THE SECOND DEGKEE. 169 
 
 165. To interpret the equation 
 
 S — KUV = 0, 
 
 <S' = being the equation to a conic. 
 The equation 
 
 S — KUV = 0, 
 
 being of the second order, represents some conic section. 
 
 To find where the straight line m= meets it, we have to 
 substitute m = in the equation S — kuv = : whence we must 
 have S = 0; 
 
 i.e. the points of intersection of the straight line m = witli 
 the conic to be investigated lie upon the given conic S; 
 
 i.e. M = is a common chord of the two conies. 
 
 So t? = is a common chord of the two conies. 
 
 Hence S — kuv = represents a conic intersecting the conic 
 /S' = in the four points which lie upon the lines m = 0, v = 0. 
 
 166. To interpret the equation 
 
 S-KU^=0. 
 
 We may regard this as a particular case of the last equation, 
 M = and v = coinciding, and so by reasoning similar to that 
 in Art. 161 we conclude that it represents a conic touching the 
 conic 8 = in two points, m = being the equation to the chord 
 of contact. 
 
 167. It will be observed that all the equations which we 
 have considered in the present chapter have been found to re- 
 present conies passing through some four fixed points. 
 
 Each equation has involved an undetermined constant k, 
 which may receive diiferent values distinguishing the different 
 conies which can be drawn through the same four points. 
 
170 ABRIDGED NOTATION 
 
 And by giving k a suitable value, we can make the equation 
 represent any conic whatever passing througli tlie four points. 
 
 For since five points determine a conic section, any conic 
 through the four points will be determined by one point more, 
 and the condition that the equation may be satisfied at this 
 point is an equation from which ic may be determined. 
 
 Hence by giving k a suitable value, the equation uv — kwx = 
 will represent any conic circumscribing the quadrilateral whose 
 opposite sides are u, v and w, x. So uv — mo' — will represent 
 any conic having u = 0, v = () as tangents, and to = as chord of 
 contact, and so in the other cases. 
 
 168. If S—Q and S'=0 represent the equations to two 
 conic sections, then will the equation S + kS'=0 represent a conic 
 section passing through all the jjoints of intersection of the first 
 two. 
 
 And by giving a suitable value to k, this equation can he made 
 to represent any conic whatever passing through those points of 
 intersection. 
 
 For the coordinates of any point of intersection of /S = and 
 S' = satisfy both the equations : i. e. they make S and S' seve- 
 rally zero : therefore they make 8 + kS' zero : i. e. they satisfy 
 the equation S + kS' = 0. Hence the locus of this equation 
 passes through every point of intersection of the given conies. 
 
 And since 8 and 8' are of the second degree, 8+ k8' = 
 is of the second degree, and therefore 8+k8'=0 represents a 
 conic section. 
 
 Further, by giving a suitable value to k this equation will 
 represent any conic passing through the four points of intersec- 
 tion of the first two. For since five points determine a conic 
 section, any conic passing through the four points of intersection 
 will be determined if one other point upon it be determined. 
 
 It is only necessary therefore to shew that by giving k 
 a suitable value, the equation S + k8' — can be made to pass 
 through any one assigned point ; which follows immediately as 
 
OF THE SKCUNU DEGREE. 171 
 
 in the last article. For if s and s' be what S and S' become 
 when the coordinates of the assigned point are substituted for 
 the current coordinates, the equation s + ks' = will be the con- 
 dition that the conic should pass through the assigned point. 
 
 s 
 Hence if we give k the value — ; , the required condition 
 
 will be fulfilled. 
 
 Exercises on Chapteu XIII. 
 
 (105) Prove that the equation 
 
 (aa + &/3 + cjY = {la. + m^ + ny) {I'a + m'/3 + n'y) 
 represents a hyperbola, and find its centre. 
 
 (106) Interpret the equations, 
 (i) k<x' = ^{act-^b^ + cy). 
 
 (ii) {h + m^ + nyf 
 = (a cos ^ + /3 cos 5 + 7 cos C) (asin^ + /3sin5 + 7sin C). 
 
 (iii) h\- + a'yS" = 2as {ay. + cy). 
 
 (107) Shew that whatever be the value of k, the locus of 
 the equation 
 
 Kx'-{x + iy=-{K-d)f 
 
 will pass through four fixed points, and find their coordinates. 
 
 (108) If s = be the equation to the straight line at infi- 
 nity, and M = 0, V =0, w = represent three other real straight 
 lines, the equation 
 
 uv +sto=0 
 
 will generally represent a hyperbola; but if u and v be parallel, 
 it will represent a parabola. 
 
172 EXERCISES ON CIIAPTEK XIII. 
 
 (109) With the notation of the last exercise the equation 
 
 u^ + sw =0 
 represents a parabola, and the equation 
 
 u' + v'' + sw = 0, 
 where u and v are not parallel, represents an ellipse. 
 
 (110) If two conies have double contact with a third, the 
 chords of contact are concurrent with a pair of common chords 
 of the first two conies, and form with them a harmonic pencil. 
 
 (111) If three conic sections have one chord common to all, 
 their three other common chords are concurrent. 
 
 (112) The straight line u — 2kv + k^w = is a tangent to 
 the conic uw = v'. 
 
 (113) The straight line u + v + w = Oisa common tangent 
 to the three conies 
 
 u^ = ivw, v' — iwu, V? = 4My. 
 
 (114) Three conic sections are drawn through a point P, 
 and each touches two sides of the triangle ABG at the extremi- 
 ties of the third side, shew that each of the tangents at P makes 
 a harmonic pencil with the straight lines joining P to the an- 
 gular points of the triangle. 
 
CHAPTER XIV. 
 
 CONICS REFERRED TO A SELF-CONJUGATE TRIANGLE. 
 
 169. We shewed in Chapter xii. that by selecting suitable 
 lines of reference any conic section might be represented by an 
 equation of the form 
 
 lal'+m^+ni'^O (1). 
 
 It is quite evident that if I, m, n are all positive, since the 
 squares a', /S", 7* are necessarily positive and cannot be all zero, 
 the expression la^ + m^ + wf must be positive and therefore 
 greater than zero. Hence the coordinates of no real point can 
 satisfy the equation, and the locus must be entirely imaginary. 
 
 If I, in, n are all negative, we may change the signs through- 
 out, and thus arrive at the case just considered. 
 
 Hence if the equation have a real locus, two of the coeflB- 
 cients must be of one sign and the remaining one of the opposite 
 sign: and therefore (by changing the signs throughout, if neces- 
 sary) we can suppose two of the coeflBcients positive and the 
 third negative. 
 
 We will suppose that I and m are positive and n negative, 
 and we may write 
 
 so that the equation becomes 
 
 LW + ]\P^'-N-'f = Q (2). 
 
174 CONICS KEFERKEI) TO A 
 
 This may be written 
 
 LW + {M^ + Ny) {31/3 - Ny) = 0, 
 Avhence we at once conclude, Art. 161, Chap. xm. that the lines 
 il//3 + Ny = 0, 
 
 are tangents, and the line 
 
 a = 
 
 their chord of contact. 
 
 That is, the side BC of the triangle of reference is the 
 chord of contact of tangents from the opposite angular point A. 
 
 Similarly, by writing the equation (2) in the form 
 ]\P0' + {La + N-y) {La - Ny) = 0, 
 we conclude that the lines 
 
 La + Ny = 0, 
 La-Ny= 0, 
 are tangents, and the line 
 
 their chord of contact. 
 
 That is, the side CA of the triangle of reference is the 
 chord of contact of tangents from the opposite angular point B. 
 
 But further, the equation (2) might be written 
 
 {La + Jf/3 V^l) {Lr - M^^J^^l) - Ny = 0, 
 
 from which form we see that the two imaginary straight lines 
 
 La + JZ/SV^ = 0, 
 
 La-M0\/~1=O, 
 
 (which both pass through the real point C) are tangents, and 
 the real straight line 
 
 7 = 
 their chord of contact. 
 
S 1-; L S-'-C< > NM Ua AT K TIM AN LE . 
 
 175 
 
 That is, the side AB of the triangle of reference is the chord 
 of contact of imaginary tangents from the opposite angular 
 point C. 
 
 Hence the conic is so related to the triangle of reference that 
 each side is the chord of contact of the (real or imaginary) tan- 
 gents from tlie opposite ver*ex. This is represented in figure 35. 
 
 170. Definitions. The chord of contact of real or imagi- 
 nary tangents from a fixed point to a conic is called the polar 
 of the point with respect to the conic. 
 
 Also a point is said to be the pole of that line which is 
 its polar. 
 
 We may therefore express the result of the last article as 
 follows : 
 
 The equation 
 
 la' + m^ + n<f = 
 
 represents a conic such that with respect to it each vertex of the 
 triangle of reference is the pole of the opposite side. 
 
 This is often briefly expressed by saying that the triangle is 
 self-conjugate with respect to the conic. 
 
176 CONICS EliFERRED TO A 
 
 171. From a given point a straight line is draion in a given 
 direction to meet the conic 
 
 lo? + m^ + nrf = 0, 
 
 it is required to find the lengths intercepted by the curve upon this 
 straight line. 
 
 Let (a', ^, y') be the given point, and \, fj,, v the sines of 
 the given direction (see Chap, vi.), then the equations to the 
 straight line are 
 
 a — a' _ /8 — /S' _ 7 — 7' 
 
 whence 
 
 a = a' + \/3, ^ = ^ +fji.p, 7=7' + vp. 
 
 If we substitute these values of a, /3, 7 which are true for 
 any point on the straight line in the equation to the conic, the 
 resulting equation, viz. 
 
 I (a + XpY + m (/3' + fipY + n{y+ vpf = 0, 
 
 ■will give the values of p at the points of intersection, that is, the 
 lengths of the intercepts required, measured from the given 
 point («', /S', 7'). 
 
 The equation is a quadratic, and may be written 
 {IX^+mfi^+nv') p^+ 2 {l\ri +miJ,^' +nvy')p + {U^-\.m^+ny"') = 0. 
 
 CoE. 1. Since this quadratic has two roots, every straight 
 line meets this — and therefore any — conic in two points (which 
 may however be distinct, coincident or imaginaiy), as we proved 
 in Chap. XII. 
 
 Cor. 2. If the point (a', /3', 7') lie on the conic, so that 
 
 ?a"' + m/3"'+»i7"' = 0, 
 
 one of the intercepts is zero, and the equation reduces to 
 
 {W + OT/i" + »M/») p + 2 (ZW + mii^ + nvy) = 0, 
 
 which therefore gives the length of the chord drawn from 
 (a', /3', 7') in the given direction. 
 
SELF-CONJUGATE TRIANGLE. 177 
 
 172. To find the equation to the tangent at any point on the 
 
 come. 
 
 Let \, /*, V be the direction sines of the tangent at (a', yS', 7'), 
 and let (a, fi, 7) be any point on the tangent, then (Chap, vi.), 
 
 and the length of the chord in this direction is given by the 
 equation 
 
 (Z\° + m/j,^ + nv') p + 2 {l\a' + m/i^' + nvy') = 0. 
 
 But since the direction is that of the tangent the length of 
 the chord must be zero, therefore 
 
 l\a.' + mfiff + nv^' = 0, 
 
 whence by the substitution of (1) we get 
 
 la.' (a - a') + m^' (/8 - ^S') + ny' (7 - 7') = 0, 
 
 or la'a + m/3'/S + W7'7 = la" + wi/S"* + ny' 
 
 = 0, 
 
 since (a, /8', 7') lies on the conic. 
 
 Hence the equation 
 
 la'a + mfi'^+ny'y = 
 
 expresses a relation among the coordinates of any point on the 
 tangent at (a', 0', 7',) and is therefore the equation to that tan- 
 gent. 
 
 173. To find the equation to the chord of contact of tangents 
 drawn from a given point to the conic, or, to find the polar of a 
 given point with respect to a conic. 
 
 Let a, ff, 7' be the coordinates of the given point and suppose 
 o", ;8", 7", a'", /3"', 7'" the coordinates of the points of contact of 
 tangents from (a, ^', 7') to the conic. 
 
 The equations to these tangents are by the last article , 
 Za"a + m0'^ + ny"y = 0, 
 and loi"'a + mr^ + ny "y=0, 
 
 w. 12 
 
178 CONICS BEFEKRED TO A 
 
 and since they pass through the given point (a', p, y) we have 
 
 Za"a'+wy8"y8' + W7"7' = 0, 
 
 Za"'o' + m/3"'/8'+m7"V = 0. 
 
 But these two equations respectively express that («", 0', 7") 
 and (a'", j8"', 7'") lie on the locus of the equation 
 
 la'oi + 7w/8'/3 + W7'7 = 0, 
 
 and this heing of the first order is -the equation to some 
 straight line. 
 
 Therefore it is the equation to the straight line joining 
 (a", ^", 7") and (a"', 0", 7'") the two points of contact, that is to 
 the chord of contact, or the polar of the point {a, ff, 7'). 
 
 It will be observed that the tangents from («', 0, 7') may be 
 imaginary, but the chord of contact or polar is always real, 
 like the polar of C in Art. 169. 
 
 174. To find the condition that any straight line whose 
 equation is given should he a tangent to the conic. 
 
 Let fa+g^ + hy = (1) 
 
 be the equation to the straight line : and suppose (a, &, 7') its 
 point of contact with the curve. 
 
 The tangent at this point is given by 
 
 la a + m/3'fi + W7'7 = 
 
 which must be identical with (1), therefore 
 
 la! _ wijS' _ ny 
 
 But (a, /S', 7') must also lie on the locus of (1) whence 
 fa! + m0 + Tvy = 0, 
 therefore eliminating a', 0, y we get 
 
 T+^ + - = 0, 
 I m n 
 
 which will be the required relation among f,g,h in order that 
 the given line may be a tangent. 
 
SELF-CONJUGATE TEUNGLE. I79 
 
 Cob. If the conic touch any one of the straight lines 
 
 it will touch them all. 
 
 175. To find the locus of the middle points of a series of 
 parallel chords in the conic whose equation is 
 
 la' + m^ + my" = 0. 
 
 Let \, fj., V be the direction sines of any one— and therefore 
 of every one— of the series of parallel chords. 
 
 And let (a', ^', y') be any point on the required locus, then 
 the chord through this point is represented by 
 
 a-a! ^^j-0^ _ y-y _ 
 A. fi ~ V ~P' 
 
 and therefore the lengths of the intercepts by the conic are given 
 by the equation 
 
 'l(a' + \py + m{^+ fipf + n (7' + vpf = 0. 
 
 But since («', /9', 7') is the middle point of the chord the 
 two values of p given by this quadratic must be equal in value 
 and opposite in sign. Therefore the coefficient of p must vanish 
 in the quadratic, and therefore we have 
 
 tka! + jn/t/S" + niTf = 
 
 a relation among the coordinates of any point (a, yS', 7') on the 
 locus. Hence the locus is the straight line whose equation is 
 
 tka. + TO/it/3 + nv^ = 0. 
 Such a straight line is called a diameter of the conic. 
 
 176. Thus far the reasoning of the present chapter will 
 apply equally whether we understand the coordinates to be 
 trilinear or triangular, except that in the latter case we must 
 modify the interpretation of \, p., v in Arts. 171, 172, 175, not 
 speaking of them as direction sines. [See (xix) and (xx) in the 
 table of formulae, pages 100, 101.] But now that we have to 
 
 12—2 
 
180 CONICS REFERRED TO A 
 
 introduce the straight line at infinity our equations will no 
 longer hold for triangular coordinates until we replace o, b, c, 
 2A, severally by unity. (Art. 87.) 
 
 177. To find the condition that the equation 
 
 la? + m^ + n'f = 
 may represent a parabola. 
 
 It is suflScient (Art. 152) to express that the line at infinity 
 must be a tangent. Therefore by Art. 174 the condition in 
 trilinear coordinates is 
 
 «' 6= c' „ 
 -T+-+- = 0, 
 t m n 
 
 or in triangular coordinates, 
 
 1 1 1 « 
 
 I m. n 
 
 178. To find the centre of the conic. 
 
 Let (a', yS', 7') be the centre ; then the lengths of the inter- 
 cepts in the direction (\, /u, v) measured from the centre are given 
 by the equation 
 
 p' (ZX" + mii^ + ni?) + 2p (?\a' + w/xyS' + nv^) 
 
 + Za"' + »8/3" + W7'= = (1). 
 
 But since all chords through the centre are bisected in the 
 centre the two roots of this quadratic must be equal in magnitude 
 and opposite in sign, therefore the coe£Scient of p must vanish, 
 
 therefore Z\a' + »n/ij8' + mi*/' = (2) 
 
 for all values of \ : /* : v, subject to the relation 
 
 a\ + J/i. + cv = 0. 
 
 Hence we must have 
 
 U__'n^ Wf 2A - • " 
 
 -7- H 1 — 
 
 L m n 
 
 which determine a', 0, 7' the coordinates of the centre. 
 
SELF-CONJUGATE TRIANGLE. 181 
 
 Obs. In triangular coordinates the centre of the conic 
 whose equation is h? + mf-\- nz* = is given by the equa- 
 tions 
 
 J__J1__J__1 1 1 
 Ix my m I m n 
 
 Cor. The coordinates of the centre are infinite if 
 
 o' J" c" „ 1 1 1 „ 
 
 -T + - + - = 0, or -+- + - = 0, 
 
 {triUnear coordinates) {triangular coordinates) 
 
 But we have already seen that when the same condition 
 holds the conic is a parabola. (Art. 177.) 
 
 Hence the centre of a parabola is at infinity. 
 179. To find the conditions that the equation 
 
 should represent a circle. 
 
 Let (a', yS', 7') be the centre, then the length of the semi- 
 diameter in any direction will be given by the equation (1) of 
 the last article, which in virtue of (2) becomes 
 
 p' (ZX* + mfi' + ni/^ + la" + jm/3" + ny" = 0. 
 
 Hence all the diameters will be equal, provided 
 
 l\' + fitn" + wv' 
 
 be constant for all directions. 
 
 But we know (Art. 71, page 77), that 
 
 \' sin 2A + fi' sin 2B + i/' sin 2 G, 
 is constant. 
 
 Hence the diameters will be all equal, provided 
 
 I m _ n 
 
 sin2.4~8in25~sin2C' 
 
 which therefore express the conditions that the conic should 
 be a circle. 
 
182 CONICS EEPEERED TO A 
 
 Hence 
 
 a' sin 2^ + iS* sin 25 + V sin 2 C= 
 
 represents the circle with respect to which the triangle of refer- 
 ence is self-conjugate. 
 
 Obs. The circle will be imaginary unless one of the coeffi- 
 cients sin 2A, sin 2B, sin 2 (7 be negative (Art. 169) ; 
 
 i. e. unless one of the angles lA, 2B, 2 (7 be greater than 180°, 
 
 i.e. unless one of the angles A, B, Cbe greater than 90°, 
 
 i.e. unless the triangle of reference be obtuse angled. 
 
 CoE. 1. The equation in triangular coordinates to the circle 
 with respect to which the triangle of reference is self-conjugate is 
 
 x^cotA + i/'cotB + z^cot G=0. 
 
 Cob, 2. In the case of the circle 
 
 a' sin 2A + ^ sin 25+ V sin 2C= 0, 
 
 the equations to give the centre become 
 
 a' sin 2^/3' sin 2B _ 7' sin 2 (7 
 a b ~ c ' 
 
 or a' cos A=0 cobB= 7' cos C, 
 
 which we have already seen Chap. 11. are the equations to the 
 point of intersection of the perpendiculars from the vertices of 
 the triangle on the opposite sides. 
 
 Hence if a triangle he self-conjvgate voith respect to a circle, its 
 perpendiculars intersect in the centre of the circle. 
 
 180. To find ike pole of any given straight line with respect 
 to the conic, whose eqwation is 
 
 la^ + m^ + ny' = 0. 
 Let fi + gfi + hy = (1) 
 
 be the given straight line whose pole is required, and suppose 
 (a', /8', 7') the pole. 
 
SELF-CONJOGATE TBIANGLE. 183 
 
 The polar of this point is given by 
 
 la' a + m/8'y3 + tvy'ry = 0, 
 
 which must be identical with (1), therefore 
 
 U__m^ _ivj__ 2A 
 f~g ~ h~ af bg ck ' 
 I m n 
 
 which determine a', /3', 7', the coordinates of the pole required. 
 
 COK. In trilinear coordinates the pole of the line at infinity 
 is given by 
 
 la _ m^ ny 2A 
 
 a b c a' V c" 
 
 -5-+- + - 
 l m, n 
 
 Hence (Art. 178) the centre is the pole of the line at infinity. 
 
 181. If we assume as the definition of the foci the well 
 known geometrical property of a conic section, that the square 
 on the semi-minor axis is equal to the rectangle contained by 
 the focal perpendiculars on any tangent, we can readily find 
 equations to determine the focus of a conic section with respect 
 to which the triangle of reference is self-conjugate. Thus : 
 
 To determine the foci of the conic whose equation is 
 
 la^ + m^^ + n'Y' = (1). 
 
 Let (oi, iSj, 7i), (a,,, yS„ 7,) be the foci required, and let k 
 be the semi-minor axis of the conic. 
 
 The tangent at any point (a', fi', 7') is 
 la'a + m^'P + 7177 = 0. 
 
 And expressing that «* is equal to the rectangle contained 
 by the product of the perpendicular distances of this line from 
 (a„ A. 7.) and (a., /3„ 7J, we get 
 
 «»= 
 
 (?g,a'4->«y3,ff+«7,7')(?«,a+w/3,^+«Y,7') 
 
 Pa"'+rri'^''+n'y''-2mn0ycosA-2nly'a'coaB-2lma'ffco3C' 
 
184 CONICS KEFEREED TO A 
 
 which is a relation among the coordinates (a', P, y') of any point 
 on the conic, and therefore, suppressing the accents, the equation 
 
 f a'+m'/S^H- «V- 2mn^y C03 A-2nlya. coa£-2lm a^coaC'"^ ' 
 is the equation to the conic and therefore is identical with (1). 
 But this equation (2) may be written 
 
 + mn^y {^^y, + /8,7, + 2«'cos A) 
 
 + nlya (yiK, + YsHj + 2«'cos B) 
 
 + Ima^ (o^iS, + a,/3,.+ 2«' cos C) = 0. 
 
 Hence equating the ratios of the coefficients of this equation 
 and the equation (1), we get 
 
 I (a,a. -/e^=m (^,^, -«") = » (7^7, - «") = t (suppose) . . . (3) , 
 
 and ^,72 + ;8s7.+ 2«'cos^=0 (4), 
 
 7ia! + 7sa, + 2«'cosjB = (5), 
 
 aA+«,^, + 2/c»cos C=0 (6). 
 
 Multiplying (5) and (6) by c and b respectively, and adding, 
 we get 
 
 «! (5/3, + C7,) + a, {bl3, + C7,) + 2^a = 0, 
 
 or Oi (2A - aa,) + a, (2A - aa^) + Sid'a = 0, 
 
 a 
 
 or ai+«, = ^ («,«,-«'), 
 
 or by equations (3), 
 
 or 
 «>+«»=S (7), 
 
 and OjajS-y+K". 
 
 Therefore a, and a, are given by the quadratic 
 
SELF-CONJUGATE TRIANGLE. 185 
 
 Similarly )8, and /8j are given by 
 
 and 7, and % are given by 
 
 «A ' » 
 
 Hence the foci of the given conic are determined by the 
 equations 
 
 lA I mA m ' wA n ' 
 
 where t may be determined as follows. 
 
 The equation (7) gives us 
 
 ax 
 
 "• + ««=7a- 
 
 hr 
 Similarly we have ^^ + fi^ = —r , 
 
 and 'y' + 'y» = S- 
 
 Multiplying by a, h, c and adding, we get 
 
 4A=T- + -+-)x, 
 dA" 
 
 and therefore 
 
 I m n 
 
 Hence the equations to determine the foci take the form 
 
 \l m n/ I I \l m nj mm 
 
 = (^ + ^ + ^)^_fA£y^4A: 
 \l m nJ n n 
 
 and the corresponding equations in triangular coordinates can 
 be immediately written down (Art. 87). 
 
186 CONICS EEFEEEED TO A 
 
 Note. If we had assumed the fact that the centre is the 
 point of bisection of the straight line joining the foci, we might 
 have written down the equation (7) and determined t at once. 
 
 For (Art 18, Cor. page 20) the coordinates of the centre 
 must be 
 
 2 ' 2 ' 2 ' 
 
 and therefore (Art. 178) 
 
 ^ f . \ '""■ to o\ w, , 2A 
 
 T+- + - 
 l m n 
 
 182. Cofi. 1. If the conic be a parabola, we have 
 (Art. 177) 
 
 o* J' c' „ 
 I m n 
 
 and the equations to determine the foci give one point at 
 infinity, and reduce for the other to 
 
 aa— A_6/8 — A 07 — A 
 I m n ' 
 
 each of which fractions must be equal to 
 
 -A 
 i + OT+ n' 
 
 Hence 
 _^ m + n n+l l+m 
 
 l + m + n' ^ l+m + n' '^ l+m + n' 
 
 or 
 
 .-_M.__£L 
 
 m + n n + l l + m' 
 
 CoE. 2. In the case of the parabola, 
 
 «• J' c" 
 since +_ + _ = o, 
 
 I m n 
 
 it follows that the coordinates of the finite focus of any parabola 
 
SELF-CONJUGATE TEIANGLE. 187 
 
 with respect to whicli the triangle of reference is self-conjugate, 
 will satisfy the equation 
 
 fl." V <? 
 
 or 
 
 aa— A h^ — A cry — A' 
 
 a" , J' c' 
 
 I I 
 
 b^ + cy — aa cy + a<x — b^ aa + b^—cy' 
 or 
 o'8in2^+/3'3in25+7'sin2 C-4 (/37sin^+7asin^+aj8sin C) =0, 
 
 which is therefore the equation to the locus of the foci of all 
 parabolas, with respect to which the triangle of reference is self- 
 conjugate. 
 
 183. Dep. The polar of a focus is a Directrix. 
 
 184. To find the equation to the directrix corresponding to 
 the finite focus of the parabola, 
 
 W + m^+'nff = 0. 
 
 The finite focus is given (Art. 182) by 
 
 aa. _ 5/3 _ cry 
 
 hence (Art. 173) its polar is represented by 
 
 Z(m + w)-+m(n + 0f + «(Z + »»)- = 0, 
 
 or 
 
 or 
 
 \,»i nJ a \n, IJ b \l ml c 
 l\b cl m\o a/ n\a bj 
 
 COK. Since 
 
 I m n 
 
 it follows that the directrix of any parabola, with respect to which 
 
188 
 
 CONICS EEPEREED TO A 
 
 the triangle of reference is self-congugate, passes through the 
 point given by 
 
 a ^ 7 
 
 or 
 
 cos A cos B cos (J ' 
 
 185. If the opposite sides of a quadrilateral he produced to 
 meet in A and B, and its interior diagonals intersect in C, then 
 the triangle ABC is self-conjugate with respect to any conic 
 circumscribing the quadrilateral. 
 
 Fig. 36. 
 
 Let PQR8 be the quadrilateral, take ABC as triangle of 
 reference, and let P be determined by the equations 
 
 la = »»/3 = wy, 
 
 then the equation to AP is 
 
 m/3 — ny = 0, 
 
 and since the pencil at ^ is harmonic, the equation to AB is 
 »n/8 + «7 = 0. 
 
SELF-CONJUGATE TRIANGLE. 189 
 
 So the equation to BP is 
 
 l7. — n'^ = 0, 
 
 and since the pencil at B is harmonic, the equation to BR is 
 
 Za + «7 = 0. 
 
 But these four lines are the sides of the quadrilateral. Hence 
 any conic circumscribing the quadrilateral must have an equa- 
 tion of the form 
 
 («iy3 — my) [m^ + ny) —k {la— ny) {la + ny) = 0, 
 
 or Ttv'^ -nY-K {Va' - nV) = 0, 
 
 or «Z V - m'0' + (1 - «) nV = ; 
 
 hence the triangle of reference is self-conjugate with respect to 
 any conic circumscribing the quadrilateral. 
 
 CoE. 1. If any number of conies intersect in four points, 
 a triangle can be found self-conjugate with respect to all of 
 them. 
 
 COE. 2. If a series of conies pass through four fixed points, 
 a suitable triangle can be found with respect to which all their 
 equations will be of the form 
 
 W + m^ + nr/^O. 
 
 ExEBCiSES ON Chaptee XIV. 
 (115) Shew that 
 
 q—r r—p p—^ 
 represents a parabola. 
 
 (lie) If AA', BB', CC the diagonals of a quadrilateral, be 
 produced to form a triangle abc, this triangle will be aelf-conju- 
 
190 
 
 EXERCISES ON CHAPTEB XIV. 
 
 gate with respect to one conic passing through BCB'C, to 
 another passing through CAC'A', and to a third passing through 
 ABAB'. 
 
 (117) Interpret the equation 
 
 {la + «i/3)» + {la. - m^y - 2nV = 0. 
 
 (118) Shew that all the lines 
 
 la±m^± ny ^/2 = 
 are tangents to the conic whose equation is 
 
 (119) Shew that the conic 
 
 u'-v' + ie'^O 
 
 circumscribes the quadrilateral whose sides in order are 
 
 u + v + w = 0, 
 
 — u + v + w = 0, 
 
 u — v + w = 0, 
 
 u + v — w = 0. 
 
 (120) Interpret the equation 
 
 a° sin 2^ + /8* sin 2(9 + 7* sin 2(7 = 0, 
 when the triangle of reference is right-angled. 
 
 (121) Shew that the diameter drawn through the point 
 (a', ^, 7') in the conic 
 
 la' + m^ + nr/=:0 
 
 is represented by the equation 
 
 a, ^, 7 
 
 f at t 
 
 a. P, 7 
 
 a h c 
 
 m' n 
 
 I 
 
 = 0. 
 
EXERCISES ON CHAPTER XIV. 191 
 
 (122) The conic referred to a self-conjugate triangle and 
 having its centre at (a', yS', 7') is represented by the equation 
 
 «' ~ iO' ^ ' — "• 
 
 a P 7 
 
 (123) The tangents at the extremities of the chord a = /cyS 
 in the conic 
 
 are given by the equation 
 
 (?/«:a+m/3)* =(?«'+»«) 7*. 
 
 (124) Find the equation to the tangents whose chord of 
 contact is 
 
 /a +.9/3 + ^7 = 0. 
 
 (125) The four straight lines of Exercise 119 are tangents 
 to the conic 
 
 «' .+ J!L+J£L=o. 
 
 m — n n — l l—m 
 
 (126) Find the equation in triangular coordinates to the 
 locus of the foci of all parabolas with respect to which the 
 triangle of reference is self-conjugate. 
 
 (127) The conditions that the point of reference A should 
 be a focus of the conic 
 
 are m = n, and J' + c' = a'. In this case the triangle of reference 
 is right-angled and the line of reference BG is the directrix 
 corresponding to the focus A. 
 
 (128) Apply the last result to shew that the distance of 
 any point on a conic from a focus bears a constant ratio to its 
 distance from the corresponding directrix. 
 
CHAPTER XV. 
 
 comes EEFEBRED TO AN INSCRIBED TRIANGLE. 
 
 186. The equation 
 
 l^'l + mrfix + no/S = 
 
 represents a conic circumscribing the triangle of reference, for it 
 is satisfied wlien any two of the coordinates are zero ; and there- 
 fore each point of reference lies upon its locus. 
 
 Further, by giving suitable values to I, m, n this equation 
 will represent any conic referred to an inscribed triangle (or 
 circumscribing the triangle of reference). For by Chap. xii. 
 every conic must have an equation of the second degree, which 
 may be written 
 
 /a' +g^ + h'f+ 2Z/Sy + Smya + 2na/3 = 0. 
 
 But if the conic pass through the point of reference A, the 
 equation must be satisfied when 
 
 Therefore (substituting these values in the equation), we get 
 
 M = 0. 
 
 Similarly, if the conic pass through B and C, we get 
 
 « = and w = 0. 
 Hence the equation reduces to 
 
 lfi<^ + mr/a. + na^S = 0, 
 where the values of I, m, n depend upon fiirther conditions. 
 
CONICS EEFERRED TO AN INSCRIBED TRIANGLE. 193 
 
 187. The equation to the conic may be written 
 
 Zy87 + a {my + n^) = 0. 
 
 Hence it is cut by the straight line nvy + n/3 = in the two 
 points where this straight line meets /3 = and y = 0, i.e. in two 
 coincident points at A. Therefore my +n^ = 0, or 
 
 m n 
 
 represents the tangent to the conic at the point of reference A. 
 
 Similarly, the tangents at the other two points of reference 
 are given by 
 
 5^+ y = 0, and ? + ^ = 0. 
 n I I m 
 
 188. From a given point a straight line is drawn in a given 
 direction to meet the conic 
 
 l^y + myoL + nayS = 0, 
 
 it is required to find the lengths intercepted hy the curve upon this 
 straight line. 
 
 Let (a', ^, y) be the given point, and \, /t, v the sines of the 
 given direction, then, as in Art. 171, the intercepts are given by 
 the equation 
 
 I (/S" + ,ip) {y + vp)+m {y + vp) (a +\p)+n («' + \p) (/3'+ p.p)= 0, 
 which may be written 
 
 {lp.v + mvK + nXp) p' + {l^y + mr/a! + Wi'/S') 
 + [\ (my' + n^)+fi (na' + ly) + v (J0 + ma!)] p = 0, 
 
 a quadratic giving two values for p, expressing the length of the 
 two intercepts. 
 
 Cor. If the point {a, /8', 7') be on the conic, so that 
 lffy' + my'a' + na'0'=O, 
 one of the intercepts is zero, and the other is given by 
 (Ifiv + mvk + w V' p+X (wi-/ + M/8') + fi {noL + ly') +vil0 + ma') = 0. 
 W. ^^ 
 
194 CONICS REFERRED TO AN INSCRIBED TRIANGLE. 
 
 189. To find the equation to the tangent at any point on the 
 conic. 
 
 Let \, fi, V be the direction sines of the tangent at (a', ^', y) 
 and let (a, ^8, 7) be any point on the tangent, then (Chap, vi.) 
 we have 
 
 ir^-l. V — '' ^^'' 
 
 and the length of the chord in this direction by the last corollary 
 is given by 
 
 {Ifiv + mvX + n\fj,) p+\ {nvf + n^') + fi [no! + Jy') + v (Z/3' + ma') = 0. 
 
 But since the direction is that of the tangent, the length of 
 the chord must be zero : therefore 
 
 \ {my' + «/3') + li {na' +ly')+v (10 + ma!) = 0, 
 or in virtue of (1), 
 
 (a - a') {my + n^) + {^-0) {na' + ly) + {y- y') {l^ + »!«')= 0, 
 or 
 
 a(w7'+n/S') +/?(««'+ V) +7(^j8'+»Ma') = 2(?/3y+»i7V+na'^'). 
 But since (a, 0, y) lies on the conic, we have 
 
 I0y' + my'a: + noL0 = (2), 
 
 and the equation becomes 
 
 a {my' + n/3') + /9 (raa + Ir/) +y{l0 + ma) = 0. 
 
 This equation expresses a relation among the coordinates of 
 any point (a, fi, 7) on the tangent at (a', /8', 7') and is therefore 
 the equation to that tangent. 
 
 Cor. The equation may be written 
 But (2) gives us 
 
 «' + ^-' + y = «' 
 
CONICS EEFERRED TO AN INSCRIBED TRIANGLE. 195 
 
 therefore the equation will take the form 
 
 a J. , ^ ^ . 2 " _n 
 «' (^^ ^^i 7'""' 
 
 a form of which we shall presently give an independent investi- 
 gation (Art. 198). 
 
 1 90. To find the equation to the chord of contact of tangents 
 drawn from a given point to the conic, or, to find the polar of a 
 given point icith respect to the conic. 
 
 Let (a', yS*, y') be the given point, then we may shew, pre- 
 cisely as in Art. 173, that the required equation is of the same 
 form as that of the tangent at a point on the curve. 
 
 That is, the equation 
 
 a {mi + wyS') + )8 {no! + Zy') + 7 {^^' + »»«') = 0, 
 which, when («', 0, y) is a point on the curve, represents the 
 tangent thereat, will, when (a', /S', 7') is a point not on the curve, 
 represent the polar of that point. 
 
 191. To find the condition that any straight line whose equa- 
 tion is given should he a tangent to the conic. 
 
 Let fa+g^ + hy = (1) 
 
 be the given equation to the straight line and suppose (a', yS', 7') 
 its point of contact with the curve. 
 
 The tangent at (a', yS', 7') is given by 
 
 o {my + wyS") + /3 {no! +ly')+y {10 + mix) = 0, 
 
 which must be identical with (1). 
 
 Therefore 
 
 my + n0 _ wa' + ly _ 10 -f wa 
 
 -^7 —j——h—' 
 
 a' _ y 
 
 or 
 
 l{lf—mg — nh) m{mg — nh-lf) n{nh — lf—mg)' 
 
 13—2 
 
196 CONICS EEFEURED TO AN INSCRIBED TRIANGLE. ' 
 
 But (a, ^, 7) must also lie on the locus of (1), whence 
 fJ + g^' + hi^O. 
 
 Hence eliminating a', /S", 7' we get 
 If {If- mg-nh) + mg {mg -rih- If) + nh {nh - If- mg) = 0, 
 
 or rf + my + n'h^- 2mngh-2nlhf-2lmfg = (2), 
 
 which is therefore the condition required. 
 
 Note. The equation of condition just obtained may be 
 written in the form 
 
 V^+ V^ + VwA = (3), 
 
 as will be seen by clearing this latter equation of radicals, when 
 it will be found to take the form (2). 
 
 192. CoE. The equation in trilinear coordinates 
 
 I0y + myoL + nafi = 0, 
 will represent a, parabola provided 
 
 \/al + y/bm + Vc« = 0. 
 And the equation in triangular coordinates 
 
 Igz + mzx + nxy = 0, 
 will represent a, parabola provided 
 
 V7+ V»i + Vw = o. 
 
 The remarks made in Art. 176 will apply here. 
 
 193. To find the locus of the middle points of a series of 
 parallel chords in the conic whose equation is 
 
 Wy + mya + Ma/3 = 0. 
 
 Let \, fi, V be the direction sines of the parallel chords, and 
 let (a, /8, 7) be the middle point of any one of them. 
 
CONICS BEFEREED TO AN INSCEIBED TEIANGLE. 197 
 
 Then (Art. 188) the lengths of the intercepts measured from 
 (a, ^, y) to the curve in the direction (\, /x, v) are given by the 
 equation 
 
 {IfLv + mv\ + mX/x) p" + (Zy87 + mya + mo/S) 
 
 + {\ (my+n^) +fi{na + ly)+v {l^ + ma)] p = 0. 
 
 But since (a, /3, 7) is the middle point of the chord, the two 
 values of p, representing the intercepts, must be equal in magni- 
 tude and opposite in sign. Therefore the coefficient of p must 
 vanish in the quadratic, and therefore 
 
 \ {my + n^)+/jL {na +ly)+v (Z/3 + ma) = 0, 
 
 or a [mv + n/j) + /3 (n\ ■\-lv) + y {Ifi + m\) = 0, 
 
 a relation among the coordinates a, /8, 7, of the middle point of 
 any one of the chords, and therefore the equation to the locus of 
 the middle points. 
 
 194. To ^nd the centre of the conic. 
 
 Let (a', /3', 7') be the centre. 
 
 Then the lengths of the intercepts measured from the centre 
 in the direction {\, /*, v) are given by the quadratic 
 
 {l/iv + mv\ + nXfi) p* + {I0y' + my a' + na'^) 
 
 + [X (jnfi' + m/3') + fi {na' + ly') + v {l^' + ma')} p = (I). 
 
 But since all chords are bisected in the centre, the two roots 
 of this quadratic must be equal in magnitude and opposite in 
 sign ; therefore the coefficient of p must vanish, and therefore 
 
 A, {my' + n^)+fi{na'+ly')+v{l0 + ma') = (2), 
 
 for all values oi\: /*; v, subject to the relation 
 a\ + bfi + cv = 0. 
 
 Hence, we must have 
 
 my' + wy3' _ na' + ly _ 10' + ma 
 
198 CONICS EEFEEBED TO AN INSCBIBED TBIANGLE. 
 a' ^ 7 
 
 or 
 
 I (la — mb — nc) m{nib — nc—la) n {nc — la — mh) 
 
 ^ 2A 
 
 ~ Pa' + ro'V + n'cr — 2mnbc — 2nlca — 2lmab ' 
 
 which determine («', y3', 7') the coordinates of the centre. 
 
 N.B. If the coordinates are triangular instead of trilinear, 
 we find that the centre of the conic whose equation is 
 
 lyz + mzx + nxy = 0, 
 is given by the equations 
 
 x' _ y _ z 
 
 l{l — m—n) m{m — n — l) n{n—l — m) 
 
 1 
 
 r + m" + n^ - 2mn - 2nl — 2lm ' 
 Cob. The centre of a parabola is at infinity. 
 
 195. To find the conditions that the equation 
 l^ + mya + na/S = 0, 
 should represent a circle. 
 
 Let (a', ^, y) be the centre, then the length of the semi- 
 diameter in any direction is given by equation (1) of the last 
 article, which in virtue of (2) reduces to 
 
 p" [Ifiv + mvK + n\/*) + I0y' + my'ci + na!^ = 0. 
 
 Hence all the diameters will be equal, provided 
 lliv + mv\ + n\/i 
 be constant for all directions. 
 
 But we know (Art. 71, page 78), that 
 
 /iti»sin.4 + j/X6injB + X^sin C 
 
CONICS EEFERRED TO AN INSCRIBED TRIANGLE. 199 
 
 is constant for all directions. Hence the diametsrs will be all 
 equal, provided 
 
 I _ 'm _ n 
 sin A sin B sin C 
 
 which therefore express the conditions that the conic should be 
 a circle. 
 
 Hence the equation 
 
 /37 sin ^ + 7a sin 5 + ayS sin (7 = 0, 
 or, a/37 + ^« + ca/S = 0, 
 
 represents the circle circumscribing the triangle of reference. 
 
 Cor. 1. In the case of this circle the equations (Art. 194) 
 to determine the centre reduce to 
 
 yS- _ 7' 
 
 cos A cos B cos C * 
 agreeing with the result of Art. ] 7, Cor. 
 
 CoR. 2. The equation in triangular coordinates to the circle 
 which circumscribes the triangle of reference is 
 
 a^ys + ¥zx + c^xy = 0. 
 
 196. To find the pole of any given straight line with respect 
 to the conic whose equation is 
 
 l^y + mya. + na/3 = 0. 
 
 Let fa+gfi+hy = (1), 
 
 be the equation to the given straight line, and suppose (a, /8', 7') 
 the pole. 
 
 The polar of this point is given hj 
 
 a {my + n/3') + /3 (na' + Jy') + 7 i^^ + »»«') = ^' 
 which must be identical with (1). 
 
200 CONICS REFERRED TO AN INSCRIBED TRIANGLE. 
 
 Therefore 
 
 nv^' + «)8' _ wa' + Jy' ZyS' + mo! 
 
 o' ^ 7' 
 
 or 
 
 Z {If— mg — nh) m {mg — nh — If) n {nh — If— mg) ' 
 which determine the coordinates of the pole required. 
 
 Cor. The pole of the line at infinity is the centre of the conic. 
 
 197. The following Theorem is attributed by Dr Salmon 
 to M. Hermes. 
 
 If (a'j ^'i 7)1 ('* "j ^'t 7") ie the coordinates of any two points 
 on the conic 
 
 l^y + myx + jiayS = 0, 
 
 I m n ^ 
 or _+ +_ = o, 
 
 a P 7 
 
 the equation to the straight line Joining them is 
 
 Ic mff , "7 _ /^ 
 aa p p 77 
 
 For this equation is satisfied at the point (a', /3', 7'), since 
 (a", /S", 7") lies on the conic, and therefore 
 
 l_ in '"'_[. 
 a P 7 
 
 So also it is satisfied at the point (a", /9", 7"), since (a, /3', 7') 
 lies on the conic. 
 
 And it is of the first degree in a, j8, 7. 
 
 Therefore it represents the straight line joining the two 
 points (a', ^, i), (a", y8", 7"). 
 
 198. Cob. Let the point (a", ff', 7") move up to and ulti- 
 mately coincide with (a', /S", 7') : then ultimately the chord" 
 becomes the tangent at (a', /3', 7) and the equation becomes 
 
 la m^ "7 _ A 
 
CONICS EEFEBRED TO AN INSCRIBED TRIANGLE. 
 
 201 
 
 the same form of the equation to the tangent at the point 
 (a, ff, y) which we deduced in the Corollary to Article 189. 
 
 199. To find the condition that three points whose coordi- 
 nates are given should lie on one conic with the three points of 
 reference. 
 
 Let (a,,^,, 7,)) («,> P^,!^, K. /^a) Vs) be the coordinates 
 of the given points. 
 
 Any conic passing through the points of reference may be 
 represented by the equation 
 
 I m n ^ 
 
 and if it pass also through the given points we must have 
 
 I m n . 
 
 -+p+- = 
 
 «i Px 7i 
 
 I m n ^ 
 Os Ps 7* 
 
 I m n ^ 
 
 - + ^ + - = 0- 
 
 Ha Ps 73 
 
 •(1), 
 
 •(2), 
 •(3), 
 .(4). 
 
 Eliminating I : m : n from the last three equations, we 
 obtain 
 
 i i i =o» 
 
 i 1 L 
 
 «,' )3,' 7, 
 
 ill 
 
 which will be the condition required. 
 
 200. Paschal's Theorem. If a heoeagon he inscribed in a 
 conic, and the pairs of opposite sides he produced to intersect, the 
 three points of intersection are collinear. 
 
202 
 
 CONICS REFERRED TO AN INSCRIBED TRIANGLE. 
 
 Let AP^BP^CP, be tlie hexagon, take ABC as triangle of 
 reference, and let (a,, ^,, 7,), (a^, /3„ 7,), (a,, ^3, 7,) be the co- 
 ordinates of P,, P,, P3 respectively. 
 
 The equation to the side AP^ is therefore 
 ^_7 
 
 and the equation to the opposite side P^G 
 
 7 _ a 
 
 7s ~ «/ 
 
 Hence these two sides intersect in the point given by 
 
 a ^ ^ ^ 7 
 a«7, ^i72 %«, 
 
 (1). 
 
 So the sides BP^, P^A intersect in the point given by 
 
 a _ i8 _ 7 
 o^aa /^saj 72«i 
 
 .(2). 
 
 and the sides CP^, P^B intersect in the point given by 
 a ^ 7 
 
 aA ^A 7A 
 
 (3). 
 
 But the three points represented by the equations (1), (2), 
 (3) are collinear (Art. 20) if 
 
 as7,, /3,7.. 7.7, 
 a,a3, ^3«5, 7,a, 
 
 aA. /3A. 7,/3. 
 
 that is, if 
 
 ill 
 
 aa' i8,' 73 
 
 i i. i 
 
 a,' /8.' 7x 
 
 1 i i 
 
 a.' ^/ 7i 
 
 = 0, 
 
CONICS EEFEEBED TO AN INSCRIBED TRIANGLE. 203 
 
 which is the condition (Art, 199) that the three points P,, Pj, P, 
 lie on the same conic with ABC. 
 
 Hence the condition that the intersections of opposite sides 
 of a hexagon should be coUinear is identical with the condition 
 that the six angular points should lie on one conic. 
 
 This proves the proposition and its converse. 
 
 201. Only one conic can he described, passing through three 
 given points and having its centre at another given point. 
 
 For if we take the first three given points as points of refer- 
 ence for triangular coordinates, the equation to the conic may be 
 written 
 
 I m n . 
 - + - + - = 0, 
 X y z 
 
 and if [x', y', a') be the coordinates of the given centre, we have 
 by Art. 194 
 
 mz' + ny' = nx' + Iz' = ly + mx', 
 whence 
 
 I m _ n 
 
 al{x'-y-z')'^y'{r/-z-x')-z{z'-x'-yy 
 
 so that the only conic satisfying the conditions will be that 
 represented by the equation 
 
 l{aS-y'-z')^t{y'-z -a;)+Uz'-al -i/) = 0. 
 X y <» 
 
 Obs. Since we have seen that a conic can be described so 
 as to fiilfil five simple conditions (such as passing through an 
 assigned point) it follows that the condition of the centre being 
 at an assigned point will count as two of these simple conditions. 
 Such a condition may be spoken of as a double condition, or a 
 condition of the second order. 
 
204 EXEECISES ON CHAPTER XV. 
 
 Exercises on Chapter XV. 
 
 (129) If \a + /i/3 + 1/7 = 
 be a tangent to the conic 
 
 I tn n ^ 
 a yS 7 ' 
 
 then the three quantities IX., m/i, nv will be either all positive or 
 all negative. 
 
 (130) If \a + /i/3+iAy = 
 
 be a tangent to the conic 
 
 I 7a n ^ 
 - + -5 + -=0, 
 a ^ 7 
 
 then will 
 
 la. + wJyS + n7 = 
 
 be a tangent to the conic 
 
 a ^ 7 
 
 (131) A conic is described so as to touch in A, B, C the 
 sides BC, C'A', A'B' of a triangle A'B'C. Shew that 
 AA, BB', CO' are concurrent, and that the straight line BC 
 is divided harmonically by the straight lines AA', BC pro- 
 duced if necessary. 
 
 (13*2) A triangle is inscribed in a conic, prove that the points 
 are coUinear in which each side intersects the tangent at the 
 opposite angle. 
 
 (133) The six points of intersection of non-corresponding 
 sides of a pair of co-polar triangles lie on one conic. 
 
 (134) If a triangle be self-conjugate with respect to a 
 series of conies which all pass through a fixed point, the centres 
 will lie on another conic which circumscribes the triangle. 
 
EXERCISES ON CHAPTER XV. 
 
 205 
 
 (135) Determine the position of ths fixed point in the last 
 exercise in order that the locus of the centres may be the circle 
 circumscribing the triangle. 
 
 (136) The normals at the points of reference to the conic 
 whose equation is 
 
 Z/S7 + ■m^a. + nayS = 
 
 will be concurrent, provided 
 
 = 0. 
 
 ', 
 
 m. 
 
 n 
 
 1 
 
 1 
 
 1 
 
 V 
 
 m' 
 
 n 
 
 1 
 
 1 
 
 1 
 
 a' 
 
 J' 
 
 c 
 
 (137) The equation 
 
 Z/37 + mrfo. + nayS = 0, 
 
 will represent a hyperbola, provided 
 
 o'Z' + JW + cV > 2 (6cOTJi + canl + cibhn). 
 
 (138) The tangents to the conic 
 
 Z)87 + wiya + wa^ = 0, 
 parallel to the line of reference BG are represented by 
 ^(/3 + 7) + (^/^±^^)'a = 0, 
 
 the coordinates being triangular. 
 
 (139) The chord of contact of the tangents 
 
 (whether the coordinates be trilinear or triangular) is 
 Z 08 - 7) + («i - n) a = 0. 
 
CHAPTEE XVI. 
 
 CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 
 
 202. The equation 
 
 ra!' + m^^ + ny-2mn^-2nlYa-2lma^ = (1) 
 
 may be written 
 
 {la + wi/3 - ny)' - Alma0 = 0, 
 
 and therefore (Art. 161) represents a conic section to which a = 
 and j3 = are tangents, and 
 
 leu + »»j8 — ny = 0, 
 the chord of contact. 
 
 Similarly, 
 
 m0 + n'y — la — 0, 
 
 is the chord of contact of tangents /8 = and 7 = 0, and 
 
 ny+la — mfi = 0, 
 
 the chord of contact of tangents 7=0 and a = 0. 
 
 Hence the equation 
 
 lV + m''^+n'''/'-2mn^y-2nlya-2lmOL^ = (1) 
 
 represents a conic section, to which the lines of reference are 
 tangents, and 
 
 — h + »i/3 + M7 = 0, 
 
 la. — 7n/8 + W7 = 0, 
 
 la + JW/8 — W7 = 0, 
 
 the chords joining the point of contact. 
 
CONICS EEFEREED TO A CIRCUMSCRIBED TRIANGLE. 207 
 
 203. It should be observed that if we write - 1 for I, the 
 equation (1) takes the form 
 
 Pa" + Tw'/S* + wV - 2mw/37 + 2nlya + llma^ = (2), 
 
 and the chorda of contact now become 
 
 Za + OTjS + jjiy = 0, 
 
 la + ?wy3 — 717 = 0, 
 
 la — myS + my = 0. 
 
 So also if the equation to the conic be written 
 
 Pa" + m'^ + wV + 2mn^y - '^'"■h/a + 2lma^ = (3) , 
 
 the chords of contact will be given by 
 
 1% + m^ — «7 = 0, 
 la + JMyS + ny = 0, 
 
 — Za + ?M;84-W7= 0; 
 and if the equation to the conic be written 
 
 Z'a' + m^/S" + wV + 2mw/37 + SraZya - 2lma^ = (4), 
 
 the chords of contact will be given by 
 
 la — »i/9 + TO7 = 0, 
 
 — Za + »»/8 + W7 = 0, 
 Za + «!)S + H7 = 0. 
 
 Hence the four equations (1), (2), (3), (4) represent conica 
 inscribed in the triangle of reference, and so related that all the 
 twelve points of contact lie three and three on the four straight 
 lines given by 
 
 i Za + WjS + »7 = 0. 
 
 This reasoning applies equally whether the coordinates be 
 regarded as trilinear or triangular. 
 
208 CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 
 
 204. The last two articles shew that every equation of the 
 form 
 
 Va + m'/S' + wV + 2mn^'f ± 2nJya + 2ljna.^ = 0, 
 
 where we take either one only or all of the doubtful signs as 
 negative, represents a conic inscribed in the triangle of reference. 
 It will be observed, that if the doubtful signs be otherwise deter- 
 mined, the first member will become a perfect square and the 
 equation will reduce to one of the forms 
 
 (Za + »w/3 + «7)'=0, 
 
 {-h. + m^+nif = 0, 
 
 {h - myS + nyf - 0, 
 
 {la + »M/3 - nyf = 0. 
 
 In each of these cases, the locus of the equation consists 
 of two coincident straight lines, the limiting form of a conic 
 section when the plane of section becomes tangential to the 
 cone along a generating line. 
 
 Such a locus will moreover meet any straight line in two 
 coincident points, and will therefore, like an inscribed conic, 
 meet each side of the triangle of reference in two coincident 
 points. It cannot however be said to touch those sides in any- 
 geometrical sense. 
 
 205. Conversely, every conic section re/erred to a circum- 
 scribed triangle will be represented by an equation of the form 
 
 Pa* + w'/S" + nV + 2mn^y ± 2nlya. ± 2lma^ = 0, 
 
 where the doubtful signs must be either all negative, or one 
 negative and two positive. 
 
 For any conic section may be represented (Art. 145) by the 
 equation 
 
 Mtt' + v/y + W7» + 2M'/8y + 2v'ya. + 2M)'a/3 = 0. 
 
 But if the triangle of reference be circumscribed, a = re- 
 presents a tangent, and therefore we must find two identical 
 
CONICS EEFEREED TO A CIRCUMSCRIBED TRIANGLE. 209 
 
 solutions when we combine ot = with the equation to the 
 conic. 
 
 Therefore the quadratic 
 
 w/S" + 2m'/37 + W7' = 0, 
 must have two equal roots. 
 
 And therefore 
 
 m'' = vio, or m' = + Vvw. 
 
 Similarly, since /3 = and 7 = are tangents, we have 
 v = ± ^/ivu, 
 and w — ± "^uv. 
 
 Hence the equation takes the form 
 
 ua? + t'/S" + vrf ± 2 Juv^y ± 2 Jwuya ± 2 Juvafi = 0, 
 or, writing P, m^, n" for m, v, w, 
 
 ZV + m'^ + wV + 2mn^y ± 2nlyu ± 2lm%P = 0. 
 
 We thus see that every conic inscribed in the triangle of 
 reference has an equation of this form ; and the doubtful signs 
 must be either all negative or only one negative, since we 
 found in the last Article that if they were otherwise determined, 
 the equation would represent two coincident straight lines, 
 
 206. It will be observed that if two of the doubtful signs 
 be positive and one negative, we can immediately make all three 
 negative without altering the rest of the equation, hy changing 
 the sign of one of the quantities I, m, n. We may therefore 
 always assume the equation to a conic referred to a circum- 
 scribed conic to be of the form 
 
 Pa" + wi'/S* + mV - 2mn/3y - 2nh{a. - 2lma.^ = 0, 
 
 where I, m, n may be positive or negative quantities. 
 
 W. 14 
 
210 CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 
 It should be noticed that the equation 
 
 + JTa± Jm^± JwY = 0, 
 
 when cleared of radicals, takes the form of the equation just 
 written down. 
 
 So the equations (2), (3), (4) of Art. 203 are the rationalised 
 forms of the equations 
 
 + 'J- la± V»?i/8 + Vny = 0, 
 
 + Via ± V-7n/3 ± ^717 = 0, 
 + Via + Vm^ ± V-W7 = 0. 
 
 Thus we may always write the equation to a conic inscribed 
 in the triangle of reference in the form 
 
 Via + Vm^ + Vny = 0, 
 
 the coefficients I, m, n being either positive or negative, and 
 double signs being understood before the radicals, 
 
 2.07. From a given point a straight line is drawn in a given 
 direction to meet the conic 
 
 Vo? + to'/S' + wV - 2>MM/87 - 2mJya - 2&na/3 = 0, 
 
 it is required to find the lengths intercepted hy the curve upon 
 this straight line. 
 
 Let (a', ^, y) be the given point and \, fi, v the sines of 
 the given direction, then, as in Art. 171, the intercepts are given 
 by the equation 
 
 V (a' + \pf+ m^ (^ + iipY + «' (7 + I'p)' - 2»in {^ + ^p) (7' + vp) 
 - 2nl (7' + vp) (a' + \p) - 2lm («' + \p) {fi! + fip) = 0, 
 a quadratic to determine p. 
 
COMICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 211 
 
 Cor. If the point (a', /3', 7') be on the conic, so that 
 Pa" + m^^^ + wV" - awmyS'y - inhi'd - 2lma'0 = 0, 
 one of the intercepts is zero, and the other is given by 
 
 {Pk' + m^fi^ + nV - imnfjLv - 2nh\ - 2lmXfj,) p 
 + 2{l\ {la' — m0 — ny) + mfi (»»/S' - ny - la!) + nv {ny - la! - wj/S')} 
 
 = 0. 
 
 In other words this equation gives the length of the chord 
 drawn from (a, /3', 7') in the direction (X, fi, v). 
 
 208. To find the equation to the tangent at any point on the 
 conic. 
 
 Let X, /I, V be the direction sines of the tangent at (a, /3', y), 
 and let (a, ^, 7) be any point on the tangent, tlien (Chap, vi.) 
 
 we have 
 
 g-g _ /3-^ _ 7-7' __ .. 
 
 -ir—;r- — -V — '' ^^>- 
 
 And the length of the chord in this direction is given by the 
 equation of the last corollary. But since the direction is that 
 of the tangent, the length of the chord is zero : therefore 
 
 1X {la' — m0 — ny') + m/j, (m/3' — ny — la') 
 
 + nv {ny — la! — m/S') = 0, 
 
 or in virtue of (1), 
 
 I (a - a') {la -m0 - ny) + &c. = 0. 
 
 But since (a', ^', y) lies on the conic, we have 
 
 Pa" + w'/3" + nV' - ^mn^y - 2nly'a' - 2lma'^' = 0, 
 which reduces the last equation to 
 
 la {la' - m^ -ny') + m^ («?/S' - ny' - la) 
 
 + ny (ny —la - wyS') = 0, 
 
 a relation among the coordinates of any point (a, ^, 7) on the 
 tangent at (a', ^, y), and therefore the equation to that 
 
 tangent. 
 
 14—2 
 
212 CONICS KEFERRED TO A CIECUMSCRIBED TEIANGLE. 
 
 209. The polar of the point (a', ^, y), or the chord of con- 
 tact of tangents from that point, may be shewn as in Art. 173 to 
 be represented by the equation 
 
 la {la -mp- ny) + m^ {m^ - ny - la) + ny {ny - la - m^') 
 
 = 0. 
 
 210. To find the condition that any straight line whose 
 equation is given should he a tangent to the conic. 
 
 Let f<x+g0 + hy=O (1) 
 
 be the given equation to the straight line, and suppose (a',^',y) 
 its point of contact with the curve. 
 
 The tangent at (a', y3', 7') is given by 
 
 la, {I'j! — OT/8' — ny) + m^ {m^ — ny — la') + ny (ny — la — m0) 
 
 = 0, 
 
 which must be identical with (1). 
 Therefore 
 
 Za' — m^ — ny m0 — ny — la ny' — la' — m^ 
 
 f ~ £ " * ' 
 
 I m. n 
 
 la' m^ ny 
 
 m n n I I m 
 
 But (a', ^, y) must also lie on the locus of (l), wherefore 
 fa[ + g^ + hi = Q. 
 
 Hence eliminating a', /S', y, we get 
 
 I \m nj m\n tj n\l mj 
 
 I m n 
 which is therefore the condition required. 
 
 I' 'in . n 
 
 or ^+ — + 7- = 0, 
 
 f 9 h ' 
 
CONICS EEFEEKED TO A CIECUMSCBIBED TEIANQLB. 213 
 
 211. CoE. The equation in trilinear coordinates 
 
 Via + VrnJiJ + V«y = 
 
 will represent a parabola, provided 
 
 I , m n 
 - + -T + - = 0. 
 a b c 
 
 And the equation in triangular coordinates 
 
 V^ + J my + Jnz = 
 will represent a parabola, provided 
 
 Z + wi + « = 0. 
 
 212. We may shew, as in Art. 175 or 193, that the locus 
 of the middle points of a series of parallel chords in the conic 
 
 VZa +Jm^ + Vn7'= 
 
 is a straight line represented by the equation 
 
 la (?X — mfi — nv) + »jyS (w/i — nv— l\) + w^ (nv —TX — m)i) = 0, 
 
 where X, /i, v are the direction sines of the given direction, or 
 (whether the coordinates be trilinear or triangular) where \, /i, v 
 are proportional to the coordinates of the point where the parallel 
 chords intersect at infinity. 
 
 213. To find the centre of the amie 
 
 Vfa + si mfi + Vw7 = 0; 
 
 Proceeding as in Art. 178 or 194, we find that the co- 
 ordinates (a, /3, 7) of the centre must satisfy the equations 
 
 - (Za'- JM/3' - ni) = ^ (m/3' -nr,' - la') = - {ny' -h'- m^, 
 
 or 
 
 bn+cm cl+an am + bl 
 
 A 
 
 Ibc + mca + nab ' 
 which express the coordinates explicitly. 
 
214 CONICS EEFEEEED TO A CIECUMSCEIBED TEIANGLE. 
 
 Obs. If the coordinates are triangular instead of trilinear, 
 we find that the centre of the conic whose equation is 
 
 'sllx + 'Jmy + Vws = 0, 
 
 is given by the equations 
 
 r I 
 
 _ y _ g _ 
 
 »4 + w n+l l + m 2{l+m+n)' 
 
 214. Cor. In order that the centre may be at the point 
 a=/3 = 7, which we have seen (Art. 16), is the centre of the in- 
 scribed circle, we must have 
 
 whence 
 
 bn + cm = cl + an = am + bl, 
 
 bcl _ cam abn 
 
 b + c — a c + a — b ~ a + b — c' 
 
 or 
 
 m n 
 
 I 
 
 cos'- cos'- cos'- 
 
 hence the equation 
 
 Va cos - + VjS cos - + V7 cos ^=0 
 ^ ^ 2 
 
 is the only equation which can represent an inscribed conic 
 having its centre at the assigned point ; therefore there is only 
 one such conic, namely, the inscribed circle, and the equation 
 
 /- -^ r- B r- C 
 
 Vacos 2+V/Scos — + V7cosj = 0, 
 
 represents that circle. 
 
 Similarly, we can shew that the escribed circles having their 
 centres at the points 
 
 -a= ^= 7, 
 
 « = -/3= 7, 
 
CONICS EEFEERED TO A CIECUMSCEIBED TRIANGLE. 215 
 
 are represented respectively hy the equations 
 
 V-acos- + v'^sin- + V7sin^ = 0, 
 
 /- • -4 , — - B ,~ f! 
 
 Vasin-+V-y3cos-+v'7sin-^ = 0, 
 
 Vasin- + \//8sin^+\rr^cos-^ = 0. 
 
 215. To find the pole of any given straight line with re- 
 spect to the conic 
 
 JloL + Jm^ + Vn7 = 0. 
 
 I^et fa+g,0 + ky = o (1) 
 
 be the equation to the given straight line, and suppose («', /3', 7') 
 the pole. 
 
 The polar of this point is given by 
 la {loL — wiyS' - ny) + w/3 (wiyS' - ny - la') + ny {ny - la - mfi') 
 
 = (2), 
 
 which must be identical with (1). 
 
 Therefore 
 
 f ^ ff A 
 
 I {la - mff - ny) m (myS' - my' - Za') ^n[n^-M- m^) ' 
 
 a' /? y' 
 
 or = — — — = - 
 
 ng + mh lh+ nf mf+ Ig ' 
 
 which determine the coordinates of the pole required. 
 
 216. If we assume the geometrical property that every 
 conic has a pair of foci situated at equal distances on opposite 
 sides of the centre, and such that the rectangle contained by the 
 perpendiculars from them upon any tangent is constant, we can 
 
216 CONICS EEFEERED TO A CIRCUMSCRIBED TRIANGLE. 
 
 readily write down equations to give the trilinear coordinates of 
 the foci of the conic 
 
 JToL + /s/jn/8 + ijnrj = 0. 
 
 For let (a, ^8, 7) be the coordinates of a focus, and let 
 (a, ;8', 7') be those of the centre. 
 
 Then, since the centre bisects the line joining the foci, the 
 sum of the two values of a is the double of a' (Art. 18. Cor.). 
 
 But since the line of reference BC is a, tangent, the rectangle 
 represented by the product of the two values of a is equal to a 
 constant, Jc' suppose. 
 
 Therefore the two values of a are the roots of the quadratic 
 
 a''-2a'a + F=0. 
 But similarly yS and 7 are given by the quadratics 
 
 ^-2^^ + I^ = 0. 
 and '/ — 277 + i' = 0. 
 
 Hence a' - 2a'a = yS* - 2^';S = 7" - 277, 
 
 or substituting the values of a', ^, 7' (Art. 213), 
 {hcl + cam + abn) a" — 2 A {bn + cm) a 
 
 = {hcl + cam + abn) /S* — 2 A (cl + an) ^ 
 = {hcl + cam + abn) 7* — 2 A {am + bl) 7, 
 /I m , n\ , 2 Aa fm n\ 
 
 or - + T + - « hr + - 
 
 \a cj a \b cj 
 
 = ^?+=)^-^(?+D 
 
 equations to determine the two values of a, fi, 7 for the 
 two foci. 
 
CONICS REFERRED TO A CIRCUMSCRIBED TRIANGLE. 217 
 Obs. In triangular coordinates, the foci of the conic 
 
 V tc + '/rny + "Juz = 0, 
 are given by 
 
 <c\ Z + OT + M / Wy l + m + n^J 
 
 _1/, l + m \ 
 -?V~l + m + nV- 
 
 217. To find the condition that it should be possible to find 
 a conic touching the three straight lines of reference and three 
 other given straight lines. 
 
 Let fa+g,^ + h,y = (1), 
 
 fa+g^ + h,ry = (2), 
 
 /,«+^a^ + *37 = (3), 
 
 be the three given straight lines. 
 
 Any conic inscribed in the triangle of reference will be repre- 
 sented by the equation 
 
 Jla + Jm^ + Jmy^O. 
 If the straight line (1) be a tangent we must have (Art. 210) 
 i + ^ + |=0 (4). 
 
 /. ffi K 
 
 Similarly if (2) and (3) be tangents, 
 
 j- + - + T- = (5), 
 
 /, 9, K 
 
 I m n ^ f„s 
 
 7 + - + T- = (6)- 
 
 /s 9i «. 
 
 Hence, eliminating l:m:n from the equations (4), (5), (6), 
 we have 
 
218 CONICS EEFEEEED TO A CIECUMSCEIBED TEIANGLE. 
 
 ILL 
 
 /,' ff,' K 
 
 L L L 
 
 A' 9: K 
 
 L L L 
 
 /»' 5'a' K 
 which will be the condition required. 
 
 = 0, 
 
 218. Brianchon's Theoeem. If a hexagon he described 
 about a conic section, the three diagonals formed hy joining oppo- 
 site angular points will be concurrent. 
 
 Let PQRPQB! be the hexagon. 
 
 Take three alternate sides, QRi BP, PQ produced, for lines 
 of reference, and let the equations of the other sides be 
 
 iQ'E),f<x+g,^ + \y = (1), 
 
 {B'P),fa+g^ + h,y = (2), 
 
 {PQ),f<x+g^ + h^y = (3). 
 
 Then P is given by 7 = 0, ^a + g^ = 0, 
 and P' is given by /9 = 0, _^a + h^y = 0, 
 
 therefore the diagonal PP is represented by the equation 
 
 so QQ' will have the equation 
 
 and BP^ will have the equation 
 
 Kfi<^ + Kff^ + KK'i = 0- 
 Hence the condition of concurrence of the three diagonals is 
 
 fJii f»9i^ /A 
 ^3/1. 9s9ii 9 A 
 
 = 0, 
 
EXERCISES ON CHAPTEE XVI. 
 
 219 
 
 or 
 
 1 
 
 1 
 
 1 
 
 /.' 
 
 f: 
 
 /s 
 
 1 
 
 1 
 
 1_ 
 
 9.' 
 
 9.' 
 
 9^ 
 
 1 
 
 1 
 
 1 
 
 K 
 
 K 
 
 *, 
 
 = 0, 
 
 which is the condition that the three straight lines (1), (2), (3) 
 should touch the same conic with the lines of reference; which 
 proves the proposition and its converse. 
 
 Exercises on Chapter XVI. 
 
 (140) The conies 
 
 Z/87 + mia. + ma/3 = 0, 
 and 7/37 + »»'7a + «'«)S = 0, 
 
 intersect in the points of reference and in the point given by 
 
 m, n 
 
 = /8 
 
 n, I 
 
 = 7 
 
 I, m 
 
 1 1 
 m, n 
 
 
 ri, X 
 
 
 X, m' 
 
 (141) The straight line Xa + /i/3 + i«y = will be cut har- 
 monically by the conies 
 
 l^'f + nrfa. + na^ = 0, 
 and r/37 + m'ya + w'a/S = 0, 
 
 provided 
 
 IX\^ + mm'fi' + nn'u' — {mn' + m'n) fiv — (nX + n'X) v\ 
 
 - (Im' + Xm) \fi = 0. 
 
 (142) The imaginary triangle whose sides are 
 
 u 4. V^w =0, v + V— 1 w = 0, w + V— 1 M = 
 is self-conjugate with respect to the conic 
 
 ^1+1 = 0. 
 u V 10 
 
220 
 
 EXERCISES ON CHAPTEE XVI. 
 
 (143) A triangle ABC is inscribed in a conic, and from 
 each angular point straight lines are drawn parallel to the oppo- 
 site sides to meet the conic again in P, Q, R; prove that QR, 
 RP, PQ are parallel to the tangents at A, B, C. 
 
 (144) The triangle whose sides are 
 
 will be self-conjugate with respect to the conic 
 
 I m n . 
 - + -5 + - = 0, 
 a P 7 
 
 provided 
 
 0, 0, 0, ?, m, n, = 0. 
 
 fi> //. fs' 9 A: 5'A> 9K 
 ffi^ 9', 9s y Kfi> KA> Va 
 
 K> h\ K' ft9x1 /i^2' fz9i 
 
 (145) Interpret the equation in trilinear coordinates 
 and find the coordinates of the foci of its locus. 
 
 (146) If a parabola touch the sides of a triangle its focus 
 will lie on the circle which circumscribes the triangle. 
 
INTRODUCTION TO CHAPTER XVII. 
 
 NOTATION, ETC. 
 
 219. Students who have not read the Differential Calculus 
 are recommended to pay particular attention to the notation 
 which we now introduce. Those who have read the Differential 
 Calculus will accept it without explanation. 
 
 Let f{x) denote any function whatever of x. Then the 
 
 df (x) 
 symbol -'— — (which must be regarded as a single expression 
 
 not capable of resolution into numerator and denominator) is 
 used to denote the expression derived from f[x) by substituting 
 for every power of x (suppose x"), the next lower power multi- 
 plied by the original index (i.e. wa;*"'), and omitting altogether 
 the terms which do not involve x. 
 
 Thus X will be replaced by a;" or 1, a? by 2x, x* by 4a;', and 
 so on. 
 
 df(x) 
 For example, if/ {x) denote x^ + Saa;" + Sa'a; + a*, then ^^^-^ 
 
 will denote 3a;' + %ax + 3n'. 
 
 So also, if / (a, yS, 7) be any function of a, /3, 7, then 
 
 df{a,P, y] ^gjjQtes the result obtained by substituting for the 
 
 da, 
 powers of a according to the law enunciated above, and neg- 
 lecting the terms in the original which do not involve a. 
 
222 INTEODUCTION TO CHAPTER XVII. 
 
 So la' denotes tte result obtained by neglecting the 
 
 terms ■which do not involve /8, and substituting for the powers 
 of /8 in the other terms. 
 
 It is usual when the abbreviation can be made without 
 ambiguity to write f- for ^ and f for ^%^) . 
 
 220. The expression -^ is called the derived function with 
 respect to a of the original expression /(a, /3, 7). So the expression 
 -^ is called the derived function with respect to y9, and -^ 
 the derived function with resjpect to 7. 
 
 Also if /(«> /8, 7) = be an equation involving a, y8, 7, then 
 
 da. 
 is called the derived equation with respect to a, and so on. 
 
 221. Let/(o, /9, 7) = ua^+ v^ + w-f+ 2m'/37 + 2f '7a + iw'aP, 
 then 
 
 ■^ = 2ucn + 2!o'/8 + 2w'7 
 
 ^=2v^+2u'y+2w'a 
 ^ = 2m)7 + 2v'a + 2m'/3 
 
 = 2 {Maa'+ ?;jS^' + loyy' + u' (fiy' + ^y) + v' (7'a + 7a') 
 
 .(i). 
 
NOTATION, ETC. 223 
 
 And this expression is not altered if a', /8', 7' be inter- 
 changed with a, /3, 7 respectively. 
 
 Hence if 
 
 da.' ' d^ ' dr,' 
 denote the derived functions of /(a, /3', 7'), then 
 'df , o'df , .df df ^df df 
 
 Again from (i) we get 
 
 + 2i>'7a + ^wa0\= 2f{a, yS, 7) . . . (iii). 
 Again, f{a + x, ^ + y, 7 + 2) 
 = M (« + «)'+« (/8 + 2/)'+ w (7 + «y+ 2m' (/3 + y) (7 + 3) 
 
 + 2t;'(7+s) (a + a;) + 2m)' (a + a;) (/S+^) 
 = Ma" + vyS° + wf + 2m'/37 + 2«'7a + 2w'a/3 
 + 2a; (wa + v'<y + w'^S) + 2j/ (v/3 + M'7 + w'a) + 2a (wy + m'/S + v'a) 
 + Ma;" + vy^ + tos' + 2m'^2! + 2i;'za5 + 2w'xy 
 
 ./(a,A7) + 4^y|+.|+/(a.,y..) (iv). 
 
 If we write Xp, iip, vp for x, y, z, the result (iv) becomes 
 f{a + \p, fi + fip, y + vp) 
 
 ^f(a,^,y)+p(x^^ + p,^^ + v^ + py{X,fi,v)....{y). 
 
 The results given in the equations (ii), (iii), (iv), (v) are of 
 the very highest importance. 
 
 222. To express the homogeneous function of the second degree 
 
 /(a, /3, 7) = ua' + «/3= + 107' + 2u'^y + 2v'ya. + iwa^ 
 
 in terms of its derived functions 
 
 df ^ ^ 
 da' rf/3 ' dy' 
 
224 
 
 INTRODUCTION TO CHAPTER XVII. 
 
 We have (Art. 221, equation iii) 
 and (equations i) 
 
 V(.,ft7)..| + 4 + 7|. 
 
 1 rlf 
 
 2 da. 
 
 I df , ,- 
 
 Therefore eliminating a, y3, 7, from (1), (2), (3), (4), we get 
 
 .(1), 
 
 •(2), 
 ■(3), 
 .(4). 
 
 da' 
 
 dj_ 
 d^' 
 
 df 
 
 ^, V, u, w 
 
 U, W , V 
 
 W , V, u 
 
 or 
 
 u, w , v' 
 
 w , V, u 
 V, U, V! 
 
 f{a,P,y)^- 
 
 ^ df df 
 ' da' d^' dy 
 
 df , , 
 
 df_ 
 d^' 
 
 df , 
 
 dy' ''' "' «' 
 
 W , V, u 
 
 ^<5-^(D'--(D' 
 
 ^^^ d/3 dr^^^^ d7 da^'^^ do. d/3' 
 
where 
 
 NOTATION, ETC. 
 
 U = vw — u', V = wu — «'", W = uv — w'^, 
 U' = v'w'—uu', V' = w'u'~vv', W = u'v'—ww', 
 
 225 
 
 and so 
 
 Ai)'M%Mt)-^^$%-^^n-^^U 
 
 U, W , V 
 W, V, u 
 
 v, w', w 
 
 Thus every homogeneous function of the second degree of 
 three quantities may be expressed as a homogeneous function of 
 the second degree of its three derived functions. 
 
 w. 
 
 15 
 
CHAPTEE XVII. 
 
 THE GENERAL EQUATION OP THE SECOND DEGEEE. 
 
 223. Having considered in the last three chapters some 
 special forms of the equation to a conic section, and thereby 
 rendered the student who reads the subject for the first time 
 familiar with the methods of treating equations of the second 
 degree, we pass on to consider the most general case, when the 
 conic is represented by an equation of the most general form. 
 
 But for the sake of the reader who needs not to be thus led 
 up to the more difficult part of the subject, but prefers to in- 
 vestigate first the most general form of the equation and thence 
 to deduce the particular cases, we shall make the present chapter 
 perfectly independent of the three preceding, so that it may be 
 read consecutively after Chapter xiii. 
 
 To this end we shall be obliged in this and the succeeding 
 chapter to repeat some explanations and definitions which have 
 been already given in the treatment of the special cases ; but, 
 our object being thus explained, the repetition will doubtless be 
 excused. 
 
 224. The equation of the second degree in its most general 
 form may be written 
 
 ua? + v^ + w'/+ 2m'/37 + 2v'yoL + 2w'afi = 0. 
 
 But when the coefficients of the several terms have not to 
 be separately discussed, we shall generally denote the first 
 member by the symbol /(a, /8, y) and write the equation 
 
 /(«,/3,7)=0. 
 
THE GENERAL EQUATION OP THE SECOND DEGREE. 227 
 
 225. To find the equation to the tangent at any point on a 
 conic section. 
 
 Let /(»,/3,7)=0 (i), 
 
 le the equation to the curve, and (a, 0, y) the coordinates of 
 the given point upon it, then any straight line through the 
 given point will be represented by the equation 
 
 «-^ = ^^ = '^-^' = p (ii), 
 
 and the lengths which the curve intercepts on this straight line 
 will be given by 
 
 /(a'+Xp, + H'P> i + vp)-(i, 
 ,/(a', 0, i) +p(x2 + M ^ + " ^)^P'f'<^' '*' "^ =0-("')- 
 
 or, 
 
 Since (a', /S', 7') lies on the curve, therefore /(«', 0, 7') = 0, 
 and one of the values of p is zero, as we should expect. But if 
 the straight line be a tangent the two values of p must be equal, 
 that is, each must be zero. 
 
 Hence the coeflBcient of p must also vanish in the quad- 
 ratic (iii), 
 
 therefore x ^+^^/ + .|< = 0, 
 
 or since \, p., v are proportional to a — a', fi — 0, y - y by (ii) 
 we get 
 
 («-a')| + (/3-^)|4-(7-7')| = 0. 
 
 = 0, 
 since (a', 0, 7') is on the curve ; therefore 
 
 
 15—2 
 
228 THE GENERAL EQUATION 
 
 a relation among the coordinates of any point on the tangent at 
 (a', ^ , y) and therefore the equation to the tangent at that 
 point. 
 
 Obs. If the equation to the conic be written 
 
 wa' + w/S" + W + 2m'/37 + iv'ya + 2w'afi = 0, 
 the equation to the tangent is 
 a {utx + w'0 + v'i) + /3 (ti/3' + mV + lo'a') + 7 (wy + v'a + u'/S') = 0. 
 
 226. Cor. The normal at (a', 0, 7') will be given by the 
 equations, (Art. 81). 
 
 g-g' /3-j8' 
 
 df df ^ df ~ df df A df ^ 
 
 -VV — -vkj cos (7— ^ cos .B -^ — -j-; 0,0^ A — -J-, cos V 
 da dp dy dp dy da. 
 
 7-7 
 
 df df T~l[f 7' 
 -^, — -^ cos £— -Hv cos A 
 ay aa, dp 
 
 if the coordinates are trilinear; or, if they be triangular, (Art. 87) 
 by the equations 
 
 g-g' ^ )8-/3' 
 
 ■y-,ct?—-To,ahcoaC—-T-jaccQaB j^b'—^.bceosA— -^,abcoaC 
 da dp dy dp dy da! 
 
 ^ 7-7 
 
 df , df n df J A ' 
 
 -j-r c — -r-, ca cos B — jh be cos A 
 dy da dp 
 
 227. On the determination of Direction. 
 
 Any two straight lines drawn in the same direction are 
 parallel and have their point of intersection at infinity. 
 
 Conversely any straight lines which intersect in a point at 
 infinity are in the same direction. 
 
OF THE SECOND DEGREE. 229 
 
 Hence every point at infinity determines a direction, and 
 every direction may be determined by assigning the point at 
 infinity in which straight lines drawn in that direction inter- 
 sect. 
 
 It has already been remarked (page 101) that if we represent 
 a straight line by equations of the forms 
 
 «-«'^/ g-^' ^7-7 ,-s 
 
 'K fj, V ' ^' 
 
 then X, fi, V will be proportional to the coordinates of the point 
 where the straight line meets infinity, or the point where a 
 system of straight lines parallel to the given one will intersect. 
 
 The quantities \, /i, v therefore determine the direction of 
 these straight lines, and we shall henceforth speak of such a 
 direction as the direction (\, //., v), where we suppose \ fi, v to 
 have such actual values as shall make each of the fractions in 
 (i), equal to the distance p between the points (a, /8, 7) and 
 («', /S', 7')- 
 
 If X, It, V all have values only proportional to these values, we 
 shall speak of the direction as the direction (X : /i : y) instead of 
 (X, fi, V). See Art. 69, ad fin. 
 
 When the direction (X : ^ : v) is spoken of, it must be borne 
 in mind that X, /t, v satisfy the relation in trilinear coordinates 
 
 oX + J/t + cv = 0, 
 
 or in triangular coordinates 
 
 X + /i + v = 0, 
 
 and if the direction (X, /t, v) be referred to, then X, ft, v satisfy 
 not only the former relation but also the non-homogeneous 
 relations of page 101 (xx). 
 
 228. Let be a point in wliich a conic section meets 
 infinity. Any straight line drawn in the direction determined 
 by meets the conic in this point at infinity and therefore in 
 one and only one finite point (Art. 150.) The number of real 
 
230 THE GENERAL EQUATION 
 
 directions in which straight lines can be drawn so as to cut a 
 conic in only one real point, will thus be the same as the number 
 of real points in which the conic meets the straight line at 
 infinity; there is therefore one such direction for a parabola, two 
 for a hyperbola, and none for an ellipse (Art. 151). Further, 
 if the tangent at lie not altogether at infinity it will be an 
 asymptote (Art. 152). Hence in the hyperbola, any straight line 
 parallel to an asymptote will meet the curve in only one real 
 point, and the two asymptotes determine the two directions of 
 such lines. 
 
 229. To determine the direction of the tangent at any point 
 on a conic section. 
 
 Let f{a, )8, 7) = be the equation to the given curve, and 
 suppose X, /J,, V the direction sines of the tangent at (a', /8', 7'). 
 
 Then the equations to the tangent will be 
 
 a-a' _ /3-j5' _ 7-7' _ 
 \ ~ (i ~ V 'P' 
 
 and the lengths which the curve intercepts on this line will be 
 given by the equation 
 
 /(a' + Xp, fi' + fjip, y' + vp) = 0, 
 or 
 
 Since (a', /3', 7') is on the curve, therefore /(a, /8', 7) =0, 
 and one of the values of p given by this quadratic is zero, as 
 we should expect. But since further the straight line touches 
 the curve at this point, we must have both values of p zero. 
 
 This equation, together with the relation 
 Xa + ftb+vc^O 
 in trilinear coordinates, or 
 
OF THE SECOND DEGREE. 
 
 231 
 
 in triangular coordinates, gives us 
 X n 
 
 df 'df 
 rf/8" d^' 
 
 ^ 
 
 df df 
 dy" dx 
 
 
 df df 
 da." d^' 
 
 h, c 
 
 
 c, a 
 
 
 a, b 
 
 in Irtlinear coordinates, or 
 
 \ /i. V 
 
 dl^_df_~ df__d£~ df _df_ 
 d^ di dri dd da. dff 
 
 in triangular coordinates, — ^which determine the ratios of X, /i, v, 
 and their actual values are immediately given by one of the 
 non-homogeneous equations of Result XX. page 101. 
 
 230. From any point there can be drawn two real or imagi- 
 nary tangents to meet any conic. 
 
 Let /(a,/3,7)=0 (1) 
 
 be the equation to the conic, and (a, 0, y) the given point. 
 
 Suppose 
 
 ?—^^^:zK^i^=p (2) 
 
 X /* V 
 
 the equations of a tangent to the curve, and suppose (a", /9", 7") 
 its point of contact. 
 
 The intercepts measured from (a", /8", 7") to the curve must 
 both be zero. 
 
 Hence the equation 
 
 f{a" + \p, 0" + fip, j" + vp)=0 
 must have both its roots p = ; 
 
 therefore /(a", ;8", 7") =0 (3) 
 
 and 
 
 ^ df df df ^ 
 
 (4). 
 
232 THE GENERAL EQUATION 
 
 But since (a", ^8", y") lies on the straight line (2), therefore 
 
 \ ft V ' 
 
 Hence the equation (4) becomes 
 
 (,.-_«.) |:,+(^"_^)^H- (,"-,') |;=o (5). 
 
 But by the property of homogeneous functions (Art. 221), 
 the equation (3) gives us 
 
 " w + ^ w^^ w ^^' 
 
 Hence, subtracting (5) and (6), 
 or, which is the same thing (Art. 221), 
 
 a" M- A. Q" M- 4. V -^ = (1^ 
 
 Hence the coordinates (a", j8", 7") of the point of contact 
 of any tangent from (a, yS', 7') to the curve, are obtained by 
 solving simultaneously the quadratic equation (3) and the sim- 
 ple equation (7). 
 
 Hence there will be two real or imaginary solutions. 
 
 Therefore from any point there can be drawn two Teal or 
 imaginary tangents to a conic section. 
 
 231. 21? find the equation to the chord of contact of tangents 
 drawn from, a givefn, point to a conic section. 
 
 Let (a', y5', 7') be the given point, and 
 /(a,A7)=0 
 the equation to the conic. 
 
OF THE SECOND DEGEEE. 233 
 
 Then if (a", y3", 7") be the point of contact of either tangent 
 from (a', ff, 7') to the curve, we have by equation (7) of Art. 230, 
 
 ° da:^'^ rf/8' + ^ ^7 "• 
 
 Hence the coordinates of either point of contact satisfy the 
 equation 
 
 «l+^|+4-° <«■ 
 
 But this is the equation to a straight line. Hence it repre- 
 sents the straight line through the two points of contact, that is, 
 the chord of contact of tangents from (a', 0, 7') to the conic. 
 
 It will be observed that the equation (8) represents a real 
 straight line wherever (a', ^, y) be situated, i. e. the chord of 
 contact is real whether the tangents be real or imaginary. 
 
 232. Def. The chord of contact of the real or imaginary 
 tangents from a fixed point to a conic is called the polar of 
 the fixed point with respect to the conic. 
 
 And the fixed point is called the pole of the straight line 
 with respect to the conic. 
 
 We have shewn in the last article that the polar of the conic 
 
 /(a, /S,7)=0, 
 
 with respect to the point (a', ^, 7'), is represented by the equa- 
 tion 
 
 df , „df^ df . 
 
 If the equation to the conic be written 
 
 wa' + wyS* + Toy' + 2m'j87 + 2w'7a + 2w'a^ = 0, 
 the equation to the polar becomes 
 
 a (wa' + Jo'/S' + «'7') + /3 («/9' + wV +«''«') + 7 (W + «'«'+ m'/3') = 0. 
 
234 THE GENERAL EQUATION 
 
 233. 2'o find the coordinates of the poh of any straight line 
 •with respect to a conic section. 
 
 Let Za + m/3 + W7 = (1) 
 
 be the equation to the straight line, and 
 
 /(a,/S,7)=0 
 
 the equation to the conic. 
 
 Suppose (a', P, y) the coordinates of the point required, 
 then the polar of (a, ^, y') with respect to the conic is given by 
 the equation 
 
 which must therefore be identical with (I). 
 
 Hence 1^ = 1^ = 1^ 
 
 I da m d^ n dr/ ' 
 
 Ma' + w';8' + i?V ^ v^ + vlrf' + w'o! _ Ky' + v'a' + u'^ ' 
 I m n ' 
 
 Therefore 
 
 (2), 
 
 or 
 
 
 a 
 
 
 I, 
 
 m, 
 
 n 
 
 w\ 
 
 V, 
 
 v! 
 
 v; 
 
 u', 
 
 w 
 
 
 /3' 
 
 I, 
 
 m, n 
 
 t 
 
 u', w 
 
 u, 
 
 w\ v' 
 
 
 7 
 
 
 I, 
 
 m, 
 
 n 
 
 u. 
 
 V,; 
 
 v'. 
 
 1 
 
 w, 
 
 V. 
 
 «', 
 
 and if the coordinates be trilinear so that 
 aa +h^ + cy' = 2A, 
 each of these fractions 
 
 
 -2A 
 
 
 0, 
 
 I, m, 
 
 n 
 
 a, 
 
 u, w', 
 
 v' 
 
 h, 
 
 w', », 
 
 u' 
 
 c, 
 
 »', «', 
 
 to 
 
 or if they be triangular so that 
 
 a' + i8' + 7' = l, 
 
OF THE SECOND DEGREE. 
 
 then each of the equal fractions in (2) 
 
 235 
 
 
 
 1 
 
 
 0, 
 
 I, 
 
 m, 
 
 n 
 
 1, 
 
 u, 
 
 w', 
 
 V 
 
 1, 
 
 w', 
 
 V, 
 
 v! 
 
 1, 
 
 v', 
 
 u', 
 
 w 
 
 Thus the actual values of the required coordinates are ex- 
 pressed. 
 
 Cob. The pole of the straight line 
 hx + mp + 717 = 
 will lie upon this straight line provided 
 
 = 0. 
 
 u, 
 
 
 I 
 
 w; 
 
 V, «', 
 
 m 
 
 V, 
 
 m', w, 
 
 n 
 
 I, 
 
 m, n, 
 
 
 
 This is therefore the condition that the straight line 
 should be a tangent to the conic. 
 
 234. If a point P lie upon the polar of a point Q, the point 
 Q will lie upon the polar of the point P. 
 
 For let (a', )8', 7^, (a", /3", 7") be any two points Pand Q, 
 
 The equations to their polars are 
 
 df , a^f . df „ 
 
 rfa' 
 
 d^'^ 'dri' 
 
 df , ^df df „ 
 
236 THE GENERAL EQUATION 
 
 If P lie on the polar of Q, we have 
 
 , df „, df , df ^ 
 
 which is the same thing (Art. 221) as 
 
 which is the condition that Q should lie on the polar of P. 
 .*. &c. Q.E.D. 
 
 235. If a straight line p pass through thepole of a straight 
 line q, the straight line q will pass through thepole of the straight 
 line p. 
 
 For if the equations to the two straight lines be 
 
 la. + m^ + M7 = {p), 
 
 ra + m'^ + n'y=0 (q), 
 
 then, by Art. 233, the equation 
 
 = 
 
 M, 
 
 w', 
 
 v', 
 
 I 
 
 W, 
 
 V, 
 
 u', 
 
 m 
 
 v, 
 
 u, 
 
 w, 
 
 n 
 
 r 
 
 m 
 
 n', 
 
 
 
 expresses equally the condition that the pole of ^ should lie on 
 gr, and that the pole of q should lie on p, which proves the 
 proposition. 
 
 236. The two preceding articles express the same proposi- 
 tion in different forms. The following corollaries follow from 
 either article. 
 
 COE. 1. If a point lie on a fixed straight line, its polar will 
 pass through a fiaxd point (viz, the pole of the fixed straight 
 line). 
 
 Or, if a series of points be colUnear, their polars are con- 
 current. 
 
OF THE SECOND DEGEEE. 
 
 237 
 
 Cor. 2. If a straight line pass through, a fixed point, its pole 
 will, lie upon a fixed straight line (viz. the polar of the fixed 
 point). 
 
 Or, if a series of lines be concurrent, their poles are col- 
 linear. 
 
 237. To find the equation to the two tangents drawn from a 
 given external point to the conic whose equation is 
 
 /(a,/S,7)=0. 
 
 Let (a', /S', 7') be the given point P, and let (a„, y8„, 7J be 
 any point Q on either tangent. 
 
 Then PQ being a tangent, passes through its own pole; or 
 P, Q and the pole of PQ are coUinear. 
 
 Therefore PQ is concurrent with the polars of P and Q, 
 (Art. 236, Cor. 1.) 
 
 But the equation to PQ is, (Art, 21) 
 
 a, P, 7 
 
 Oo> ^o> 7o 
 
 = 0, 
 
 a', ^', 7' 
 
 and the equations to the polars of Q and P are, (Art. 231) 
 df , oAf ^ df ^ 
 ^a„ aPo »7o 
 
 '^dd^'^ d^'^^ dri "' 
 
 and therefore by the condition of concurrence, (Art. 26) 
 
 = 0, 
 
 ^o> 7o 
 
 ^, 7 ' 
 
 7o, «0 
 
 7', a' 
 
 ) 
 
 a', /3' 
 
 
 
 da." 
 
 df 
 
 
 
238 
 
 THE GENERAL EQUATION 
 
 a relation among the coordinates of any point (a,, j8„ 7„) on 
 either tangent. Hence, suppressing the subscripts, we have the 
 equation to the two tangents 
 
 ys, 7 
 
 
 7, a 
 
 
 a, i8 
 
 ^, 7 
 
 > 
 
 7', a' 
 
 J 
 
 a', /3' 
 
 da' 
 do!' 
 
 d^' 
 
 M 
 
 d/3 ' 
 
 ^7 
 
 = 0, 
 
 This equation may be written 
 
 or in virtue of Art. 221, 
 
 the form in which the equation is commonly quoted. 
 
 238. The following is another method of obtaining the 
 equation in the form just written. 
 
 To find the equation of the pair of tangents drawn from a 
 given external point to the conic whose equation is 
 
 /(a,A7)=0 (1). 
 
 Let (a, 0, y) be the external point : and suppose 
 a-a' _y 3-^' _ y_y _ 
 \ fjk V 
 
 to be one of the tangents. 
 
 Then the equation of the intercepts 
 
 /(a' + V, + /ip, y'+vp)=0 
 must have two equal roots. 
 
 (2), 
 
OF THE SECOND DEQREE. 239 
 
 Therefore 
 
 or substituting the equations (2), and remembering that 
 we get 
 
 = 4/(a',^, 7') /(«-«', y3-/9', 7-7) 
 
 = 4/(a',^,7'){/(a,^,7)-(a|H-^f, + 7f)+/(a',^,7')}. 
 therefore 
 
 a homogeneous equation of the second order connecting the 
 coordinates of any point (a, )8, 7) on either of the tangents from 
 {a, /8', 7') to the curve, and therefore the equation to the two 
 tangents from that point, which was required. 
 
 239. To find the locus of the middle points of a series of 
 parallel chords in the conic whose equation is 
 
 /(a,/3,7) = 0. 
 
 Let (X, /i, v) be the point of intersection at infinity of all the 
 parallel chords : and let (a', ff, y) be the middle point of one 
 of them. 
 
 Then the equation to this chord is 
 
 a_a'_^-^'_7-7'_ 
 
 ~\ jT- ir-p (^)' 
 
 and the lengths of the intercepts cut off by the curve are given 
 by the quadratic 
 
 /(a' + V, ^ + ^p, 7' + z/p)=0 (2). 
 
240 THE GENERAL EQUATION 
 
 But since these intercepts are measured from the middle 
 point of the chord, they must be equal in magnitude and oppo- 
 site in sign : therefore the coeflScient of p in the quadratic (2) 
 must vanish : therefore we have 
 
 »'l+^|+''f-« («• 
 
 an equation connecting the coordinates (a', ^, y) of the middle 
 point of any one of the chords, and therefore (accents sup- 
 pressed,) the equation to the locus of the middle points. Since 
 the equation (3) is of the first degree, the locus of the middle 
 points of any system of parallel chords in a conic section is a 
 straight line. 
 
 240. Dep. The locus of the middle points of a series of 
 parallel chords is called a diameter. 
 
 One of these chords in its limiting and evanescent position 
 will become the tangent at the extremity of the diameter. 
 Hence the tangent at the extremity of a diameter is parallel 
 to the chords which the diameter bisects. 
 
 Since all chords are bisected in the centre all diameters must 
 pass through the centre, and every straight line through the 
 centre is a diameter. 
 
 Dep. The diameter parallel to a system of parallel chords 
 is said to be conjugate to the diameter which bisects those 
 chords. Some properties of conjugate diameters will be found 
 investigated in Chap, xvili. 
 
 241. As a particular case of Article 239 we may observe 
 that the diameters bisecting chords parallel to the lines of re- 
 ference are represented in trilinear coordinates by the equations 
 
 b dfi c dy' c dy a dn' a da, b d^' 
 and in triangular coordinates by the equations 
 
 d^ dy' df da' da~ d^' 
 
OF THE SECOND DEGREBi 241 
 
 Hence the centre, being the point of concurrence of dia- 
 meters is represented in trilinear coordinates by the equations 
 
 a da h d^~ c d'i'' 
 and in triangular coordinates by the equations 
 
 dx d$ &f' 
 
 Or, we may establish these equations more independently as 
 follows. 
 
 242. To find the centre of the conic whose equation is 
 /(c-',/3,7)=0. 
 
 Let («', y9', y') be the centre. Then since all chords through 
 the centre are bisected in the centre, the roots of the equation 
 
 /(a' + Xp, /3' + /tp, y' + vp)=0 
 
 must be equal and opposite whatever be the direction {X, fi, v). 
 
 „ ^df df df 
 
 Hence x^,+,.^+.^, 
 
 must be zero for all values of X, fi, v, subject to the relation 
 
 a\ + bfj, + cv=: 0, [trilinear 
 
 or \ + /I + V = 0. [triangular 
 
 Hence the centte is given by the equations 
 
 1 df^l df__l ^ 
 a da b d^' c dy ' 
 
 [trilinear 
 
 df df df u- 1 
 
 di^W^H- [tru^ngular 
 
 Comparing these equations with those of Art. 233, we observe 
 that the centre is the pole of the straight line at infinity (which 
 we might have inferred a, priori, from the fact that every diameter 
 is the chord of contact of tangents drawn from a point at 
 infinity). 
 
 W. 16 
 
242 
 
 THE GENERAL EQUATION 
 
 The equations can be expressed (as in the article referred 
 to) so as to give the coordinates explicitly. 
 
 Thus we shall have in trilinear coordinates, 
 
 /3' 
 
 -2A 
 
 W, V, u' 
 
 ~" 
 
 v, u', w 
 
 
 u, 
 
 w', 
 
 V 
 
 
 M, 
 
 w', v, a 
 
 v, u', to 
 
 
 u, w', v' 
 
 
 w', 
 
 V, 
 
 u' 
 
 
 W', 
 
 V, u , h 
 
 a, h, c 
 
 
 a, h, c 
 
 
 a, 
 
 h, 
 
 c 
 
 
 v, 
 
 u', w, c 
 
 
 
 
 
 a, 
 
 I, c, 
 
 and in triangular coordinates, 
 
 
 
 
 
 a' 
 
 
 ^' 
 
 
 
 1 
 
 7 
 
 
 
 
 -1 
 
 W, V, 11 
 
 
 v, u, w 
 
 
 M, 
 
 v=', 
 
 v' 
 
 
 M, 
 
 ' ' t 
 
 W, V, 1 
 
 v', u, w 
 
 
 u, w', v 
 
 
 w' 
 
 V, 
 
 u 
 
 
 r 
 
 w 
 
 V, u, 1 
 
 1, 1, 1 
 
 
 1, 1, 1 
 
 
 1, 
 
 1, 
 
 1 
 
 
 ■t 
 V, 
 
 1. 
 
 «', w, 1 
 1, 1, 
 
 giving the actual values of the coordinates of the centre. 
 
 243. To find the length of the semi-diameter drawn in any 
 given direction in the conic 
 
 /(a,/3,7)=0. 
 
 Let (a, /3, 7) be the centre of the conic, and let \, fi, v define 
 the given direction, so that 
 
 a — tt _ y3 — /3 _ 7 — 7 _ 
 
 are the equations to the diameter. 
 
 The lengths of the intercepts are given by the equation 
 f{a + \p, P + fip, y + vp) = 0. 
 
 But since (a, /3, 7) is the centre, the two values are equal 
 and of opposite sign, and therefore 
 
 /(a,^,7) 
 '' f{\, /*, p) ' 
 
 which gives the square on the semi-diameter required. 
 
OF THE SECOND DEGREE. 243 
 
 244. To find the conditions that the general equation of the 
 second degree should represent a circle. 
 
 Let the equation be written 
 
 then if (a, )8, 7) be the centre, the semi-diameter in direction 
 (\, fi, v) is given by the equation 
 
 If the conic be a circle all diameters are equal, and therefore 
 /(\, fi, v) ia constant. But it will be sufficient to express 
 that diameters in three different directions are equal, and for 
 simplicity we will select the directions of the three lines of 
 reference. 
 
 For the direction BG we have in trilinear coordinates 
 \ = 0, ^ = ± ain (7, v = + sin J?, 
 
 and symmetrical values for the directions of the other lines of 
 reference. 
 
 Hence we must have 
 /(O, sin C, - sxrB) =/(-ainC, 0, sin^) =/(sin J5, - sin^, 0), 
 
 or /(O, c, ^l)=f[-c, 0, a)=/(5, -a, 0). 
 
 Or, if the coordinates be triangular, the direction BC is 
 given by 
 
 and symmetrical values may be written down for the other direc- 
 tions, so that the conditions become 
 
 m 1. -1) /(- 1, 0, i) _ /(i. -1, 0) 
 
 16—2 
 
244 THE GENERAL EQUATION 
 
 Obs. If the equation be written in the form 
 
 Mi' + v/y + W7' + 2m'/37 + 2i''7;( + 2«;'a/3 = 0, 
 
 the conditions that it should represent a circle become 
 
 v(? + wW — 2u'bc = wa' + tw;' - 2v'ca = mJ" + wa" — 2tc'ab 
 
 in ti-ilinear coordinates, and in triangular coordinates they be- 
 come 
 
 v + w — '2u'_ w + u — ^v _ u+v~ Iw 
 a^ " &» ~ c' • 
 
 Or the conditions in trilinear coordinates will be given by 
 any two of the equations 
 
 V w 2m' w u 2v' u V 2 It"' 
 
 W^?~hE ?"^^~m ^"'"5"'""^ 
 
 a" ^9 7 
 
 M U V W V v w' u w w' v! v' 
 
 I _-j I . 
 
 _ fl* be ca ab b ' ca ab be _ <? ab he ca 
 he cos A ca cos B ah cos C ' 
 
 and in triangular coordinates by any two of the equations 
 
 v+ w — 2u' _w +u — 2v' _ M + « — 2w' 
 a' P ? 
 
 u + u —v —w' v + v — w — u' w + w' — u' — v' 
 
 be cos A ca cos B ab cos C 
 
 The student will not be surprised if by other methods he 
 obtains these conditions under other forms. In a case where 
 only one condition is arrived at the equation expressing that 
 condition must be identically the same however it be obtained, 
 but a system of two simultaneous equations can always be re- 
 placed by any two of the equations formed by combining them. 
 
 24.5. To find the condition that the general equation of the 
 second degree 
 
 ua? + v^+wf+ 2u'^y + 2v'yx + 2w'afi = 
 may represent two straight lines. 
 
UF THE SECUND DEGREE. 
 
 245 
 
 Let the equation be written 
 
 /(a,/S,7)=0, 
 and suppose it represents two straight lines so that 
 
 /{a, A 7) = (^a + »n/3 + ny) (fa + m'fi + n'7), 
 
 then -/ = I {I'a + m'/S + n'y) + r(h + m^ + ny). 
 
 So 28 - '" (^'* + '"''^ + "'"y) + "*' (^" + '»/3 + «7) > 
 
 and 
 
 df 
 dy 
 
 = n (I'a + m'/3 + n'y) + n [h + »ny8 + ny). 
 
 Hence (Art. 88) the equations 
 
 da "' rf/3 "' dy~^ 
 
 represent three straight lines passing through the point of inter- 
 section of the straight lines 
 
 la + mfi + Mfy = 0, 
 
 and I'a + m'yS + n'y = ; 
 
 that is, the three straight lines 
 
 £^=ua + w'0 + v'y=O, 
 
 dfi 
 
 = to'a + v^ + u'y = 0, 
 
 ■4- =v'a + m'/9 + m)7 = 0, 
 
 are concurrent; and therefore (Art. 26), 
 
 u, w', v i = 0, 
 w', V, u' 
 v, u', w 
 
 which will be the required condition. 
 
246 
 
 THE GENERAL EQUATION 
 
 246. To find the equation to the common chords of two amies 
 whose equations are given. 
 
 Let F{ai, /8, 7) a Ud^ + V^+ Wrf 
 
 + 2 Z7/37 + 2 F'7a + 2 W'afi = 0, 
 and / (a, /8, 7) = wa" -^-v^+vrf 
 
 + 2m'/S7 + 2t)'7a + 2M;'a/8 = 0, 
 be the two conies. 
 
 Any pair of common chords constitute a locus of the second 
 order passing through points of intersection of the two conies, 
 and must therefore be represented by an equation of the form 
 
 F{a, A 7) + «/(«. A 7) = (1), 
 
 where k must be so determined that this equation may satisfy 
 the condition of representing two straight lines. 
 
 That is, K must be determined by the equation (Art. 245) 
 
 = 0. 
 
 •(2), 
 
 U +«M, W'+KW', V'+icv' 
 W' + KW', V +KV, U' + KU' 
 
 V +KV', V +KU', W+KW 
 
 a cubic equation giving three values of k for the three pairs of 
 common chords (Art. 163). 
 
 ObS. The equation (2) may be written 
 U, W V 
 W, V, U' 
 
 r, u, w 
 
 ■\-K{u{yW- U") + v{WJJ- F'=) + w {UV- W") 
 + 2u'{rW-UU')+2v'{W'U'-VV') + 2w'{U'V'-WW')} 
 + /c' { U{vw - u") + V{wu - v'^) + W{uv- m") 
 
 + 2 tr {v'w - uu') + 2 F' (wV - W) + 2 IF' iuv' - ww) } 
 
 + «" 
 
 M, W , V 
 
 W, V, II 
 
 u, u, w 
 
 = 0. 
 
OF THE SECOND DEGREE. 
 
 247 
 
 247. Cob. 1. In the particular case when the first conic 
 consists of the two coincident straight lines 
 
 the equation for k reduces to 
 
 M, 
 
 w', v\ I 
 
 - 
 
 w', 
 
 V, u', m 
 
 
 v', 
 
 u', w, n 
 
 
 I, 
 
 m, n, 
 
 
 M, 
 
 w', v' 
 
 w', 
 
 V, u 
 
 V, 
 
 u, w 
 
 ic = 0. 
 
 Hence the tangents whose chord of contact is 
 
 la + m^ + wy = 0, 
 have the equation 
 
 u, w, v\ I 
 w, V, u', m 
 v', u, to, n 
 I, m, n, 
 
 /(«, ^. 7) + 
 
 u, 
 
 w. 
 
 »' 
 
 w', 
 
 ■0, 
 
 u 
 
 V, 
 
 u, 
 
 w 
 
 {la + m0 + nrif= 0. 
 
 CoE. 2. Since the asymptotes are the two tangents whose 
 chord of contact is at infinity they will he represented hy the 
 equation 
 
 «, 
 
 w, 
 
 V, 
 
 a 
 
 w, 
 
 V, 
 
 u, 
 
 b 
 
 t 
 
 t 
 
 u 
 
 w, 
 
 c 
 
 a, 
 
 b, 
 
 c, 
 
 
 
 /(«.A7) + 
 
 U, W, V 
 W, V, u 
 
 v, u, w 
 
 (aa + J/8 + C7)==0, 
 
 if the coordinates are trtlinear, and by the equation 
 
 u, w', v, 1 /(a, ^, 7) + 
 
 w, V, u', 1 
 
 v, u, w, 1 
 
 1, 1, 1, 
 
 if the coordinates are triangular. 
 
 u, w , V 
 w' , V, u 
 v', u', w 
 
 (a + /8 + 7r = 0, 
 
248 
 
 THE GENERAL EQUATION 
 
 M, 
 
 w', 
 
 ■0, 
 
 a 
 
 = 0, 
 
 or 
 
 «>', 
 
 «, 
 
 u', 
 
 h 
 
 
 
 V, 
 
 u', 
 
 w, 
 
 c 
 
 
 
 a, 
 
 b, 
 
 c, 
 
 
 
 
 
 Cob. 3. If the conic be a parabola the asymptotes lie alto- 
 gether at infinity : therefore in the equations of the last corollary 
 we must have 
 
 u, w', v', 1 =0, 
 
 w', V, u, 1 
 
 v', u, w, 1 
 
 1, 1, 1, 
 
 according as the coordinates are tnlinear or triangular. These 
 will therefore be the respective forms of the condition that the 
 equation sliould represent a parabola. 
 
 But we shall arrive at this result more directly in the 
 next article. 
 
 Cor. 4. The asymptotes will be at right angles to one 
 another (Art. 49, Cor. 2) provided 
 
 M + y + «j — 2m' cos ^ — 2t;' cos .B — 2w' cos (7=0, 
 
 when the coordinates are trilinear. 
 
 This is therefore the condition that the general equation of 
 the second degree should represent a rectangular hyperbola. 
 
 When the coordinates are triangular this condition becomes 
 
 wa" + vb* + wc" — iu'bc cos A — 2v'ca cos B — 2w'ab cos C = 0, 
 
 or a'{u + u' — v' — w'} + b'(v + v' — w — u) 
 
 + c'{w + w'-u'-v')=0. 
 
 248. To find the condition that the general equation of the 
 second degree 
 
 ua? + v^ + 1*7' + 2m'/37 + ^v'-ya. + 2zo'a/3 = 
 
 should represent a hyperbola, parabola, or ellipse. 
 
 I. Suppose tlie coordinates are trilinear. We shall find the 
 coordinates of the points where the locus meets infinity by 
 solving the given equation simultaneously with the equation 
 
 oa + J/S + C7 = 0. 
 
OF THE SECOND DEGREE. 
 
 249 
 
 Eliminating 7 we get 
 c^{uoi' + i;/3»+ 2M;'a/3) - 2c (aa + 1^) (v'a + m'/3) + w{a<x + f/3)' = 0, 
 or {wa" + uc' - 2v'ac) 0." + {vc' + wV - '2u'hc) 0' 
 
 + [w'<? + wdb — v'hc — u'ac) 2a/3 = 0, 
 a quadratic whose roots are unequal, equal, or imaginary, accord- 
 
 > 
 
 = 0. 
 < 
 
 {wdb + w'<? — v'hc — u'acY = (i»a' + uc- — 2v'ac) {vc' + wh^ — 2u'hc), 
 
 that is, according as 
 
 M, w , v , a 
 
 w, V, u', h 
 
 v, m', w, c 
 
 a, h, c, 
 
 Hence the given equation represents a hyperbola, parabola, 
 or ellipse, according as the determinant just written is positive, 
 zero, or negative. 
 
 II. Suppose the coordinates are triangular. The reason- 
 ing will be the same as in the other case, with unity substi- 
 tuted for a, b, c severally. Hence the equation will represent 
 a hyperbola, parabola, or ellipse, according as 
 
 u, w', v', 1 
 
 w', V, u', 1 
 
 v', u, w, 1 
 
 1, 1, 1, 
 is positive, zero, or negative. 
 
 249. We may arrange the proof of the last article in a 
 diflferent form as follows : 
 
 Let 
 
 a — a'_j8 — ;8'_7 — 7'_ 
 
 = P 
 
 be a straight line drawn from any point (a', 13', y) so as to meet 
 
250 
 
 THE GENERAL EQUATION 
 
 the conic at an infinite distance. Then since one of the inter- 
 cepts on this line is infinite, we must have 
 
 f(X,,i,v) = 0. 
 
 Therefore the conic will have one, two, or no directions in 
 which one of the radii from a finite point is infinite, according 
 as the equation 
 
 f{\, ii,v) = 
 
 gives real and equal, real and unequal, or unreal solutions. 
 
 Eliminating v by means of the relation 
 
 dX + bfL-\- cv = 0, \trilinear 
 
 we get 
 
 X' {wa^ + V.& - 2v'ca) + /a" {v(? + wV - 2m'Jc) 
 
 + 2X)tt {wdb + w'(? — u'ac —v'bc) — (1). 
 
 Hence the conic is a hyperbola, parabola, or ellipse, (Art. 
 228) according as 
 
 (loaS + w'c" — u'ac — v'bc) 
 
 > 
 
 = (joa' + uc^ — 2v'ca) (vo^ + wV — 2u'bc), 
 
 < 
 
 that is, according as 
 
 > 
 = 0, 
 
 < 
 
 M, w, V, a 
 
 w', V, w', b 
 
 v, u, to, c 
 
 a, b, c, 
 
 as in the previous article. 
 
 Obs. If the coordinates are triangular the equation (1) takes 
 the form 
 
 \' {w + u- 2 w') + fj? {v + w- 2m') 
 
 + 2\fjL {w + w -u'- v) = 0, 
 
OF THE SECOND DEGREE. 
 
 251 
 
 and the final condition becomes 
 
 u. 
 
 w, 
 
 ■0, 
 
 1 
 
 t 
 10, 
 
 V, 
 
 v!, 
 
 1 
 
 t 
 
 u. 
 
 w, 
 
 1 
 
 0. 
 
 1, 1, 1, 
 
 COE. By reference to Art. 242, we see that if the conic is a 
 parabola the centre is at infinity, and the diameters are therefore 
 parallel. Hence the following proposition arises, 
 
 250. To find the direction of the diameters of the parabola 
 ua^ + v^ + wi'+ 2m'/37 + 2v'rfa + 2M)'a/3 = 0. 
 
 Let (\, /t, v) be the required direction. The equations con- 
 necting \, fjL, V may be expressed in a variety of forms derivable 
 from one another in virtue of the relation among the coefficients 
 expressing the condition that the conic is a parabola. 
 
 But one of the most useful forms may be obtained as 
 follows : 
 
 One of the diameters is represented by the equation in tri- 
 linear coordinates 
 
 1 df^^l df^ 
 a da b d^' 
 
 ua + w'B + v'7 w'oL +vB + mW 
 
 or = T — , 
 
 a 
 
 Now \, fi, V are proportional to the coordinates of the point 
 where the diameter meets the straight line at infinity. Hence 
 we have 
 
 vX + w'/i + v'v w'\ + «/i + u'v 
 a " h ' 
 
 and o\ + 5/* + cv = 0, 
 
 whence eliminating v, we get 
 
 X {uic + u'a^ — v'ab — w'ac) = fi {vca + v'V — w'hc — u'ab), 
 and by symmetry = v {wab + w'<^ — u'ca — v'bc), 
 
 which determine the ratios \ : fi : v required. 
 
252 THE GENEKAL EQUATIUN OF THE SECOND DEGREE. 
 
 Obs. In triangular coordinates the result will become 
 ^[u + u' —v —w')= n,{v + v' — w — u) = v{w + w' — u' — v). 
 
 251. CoE. 1. The equations to the diameter of the parabola, 
 
 uo,^ + D/S" + w-f + 2u'^y + 2v'yjL + 2w'a0 = 0, 
 
 through the point (a, ^, 7') 
 
 a-a' /S-yS" 7-7' 
 are -^— = = , 
 
 where X, /j., v are the reciprocals of 
 
 ■ubc + ua? — v'ah — w'ac, vca + v'b" — w'bc — u'ah, 
 wab + V3& — u'ca — v'hc, 
 if the coordinates are trilinear, and the reciprocals of 
 
 u-\-ri —v —w', v + v' — w' — u, w + w —u — v, 
 if the coordinates are triangular. 
 
 CoE. 2. In the particular case of the parabola inscribed in 
 the triangle of reference, and represented by the equation 
 
 Po." + wi'/? + wV - 2OTW/37 - 2w?ya - 2lmafi = 0, 
 
 , I m n r •!• 
 
 where - + t- + - = 0, \trthnear 
 
 or l + m + n = 0, {triangular 
 
 the equations to the diameter through (a, /3', 7') reduce to 
 
 a-a'_^-fi' 7-7' 
 
 "7 m ~ ' [tnUnear 
 
 a' ¥ ? 
 
 a-oi' _ ^-ff 7-7' ^ . 
 
 or — ^ ^^ ~ . itriangular 
 
 252. It will be observed that everything in these chapters 
 applies equally whether the coordinates are trilinear or triangu- 
 lar, except when a restriction is specially made. 
 
EXERCISES ON CHAPTER XVII. 253 
 
 Exercises on Chapter XVII. 
 
 (147) When a conic breaks up into two right lines, the 
 polar of any point whatever passes through the intersection of 
 the right lines. 
 
 (148) A point moves so that the sum of the squares of its 
 distances from n given straight lines is constant. Shew that it 
 will describe a conic section. 
 
 (149) If all but one of the straight lines in the last exercise 
 be parallel, this one will be a diameter of the conic, and the con- 
 jugate diameter will be parallel to the other straight lines. 
 
 (150) If the straight lines in Exercise 148 consist of two 
 groups of parallel straight lines they will be parallel to a pair of 
 conjugate diameters in the conic. 
 
 (151) If two conies have double contact, any tangent to the 
 one is cut harmonically at its point of contact, the points where 
 it meets the other, and where it meets the chord of contact. 
 
 (152) A point moves along a fixed line ; find the locus of 
 the intersection of its polars with regard to two fixed conies. 
 
 (153) Given a self'conjugate triangle with regard to a 
 conic; if one chord of intersection with a fixed conic pass 
 through a fixed point, the other will envelope a conic. 
 
 (154) Shew that in order to form the equations of the lines 
 joining to (a', /3', 7') all the points of intersection of two curves, 
 ■we have only to substitute la. + ma.', Z/3 + m^', Ir/ + mr/ in both 
 equations, and eliminate I : m from the resulting equations. 
 
 (155) The polars of the two circular points at infinity with 
 respect to the conic f{a, /8, 7) = are represented in trilinear 
 coordinates by the equation 
 
254 EXERCISES ON CHAPTER XVII. 
 
 (156) The locus of a point which moves so as to be always 
 at a constant distance from its polar with respect to a conic is a 
 curve of the fourth order, having four asymptotes parallel two 
 and two to those of the conic, and cutting the conic in four 
 points lying on the polars of the circular points at infinity. 
 
 (157) A conic circumscribes the triangle ABC. Any conic 
 is described having double contact with this, and such that the 
 bisector of the angle C is the chord of contact. Prove that the 
 straight line in which this latter conic cuts OB and CA meets 
 ABma. fixed point. 
 
 (158) Straight lines are drawn through a fixed point ; shew 
 that the locus of the middle points of the portions of them inter- 
 cepted between two fixed straight lines is a hyperbola whose 
 asymptotes are parallel to those fixed lines. 
 
 (159) If a conic pass through the three points of reference, 
 and if one of its chords of intersection with a conic given by the 
 general equation be \a + /it/S + i^ = 0, the other will be 
 
 u V n i*> 
 
 -a+-/S + -7 = 0. 
 
 \ /A V 
 
 (160) A conic section touches the sides of a triangle ABC 
 in the points a,b,C; and the straight lines Aa, Bb, Oc intersect 
 the conic in a, V , c ; shew that the lines Aa, Bb, Cc pass 
 respectively through the intersections of Be and Cb', Ca' and 
 Ac, AV and Bd\ and the intersections of the lines ah and dV , 
 be and Vc , ca and cd, lie respectively in AB, BC, CA. 
 
 (161) If two triangles circumscribe a conic their angular 
 points lie in another conic. 
 
 (162) If with a given point as centre an ellipse can be 
 described so as to pass through the angular points of a triangle, 
 then with the same point as centre another ellipse can be de- 
 scribed so as to touch the sides of the triangle. 
 
CHAPTEE XVIII. 
 
 THE GENERAL EQUATION OF THE SECOND DEGREE CONTINUEp, 
 
 253. The determinant 
 
 u, 
 
 w', 
 
 t 
 V 
 
 w, 
 
 V, 
 
 u' 
 
 v', 
 
 u', 
 
 w 
 
 is called tlie Discriminant of the function 
 
 wa" + v^ + W7* + 2u'l3y + 2v'7a + 2w'a0, 
 and will be conveniently denoted by the letter H. 
 
 The minors of this determinant with respect to the terms 
 u, V, w, u, v', td will be represented by the letters U, V, W, 
 U', V, W respectively, so that 
 
 U= vw — u^, F= wu — w", W= uv — w'', 
 
 U' = i/w' — mm', V s w'u — vv, W = mV — WW. 
 
 254. If the function 
 
 mo" + v^ + vrf + 2m'/37 + Zw'va + 2M)'a/3 
 
 be denoted hyf{a, yS, y), it will be convenient and suggestive to 
 use F{a, yS, 7) to denote the function 
 
 Ua' +V0'+Wy'' + 2 U'^y + 2 V'yx + 2 Wa^. 
 Thus by Art. 222 we have 
 
256 THE GENERAL EQUATION OF THE 
 
 255. The determinant 
 
 w, w', V, f 
 w, V, u, g 
 v , u, w h 
 
 f. ff, K 
 
 (where /, g, h are the coefficients of a, /S, y in the identical rela- 
 tion /i +_^/3 + ^7= 1, connecting the coordinates of any point) is 
 called the bordered Discriminant of the fanction 
 
 Ma' + v^ + My" + 2m'/37 + 2v'ya + Sto'a/S, 
 
 and will be denoted by the letter K. 
 
 In triangular coordinates f, g, h are each unity, and we 
 have 
 
 K= u, w, v, 1 
 
 w', V, u, 1 
 
 v , u', w, 1 
 
 1, 1, 1, 
 
 or -K^Fil,!,!). 
 
 In trilinear coordinates /, g, h become 
 
 a b c 
 2A' 2A' 2A' 
 
 and we get 
 
 K= 
 
 4A* 
 
 u, w, V, a 
 w' , V, u, b 
 v, m', w, c 
 
 a, b, c, i 
 
 or 
 
 -K = 
 
 j^iF{a, b, c). 
 
SECOND DEQKEE CONTINUED. 
 
 257 
 
 256. The minors of the bordered discriminant with respect 
 to the terms /, g, h will be denoted by A, B, C, so that we 
 have in trilinear coordinates 
 
 A = 
 
 w\ 
 
 «. 
 
 u' 
 
 v', 
 
 w. 
 
 to 
 
 ffl, 
 
 h, 
 
 c 
 
 B = 
 
 ■0, u. 
 
 w 
 
 u, w , V 
 a, b, c 
 
 2A » "- 2A 
 
 and in triangular coordinates, 
 A = 
 
 w', 
 
 V, u' 
 
 , B = 
 
 t 
 V, 
 
 u, w 
 
 
 1, 
 
 1, 1 
 
 
 c= 
 
 u, to', 
 
 r 
 V 
 
 W, V, 
 
 u' 
 
 a, b, 
 
 c 
 
 
 u, w 
 
 . c= 
 
 u, 
 
 w, v' 
 
 
 1, 
 
 1, 1 
 
 
 2A 
 
 u, w, 
 w', V, 
 
 1, 1, 1 
 
 257. It should be observed that in trilinear coordinates K 
 has — 2 linear dimensions and A, B, C have each "- 1, while H 
 is of zero dimensions. 
 
 But in triangular coordinates all these functions are of zero 
 dimensions, giving a great advantage to the triangular system. 
 
 It will be seen that the expressions in triangular coordinates 
 throughout this chapter will be mostly derivable from the ex- 
 pressions in trilinear coordinates by writing 1 for 2A or ^ for A. 
 So, conversely, the expressions in trilinear coordinates may often 
 be derived &om the corresponding expressions in triangular coor- 
 dinates by multiplying each term by such a power of 2A as will 
 produce homogeneity. 
 
 258. The student can easily verify the following results, 
 which it is convenient to collect here for future reference^ 
 
 I. For the conic whose equation is 
 
 H~ Imn, 
 K—— {mn + nl+ Im), 
 
 we 
 
 have 
 
 and 
 
 r=- 
 
 a'mn + Vnl + <?lm 
 
 w. 
 
 [trtangular 
 \trilinear 
 17 
 
258 
 
 THE GENERAL EQUATION OF THE 
 
 II, For the conic whose equation is 
 Z/97 + mrfi. + ni/9 = 0, 
 
 we have 
 
 and 
 
 11= - Imn, 
 4 
 
 K =2 {P + ni' + n' — 2mn — 2nl — 2lm) , [trianffular 
 
 K= 
 
 aH'+b^iti' + C'li' - 2hcmn - 2canl - -lahhn 
 
 III. For the conic whose equation ia 
 
 VZa+v'^ + V«7 = 0, 
 
 we have 
 
 H= - 4P»iV, 
 
 iK= — ilmn {I + m + n), 
 K= — -rj- {Ihc + mca + ndb). 
 
 IV. For the conic whose equation is 
 a»+2A;/37 = 0, 
 
 and 
 
 we have 
 
 and 
 
 H=-ls', 
 iK^Te+2k, 
 
 \K = 
 
 [trilmear 
 
 \triangular 
 \trilinear 
 
 4A' 
 
 \tr{g,ngular 
 [trilinear 
 
 259. To erpress the equation of the straight line at infinity 
 in terms of the derived functions 
 
 d£ df df 
 da.' d^' d-i 
 when 
 
 fia, ^, 7) 5 Mi" +v^+ vyf + 2w'/37 + 2w'7a + iwa^. 
 
 ,df df , df . 
 be the equation required. 
 
SECOND DEGREE CONTINUED. 259 
 
 The first member may be written 
 
 {lu + mw'+ nv) 2a + {Iw + mv + nu) 2^8 + {W + jn«' + nw) 2% 
 which must be identical with 
 
 27i;(aa + &/3+C7), 
 if the coordinates be trilinear, where k is some constant. 
 
 We have, therefore, 
 
 lit + mw + nv = ak, 
 
 Iw + mv + nil = bk, 
 
 Iv + mu' + nw = ck, 
 
 , I m n 2Ak 
 
 whence A^B=0 = -H' 
 
 and therefore the equation to the straight line at Infinity 
 becomes 
 
 which holds equally whether the coordinates be trilinear or tri- 
 angular. 
 
 260. CoE. 1. The identical relation connecting the trili- 
 near coordinates of any finite point becomes 
 
 = 2^: 
 
 So also in triangular coordinates, 
 
 261. Cor. 2. The result of the last Cor. may be written 
 (by Art. 221, equation 11), 
 
 'L-^rB^-<w''^' 
 
 17-2 
 
260 
 
 THE GENERAL EQUATION OF THE 
 
 which being a relation among the coordinates of any point 
 irhatever must be identical with 
 
 or a + )9 + 7 = 1. 
 
 Hence in trilinear coordinates, 
 
 §£_aE 6£_bH_ 
 d^~ A ' dB~ A ' 
 
 and in triangular coordinates, 
 
 dC~ A ' 
 
 df_d£_^_„ 
 dA~dB~dG~ 
 
 262. Given 
 
 to easpress L, M, N explicitly in terms of I, m, n. 
 Snbstitnting their values for 
 
 ^ ^ ^ 
 da' d^' rfy' 
 
 and equating coefficients of a, /3, 7, we obtain 
 
 1=2 (uL +ii/M+v'N)-i 
 
 m = 2(w'L+ vM +«'^> 
 
 n = 2 {v'L + uM + wN)\ 
 
 and solving for L, M, N, we obtain 
 
 L=~{JJl + Wm + Vn) 
 
 M^—iWl+Vm + Un) 
 
 N = ~iV'l+U'm+Wn) 
 the required results. 
 
 [trilinear 
 [triangular 
 
 •(1), 
 
 •(2}. 
 
SECOND DEQBEE CONTINUED. 261 
 
 263. By reference to Art. 242 it will be seen that the 
 coordinates of the centre of the conic whose equation is 
 /(a. A 7) =0, are 
 
 _A _B C 
 
 K' K' K 
 
 whether the coordinates be trilinear or triangular. 
 
 Hence in trilinear coordinates, (Art. 261) 
 
 a da. h d(i c d^ KA.' 
 and in triangular coordinates 
 
 di d^ dy K ' 
 
 264. By Art. 245 the condition that the equation /(a, /8, 7)=0 
 should represent two straight lines is 
 
 21=0. 
 
 265. By Art. 247, Cor. 2, the asymptotes of the conic 
 represented by the general equation of the second degree 
 f{a, /8, 7) = are given by 
 
 /(a,i9,7)+f=0. 
 
 266. By Art. 247, Cor. 3, the condition that the equation 
 /(a, yS, 7) = should represent a parabola is 
 
 jr=o. 
 
 By Art. 248 the equation will represent an ellipse or a 
 hyperbola according as K is positive or negative. 
 
 267. We found in Art 247, Cor. 4, the condition that the 
 equation/ (a, /8, 7) =0 should represent a rectangular hyperbola. 
 
 We shall write this condition 
 
 ^=0, 
 
262 THE GENERAL EQUATION OF THE 
 
 SO that E represents tlie function 
 
 ■u + v + w — 2u' cos A — 2»' cos B — 2w' cos C 
 
 if the coordinates are trilinear, or 
 
 a* (u + u — v' — w') + 7'* {v + v' — w' — u) + c" (w + w —u— v) 
 
 2A 
 if they are triangular. 
 
 268. If {i, /3, 7) he the centre of the conic whose equation is 
 /(a, jQ, 7) = 0, then will 
 
 /(a, ^,7)=-^. 
 For by Art. 261, we have identically 
 
 da. rf/9 dy 
 And by Art. 263, 
 
 A = -Ki, B = -K^. C = -Ky. 
 Hence 
 
 and therefore (Art. 221, equation 3) 
 
 /(a,A7) = -:^- 
 
 COE. It follows that 
 
 f{A,B, C) = -HK. 
 
 269. To find the equation to the diameter throvgh a given 
 point. 
 
 Let (a', ff, 7) be the given point ; then since the diameter 
 joins tliis point to the centre of the conic its equation must 
 be (Art. 21) 
 
 !«, /8, 7 
 
 U', ^', i 
 
 ' A, B, C 
 
SECOND DEaREE CONTINUED. 
 
 26* 
 
 Cor. 1. The equation in terms of |J, %, j^ , will be 
 (Art. 262) * ^ 
 
 u, w, v 
 
 A, B, C 
 
 da. 
 
 + 
 
 W, V, U' 
 A, B, C 
 
 df 
 
 d^ 
 
 + 
 
 V, U', w 
 A, B, C 
 
 dy 
 
 Cor. 2. Tlie general equation to a diameter may be 
 written 
 
 a, /3, 
 A, B, C 
 
 = 0. 
 
 or 
 
 u, w, v 
 
 A, B, C 
 
 da 
 
 W, V, TJ'\d£^ 
 A, B, C 1"/^ 
 
 V, U\ W 
 A, B, C 
 
 dy 
 
 = 0. 
 
 270. T/ie polar of any point on a diameter is parallel to 
 the tangent at the extremity of the diameter. 
 
 For let \, /i, V be proportional to the coordinates of a point 
 on a diameter, then its polar is given by 
 
 ^ df df df ^ 
 
 •(1) 
 
 and the equation to the diameter can be written 
 
 a, ^, 7 j = 0. 
 \, /tt, V \ 
 A, B, C\ 
 
 Now let («', ^, y) be the coordinates of the extremity of the 
 diameter, then we have 
 
 «'. ^, 7 =0. 
 
 X, /A, V 
 
 A, B, C 
 
264 
 
 THE GENERAL EQUATION OF THE 
 
 wliicli expresses that the straight line (1) is parallel to the 
 straight line 
 
 which is the tangent at (a', 0, y). 
 Thus the proposition is established. 
 
 271. To find the condition that the diameter 
 
 = 
 
 a. /3, 7 
 A, B, C 
 
 \, (I, V 
 
 should he conjugate to the diameter 
 
 a, /3, 7 =0. 
 A, B, C 
 \, n, V 
 
 ' The first equation (1) may be written 
 
 df 
 
 •(1) 
 
 .(2). 
 
 u, w, r 
 
 A, B, C 
 
 da 
 
 W, V, V 
 A, B, C 
 
 dp 
 
 v. U, W 
 A, B, C 
 X, /*, V 
 
 dy 
 
 = (3), 
 
 and the second bisects chords parallel to the straight line 
 
 .(4). 
 
 ^.df ^ ,df ^ ,df ^ 
 
 '^5; + '* ^ + "5^ = ^ 
 
 If the diameters be conjugate these equations (3) and (4) 
 must represent parallel straight lines ; hence we must have 
 
 A, B, C 
 \, n, V 
 
 A, 
 
 X', 
 
 W, V, U' 
 A, B, C, 
 
 B. 
 
 r, w, w 
 
 A, B, C 
 
 c 
 
 I 
 V 
 
 = 0. 
 
SECOND DEGREE CONTINUED. 
 Or, if I, m, n denote the determinants 
 
 265 
 
 B, C 
 
 /I, V 
 
 C, A 
 
 V, \ 
 
 A, B 
 
 and I', m', n' the determinants 
 
 B, C 
 
 
 C, A 
 
 
 A, B 
 
 t f 
 
 > 
 
 v', X' 
 
 > 
 
 X', M 
 
 the condition takes the form 
 
 UIJ^+ Vmm'+Wnn 
 + XT (mri + m'n) + V {nT + n'T) + W{J.m' + I'm) = 0. 
 
 Cor. 1. From the symmetry of this result we infer that if 
 one diameter he conjugate to a second, the second is also conju- 
 gate to the first. 
 
 Cor. 2. The equations 
 
 la + Mi)3 + W7 = 0, 
 fa + jre'/3 + m'7=0, 
 
 will represent a pair of conjugate diameters of the conic 
 /(a,^,7)=0, 
 
 provided 
 
 Al-\-Bm+ Cn = 0, 
 
 Al' + Bm'+Cn' = 0, 
 
 UU + Vmm + Wnn 
 
 + U {mn + m'n) + V {nV + n'l) + W (?m' + Im) = 0. 
 
 272. To find the equation to the diameter parallel to the 
 tangent at a given point. 
 
 Let (a, ^', 7') be the given point, (a, /3. 7) the centre, and 
 (a, /9, 7) any point on the diameter whose equation is required. 
 
266 THE GENERAL EQUATION OF THE 
 
 The tangent at (a, yS', 7') is given hy 
 df r,df df ^ 
 
 and the two points {x, /8, 7), {j., /3, 7) are equidistant from this 
 tangent, tlierefore 
 
 df ^df df - df -^ df - df 
 
 IJT 
 = _ (Arts. 261 and 263), 
 
 which can be rendered homogeneous as in Art. 10. 
 
 273. To establish equations determining the foci of the conic 
 
 /UA7) = 0. 
 
 [Def. Tlie foci are a pair of points, equidistant on opposite 
 sides of the centre, such tliat the rectangle contained by the per- 
 ])endiculars i'rom them on any tangent is constant.] 
 
 Let (x,, /3,, 7,), and (2,, /3j, 7,) be the foci, and let (a, /3', 7') 
 be any point on the conic. 
 
 The tangent at this point is 
 
 df ^ df df „ 
 
 and the rectangle contained by the perpendiculars upon it from 
 («„ A.7,). («,, /3», 7,) is, (Art. 46) 
 
 / df ^. df ^ df\f df . df df\ 
 
 
 (df-^ dl df]^ 
 
 But by definition this is equal to a constant area (A*, sup- 
 pose), hence we obtain 
 
SECOND DEGREE CONTINUED. 267 
 
 + (7,«, + 7A + 2A'cos5)J^, ^ 
 
 + (7^/3, + a,5, + 2i' cos C) ^ ^^. = 0. 
 
 Now til is IS a relation between the coordinates of any point 
 whatever on the conic, and must therefore (accents suppressed) 
 Ije the equation to the conic, and identical with the given equa- 
 tion which may be written (Art. 222) 
 
 Hence we have 
 
 ^A-^-' _ A^,-^-' _ 7.7.-F 
 U l^ IV 
 
 ^ y3i7» + ^i7 i + 2^''co3J . ^ 7,g,+ 7a«.+ 2^'cos^ 
 
 2 U' a y 
 2ir ^^'' 
 
 five equations, which with the two relations 
 aa.+ J^, + C7,= 2A) 
 aa,+ Z'/3,+ C7,= 2Aj ^^^' 
 
 determine the seven quantities a,, /S,, y^, a,, ;8j,, 7^ and k. 
 Each of the equal fractions in (1) is equal to 
 
 (m, + 5j8, + cy.) (aa, + &/3 , + cy,) 
 Ua' + F6' + TFc- +2U'bc + 2V'ca + 2 W 'ab 
 
 = i^/^i , = - 4 (Art. 255.) 
 J^(a, b, c) K ^ ' 
 
268 THE GENERAL EQUATION OP THE 
 
 Hence the system of equations (1) may be written 
 
 — 2 cos A — 2 cos B 
 
 -2cosC ^ '' 
 
 But instead of using all these equations which are somewhat 
 
 complicated we may combine some of them with the simpler 
 
 relations 
 
 2^ o . o 25 ^ 20 
 
 Oj + Ofj = - -g. , ^,+/if, = - -gr, 7,+7, = - -^ 
 
 expressing the fact that the centre bisects the line joining 
 the foci. (Art. 18. Cor.) 
 
 Thus we have 
 
 - A\a, = 2Aa, + Ka^, - ITyS./S, = 25/3. + K^^, 
 -Ky,y,= 2Cy,+ Ky,' 
 
 so that the equations (3) give 
 
 Ka,' + 2Aa, - Z7= JT/S.' +25/3, - F= Ky,' + 2Cy,-W 
 
 which with the identical relation connecting the coordinates 
 of any point will be sufficient to determine the coordinates 
 
 («„ /3i. 7i)- 
 
 There will generally be two imaginary solutions as well as 
 the two real ones, indicating two imaginary points having the 
 property enunciated of the two real ones. 
 
 CoE. If the conic be a parabola, K= and the equations 
 reduce to 
 
 2^a, - U= 25^, - V= 2CV, - W. 
 
 Obs. The equations in the form in which we have written 
 them hold equally whether the coordinates be trilinear or 
 
SECOND DEGEKE CONTINUED. 
 
 269 
 
 triangular. If we write K, A, B, O, at lengtli, as functions of 
 U, V, W, &c. the equations will take the form 
 
 a* ( Ua" + FJ' + ITc' + 2 U'bc + 2V'ca + 2 W'ah) 
 
 -4Aa{JlF' + cr' + a!7)+4A»D" 
 = /3" ( Ua'+ Vb'+ Wc''+ 2 U'bc + 2 F'co + 2 Wab) 
 
 -4A/S(cCr+aTr'+6F) + 4A''F 
 = '/{Ua'+ F6'+ Wc'+ 2U'bc + 2V'ca + 2 W'ah) 
 
 -iAy{aV'+bU'+cW) + iAW 
 
 for trilinear coordinates, and for triangular coordinates the form 
 
 a'{U+V+W+2U'+2V'+2W')-2{V+W'+Zr)a+U 
 
 _ ^{u+v+w+2U'+2V'+2'fr)-2{w"+u'+v)0+v 
 
 - 
 
 7'(t7-+F+TF+2£r'+2F'+2TF')-2(tr+F'+TF)y+TF 
 
 274. Definitions. A point on a conic at which the tangent 
 is at right angles to the diameter is called a vertex, and the 
 diameter through a vertex is called an axis. 
 
 275. To find equations to determine the vertices of the conic 
 whose equation is 
 
 /(a,/8,7)=0. 
 
 Let (o'j /8', 7') be the yertex ; then the tangent 
 
 
 is at right angles to the diameter 
 
 a, A 7 
 a. P. 7 
 A, B, C 
 
 = 0. 
 
27Q THE GENEHAL EQUATION OP THE 
 
 Hence if the coordinates be trilinear (Art. 49, Cor. 1). 
 
 df_ 
 
 A, 
 
 /S', 
 B, 
 
 7 
 G 
 
 1, —cos G, —coaB 
 df 
 
 If 
 
 «', /3', 7 
 A, B, G 
 — cos C, 1, — cos A 
 
 + 
 
 dy 
 
 «', ^', 7 
 
 — cosB, —cos J, 1 
 
 = 0. 
 
 or if they be triangular, 
 
 
 J, B, 
 
 y 
 
 G 
 
 
 a, — & cos C, 
 
 — c cos -4 
 
 
 
 r 
 
 7 
 
 G 
 
 
 — a cos C, h, 
 
 — c cos ^ 
 
 + c 
 
 
 a, /3, 7 
 
 A, B. G 
 
 — acoaB, —hcoaA, c 
 
 = 0. 
 
 (1). 
 
 ■0). 
 
 This equation, together with the relation 
 /(a',/3',7') = 0, 
 
 will be sufficient to determine the ratios of the coordinates. 
 And since each equation is of the second order there will be four 
 solutions indicating four vertices. 
 
 276. To find the equation to the axes of the conic. 
 
 The equation (1) of the last article is a relation among the 
 coordinates a', )8', y' of any vertex of the conic. 
 
SECOND DEGREE CONTINUED. 271 
 
 Therefore if we suppress the accents it will represent a 
 locus of the second order passing through the four vertices. 
 But it is satisfied also at the point 
 
 A B a' 
 
 that is, at the centre. Hence it will represent a locus of the 
 second order passing througli the four vertices and the centre. 
 But througli tliese five points there can be only one locus of tiio 
 second order (Art. 147), and the two axes constitute such a locus. 
 Hence the equation will represent the axes. 
 
 277. The equation to the axes may be directly obtained 
 in another form which is sometimes useful, by the following 
 method, which depends upon the property tliat a conic is sym- 
 metrical with respect to an axis, and therefore the two tangents 
 from any point on an axis are equal in length. 
 
 Suppose the coordinates trilinear. 
 
 Let \, fi, V, \', fi, V be the direction sines of the two tan- 
 gents drawn from a point (i, /3', 7') to the conic. 
 
 The length of the first tangent is givcTi by the equation 
 
 /(a' + X/3, ^' + fip, 7' + vp) = 0, 
 
 and the roots of this equation must be equal, therefore 
 
 Similarly, 
 
 But if («', /8', 7') be any point on either axis of the conic the 
 two tangents are equal, and therefore 
 
 /(\, /x, r) =/(X', /i', .'') (1). 
 
 consequently 
 
 
 ). 
 
272 TU£ GENEBAL EQUATION OF THE 
 
 Also by the identical relations which exist among the direc- 
 tion sines of any straight line (Art. 70), 
 
 (aX + i/* + ci/)' = (a\'+V + c«'T (3)' 
 
 /[4'+ 1/'+ a^vcos^ = fji'+v'^ + ifi'vcosA (4), 
 
 i^ + X''+2«/\cos£=i'"+\'"+2y'\'co3 5 (5), 
 
 \'+^»+2X/*co3(7=\" + /*"+2XVco3a (6). 
 
 Eliminating the six quantities 
 X' - X^ fi' - fi", v" - v'^, (IV - fiv, vK - i/'X', X/i - X>' 
 
 from these six equations, and suppressing the accents on the 
 coordinates, we obtain 
 
 r^V /^' /^' df_^ dfdf iidf_ 
 
 \da) ' U/3/ ' \dy) ' d^ dy' dy da' da rf/3 
 
 «', 
 
 h\ 
 
 c\ 
 
 be, 
 
 ca, 
 
 ab 
 
 «. 
 
 V, 
 
 to, 
 
 «', 
 
 V, 
 
 w' 
 
 0, 
 
 1, 
 
 1, 
 
 cos^. 
 
 0. 
 
 
 
 1, 
 
 0, 
 
 1, 
 
 0, 
 
 cos£, 
 
 
 
 1, 
 
 1, 
 
 0. 
 
 0, 
 
 0, 
 
 cos C 
 
 = 0, 
 
 a relation of the second order between the coordinates of any 
 point on either axis, and therefore the equation to the axes, the 
 coordinates being trilinear. 
 
 278. Def, Two conies are said to be similar and similarly 
 situated when the lengths of any two parallel diameters are to 
 one another in a constant ratio. 
 
 279. The conies represented by the equations 
 
 /{a, fi, y) + Qa + mfi + ny) {aa + b^ + cy) = 0, 
 are similar and similarly situated. 
 
SECOND DEGREE CONTINUED. 273 
 
 Let (a, /S', 7'), (a", /3", 7") be the two centres, then (Art. 213) 
 the squares on the semi-diameters in direction \, /a, v are 
 
 / («', ^, 7') ^^^ /(a",r,7")+2A(?a" + >»;3"+'»7") 
 
 since a\ + J/x. + ci/ = 0. 
 
 Hence these semi-diameters are to one another in the ratio 
 f /(«'. ^'. 7 ) 
 
 l/(«", ^", 7") + 2A (Za" + m^' + «7")| ' 
 
 which is independent of \, /i, v, and therefore constant for all 
 directions. 
 
 Hence if o- = be the straight line at infinity, the equations 
 
 /(a, A 7) = 0, 
 
 and /(a, /3, 7) +o-(Za + »n^ + W7) =0, 
 
 represent similar and similarly situated conies. 
 
 Cor. Two similar and similarly situated conies meet the 
 line at infinity in the same points. Hence their asymptotes are 
 parallel. 
 
 280. Def. Two concentric conies are said to be conjugate 
 ■when the polars with respect to them of any point are parallel, 
 and equidistant from the common centre. 
 
 281. If fi?-, /3, 7) = and f{a., A 7) =0 ie the equations 
 to a pair of conjugate conies, and if H, W le their discriminants 
 and K, K' their bordered discriminants, then will 
 
 f/(a,/3,7)+§/'(«.A7) = -2. 
 
 For let (a, /3, 7) be the common centre, and («', ^, 7) any 
 other point. 
 
 w. 18 
 
274 THE GENERAL EQUATION OF THE 
 
 The polars of this point are 
 
 df ^ df df „ 
 
 Now if these be parallel and equidistant from the centre 
 they will cut off equal and opposite intercepts from any straight 
 line drawn from the centre. Such a straight line is repre- 
 sented by 
 
 g— a_ i8-;3 _ 7 — 7 _ 
 and the intercepts are given by 
 
 and (« + Xp)^^+(^ + /.p)^ + (7 + z^p)^ = 0. 
 
 But by Art. 260 we Lave 
 
 -df -^df -df E -df ^df -df H' 
 
 in virtue of which, the condition that the intercepts should be 
 equal and opposite in sign becomes 
 
 K/^ df ^ df ^ df\ , K'/^ df , df , df\ „ 
 
 which must be true for all values of \, /i, v subject to the 
 relation 
 
 dX + hii + cv= 0, \triUnear 
 
 or \ + /* + v = 0. \triaTigular 
 
 But this relation will be satisfied if X, /t, v be proportional to 
 a'-a, ^-t, y'-y. 
 
or 
 
 SECOND DEGEEE CONTINUED, 275 
 
 Hence we obtain 
 
 4'{(=''-«)|'+(^'-^)f'-(v'-7)|;}=o, 
 
 (Art. 260) J/(a', ^', i) + ^,f («', ^', 7') = - 2. 
 
 But (a', /y, 7') is any point whatever; hence, suppressing the 
 accents, 
 
 282. Cor. 1. If /(«, /9, 7) = be the equation to any 
 conic, its conjugate is represented by the equation 
 
 /(a, /3, 7) + -^ (a + ^ + 7)"= 0. {triangular 
 
 Cor. 2. The squares on the semi-diameters of the two 
 curves in direction (X, /x, v) are 
 
 -f^^A^ and J±±2lli 
 which are equal and of opposite sign, since 
 
 /(a. A i)=-K' 
 
 Hence the diameters are in the ratio v— 1:1. 
 
 Therefore a conic and its conjugaie are similar , similarly 
 situated, and concentric conies whose linear dimjensions are as 
 'J~l : 1. 
 
 It follows that the conjugate of an ellipse is wholly imagi- 
 nary. In the hyperbola, any central radius which meets the 
 curve in real points meets the conjugate in imaginary points and 
 vice versa. 
 
 18-2 
 
276 TaE GENERAL EQUATION OP THE 
 
 283. To find the maximum or minimum value of/{\, /i, v) 
 where X, /*, v are subject to the relations {Art. 71), 
 
 X°a cos ^ + y[t°& cos J5 + v'c cos C = asiii5sm C. (1), 
 
 and \a + fib + vc = ^ (2), 
 
 the coordinates heing trilinear. 
 
 Let ^ be any maximum or minimum value of /(X, /i, v). 
 Ef^uating to zero the differential of the given function, we have 
 
 l»''+|«''4^-'' (*>■ 
 
 and from (1) and (2) 
 
 XSX . a cos ^ + /LiS/t . J cos 5 + i/Sv . c cos (7=0 (4), 
 
 oSX + ZiS^ + c8j/ = (5), 
 
 Multiplying (4) and (5) by undetermined constants 2h, 2lc 
 and adding to (3), and equating the coefficients of the differen- 
 tials to zero, we obtain 
 
 ;^ + 2iXacos^ + 24'a = (6), 
 
 ^+2kfibcosB+ 2k'b = (7), 
 
 -^+ 2kvccos C+ 2A'c = (8). 
 
 Multiplying these by X, fi, v respectively and adding, we get 
 in virtue of (1) and (2), and by Art. 221, 
 
 ^ + iasin^sin C=0, or '<== — Ta^ W- 
 
 But the equations (6), (7), (8) may be written 
 (m + ha cos ^) X + w'lL + vv + lea = 0, 
 tc'X + (» + M cos 5) /u+ uv + yb = 0, 
 v\ + ufi + (w + X;c cos B)v-\- Ic'c = 0, 
 
SECOND DEGREE CONTINUED. 
 
 which equations, together with the relation 
 
 aX + h/jL + cv = 0, 
 give upon the elimination oi \ : /i : v : k', 
 u + ka cos a, w', v', 
 
 w, v + kh cos B u' 
 
 v, u', w + kccoa G, 
 
 a, b, c, 
 
 a quadratic to determine k. 
 
 This quadratic may be written 
 
 i^^K-Ikbck - lA'i' = 0, 
 which by the substitution of (9) gives us 
 
 16A*ir 
 
 277 
 
 = 0, 
 
 f^^-E<^ = 
 
 aWe' 
 
 a quadratic equation, one of whose roots will be tlie maxlmu'-n 
 and the other the minimum value of/(\, fi, v). 
 
 284. To find the maximum or minimum value off{\, /j,, v) 
 where \, fi, v are subject to the relations 
 
 T^hccosA + fi^ca cos B+ v^ab cos (7=1 (I), 
 
 and X + /i + v = (2), 
 
 the coordinates being triangular. 
 
 Let ^ be a maximum or minimum value of/(\, fi, v). 
 
 Equating to zero the differential of the given function, we 
 
 have 
 
 ak dfj, dv 
 
 .(3), 
 
 and from (1) and (2), 
 
 \S\. 5c cos ^ + fJiZiJu.ca cos B+vhv.cJ> cos (7=0.. ..(4), 
 S\ + S/* + Si/ =0....(5). 
 
278 
 
 THE GENERAL EQUATION OP THE 
 
 Multiplying (4) and (5) by undetermined constants 2Jc, 2k', 
 and adding to (3), and equating the coefficients of the differentials 
 to zero, we obtain 
 
 ^ + 2k\hc cos A + 2k' = 
 
 + ihiica cos 5 + 24' = 
 
 if+2h>ahcoaC+2k' = 0. 
 av 
 
 .(6), 
 •(7), 
 .(8). 
 
 Multiplying these by \, fo, v respectively and adding, we 
 get in virtue of (1) and (2), 
 
 ^ + A; = (9). 
 
 So that the equations (6), (7), (8) may be written 
 (m — </)Jc cos -4) \ + w'/i + v'v-\-Tc = 0, 
 w'\ + (« — <^a cos B)fi+ u'v + k' = 0, 
 v'\ + u'fi + (w — ^ab cos G)v+k' = 0, 
 which equations, together with the relation 
 
 X + /i + v=0, 
 give upon the elimination oi\ : /i -.v. k', 
 
 u — ^bc cos A, w', v', 1 =0, 
 
 w', V — jtca cos B, u', 1 
 
 v', u', w — ^xjh cos 0, 1 
 
 1. 1, 1, 
 
 a quadratic to determine 0. 
 
 This quadratic may be written 
 
 4Ay-2A£<|. = ir, 
 
 one of the roots of which will be the maximum, and the other 
 the minimum value of/(\, fi, v). 
 
SECOND DEGREE CONTINUED. 279 
 
 286. To find the lengths of the axes of the conic whose equa- 
 tion is 
 
 /(a. /8, 7) = Ma" +v^+wrf+ 2u'^ + 2v'ya + 2«;'a/3 = 0. 
 
 Let (a, A 7) be the coordinates of the centre. Then the 
 length of the semi-diameter in the direction (\, ft, v) is given by 
 
 . ■/•(«. ^, 7) 
 P - /(X, ft, V) ' 
 
 and this must be a maximum or minimum. 
 
 Hence f(\, fi, v) must be a minimum or maximum, and 
 must therefore have one of the values of ^ given by the qua- 
 dratic of Art. 283 for trilinear coordinates, and Art. 284 for tri- 
 angular coordinates. 
 
 Therefore 
 
 or 
 
 „« 
 
 n 
 
 «, /3, 7) 
 
 p ■ 
 
 
 «/> 
 
 = 
 
 'k^ ' 
 
 ;Art. 268), 
 
 4>- 
 
 H 
 -Kp^' 
 
 
 Now suppose the coordinates are 
 
 trilinear, or triangular, 
 
 so that 
 
 16A* I i:<f>_K 
 
 Then, substituting the value of <f> given above. 
 
280 THE GENERAL EQUATION OF THE 
 
 a quadratic equation, giving two values for p', expressing the 
 squares on the two semi-axes. 
 
 Hence, if ^ and 33 denote these two semi-axes, we have 
 
 and \ 
 
 ^+^=-16A^Z^- ^+33--2A-^. 
 
 286. COE, Considering the quadratic in p°, we observe that 
 the two values of p" will be of opposite sign, and therefore one 
 value of + p real and the other imaginary if K> 0, — the condi- 
 tion that the conic should be a hyperbola. 
 
 The two values of p' will be of the same sign, and there- 
 fore hoth values of + p real or hoih imaginary if K<0, — the con- 
 dition that the conic should be an ellipse. 
 
 And further, the ellipse will be real when E and H arc of 
 opposite sign, and imaginary when they are of the same sign. 
 
 Again, both the values of p", and therefore both values of 
 + p will be infinite if K= 0, — ^the condition that the conic should 
 be a parabola. 
 
 Similarly, if J^= 0, the values of p" are equal and of oppo- 
 site sign, and the conic is a rectangular hyperbola, as we saw in 
 Art. 247, Cor. 4. 
 
 So, if M= 0, the two values of p° are zero, and the conic 
 degenerates into two straight lines, as we proved in Art. 245. 
 
 287. To find the length of the latus rectum in a parabola 
 represented by the equation 
 
 uo? + r/? + rc^" + 2m'/37 + 2w'7a + 2w'a^ = 0. 
 
SECOND DEGREE CONTINUED. 281 
 
 In the general case, we have hy the preceding articles, 
 16A* K"' !^+5W--JA-^, 
 
 and 
 
 (2A)' ^ . »* *> \.^^) _g-. • 
 
 Therefore by division, I 
 
 \MV^\M)~ (2A)*^i' jW"^Vg(V~ (2A)4lfi' 
 
 In the case of the parabola (when Jr=0), one of the frac- 
 tions 
 
 g^a ana ^5 , 
 
 is the reciprocal of the semi-latus rectum, and the other is zero. 
 Hence if 31 be the semi-latus rectum of the parabola, we obtain 
 
 -(2Ayff I ^,_2Ag 
 
 288. To find the area of an ellipse whose equation is given. 
 
 Let 
 
 ua' + vj^ +iiy/ + 2u'^y + 2vyct + 2m' a/3 = 
 
 represent an ellipse whose semi-axes are ^ and 33. 
 
 Then the area = tt^IS 
 
 TTobcH 
 
 [trilinear 
 
 or = ■; -r, • \triangular 
 
 2irAH 
 
 (-A')3 
 
 289. To find the equation to the greatest ellipse which can 
 he inscribed in the triangle of reference. 
 
 Using triangular coordinates, let the equation to the ellipse 
 
 be _ _ 
 
 v7a + ^/m^ + 'Jny = 0. 
 
282 THE GENERAL EQUATION OF THE 
 
 Ihe area = ; — — , ; 
 
 therefore (area)" cc -=3 . 
 
 But ir= - 4f mV, 
 
 K= —Almn {l+m + n). 
 
 Therefore (area)''oc 77 rj. 
 
 ^ ' [l + m + n) 
 
 But 75 r^ will have either a maximum or minimum 
 
 (l + m + n)' 
 
 value (by symmetry) when l = m = n, and minima values occur 
 when 
 
 Z = or «i = or w = 0, 
 
 leading to the inference that l = m=n produces a maximum. 
 Hence the equation of the maximum ellipse will be 
 
 or o''+/S''+7'-2/87-27a-2a/3=0. 
 
 Obs. In trilinear coordinates the equation will be 
 'Jaoi + VJi§'+ Vc7 = 0, 
 or aV + &';8' + cV - 2Jc/37 - ica^ix - 2a5a/3 = 0. 
 
 COE. Similar reasoning shews that the least ellipse circum- 
 scribing the triangle of reference is represented in triangular 
 coordinates by the equation 
 
 /37 + 7a + a/S = 0, 
 
 and in trilinear coordinates by the equation 
 
 a c 
 
 290. If H, H' be the discriminants, K, K' the bordered 
 discriminants of the equations to two conies, and ^ = 0, E' =0 
 the conditions that the equations should represent rectangular 
 hyperbolas: then 
 
SECOND DEGREE CONTINUED. 283 
 
 (I) If the conies he similar, K, K' will be in the duplicate 
 ratio of E, E', and 
 
 (II) If the conies he also similarly situated, we shall have 
 K=K' and E = E', and the linear dimensions of the two conies 
 will he in the subduplicate ratio ofH,: H'. 
 
 Suppose the conies are similar, and that the linear dimen- 
 sions of the first are to those of the second as 1 : a;. 
 
 Then since the sum of the squares on the axes must be in 
 the duplicate ratio of the linear dimensions, therefore (Art. 285) 
 
 zri ■ ^'t — i • •*- V-"-/' 
 
 But the rectangle contained by the axes must also be in the 
 duplicate ratio of the linear dimensions, and therefore (Art. 285) 
 
 4:^=1:«'' (2). 
 
 Comparing (1) and (2), we obtain 
 E__E^ 
 
 or K:K' = E':E" (3). 
 
 Next, suppose that the conies are not only similar but also 
 similarly situated, then the squares on parallel diameters must 
 be in the duplicate ratio of the linear dimensions ; therefore we 
 have, (in virtue of Arts. 279, and 268), 
 
 §■■§-'■■" »• 
 
 Hence comparing (2) and (4), we have 
 
 K=K' (5), 
 
 and therefore in virtue of (3), 
 
 E=E' (6), 
 
284 THE GENEEAL EQUATION OP THE SECOND DEGREE, 
 and from (2), 
 
 or H^:H'-' = -[ : x ; (7). 
 
 Thus the propositions are proved. 
 
 291. Cor. If H and H! he of opposite signs the ratio of 
 the linear dimensions is an imaginary quantity, and therefore in 
 whatever direction diameters can be drawn, terminated in real 
 points in one of the conies, parallel diameters in the other conic 
 will meet the conic in imaginary points, and vice versa. An 
 instance is aiforded in a pair of conjugate hyperbolas. 
 
 292. We will conclude this chapter by observing that tlie 
 equation to any two straight lines satisfies the condition of 
 representing a conic section. 
 
 If the two straight lines are imaginary but intersect in a real 
 point, the equation may be regarded as representing an indefi- 
 nitely small ellipse, and the two imaginary straight lines will be 
 the ultimate form of the imaginary branches of the ellipse 
 which have now coincided with the asymptotes. 
 
 If the straight lines be real they may be regarded as the 
 limiting case of the hyperbola, when the imaginary part con- 
 necting the two real branches has become evanescent. 
 
 We may state the result as follows : 
 
 If the real part of a conic degenerates into a point, the ima- 
 ginary part will become two straight liries coincident with the 
 asymptotes, and if the imaginary part becom.es evanescent, the 
 real part will become two straight lines coincident with the 
 asymptotes. 
 
 These cases of conic sections will be evidently obtained by 
 cutting a cone by a plane passing through the vertex, the first 
 case occurring when the plane passes between the two sheets of 
 the cone, the second when it intersects the sheets. 
 
EXERCISES ON CHAPTER XVIII. 
 
 285 
 
 (163) 
 
 (164) 
 
 Exercises on Chapter XVIII. 
 With the notation of Art. 254, shew that 
 
 /(|.|.D-^(«.A,). 
 
 If /(«, A 7) =0 he the equation to a conic in trili- 
 near coordinates, the polars with respect to it of the points of 
 reference will form a triangle whose area is 
 
 1 ahcH'' 
 8 ABC6? ' 
 
 Verify the result in the particular case of the conic repre- 
 sented by 
 
 _ (165) If (flj, A, 7,), («„ ^„ 7^), (a„ ^3, 73) be any three 
 points, their polars with respect to the conic /(a, /3, 7) = form 
 a triangle whose area is 
 
 ahcIT 
 
 8A' 
 
 «i. A' 7, 
 
 «3' A. 7a 
 
 A, B, 
 
 
 A, B, G 
 
 
 ^, B, C 
 
 "2. /S2. 72 
 
 
 aa> A. 73 
 
 
 «.. A. 7i 
 
 «a> A. 73 
 
 
 "!» /5., 7l 
 
 
 "2. /32> 72 
 
 (166) Each of a series of parallel chords in a conic is 
 divided so that the rectangle under the segments is constant. 
 Shew that the points of section lie upon a similar, similarly 
 situated and concentric conic. 
 
 (167) Each of a series of parallel chords in a conic is 
 divided so that the algebraical sum of the reciprocals of the seg- 
 ments is constant. Shew that the points of section lie upon a 
 similar and similarly situated conic which cuts the original 
 conic at the extremities of the diameter bisecting the parallel 
 chords. 
 
 (168) A straight line is drawn from the focus of a conic 
 to meet the tangent at a constant angle ; find the locus of the 
 
286 EXERCISES ON CHAPTER XVIII. 
 
 point of intersection, and shew that, in the case of the parabola, 
 the locus will always touch it, but in the case of the other two 
 curves it will only touch them (in one or two points) under cer- 
 tain conditions. 
 
 (169) If A', B', C, K' be the values of the functions 
 A, B, C, K (Arts. 255, 256) for the equation 
 
 /(«, /S, 7) + 2 (aa +h^ + cri) (?a + w/S + J17) = 0, 
 
 shew that 
 
 A—A'=l {c'v + Vw — 2hcu') — m {dbw — acu — hcv + cW) 
 
 — n {acv — abu' + 5V — hew), 
 
 with similar expressions for B — B' and C — C, the coordinates 
 being trilinear. Hence deduce K = K', 
 
 (170) With the notation of the last exercise, shew that 
 I iA-A')+m{B-B')+n{C- C) =F, 
 
 where 
 
 
 h, c 
 
 c, a j , \ a, b 
 n, I \ \ I, m 
 
 Hence the result of the last exercise may be written 
 
 A-A'='/4, B-B' = lf:, C-C'='-f^. 
 2 dl ' 2 dm' 2 dn 
 
 (171) If H' be the discriminant of the equation 
 /(a, /3, 7) + 2 {la. + m/3 + ivy) {aa + J/3 + 07) = 0, 
 
 then 
 
 S'-H=l{A + A') + m{B + B') + n{C+C'). 
 
 (172) If be the angle between the asymptotes of the 
 conic /{a, /9, 7) = 0, then 
 
 . .^ 8A^ . , . „ . ^ 
 sm ff = —^ sm A sm B sm C, [trilinear 
 
 01" sm'^ = -p2 • [triangular 
 
CHAPTER XIX. 
 
 CIRCLES. 
 
 293. To find the equation in trilinear coordinates to a circle 
 whose radius and centre are given. 
 
 Let r be the given radius and a, /3', 7' the coordinates of the 
 given centre. 
 
 And let a denote the expression 
 
 aa + b^+ cy 
 2A 
 
 so that ff = is the equation to the straight line at infinity. 
 
 Let (a, /3, 7) be any point on the circle : then since r is the 
 distance between the points (a, ^, 7) and (a', y3', 7') we have 
 (Art. 72), 
 
 (a - a')' sin 2^ + (/3 - 0y sin 25 + (7 - 7')' sin 2 C 
 
 = 2r' sin A sin B sin C, 
 
 an equation which may be written in the homogeneous form 
 
 a' sin 2A + ^ sin 25 + 7* sin 2 (7 
 
 — 2a- (aa' sin 2 A + 00 sin 2B+ 77' sin 2 C) 
 
 + <r''(a'=sin2^+/S'sin25+7"sin2C-2r'sin^sin5sin C) =0, 
 
 a relation among the coordinates of any point («', /8', 7') on the 
 circle and therefore the equation to the circle. 
 
288 CIRCLES. 
 
 Obs. In triangular coordinates, if a denote a + ^ + 7 the 
 equation to the circle at a distance r from the centre (a, ^, y) 
 will become 
 
 a" cot^ + ^cotB + y^ cot C 
 
 - 2a (aa' cot A + ;S/3' cot B+yy' cot G) 
 
 + o^ (a." cot A + 13" cot B + y" cot C- ^) = <>• 
 
 294. If -we render the equation in trilinear coordinates, of 
 the last article, homogeneous with respect to the coordinates 
 (a, yS', 7') of the centre, it may be written 
 
 a» {^" + y" + 2/3'7' cos A-r^ sin' A) 
 +. ^ (7" + a" + 27 a' cos B-r^ sin' B) 
 + 7' (a" + ;8" + 2a'/3' cos C - r" sin' C) 
 + 2/S7{(a"-Osin-Bsin C - (/3' + a cos C) (7 + 0' cos 5)} 
 + 27a{(/S''-r') sin sin ^ - (7 4-/8' cos -4) (a' +/3' cos C)} 
 + 2o;8{(7''-0 sin^sin^- (a' + y cos^) (/3'+7'cos^)} =0. 
 
 295. Refen-ing to the equations of Art. 293 it will be 
 observed that the radius r is only introduced in the coefficient of 
 cr'. Hence it follows that 
 
 If = be the equation to any circle, any concentric circle 
 ill have the equation + ka^ = 0, where k is some constant. 
 
 296. Again, referring to the same equation, the tei-ms which 
 are independent of <t are free from a', )8', 7' and r, they will 
 therefore be the same for all circles. Hence 
 
 If O—^be the eqtiation to any circle, any other circle will be 
 represented by the equation O + ua- = 0, where u = is the equa- 
 tion to some straight line. 
 
 This is equivalent to saying that any two circles have a com- 
 mon chord at infinity, or meet the line at infinity in the same 
 two points, a property which is exhibited more explicitly in 
 the next article. 
 
CIRCLES. 289 
 
 297. Every circle passes through the two imaginary points 
 called the circular points at infinity. (See the definition 
 Art. 109.) 
 
 For any circle is represented (Art. 293) by the equation 
 
 o' sin 2A+^ sin 2B + <fain2G 
 
 - 2a {oLT.' sin 2 A + 0ff sin 2^+77' sin 2 C) 
 
 + 0-'(a"'sin2^+/3"'sin25+7"'sin2C-2r'sin^sin5sin C) =0. 
 
 And substituting <r = 0, we find the points of intersection 
 with the line at infinity to be determined by 
 
 o" sin 2^ + /S" sin 25 + y sin 2 a= ; 
 
 and we shewed in Art. 108 that this equation determined on the 
 straight line at infinity the same two points as any equation of 
 the system 
 
 fi' + r' + 20ycosA = O, 
 7° + a" + 27a cos 5=0, 
 
 a» + /3='+2ay3cos (7=0, 
 
 the loci of which intersect in the two circular points; which 
 proves the proposition. 
 
 298. Conversely, every conic which passes through the two 
 circular points is a circle. 
 
 For if not, take any three points P, Q, R on the conic, and 
 describe a circle about the triangle PQR : then the conic and 
 the circle intersect in the points P, Q, R as well as in the two 
 circular points, i.e. they intersect in five points, which is impos- 
 sible since two loci of the second order can only intersect in four 
 real or imaginary points (Art. 162). Hence no conic but a circle 
 can pass through the two circular points. 
 
 299. If = 0, 0' = Q he thb equations to two circles, then 
 will kO+k'0' = also represent a circle. 
 
 For by Art 168 the last equation represents a locus passing 
 through all the points of intersection of the first two. But, 
 by Art. 297, the loci of the first two equations intersect in the 
 
 w. 19 
 
290 CIRCLES. 
 
 circular points, therefore the locus of the third passes through 
 these two points, and therefore the locus is a circle (Art. 298). 
 
 300. Since the straight line at infinity meets a circle in the 
 two circular points, it cannot meet it in any other point. Hence 
 the other two points of intersection of any two circles are not at 
 infinity. They may be real or imaginary, but must be finite. 
 
 Def. The straight line joining the two finite points of in- 
 tersection of two circles is called the Radical axis of the two 
 circles. 
 
 301. Any two real circles, whether they intersect in, real or 
 imaginary points, have a real radical axis. 
 
 For let = be the equation to one of the circles. Then 
 the other may be represented by the equation 
 
 + MO- = 0, 
 where m = is the equation to a real straight line (Art. 296). 
 
 But (Art. 165) m = 0, <7 = are a pair of common chords of 
 the two circles, and we know that the two points of intersection 
 at infinity lie on the locus of o- = 0, therefore m = joins the two 
 finite points, or m = is the radical axis. 
 
 Hence any two circles have a real radical axis. 
 
 COK. We see at once (from symmetry or otherwise) that 
 the radical axis of two circles is at right angles to the straight 
 line joining the centres. 
 
 302. The three radical axes of three circles are concurrent. 
 For if the three circles be represented by the equations 
 
 + MO- = 0, 
 
 + wo- = 0, 
 their radical axes will have the equations 
 
 »— M> = 0, w — M = 0, M— u = 0, 
 and therefore all pass through the point given by 
 
 u = v = w. 
 
CIRCLES. 291 
 
 303. If a system of circles have a common radical axis, 
 ihepolars with respect to them of any ficed point are concurrent. 
 
 Let = be the equation to any circle of the system, and 
 let M = be the equation to the common radical axis. 
 
 Then, by giving different values to h, every circle in the sys- 
 tem will be represented by the equation 
 
 O + ^M<r = (1). 
 
 Now let (a', /9', 7') be the fixed point, and let O, u', and a' 
 denote what 0, u, and <t become when a', ^, 7' are substituted 
 for o, A 7. 
 
 Then the polar of the point («', yS', 7') with respect to the 
 circle (1) will be represented by 
 
 dO ndO' dO . I , , I \ „ /„v 
 
 «^ + ^^+^^+*(""+""^=<' (')' 
 
 and it will therefore pass through the point given by 
 
 ^f^\a^O' da „ ,„. 
 
 «57 + ^^+T'^=« (^) 
 
 and Ma-' + M'o-=iO (4), 
 
 whatever be the value of h. 
 
 Hence the polars of the point («', , 7') with respect to all 
 the conies are concurrent. 
 
 304. For convenience of reference we will now recapitulate 
 the results arrived at in the foregoing chapters respecting some 
 particular cases of circles (see Chapters xiv. xv. xvi.). 
 
 I. The circle with respect to which the triangle of reference is 
 self-conjugate is represented (Art. 179) by the equation in trili- 
 near coordinates 
 
 a" sin 2A + /3' sin 25 + 7" sin 2 (7= 0, 
 
 which is obviously a particular case of the equation of Art. 293. 
 
 The centre of this circle is given by 
 
 a cos ^ = /8 cos .B = 7 cos C, 
 
 19—2 
 
292 CIRCLES. 
 
 and is the point of intersection of perpendiculars from the angu- 
 lar points of the triangle of reference on the opposite sides. 
 
 In triangular coordinates the same circle is represented by 
 the equation 
 
 a' cotA + ^ cotB + 'Z' cot (7 = 0, 
 
 and its centre is given by 
 
 a cot ^ = )8 cot 5 = 7 cot C. 
 
 II. The circle passing through the three points of reference 
 is given (Art. 195) by the equation in trilinear coordinates 
 
 a/37 + ^7^ + cayS = 0, 
 
 and its centre is given by the equations 
 
 a = ^ = 7 
 QosA cosB cos C 
 
 The same circle is represented in triangular coordiiaates by 
 the equation 
 
 a';87 + JV + c'ai8 = 0, 
 
 and its centre is given by 
 
 . « ^ /3 7 
 
 sin 2 A sin 2B sin 2 O ' 
 
 III. The circle inscribed in the triangle of reference is repre- 
 sented (Art. 214) by the equation in trilinear coordinates 
 
 I- A ra. B I- C 
 Va cos — + V/S cos — + V7 cos — = 0, 
 
 and its centre is given by 
 
 a = ;S = 7. 
 
 In triangular coordinates the same circle is represented by 
 the equation 
 
 Y a cot- + sj fi cot - + ^7 cot - =0, 
 and the centre is given by 
 
 - = ^ = 2 
 a h c' 
 
CIRCLES. 293 
 
 IV. The circles escribed to the triangle of reference (i, e. 
 touching one side of the triangle and the other sides produced) 
 are represented (Art. 214) by the equations 
 
 V- a cos — + ^/^ sm— + V7 sin- = 0, 
 
 •J a sm — + n/- y3 cos — + V 7 sm -j = 0, 
 
 ^ . A ,^ . B I . G ^ 
 
 V o sm — + v/3 sm — + V— 7 sm — = 0, 
 
 ^ A A 
 
 and their centres are given respectively by 
 
 -a= ^= 7, 
 a=_/3= 7, 
 a= /3=-7. 
 
 The same circles are represented in triangular coordinates by 
 the equations 
 
 ^ - a cot - + y /3 tan - + y 7 tan - = 0, 
 sj a. tan - + y' - /8 cot - + /y/7 tan -= 0, 
 
 y' a tan - + y ^ tan - + y - 7 cot - = 0, 
 and their centres are given respectively by 
 
 a_ 
 a 
 
 /3 7 
 b" c 
 
 a. 
 
 /3 7 
 
 a 
 
 b~ c 
 
 a_ 
 a 
 
 y3 7 
 
 b c 
 
294 CIKCLES. 
 
 305. It should be observed that 
 a* sin 2^ + ;S* sin 2jP + 7* sin 2 C 
 
 + 2(/37siii-4 + 7a sin 5+ a/3 sin 0) 
 
 = 2 (a sin ^ + /8 sin 5 + 7 sin C) (a cos -4 +/8 cos 5 + 7 cos C). 
 
 Hence the straight line represented in trilinear coordinates 
 hj the equation 
 
 acos^ + ;8cosjBH-7Cos (7=0, 
 
 is the radical axis of the circle circumscribing the triangle of 
 reference, and the circle with respect to which that triangle is 
 self-conjugate. 
 
 306. To find the equation to the circle which passes through 
 the middle points of the sides of the triangle of reference. 
 
 Let the equation be (Art. 293) 
 
 a' sin 2^ + /S* sin 2^ + y sin 2 C 
 
 = (?a + jw/8 + nrf) (a sin .4 + yS sin 5 + 7 sin C). 
 
 Since the circle passes through the middle point of BC it 
 must be satisfied bj 
 
 a = 0, /SsinjB=7sin C, 
 hence we get 
 
 cot5+cotC = -^+- " 
 
 sin B sin £7 ' 
 or sin .4 = 7w sin C+n, sin B. 
 
 Similarly, since the circle bisects CA and AB, 
 sin 5= M sin .4 + I sin C, 
 sin C= Z sin J5 + w sin .4, 
 these equations give 
 
 1 = cobA, m = cosB, « = cosC. 
 Hence the required equation is 
 a' sin 2^ + yS* sin 25 + 7* sin 2 (7 
 = (a cos.4 + /8 cos £+7 cos C){a BinA+ fi sin 5+ 7 sin C)....{1), 
 
CIBCLES. 295 
 
 or, in virtue of the identity of Art. 305, 
 a* sin 2^ + /S" sin 25 + 7" sin 2 (7 
 
 = 2 ((87 sin -4 + 7a sin j5 + 0)8 sin (7) (2), 
 
 or again, by comparison of (1) and (2), 
 2 (fiy sin ^ + 7a sin 5 + a^ sin C) 
 ^ (acos^ + /8cos5 + 7 cos C)(asin^+ /8sin5 + 7sin (7). ..(3). 
 
 All these three forms of the equation are useful. The second 
 shews that the circle passes through the points of intersection of 
 the circumscribed circle, and the circle with respect to which 
 the triangle of reference is self-conjugate: the first and third 
 shew that its radical axis with respect to either of these circles 
 is the straight line 
 
 acos^+j8cos5+7Cos (7=0. 
 
 307. To find where the circle which bisects the sides of the 
 triangle of reference meets those sides again. 
 
 The equation to the circle is 
 
 a' sin 2^ +/? sin 25 + y sin 2 (7 
 
 — 2^7 sin A - 27a sin 5 — 2a/3 sin (7 = 0. 
 
 Putting a = 0, we get 
 
 /? sin 25 + 7* sin 2 (7 - 2/37 sin ^ = 0, 
 or 
 
 /?" sin 5 cos 5 + 7* sin CcoaC — ^y (sin 5cos C + sin (7 cos 5) = 0, 
 or (y3 sia 5 — 7 sin (7) (/3 cos 5 — 7 cos B) = 0, 
 
 shewing that the circle meets BC in the two points deter- 
 miaed by 
 
 a = 0, /3 sin 5 = 7 sin (7, 
 
 and a = 0, /3 cos 5= 7 cos C, 
 
 that is (Arts. 14, 15), in the middle point of BG, and in the foot 
 ©f the perpendicular from A. And similarly for the other sides. 
 
296 
 
 CIRCLES. 
 
 Hence the circle which bisects the sides of the triangle of reference 
 passes also through the feet of the perpendiculars from the oppo- 
 site angles. 
 
 308. But this circle has other significant properties with 
 respect to the triangle. 
 
 Thus, let a:, B', C (fig. 37) be the middle points of the sides 
 of the triangle ABC, and let AA", BE', CC" be the perpendi- 
 culars from the opposite angles, and their point of intersection. 
 Bisect OA, OB, OG in a, b, c respectively. 
 
 Fig. 37. 
 
 Then, since A", B', C" are the feet of the perpendiculars 
 from the angular points on the sides of the triangle OBC, there- 
 fore by the last article the circle through A!', B' , C" will bisect 
 the sides of the triangle OBC, and therefore will pass through 
 b and c. Similarly it will pass through a. Hence the same 
 circle which passes through A', B, C, A", B', C" passes also 
 through a, b, c. In consequence of all these nine points lying on 
 
CIRCLES. 297 
 
 its circumference, this circle is called the nine-points' circle 
 of the triangle ABC. 
 
 309. It is easily seen that the property last proved is only 
 a particular case of the following more general theorem. 
 
 Any straight line drawn from 0, the point of intersection of 
 the perpendiculars of the triangle ABC to meet the circumference 
 of the circumscribed circle is bisected by the nine-points^ circle. 
 
 310. Any circle may be represented (Art. 296) by the 
 equation 
 
 ay97 + frya + caj8 + (Za + m/3 + np/) (oa + J/9 + cy) = 0...(1). 
 
 Hence if 
 
 Ma'+ v^+w'/ + 2u'^y + 2«'7a + 2w'a/S= (2), 
 
 be the equation to a circle, this equation must be identical 
 with 
 
 k (a^7 + by% + ca/3) + (^ + ^ + *^j (aa + 6y8 + C7) = 0...(3). 
 
 311. From the result stated in the last article we may 
 readily deduce the conditions (already established by a diflferent 
 method in Art. 244), that the general equation of the second 
 degree should represent a circle. 
 
 For, comparing the equations (2) and (3), and equating 
 coefficients, we get 
 
 „ , , vc wb 
 
 2m =A;a+-j-H , 
 
 b c 
 
 „ , ,, wa uc 
 2v =Tcb + — + — , 
 c a 
 
 „ , , ub va 
 
 a b 
 
 or 
 
 — kabc = vc^ + wV — 2u'bc = wa' + mc' — 2v'ca = uP + va^ — 2u'ab, 
 
 the same conditions as we found in the article referred to. 
 
298 CIBCLES. 
 
 312. The equation in trilinear coordinates to the circle 
 inscribed in the triangle of reference may be written 
 
 a' cos* j+^eos*^ + '/ cos* j - 2^7 cos' | cos= -^-&c. = 0. 
 
 Hence (Art 310) it must take the form 
 
 k (a/3y + frya + ca^ 
 
 + f-cos*^ + fcos*f + !?^cos*-^)(aa + 5/3 + C7)=0, 
 \a 2 6 2 c a / 
 
 where (by Art. 311), 
 
 7? /^ no 
 
 - dbch = c" cos* "2 + *' cos* -^ + 2hc cos* - cos* j 
 
 = (ccos''- + 6cos'2-l 
 
 s% 
 
 s denoting as in trigonometry 5 (a + J + c) ; therefore the equa- 
 
 tion to the circle becomes 
 
 aySy + hyoi + ca/8 
 
 a&c/a iA B 4-^,7 4 <7\ , , 7-3 1 ~\ 
 = ^U*^°' 2 +f "°' 2 +r°' 2-j («« + */3 + C7) 
 
 = ^ La (s - a)" + JyS (s - J)' + C7 (s - c)'J (aa + JiS + C7) . 
 
 By a similar method we may shew that the equations to the 
 escribed circle opposite to A may be written 
 
 a/87 + 57a + ca/3 
 
 ale fa. ^A 8 . tB ^ r^ . tC\ , , ,0 , » 
 = (F3^4a*=°' 2-+jSm- + isin*-J(aa + J^ + c7), 
 
 or a/37 + 67a + ca^ 
 
 = ^ W +h^{s- cy + C7 (s - J) j (oa + S^S + 07), 
 
CIBCLES. 299 
 
 and by symmetry the other two escribed circles may be repre- 
 sented by the equations 
 
 o/Sy + 57a + ca/3 
 = ^ jaa (s - c)" + J/3s' +cy{s- a)4 {act + &/3 + cy), 
 and 0^7 + Jya + caj8 
 
 = ^ jaa (s - hy +b^{s- ay + oys'j (aa + J/3 + 07). 
 
 313. Cor. The radical axes of the circumscribed circle 
 and the several inscribed (or escribed) circles are represented in 
 trilinear coordinates by the equations 
 
 aa{s-ay + b^{s-iy+cy{s-cy = 0, 
 
 aas'-i- b^ {s-cy+ ffy {s-by = 0, 
 
 aa (s - c)' + 5/3/ + C7 (s - a)' = 0, 
 
 aa{s-by+b^{s-ay + cys' = 0, 
 
 and therefore in triangular coordinates by the equations 
 
 a{s-ay+fi{s-by + yis-cy='0, 
 
 as^ + ^{s-cy + y{s-by=0, 
 
 a(s-c)' + /3s' + 7(s-o)' = 0, 
 
 a {s -by + ^{s -ay + ys' = 0. 
 
 314. The equation to the nine-points' circle may be writ- 
 ten (Art. 306) 
 
 0/87 + frya+ca/S 
 
 = - (a cos ^ + /3 cos 5 + 7 cos C) (aa + 5jS + 07), 
 
 and the equation to the inscribed circle is 
 ay37 + by a, + caff 
 
 = ^La{s-ay+bff{s-hy + cy{s-cy\{aa + hff + cy), 
 
300 CIRCLES, 
 
 therefore their radical axis is represented by 
 
 ajcos^-- i^l +^|cos5- -L_^|+y|cos C ^^^-j 
 
 = 0, 
 or 
 
 ^(c-a)(«-6)+£(a-6)(J-c) + J(J-c)(c-a)=0, 
 
 ^ + ^^+-^ = (1). 
 
 5_c c—a a—o 
 
 Similarly the radical axes of the nine-points' circle with 
 each of the several escribed circles will be seen to have the 
 equations 
 
 -£L + _M-_^ = o (2) 
 
 0—c c+a a+o ^ 
 
 -vVc^^L^^r' (^)' 
 
 - _^ + ^ = (4). 
 
 315. It will be observed that the equation (1) of the last 
 article satisfies tlie condition (Art. 210) of tangency to the in- 
 scribed circle. 
 
 Hence the radical axis of the nine-points' circle and the 
 inscribed circle is a tangent to the latter, and therefore the 
 circles touch one another, and the radical axis is the tangent at 
 the point of contact. 
 
 So the equations (2), (3), (4) of the last article will be seen 
 to represent tangents to the escribed circles; hence the nine- 
 points' circle touches these circles also. 
 
 That is, the nine-points' circle of any triangle touches all the 
 four circles which touch the sides of the triangle, or the sides pro- 
 duced. 
 
CIRCLES. 301 
 
 316. It should be observed that the trilinear equation 
 
 (oa + 6/3 + oyf = 
 satisfies all the conditions of representing a circle. 
 
 Its radius is infinite, and the equations to give the centre 
 reduce to 
 
 0' 
 
 shewing that the centre is indeterminate. 
 
 It represents therefore the circle spoken of in Art. 38, 
 being the limiting form of a circle drawn from any centre at a 
 distance which is indefinitely increased. 
 
 Every point at infinity lies upon the locus, which indeed is 
 geometrically identical with the straight line at infinity as ex- 
 plained in Art. 38, but analytically it must rather be regarded 
 as equivalent to two straight lines, both lying altogether at 
 infinity. 
 
 We shall have much more to say about this circle in the 
 chapters on tangential coordinates and polar reciprocals. 
 
 317. The equation 
 
 (aa + 5/3 + C7) (Za + TO/3 + W7) = (1), 
 
 also satisfies the conditions of representing a circle. 
 To explain this, consider any straight line 
 
 Za + 9n/3 + W7 = (2), 
 
 and suppose a circle is described from any centre so as to 
 touch this straight line. 
 
 If the distance of the centre from the straight line be in- 
 creased, the circumference in contact with the straight line 
 becomes less curved and tends to coincide with the straight line ; 
 and by increasing the radius indefinitely the circumference will 
 coincide as nearly as we please with the straight line. Hence 
 ultimately the part of the circle remaining in finite space coin- 
 cides with the straight line, while there is still another part, at 
 
302 CIRCLES. 
 
 an infinite distance, which since its curvature is indefinitely 
 small will ultimately coincide with the straight line at infinity. 
 
 Hence the equation (1) which we know represents the 
 straight line (2) and the straight line at infinity, represents the 
 ultimate form of an infinite circle : — ^which accounts for its satis- 
 fying the circular criteria. 
 
 318. Before we leave the subject of circles, we ought to 
 observe as a particular case of Art. 292 that the equation to the 
 straight lines joining any real point to the circular points at 
 infinity satisfies all the conditions of representing a circle. 
 
 To take the very simplest case in ordinary Cartesian coordi- 
 nates, the equation ai'+y'=0 may be said to represent two 
 imaginary straight lines through the origin, or to represent an 
 indefinitely small circle at the origin. 
 
 So in trilinear coordinates the equation 
 
 /3' + 7" + 2/37 cos ^ = 0, 
 
 represents the two imaginary straight lines joining the point of 
 reference A to the circular points at infinity, but it is also the 
 equation to the indefinitely small circle round A. 
 
 Perhaps this circumstance is best represented by the follow- 
 ing statement : 
 
 WJien the radivs of a circle is indefinitely decreased the real 
 part of the circle degenerates into a point, and the imaginary 
 branches into two straight lines joining that point to the two cir- 
 cular points at infinity. 
 
 319. To find the equation to an indefinitely small circle at 
 the point {a!, /3', 7'). 
 
 The equations to the straight lines joining this point to the 
 circular points are by Arts. 21 and 110, 
 
 -1, cos C7+>/^lsin f7, cos 5 + ,/- 1 sin 5 
 «. A 7 
 
 a', 0, 7' 
 
 = 0, 
 
CIRCXES. 
 
 303 
 
 or if A, B, C denote respectively the determinants 
 
 A 7 
 
 ) 
 
 7, a 
 
 1 
 
 OL, /9 1 
 
 y3', 7' 
 
 
 7. a' 
 
 
 «', /y 
 
 -A+BcoaC+ OcosB±J^l{Bsm C- OsmB) = 0, 
 and therefore the two straight lines are given hy 
 
 {A-B cos C- Ccos 5)' + (5sin C-Gain By = 0, 
 or {^, 5, CY=0. 
 
 But by the last article these two imaginary straight lines 
 coincide with the indefinitely small circle at (a', y8', y"), therefore 
 that circle is given by the equation 
 
 7> a . a, /3 
 
 7, a' a', /3' 
 
 (I /3'. i I 
 
 }■-■ 
 
 320. CoE. 1. A circle such that (a, y8', 7') is the pole of 
 the straight line 
 
 ?(a + a') + wi(/3 + /3') + w(7 + 7')=0, 
 is represented by the equation 
 
 n /3, 7 
 II ^', 7 
 
 7, a 
 7, a 
 
 
 + fc(fa + w^ + W7) (aa + &/8 + C7) = 0. 
 
 321. CoE. 2. A particular case of considerable importance 
 is that of the indefinitely small circle at the intersection of the 
 perpendiculars of the triangle of reference. 
 
 This point is given by the equations 
 
 a cos ^ = /8 cos i? = 7 cos (7, 
 
 and the equation to the circle reduces to 
 
 {aacos^, b^cosB, 07 cos Cy=0, 
 
 or in triangular coordinates 
 
 {a cos A, ^ cos B, 7 cos G}' = 0. 
 
304 EXKKCISE3 ON CHAPTEE XIX. 
 
 322. Cor. 3. The equation to any circle may be -written 
 
 {aa cos A, S/3 cos £, cy cos 'CY= (la + m^+ ny) (aa. + bl3 + cy) 
 
 in trillnear coordinates, or 
 
 {a cos A, /8 cos B, 7 cos CY= (lot. + j»/3 + W7) (a +fi + 7) 
 
 in triangular coordinates, where 
 
 Z(a + o')+jn(/e + /3')+M(7 + 7')=0 
 
 represents the polar of the point of intersection of the perpendi- 
 culars from the angular points of the triangle of reference on the 
 opposite sides. 
 
 Exercises on Chapter XIX. 
 
 (173) The equation in triangular coordinates to the circle 
 whose centre is at the point of reference A and whose radius ia 
 r may be written 
 
 (/3 + 7) (c^^ + Fy) - «'/37 = r» (a + /8 + 7)^ 
 or {a^^y + yya. + <?a^) - {a + ^ + y){c'fi + b'y) +r'{a+0+yy=O. 
 
 (174) The circles described on the sides of the triangle of 
 reference as diameters are represented in trilinear coordinates 
 by the equations 
 
 /37=a(acos^ — /3cos5— 7C0S C), 
 7a = /3(/3cos£ — 7COS 0—acosA), 
 0/3 = 7 (7 cos C— a COS ^ — ;S cos E) . 
 Deduce Euclid, iii. 31. 
 
 (175) The polar of the centre of one circle with respect to 
 another circle is parallel to the radical axis of the two circles. 
 
 (176) The distance of a point from its polar with respect 
 to a circle is double of its distance from the radical axis of that 
 circle and an indefinitely small circle described about the point 
 as centre. 
 
EXERCISES ON CHAPTER XIX. 305 
 
 (177) A circle is described through the centres of three 
 other circles: shew that the radical axes of the new circle with 
 each of the others will be concurrent provided 
 
 r^ sin 2^j + r,' sin 25, + r," sin 20, = 4A, 
 
 where »•,, r,. r, are the radii of the original circles, and 6^,6^, 6^ 
 the angles which each pair of centres subtend at the remaining 
 centre, and A the area of the triangle formed bj the centres. 
 
 (178) The equation of the circle which passes through the 
 three centres of the escribed circles of the triangle of reference is 
 
 aa'' + J/S" + C7» + (a + J + c) (/Sv + 7a + a/9) = 0. 
 
 (179) The equations to the three circles which pass through 
 the centre of the circle inscribed in the triangle of reference and 
 through the centres of two of the escribed circles are 
 
 5/S' + C7'-aa''+(5 + c-a)(^7-7a-«/S)=0, 
 07* + oa" - J/S" + (c + a - 6) (7a - 0/3 - /37) = 0, 
 oa" + iiS' - C7» + (a + J - c) (a/3 - /37 - 7a) = 0, 
 
 the coordinates being trilinear. 
 
 (180) Circles are drawn through the angular points of a 
 triangle, through the centres of the three escribed circles, and 
 through the centres of the inscribed and each two of the escribed 
 circles; shew that the radical axes of these circles will meet the 
 sides of the triangle at the points where they are cut by the 
 straight lines bisecting the angles. 
 
 (181) Find the equations of the circles whose diameters are 
 (i) the perpendiculars drawn from the angles of a triangle to the 
 opposite sides, (ii) the lines bisecting the angles and terminated 
 by the opposite sides, and (iii) the lines joining the angles with 
 the middle points of the opposite sides. 
 
 (182) Find the locus of a point such that, if parallels be 
 drawn through it to the three sides of a triangle, the sum of the 
 rectangles under the three pairs of intercepts on each line respec- 
 
 W. 20 
 
306 EXEECISES ON CHAPTER XIX. 
 
 tively, between the point and the two sides which it meets, shall 
 be equal to a given rectangle. 
 
 (183) From a point in the plane of a given triangle draw 
 perpendiculars to its sides, and find the locus of this point, when 
 the area of the triangle formed by joining the feet of the perpen- 
 diculars is constant. 
 
 (184) The four tangents to the nine-points' circle of any 
 triangle at the points where it touches the inscribed and escribed 
 circles, all touch the maximum ellipse which can be inscribed in 
 the triangle. 
 
 (185) If /(a, jS, 7) =0 be the trilinear equation to a circle, 
 the equation 
 
 <df dl dlY_ 
 V«' rf/3' driS 
 
 will represent an indefinitely small concentric circle. 
 
 (186) If /(a, /3, 7) = be the trilinear equation to a circle 
 at distance r from the centre (a, yS', 7'), then will 
 
 1 _ a"-r' _ /y'-r' 7"-r' 
 K~ U ~ V W~ 
 
 _ /3'7 -f-r'cos^ _ 7V + r°cos^ _ a';3'-f-r'cos(7 
 
 u' v w • 
 
CHAPTER XX. 
 
 QUADEILINEAE COORDINATES. 
 
 323. In investigating theorems whicli involve four straight 
 lines symmetrically, it is often advantageous to take them all as 
 lines of reference using a system of four coordinates. 
 
 In the present chapter we shall exhibit and exemplify the 
 use of such four coordinates. 
 
 324. Let a', ^, 7' be trilinear coordinates of a point 0, and 
 let S' be its distance from the straight line whose equation is 
 
 loL + m^ + M7 = ; 
 then (Art. 46), 
 
 la! + m^ + ny = {I, m, n] S', 
 
 or Za' + «i/S' + W7' + rS'=0 (1). 
 
 Also, if a, h, c, d be the sides of the quadrilateral formed by 
 the three lines of reference and the other line, we have 
 
 aa! + h0 + cy + dh' = twice the area of the quadrilateral, 
 
 = 8, (suppose) (2). 
 
 Thus if a', jS', 7', S be the perpendiculars from any point on 
 the four sides of a quadrilateral, they are connected by the 
 relations 
 
 Za' + OT/3' + K7' + rS'=0, 
 
 aa' + &/3' + C7' + c?S'=/S', 
 
 where I, m, n, r, a, b, c, d, S are constant functions of the sides 
 and angles of the quadrilateral. 
 
 20—2 
 
308 QUADEILINEAR COORDINATES. 
 
 325. The perpendiculars a', /?, 7', B' might be used as the 
 quadrilinear coordinates of the point 0: but it will generally 
 be found preferable to use certain multiples of these perpen- 
 diculars instead of the perpendiculars themselves to express the 
 position of the point. In the few cases when it is advantageous 
 to use the perpendiculars we shall expressly call them perpen- 
 dicular coordinates. In all other cases when we speak of 
 quadrilinear coordinates, the system set forth in the next 
 article must be understood to be referred to. 
 
 326. Let 1(1= a, m^' = ^, ny' = y, r8' = 8; then a, /9, 7, S 
 being constant multiples of the perpendiculars, a', /S', 7', S' may 
 be used as coordinates to express the position of the point 
 with respect to the lines of reference. (See Art. 105.) We shall 
 call a, /3, 7, S the quadrilinear coordinates of the point 0. 
 
 Also, referring to the notation of Art. 324, let 
 
 m~ ' mS~ ' nS~ ' rS~ ' 
 
 then the equations (1) and (2) of that article become 
 
 a + /3 + 7 + S = (1), 
 
 Aa + B^+Cy+D8=l (2), 
 
 two equations which will always connect the quadrilinear co- 
 ordinates of any point. 
 
 327. The equation (1) of the last article plays a very 
 important part in the manipulation of equations in quadrilinear 
 coordinates. It will be well to consider at once one or two 
 cases of its application, 
 
 I. It enables us, at any time, to reduce our quadrilinear 
 equations to trilinear, by eliminating one of the variables 
 (8 suppose) by the substitution 
 
 8 (a + y8 + 7). 
 
 And since this transformation cannot affect the degree of an 
 
QUADEILINEAE COORDINATES. 309 
 
 equation, it follows that, as in trilinear coordinates, an equa- 
 tion of the first degree represents a straight line, of the second 
 a conic section, &c. 
 
 II. It gives us the power of eliminating, from any equation, 
 terms of any particular argument that we may wish to be free 
 from. For instance, we may without loss of generality eliminate 
 from the general equation of the second degree, the terms whose 
 arguments are a', ^, </, h", by substituting 
 
 a''=-(a/3-l-a7 + aS), 
 
 7'=-(7S + 7a + 7;8). 
 S^ = -(Sa + S/3 + S7). 
 
 Hence there is no objection to our assuming the general 
 equation of the second degree of the form 
 
 X/37 + X'aS + fiTiO. + /i'ySS + va/3 + 1^78 = 0, 
 
 or we may omit some of these terms and retain some of the 
 others. 
 
 III. It must be observed that the same locus can be repre- 
 sented by an indefinite number of different equations, formed 
 from one another by substitutions of the equation (1). Hence we 
 are not at liberty to infer that because two different equations 
 
 la. + «i/3 + «7 + rS = 0, 
 and Ta + W/3 + ny + r'h = 0, 
 
 (suppose) represent the same straight line, the coefficients of like 
 terms in the two are proportional : we may infer no more than 
 that the two equations, together with 
 
 a + /3 + 7-t-S = 0, 
 form a system of only two independent equations. 
 
 Thus too, the equation to the straight Ime at infinity is not 
 restricted to the form 
 
 Aa + B^+ C7 + 2)S = 0, 
 
310 
 
 QUADBILINEAE COOKDINATES. 
 
 but, more generally, that straight line is represented by any 
 equation of the form 
 
 {A + k)OL+ (/3+ «) ^+ {C+k) ry + (D + «) S = 0, 
 
 K being any constant. 
 
 328. We shall assume the following construction through- 
 out the present chapter. 
 
 Fig. 38. 
 
 Let AA', BB', CC be the three diagonals of the complete 
 quadrilateral formed by the lines of reference, and let ABC, 
 ABC, ABC, A' EC be the lines respectively denoted by 
 
 a=0, /8 = 0, 7 = 0, 8 = 0. 
 
 Or, in other words, let ABC be the triangle formed by the 
 lines 
 
 a = 0, ^=0, 7 = 0, 
 
 and let A', B, C be the points where these lines are re- 
 spectively cut by the fourth line (S = 0). 
 
 Also let BB, CC intersect in a, CC, AA! in I, and 
 AA', BB' in c, so that dbc is the triangle formed by the diago- 
 nals of the parallelogram. 
 
QUADEILINEAB COORDINATES. 311 
 
 329. To interpret the constants A, B, C, Dm. equation (2) 
 of Article 326, it is not necessary to trace the course by which 
 they were introduced. The following interpretation is deduced 
 immediately from the equations and is sufficient. 
 
 In virtue of the relation 
 
 a+/3 + 7 + 5 = (1) 
 
 the equation 
 
 ula + J5/9+ CV + 2)S=1 (2) 
 
 may be written 
 
 (^-D)a+(J5-Z>);S+(G-J>)7 = l (3). 
 
 But a, /3, 7 are multiples of the trilinear coordinates of the 
 point ; hence the equation (3) expresses implicitly an identical 
 relation amongst the trilinear coordinates of the point, and there- 
 fore must be equivalent to the other forms in which the identical 
 relation can be written (Art. 84), therefore the three terms 
 
 {A-D)^, {B-D)^, {C-I))y, 
 
 must represent respectively the ratios of the triangles OBG, 
 OCA, OAB to the whole triangle ABC. 
 
 So by eliminating a we should prove that 
 
 {B-A)^, {0-A)y, {D-A)S, 
 
 represent the ratios of the triangles OAB', OAC, OB'C to 
 the whole triangle AB' C. 
 
 Similarly, 
 
 {A-B)a, {0-B)y, {D-B)8, 
 
 represent the ratios of the triangles 0^'5, OBC, 00'^' to the 
 whole triangle O'BC. 
 
 And (A-C)oi, {B-C)^, {D-C)h, 
 
 represent the ratios of the triangles OA' O, OB'C, OA'F to the 
 whole triangle AB' C. 
 
 The student will observe that these results hold while 
 a yS, 7, 5 are the quadrilinear coordinates of any point whatever. 
 
312 
 
 QUADEILINEAR COORDINATES. 
 
 330. It will be easily seen that the coordinates of the six 
 angular points of reference are as follows : 
 
 of ^ . 
 oi JB ., 
 of C . 
 oi A' . 
 of B" . 
 of C. 
 
 ,a = -S = 
 
 -B 
 
 •/8 = -7 
 
 T C-D 
 
 B-G 
 
 , ^ = 7 = 0; 
 
 7 = a = 0; 
 
 , a = /3 = 0; 
 
 A-B 
 
 /3 = S = 0; 
 7 = 8 = 0. 
 
 Cor. The coordinates of the middle points of the diagonal 
 A A will be (by Art. 18), 
 
 ^ = 1 
 
 1 X- -^ 
 
 '-^{A-D)' '^~2{B-C)' '^~2{B-G)' "~2{A-D)' 
 so the middle point of BB is given by 
 
 -' ..»- ' 
 
 S = 
 
 -1 
 
 2{C-A)' *" 2[B-D)' "^ 2((7-^)- " 2{B-D)' 
 and the middle point of CC is given by 
 
 -1 
 
 -1 
 
 ''-2{A-B)- ^~2{A-B)' '^ 2{€-D)' ^~2{G-D)- 
 
 331. We proceed to interpret some ©f the simplest equa- 
 tions connecting quadrilinear coordinates. 
 
 In virtue of equation (1) of Art. 326, the equations 
 /3 + 7 = and a + 8 = (1) 
 
 must be identically equivalent, and therefore represent the same 
 straight line. But from their form, the first represents a straight 
 
QUADKILINEAE COORDINATES. 313 
 
 line through A, and the second represents a straight line through 
 A'. Hence either equation must represent the diagonal AA'. 
 
 Similarly either of the equations 
 
 7 + a = and /3+S = (2) 
 
 represents the diagonal BB', and either of the equations 
 
 a + /3 = and 7 + 8 = (3) 
 
 represents the diagonal CC 
 
 Again, the locus of the equation yS — 7 = must pass through 
 the intersection of BB', CC as well as through the point A. 
 
 Hence /3 — 7 = represents Aa. 
 
 So 7-a = Bb, 
 
 a-/3 = Cc, 
 
 a-8 = A'a, 
 
 /3-S = B'b, 
 
 and 7 — S = C'c. 
 
 332. The equations in the last article immediately lead us 
 to some of the most important harmonic properties of a quadri- 
 lateral. 
 
 Thus from the form of the equations we observe (Art. 129) 
 that the lines AA' and Aa divide the angle at A harmonically. 
 So also the pencils 
 
 [B.AB'Cb], [CBCAc] 
 
 as well as the pencils 
 
 {A'.CAB'a}, [B.ABG'b], {G'.BCA'c} 
 
 are harmonic. 
 
 333. From the form of the equations in Art. 331, we 
 observe that the straight lines 
 
 Aa, B'b, C'c 
 
314 QUADBIUNEAB COOEDINATES. 
 
 are concurrent. So 
 
 A'a, Bb, C'c, 
 
 A'a, B'l, Cc, 
 
 Aa, Bb, Cc, 
 
 are concurrent systems. 
 
 Let a (Fig. 39) be the point of intersection of the first set, 
 j8, 7, S those of the other sets respectively, 
 
 Fig. 39. 
 5> 
 
 then the points a, /8, y, S are given respectively by the systems 
 of equations 
 
 /3=7=S, 
 
 7 = S = a, 
 
 .a=/8 = 7. 
 
 334. To find the condition that three points whose quadrUinear 
 coordinates are given should be collinear. 
 
 Let (a', ^', 7, 8'), (a", /3", y", B"), (a'", /3"', 7'", 8'"), be the 
 three points. 
 
QUADBILINEAR COOBDINATES. 
 
 315 
 
 Suppose they lie upon a straight line whose equation free 
 from 8 is 
 
 Then 
 
 therefore 
 
 la +mfi +wy =0. 
 la' +m^ +ny' =0, 
 la" +m^' +wy"=0, 
 la"'+m^"' + ny"'=0, 
 
 .(1). 
 
 a . P . 7 
 
 So if we had written the equation to the straight line free 
 from 7 we should have found the condition in the form 
 
 = 0, 
 
 «', /S', 
 
 s- 
 
 a", /3". 
 
 S" 
 
 «'", /8"', 
 
 8'" 
 
 which is obviously identical with (1), since 
 
 o' +/3' +7 +S' =0, 
 a" +^" + 7" + S"=0, 
 a"' + /3"' + 7"'+S"'=0. 
 
 Thus it will be conyenient to write the condition in the form 
 
 «', /3', 7', S' 
 a", ^'. 7", S" 
 a , ^ , 7 , 6 
 
 = 0, 
 
 the unequal determinant denoting that we may take any three of 
 the four columns to form a determinant equal to zero. 
 
 335. The middle points of the three diagonals of a quadri- 
 lateral are collinear. 
 
 Taking the sides of the quadrilateral as lines of reference 
 and using the coordinates of the middle points obtained in 
 
316 
 
 QUADEILINEAR COORDINATES. 
 
 Art. 330, the condition that the points should be coUinear will 
 
 he (by the last article) 
 
 1 
 
 1 
 
 -1 
 
 or 
 
 A- 
 
 -B' 
 
 B 
 
 -G' 
 
 B- 
 
 -C 
 
 
 I 
 
 
 1 
 
 A- 
 
 1 
 
 A- 
 
 -C" 
 
 B 
 
 -D' 
 
 -C 
 
 
 I 
 
 A 
 
 1 
 -B' 
 
 ] 
 
 
 A- 
 
 -B' 
 
 6- 
 
 ■D 
 
 B- 
 
 -0, 
 
 A- 
 
 -D, 
 
 D- 
 
 -A 
 
 D- 
 
 -B, 
 
 C- 
 
 -A 
 
 B- 
 
 -D 
 
 C- 
 
 ■D, 
 
 D- 
 
 -G, 
 
 A- 
 
 -B 
 
 = 0, 
 
 = 0, 
 
 which is seen to be an identity by the addition of its rows. 
 .'. &C. Q.E.D. 
 
 336. It may be shewn, as in trilinear coordinates, that the 
 polar of the point (a, /3', 7', S') with respect to a conic whose 
 equation is 
 
 /(a, /3, 7, a)=0, 
 
 is represented by the equation 
 
 Obs. If (a', /3', 7', S) lie upon the conic this equation will 
 represent the tangent thereat. 
 
 337. Kefei-riug to Art. 159 we observe that the general 
 equation in quadrilinear coordinates to a conic passing through 
 the four points B, B , C, C", is 
 
 /S7 + iaS = 0. 
 
 So the general equation to a conic through (7, C , A, A', is 
 
 and to a conic through A, A', B, B' 
 
 c/3 + Z;yS = 0. 
 
QUADRILINEAR COOEDINATES. 317 
 
 338. As an example we may prove the well known 
 theorem : 
 
 If a system of conies pass ihrougli four fixed points, the polars 
 with respect to them of any fixed point are concurrent. 
 
 Take the four iixed points as the angular points B, C, B, C 
 of the quadrilateral of reference. 
 
 Then the equation to any conic in the system may he 
 written 
 
 Let (a', /3', 7', S') be the fixed point, then the polar is given 
 by the equation 
 
 /37' + /3'7 + ^(aS' + a'8) = 0, 
 and therefore it passes always through the point determined by 
 
 1 + ^=0, and ^, + 1 = 0, 
 p 7 a o 
 
 a fixed point since a', ^', 7', S' are constant. 
 
 Hence if a system of conies pass through four fixed points 
 the polars with respect to them of any fixed point are con- 
 current. 
 
 CoE. Let two of the four points be the circular points at 
 infinity, then the theorem reduces to the following, which we 
 proved otherwise in Art. 303. 
 
 If a system of circles have a common radical axis the polars 
 with respect to them of any fixed point are concurrent. 
 
 339. To find the equations to the tangents at B, C, B', C to 
 the conic whose equation is 
 
 The coordinates of 5 are (Art. 329) 
 
 a=7 = 0, /3 = -S = 53^. 
 
318 QUADEILINEAE COOEDINATES. 
 
 Hence, applying Art. 336 the equation to the tangent at B 
 will be 
 
 So the tangent at B' will be given by 
 Similarly, the tangents at C, C will be given by 
 
 y-Jc8 = 0. 
 
 340. Cob. From the form of the equations to the tangents 
 we observe, that in ant/ conic passing through B, C, B", C, 
 
 the tangents at B and C) . . 
 
 ° „ „,[■ meet on Aa; 
 
 ,\ ^'«; 
 
 B ... C] 
 B' ... C 
 
 B ... B'X 
 
 ... c] 
 
 AA. 
 
 A great number of well known properties follow from these 
 results. We will enunciate two of them. 
 
 I. If a quadrilateral {BCB'C) he inscribed in a conic, 
 and another quadrilateral {^arfh) he described touching the conic 
 in the angular points of the former one, the four interior diagonals 
 of the two quadrilaterals meet in one point (a) and the two exterior 
 diagonals coincide {AA'). 
 
 II. If a quadrilateral be inscribed in a conic, the points of 
 intersection of opposite sides and the points of intersection of the 
 tangents at opposite angles are collinear. (Camb. Math. Tripos, 
 1847.) 
 
 341. The following proposition exemplifies the use oi per- 
 pendicular quadrilinear coordinates (Art. 325). 
 
 To shew that the circles circumscribing the triangles AEG', 
 A'BC, A!B 0, ABC pass all through one point. 
 
QUADEILINEAR COORDINATES. 
 
 819 
 
 In perpendicular coordinates, all the distances being positive 
 towards the interior of the quadrilateral BGBC (fig. 39), the 
 equations to the four circles will be (Art. 195) 
 
 i^AB'C-), «^ + !ii^-!i^ = o, 
 
 ^A'BG\ 5ilL^'_!i^ + %? = o, 
 a 7 ' 
 
 I A>T»n\ sin5' , sin^' sin (7 . 
 {ABC), --- + _^ + _g_ = o, 
 
 {ABC), !!^ + ^ + !iL^=o. 
 
 And these equations will be satisfied by the same values of 
 <*> A 7i ^) provided 
 
 .(1). 
 
 0, sin C, 
 
 sin E, 
 
 — Bin A 
 
 = 
 
 sin C", 0, 
 
 — sin A', 
 
 sin J5 
 
 
 sin 5', sin^', 
 
 0, 
 
 sinC 
 
 
 sin^, sin 5, 
 
 sin C, 
 
 
 
 
 But if a, h. c, d denote the sides BC, CB' , C'B, EC 
 respectively of the closed figure BCB'C, we obtain by project- 
 ing the sides upon lines at right angles to each of them in order 
 
 — J sin C +csmB + d sin A' = 0, 
 
 — a sin C + ca\nA + dsinB' = 0, 
 
 — a sin jB + J sin ^ + rf sin C — 0, 
 asin^'- 6sin£'+c sin C" = 0, 
 
 whence eliminating a'.h '. c\ d, 
 
 0, sinC^, sin5, sin^' =0 (2). 
 
 sin C, 0, sin A, sin ^ 
 
 sin B, — sin A, 0, sin C" 
 — sin A\ sin S , sin C , 
 
320 QUADRILINEAE COOEDINATES. 
 
 The equations (1) and (2) are the same, each of the determi- 
 nants being eqnal to 
 
 (sin A sin A — sin .B sin .B' + sin C sin C")". 
 
 Hence the condition (1) is satisfied, and therefore the four 
 circles meet in a point. 
 
 342. To find the anharmouic ratio of the range in which 
 the lines of reference are cut by a given line. 
 
 Let 1% + wi/3 + M7 + rS = 
 
 be the given line, and let it meet a = 0, )8 = 0, 7 = 0, 8=0 in 
 i, M, N, B. 
 
 Then the equations to J3L, BM, BN, BB will be 
 a = 0, (Z — r) a + (m — r) 7 = 0, 
 7 = 0, (Z — w?) a + (n — »») 7 = 0. 
 
 Hence {LMNB] = \i_J^\i_J^ • 
 
 343. To find the anharmomc ratio of the pencil formed 
 hy joining any given point to the four points of reference 
 
 B, G, B, c: 
 
 Let (a', 0, 7, S*) be the given point 0. Then the four 
 straight lines are represented by 
 
 {OB), 
 
 7 
 
 7'" 
 
 -?-». 
 
 {OB), 
 
 /3 
 0'' 
 
 a „ 
 
 -g. = 0, 
 
 {00), 
 
 a 
 a'~ 
 
 -| = o. 
 
 {OC), 
 
 S 
 
 -^ = 0. 
 7 
 
EXERCISES ON CHAPTER XX. 321 
 
 Now let „ = :y_« andr; = |,-i 
 
 Then the equations to the four straight lines become 
 
 Hence {O.BCB'C'}^^. 
 
 PI 
 
 344. CoE. If lie upon the conic 
 ^7 + KoZ = 0, 
 
 /3'7' + *:a'S' = 0, 
 
 so that 
 then we have 
 
 K 
 
 or, the anJiarmonic ratio of the pencil formed hy joining any 
 point on a conic to four fixed points on the same is constant. 
 
 And conversely, if the anharmonic ratio of the pencil formed 
 hy joining a variable point to four fixed points is constant, the locics 
 of the variable point is a conic passing through the four fixed 
 points. 
 
 Exercises on Chapter XX. 
 
 (187) The four coordinates of a point cannot be all positive 
 in the ordinary system of quadrilinear coordinates. But in per- 
 pendicular coordinates the four coordinates of a point may be of 
 the same sign. 
 
 (188) The general equation to a conic inscribed in the qua- 
 drilateral of reference may be wi-itten 
 
 {fi - vf (/37 + aS) +(y- Xf (7a + ^B) + {\- /.)' {a^ 4- 7^) = 0. 
 w. 21 
 
322 
 
 EXERCISES ON CHAPTER XX. 
 
 (189) The equation 
 X/87 + X'aS + iiya. + /^SS + va^ + y'78 = 
 
 will represent a parabola, provided 
 
 0, 1, 1, 1, 1, =0. 
 
 1, 0, \, n, V, A 
 1, \, 0, v, /J,', B 
 1, /i, v, 0, V, C 
 1, V, /, \', 0, D 
 0, ^, 5, C, D, 
 
 (190) The equation 
 
 /37 + KaS = 
 
 will represent a parabola, provided 
 
 ^{B-CY + K{[A-Bf+{G-DY+[A-CY+{B-Df] 
 
 + {A-Dy = 0. 
 
 (191) Through any four points on a parabola another para- 
 bola can be drawn unless the four points lie on two parallel 
 straight lines. 
 
 (192) The general equation of a conic circumscribing the 
 triangle formed by the three diagonals of the quadrilateral of 
 reference may be written 
 
 \ (/37 - aS) + /i (7a - ^S) + !> (a/3 - 7S) = 0. 
 
 (193) The equation to the parabola inscribed in the quadri- 
 lateral of reference is 
 
 {B -C)'{A- BY (JSy + aS) + {C-AY {B - Df (7a + ^S) 
 
 + {A-By{G-By{a^ + yB) = 0. 
 
 (194) The conic passing through the four points of reference 
 B, C, B, C and through the fifth point (a', ^', 7', 8'), is repre- 
 sented by the equation 
 
 and its tangent at the point (a, 0, 7', S') is represented by 
 a' ff y'^S "• 
 
EXERCISES ON CHAPTEE XX. 323 
 
 (195) If two conies circumscribing GAG' A, aai ABA' B', 
 intersect in a point 0, the tangents at divide the angle be- 
 tween OA, OA' harmonically. 
 
 (196) If the point of intersection of a pair of common 
 chords of two conies be joined to the points of contact of a 
 common tangent, the pencil thus formed is harmonic. 
 
 (197) If four common tangents be drawn to a pair of conies 
 which intersect in real points, and if the four points of contact 
 with one of the conies be joined in all possible ways by straight 
 lines, the three points of intersection of these straight lines 
 coincide with the points of intersection of the six common chords 
 of the two conies. 
 
 21—2 
 
CHAPTER XXI. 
 
 CERTAIN CONICS RELATED TO A QUADRILATERAL. 
 
 345. We shall use the term tetragram to describe the figure 
 contained by four straight lines indefinitely produced, and not 
 regarded in any particular order. 
 
 We shall use the word quadrilateral when we speak of the 
 four-sided figure contained by four straight lines taken in a par- 
 ticular order. 
 
 Thus a tetragram has three diagonals, but a quadrilateral 
 has two proper diagonals and an exterior diagonal. 
 
 Thus any four straight lines a, fi, 7, S forming a tetragram, 
 form three quadrilaterals 'according to the order in which we 
 take the sides, viz. : 
 
 (1) ya^B, with a opposite to S, 
 
 (2) 0^378, with /3 opposite to S, 
 
 (3) ^yaS, with 7 opposite to S. 
 
 One of these quadrilaterals will generally be proper, another 
 sectant, and the third re-e,ntrant. 
 
 Thus, retaining the construction of the last chapter (Fig. 39, 
 page 314) the four straight Hnes ABC, EGA, CAB, A' EC 
 form one tetragram, but they form three quadrilaterals, viz. : 
 
 (1) BCEG'B,^xo^r, 
 
 (2) ACA'CA, sectant, 
 
 (3) BCECB, re-entrant. 
 
CERTAIN COXICS BELATED TO A QUADRILATERAL. 325 
 
 346. Definitions. Amongst the conica passing through 
 the four points B, C, B', C there is one which touches at 
 B, C, B', C the four straight lines Bh, Cc, Eb, G'c : this is 
 called the critical circumscribed conic of the quadrilateral 
 B, C, B', C". 
 
 So the critical circumscribed conic of the quadrilateral 
 CAC'A' touches the four straight lines Cc, Aa, Cc, A'a, and 
 the critical circumscribed conic of the quadrilateral ABA'B' 
 touches the four straight lines Aa, Bb, A'a, Bb. 
 
 Amongst the conies inscribed in the quadrilateral BOB C 
 there is one whose points of contact lie on the two chords Aa, 
 A'a: this is called the critical inscribed conic of the quadri- 
 lateral 5 CJ5'C'. 
 
 So the critical inscribed conic of the quadrilateral CA C'A' 
 has Bb, BV as chords of contact, and the critical inscribed conic 
 of the quadrilateral ABAB has Gc, Cd as chords of contact. 
 
 Obs. The critical circumscribed conic of a square is the 
 circumscribed circle, and the critical inscribed conic is the in- 
 scribed circle. 
 
 347. It follows from the definitions that the critical cir- 
 cumscribed conic of any quadrilateral is the critical inscribed 
 conic of the quadrilateral formed by the tangents at the angular 
 points. 
 
 And (similarly) the critical inscribed conic of any quadri- 
 lateral is the critical circumscribed conic of the quadrilateral 
 formed by joining its points of contact. 
 
 348. To shew the existence of a critical circumscribed conic 
 with respect to any quadrilateral. 
 
 Let BCB'C be the quadrilateral, and let 
 
 )87 + /caS = 
 
 be any circumscribed conic. Then the tangents at the angular 
 points are given by 
 
 7 — Ka = 0, /3 — /ifa = 0, 7 — /cS = 0, jS — «:S = 0, 
 
326 CERTAIN CONICS 
 
 (Art. 339). Therefore in the particular case when ic = l, the 
 equations to these tangents are 
 
 which (Art. 331) represent the lines 
 
 £h, Co, Bl, C'c. 
 
 Therefore the conic 
 
 /37 + aS = 
 
 represents the critical circumscribed conic of the quadrilateral : 
 and therefore there is such a conic with respect to any quadri- 
 lateral. 
 
 Cor. 1. The three equations 
 
 7a + ,8S = 0, 
 0/3 + 78 = 0, 
 
 represent the critical circumscribed conies of the three quadri- 
 laterals BCB'C, OAC'A', ABA'B. 
 
 349. Cor. 2. Since (Art. 346) the critical inscribed conic of 
 any quadrilateral is the critical circumscribed conic of the qua- 
 drilateral formed by joining the points of contact assigned in the 
 definition, it follows that there always exists a critical inscribed 
 conic with respect to any quadrilateral, 
 
 350. To find the equation to the critical inscribed conic of 
 the quadrilateral BOB C. 
 
 Since /S = 0, 7 = represent tangents, and a— 8 = their 
 chord of contact (Def.), the equation to the conic must be (Art. 
 161) of the form 
 
 (a-S)'=«/37, 
 
 or {a+Sy = K^y + iaB (1). 
 
RELATED TO A QDADRILATERAL. 327 
 
 Similarly, since a = 0, 8 = represent tangents, and 
 yS— 7=0 their chords of contact (Def.), the equation to the 
 conic must be of the form 
 
 {^ + yy = i^y + K'aS (2). 
 
 But a + y3 + 7 + S = 0, 
 
 and therefore (a + S)" = (/3 + 7)'. 
 
 Hence the equations (1) and (2) will be identical if a;=k' = 4, 
 or the equation 
 
 (a + S)^ = 4(/37 + aS) (3), 
 
 or + 7)» = 4(/37 + aS) (4), 
 
 represents a conic fulfilling all the conditions required by the 
 definition. 
 
 Hence, either of these equations, or any other equation 
 obtained by combining them with the equation 
 
 a + /3+7 + 8 = 0, 
 
 will represent the critical inscribed conic of the quadrilateral 
 BCFC. 
 
 351. Cor. Comparing the equation (3) or (4) with the equa- 
 tion to the critical circumscribed conic (Art. 348) 
 
 /37 + aS = 0, 
 
 we observe that tJie critical inscribed and circumscribed conies 
 with respect to the same quadrilateral have double contact with 
 one another, the chord of contact being the diagonal joining the 
 intersections of opposite sides of the quadrilateral. 
 
 We shall however see immediately (Art. 353) that the points 
 of contact are imaginary whenever the quadrilateral is real. 
 
 352. One form of the equation to the critical inscribed 
 conic of the quadrilateral BCBG', obtained from equations (3) 
 and (4) of the last article by addition and transposition, is 
 
 a^ + ^+7'+8'=6(/S7 + a8) (1). 
 
328 CEKTAIN CONICS 
 
 So the critical inscribed conies of the quadrilaterals CA C'A 
 and ABA'B are 
 
 a' + /8= + 7' + a==6(7a + /38) (2), 
 
 and a'' + )8W+8' = 6(a/3 + 7S) (3). 
 
 But the critical inscribed conies of the same quadrilaterals 
 are given by the equations 
 
 y37 + a8 = (4), 
 
 7a + /8a=0 (5), 
 
 a^ + 78 = (6), 
 
 hence the equation 
 
 a= + ^' + 7+«'=0 (7), 
 
 represents a conic passing through the points of intersection of 
 the critical inscribed and circumscribed conies of each quadri- 
 lateral. 
 
 But the critical inscribed and circumscribed conies of each 
 quadrilateral have double contact. Hence the conic represented 
 by equation (7) has double contact vrith all those six conies in 
 the six points where the three inscribed touch the corresponding 
 circumscribed conies. 
 
 353. The conic represented by the equation 
 
 is necessarily imaginary when the quadrilateral is real, since 
 each term is essentially positive. 
 
 Hence the six points of contact of the inscribed and circum- 
 scribed critical conies (which may be conveniently termed the 
 critical points of the tetragram) are imaginary, since they lie 
 upon the imaginary conic 
 
 a'+/3' + 7»+6'=o. 
 
RKLATED TO A QUADHILATEEAL. 329 
 
 354. To find where the conic whose equation is 
 
 cuts the lines of reference. 
 
 To find where the conic cuts the line A' EC we have to nut 
 8 = 0, 
 
 therefore a* + /3' + 7= = (i), 
 
 but when 8 = 0, a + /3 + 7=0 (2). 
 
 Hence, eliminating a, 
 
 /?'+2)Sy + V = 0. 
 
 Hence the two points divide EC, so that the anharmonic 
 ratio of the section is unity (Art. 123). 
 
 But if we eliminate /S between (1) and (2), we find that the 
 two points are given by 
 
 y'+27a + a'=0, 
 
 and therefore they divide CA' so that the anharmonic ratio of 
 the section is unity. 
 
 Similarly, by eliminating 7 we may shew that the same two 
 points divide A'E so that the anharmonic ratio of the section is 
 unity. 
 
 Under these circumstances the points are said to form with 
 A', E, C an equi-anharmonic system. 
 
 And, similarly, each of the other sides of the tetragram can 
 be shewn to be cut by the conic in two points forming with the 
 three vertices in the same side an equi-anharmonic system. 
 
 355. The conic 
 
 «'+/? + 7*+ S' = 
 
 is thus seen to cut each of the diagonals of the tetragram in the 
 two critical points on that diagonal, and to cut each of the three 
 sides in two points which form with the three vertices in that 
 
330 CERTAIN CONICS 
 
 side an equi-anharmonic system. On account of these properties 
 the conic has been named by Professor Cremona (who seems to 
 have been the first to discover and investigate it) thefourteen- 
 points' conic of the tetragram. 
 
 356. It is easily seen that the straight lines 
 
 Aa, Eb, G'c, Aa, Bb, Cc, 
 
 are the polars of the points A, B, C, A', B', C, with respect to 
 the fourteen-points' conic. 
 
 It follows that 
 
 AA', BB, CC 
 
 are the polars of the points 
 
 a, b, c, 
 
 and the lines of reference 
 
 A'BC, AEG, AEG', A'EG', 
 
 are the polars of the points 
 
 a, ^, 7, 8. 
 
 It follows that the tangents from the seven points a, b, c, 
 a, /3, 7, S touch the conic in the fourteen points from which it 
 derives its name. 
 
 357. If the equations to four straight lines in any system of 
 coordinates be 
 
 + M + «+w = (1), 
 
 the fourteen-points^ conic of the tetragram which they form will 
 be represented by 
 
 m' + v''+w'' = (2). 
 
 For if we wiite 
 
 a = — M + u + tc'" 
 
 /9= u — v + w 
 
 > (3), 
 
 h =. — u — v — w^ 
 we have 
 
 a + /3 + 7 + S = 0, 
 
BELATED TO A QUADRILATERAL. 331 
 
 and therefore the equations (3) will be the relations by which to 
 transform to quadrilinear. coordinates, having the four given lines 
 as lines of reference. 
 
 But in quadrilinear coordinates the fourteen-points' conic is 
 given by 
 
 therefore by (3) it will be represented in the original coordi- 
 nates by 
 
 {-u+v + wy+{u-v + wY+{u + v-wy+{u + v+wy=o, 
 
 or u^ + v' + v^=0. Q. E.D. 
 
 Exercises on Chapter XXI. 
 
 (198) The critical circumscribed conies with respect to the 
 quadrilaterals CAC'A', ABAE have double contact at A 
 and A. 
 
 (199) The chords of contact Aa, Aa of the critical in- 
 scribed conic of the quadrilateral BCEC are common chords 
 of the other two critical inscribed conies of the same tetragram. 
 
 (200) Shew that the conies 
 
 have three points common to them all, two of which also lie on 
 the fourteen-points' conic of the tetragram of reference. 
 
CHAPTER XXII. 
 
 TANGENTIAL COORDINATES. THE STKAIGHT LINE AND POINT. 
 
 358. Let us agree to determine a straight line by its perpen- 
 dicular distances p, q, r from the three points of reference, just 
 as hitherto we have determined a point by its perpendicular dis- 
 tances from the three lines of reference. 
 
 We may with propriety speak of these quantities p, q, r as the 
 coordinates of the line. Thus we shall use the symbol {p, q, r) 
 to denote the line whose coordinates are p, q, r, or which lies at 
 perpendicular distances p, q, r from the points of reference. 
 
 Such coordinates are called tangential coordinates. 
 
 359. When we commenced with the coordinates of a point, 
 a straight line was determined by passing through, two points; 
 so now, when we commence with the coordinates of a straight 
 line, a point will be determined as lying on two straight 
 lines. 
 
 Again, as we formerly defined the equation of a line as an 
 equation satisfied by the coordinates of all points on the line, 
 so now we shall define the equation to a point as an equation 
 satisfied by the coordinates of all straight lines passing through 
 the point. 
 
TANGENTIAL COORDINATES. 
 Fig. «. 
 
 333 
 
 360. We have seen (Chap, v.) that itj), g, r be th« perpen- 
 dicular distances of a straight line from the points of reference, 
 and a, /8, 7 the perpendicular distances of a point in the line 
 from the lines of reference, then will 
 
 apa. + hq^ + cr^ = 0. 
 
 Further, if the straight line he determined by the quantities 
 f, q, r being given, this equation constitutes a relation among 
 the coordinates of any point upon the line, and is therefore the 
 equation to the line. 
 
 But if instead of p, q, r being known quantities entering 
 into the coefficients of the equation which connects the variables 
 a, /3, y, these latter be known (as being the coordinates of a 
 fixed point 0), then the same equation 
 
 atKp + h^q + c^r — 
 
 will constitute a relation among p, q, r which will hold for any 
 straight line passing through the point 0, and will therefore be 
 the equation to the point according to the definition of the last 
 article. 
 
334 TANGENTIAL COORDINATES. 
 
 That is, any point whose trilinear coordinates are a, /3, y is 
 represented in tangential coordinates hy the equation 
 
 a'jp + JySg' + cyr = 0, 
 
 the triangle of reference heing the same for both systems. 
 
 Cor. It follows that in tangential coordinates every point 
 has an equation of the first degree, and every equation of the 
 first degree represents a point. 
 
 361. Of course the distances p, q, r will he regarded as of 
 the same algebraical sign when they are all on the same side of 
 the line on which they are let fall, and any two will be of oppo- 
 site sign when they are on opposite sides of the line. 
 
 But it is never necessary to determine which side of the line 
 shall be the positive side and which the negative, nor to give 
 any one of the coordinates by itself any absolute sign, since all 
 our equations in tangential coordinates are either homogeneous, 
 or if their terms be of different orders they are at least all of 
 even orders or all of odd, so that a change in the absolute signs 
 of p, q, r would have no effect. This is a direct consequence 
 of the circumstance that the coordinates p, q, r of any line are 
 identically connected by a relation of the second order, 
 
 a'(p-q) {p-r)+b'{q-r) {q-p) +e'{r -p) {r-q)=iA\ 
 (Art. 74) and not like the coordinates of a point by a simple 
 equation. 
 
 362. To find the equation in trilinear coordinates to the 
 straight line whose tangential coordinates are p, q, r. 
 
 Since p, q, r are the perpendicular distances of the straight 
 line from the points of reference, therefore by Chap, v., the 
 equation to the straight line is 
 
 apy. + Jg/S + cry = 0. 
 
 Cob. The equation to the same straight line in triangular 
 coordinates is 
 
 paL + q^ + ry=0. 
 
THE STEA.IGHT LINE AND POINT. 
 
 335 
 
 363. To find the trilinear coordinates of the point of inter- 
 section of two straight lines whose tangential coordinates are 
 given. 
 
 Let the given coordinates of the two lines be (/>,, ^j, r,), 
 (i'2' iv O' ^°^ suppose (a, j8, 7) the trilinear coordinates of 
 their point of intersection. 
 
 Then since (a, P, 7) is a point on a straight line whose per- 
 pendicular distances from the points of reference are p,, q^, r^, 
 therefore (Chap, v.) 
 
 apitt + hqfi + cri7 = 0. 
 
 Similarly, 
 
 ap^OL + hqfi + crj,7 = 
 
 0. 
 
 
 Therefore we have 
 
 
 aa 
 
 
 i^ 
 
 
 C7 
 
 
 2.. '•i 
 
 
 n. P, 
 
 
 i>i. ?. 
 
 
 ?2. '•j 
 
 
 "•i'Fi 
 
 
 i'a. ?i! 
 
 equations which determine the ratios of the coordinates required. 
 
 Cor. In virtue of Art. 360 it follows that the tangential 
 equation to the same point is 
 
 p, q, r =0, 
 P.i> ?i. »•. 
 
 a result which we presently establish (Art. 366) without reference 
 to the trilinear system. 
 
 364. To find the coordinates of the straight line joining two 
 points whose equations are given. 
 
 Let Ip + mq + nr = 0, 
 
 and I'jP + i^q + «'»■ = 0, 
 
 be the equations to the two points. 
 
336 
 
 TANGENTIAL COORDINATES. 
 
 Then the coordinates of the straight line joining them must 
 satisfy both equations (Art. 359), and therefore their ratios are 
 given by 
 
 p 
 
 
 S 
 
 
 r 
 
 m, n 
 
 
 n, I 
 
 
 I, m 
 
 1 1 
 m , n 
 
 
 nW 
 
 
 l',m 
 
 To find the absolute values of these coordinates we shall 
 have to substitute their ratios in the relation which we found 
 in Chap. Tl. (Arts. 73, 74) connecting the perpendiculars upon 
 any straight line. That relation, as we there shewed, can be 
 written in any of the various forms, 
 
 a'^* + Ftf + c^r' — ibcqr cos A - 2carp cos B 
 
 -2abpq COS (7 =4 A', 
 
 «' iP -i){p-r) + h'{q - r) {q -p) + c= {r-p) {r - q) = 4A'. 
 {q-ry cotA+ (r -pf coiB+ {p - qY cot C= 2A, 
 or with the notation of Art. 46, it may be written 
 
 {ap, hq, crY = iA\ 
 the form in which we shall generally quote it. 
 
 365. It appears from the foregoing article that by solving 
 together the equations of any two points we may determine the 
 coordinates of the straight line joining them. 
 
 Hence any two equations of the first degree taken simnl- 
 taneoasly will determine a straight line, viz. the straight line 
 joining the two points which the equations represent separately. 
 Therefore two equations may be spoken of as the equations of a 
 straight line. For example, the straight line {p', q, r') may be 
 said to be given by the equations 
 
 ■£--2. andi=- 
 p q q r 
 
 or £ = 2. = !L 
 
 p q r 
 
THE STRAIGHT LINE AND POINT. 
 
 337 
 
 " 366. To find the eqvMtion of the point of intersection of two 
 straight lines whose coordinates are given. 
 
 In other wprds, to find the relation among the perpendiculars 
 p, q, r from the points of reference upon any straight line pass- 
 ing through the point of intersection of the given straight lines. 
 
 Let {p^, g-,, r,), {p^, q^, r^ be the given straight lines, and 
 suppose 
 
 lp+mq-\-nr = (i (1), 
 
 the equation of their point of intersection 0. 
 
 Then this equation expresses a relation satisfied by the coor- 
 dinates of any straight line passing through 0, 
 
 But {p^, q^, r,) passes through 0, therefore 
 
 lpi + mqi + nr^ = (2). 
 
 Similarly, {p^, q^, r,) passes through 0, and therefore 
 
 lp^+mq^+nr^ = (3). 
 
 Hence eliminating I: m : n from the equations (1), (2), (3), 
 we get 
 
 p, q, r =0, 
 
 V^.' ?!> '•i 
 
 a relation among the coordinates p, q, r, and therefore the equa- 
 tion of the point 0. 
 
 Cor. 1. The equation just obtained will not be affected if 
 Pa Si7 ^i or j»,, ?,, »•, be multiplied by any constant ratio. Hence 
 if the coordinates of one straight line be only proportional to 
 three given quantities ^,, q^, r^, and those of another straight 
 line proportional top,, q^, r^, the equation of their point of in- 
 tersection is still 
 
 P> 2. »• 
 P,' 9u '•i 
 
 = 0. 
 
 W. 
 
 22 
 
338 
 
 TANGENTIAL COORDINATES. 
 
 367. To find the eqimtton of the point where a given straight 
 line meets the line at infinity. 
 
 Let (^jj, jp rj be the given straight line, then 
 
 will be a parallel straight line (fig. 41), and these will therefore 
 intersect in the point required. 
 
 Q' P' R' 
 
 Hence by the last article the eqnation required is 
 
 2»i ?» f 
 Px-Vh, 2i + /i, »"i + A 
 
 = 0, 
 
 or 
 
 hf A, h 
 
 = 0, 
 
 or 
 
 1, 1, 1 
 
 = 0. 
 
THE STRAIGHT LINE AND POINT. 339 
 
 368. Cor. The last equation is satisfied ii p = q = r, for 
 then the first and third rows of the determinant become identical. 
 
 Hence all points at infinity lie upon the straight line given 
 by p = 2=r. 
 
 Hence jp=:q = r 
 
 are the equations of the straight line at infinity. 
 
 369. To find the coordinates of a straight line passing 
 through a given point and parallel to a given straight line. 
 
 Let lp + mq + nr = (1) 
 
 be the equation to the given point, and let {p^, q^i rj be the 
 given straight line. 
 
 The coordinates of any straight line parallel to (;?,, q^, r,) 
 J be written (fig. 41) p^ +h, q^ + h, r^ + h. 
 
 If this straight line pass through the point (1) we must 
 'P. 
 
 Ip, + mq^ +wr,+ (l + m + n) h = 0, 
 
 therefore A = - V"^'^"'' ? 
 
 l + m + n 
 
 therefore , p, + h = ^ (P^y^+-(P^-^.) , 
 ^'^ l + m + n 
 
 + j,^ ri{q,-r,)+l{q,-p,) 
 ^' l+m+n ' 
 
 * l + m + n ' 
 
 which are therefore the coordinates required. 
 
 370. Tofnd the distance of the point whose equation is 
 
 Ip + mq + nr = (1) 
 
 fi-om the straight line whose coordinates are {p', q, r). 
 
 22—2 
 
 may 
 
 I 
 have 
 
340 TANGENTIAL COOEDINATES. 
 
 Let h be the distance, then the line parallel to {p, q, r) 
 through the given point will have the coordinates 
 
 (/ ±h, q ± 7i, r ± h). 
 
 These must satisfy the equation (1), therefore 
 
 Ip' + mq +nr''± {l + m + n)h = 0, 
 
 ■ J . Ip' + mq' + nr 
 
 or A = + -^-y- — ^- . 
 
 I + 7n + n 
 
 371. To find the equation to a point which divides in a 
 given ratio the straight line joining two given points. 
 
 Let 1 : i he the given ratio, and 
 
 Ip + mq + nr = 0, and Tp + m'q + n'r = 0, 
 
 the equations to the given points. 
 
 Suppose jj, q, r the coordinates of any straight line through 
 the required point. Then the perpendicular distances of the 
 given points from this line are in the given ratio. 
 
 Therefore by, the last article, 
 
 Ip + mq + nr _ I'p + m'q + n'r _ , 
 - l+m, + n. • "•■ i' + m'+w' -^•'^' : 
 
 the two expressions for the distance having opposite signs, since 
 the two points a,x& on opposite sides of the straight line. 
 
 Therefore 
 
 J Ip -}- mq ■+ nr l'p + m'q + n'r _ ' 
 l + m + n I + m +n 
 
 a relation among the coordinates of any straight line through the 
 required point, and therefore the equation to the required point. 
 
 Cob. The middle point between the points 
 
 lp + mq + nr = 0, a,nd I'p + m'q +_n'r = 0, 
 
 is given by the equation 
 
 lp + niq + nr Tp + m'q + n'r _ 
 t + m + n P + m'+n"^ ~ 
 
THE STRAIGHT LINE AND POINT. 341 
 
 372. The principles of abridged notation explained in 
 Chapter Ylli. for trilinear coordinates are equally applicable to 
 tangential coordinates. 
 
 As we there used m = 0, v = Q, to=0 to represent equations 
 to straight lines expressed in their most general form, so now we 
 shall use the same expressions to denote the most general forms 
 of the equation to the point in tangential coordinates. 
 
 373. 1/ u=Q, V = he equations to two points in tangentia 
 coordinates, then 
 
 M + ««? = 0, 
 
 {lohere k is an arbitrary constant) will represent a point lying on 
 the straight line joining the two points. 
 
 For \ip, q, r be the coordinates of this point they satisfy the 
 equations m = and v = ; that is, their substitution makes u 
 and V severally vanish, therefore it must make u 4- kv vanish ; 
 that is, p, q, r satisfy the equation 
 
 U + KV = 0, 
 
 and therefore this equation represents a point on the line 
 {p, q, r). Q.E.D. 
 
 374. If the line Joining the points m = and v =0 be divided 
 hy the points u + kv=0, and u + k'v = 0, the anharmontc ratio 
 of the section is k : k. 
 
 Let A, B be the two points represented by m = and, v = 0, 
 and P, Q the two points represented hju + KV=0 and u + k'v = 0. 
 
 Let {p'j q, r) be the coordinates of any straight line what- 
 ever, and let u, v be what u, v become when p', q', r are 
 written for p, q, r, and let m, n be what u, v become when 
 unity is written for each of these letters p, q, r. 
 
 Then the perpendicular distances of the points A, B, P, Q 
 from the straight line (p, q, r) are respectively 
 
 U V «' + KV' U + KV 
 
 111 n m 4- K7i m + Kn 
 
342 TANGENTIAL COORDINATES. 
 
 and therefore the distances AP, A Q, BP, BQ are proportional 
 (by similar triangles) to the differences 
 
 U u' + KV' U U+KV' V w' + KV V U + KV 
 
 m «» + «»'»» m-\-K'n' n m + icn^ n m + «'» 
 
 therefore 
 
 fu 11 + K.v\ fv u+k'v' \ 
 (,<r>T>/^i AP.BQ \m m + Kn/'\n m+jcnj 
 
 ^^^^^^ ^aqTbp^ '-' -' • -•-'^ '-■' -•' 
 
 AQ.BP~ /u u' + k'v' \ fv' u -TotA 
 V»» m + KtiJ '\n m + Kn/ 
 
 K (rait' — mv) {mv — nu!) 
 K [nu — mv'){mv' — nu) ' 
 
 or {APBQ}=-, . Q.E.D. 
 
 375. To find the anharmonic ratio of the range of the four 
 points whose equations are 
 
 M + /f V = 0, M + Xt) = 0, u-\- fiv = 0, M + iw = 0. 
 
 The proof of Article 125 (p. 137) applies verbatim. Thus we 
 find that the anharmonic ratio required is 
 
 {K-\)(p.-v) 
 (k - v) (/* - \) ■ 
 
 376. It follows, as in Art. 123, that the line joining the 
 points M = 0, V = 0, is divided by the two points 
 
 Zm* + 2muv + m? = ; 
 so that the anharmonic ratio is 
 
 {m+Jni'-lnY 
 In 
 
EXEECISES ON CHAPTEB XXIL 343 
 
 ExEECiSES ON Chaptee XXII. 
 
 (201) The coordinates of the line of reference BC are 
 
 2A 
 
 (202) The coordinates of the perpendicular from A on BC 
 are 
 
 0, +5co3C, +ccos5. 
 
 (203) The coordinates of the straight line through A paral- 
 lel to BC are 
 
 2A 2A 
 a a 
 
 (204) The straight line joining A to the middle point of 
 BC is given by 
 
 p = 0, q + r = Q. 
 
 (205) The equation q^-r = Q represents the middle point 
 of the side BO of the triangle of reference. 
 
 (206) The equition j tan JB + r tan (7 = represents the 
 foot of the perpendicular from the point of reference A upon BG. 
 
 (207) The equation mq + nr = (i represents a point P in the 
 line BC such that BP : PC = n : m. 
 
 (208) The equation mc[ — nr = (i represents a point P in the 
 line BC produced, such that PB : PC = n:m. 
 
 (209) The equation q—r = represents the point of inter- 
 section (at infinity) of straight lines parallel to BC._ 
 
 (210) The equation p + q + r — represents the point of 
 intersection of the straight lines which join the angular points of 
 the triangle of reference to the middle points of the opposite 
 sides. 
 
344 EXERCISES ON CHAPTEE XXII. 
 
 (211) The equation 
 
 J) tam A + q tan B + r ta,n C=0 
 
 represents the point of intersectioti of the perpendiculars from 
 the angular points on the opposite sides of the triangle of 
 reference. 
 
 (212) The equations 
 
 + psin^ + gsin^ + rsin C=0 
 
 represent the centres of the inscribed and escribed circles of the 
 triangle of reference. 
 
 (213) The equation 
 
 p sin 2A + q sin 25+ r sin 2 C= 
 
 represents the centre of the circle passing through the points of 
 reference. 
 
 (214) The equation 
 
 (q + r) sin 2 A + (r + p) sin 2B+{p + q) sin 2 C= 
 
 represents the centre of the nine-points' circle of the triangle of 
 reference. 
 
 (215) Apply tangential coordinates to shew that the middle 
 points of the three diagonals of a complete quadrilateral are 
 coUinear. 
 
 (216) The straight line jxiining the points 
 
 Ip + rriq + nr = 0, Tp + m'q + nV = 
 
 is divided harmonically in the points 
 
 Ip + mq + «r Tp-\- m'q + n'r 
 
 — ; 1- AC -^S • ;— = 0. 
 
 l + m + n I +TO +n 
 
CHAPTEE XXIir. 
 
 TANGENTIAL COORDINATES. CONIC SECTIONS. 
 
 377. Definition.. The equation to a curve in tangential 
 coordinates is a relation among the coordinates of any straight 
 line which touches the curve. 
 
 The equation to a curve is therefore satisfied by the coor- 
 dinates of any tangent to the curve; and any sti-aight line whose 
 coordinates satisfy the equation is a tangent' to the curve. 
 
 378. We have already seen that the identical relation con- 
 necting the coordinates of any straight line may be written in 
 any of the forms 
 
 {q - ry cot J + (r -j))' cot S,+ {p- qY cot G= 2A, 
 a^ {p - q){p-r) + h' {q-r){q-p) +c' [r -p)[r- q) ^ 'i.^\ 
 or [ap, iq, crY = 4A^ 
 
 It should be noticed, that if 
 
 = [ap, bq, crY, 
 
 then -T- = 2a {ap — bq cos C— cr cos B), 
 
 -r- = 2b {bq — cr cos A — ap eos C), 
 
 -J- = 2c {cr — ap cos B—bq cos A), 
 
 dO dO dO „ 
 and -r- + -y--\- -j- = 0. 
 
 dp dq dr 
 
346 TAKGENTIAL COORDINATES. 
 
 379. To find the equation to the circle whose centre is at the 
 point 
 
 lja + mg[ + nr = 0, 
 and whose radius is p. 
 
 Let p, q, rhe the coordinates of any tangent to the circle. 
 Then since p is the distance of the tjingent firom the centre, we 
 have (Art. 370) 
 
 _,lp + inq + nr 
 f~^ l+m+n ' 
 
 and rendering this homogeneous by the relation 
 [ap, hq, a-Y = AA% 
 
 „ A^* /lp + mq + nr\' 
 we get {ap, hq, cr] = y( i+^ + ^ ) . 
 
 a relation among the coordinates of any tangent and tho^fore the 
 equation to the circle. 
 
 380. The general equation to a circle is therefore 
 
 {ap, hq, cr]' ={\p + fiq + w)', 
 
 and its radius is 
 
 2A 
 \ + p, + v' 
 
 and the equation to its centre is 
 
 "Kp + fiq + vr = 0. 
 
 For comparing the equation just written down with the form 
 which we investigated, we have 
 
 ^ , , 2A Ip+mg+nr 
 
 ^ p l+m+n ' 
 
 therefore 
 
 Xp I fip _ m vp 
 
 2A l + m + n' 2A l + m+n^ 2A l + m + n' 
 and by addition, 
 
 (\ + ^ + v)p 2A 
 
 2A ~^' °' P~\ + ^i + v' 
 
CONIC SECTIONS. 347 
 
 381. A particular case of the equation to a circle occurs 
 when 'K = /i = v = 0, or when p = co . 
 
 In this case the equation takes the form 
 {ap, hq, cr}» = 0, 
 or a" {p-q) {p-r)+h*{q-r){q-p) +e (r- p) {r -^) = 0. . .(1), 
 
 which is evidently satisfied when p = q = r, shewing that any 
 straight line lying altogether at infinity is a tangent. 
 
 But since the coordinates p, q, r of any finite straight line 
 satisfy the relation 
 
 a*{p-q){p-r)-if¥{q-r){g^-p)-\-c\r-p){r-q) = 4:^\ 
 
 which is inconsistent with (1), we see that no finite straight line 
 is a tangent to the circle. 
 
 The circle is in fact that described in Article 38, and would 
 be represented in trilinear coordinates by the equation 
 
 (aa + h^ + cyY = 0. 
 
 The centre is given by Op+ Oq+ Or = 0, and is indeter- 
 minate : the radius p is infinite. 
 
 We shall speak of this circle briefly as the great circle. 
 
 382. Some writers speak of the equation 
 a'{p-q){p-r) + F{q-r){q-p) + d'{r-p){r-q)=0, 
 
 as representing only the two circular points at infinity: and some 
 correct results are deduced from giving it this interpretation. 
 
 The discrepancy is precisely analogous to that which attaches 
 to the interpretation of the trilinear equation 
 
 ^3* + 7" + 2/37 cos -4 = 0, 
 
 or to the Cartesian equation 
 
 x» + /=0. 
 
 It has already been pointed out (Art. 318) that either of 
 these equations represents two imaginary straight lines intersect- 
 
348 TANGENTIAL COORDINATES. 
 
 ing in a real point, but is also the limiting form of the equa- 
 tion to an evanescent circle at that real point. We explained 
 that a complete description of the locus of such an equation of 
 the second order must recognise the fact that when the real part 
 of a conic section degenerates into a point, the imaginary 
 branches become two straight lines through the point ; and the 
 equation to any two imaginary straight lines intersecting in a 
 real point — so soon as it is regarded as representing a locus 
 of the second order at all — must be regarded as representing the 
 ultimate conic evanescent at the real point and having the two 
 straight lines as imaginary branches. 
 
 In the' present case we have to deal with the ultimate conic 
 at the opposite limit. Instead of the diameters becoming inde-; 
 iinitely small they have become indefinitely great : but as before 
 the asymptotes are imaginary, and in the limit the imaginary 
 branches of tlie curve coincide with them. And just as in the 
 former case, the equation to the conic could in a partial view be 
 regarded as only representing the imaginary asymptotes, so 
 in this case the tangential equation to the conic maybe regarded 
 as representing only the two circular points at infinity, which are 
 at the same time the points of contact of the asymptotes and 
 their, polars with respect to the curve. 
 
 We must again refer to the chapter on reciprocal polars, 
 where this point is rnors fully discussed. 
 
 383. To find the equation to the conic section whose foci are 
 
 at the points 
 
 Ip + mq + nr = 0, 
 
 Up + m'q + n'r = 0, 
 
 and whose conjugate or minor axis is 2p. 
 
 Let {p, q, r) be any tangent to the conic ; then since p' is 
 equal to the rectangle under the focal perpendiculars on any tan- 
 gent, we have 
 
 i_lp + mq + nr Tp + m'q + wV 
 ^ I +.m + « ? + »»' + «' ' 
 
CONIC SECTIONS. 349 
 
 and, rendering this homogeneous by the relation 
 
 {ap, hq, crY = 4A', 
 we get 
 
 a relation among the coordinates of any tangent, and therefore 
 the equation to the conic. 
 
 384. The general equation to a conic may therefore he 
 written 
 
 up' + v^ + wr' + 2u'qr + 2v'rp + 2w'j)q = 0, 
 
 and the foci are given by the equation 
 
 up' + V(^ + wr' H- 2uqr + 2wVp + 2w'pq + k [ap, hq. cr j" = 0, 
 
 ■where k is. to be so determined that the left-hand member of 
 this equation may be resolvable into two factors. 
 
 385. Obs. The equation to give k is 
 
 u + ha', w' — kah cos C, v — kca cos B =0. 
 w' — kah cos G, v-\- k¥, v! — kbc -cos A 
 v' — kca cos B, u' — khc cos A, w + Ac" 
 
 The coeflScient of k" in this cubic vanishes, and the equation 
 reduces to a quadratic giving two values for k, indicating two 
 pairs of foci. One will be a real pair, the other an imaginary 
 pair. 
 
 Or, viewing the equation for k in a, more general aspect, it 
 has three roots, one of which is infinite. There will therefore be 
 three pairs of foci, the two pairs just spoken of and another pair 
 represented by the equation 
 
 {ap, hq, crY = 0, 
 
 to which we must in this case give its partial interpretation, as 
 representing the two circular points. 
 
350 TANGENTIAL COORDINATES. 
 
 Hence every conic may be said to have six foci, two coin- 
 ciding with the circular points, two real ones whose geometrical 
 properties are known, and two other imaginary ones. 
 
 When we speak of the four foci of a conic, it will be under- 
 stood that we neglect the two circular points which arise from 
 the interpretation of the evanescent term in the cubic for k. 
 
 386. To find the coordinates of the tangents drawn from a 
 given point to a given conic, we have only to solve simultaneously 
 the equations to the point and the conic and we shall get two 
 solutions for the ratios of the coordinates of the tangents required. 
 
 387. To find the condition that a given point may lie upon 
 a conic, we must construct the equation for the coordinates of 
 the two tangents from the point, and express the condition that 
 the quadratic thus constructed may have equal roots. 
 
 388. The imaginary tangents drawn to a circle from its 
 centre touch all concentric circles and the great circle at infinity. 
 
 For let \j> + /tj 4- w = (1) 
 
 be the centre : then by giving diflferent values to Te, the equa- 
 tion 
 
 {ap, Iq, crY + h(^p + fiq + vr)'' = (2) 
 
 will represent any circle in the concentric series, and the coordi- 
 nates of the tangents from the centre are obtained by solving 
 simultaneously equations (1) and (2). Hence they are given by 
 
 {\p + liq-\-vrY=Q\ 
 and {ap, bg, cr}' 
 
 which are Independent of k, shewing that the same imaginary 
 tangents touch all the concentric circles. 
 
 But the equation 
 
 {ap, iq, crY = 
 
 represents the great circle, and therefore the equations (3) deter- 
 mine the coordinates of the tangents from the given centre to the 
 great circle. Hence this circle has imaginary tangents in com- 
 mon with any concentric series. 
 
 i::S '')■ 
 
CONIC SECTIONS, 351 
 
 38?. Coil The four common tangents to the great circle 
 and any other coincide two and two, for they coincide with the 
 two tangents to the latter circle from its centre. 
 
 390. The common tangents to the great circle cmd any coma 
 intersect, two and two, in the foci of the conic. 
 
 Let {op, hq, cr^ = (p^ + fiq + vr) (\'p + fiq + v'r). ...{!) 
 be the equation to any conic. 
 
 The common tangents to this conic and the great circle will 
 be obtained by solving together the equation (!) and the equa- 
 tion 
 
 {op, Iq, crY=0 (2). 
 
 These tangents are therefore four in number. 
 
 From the equations (I) and (2) we obtain 
 
 {\j>+ fiq + vr) {\'j) + fi'q + v'r) = 0. 
 
 Hence the four tangents pass through one or other of the 
 points represented by this equation, i. e. through one or other of 
 the foci of the conic 
 
 Therefore, &c. Q. E, d. 
 
 391. Cob. 1. We may adopt the following definition of 
 the foci of a conic. 
 
 The four common tangents to any conic and the great circle 
 at inanity intersect in six points which are called the foci of 
 the conic 
 
 Two of these six foci are the circular points, as we saw in 
 Art 385. Hence every real or imaginary tangent to the great 
 circle passes through one or other of the circular points. 
 
 392. Cor. 2. The common tangents to two confocal conies 
 pass two and two through the foci and touch the great circle at 
 iufinily. 
 
352 TANGENTIAL COORDINATES. 
 
 393. To find the equation to the centre of the conic whose 
 tangential equation is 
 
 f{p, q, r)=0. 
 
 Let {p, q, r') be the coordinates of any diameter, ajid sup- 
 pose {p +h, q + h, r +h) a parallel tangent. Then, since 
 Ip' + h, q + h, r + A) is a tangent these coordinates must satisfy 
 the equation to the curve, therefore 
 
 f{p "rh, q+h, V + h)=(i, 
 an equation to determine h. 
 
 We may write it 
 
 and since the two values of h must be equal and of opposite 
 sign we have • 
 
 dp dq dr 
 
 But (p'j g', r) is any diameter: therefore every diameter 
 passes through the point whose equation ia 
 
 dp dq ' dr. ' 
 therefore this is the equation to the centre which was required. 
 
 394. CoE. 1. If the equation to the conic be written 
 
 [ap, hq, cry. + Jc{lp + mq + nr) (JL'p + m'q + n'r) s= 0, 
 
 the equation to the centre becomes 
 
 Ip + mq + nr Vp + mJq + n'r _ 
 l + m + n r + wi' + n' * 
 
 a result which we might have inferred a priori from the pro- 
 perty that the centre bisects the line joining the foci. 
 
CONIC SECTIONS. 353 
 
 395. CoE. 2. If we write the equation to the conic In the 
 general form 
 
 up^ + wg' + vyr^ + ^u'qr + iv'rp + 2w'pq = 0, 
 
 the equation to the centre takes the form 
 
 iip + vq-\-wr = 0, 
 
 where u^u + v' + w, v = v-{-w' + u', w~w + u' + v', 
 
 396. Cor. 3. If f{p, q, r)=0 represent a circle this equa- 
 tion must (Art. 380) be identical with 
 
 {ap, bq, cry — k {up + vq + lory = 0. 
 
 Hence we must have (see Prolegomenon,) 
 
 u, V, w, 1 = 0, 
 
 u", v^, M>*, u + v + w 
 a', b\ c^ 
 
 which, therefore, express the conditions that the general equa- 
 tion of the second degree should represent a circle. 
 
 397. To find the coordinates of tJie diameter parallel to a 
 given straight line. 
 
 Let {p', q', 1^) be the given straight line, and suppose 
 
 {p' + h, q+h, r' + h), 
 
 the parallel diameter. Then these coordinates must satisfy the 
 equation to the centre, therefore 
 
 u (p' + h) +v {q + h) + w (r + A) = 0, 
 
 J _ up' + vq + wr _ 
 -*. "> — — , - , =- J 
 
 hence the required coordinates are 
 
 i{p^qy-¥w{p-r) w(q'-r')+u{^-p') uir'-p')+v{r-^)- 
 u+v+w ' u+v+w ' v+v+w 
 
 W. 23 
 
354 TANGENTIAL COORDINATES. 
 
 398. To find the condition that the equation 
 
 f ipj S> = W^ + ^9' + ^'^ + 2m' gr + 2v'rp + 2w'pq = 0, 
 
 should represent a parabola. 
 
 The necessary and sufficient condition is that the line at 
 infinity should be a tangent. 
 
 Therefore p = qz=r must satisfy the equation. 
 
 Therefore /(I, 1, 1) = 0, 
 
 or !< + r + w + 2m' + 2v' + 2m)' = 0, 
 
 or M + i; + io = 0. 
 
 the required condition. 
 
 399. Cor. If the equation to the conic be written 
 {ap, bq, erf + (Xp + iiq + vr) (k'p + fi'q + v'r) = 0, 
 
 the condition becomes 
 
 {\ + fi + v) (\' + /i' + j/')=0, 
 shewing that a conic is a parabola if either focus lie at infinity. 
 
 400. To interpret the equation 
 
 df df df ^ 
 
 with respect to the conic f{p, q, r) =0. 
 
 I. Suppose that the straight line {p, q, r') is a tangent to 
 the conic. 
 
 Then fip',s',r') = 0, 
 
 or, as we may write it, 
 
 ^ dp'^^ di^ dr'^' 
 
 which shews that the equation (1) represents some point on the 
 tangent (p', j', r'). 
 
CONIC SECTIONS. 359 
 
 Now let (j>", g'", r") be the otlier tangent from this point. 
 Then since it passes through the point (1), we have 
 
 , df , df , df ^ 
 which shews that {p, q, r') passes through the point given by 
 
 ^|+»f'+^^=» <^'- 
 
 But since (/>", q', r") is a tangent, we have 
 /(p", 2". = 0, 
 
 ^'^ P dp"^^ dq"^"^ dr" "' 
 
 which shews that (p", q , r") also passes through the point (2). 
 
 Hence the point (2) is the point of intersection of the tan^ 
 gents {p', q, r) and {p", q", r") ;. that Is, it coincides with the 
 point (1), therefore the equations (1) and (2) are identical. 
 
 Therefore, 
 
 _^ ^ df 
 dxi _ dd _ dr 
 
 dp' dq dr" 
 
 of which a solution (and since they are simple equations, the 
 only solution) is evidently 
 
 V -L-lL 
 
 or the tangents {p , q', r), {p", q", r") coincide. Hence the 
 given equation represents the point of contact of the tangent 
 
 (i>'. i'. r'). 
 
 23—2 
 
356 
 
 TANGENTIAL COORDINATES. 
 
 But II. suppose {p, g", r') be not a tangent, then let 
 (i'u Si' ''i)> iPi' ?«> '"s) ^® *^^ tangents at the points where 
 ip> 9' > *■ ) ™eeta the conic. 
 
 Then by the Case I. their points of contact are given by the 
 
 equations 
 
 df df df ^ 
 P:r- + q:r- + r-f = o, 
 
 ^Pi <%. «?»•. 
 
 df df ^^^Q 
 ^ dp^ " dq^ dr^ 
 
 And since these points both lie upon {p, c[, r), we have 
 
 ^ dp^ ^ dc[i dr^ 
 
 . •df^,df ,df 
 
 and » J — Vq-r-+'rj- = ^, 
 
 ^ dp, dg^ d'^t 
 
 df df df „ 
 df df df ^ 
 
 which- shew that (j),, j,, rj, (^,, q,, r,) pass through the 
 point given by 
 
 ¥ df df „ 
 
 that is, the equation 
 
 df df df ^ 
 
 represents the point of intersection of tangents at the extremities 
 of the chord (p', q', r'). 
 
 Therefore always — 
 
 The pole of the straight line (p', q', /) is represented hy the 
 equation 
 
 df_ df §f_Q 
 ■^ dp' ^ dq dr ~ 
 
CONIC SECTIONS. 357 
 
 401. COE. 1. If the equation to the conic be written 
 ■up^ + v^ -\-wr^+ luqr + ivrp + Iw'pq^ = 0, 
 
 the pole of the line (p', c[, r) is given by 
 
 p {up + wg + v'r) + 2 (wg' + mV + !«^') + r {wr + v'/ + u^) = 0. 
 
 402. CoH. 2. With respect to the great circle the pole of 
 the straight line {p, q, r') is 
 
 ap {ap — bq cos C — cr cos B) + bq {bq — cr' cos A — ap cos C) 
 
 + cr (cr' — ap cos B — bq cos A) = 0, 
 
 which is satisfied if ^ = g = r. 
 
 Hence the pole of any straight line with respect to the great 
 circle is at infinity. 
 
 403. Cor. 3. The equation of the last corollary becomes 
 indeterminate if p' =q =r'. 
 
 Hence the pole of the straight line at infinity with respect 
 to the great circle is indeterminate, as we shewed otherwise in 
 Art. 381. 
 
 It also follows from Cor. 2, in virtue of Art. 234, that the 
 polar of any finite point with respect ■ to the great circle is the 
 straight line at infinity. 
 
 404. To find the coordinates of a diameter of a conic con- 
 jugate to a given diameter. 
 
 Let f{p, q, r) = be the given conic, and {p', q, r) the 
 given diameter. 
 
 Let {p' + h, g' + h, r + h) and {p' — h, q' —h, r — h) be 
 the parallel tangents. 
 
358 TANGENTIAL COORDINATES. 
 
 Then their points of contact (or poles) are given by the 
 equations 
 
 " dp ^ dq dr \dp dq dr) ' 
 
 and ^'f + ,f + r'f-k(f + f+f)=0, 
 
 ^ dp ^ dq dr \dp dq dr/ 
 
 The conjugate diameter joins these two points: hence its 
 coordinates will be obtained by solving together these two equa- 
 tions. Hence the coordinates are given by 
 
 ,df ,df_ /^ = o 
 " dp ^ dq dr 
 
 and f + §^+f = 
 
 dp d^ dr 
 
 y -•(!), 
 
 d^ ^ ^ 
 >> <^y> _ dq _ dr . . 
 
 q—r r — jj ~ p — q ^ '' 
 
 The equations (2), with the identical relation (Art. 364), 
 determine the coordinates p, q, r required. 
 
 COE. The first of the equations (1) shews that the con- 
 jugate diameter passes through the pole (at infinity) of the 
 original diameter. Hence we might express the definition of 
 conjugate diameters thus : 
 
 Two diameters of a conic are said to he corrugate when each 
 passes through the pole of the other. 
 
 405. If the equation to the conic be written in the form 
 up^ + vq' + wr^ + 2uqr + "iv'rp + ^wpq — 0, 
 
 the equations to determine the diameter conjugate to a diameter 
 {p, q', r) become 
 
 up + v'r + w'q _vq + w'p + u'r _ wr +u'q + v'p 
 q —r r' — p' ~ P ~i' ' 
 
CONIC SECTIONS. 
 
 359 
 
 or 
 
 p 
 
 
 q-r, w\ 
 
 v' 
 
 r'-p, V, 
 
 u 
 
 r / / 
 
 p-q, u, 
 
 w 
 
 
 2 
 
 i 
 
 -r, v', u 
 
 r' 
 
 -p', u', to' 
 
 1 
 
 P 
 
 -q, W, v 
 
 
 r 
 
 
 
 i 
 
 -r, 
 
 M, 
 
 w' 
 
 r 
 
 -P, 
 
 w. 
 
 V 
 
 p' 
 
 -3'. 
 
 v, 
 
 u 
 
 Cor. If the equation /(p, g, r) =0 represent a parabola, 
 {jp, q, r) will be a diameter, provided 
 
 q —r _r —p _p —q 
 u V w 
 
 406. To find the asymptotes of the conic 
 
 f{p,q,r)=0. 
 
 Let [p'f q, r) be an asymptote. 
 
 Then, since (/>', q, r) is a tangent whose point of contact is 
 at infinity, these coordinates must satisfy the equation to the 
 conic, and the coordinates of infinity must satisfy the equation 
 to the pole of this tangent. 
 
 Hence 
 
 and 
 
 f{p',q',r) = 0, 
 dp' dq dr 
 
 The first of these is a quadratic, and the second is a simple 
 equation ; the coordinates of the two asymptotes will therefore 
 be obtained by solving them together. 
 
 COE. It appears therefore that the coordinates of the asymp- 
 totes of a conic are obtained by solving together the equation to 
 the conic and the equation to its centre. Hence (Art. 386) the 
 asymptotes are the tangents to the curve from its centre. 
 
 407. To shew that the equation 
 
 Iqr + mrp + npq = 
 represents a conic inscribed in the triangle of reference. 
 
360 TANGENTIAL COOKDINATES. 
 
 The equation is satisfied if g = 0, r = are satisfied. 
 
 But these equations represent the side BC. Hence BG is a 
 tangent to the conic. So the other sides are tangents. 
 
 Therefore &c. Q. E. D, 
 
 408. To shew that the triangle of reference is self-conjugate 
 with respect to the conic 
 
 Ip' + m^ + nr' = 0. 
 
 By Art. 400, the equation to the pole of the line (p, q', r") is 
 Ipp' + mqq + nrr = 0. 
 
 Hence, putting q =0, r = 0, the pole of the side BC oi the 
 triangle of reference is given by 
 
 p = 0, 
 
 that is, it is the point A. 
 
 Hence each side of the triangle of reference is the polar of 
 the opposite angular point. 
 
 Therefore &c. Q. E. D. 
 
 409. To find the general equation to a conic circumscribing 
 the triangle of reference. 
 
 Let up* + vq^ + wr* + 2u'qr + 2v'rp + 2w'pq = 
 
 be the equation of a conic passing through the points of re- 
 ference. 
 
 The tangents from j? = are given by 
 
 v^ + wi^ + ^tiqr = 0, 
 
 and these must be coincident ; 
 
 therefore m' = ± Vww, 
 
 so r' = + VwM, 
 
 and m' = + Vm». 
 
CONIC SECTIONS. 361 
 
 Hence writing P, m^, n' for u, v, w, the equation becomes 
 
 Vp^ + m'j' + nV° + ^mnqr + 'inlrp f Umpq = 0, 
 
 and, as in Art. 205, the doubtful signs must be taken either all 
 negative or only one negative, or else the equation would 
 degenerate into two simple equations. 
 
 Hence the general equation of a conic circumscribing the 
 triangle of reference may be written 
 
 yip + ^l7nq + Vwr = 0. 
 
 410. If 8=0 he the equation to a conic, and m = 0, v=0 
 the equations to two points, it is required to interpret the equation 
 
 S + KUV = 0, 
 
 where k is an arbitrary constant. 
 
 Let {p, q, r) be one of the tangents from the point w = to 
 the conic S=0, 
 
 Then {p, q, r) satisfy both the equations 
 8=0 and w=0, 
 
 and therefore {p, q, r) satisfy the equation 
 
 S + KU\> = 0. 
 
 Hence either tangent from m = to 8 = is a tangent to 
 the conic 
 
 S+KUV= 0. 
 
 Similarly, either tangent from « = Otofi' = is a tangent to 
 the same conic. 
 
 Hence the equation represents a conic section, so related to 
 the given conic that two of the common tangents intersect in 
 (m = 0), and the other two in,(w = 0). 
 
362 TANGENTIAL COORDINATES. 
 
 411. To interpret the equation 
 
 UV + KWX = 0, 
 
 where m = 0, v = 0, to = 0, x = 
 
 are the equations to points. 
 
 Suppose [p, q, r) the straight line joining the points 
 M = 0, w = 0, 
 
 then these coordinates satisfy both the equations 
 M = 0, M) = 0, 
 
 and therefore satisfy the equation 
 
 UV + KWX = 0. 
 
 But this equation being of the second order represents a 
 conic section. Hence it represents a conic section touching the 
 straight line joining the points 
 
 M = 0, w = 0. 
 
 Similarly, the conic touches the line joining 
 
 M = 0, a; = 0, 
 
 and the line joining 
 
 r = 0, to = 0, 
 
 and the line joining 
 
 v = 0, a; = 0. 
 
 Hence it represents a conic inscribed in the quadrilateral 
 whose angular points are 
 
 M=0, w = Q, v = 0, x = 
 in order. 
 
 41 2. To interpret the equation 
 
 UV + KW^ = 0. 
 
 As in the last case, this is a conic touching the lines joining, 
 the points (m = 0, to = 0) and (« = 0, w= 0). 
 
CONIC SECTIONS. 363 
 
 Moreover, the tangents from m = to the curve, are given by 
 
 M = 
 
 ,2, 
 
 1 
 
 and therefore are coincident. Hence {u = 0) lies on the curve. 
 
 Similarly, (v = 0) lies on the curve. 
 
 Hence the equation represents a conic section passing 
 through the points m=0, v=^0, and whose tangents at those 
 points intersect in w = 0. 
 
 413. To interpret the eqttation 
 
 Ivw + mwu + nuv = 0, 
 where m = 0, v = Q, w =0 are the equations of three points. 
 
 Being of the second order the equation represents some conic. 
 The equation is satisfied when v = and w = 0. 
 Hence the straight line joining v = and w = is a tangent 
 to the conic. 
 
 Similarly, 'he straight line joining w= and m = 0, and the 
 straight line joining u = 0, v = 0, are tangents. 
 
 Hence the equation represents a conic inscribed in the tri- 
 angle whose angular points are u = 0, v = 0, w = 0. 
 
 414. By comparison with Art. 408, it will be seen that the- 
 equation 
 
 lu^ + mv" + nw^ = 0, 
 
 represents a conic, with respect to which the triangle formed by 
 joining the points 
 
 M = 0, « = 0, w = 0, 
 
 is self-conjugate. 
 
 So it may be shewn as in Art. 409, that the equation 
 Jlu + ijmv + Jrm = 0, 
 represents a conic circumscribing the same triangle. 
 
364 EXEECISES ON CHAPTEH XXIII. 
 
 415. It will be necessary for the student to distinguish 
 between a curve of the w*" order and a curve of the w"" class. 
 The following definitions are usually given. 
 
 Def. 1. A curve is said to be of the w"" order when any 
 straight line meets it in n real or imaginary points. 
 
 Def. 2. A curve is said to be of the w'" class when from 
 any point there can be drawn to it n real or imaginary tan- 
 gents. 
 
 A curve of the m* order will therefore be represented by an 
 equation of the w* degree in trilinear or triangular coordinates, 
 and a cui-ve of the n* class will be represented by an equation of 
 the m"" degree in tangential coordinates. 
 
 We have shewn that every conic section is both of the second 
 order (Art. 145) and of the second class (Art. 230). 
 
 Exercises on Chapter XXIII. 
 
 (217) The equation 
 
 /tan^ + ^tan^ + r'tan 0=0, 
 
 represents the circle with respect to which the triangle of refer- 
 ence is self-conjugate. 
 
 (218) The circle circumscribing the triangle of reference 
 has the equation 
 
 '/pain A + \/qainB+'/r8in C=0. 
 
 (219) The circles escribed to the triangle of reference are 
 given by the equations 
 
 — 8qr+{s—c)rp+(8—h)pq=0, 
 
 (« — c)qr — srp + {s — a)pq = 0, 
 
 (s — h) qr + {3 — a) rp — spq = 0. 
 
EXEECISES ON CHAPTER XXIII. 365 
 
 (220) The circle inscribed in the triangle of reference is 
 given hy 
 
 {s — a)qr+{s — b)rp+ {s— c) pq = 0. 
 
 (221) The equation to the nine-points' circle of the triangle 
 of reference is 
 
 {ap, hq, crY= {op cos {B-C) +hq cob {G- A) +cr cos {A -B)Y, 
 
 or a'Jq + r + bs/r +p + c \p^q = 0. 
 
 (222) The general equation to a conic bisecting the sides of 
 the triangle of reference is 
 
 {m - nYp' +{n- I)' ^+ (?- »n) V + 2 {Pqr + m^rp + n'pq) 
 
 = 2 (mn + nl+ Im) {qr + rp +pq). 
 
 (223) The conic which touches the sides of the triangle of 
 reference at their middle points has the equation 
 
 qr + rp +J>q = 0. 
 
 (224) The point ^ + gq + hr = 0, lies on the conic 
 
 lp' + mq'' + nr'' = 0, 
 
 P ff" A" 
 provided T+l + « = «' 
 
 (225) The point fp+ffq + hr=0, lies on the conic 
 
 Iqr + mrp + npq = 0, 
 provided V^ + V»w^ + VwA = 0. 
 
 (226) The point fp+gq + hr = 0, lies on the conic 
 
 V^+ Vrng + Vwr = 0, 
 
 I m n ^ 
 provided f'^'^'^l" 
 
366 EXERCISES ON CHAPTER XXIII. 
 
 (227) The six straight lines joining the non-corresponding 
 vertices of two co-polar triangles tonch one conic. 
 
 (228) The points given by the equation 
 
 l{q + r) + (Vm + v'n)'^ = 0, 
 
 lie upon the conic Iqr + mrp + npq = 0, and the pole of the chord 
 joining them is given by the equation 
 
 l(q-r) + (m — n)p = 0. 
 
 229) Shew that the conic 
 
 VZp + Vwtg' + Vnr = 0, 
 is inscribed in the triangle whose angular points are 
 
 mq + n7 — lj) = 0, nr+lp— mq = 0, lp + mq — nr = Q. 
 
 (230) If conies are inscribed in a quadrilateral the poles of 
 any fixed straight line lie on another straight line. 
 
 (231) 
 
 Shew tha,t the conic 
 
 = 0, 
 
 
 
 
 is inscribed in the quadrilateral whose 
 
 angular 
 
 points in 
 
 order 
 
 are 
 
 u+v+w= 
 
 —u+v+w= 
 
 u—v+w= 
 
 u+v—w= 
 
 = 0, 
 
 0, 
 = 0. 
 
 
 
 
 (232) 
 
 The conic" 
 
 
 
 
 
 
 
 
 = 0, 
 
 
 
 ■m 
 ■circumscribes the same quadrilateral. 
 
EXERCISES ON CHAPTER XXIII. 367 
 
 (233) The equation 
 
 {l" - mn) (/ - qr) + (w* - nl) (^ - rp) + (re' - Im) {r' - pq) = 0, 
 represents a parabola, passing through the points 
 
 mp + nq + lr = and np + lq + mr = 0, 
 
 and touching the straight lines joining these points to the point 
 
 lp + mq + nr = 0. 
 
 (234) The points of contact of tangents from the point 
 lp + mq + nr = to the conic /(^, q, r)=Q, are given by the 
 equation 
 
 M, w, v', I 
 w', V, u, m 
 v, 11 , w, n 
 I, m, n, 
 
 /(P) q, r) + 
 
 u, w, V 
 w', V, u' 
 v', vl, w 
 
 {lp+mq + nry=0. 
 
 (235) Two conies have double contact and the common 
 tangents intersect in 0. If P be any point on the exterior conic 
 the tangent at P and the tangents from P to the interior conic 
 form with the straight line OF, a harmonic penciL 
 
CHAPTER XXIV. 
 
 POLAR EECIPKOCALS. 
 
 416. If we refer to the proof of Pascal's Theorem in 
 Art. 200, and to that of Brianchon's Theorem in Art. 218, 
 we shall observe that by interpreting the coordinates as tan- 
 gential instead of trilinear in the proof of either theorem, we 
 should obtain a proof of the other. 
 
 And so in many other cases, the same equations being 
 written down and the same eliminations or other processes 
 being performed, we shall arrive by the selfsame work at two 
 different theorems, differing by the interpretation which we 
 give to the coordinates and to the equations into which they 
 enter. 
 
 This is the strict analytical method of applying the prin- 
 ciple of duality, the principle by which every theorem concern- 
 ing the configuration of points has another theorem correspond- 
 ing to it concerning the configuration of straight lines. And 
 in working with equations either in trilinear or tangential 
 coordinates, we ought always to be on the watch for propei- 
 ties which may be suggested by supposing our coordinates to 
 belong to the opposite system. 
 
 But when we use geometi-ical methods, and arrive at pro- 
 perties of points or lines without the aid of equations, we have 
 not generally any symbols capable of a double interpretation 
 by which we may take advantage of the principle of duality. 
 In this case, therefore, since we cannot obtain a double result 
 by a double interpretation of symbolical expressions, it is useful 
 
POLAR EECIPKOCALS. 369 
 
 to consider by what means we can transform a single result so 
 as to arrive at the corresponding theorem. 
 
 The method by which we can most directly eflfect this trans- 
 formation is called the method of Polar Reciprocals. As a 
 geometrical method it does not strictly enter into the scope of 
 the present work, and therefore we shall not greatly enlarge 
 upon its application. We shall, however, explain the funda- 
 mental principles upon which these transformations are made, 
 both because we shall thereby obtain an opportunity of exhibit- 
 ing the significance of many of the equations in tangential 
 coordinates, and because the nomenclature which the method 
 introduces is often employed in the statement of propositions of 
 importance in the analytical methods. 
 
 417. As an example of the double interpretation of results to 
 which we have just referred, we will arrange in parallel columns 
 two important propositions connected together by the principle 
 of duality, and give their common method of proof, using 
 trilinear coordinates for the one proposition and tangential for the 
 other. To shew the identity of the work, we will use the same 
 letters x, y, z to represent triangular coordinates in the first 
 column and tangential coordinates in the second. 
 
 If two triangles he inscribed- 1 If two triangles circumscribe 
 in one conic, their sides will \ one conic, their angular points 
 touch one conic. \ will lie on one conic. 
 
 Take one of the triangles as triangle of reference, and let 
 (a^i, Vv 2x), (a^^, y-i, zj. K> 3^8 . ^s) be the 
 
 angular points | sides 
 
 of the other. And let the conic have the equation 
 
 .,. -, I m n . r .7 
 
 tnlinear] - H h - = Q, [taixjential 
 
 X y z 
 
 then the equations to the 
 
 sides \ angular points 
 
 W. 24 
 
370 
 
 POLAR EECIPKOCALS. 
 
 of the second triangle will be 
 
 Ix my m 
 +-^ + = 0, 
 
 ^^i y^Vz 22^3 
 
 Ix my nz „ 
 + — ^+ = 0, 
 
 *3^, ViVl ^S^l 
 
 Ix my nz . 
 - — + — ^- + = 0. 
 
 '^l^i VlVi ^1^2 
 
 Now the equation 
 
 V\a; + V/xy + Vvz = 
 represents any conic 
 
 inscribed in | circumscribing 
 
 the first triangle (the triangle of reference) and it will also 
 
 be inscribed in \ circumscribe 
 
 the second triangle provided \, fi, v be determined so as to 
 satisfy the equations 
 
 I m n ' 
 
 I m n ' 
 
 I 7n n ' 
 
 which are consistent equations provided 
 
 = 0. 
 
 1 
 
 1 
 
 1 
 
 < 
 
 !/.' 
 
 ». 
 
 1 
 
 1 
 
 1 
 
 < 
 
 y.' 
 
 «! 
 
 1 
 
 1 
 
 1 
 
 < 
 
 ys' 
 
 «a 
 
POLAR RECIPROCALS. 371 
 
 And this is seen to be identically satisfied, since by hypo- 
 thesis 
 
 i. + f^ + £ = o, 1 + ^ + ^ = 0, i + - + !L = o. 
 
 *i Vx «i a;, jr, a, a;, y^ z^ 
 
 Therefore, &c. q.e.d. 
 
 418. Let the points P^, P^, P, be the poles of the 
 
 straight lines p^, p^, p^ respectively, with respect to a 
 
 conic 0. 
 
 If the points P,, P^, P, all lie upon one straight line, 
 
 we know that the straight lines p^, p^, p^ will all pass 
 
 through one point. But if otherwise, the points P^, P,, P^ 
 
 Fig. 42. 
 
 t 
 I 
 i 
 
 / ^2 
 
 7" ' - 
 
 ■o%v ]/ / 
 
 in order may be regarded as the angular points of a polygon, 
 
 and the straight Imes p^, p^, p^ in order may be regarded 
 
 as the sides of another polygon. 
 
 This second polygon is called the reciprocal of the first 
 polygon with respect to the conic 0. 
 
 24—2 
 
372 POLAR RECIPROCALS. 
 
 419. If with respect to a conic, the reciprocal of the polygon 
 G he the polygon g ; the reciprocal of the polygon g will be the 
 polygon G. 
 
 For let Pi, Pj be any two adjacent angular points of the 
 polygon G, and p^, p^ their polars : then since g is the recipro- 
 cal of G, pi, p^ are sides of the polygon g. 
 
 Let the sides p^, p^ intersect in Q, and let the polar of 
 Q be q, then since p„ p^ intersect in Q, their poles Pj, Pj lie on 
 q the polar of Q. 
 
 Hence Q is an angular point of the second polygon, and its 
 polar 2 is a side of tlie first polygon. 
 
 Therefore the polars of all the angular points of the second 
 polygon are sides of the first. Therefore the reciprocal of the 
 second polygon is the first polygon. 
 
 Therefore, &c. Q. E. D. 
 
 420. If the number of angular points P„ Pj,, P, of the 
 
 first polygon be indefinitely increased, so that the polygon 
 becomes ultimately a curve, the number of sides of the second 
 polygon will likewise increase indefinitely, so that it will also 
 become ultimately a curve. And if we regard any of the points 
 
 Pj, Pj on the first curve, the corresponding straight lines 
 
 Py, p^ are tangents to the second curve. 
 
 So if n points on cither curve lie upon a straight line, then 
 will n tangents to the other curve puss through a point. 
 
 Consequently if one curve he of the m"" order and n"" class, 
 the other will he of the n"* order and m**" class. 
 
 421. It follows from Art. 419 that if any curvilinear or 
 other locus F be the reciprocal of another locus / with respect 
 to a conic 0, the locus / will also be the reciprocal of the locus 
 F. The two loci are said to correspond to each other with 
 respect to the conic 0. 
 
 It is convenient to speak of the centre of the conic as the 
 centre of reciprocation. 
 
POLAK EECIPROCALS. 
 
 373 
 
 422. The following theorems follow immediately from the 
 principles we have laid down. 
 
 A point cor; 
 
 (ii)-. 
 The point of intersection of 
 two straight lines, 
 (iii). 
 Collinear points. 
 
 (iv). 
 A polygon of n sides. 
 
 (V). 
 The angular points of a 
 polygon. 
 
 (vi). 
 A curve of the w* order 
 and n* class. 
 
 (vii). 
 A point on a curve. 
 
 (viii). 
 The point of contact of a 
 tangent. 
 
 A chord joining two points. 
 
 (x). 
 The chord of contact of two 
 tangents. 
 
 (xi). 
 A curve inscribed in a poly- 
 gon. 
 
 (xii). 
 A point of intersection of 
 two curves. 
 
 (i)- 
 to a straight line. 
 
 (ii). 
 The straight line joining 
 the corresponding points. 
 
 (iii). _ 
 Concm-rent straight lines. 
 
 (iv). 
 A polygon of n sides. 
 
 (v). 
 The sides of the correspond- 
 ing polygon. 
 
 (vi). 
 A curve of the m*"" class and 
 re"" order. 
 
 (vii). 
 A tangent to the corre- 
 sponding curve. 
 
 (viii). 
 The tangent at the cor- 
 responding point, 
 (ix). 
 The point of intersection of 
 the corresponding tangents. 
 
 The point of intersection of 
 tangents at the corresponding 
 points. 
 
 (xi). 
 A curve circumscribing the 
 corresponding polygon, 
 (xii). 
 A common tangent to two 
 curves. 
 
374 
 
 POLAR RECIPROCALS. 
 
 corresponds to 
 
 (xiii). 
 
 Two curves wliicli toucli s 
 
 one another. ; 
 
 i. e. Which have a common \ 
 
 point and the same tangent ; 
 
 thereat. 
 
 (xiv). 
 Two curves having double 
 contact. 
 
 (XV). 
 
 The chord of contact. 
 
 (xvi). 
 A double point on a curve*, 
 i.e. A point at which there 
 are two tangents. 
 
 (xvii). 
 A point of osculation*. 
 
 (xviii). 
 A point Q in which the 
 tangent at P cuts the curve, 
 (xix). 
 A point of inflexion*. 
 Obtained from the last case 
 by making Q coincide with P. 
 
 (XX). 
 
 A curve having r points of 
 inflexion. 
 
 (xxi). 
 The straight line at infinity. 
 
 (xxii). 
 A point at infinity. 
 
 (xiii). 
 Two curves which touch 
 one another. 
 
 i. e. Which have a common 
 tangent and the same point of 
 contact. 
 
 (xiv). 
 Two curves having double 
 contact. 
 
 . ^^^\ 
 The point of intersection of 
 
 the common tangents. 
 
 (xvi). 
 
 A double tangent* to the 
 
 corresponding curve. 
 
 i.e. A tangent having two 
 
 points of contact. 
 
 (xvii). 
 
 A point of osculation. 
 
 (xviii). 
 
 I A tangent q drawn from the 
 
 \ point of contact of a tangent^. 
 
 1 (xix). 
 
 I The tangent at a point of 
 
 I inflexion. 
 
 l Obtained from the last case by 
 
 making q coincide with p. 
 
 (XX). 
 
 A curve having r points of 
 inflexion. 
 
 (xxi). 
 \ The centre of reciprocation. 
 I (xxii). 
 
 I A straight line through the 
 \ centre of reciprocation. 
 
 * See the Definitions infr& Chap. xxvi. 
 
POLAR RECIPKOCALS. 
 
 375 
 
 (xxiii). 
 
 An asymptote. 
 
 i.e. A tangent at infinity. 
 
 (xxiv). 
 Parallel straight lines. 
 
 corresponds to (xxiii). 
 
 The point of contact of a 
 tangent from the centre of re- 
 ciprocation. 
 
 (xxiv). 
 Points coUinear with the 
 centre of reciprocation. 
 
 423. The foregoing properties apply to all curves whatso- 
 ever : we proceed now to state some which apply to conic 
 sections in particular. 
 
 Since a conic section is of the second order and of the 
 second class (Art. 415), it follows immediately from (vi) that 
 
 (xxvi). (xxvi). 
 
 A conic section corresponds to a conic section. 
 
 (xxvii). \ (xxvii). 
 
 The pole of a straight line \ The polar of the correspond- 
 with respect to any conic. \ ing point with respect to the 
 
 This follows from (x). 
 (xxviii). 
 
 The centre of a conic. 
 
 This follows from the preced- 
 ing by supposing the line to be 
 at infinity. See (xxi). 
 
 (xxix). 
 Parallel tangents. 
 See (xxiv). 
 
 (xxx).^ 
 Concentric conies. 
 
 (xxxi). 
 A pair of conjugate diame- 
 ters in a conic. 
 
 corresponding conic. 
 (xxviii). 
 The chord of contact of tan- 
 gents from the centre of reci- 
 procation to the corresponding 
 conic. 
 
 (xxix). 
 The extremities of a chord 
 through the centre of recipro- 
 cation. 
 
 (xxx). 
 Conies with respect to which 
 the polars of the centre of reci- 
 \ procation coincide. 
 s (xxxi). 
 
 ; Two points which with the 
 \ centre of reciprocation form a 
 
 i. e. Two lines each of which j self-conjugate triangle Avith re- 
 is the polar of the point where > spect to the corresponding 
 the other meets infinity. ) conic. 
 
376 
 
 POLAR RECIPROCALS. 
 
 (xxxii). corresponds to (xxxii). 
 
 The points where a conic s The tangents to the cor- 
 meets the straight line at in- respotiding conic from the cen- 
 
 finity. 
 
 (xxxiii). 
 A hyperbola. 
 
 tre of reciprocation, 
 (xxxiii). 
 A conic having its convex- 
 
 i. e. A conic meeting the ity towards the centre of reci- 
 straight line at infinity in real i procation. 
 
 points. 
 
 i. e. Having real tangents from 
 that point. 
 
 (xxxiv). 
 
 A conic having its con- 
 
 (xxxiv). 
 An ellipse. 
 
 i. e. A conic meeting the S cavity towards the centre of 
 straight line at infinity in ima- reciprocation, 
 
 i. e. Having imaginary tan- 
 gents from that point, 
 (xxxv). 
 A conic passing through the 
 
 ginary points. 
 
 (xxxv). 
 A parabola. 
 
 i. e. A conic meeting the \ centre of reciprocation. 
 
 straight line at infinity in coin- 
 cident points. 
 
 i. e. Having coincident tan- 
 gents from that point. 
 
 424. Given the equation of a curve in triangular coordi- 
 nates, it is required to find the equation to its polar reciprocal 
 with reference to the conic 
 
 la:' + m^ + n'f=0 (1). 
 
 Let /(a, yS, 7) = 
 
 be the equation to the given curve : let («', /8', 7') be any point 
 upon it so that 
 
 /(a',/3',7')=0 (2), 
 
 and let p, q, r be the tangential coordinates of the straight line 
 corresponding to (a /8', 7'). 
 
 Then the equation to this straight line in triangular coordi- 
 nates is 
 
 P'x+q^+ry = 0. 
 
POLAR RECIPROCALS. 377 
 
 But since it is the polar of (a', ^, 7') with respect to (1), its 
 equation may be written 
 
 Za'a + W/S'/S + wy'r^ = 0, 
 
 therefore l^^m^^ni 
 
 p q r 
 
 and substituting in (2), we get 
 
 f\l ' m' nj~^' 
 
 a relation amongst p, q, r, and therefore the tangential equation 
 of the curve reciprocal to the given one. 
 
 CoK. 1. The conic of reciprocation being its own reciprocal 
 is represented in tangential coordinates by the equation 
 
 3 2 2 
 
 I m n 
 
 The centre of reciprocation is given in triangular coordinates 
 by the equations 
 
 la =■ mfi = ny, 
 and in tangential coordinates by the equation 
 
 ^ + ^ + ^ = 0. 
 I m n 
 
 Cob. 2. The reciprocal of the curve whose tangential equa- 
 tion is 
 
 f(p, 2> »■) = 
 is represented in triangular coordinates by the equation 
 f{h, wi/3, «7) = 0. 
 
 CoK. 3. With respect to the conic 
 
 the equations 
 
 /(a,/8,7)=0, a.jiif(j>,q,r)=0 
 
 represent corresponding curves. 
 
378 POLAR RECIPROCALS. 
 
 But it must be borne in mind that if the lines of refer- 
 ence are real, the conic of reciprocation is here imaginary. 
 The centre of reciprocation is however the real point 
 
 a = /3 = 7, or p + q + r = 0, 
 
 viz. the centre of gravity of the triangle. 
 
 425. We are now in a position to illustrate the apparent 
 discrepancy alluded to in Art. 382 as to the interpretation of tlie 
 equation in tangential coordinates 
 
 {ap, bq, erf = 0. 
 
 We have seen (Art. Ill) that the circular points at infinity 
 are given in trilinear coordinates by the equations 
 
 a 7 
 
 — 1 cos (7 + J— 1 sin C cosB ± J— 1 sin B 
 Their polars with respect to the conic 
 
 /(«, ^, 7) = 
 are therefore represented by the equations 
 
 °'' \da' d^' ^|-^- 
 
 So in triangular coordinates the polars of the circular points 
 at infinity with respect to the conic 
 
 /(a,A7)=0, 
 
 are given by the equation 
 
 ( df ,df dff „ 
 
POLAR RECIPROCALS. 379 
 
 Consider the particular case of the conic whose equation is 
 
 la? + m^ + wf = 0, [triangular 
 
 and whose centre is given (Art. 178) by the equation 
 la. = ot/3 = «7. 
 
 The polars with respect to it of the circular points are repre- 
 sented by the equation 
 
 [laa, mh^, nc^f = 0. 
 
 But since the circular points are imaginary points at infinity, 
 their polars must be imaginary and pass through the centre of 
 the conic. 
 
 Hence the equation 
 
 [laa, mh^, ncyf = 
 
 represents two imaginary straight lines intersecting in the real 
 point 
 
 la. = m/S = ny. 
 
 But it follows from Art. 292 that the same equation may be 
 more completely viewed as representing an evanescent conic 
 section, whose real branch has degenerated into the point 
 
 la. = m^ = ny, 
 
 and whose imaginary branches have become two imaginary 
 straight lines. 
 
 Now suppose we reciprocate the locus of this equation 
 {laa, mh^, ncr^Y = 
 with respect to the conic 
 
 la^ + m^ + 7iy'' = 0. 
 
 If we interpret the locus as two imaginary straight lines (the 
 polars of the circular points) it will reciprocate into two imagi- 
 nary points (the circular points themselves). On the other 
 hand, if we regard the locus as a conic (evanescent at tlie 
 
380 POLAR RECIPROCALS. 
 
 centre of reciprocation) it will reciprocate into a conic (the great 
 circle at infinity). * 
 
 Now by Art. 424, the equation to the reciprocal is 
 {op,, bq, cr}' = 0. 
 
 Hence the ambiguity in the interpretation of this equation is 
 accounted for, and is seen to be the direct result of the ambi- 
 guity in the interpretation of any equation to an evanescent 
 conic, which may always be regarded as equally representing 
 two imaginary straight lines. 
 
 426. We can now continue our table as follows : 
 
 (xxxvi). (xxxvi). 
 
 Infinity corresponds to the centre of reciprocation. 
 
 (xxxvii). I (xxxvii). 
 
 The great circle at infinity. \ An evanescent conic at the 
 
 centre of reciprocation, 
 (xxxviii). I (xxxviii). 
 
 The circular points at in- ! The polars of the circular 
 finity. i points at infinity with respect 
 
 to the evanescent conic, 
 (xxxix). I (xxxix). 
 
 The foci of a conic. The chords joining the four 
 
 points in which the correspond- 
 ing conic is cut by the lines 
 corresponding to the circular 
 points. 
 
 427. If a pencil of straight lines be reciprocated into a 
 range of points, the anharmonic ratio of the range is the same as 
 that of the pencil. 
 
 Take the lines of reference so that the conic of reciprocation 
 may have the equation 
 
 a'-l-;8' + 7'=0. 
 
POLAR RECIPROCALS. 381 
 
 And let m = 0, u + kv=0, v = 0, u + k'v = 0, be the equa- 
 tions to the straiglit lines forming the pencil, then the anhar- 
 
 monic ratio is — . (Art. 124). 
 
 But the same equations with p, q, r written for a, /8, 7 will 
 represent in tangential coordinates the range of four points. 
 (Art. 424, Cor. 3). 
 
 Therefore (Art. 374) the anliarmonic ratio -of the range is 
 
 —, , the same as that of the pencil. Q. e. d. 
 
 428. Cor. 1. If four straight lines a, b, c, d (not neces- 
 sarily concurrent) be cut by another straight line p in four 
 points forming a range whose anharmonic ratio is k, the points 
 corresponding to a, h, c, d being joined to the point correspond- 
 ing to p will form a pencil whose range is also k: and con- 
 versely. 
 
 For by Art. 422 (ii), the points in the range correspond to tlie 
 lines in the pencil. 
 
 Note. If a, b, c, d, p represent straight lines, it is often 
 convenient to use the symbol [p. abed] to denote the anhar- 
 monic ratio of the points in which the straight lines a, b, c, d 
 intersect the straight Wnep. 
 
 429. Cor. 2. If p, a, b, c, d be the tangents to a conic at 
 the points P, A, B, 0, D respectively, then will 
 
 {p.<J)cd] = {P.ABCD}. 
 
 430. With respect to a circle, any circle reciprocates into a 
 conic, having a focus at the centre of reciprocation. 
 
 Take a triangle self-conjugate with respect to the circle of 
 reciprocation as triangle of reference. 
 
 Then in triangular coordinates the circle of reciprocation has 
 the equation (Art. 179), 
 
 a''cot^ + /S''cot.B + 7'cot(7 = (1), 
 
382 POLAR RECIPROCALS. 
 
 and any other circle may be represented by the equation (Art. 
 322), 
 
 {acos^, /3cos£, 7C0sC}'' = (Za + »i/S + n7)(a+/3+7)...(2). 
 
 Now by Art. 424 the reciprocal of the circle (2) with respftct 
 to the circle (1) is represented in tangential coordinates by the 
 equation 
 
 {j> sin A , q sin B, r sin Cf 
 
 = (Ip tan A + mqta,nA+ nr tan A) {p tSinA + qtanB + r tan C), 
 
 or [ap, iq, crY 
 
 = . , (?jptan^+»ng'tan5+wtan G){p tan^+g'tan B+r t&n C). 
 
 But (Art. 383) this represents a conic whose foci are given by 
 
 Ip tan A + mq tan B + nr tan C = 0, 
 
 and pisjciA-\-qtas.B+rt2txiC=0; 
 
 the latter of which is the equation to the centre of recipro- 
 cation. 
 
 Hence the reciprocal of a circle with respect to a circle is a 
 conic, having a focus at the centre of reciprocation. 
 
 Cor. Conversely, any conic reciprocated with respect to a 
 circle having a focus as centre, corresponds to a circle. 
 
 431. We can tabulate our results as follows: 
 
 Reciprocal Loci with respect to a Circle. 
 
 (xl). X (xl). 
 
 A hyperbola having the | A circle having the centre 
 
 centre of reciprocation as fo- \ of reciprocation without it. 
 cus. \ See (xxxiii). 
 
POLAR RECIPROCALS. 
 
 383 
 
 (xli). 
 
 An ellipse having the centre 
 of reciprocation as focus. 
 
 (xlil). _ 
 A parabola having the cen- 
 tre of reciprocation as focus. 
 
 (xliii). 
 The directrix of the conic. 
 
 (xliv). 
 The great circle at infinity. 
 
 . (^^^^- 
 The circular points at in- 
 finity. 
 
 (xlvi). 
 The foci of a conic. 
 See Art. 391. 
 
 (xlvii). 
 The focus of reciprocation. 
 
 (xli). 
 A circle having the centre 
 of reciprocation within it. 
 (See xxxiv). 
 
 (xlii). 
 A circle passing through the 
 centre of reciprocation. 
 See (xxxv). 
 
 (xliii). 
 The centre of the circle. 
 
 (xliv). 
 The evanescent circle at the 
 centre of reciprocation. 
 (xlv). 
 The straight lines joining 
 the centre of reciprocation to 
 the circular points, 
 (xlvi). 
 The chords joining the four 
 points in which the correspond- 
 ing conic is cut by radii from 
 the centre of reciprocation to 
 the circular points, 
 (xlvii). 
 The straight line at infinity. 
 
 432. Observing that the polar of a point P with respect 
 to a circle whose centre is 0, is the common chord of that 
 circle, and the circle on OP as diameter, and is therefore at 
 right angles to OP, it follows that the angle which two points 
 subtend at the centre is equal to the angle between their polars. 
 
 Hence, when we reciprocate with respect to a circle, the 
 angle between two straight lines is equal to the angle xohich the 
 corresponding points subtend at the centre of reciprocation. 
 
 433. Moreover the distances of a point and its polar from 
 the centre of the circle contain a rectangle equal to the square 
 on the radius. 
 
384 POLAR RECIPROCALS. 
 
 Hence, when we reciprocate with respect to a circle, the 
 distances of different points from the centre of t^ciprocation are 
 inversely proportional to the distances of the corresponding lines. 
 
 434. By the aid of this property it is easy to calculate the 
 magnitude of the conic corresponding to any circle with respect 
 to another circle. 
 
 For let Ic he the radius of the circle of reciprocation, r the 
 radius of the circle to he reciprocated, and h the distance be- 
 tween their centres. And suppose a and h the semi-axes of 
 the conic, and e its excentricity. 
 
 By symmetry, the line joining the centres of the circles 
 must be the axis of the conic, and the perpendiculars on the 
 tangents at the vertices lie along this line. 
 
 We have, therefore, 
 
 a (1 + e) = y , and a (1 - e) = — -j- , 
 
 ^ ' r — h ^ r-\- h 
 
 k^r P h 
 
 whence, ^= i^^k'" ^ = V(r'-A')' ^^r' 
 
 435. It thus appears that the excentricity of the reciprocal 
 conic is independent of the radius of the circle of reciprocation. 
 The magnitude of this circle therefore only affects the magni- 
 tude, not the foiTii of the resulting figure. Thus it happens 
 in many cases that the magnitude of the circle of reciprocation 
 does not affect a proposition, and it is therefore often con- 
 venient to speak briefly of reciprocation with respect to a point 0, 
 when we mean reciprocation with respect to a circle drawn at an 
 undefined distance from the centre 0. 
 
 We will now give some examples of the manner in which 
 the method of polar reciprocals is applied in tiie solution of 
 problems. 
 
 436. Four fxed tangents are drawn to a conic: to prove that 
 the anharmonic ratio of the points in which they are cut by any 
 variable tangent is constant. 
 
POLAR EECIPROCALS. 385 
 
 Let a, h, c, d denote four fixed tangents to a conic, and let 
 p and q be any other tangents. Reciprocate the figure with 
 respect to a, focus.- then the tangents a, b, c, d correspond to four 
 fixed points A, B, C, D on & circle, and p, q to any other points 
 P, Q on the same circle. 
 
 Now, by Eud. ili. 21, the chords joining A, B, C, D sub- 
 tend the same angles at P as at Q. 
 
 Hence, {P. ABOD] = {Q. ABCD] ; 
 
 therefore by Art. 428, 
 
 [p . abed] — \q . ahcd}. Q. E. V. 
 
 437. Four Jioced points are taken on a conic; to prove that 
 the anharmonic ratio of the pencil Joining them to any variable 
 point on the same conic is constant. 
 
 Let A, B, G, D denote four fixed points on a conic, and let 
 P, Q be any other points. Reciprocate the figure with respect 
 to any point; then the points A, B, G, D, P, Q correspond to 
 tangents a, b, c, d, p, q to another conic, and therefore by the 
 last proposition 
 
 \f . ahcd\ = [q . abed]. 
 
 Hence by Art. 428, 
 
 {P. ABOD] = {Q. ABCD}. q.e.d. 
 
 438. An ellipse is inscribed in a quadrilateral: to prove 
 that any two opposite sides subtend supplementary angles at 
 either focus. 
 
 Reciprocate the whole figure with respect to a circle having 
 the focus as centre. Then, by Art. 430, the conic corresponds 
 to a circle, and the circumscribed quadrilateral to an inscribed 
 quadrilateral. By Eucl. ill. 22, any two opposite angles of this 
 quadrilateral are equal to two right angles. Hence, by Art. 
 432, any two opposite sides of the corresponding quadrilateral 
 subtend at the centre of reciprocation angles which are together 
 equal to two right angles. Hence the proposition is proved. 
 w. 25 
 
386 
 
 POLAR RECIPROCALS. 
 
 439. The following "corresponding theorems" will suffice 
 to shew how the principal properties of conic sections may be 
 deduced from the simplest properties of the circle by the method 
 of polar reciprocals : 
 
 Two tangents to a circle are 
 equally inclined to their chord of 
 contact. 
 
 Two tangents to a circle are 
 equally inclined to the diameter 
 through their point of inter- 
 section. 
 
 Parallel tangents to a 
 
 circle 
 
 touch it at the extremities of a 
 diameter. 
 
 A chord which subtends a 
 right angle at a fixed point on a 
 circle passes through the centre. 
 
 In any circle the sum of the 
 perpendiculars from a fixed point 
 on a pair of parallel tangents is 
 constant. 
 
 If chords of a circle be drawn 
 through a fixed point, the rect- 
 angle contained by the segments 
 is constant. 
 
 Two tangents to a conic mea- 
 sured from their point of inter- 
 section subtend equal angles at 
 a focus. 
 
 The segments of any chord of 
 a conic, measured from the direc- 
 trix subtend equal angles at a 
 focus. 
 
 Tangents at the extremities 
 of a focal chord intersect in the 
 directrix. 
 
 Tangents to a parabola at right 
 angles to one another intersect on 
 the directrix. 
 
 In any conic the sum of the 
 reciprocals of the segments of any 
 focal chord is constant. 
 
 \ The rectangle contained by 
 \ the perpendiculars from the focus 
 ^ of a conic on a pair of parallel 
 I tangents is constant. 
 
 440. The following corresponding theorems illustrate the 
 nature of the great circle at infinity: 
 
 All real points on an evanes- 
 cent conic coincide. 
 
 All real tangents to an evan- 
 escent conic meet in a point. 
 
 All real tangents to the great 
 circle coincide with the straight 
 line at infinity. 
 
 All real points on the gi-eat 
 circle lie on the straight line at 
 infinity. 
 
POLAU EECIPHOCALS, 
 
 387 
 
 All imaginary points on an 
 evanescent conic lie on one of two 
 imaginary straight lines. 
 
 441. The following will 
 theorems : 
 
 Pascal's Theorem. If a hex- 
 agon be inscribed in a conic the 
 points of intersection of opposite 
 sides are collinear. 
 
 If a quadrilateral circumscribe 
 a conic, the intersections of its 
 opposite sides and of its diago- 
 nals will be the vertices of a self- 
 conjugate triangle. 
 
 If two triangles be polar reci- 
 procals with resjiect to any conic, 
 the intersections of the correspond- 
 ing sides lie on a straight line. 
 
 All imaginary tangents to the 
 gi-eat circle pass through one of 
 two imaginary points (viz. the 
 { circular points). 
 
 be also seen to be reciprocal 
 
 Brianchon's Theorem. If a 
 hexagon circumscribe a conic the 
 straight lines joining opposite ver- 
 tices are concurrent. 
 
 If a quadrilateral be inscribed 
 in a conic, the intersections of its 
 opposite sides and of its diagonals 
 will be the vertices of a self-con- 
 jugate triangle. 
 
 If two triangles be polar reci- 
 procals with respect to any conic, 
 the straight lines which join their 
 corresponding vertices meet in a 
 point. 
 
 ExEBCiSES ON Chapter XXIV. 
 
 (236) If a conic touch the sides BO, CA, AB of a triangle 
 in the points A, B, C, then at either focus BC, CA, AB sub- 
 tend equal angles : so also do CA, AB', BC, and so do AB, 
 EC,GA'. 
 
 (237) If two tangents to a parabola meet the directrix in 
 Z, Z', and if S be the focus, the angle ZSZ' or its supplement is 
 double of the angle between the tangents. 
 
 (238) In the plane of the figure 38 (page 310) any point 
 is taken, and through A, A', a straight lines AP, A'F, ap are 
 drawn so as to make the pencils 
 
 {A.BPCO}, {A'.BP'CO}, {a,hpcO\ 
 harmonic. Shew that these three straight lines are concurrent. 
 
 25—2 
 
388 EXERCISES ON CHAPTER XXIV. 
 
 (239) If two conies have a common focus and directrix, the 
 tangent and focal radius at any point on the exterior conic divide 
 harmonically the tangents from that point to the interior conic. 
 
 (240) With a given point as focus four conies can be 
 drawn so as to pass through three given points, and another 
 conic can be described having the same focus and touching the 
 first four conies. 
 
 (241) S is the common focus of two conies, and iS, , S^ are 
 the poles with respect to either of the directrices of the other. 
 Shew that 8, 8^ , 8^ are coUinear. 
 
 (242) Four conies are described each touching the three 
 sides of one of the four triangles ABC, BCD, CAD, ABD, and 
 all having a common focus S : shew that they all have a common 
 tangent. 
 
 (243) The reciprocal of a parabola with regard to a point 
 on the directrix is an equilateral hyperbola. 
 
 (244) The intersection of perpendiculars of a triangle cir- 
 cumscribing a parabola is a point on the directrix. 
 
 (245) The intersection of perpendiculars of a triangle in- 
 scribed in an equilateral hyperbola lies on the curve. 
 
 (246) The tangents from any point to two confocal conies 
 are equally inclined to each other. 
 
 (247) The locus of the pole of a fixed line with regard to a 
 series of confocal conies is a straight line. 
 
 (248) On a fixed tangent to a conic are taken a fixed point 
 A and two moveable points P, Q, such that AP, A Q subtend 
 equal angles at a fixed point 0. From P, Q are drawn two 
 other tangents to the conic, prove that the locus of their point of 
 intersection is a straight line. 
 
 (249) Chords are drawn to a conic, subtending a right 
 angle at a fixed point; prove that they all touch a conic, of 
 which that point is a focus. 
 
EXERCISES OS CHAPTER XXIV. 389 
 
 (250) Prove that two ellipses which have a common focus 
 cannot intersect in more than two points. 
 
 (251) OA, OB are common tangents to two conies which 
 have a common focus S, and A SB is a focal chord. Shew that 
 if the second tangents from A and B to one conic meet in O, 
 and those to the other conic meet in D, then C, D, S are colli- 
 near. 
 
 (252) If two conies circumscribe a quadrilateral and have 
 double contact with another conic, the tangents at the extremi- 
 ties of the chords of contact intersect in two points which 
 divide harmonically one of the diagonals of the quadrilateral. 
 
 (253) Three conic sections have a common tangent, and 
 each touches two sides of the triangle ABC at the extremities 
 of the third side ; shew that if the sides of this triangle meet 
 the common tangent in A', B, C, each of the points of contact 
 of that tangent will form with A!, E, C a harmonic range. 
 
 (254) A triangle ABG is inscribed in a conic, and the tan- 
 gents at the angular points A, B, C are produced to meet the 
 opposite sides in P, Q, B. From these points other tangents 
 are drawn to touch the conic in A', B", C Shew that if the 
 tangents at ^, .B, C form a triangle aic, and the tangents at 
 A', B, C" form a triangle a'b'c, then A, a, a' are coUinear, so 
 are B, b, V, and so are C, c, c. 
 
CHAPTER XXV. 
 
 CONICS DETEUMINED BY ASSIGNED CONDITIONS. 
 
 442. We shewed in Art. 147 that a conic can generally be 
 found to satisfy five simple conditions, each condition giving 
 rise to an equation coiuiecting the coeiBcicnts in tlic general 
 equation to a conic. It will, however, be observed that if any 
 of these equations are of the second or a higher order, we shall 
 have two or more solutions indicating two or more conies satis- 
 fying the given conditions. 
 
 Again, in Art. 201 we gave an example of a double con- 
 dition, when we shewed that if the centre of a conic be assigned 
 this is equivalent to two simple conditions being given: and 
 it will presently be seen that conditions may occur equivalent 
 to three or foui- or five simple conditions. 
 
 In order therefore that we may in all cases be able to judge 
 of the sufficiency of any assigned conditions to determine a conic, 
 it will be desirable 
 
 (1°) To determine what conditions shall be regarded as simple 
 conditions, classifying them according to the nature of the 
 relations to which they give rise, among the coefficients of the 
 general equation. 
 
 (2°) To consider how many conies can be drawn to fulfil 
 five simple conditions when the classes of those conditions are 
 assigned, and 
 
CONIGS DETERMINED BY ASSIGNED CONDITIONS. 391 
 
 (3°) To analyse more complicated conditions, and to deter- 
 mine to how many simple conditions they are equivalent, assign- 
 ing the class of those simple conditions. 
 
 443. We shall only find it necessary to make two classes 
 of simple conditions, which we shall distinguish as point-condi- 
 tions and line-conditions. We shall find that all other conditions 
 of common occurrence may be regarded either as particular cases 
 of these two, or as made up of repetitions of them. 
 
 444. Def. We shall call two points conjugate with re- 
 spect to a conic when each lies on the polar of the other, and we 
 shall call two straight lines conjugate when each passes through 
 the pole of the other. 
 
 445. Let 
 
 /(a, (S, 7) = Ma' 4- v/S' + w-f + 2«'/37 + SwVa + 2M)'a/3 = 
 be the equation in trilinear coordinates to a conic section, and let 
 (a„/3„7,), (as,^2'72) ^^ ^^7 ^^ points in the same plane. 
 
 The equation 
 Ma^fXj + v^fi^ + t07i7a + "' (^i72 + /SaYi) + «' (7i«2 + iP-b 
 
 may be written in either of the forms 
 
 df r, df df ^ 
 
 "' ''-di.-^^^w^^^^wr ' 
 
 and expresses the condition (Art. 232) that each of the points 
 (a„ ^„ 7,), (a,, /8,, 7^) lies on the polar of the other with respect 
 to the conic, or that the two points are conjugate with respect to 
 the conic. 
 
 It will be observed that when the two points (a„ /3,, 7,), 
 K, /Sj. 72) are given, the condition that they should be conjugate 
 furnishes us with a simple equation, connecting the six co- 
 efficients in the general equation to a conic. Five such con- 
 
392 
 
 CONICS DETERMINED BY ASSIGNED CONDITIONS. 
 
 ditions will therefore snflSce to determine the five ratios of the 
 coefficients in the equation, and therefore to determine the 
 conic. 
 
 446. Let l^oi + mfi + w,7 = 0, 
 
 and l^a. + mfi + n^ = 0, 
 
 be two straight lines. Then the equation 
 
 = 
 
 u, 
 
 to, v, Z, 
 
 w', 
 
 V, u, m 
 
 t 
 V, 
 
 u, w, n, 
 
 h> 
 
 »»„ w„ 
 
 will express the condition (Art. 233) that each passes through 
 the pole of the other, or that the two straight lines are conjugate 
 with respect to the conic. 
 
 It will be seen that this equation is a quadratic in u, v, w, 
 u, v, w. Hence when two straight lines are given, the con- 
 dition that they should be conjugate furnishes us with a quad- 
 ratic equation connecting the six coefficients in the general 
 equation to the conic. Therefore if a condition such as this 
 be substituted for one of the conditions in the case last con- 
 sidered, there will be an ambiguity in the determination of the 
 coefficients of the trilinear equation unless it happen that the 
 quadratic have equal roots. 
 
 447. Let 
 f{p, q, r)= up^ + vc^ + wr^ 4 iu'qr + 2t;'rp + 2w'pq = 0, 
 
 be the equation to a conic section in tangential coordinates, and 
 let {p^, gj, rj, {p^, q^, r,) be any two straight lines in the same 
 plane. 
 
 The equation 
 
 »PtP, + «?. i> + «"■. »■» + «' C?/, + ?,»•,)+«' (»•, Pi + r^pd 
 
 + w'{p,q,+p,q,) = o, 
 may be written in either of the forms 
 
CONICS DETEKMINED BY ASSIGNED CONDITIONS. 
 
 393 
 
 or 
 
 df , df df ^ 
 df df df ^ 
 
 and expresses the condition (Art. 232) that each of the straight 
 lines ( jOj, q^, rj, (p,, q^, r^ passes through the pole of the other 
 with respect to the conic, or that the two straight lines are con- 
 jugate with respect to the conic. 
 
 It will be observed that when the two straight lines (Pi , g", , O , 
 (^j, g-,, r^) are given, the condition that they should be con- 
 jugate furnishes us with a simple equation connecting the six 
 coefficients in the general equation to a conic. Five such con- 
 ditions will therefore suffice to determine the five ratios of the six 
 coefficients in the equation, and therefore to determine the conic. 
 
 448. Let ^1 1' + ''"'li + ^i*" = *^i 
 
 and l^p + »n,2 + '"■^^ — 0> 
 
 be two points. Then the equation 
 
 = 
 
 u, 
 
 w, 
 
 V, 
 
 'x 
 
 w', 
 
 V, 
 
 u, 
 
 m, 
 
 t 
 V, 
 
 t 
 
 u. 
 
 w, 
 
 «i 
 
 h, 
 
 m^, 
 
 «., 
 
 
 
 will express the condition (Art. 233) that each lies on the polar 
 of the other, or that the two points are conjugate with respect 
 to the conic. 
 
 It will be seen that this equation is a quadratic in u, v, w, 
 u, v, w. Hence when two points are given, the condition that 
 they should be conjugate furnishes us with a quadratic equation 
 connecting the six coefficients in the general equation to a conic. 
 Therefore if a condition such as this be substituted for one of the 
 conditions in the case last considered, there will be an ambiguity 
 in the determination of the coefficients of the tangential equation 
 unless it happen that the quadratic have equal roots. 
 
394 CONICS DETERMINED BY ASSIGNED CONDITIONS. 
 
 449. Def. The condition that a conic be such that with 
 respect to it two given points are conjugate, is called a jiomt- 
 condition. 
 
 The condition that a conic be such that with respect to it 
 two given straight lines are conjugate, is called a line-con- 
 dition. 
 
 450. To fulfil five point-conditions there can he drawn one 
 and only one conic. 
 
 For, using trilinear coordinates, each of the five conditions 
 will furnish us with a simple equation (Art. 445) connecting the 
 coefficients of the general equation to a conic. These five equa- 
 tions will determine the five ratios of the coefficients without 
 ambiguity, and therefore will determine one and only one conic 
 fulfilling the given conditions. 
 
 451. To fulfil four point-conditions and one line-condition 
 there cannot he drawn more than two conies. 
 
 For, using trilinear coordinates, each of the four point- 
 conditions will furnish us with a simple equation (Art. 445) 
 connecting the coefficients of the general equation. And the 
 line-condition will furnish us with a fifth equation, a quadratic 
 (Art. 446), connecting the same coefficients. These five equa- 
 tions will determine the five ratios of the coefficients, but since 
 one is a quadratic there will in general be two solutions, indi- 
 cating two conies fulfilling the given conditions. 
 
 452. To fulfil three point-conditions and two line-conditions 
 there cannot he drawn more than four conies. 
 
 For, using trilinear coordinates, each of the three point-con- 
 ditions will furnish us with a simple equation (Art. 445) con- 
 necting the coefficients of the general equation and the two line- 
 conditions will furnish us with two more equations, quadratics 
 (Art. 446), connecting the same coefficients. These five equa- 
 tions will determine the five ratios of the coefficients, but since 
 two are quadratics there will in general be four solutions, indi- 
 cating four conies fulfilling the given conditions. 
 
COMCS DETERMINED BY ASSIGNED CONDITIONS. 395 
 
 453. To fulfil three line-conditions and two point-conditions 
 there cannot he drawn more than four conies. 
 
 For, using tangential coordinates, each of the three line^ 
 conditions will furnish us with a simple equation (Art. 4i7) 
 connecting the coefficients of the general equation, and the two 
 point-conditions will furnish us with two more equations, 
 quadratics (Art. 448), connecting the same coefficients. These 
 five equations will determine the five ratios of the coefficients, 
 but since two are quadratics there will in general he four 
 solutions, indicating four conies fulfilling the given conditions. 
 
 454. To fidfil four line-conditions and one point-condition 
 there cannot he drawn more than tioo conies. 
 
 For, using tangential coordinates, each of the four line-con- 
 ditions will furnish us with a simple equation (Art. 447) con- 
 necting the coefficients of the general equation. And the point- 
 condition will furnish us with a fifth equation, a quadratic (Art. 
 448), connecting the same coefficients. These five equations will 
 determine the five ratios of the coefficients, but since one is 
 a quadratic there will in general be two solutions, indicating two 
 conies fulfilling the given conditions. 
 
 455. To fulfil five line-conditions there can he dravm one and 
 only one conic. 
 
 For, using tangential coordinates, each of the five con- 
 ditions will furnish us with a simple equation (Art. 447) con- 
 necting the coefficients of the general equation to a conic. 
 These five equations will determine the five ratios of the coeffi- 
 cients without ambiguity, and therefore will determine one and 
 only one conic fulfilling the given conditions. 
 
 456. It remains that we should analyse the conditions most 
 usually assigned, and determine to how many point- or line- 
 conditions they may severally be equivalent. We shall then be 
 able to apply the five preceding articles to determine how many 
 conies (at most) can be drawn in cases where such conditions 
 are given. 
 
396 CONICS DETERMINED BY ASSIGNED CONDITIONS. 
 
 I. Given a point on a conic. 
 
 Since a point on a conic lies on its own polar it is conjugate 
 to itself. This therefore is equivalent to one point-condition. 
 
 II. Given a tangent to a conic. 
 
 Since a tangent to a conic passes through its own pole it is 
 conjugate to itself. This therefore is equivalent to one line-con- 
 dition, 
 
 III. Given a diameter. 
 
 Any diameter passes through the centre, which is the pole of 
 the straight line at infinity. Hence a diameter and the straight 
 line at infinity are conjugate lines. This therefore is equivalent 
 to one line-condition. 
 
 IV. Let a given point he the pole of a given straight line 
 with respect to a conic. 
 
 Let F be the given point and QE the given straight line. 
 Then the polar of P passes through Q, which is one point-con- 
 dition ; and the polar of P passes through B, which is another. 
 Hence the data are equivalent to two point-conditions. 
 
 Or we may reason thus : the pole of QR lies on PQ, which 
 is one line-condition, and the pole of QB lies on PB, which is 
 another. Hence the data are equivalent to two line-conditions. 
 
 Therefore the pole of a given straight line being given may 
 be regarded as equivalent to two point-conditions or two line-con- 
 
 V. Given a point on a conic and the tangent thereat. 
 
 This is a particular instance of the last case, the given pole 
 lying on the given polar. It is therefore equivalent to two 
 point-conditions or two line-conditions. 
 
 VI. Given an asymptote. 
 
 This is an instance of the last case, the given point being at 
 
CONICS DETEEMINED BY ASSIGNED CONDITIONS. 397 
 
 infinity. It is therefore equivalent to two point-conditions or two 
 line-conditions. 
 
 VII. Given the direction of an asymptote. 
 
 In this case one of the points in which the conic meets the 
 straight line at infinity is given. It is therefore an instance 
 of (i) and is equivalent to one point-condition. 
 
 VIII. Given that the conic is a parabola. 
 
 Or that the line at infinity is a tangent. This is an in- 
 stance of (ii) and is therefore equivalent to one line-condition. 
 
 IX. Given that the conic is a circle, 
 
 Or that it passes through the two circular points. By (i) 
 this is equivalent to two point-conditions. 
 
 X. Given the centre. 
 
 The centre is the pole of the straight line at infinity; hence 
 this case is an instance of (iv) and is therefore equivalent to two 
 point-conditions or two line-conditions. 
 
 XL Given a self-conjugate triangle. 
 
 A triangle is self-conjugate if each pair of angular points are 
 conjugate. Hence this case is equivalent to three point-con- 
 ditions. 
 
 Or again, a triangle is self-conjugate if each pair of sides 
 are conjugate lines. Hence it is equivalent to three line-con- 
 ditions. 
 
 Therefore a self-conjugate triangle being given, constitutes 
 three point-conditions or three line-conditions. 
 
 XII. Given in position {not in magnitude) a pair of con- 
 jugate diameters. 
 
 A pair of conjugate diameters form with the straight line 
 at infinity a self-conjugate triangle. Hence this is an instance 
 of (xi) and is equivalent to three point-conditions or three line- 
 conditions. 
 
398 CONICS DETERMINED BY ASSiaNED CONDITIONS. 
 
 XIII. Given the directions of a pair of conjugate diameters. 
 
 The points wliere any lines in these directions meet the 
 line at infinity arc conjugate points. Hence this is equivalent 
 to one point-condition. 
 
 XIY. Given in position an axis. 
 
 Tlie axis is a diameter, and this being given is equivalent 
 to one line-condition. But the direction of the conjugate dia- 
 meter is known to be at right angles to this, which gives by (xiil) 
 a point-condition. Therefore that an axis be given in position 
 is equivalent to one point-condition and one line-condition, 
 
 XV. Given in position the two axes. 
 
 This is no more than a case of (xiv) and is equivalent to 
 three point-conditions or three line-conditions. 
 
 XVI. Given a focus. 
 
 The two tangents from the given point to the great circle at 
 infinity are tangents to the conic. Hence two tangents are 
 given, and therefore by (ii) the data are equivalent to two 
 line-conditions. 
 
 XVII. Given a similar and similarly situated conic. 
 
 Since .similar and similarly situated conies are those which 
 meet the straight line at infinity in the same points, this is equi- 
 valent to two points being given. Hence by (i) it may be 
 treated as two point-conditions. 
 
 Aol. When a conic has to be drawn subject to conditions 
 having reference to anotlier conic, we may often estimate the 
 value of the conditions by considering the particular case in 
 which the latter conic reduces to two straight lines. Thus : 
 
 XVIII. Given a conic having double contact with the re- 
 quired one. 
 
 Consider the case when the given conic reduces to two 
 straight lines. Then we have two tangents given, furnishing 
 
CONICS DETERMINED BY ASSIGNED CONDITIONS. 399 
 
 two line-conditions. Hence we may infer that generally a conic 
 having double contact with the required one being given is 
 equivalent to two line-conditions. 
 
 The following are examples of the application of our results : 
 
 458. Only one parabola can he inscribed in a given qua- 
 drilateral. 
 
 That the required conic is a parabola is one line-condition 
 (viil) ; that it touch the sides of the quadrilateral gives four 
 more. Hence we have five line-conditions, and therefore (Art. 
 455) the conic is absolutely determined. 
 
 459. Not more than two parabolas can he described about a 
 given quadrilateral. 
 
 That the required conic is a parabola is a line-condition 
 (viii); that it circumscribe the quadrilateral gives four point- 
 conditions (i). Hence (Art. 451) not more than two solutions 
 are possible. 
 
 460. Two conies can generally he described with given fod 
 and passing through a given point. 
 
 For the foci give four line-conditions (xvi) ; and the point 
 gives a point-condition (i). Hence (Art. 454) there will gene- 
 rally be two solutions. 
 
 461. Only one conic can be described with given foci so as to 
 touch a given straight line. 
 
 For the foci give four line-conditions (xvi), and the tangent 
 gives a fifth (ii). Hence (Art. 455) there is only one solution. 
 
 462. Only one conic can be described with a given centre, 
 with respect to which a given triangle shall he self-conjugate. 
 
 For the self-conjugate triangle may be regarded as giving 
 three point-conditions (xi), and the given centre as giving two 
 more (x). Hence (Art. 450) there will be only one solution. 
 
400 EXERCISES ON CHAPTER XXV. 
 
 Exercises on Chapter XXV. 
 
 (255) Given two tangents and their chord of contact, shew 
 that only one conic can be described so as to touch a given 
 straight line. 
 
 (256) Given two tangents and their chord of contact, shew 
 that only one conic can be described so as to pass through a 
 given point. 
 
 (257) Two confocal conies cannot have a common tangent. 
 
 (258) Three confocal conies cannot have a common point. 
 
 (259) Two concentric conies cannot circumscribe the same 
 triangle. 
 
 (260) Two concentric conies cannot be inscribed in the same 
 triangle. 
 
 (261) Only two conies can be described about a triangle 
 having an axis in a given straight line. 
 
 (262) Only two conies can be inscribed in a triangle and 
 have an axis in a given straight line. 
 
 (263) Four circles can generally be described through a 
 given point so as to have double contact with a given conic. 
 
 (264) Four circles can generally be described so as to touch 
 a given straight line and have double contact with a given 
 conic. 
 
 (265) Only one conic can be described having double con- 
 tact with a given conic, and such that a given triangle is self- 
 conjugate with respect to it. 
 
 (266) One conic can generally be inscribed in a given quad- 
 rilateral so as to have its centre on a given straight line. 
 
CHAPTER XXVI. 
 
 EQUATIONS OF THE THIRD DEGREE. 
 
 463. Definitions. The curve determined by an equation 
 of the third degree in trilinear coordinates, or in any other 
 system in which a point is represented by coordinates, is called 
 a cubic curve, or a cubic locus. 
 
 The curve determined by an equation of the third degree 
 in tangential coordinates where a straight line is represented 
 by coordinates, is called a cubic envelope. 
 
 464. Every straight litie meets a cubic locus in three points, 
 real or imaginary, coincident or distinct: and from every point 
 there can be drawn to a cubic envelope three tangents real or ima- 
 ginary, coincident or distinct. 
 
 For to find the points of intersection (or the tangents) we 
 have to solve together the equation to the given, straight line (or 
 the given point), which is of the first degree, and the equation to 
 the curve, which is of the third degree. Hence we shall have 
 three solutions real or imaginary, equal or unequal. 
 
 All the solutions however cannot be imaginary, since ima- 
 ginary roots enter into an equation by pairs. One at least must 
 be real, and the other two either both real or both imaginary. 
 Hence every straight line meets any cubic locus in one or three 
 real points, and from every point there can be drawn either one 
 or three real tangents to any cubic envelope. 
 
 Cor. By Art. 415, a cubic locus is a curve of the third 
 order, and a cubic envelope is a curve of the third da,ss. 
 W. 26 
 
402 equations of the third degree. 
 
 465. Definitions. 
 
 I. A point in which two branches of a curve intersect, or at 
 which there are two distinct tangents, is called a double point. 
 
 II. A point in which more than two branches intersect, or 
 at which there are more than two distinct tangents, is called a 
 multiple point, 
 
 A multiple point is said to be of the n*" order when n branches 
 intersect in it, or when n tangents can be drawn at it. 
 
 III. When a closed branch of a curve becomes indefinitely 
 small so as to constitute an isolated point satisfying the condi- 
 tions of a point on the curve, it is called a conjugate point. From 
 the consideration that a conjugate point is an indefinitely small 
 oval, it follows that any straight line through it must be regarded 
 as the ultimate position of a chord of the oval. Any such 
 straight line will therefore satisfy the condition of meeting the 
 curve in two coincident points. 
 
 IV. A cusp is a point on a curve at which two branches 
 meet a common tangent and stop at that point. Any straight 
 line through a cusp must be regarded as cutting both branches 
 at the cusp, and therefore satisfies the condition of meeting the 
 curve in two coincident points. 
 
 V. If the two branches having the common tangent be con- 
 tinued through the point, then the point is called a point of oscu- 
 lation. 
 
 VI. A point at which a curve crosses its tangent, is called a 
 point of inflexion. 
 
 If P be a point of inflexion and Q be another point on the 
 curve very near to P, the straight line QP being produced 
 through P, will meet the curve again in another point Q, very 
 near to P. If this straight line turn about the fixed point P 
 until it ultimately coincide with the tangent, since it must ulti- 
 mately be a tangent to the branch on each side of the point P, it 
 follows that as Q approaches P so also will Q', and that they will 
 
EQUATIONS OP THE THIRD DEGREE. 403 
 
 both simultaneously arrive at coincidence with P. Hence the 
 tangent at a point of inflexion may be regarded as meeting the 
 curve in three coincident points. 
 
 All the foregoing are often classed together as singular points. 
 
 VII. A double tangent to a curve is a tangent which 
 touches the curve in two distinct points. 
 
 466. It will be observed from the definitions in the last 
 article, that a double point, cusp, and conjugate point are 
 marked by the same property, that any straight line through 
 such a point meets the curve in two coincident points, and that 
 a tangent thereat meets the curve in three coincident points. 
 But they are distinguished by the property that the two tangents 
 at a double point axe distinct, at a cusp — coincident, and at a 
 conjugate point — imaginary. 
 
 Again, a cusp and a point of inflexion are both characterised 
 by the property that the tangent at such a point meets the curve 
 in three coincident points, but they are distinguished by the fact 
 that a straight line other than the tangent meets the curve in 
 only one point at a point of inflexion, but in two points at 
 a cusp. 
 
 46.7. A cubic curve cannot have more than one double point, 
 cusp or conjugate point. 
 
 For, If possible, let it have two such points F and Q, and 
 join them by a straight line. Then this straight line cuts the 
 curve in two coincident points at P, and in two coincident points 
 at Q (Art. 465), i. e. in four points altogether, 
 
 Which is impossible (Art. 464). 
 
 468. A cubic curve cannot have a double tangent. 
 
 H'or such a tangent, touching at P and at Q, would meet the 
 curve in two coincident points at P and in two coincident points 
 at Q, i. e. in four points altogether. 
 
 Which is impossible (Art. 464). 
 
 26—2 
 
404 EQUATIONS or THE THIRD DEGREE. 
 
 469. A cubic curve cannot have a point of osculation. 
 
 For the tangent at a point of osculation, touching both 
 branches of the curve, would meet it altogether in four points. 
 
 Which is impossible (Art. 464). 
 
 470. The general homogeneous equation of the third degree 
 in three coordinates consists of ten terms, viz. the three terms 
 whose arguments are 
 
 /»8 ».' »' 
 
 ^, y > «> 
 
 the six, jc^y, afa ; y'z, i/'x ; s'a;, s'y ; 
 
 and the one, xyz. 
 
 If the coefficient of any one of these terms be arbitrarily 
 assigned, those of the remaining nine Will be undetermined 
 constants. 
 
 Hence the general equation of the third degree involves nine 
 undetermined constants, and can therefore generally be made to 
 satisfy nine independent conditions. 
 
 Hence a curve represented by an equation of the third 
 degree can generally be drawn through nine given points, or 
 otherwise made to satisfy nine given conditions. 
 
 471. If the nine conditions be given, the equation to 
 the curve can generally be determined. If any less number 
 (r suppose) of conditions be given, a series of curves can gene- 
 rally be drawn to satisfy them, and their general equation will 
 involve the complementary number (9 — r) of undetermined con- 
 stants. 
 
 For example, we shall shew in the next article that the 
 general equation in trilinear coordinates to a curve of the third 
 order, circumscribing the triangle of reference ABC, and whose 
 tangents at A, B, C are represented by the equations 
 
 M = 0, t) = 0, w = 0, 
 
 respectively, has for its equation 
 
 0/87 + lua? + mv^ + nvyf = 0, 
 
EQUATIONS OF THE THIRD DEGREE. 405 
 
 which involves three undetermined constants I, m, n, the num- 
 ber of given conditions having been six. 
 
 472. The general equation of the third degree in trilinear 
 coordinates may be written 
 
 a/87 + a' (i,a + m,/3 + n^i) +^{lji + m^ + n^y) 
 
 + '/{l,a + m^ + n^y)=0. 
 
 If we take three points on the curve as the angular points 
 of the triangle of reference, then since the equation must be satis- 
 fied by any of the systems 
 
 (^=0,7 = 0), (7 = 0, a=0), (a=0, /3 = 0), 
 we obtain 
 
 Z, = 0, »i,= 0, M3 = 0, 
 
 and the equation reduces to 
 
 0^7 + a' («t,;8 + n,7) + j8= («,7 + Z,a) + 7* {l^a + mfi) = 0. 
 
 The tangent to the locus of this equation at the point A 
 (/3 = 0, 7 = 0) is readily seen to be given by the equation 
 
 jw,^ + n^ = 0. 
 Similarly, n,7 + Iji =0 
 
 and ZjK + OT,/3= 
 
 represent the tangents at the points B and C. 
 Hence the equation 
 
 a/87 + ^^"^ + ^v0' + nwf = 
 
 represents a cubic touching at the points of reference the straight 
 
 lines 
 
 M = 0, v=0, w = Q. 
 
 Or, more generally, if 
 
 x = 0, y = 0, 3 = 
 represent any equations to straight lines, then 
 osyz + lua? + mvy* + nws' = 
 
406 EQUATIONS OF THE THIRD DEGREEi 
 
 is the general equation of a cubic to which 
 
 u—0, v = 0, w = 
 are tangents, and 
 
 x = 0, y = 0, s = 
 
 the chords of contact. 
 
 473. Similarly, in tangential coordinates, 
 
 xye + lux^ + mvy' + nwz' = 
 is the general equation to a curre of the third class, on which 
 
 M = 0, D = 0, 10=0 
 
 ;re the points of contact of tangents intersecting in the points 
 x=0, y = 0, s=0. 
 
 474. To find the general equation in triUnear coordinates to 
 a cubic curve having a double point, cusp, or conjugate point at 
 one of the points of reference. 
 
 The general equation to a cubic curve may be written 
 
 a/37 + a" (^i* + »w./3 + «.7) + ^ (Js + "^fi + «,7) 
 
 If the point -4 be a double point, cusp, or conjugate point, 
 any straight line through A must meet the cubic in two coinci- 
 dent points at A. Any such straight line may be represented 
 by the equation 
 
 /3 = «7- 
 
 Hence, substituting for /3 in the general equation, the result- 
 ing equation must have two roots 7 = 0. Hence the terms in- 
 volving a' and a^ must vanish, and therefore we must have 
 
 Z, = 0, Wj = 0, n, = ; 
 
 these are therefore the conditions that the locus of the equation 
 (1) should have a double point, cusp, or conjugate point at A. 
 
EQUATIONS OP THE THIllD DEGREE. 407 
 
 "When these conditions are satisfied we may express the con- 
 stants differently, and write the equations 
 
 a (//S' + 9^1 + ^7') + W + rti^'i + n^rf + rv' = 0, 
 
 which is therefore the general equation to a cubic having a 
 double point, cusp, or conjugate point given by 
 
 = 0, 7=0. 
 
 475. To find the equation to the tangents at ike double point 
 or cusp to the cubic curve whose equation is 
 
 a (J^ +g^y + h'f) + l^ + m^'^+n^'f + ^7' = (1). 
 
 Let /3 = «7 be a tangent at A, then substituting in the equa- 
 tion, the resulting equation 
 
 a.'f{fi£' + ffK+h) +'/{Ik'' + iuk" + nK + r) = 
 
 must have all three roots equal, (y = 0). 
 
 Hence //c' +gK + h = 0, 
 
 giving the two values for k corresponding to the two tangents. 
 
 The equation to the tangents is therefore 
 //3' + ^y87 + V=0. 
 
 If the two roots of the quadratic be equal, the point A will 
 be a cusp : if they be real and unequal it will be a double point : 
 if they be imaginary it will be a conjugate point. 
 
 COK. The equation 
 
 a (//S" + ff^y + hrf) + Zj8' + «i/3''7 + n^i' + ry' = 
 
 represents a cubic having at ^ a double point, conjugate point, 
 or cusp, according as g' — 4/A is positive, negative, or zero. 
 
 476. Every curve of the third order which has a cusp is also 
 of the third class. 
 
 Let there be a curve of the third order having a cusp A, and 
 let B be any point whatever in its plane. We have to shew that 
 only three tangents can be drawn from B to the curve. 
 
408 EQUATIONS OF THE THIRD DEGREE. 
 
 Let BG be one of the tangents from the point B and let it 
 meet the tangent at the cusp in C. Then if we refer the cubic 
 to the triangle ABC its equation may be written (Art. 475) 
 
 0.^+ l^ + m^rf + n^rf + rv' = 0. 
 
 But since a = is a tangent the equation 
 
 7/3' + ni/3'7 + n^rf + r^ = 0, 
 
 must have two of its roots equal, and may therefore be written 
 
 Z()3 + /^)(/3 + ^r = 0. 
 
 Hence the equation to the cubic may be written 
 
 ay3'+Z(y9 + ^)(/9 + 1^)^ = 0. 
 
 Now let a = Ky be any tangent from B to the curve. Sub- 
 stituting for a in the equation to the curve the resulting equa- 
 tion 
 
 must have two of its roots equal. 
 
 The condition that this should be the case will be found 
 to be 
 
 4/««' + ?«' (8^' + 20fw -v*)+ 4i!'/c (/* - 1;)' = 0, 
 
 a cubic equation giving three values of k of which one is zero. 
 Then there are three tangents from B to the curve, one of which 
 is the known tangent a = 0. 
 
 Hence from any point, only three tangents can be drawn to 
 a cubic which has a cusp, q.e.d. 
 
 477. If a cubic curve have three real points of inflexion, the 
 tangents at which do not meet in a point, we may take those 
 tangents as lines of reference for trilinear coordinates. 
 
 Each line of reference will now meet the cubic in three 
 coincident points ; therefore if we substitute a = in the equa- 
 tion to the cubic the resulting equation must have three equal 
 roots ; that is, the terms free from a must form a perfect cube, 
 {m^ + n'y)' suppose. So the terms free from j9 must form a 
 
EQUATIONS OP THE THIED DEGREE. 409 
 
 perfect cube, which (since the coefficient of y' is ah-eady known 
 to be «') may be written {ny + h)\ Similarly the terms free 
 from 7 most be {la. + m^f. Hence the equation may be written 
 {m^ + «7)» + (^ + fa)> + (j^ + m^y_ i^^ -m^-rir/ + hafiy = 0, 
 or, expressing the constant A differently, 
 
 {h + jw/3 + wyY + kot^y = Q. 
 
 (The argument would not hold if one or more of the points 
 of mflexion were imaginary, as in such case diferent cube roots 
 of n', &c. might be involved.) 
 
 478. Cob. The points of contact of the tangents, or the 
 points of inflexion themselves are given by 
 
 (a = 0, OT/3 + n7 = 0), {^ = 0, ny + h = 0), (y = 0, Za + m^ = 0). 
 
 Hence they all lie on the straight line 
 
 lot + »«j8 + wy = 0. 
 
 Therefore, if p, q, r he the tangents at three real points of in- 
 flexion P, Q, R on a cubic, then either p, q, r are concurrent, 
 or P, Q, R are collinear. 
 
 479. If a cubic curve have three real points of inflexion they 
 will be collinear. 
 
 For if not we may take them as points of reference for 
 trilinear coordinates. Then since the tangents at the three 
 points are concurrent (Art. 478) we may represent them by the 
 equations 
 
 m^ — ny = 0, ny — Za = 0, la — m^ = 0. 
 
 Hence the equation to the cubic may be written (Art. 472) 
 
 Imna^y + Tj'a." (w/3 - ny) + fim^^ (ny - la) + imV {h - m^) = 0. 
 
 And since the line m^ — ny = Q is a tangent at a point of in- 
 flexion, it meets the cubic in three coincident points ; therefore 
 we must have 
 
 1 - ^ + V = 0, 
 
410 EQUATIONS OF THE THIRD DEGREE. 
 
 Similarly, since «7 — Za = and la — jw/3 = are tangents 
 at points of inflexion, 
 
 l-i;4-\ = 0, 
 
 l-X + ^ = 0. 
 
 But these equations are inconsistent, as we find by adding 
 them together. 
 
 Hence the points of reference cannot be points of inflexion. 
 Therefore &c. Q. E. D. 
 
 480. The theorems of the following articles, being expressed 
 in a most general form in abridged notation, will be found very 
 useful in interpreting equations of the third degree, and will 
 often enable us to recognise by simple inspection the existence 
 of singular points. 
 
 481. If M = 0, v = 0, w=0, x = 0, y = 0, s = 
 are the equations of six straight lines, then the equation 
 
 uvw = hct/s 
 
 will represent a cubic locus passing through the nine points given 
 hy the intersection of the straight lines 
 
 (m=>=0, a! = 0), (m = 0, 2^ = 0), (u = 0, 3=0), 
 
 (« = 0, a! = 0), (» = 0, 3, = 0), (v = o, g = o), 
 
 (w = 0, 35 = 0), (w = 0, ^ = 0), (w = 0, a = 0); 
 
 and h can be determined so as to make the equation represent 
 a cubic passing through any tenth point. 
 
 The proof follows immediately aa in the cotresponding pro- 
 position respecting conies, Art. 159. 
 
 But we may observe with respect to our result that it is 
 only because the first nine points lie three and three on six 
 straight lines that we are able to describe a cubic passing 
 through a tenth point. If the first nine points had been un- 
 connected and perfectly general they would have sufficed to 
 determine the cubic absolutely, as we fihewed in Art. 470. 
 
EQUATIONS OF THE ThIED DtlOiiEB. 411 
 
 482. 80 if 
 
 M = 0, r = 0, w = 0, x = 0, y = 0, z = 
 represent points in tangential coordinates, the equation 
 
 uvw = hxys 
 represents a cubic envelope touching the nine straight lines 
 
 (w = 0, x=0)j {u=0, y = 0), (m = 0, s = 0), 
 (v = 0,x=0), {v=0,t/ = 0), (v = 0, » = 0), 
 {w=0,x = 0)t (^ = 0,^ = 0), (w = 0, s = 0); 
 
 and k can be determined so as to make the Equation represent a 
 cubic envelope touching any tenth straight line. 
 
 483. Consider the equation 
 
 u'to = kxys. [trUinear 
 
 This is a particular case of the equ&tion of Art. 481, the 
 straight lines m = and v = being coincident The equation 
 represents a cubic locus to which the straight lines 
 
 a! = 0, y = 0, a=0 
 
 are tangents, their points of contact lying all on the straight line 
 M = 0, and the other points where they meet the curve lying on 
 the straight line w = 0. 
 
 484 So the equation 
 
 m'w = kxyz {tangential 
 
 represents a cubic envelope passing through the points 
 a; = 0, y = 0, = 0, 
 
 and touching at those points the straight lines 
 
 (m = 0, a; = 0), (m = 0, 2^ = 0), (m = 0, «=0), 
 
 and also touching the straight lines 
 
 (w = 0, a; = 0), (w = 0, y = 0), (w = 0, 8 = 0), 
 
412 EQUATIONS OF THE THIRD DEGREE. 
 
 485. Consider the equation 
 
 w' = Icxyz, \triltnear 
 
 This is a particular case of the last equation, the straight 
 lines M = and w = being coincident. The equation repre- 
 sents a cubic locus having three points of inflexion in the 
 straight line u = 0, the tangents at those points of inflexion being 
 
 given by 
 
 a; = 0, y = 0, = 0. 
 
 For each of the straight lines 
 
 a; = 0, 2^ = 0, z = 
 meets the curve in three coincident points determined by m' = 0. 
 
 486. So the equation 
 
 m' = Icxyz [tangential 
 
 represents a cubic envelope having points of inflexion at 
 
 a; = 0, y=0, 8 = 0, 
 the tangents at these points intersecting in the point u = 0. 
 
 487. Consider the equation 
 
 m' = Tex*y. [trilinear 
 
 This equation represents a cubic locus in which a: = is the 
 tangent at a cusp, y = the tangent at a point of inflexion, and 
 M = the chord of contact. 
 
 For a; = 0, y = both meet the cubic in three coincident 
 points on the line m = 0, but m = cuts it in two points on 
 aj = 0, and in only one on y = 0. Hence (m = 0, a; = 0) must be 
 a cusp, and (m = 0, y = 0) a point of inflexion. 
 
 488. So the equation 
 
 m' = Jca^y [tangential 
 
 represents a cubic envelope having a cusp at a; = 0, and a point 
 of inflexion at y = 0, the tangents at these points intersecting 
 in M = 0. 
 
EQUATIONS OF THE THIRD DEGREE. 413 
 
 489. Consider the equation 
 
 uv* = Tcxy'. [tnUnear 
 
 The straight lines m = and a; = are tangents : their points 
 of contact lying respectively on y = and v = 0, and their 
 point of intersection also lying on the cubic, and there is a sin- 
 gular point at the intersection of v = and y=0. 
 
 For V = meets the curve in two incident points lying on 
 y = 0, and y = meets it in two coincident points lying on 
 « = 0. Hence at the point of intersection {y = 0, t> = 0) both lines 
 satisfy the condition of meeting the cubic in two coincident 
 points ; hence this point must be a double point, cusp, or conju- 
 gate point. 
 
 490. So the equation 
 
 uv' = Jcxy' [tangential 
 
 represents a cubic envelope to which the straight line {v=0, y=0) 
 is a double tangent, and the points u = 0, x = are points of 
 contact of tangents from y = and v = 0. 
 
 491. The cubic represented by the equation 
 
 x*+y'' + s' + 3kayz=0, 
 
 deserves special attention, as an example of a curve free from 
 double points, cusps and conjugate points. 
 
 The straight lines x — 0,y = 0, s = meet the cubic in nine 
 points, lying by threes on twelve straight lines. 
 
 The straight line x = meets the cubic in the points given 
 
 by y + 3' = o, 
 
 that is (if * denote one of the imaginary cube roots of unity), in 
 the three points 
 
 {x = 0,y+z = 0),{x = Q, y + iz^Q), (a; = 0, 3^ + t'« = 0). 
 We will call these points respectively X, X', X". 
 
414 EQUATIONS OP THE THIRD DEGREE. 
 
 So the straight lines y — 0, z = meet the cubic in the 
 points 
 
 (y = 0, s+aj = 0), (y=0, z + tx = 0), (3r=0, z + t^x = 0), 
 
 (a = 0, x + y = 0), (« = 0, a; + iy = 0), (z = Q, a3 + i>=0), 
 
 which we will denote loj the letters Y, Y', Y", Z, Z\ Z" re- 
 spectively. 
 
 Now it is easily seen that 
 
 X, r, Z lie on X + y +« =0, 
 
 X\ Y\ Z' X +tif +t'z = 0, 
 
 X", Y", Z" X +t> + w=0, 
 
 X, F', Z" {x+y +z =0, 
 
 X, Y", Z' t^x + 1/ +z =0, 
 
 X", Y, Z' X +iy +z =0, 
 
 X', Y, Z' X +ey-\rz =0, 
 
 X', Y', Z X +y +{z =0, 
 
 X\ Y', Z X ^y +r«=0. 
 
 And we know that X, X', X" lie on the straight line a; = ; 
 Y, Y', Y" on 3^=0; Z, Z, Z" on z=Q. 
 
 Hence the nine points lie by threes on twelve straight lines. 
 
 Q. E. D. 
 
 492. Thes^ nine points are points of inflexion. 
 
 Let the tangent at the point a; = 0, y + w =0, be y + w = /mj. 
 
 Then the equation 
 
 {y + izY + li* {i/' + z')+3k^L*yz (y + ta).= (1) 
 
 must have two equal roots (y + iz = 0). 
 
 Hence y + iz=0 must satisfy the equation 
 
 (y + izY + fi* if - iyz + iV) + Sk/jt'yz = 0, 
 whence /* = i^Je and this equation reduces to 
 
 shewing that all the three roots of (1) are equal. 
 
EQUATIONS OP THE THIBD DEGBEE. 415 
 
 Hence y + {z = {'kx 
 
 represents a tangent meeting the curve in three coincident 
 points af the point (aj = 0, y + is=0). Hence there is a point 
 of inflexion. Thus all the nine points 
 
 X, x; x", 
 
 Y, T. Y". 
 Z, Z\ Z', 
 
 are points of inflexion, and the tangents are given respectively 
 by the equations 
 
 y + z — kx, y + iz = t'kx, y + Pz = ikx, 
 
 z + x = Jcy, z + tx= i'ky, z + i'x = iky, 
 
 ■ x + y = kz, x + iy = i'kz, x + t'y = ikz. 
 
 ON THE INFINITE BRANCHES OP CUBIC CUBVES. 
 
 493. Since every straight line meets a curve of the third 
 order in either one or three real points, the straight line at in- 
 finity meets it in one or three real points. 
 
 And since any system of parallel straight lines meet the line 
 at infinity in one point, there is always at least one system 
 of parallel straight lines Trhich meet the curve on the line at 
 infinity, and therefore only meet it in two other points (real or 
 imaginary), and there may be three such directions or systems of 
 parallel straight lines. 
 
 If P be the point in which such a system of parallel straight 
 lines intersect at infinity, one of these straight lines through P 
 will generally be the tangent at P and therefore an asymptote. 
 Hence a straight line which meets a cubic in only two finite 
 points Is generally parallel to an asymptote. 
 
 We say generally, because It may happen that the tangent 
 at P at infinity lies altogether at Infinity. In this case lines in 
 
416 EQUATIONS OF THE THIRD DEGREE. 
 
 the direction P will meet the curve in two finite points, but will 
 not be parallel to an asymptote, except in the sense in which all 
 straight lines are parallel to the straight line at infinity. 
 
 It follows that there can generally be one asymptote drawn 
 to a cubic curve, and that there may be as many as three asymp- 
 totes. 
 
 The only cases in which there can be no asymptote will 
 occur when the straight line at infinity meets the curve in three 
 coincident points, at a cusp or a point of inflexion. (See Arts. 
 502—504.) 
 
 494. All possible cases may be analysed according to the 
 nature of the three points in which the straight line at infinity 
 cuts the cubic. 
 
 I. ff two of these points he imaginary and one real, there 
 will be two imaginary and one real asymptote. 
 
 II. If all the points he real and distinct, they will determine 
 the direction of three asymptotes. 
 
 III. If all the points he coincident, either the straight 
 line at infinity is a tangent at a point of inflexion or a cusp, 
 and there is no asymptote, or else it is one of the tangents at 
 a double point, in which case the other tangent at the doable 
 point is an asymptote. 
 
 IV. If two of the points he coincident at P, the third point 
 will always determine the direction of the only asymptote, and 
 unless P be a singular point, the straight line at infinity will be 
 the tangent at P, and all straight lines in direction P cut the 
 curve in two finite points. 
 
 It may happen however that P is a double point, or a conju- 
 gate point, in which case all straight lines in direction P, cutting 
 the curve in two coincident points at infinity, will cut it in only 
 one finite point. 
 
KQUATIONS OF THE THIKD DEGEEE. 417 
 
 We proceed to consider some typical examples of all these 
 cases. We shall use abridged notation throughout, each of the 
 symbols u, v, w, x, y, z denoting expressions of the most general 
 form which, when equated to zero, represent straight lines ; and 
 we shall use o- = to denote the equation to the straight line at 
 infinity. 
 
 495. Consider the equation 
 
 xiva- = kxcyz. 
 
 Each of the straight lines a; = 0, y = 0, z = meets the locus 
 in two finite points lying on the straight lines m = 0, -y = and 
 in one point at infinity. Hence the asymptotes are parallel to 
 the straight lines a; = 0, y = 0, z = 0. 
 
 496. Consider the equation 
 
 mV = Texyz. 
 
 The straight lines a;=0, ^ = 0, 3 = are now tangents paral- 
 lel to the asymptotes, their points of contact lying in the straight 
 line M = 0. 
 
 497. Consider the equation 
 
 ua^ = kxyz. 
 
 Each of the straight lines a; = 0, y = ^, z = meets the locus 
 in two points at infinity and in one point on the straight line 
 M = 0. Hence the equation represents a cubic having x = 0, 
 y = 0, a = as asymptotes, the points in which they cut the 
 curve again lying on the straight line m = 0. 
 
 498. Consider the equation 
 
 a^ = kxyz. 
 
 This is a particular cas§ of the last equation, m = now 
 coinciding with the line at infinity. It therefore represents a 
 cubic having three asymptotes which do not cut the curve in 
 any finite points, the asymptotes being tangents at points of 
 inflexion. 
 
 w. 27 
 
418 EQUATIONS OF THE THIRD DERKEE. 
 
 499. Consider the equation 
 
 uv<T = ka?y. 
 
 The straight line at infinity meets the locus of this equation 
 in two coincident points on a; = 0, and in a third point on y = 0. 
 The locus further cuts a; = in two finite points lying on m = 
 and v = 0. Hence the straight line at infinity is a tangent at 
 the point given by a; = 0, and there is only one asymptote, its 
 direction being given by ^ = 0. 
 
 If y = mu and y = nv be the straight lines through (m = 0, 
 y = 0) and (v = 0, y = 0) parallel to the straight line a; = 0, 
 so that 
 
 y — mu = mx + /xcr, 
 
 and y — nu = nx -(- va 
 
 identically, then the parabola whose equation is 
 
 ii'mVat? = kmna- {mu + nv — y) + m'n'o^ 
 
 will be found to meet the cubic in five coincident points at 
 infinity. 
 
 This parabola having five-pointic contact with the cubic at 
 infinity will serve the purpose of an asymptote in approximating 
 to the form of that infinite branch of the cubic to which the 
 linear asymptote is not an approximation. 
 
 Such a parabola is called a Parabolic asymptote. 
 
 500. Consider the equation 
 
 ua' = ka?y. 
 
 In this case also, the straight line at infinity meets the curve 
 in three real points two of which are coincident. As before the 
 distinct point at infinity determines an asymptote (its equation 
 now being y = 0), but the two coincident points now indicate 
 a singular point. Thus the straight line at infinity will not be 
 itself a tangent, but the two coincident points upon it will be 
 
EQUATIONS OF THE THIRD DEGREE. 419 
 
 the points of contact of two tangents real or imaginary accord- 
 ing as the singular point is a double point or conjugate point. 
 
 If u = iJby be the equation to the straight line through (m = 0, 
 y = 0) parallel to a; = 0, it is easily seen that the straight lines 
 x = ± Jfia- are the two tangents at the singular point, forming 
 two real or imaginary asymptotes according as /it is positive or 
 
 We have spoken of the singular point either as a double 
 point or a conjugate point. If it happen to be a cusp the two 
 
 straight lines a; = + J iia must coincide and fj, must be either 
 zero or infinite, i. e. either m = or y = 0, must represent a 
 straight line parallel to x = 0. These two cases we proceed to 
 consider separately. 
 
 501. Consider the equation 
 
 {x + tr) a" = kjc'y. 
 
 This is a partici^lar instance of the last case, the straight 
 line M = being now parallel to x= 0. As we saw in con- 
 sidering the former case, the straight line at infinity meets the 
 curve in a cusp at the point determined by a; = 0, and in a dis- 
 tinct point at y = 0. We have now /* = 0, shewing that there 
 are two coincident asymptotes represented by the equation 
 x = 0, tangents at the cusp, as well as a distinct asymptote 
 y = 0. 
 
 502. Consider the equation 
 
 Mtr* = kx' {x + a). 
 
 This is a particular instance of the case of Art. 500, the 
 straight line y = being now parallel to x = 0. The three 
 points in which the curve meets the straight line at infinity are 
 now coincident, that line being itself the tangent at the cusp. 
 There is therefore no asymptote, but there will be two branches, 
 each touching the straight line at infinity at the cusp. 
 
 27—2 
 
420 EQUATIONS OP THE THIRD DEGREE. 
 
 503. Consider the equation 
 
 uvcr = Ten?. 
 
 We observe that w = 0, « = 0, o- = are the tangents at 
 three points of inflexion lying on the straight line x = Q. There 
 is no asymptote, since the straight line at infinity is a tangent 
 at the point of inflexion at infinity. 
 
 504 The equation 
 
 u'cr = kx" 
 
 represents a curve, only differing from the last in that m = and 
 v = coincide, and the two finite points of inflexion Jiave 
 coalesced into a cusp. 
 
 505. The examples which we have given will suffice to 
 shew the student how to analyse any cubic whose equation is 
 given, as to its infinite branches and its singular points. We 
 have exhibited types of curves having every variety of asymp- 
 tote and every variety of singular point, and other examples 
 in which different combinations of asymptotes and singular 
 points occur may be analysed by analogous methods. 
 
 506. We will conclude with the following proposition 
 which is very important. 
 
 All cubtcs which pass through eight fixed points pass also 
 through a ninth. 
 
 Let ^ (a, /8, 7) = and 1^ (a, 13, y) =0 be the equations to 
 two cubics which pass through eight given points. Since nine 
 points determine a cubic, any other cubic through the eight 
 points will generally be determined if another point upon it be 
 
 Consider the cubic which passes through the eight given 
 points, and the ninth point (a, ^, y). The equation 
 
 «^ (g. /3, 7) ^ -f («. /3, 7) ■ 
 
 .^(a', ^, 7') t(a',^',7') ^'i 
 
EXERCISES ON CHAPTER XXVI. 421 
 
 must represent it. For since the functions <^ (a, /9, 7) and 
 ■\fr (a, l3, 7) are of the third degree, this equation is of the third 
 degree and therefore represents some cubic. But it is satisfied 
 at the point (a, /3, 7), and also at all points of intersection of 
 the two cubics <f) (a, /8, 7) = and yjr (a, ^, 7) = 0, and therefore 
 at the eight given points. But the two cubics <f> (a, ^, y) =0 
 and i/r (a, y3, 7) = intersect A nine points altogether. Hence 
 the locus of the equation (1) passes not only through the as- 
 signed point (a', /3', 7') and the eight given points, but it passes 
 also through a ninth fixed point on each of the original cubics. 
 Therefore all cubics through eight fixed points pass also through 
 a ninth. 
 
 Exercises on Chapter XXVI. 
 
 (267) The only cubic having three double points consists 
 of three straight lines forming a triangle. 
 
 (268) The only cubic having two double points consists of 
 a straight line intersecting a conic. 
 
 (269) Shew that the straight lines m = 0, v = are tan- 
 gents to the cubic 
 
 uvw = ku^ + k'v^ 
 
 at a double point. 
 
 (270) Shew that the equation 
 
 uvw = kv? + lev" {tangential 
 
 represents a curve having a double tangent. 
 
 (271) The cubic 
 
 u^v = hu^w + k'v'w 
 
 has a conjugate point or a double point according as k, k' are of 
 the same or of opposite signs. 
 
422 EXERCISES ON CHAPTER XXVI. 
 
 (272) Find the six points of intersection of the conic 
 
 and the cubic 
 
 P + ni' + if inn -\-nl+ Im ' 
 P + m^ + n^ Imn 
 
 (273) If a triangle be inscribed in a cubic so as to have its 
 sides parallel to the asymptotes of the cubic ; and if the tan- 
 gents at B and C intersect in P, those at C, -4 in Q; those at 
 A, B in E; the three straight lines PA, QB, EC will be con- 
 current. 
 
 (274) If a cubic curve consist of three equal and sym- 
 metrical branches having contact with tliree asymptotes which 
 form an equilateral triangle, the algebraical sum of the recipro- 
 cals of the perpendiculars from any point on the three sides of 
 the equilateral triangle formed by joining the vertices of the 
 three branches, is constant. 
 
 (275) If two cubics touch one another in three coUinear 
 points, their other points of intersection will be collinear. 
 
 (276) If a series of conies have the same three asymptotes, 
 the points of intersection of any two lie on a straight line, and 
 all these straight lines are concurrent. 
 
 (277) If on any cubic the points P, F , P" be collinear, 
 and Q, Q, Q" be collinear, and E, E', E" be collinear, and if 
 also P, Q, E be collinear and P, Q', E' be collinear, then will 
 P", Q", R' be also collinear. 
 
 (278) If a cubic have three asymptotes, one of which cuts 
 it in a finite point, another must do so also. 
 
 (279) If two and only two asymptotes cut the curve in 
 finite points, the other asymptote is parallel to the straight line 
 joining those points of intersection. 
 
EXERCISES ON CHAPTER XXVI. 423 
 
 (280) If a curve of the third order have a double point A, 
 and be cut by any straight line in B, G,B; and if when ABG 
 is taken as the triangle of reference, the tangents at A are 
 represented by the equation 
 
 and the tangents at B and G by the equations 
 
 Pa + JV-y=0, and J)i/3 + i?a = 0, 
 shew that the equation to the straight line AD is 
 
 iV/3 + M7=0, 
 and find the equation to the curve. 
 
 (281) Shew that the cubic 
 
 a?y = o- (Zar* + inf + n<r^') 
 has a parabolic asymptote whose equation is 
 
 s^ = m<7y. 
 
 (282) Shew that the cubic which circumscribes the triangle 
 of reference and passes through the six points in which the two 
 straight lines 
 
 \a + /i;S + 1/7 = 0, ? + ^+T^ = 
 
 K ft, V 
 
 intersect the three straight lines 
 
 la. + 9W/3 + »i7 = 0, jna + w/8 + Jy = 0, wa + Z/3 + OT7 = 0, 
 
 (the coordinates being triangular) has its asymptotes parallel to 
 the last three straight lines. 
 
 (283) The cubic which circumscribes the triangle of refer- 
 ence and touches the straight lines 
 
 ?'a = ;8 + 7, »i''/8 = 7 + a, wV=a + y3 
 
 at points lying on the straight line 
 
 Ifj. + m^ + M7 = 
 
 (the coordinates being triangular) has its asjrmptotes parallel to 
 the given tangents. 
 
424 
 
 EXERCISES ON CHAPTER XXVI. 
 
 (284) If (\, fi, v) be a point on the curve 
 
 Ix (/ + z') + my (z" + ar") + W2 (a;' + f) + toj^a = 0, 
 
 the point ( - , - , - ) will also lie upon it. 
 
 Shew that the tangents at these two points intersect on the 
 curve. 
 
 (285) If (\, n, v) be a point on the cubic 
 
 [x^y + a;/ + y'^z + yz^ + z'x + zx^) + ikxyz = 0, 
 then all the twelve points 
 (\, /i, v), (iM, V, X), {v, X, m), {\ V, fi), ifJ', \ v), {v, fi, \), 
 
 fi i 1^ fl i IWl 1 i^ 
 
 lie upon it. 
 
 (286) li x = 0, y=0, s = be the equation of any three 
 straight lines, and if 
 
 P= x' + f + z\ 
 
 Q = yz^ + y^z + sa;" + z^x + as/ + a;"?/, 
 
 B, = xyz, 
 
 and if P', 9', P' denote what P, ^, R become when any con- 
 stant quantities a, &, c are substituted for x, y, z, shew that the 
 equation 
 
 P, Q, R =0, 
 P, Q', R' 
 
 \, /ji, V \ 
 
 (where \, fi, v are arbitrary), represents a series of cubic cui-ves 
 passing through nine fixed points of which six lie upon a conic 
 and the other three upon a straight line. 
 
EXERCISES ON CHAPTER XXVI. 425 
 
 (287) Find the values of \, /i, v in the last exercise, in 
 order that the cubic may break up into the conic and the 
 straight line. 
 
 (288) In curves of the third order the locus of the middle 
 points of chords parallel to an asymptote which does not cut the 
 curve is a straight line. 
 
 (289) In curves of the third order the locus of the middle 
 points of chords parallel to an asymptote which cuts the curve 
 is a hyperbola. 
 
 (290) The tangents to a cubic at three coUinear points will 
 meet the cubic again in coUinear points. 
 
 (291) If a cubic have three asymptotes which do not meet 
 in a point, its equation referred to" the asymptotes will be in tri- 
 angular coordinates 
 
 «y87 = (?;-. + wiyS + ni) (a + /8 + 7)'- 
 
 (292) A cubic circumscribes a triangle ABC, and cuts the 
 sides BC, GA, AB again in A', B', C respectively. Shew that 
 if the chords A^H , BB , CC are concurrent, so also are the tan- 
 gents aX A, B, C. 
 
 (293) The general equation to a cubic touching the conic 
 
 i/3y + mya + wa/3 = 
 in the three points of reference is 
 
 ka^y + (X« + fj.^ + vy) (l^y + myx -\- na^) = 0. 
 
 (294) If a conic touch a cubic in three points the three 
 chords of contact will cut the cubic again in coUinear points. 
 
 (295) The general equation in triangular coordinates to a 
 cubic which cuts each of the sides of the triangle of reference in 
 only two finite points, viz. the points which lie on the nine- 
 points' circle, is 
 
 Jfc,^7-l-acot^ (a'-/3'- 7') + y3 cot 5 (/3= - 7" - a=) 
 
 + 7CotC(7'-a^-/3=)=0. 
 
INTEODUCTION TO CHAPTER XXVII. 
 
 GENERAL PROPERTIES OP HOMOGENEOUS FUNCTIONS. 
 
 507. In the following chapter a knowledge of the principles 
 of the Differential Calculus will be required on the part of the 
 student. 
 
 We call attention at once to some of the principal results 
 which we shall assume, referring to treatises on the Differential 
 Calculus for the discussion and proof 
 
 508. If /(a, /8, 7) he a homogeneous function q/" a, /S, 7 of 
 the m* degree, then will 
 
 COK. 1. Since -^ , -5^ , ^ are themselves homogeneous 
 functions of the w"" degree, we have 
 
 cPf , .d'f , dJ'f , ,s df 
 dPf „ d^f dj , ,.df 
 
GENERAL PHOPEETIES OF HOMOGENEOUS FUNCTIONS. 427 
 
 CoE. 2. By the elimination of a, /3, 7 from the last four 
 equations, we obtain 
 
 ^i/C^'^.v). 
 
 da.' 
 d^' 
 
 d^ d£ dj_ 
 
 da.' dfi' «?7 
 
 ^ _^ ^'/ 
 
 <?y <fy dy 
 
 
 COE. 3. If / (a, jS, 7) = be a homogeneous equation of 
 any degree, then will 
 
 df ^df df „ 
 
 da. 
 
 and 
 
 0, 
 
 df 
 da' 
 
 df 
 
 d^' 
 
 df 
 dy 
 
 da. '■ 
 
 dJ'f 
 da?' 
 
 dj 
 
 da.d^' 
 
 d^f 
 dyda. 
 
 df 
 d^' 
 
 d^.f 
 dadfi 
 
 d^f 
 
 ' d0" 
 
 df 
 dfidy 
 
 d± 
 
 d^f 
 
 d^f 
 
 d'f 
 
 = 0. 
 
 ^7' dyda' dfidy' drf 
 
 509. The following notation is very convenient. 
 The symbol 
 
 denotes the same thing as 
 
 4a + '^| + ''f' 
 
 where -f , -tt,, -/are the derived functions of /(a, /3, 7) with 
 da dp ay 
 
 respect to a, yS, 7. 
 
428 GENERAL PROPERTIES OF HOMOGENEOUS FUNCTIONS. 
 The symbol 
 
 or 
 
 d' , d' ^ (T 
 
 
 denotes the same thinsr as 
 
 dy 
 
 dY dY 
 where ^4, , -tA-,- &c. are the second derived functions of 
 rfa dp drf 
 
 /(a, /3, 7). 
 Similarly, 
 
 ^Ij+^+^DV^"'^'^) 
 
 denotes the expression obtained by expanding 
 
 (^ d d ^ dy 
 
 as if each of the expressions >--y- , l^;jo) " j- w^ere an alge- 
 braical term, and then replacing every such term as 
 
 which occurs, by 
 
 X""^ VV -, ^ 
 
GENERAL PEOPEETIES OF HOMOGENEOUS FUNCTIONS. 429 
 
 So also 
 
 L d d d\ f^, d , d , d\ ., „ , 
 
 will denote 
 
 ,,,<f/" ,d^f .dj , , , , d'f , „ 
 
 610. If f{a., /8, 7) he a homogeneous function of a., 0, y of 
 the (^ + 2)* degree, then 
 
 \ { d d dV ,, a s 
 
 COK. As a particular case, ii f(a, /3, 7) be a homogeneous 
 function of a, y3, 7 of the «*'• degree, then 
 
 and 
 
 
 511. Iff{tx, (8, 7) Je a homogeneous Junction of a, /3, 7 q/" 
 «Ae n*** degree, then will 
 
 f{ai + x, ^ + y, 7 + ^) 
 
 lid d dV., _ , 
 
 + &c.+f{x,i/, z). 
 
430 GENERAL PROPEKTIES OF HOMOGENEOUS FUNCTIONS. 
 
 Cor. So we have 
 
 /(a + V> ^ + 1^9, 1 + vp) 
 
 + &C. 
 
 This last expansion is of the greatest utility, and will be 
 frequently applied in the following chapter. 
 
CHAPTER XXVII. 
 
 THE GENERAL EQUATION OF THE W* DEGREE. 
 
 512. It is our purpose, in this concluding chapter, to ex- 
 hibit in its most general form the method by which the investi- 
 gation of the fundamental properties of any curve must be 
 carried on, when the curve is presented under an equation in tri- 
 linear coordinates of any degree whatever. We shall obtain 
 equations to give the direction of the curve at any point, to 
 determine the points of inflexion and the singular points, the 
 tangents at the singular points and the asymptotes, and the 
 curvature at any point whatever ; but we shall not attempt any 
 detailed discussion of the properties of the several classes of 
 curves in general, as such a discussion properly demands by its 
 magnitude to be treated by itself, and from its intricacy cannot 
 with propriety find a place in an elementary treatise such as 
 the present. 
 
 513. Let /(a,i8,7) = (1) 
 
 represent the general homogeneous equation of the m* degree 
 in trilinear coordinates. 
 
 And let (a', yS', 7') be any point whatever, and let 
 a — a' _ ;8 — /3' _ 7 — 7' 
 
 /* 
 
 = P (2) 
 
 represent any straight line drawn through the point (a', /8', 7') to 
 meet the locus of the equation (1). 
 
432 THE GENERAL EQUATION OF THE »l"' DEGREE. 
 
 The lengths of the intercepts measured from (a, /3', 7') are 
 given by 
 
 /(«' + Xp, ^' + lip, 7' + vp) = 0. 
 
 And if [a', /S", 7") be any other point on the same Hue, the 
 intercepts measured from (a", ;8", 7") are given by 
 
 f{a" + \p, ^" + fip, y" + vp) = 0. 
 This equation may be written 
 
 /(a", ^", 7") +P ('^ ^ + /^ ^ + " ^) + ^"§^«^ P°^^^"^ of p = 0. 
 
 Now suppose the straight line is a tangent to the curve, and 
 that (a", /3", 7") is the point of contact, then two of the roots 
 of the last equation must be zero, and we have 
 
 fW',^",y") = o (3) 
 
 -^ ^S + ^# + ''^=^ w- 
 
 But since (a", /S", 7") lies on the locus of the equation (2), 
 we have 
 
 o-"-a.' ^ 0'-0 ^ y"-y ' 
 
 in virtue of which, the equation (4) becomes 
 
 (.'._a')|:, + (r-^'),f+(7"-7')f = (5). 
 
 But from (3), by the property of homogeneous functions 
 (Art. 508), We have 
 
 hentie the equation (5) becomes 
 
 «j;+^'J^+7'S = (6), 
 
THE GENERAL EQUATION OP THE «*•» DEGREE. 
 
 433 
 
 But (a", ^", 7") is the point of contact of any tangent from 
 (a , ^', 7') to the curve ; hence all the points of contact of tan- 
 gents from (a!, ^, 7') to the locus of the equation 
 
 /(a,/3,7) = 
 
 lie upon the locus of the equation 
 
 •{')• 
 
 »l+^|+''|-»' 
 
 But again, since (a', 0', 7') is any point upon the tangent at 
 (a", ;8", 7"), therefore the equation (6) may be also read as 
 stating that the equation to the tangent at any point {<x' , )8", 7") 
 on ike locus of the equation 
 
 f{a,0,y) = O 
 
 is represented by the equation 
 
 „ *¥ ,a ^f ,^df _n 
 
 .(8). 
 
 514. If X, fjL, V he the direction sines of the tangent at any 
 point (a, 0, 7') 071 the curve f [a, 0, 7), then will 
 
 d0' dy 
 sin B, sin C 
 
 df 
 
 df 
 
 do; 
 
 sinC, 
 
 sin^ 
 
 df df_ 
 c?a" d0 
 
 sin A, BvaB 
 
 1 
 
 (^ df m 
 
 \dd' did!' dyj 
 
 For by equation (4) of the last article, we have 
 , df df df „ 
 
 and by the identical relation (Chap, vi.), 
 
 \ sin ^ 4- /i sin 5 + i; sin C = 0. 
 w. 
 
 28 
 
434 THE GENERAL EQUATION OP THE «"" DEGREE. 
 
 Therefore 
 
 df 
 
 df 
 
 rf/3" 
 
 dy 
 
 sinjB, 
 
 sin C 
 
 sin (7, sin^ 
 
 dd' d^ 
 sin j1, sin 5 
 
 .(I). 
 
 But /i" + v" + i/iv cos A = sinM, 
 
 whence each of the equal fractions in (1) 
 
 ~M df dfy 
 
 the coordinates being trilinear. 
 Therefore, &c. Q. E. D. 
 
 515. The locus of the equation (7) of Art. 513, 
 
 'df,ffdf,df_ 
 "di + ^W'^^d^'' 
 
 containing all the points of contact of tangents from (a', /8', 7') 
 to the curve represented by the equation 
 
 /(=, A 7) = 
 
 may be conveniently called the Jirst polar curve of the point 
 (a', /S', y') with respect to the original curve. 
 
 It will be observed that this polar curve is of an order lower 
 by unity than the original curve, and when the original curve 
 is a conic section, the first polar curve of any point, becomes 
 the straight line which we have been accustomed in previous 
 chapters to speak of as the polar of that point with respect to the 
 conic. 
 
 516. If we take the equation to the first polar of the point 
 (a, /S", 7') with respect to the given curve, and form the equa- 
 tion to the first polar of the same point with respect to this new 
 curve, the locus of this equation is called the second polar of 
 the point (a', /S', 7') with respect to the given curve. Similarly 
 
THE GENERAL EQUATION OF THE W** DEGREE. 435 
 
 the polar with respect to the second polar is called the third 
 polar, and so on. 
 
 The equation to the first polar of the point (a, /8', 7') being 
 
 "1+^|h-^'|-« «. 
 
 the equation to the second polar wiU be 
 
 or, as we may write it, 
 
 So the equation to the third polar may be written 
 
 (4«+^'|+^'|)V(«'^'^)=« (^)' 
 
 and the equation to the r"" polar, 
 
 517. It has been observed that the first polar is of an order 
 one less than that of the original cm've. So each polar is of an 
 order one less than that of the preceding polar, and therefore, 
 the original curve being of the m"* order, its (re — 1)'*' polar will 
 be a straight line represented by the equation 
 
 518. As our condition of tangency we have simply express- 
 ed that a line should meet a curve in two coincident points. 
 It is obvious that this condition will be satisfied by any line 
 through a cusp, multiple point, or conjugate point. 
 
 Hence every cusp, multiple point, or conjugate point in any 
 curve will lie upon the polar curve of any point whatever. 
 
 28—2 
 
436 THE GENERAL EQUATION OF THE w"" DEGREE. 
 
 Therefore the cusps, multiple points, and conjugate points 
 of the curve whose equation is 
 
 /(a,^,7)=0 
 lie upon the curve 
 
 where the ratios a. : /3' : 7' may have any values whatever. 
 
 Hence the coordinates of all these singular points must 
 satisfy simultaneously the equations 
 
 519. If a 'point lie upon a fixed straight line, its first polar 
 with respect to a curve of the n*^ order will pass through {n — 1)" 
 fixed points. 
 
 Let the point (a', yS', 7') lie upon the straight lin« 
 
 la. + »w/8 + W7 = 0. 
 
 Its first polar curve with respect to the curve / (a, ^S, 7) = 0, 
 is represented by 
 
 Therefore, since 
 
 la! + m^' + n<y' = 0, 
 
 the polar curve passes through the points given by 
 
 1 df^-}_ ^^1 _^_ 
 I da m rf/3 n dy ' 
 
 And these equations represent the intersection of two curves 
 each of the (« — l)"" order, and therefore give (n — l)" points. 
 Therefore, &c. Q.e.d. 
 
 Cor. The polar curve of any point at infinity passes through 
 the (ji — 1)" points given by 
 
 1 ^^1 df^l df 
 a da b d0 c dy' 
 
THE GENERAL EQUATION OF THE n"" DEGREE. 437 
 
 520. Of the tangents at singular points. 
 
 If a point (a', ff, 7') can be found whose coordinates satisfy 
 simultaneously the three equations 
 
 |-». |=». f-» «- 
 
 that point is, as we have seen, a douhle point, cusp or conjugate 
 point, and any straight line through it cuts the curve in two 
 coincident points. 
 
 The equation to a straight line meeting the curve in two 
 coincident points at {a', 0, 7'), viz. 
 
 becomes indeterminate, and in order to find a true tangent at 
 this point, we must express that such a line meets the curve in 
 three coincident points. 
 
 Thus if 
 
 a-a' /S-/? 7-7' 
 
 = P (2) 
 
 be any straight line through (a, /3', 7') the equation to determine 
 the intercepts on this line may be written 
 
 + &c. = (3), 
 
 of which the first two terms vanish since /(«', /8', 7') and its first 
 derived functions are zero. 
 
438 THE GENERAL EQUATION OP THE n^ DEGREE. 
 
 Hence in order that tlie straight line may be a tangent, 
 X, fi, V must be such as to make the third coefficient vanish and 
 thus make three of the roots zero. 
 
 So we must have 
 
 or in virtue of (2) 
 
 {(«-«') ^'+ (/3-/3') ^'+ (7-7')^.|V(a'> ^'7') = 0...(4). 
 
 Now if we expand the first member of this equation, the 
 terms of the second order in a, fi, 7 are 
 
 The coefficient of a is 
 
 which (since -5^ is a homogeneous function of the («— !)"■ de- 
 gree) becomes 
 
 and vanishes in virtue of (1). 
 
 So the coefficients of /3 and 7 vanish, and the remaining 
 terms are 
 
 which similarly vanish. 
 
THE GENERAL EQUATION OF THE w"' DEGREE. 439 
 
 So the whole equation (4) reduces to 
 
 («i+^|^+'y|)V(«''^''^')=« (^)' 
 
 a relation among the coordinates of a point on any tangent at 
 (a', ^, y), and therefore the equation to the two tangents (real 
 or imaginary). 
 
 521. That the equation just obtained does really represent 
 two straight lines is immediately seen by applying the criterion 
 of Art. 245. 
 
 That criterion requires that 
 
 d^f <Pf 
 
 = 0. 
 
 .(6). 
 
 d</' ' da.'d^' dy'da' 
 
 dy dj _dy_ 
 
 ddd^' eZ/3'' ' d0dy' 
 
 dy'd^" d^dy" dy'^ 
 
 Now, by multiplying the three rows of the determinant by 
 a, ^, y, and adding, we obtain 
 
 f , d , a> ^ , ' el\ df ( , d , o, d , d\ df 
 ( , d , ly d , , d\ df 
 or by the property of homogeneous functions (Art. 506) 
 
 ('^-^)|' ('^-^)|' ('^-i)!" 
 
 each of which terms vanishes in virtue of (1). 
 
 Hence the condition (6) is satisfied, and therefore the equa- 
 tion (5) represents two straight lines. 
 
440 THE GENERAL EQUATION OF THE M* DEGREE, 
 
 522. To distinguish between a double point, cttsp or con- 
 jugate point. 
 
 If (a, /3', 7') be a double point, cusp or conjugate point, the 
 real or imaginary tangents thereat are given by 
 
 According as the point is a double point, cusp or conjugate 
 point, the tangents will be real and distinct, real and coincident, 
 or imaginary ; and therefore they will meet any straight line in 
 real and distinct, real and coincident, or imaginary points. 
 
 Consider the straight line a = : it meets the tangents in the 
 points given by 
 
 which are real, coincident, or imaginary, according as 
 
 [d^'dy'J d^' d^'-'' 
 
 Hence if (a', ^', 7') be a double point, 
 
 dy dy 
 
 dfi" dfidy 
 
 
 dy dy 
 
 dy^ ' dy dct 
 
 
 dy d'f 
 
 da" dad^ 
 
 d'f dy 
 
 d^dy' dy' 
 
 
 dy dy 
 
 dj da ' do.' 
 
 
 dy d'f 
 doid^' d^ 
 
 will be negative : if it be a cusp, they will vanish : if it be a 
 conjugate point, they will be positive. 
 
 523. Cor. If («', /S', 7') be a cusp, the tangent thereat is 
 represented by any one of the equations 
 
 drf - cPf dy „ 
 
 dy ^ d^f cF/ „ 
 
THE GENERAL EQUATION OF THE «"» DEGREE. 441 
 
 which are identical since 
 
 d^f d^f 
 
 d^^' d^'dy' 
 
 dy dr 
 
 d/Sdy" dy" 
 
 dy"' i^'da' 
 
 d'f dy 
 
 dy'du ' da." 
 
 ^ <^y 
 
 du" du'd^' 
 
 d'f dy 
 da'd^'' d/3" 
 
 = 0. 
 
 524. At a cusp, the first polar of any point whatever touches 
 the curve. 
 
 cusp. 
 
 Let {I, m, n) be any point whatever, and let (a', yS', 7') be a 
 
 The first polar of the point {I, m, n) is given hj 
 jdf df df ^ 
 
 Tlie tangent to this curve at the point (a, yS', 7') is 
 
 /, df , dy d'f \ 
 
 But the equations 
 
 dy _ dy d^f „ 
 
 are any one of them the equation to the tangent at the cusp. 
 Therefore the equation (1) which is derived from them by addi- 
 tion represents the same tangent. 
 
442 THE GENERAL EQUATION OP THE n"* DEGREE. 
 
 Hence at the cusp the first polaxs of all points touch one 
 another and touch the curve. 
 
 525. To determine the points of inflexion on a curve of the 
 n"" order. 
 
 Let (a', /S', 7') be a point of inflexion, and \, /u., v the direc- 
 tion sines of the tangent thereat. 
 
 The equation giving the lengths of the intercepts on this 
 tangent is 
 
 /(«',/3',7)+p(^^. + A^|^ + ''|)/(«,^',7) 
 
 + &c. = 0. 
 
 Three of the roots of this equation must be evanescent. We 
 have therefore 
 
 as well as 
 
 ^ df df df . 
 
 .(2). 
 
 The equations (1) and (2) determine the ratios of \, fj,, v, the 
 direction sines of the point of inflexion, the two roots of the 
 resulting quadratic being equal, since we have (Art. 506, Cor. 3) 
 
 jy dy d'f df_ 
 
 dd^' d/d^' dy'da'^' da' 
 
 df df df df 
 da'd^' <Z/3"" d^'di' d^' 
 
 d^f ' ^f d^ df 
 di'dr d^dri" &i'" di 
 
 df 
 
 df df 
 
 da: ' d/8' ' &i' ' 
 
 = 0. 
 
THE GENERAL EQUATION OF THE n^ DEGREE. 
 
 443 
 
 But X, fi, V must satisfy the relation 
 
 a\ + J/i + ci/=0 (3), 
 
 (the coordinates being trilinear). Hence we have three equa- 
 tions (1), (2), (3) from which to eliminate the ratios of "K, fi, v; 
 and the resulting equation will constitute the condition amongst 
 the coordinates (a', ^, y) in order that they may represent a 
 point of inflexion. 
 
 But instead of performing the elimination directly, we may 
 write down the result at once from indirect considerations. For 
 we have seen that the equations (1) and (2) lead to a quadratic 
 having equal roots. Now \, fi, v might as well be detennined 
 from the equations (1) and (3), hence these also must lead to a 
 quadratic having equal roots. 
 
 Hence 
 
 cPf d'f 
 
 ^ 
 
 d'f ^ dj 
 
 dd'd^' d^'" 
 
 _dFf_ (T/ 
 dddy ' dfi'dy 
 
 a, b, c. 
 
 d^&f" 
 
 d^ 
 dy"' 
 
 = 0, 
 
 which must be one form of the equation resulting from the 
 elimination. 
 
 The points of inflexion will be obtained by solving this 
 equation simultaneously with the equation 
 
 /(a',^,7')=0- 
 
 And of these two equations, the one is of the 3 (w — 2)"' degree, 
 and the other of the n"*- 
 
 Hence there will be in general 3n {n — 2) points of inflexion 
 on a curve of the n"" order. 
 
 526. 0/ multiple points and the tangents thereat. 
 
 We have seen that the intercepts which the curve makes 
 on the straight line drawn from (a, /3', 7') in the direction 
 \, fjb, V are given by the equation 
 
444 THE GENERAL EQUATION OF THE w"* DEGREE. 
 
 + &c. = (1). 
 
 Now suppose («', j8', 7') is a point such that /(a', /3', 7') and 
 its diflferential coefficients with respect to a, /8', 7' up to those of 
 the (r — 1)''' order inclusive, all vanish identically. 
 
 The first r terms in the equation (1) will disappear, and the 
 equation will take the form 
 
 + higher powers of p = 0, 
 
 shewing that any straight line through the point (a', /8', 7') 
 meets the curve in r coincident points thereat. Such a point is 
 called a multiple point of the r^ order, the double points consi- 
 dered in the preceding articles constituting the particular case 
 when r = 2. 
 
 The tangents at a multiple point of the r"" order will meet 
 the curve in r + 1 coincident points ; and, as in Art. 520, it will 
 be seen that their directions are given by the equation 
 
 and the equation to the r tangents themselves is 
 
 527. To find the directions of the asymptotes of the curve 
 whose equation is 
 
 /(«,A7)=0 (1). 
 
THE GENERAL EQUATION OF THE n"" DEGREE. 445 
 
 Let X, (J,, V be the direction sines of an asymptote. Then if 
 (a'j /S', 7') be any point whatever, one of the roots of the equa- 
 tion 
 
 /(a' + V. /3' + W i + vp) = (2) 
 
 must be infinite, and therefore 
 
 f{\ (L, i;) =0 (3). 
 
 Solving this equation simultaneously with the identical relation 
 a\ + 5/t + «< = 0, \trilinear 
 
 \ + /i+i/ = 0, \triangular 
 
 we obtain the ratios \ : (jl : v, determining the direction of an 
 asymptote. 
 
 Cor. Since the equation (3) is of the n"" order there will in 
 general be n solutions determining the directions of n asymp- 
 totes. 
 
 528. To find the equation to an asymptote of the samo 
 curve. 
 
 Let \ fi, V be the direction sines of an asymptote deter- 
 mined as in the last article, and suppose (a', /3', 7') any point on 
 the asymptote. Then two of the radii from this point in the 
 direction \, ft, v must be infinite, and therefore the equation 
 
 /(a'-hX/), ^' + w 7' + ''p)=0 (2) 
 
 must have two infinite roots. 
 
 Hence we must not only have 
 
 /(\, /i, V) = 0, 
 but also 
 
 «l+^|+-f-» » 
 
 Now this is a relation among the coordinates (a', /S', 7') of any 
 point on the asymptote. Hence, suppressing the accents, the 
 equation to the asymptote is 
 
 dX dfi dv 
 
446 THE GENERAL EQUATION OP THE w"" DEGREE. 
 
 529. To find the radius of curvature at any point on a 
 plane curve. 
 
 Let P be the given point, (a, /3, 7) its coordinates, and 
 
 /(«,/3,7)=0 (1) 
 
 the equation to the given curve. 
 
 Let Q be a point on the curve near to P, and di-aw QN per- 
 pendicular on the tangent at P. 
 
 Fig. 43. 
 r> 
 
 N 
 
 Let \, fi, V be the direction sines of the straight line PQ, 
 and let PQ=Bs, so that the coordinates of Q are 
 
 a'+\Ss, ^'+fiSs, y' + vBs. 
 Then we have (Art. 46) 
 
 QN= 
 
 (a' + X8*) ^ + 08' + mS.) ^, + (7' + pSs) ^ 
 
 Va" of/S" dy] 
 
 but (a', /3', 7') lies on the locus of the given equation (1), and 
 therefore 
 
 , nAT V <^« d^ dr /l 
 
 ^^""^ ^^ J^ Jl df\ • 
 
 \da" d^' dy'} 
 
 Now if p be the radius of curvature at P, and if Q move up 
 to and ultimately coincide with P, we have 
 
THE GENERAL EQUATION OF THE n^ DEGEEE. 447 
 
 .(3). 
 
 therefore 
 
 (df df df^ 
 .. W'd0'dy' 
 
 But since Q lies on the given curve, we have 
 /(a' + \Ss, /S' + /iSs, 7' + i;Ss)=0, 
 
 + higher powers of Ss = 0. 
 But we have /{a', ^, y') = 0, 
 
 therefore 
 
 + higher powers of Ss = 0. 
 
 Hence, substituting in (3) and diminishing Bs indefinitely 
 
 (^ dl df) 
 Va" d^' dy'\ 
 
 P = — 
 
 (^^ + '^i+''^)>(«''^''^')' 
 
 But in the limit X, /i, v are the direction sines of the tangent 
 at (a, y3', 70, and therefore (Art. 514), 
 
 df df 
 sin B, sin C 
 
 
 df df 
 dy' ' da' 
 
 sin C, sin A 
 
 
 df df 
 dd ' dii 
 
 sin A, sin B 
 
 1 
 
 
 
 [df df a 
 da" d^' d 
 
 '^ 
 
448 THE GENERAL EQUATION OF THE m* DEGEEE. 
 
 Hence, if L, M, N denote the determinants 
 
 d^' dy 
 smB, sin C 
 we obtain 
 
 P = - 
 
 > 
 
 df df 
 dy" d% 
 
 » 
 
 df df 
 dil' d0 
 
 
 sin C, sin A 
 
 
 sin A, siaB 
 
 {df df 
 
 Ida." d0 
 
 dfV 
 ' d-^S 
 
 (^£:'+^i'+^^)>'^''^')" 
 
 This result in a slightly different form has been recently 
 published by Mr Walton in the Quarterly Journal of Mathematics, 
 Vol. VIII. p. 41 (June 1866), in a paper on Curvature, to which 
 the reader is referred. 
 
 530. To determine the coordinates of the centre of curva- 
 ture at any point on a plane curve. 
 
 Let (a', /8', 7') be the given point, and (a, /8, 7) the centre of 
 curvature. 
 
 The equations to the normal are (Art. 226) 
 
 a — a 
 
 /9-i8' 
 
 ^,__Z,cosC-^cos5 ^,_^cos^-^,cosC 
 
 7-7' 
 
 ~ df df ~^ --P- 
 XT' - jT' COS .B- ^ cos -4 
 
 dr/ da! 
 
 d/3' 
 
 And if p represent the length of the radius of curvature, 
 the point (a, /3, 7) lying on the normal at the distance p from 
 the point (a', 0, y) must be the centre of curvature. 
 
 Hence, we have 
 a -a' 
 
 J-0 
 
 f'-I-(7-f,cos^"^-^eos^-.^cosa 
 
 da! d0 
 
 dy 
 
 d0 dy' 
 
 da.' 
 
THE GENERAL EQUATION OF THE n*^ DEGREE. 
 
 7-7 
 
 Clf df^ 
 
 J~'~J~> ^^^ ^~'9ai COS^ 
 
 
 Va' d^' drj] 
 
 K^+^|^+^|^}/(«''-^'^')' 
 
 where L. M, iV denote the determinants 
 
 a ^ d£ d£ 
 
 d/3" 0(7' ' di' dj: 
 
 sin B, sin C sin C, sin ^ 
 
 ^ df 
 sin ^, sin B 
 
 449 
 
 as in the last article. 
 
 These equations determine the coordinates (ot, /3, 7) required. 
 
 531. A curve of the w* order can generally he found to satisfy 
 — ^-- — - simple conditions. 
 
 The homogeneous equation of the n^'^ degree in its most 
 general form may be written 
 
 a" + a"- (a,/8 + ^,7) + 0.'-^ {\^ + ^^7 + 6,7') 
 
 + a"-» {c,^^ + c^7 + c^7= + C.7') + &c. = 0. 
 
 Whence we ohserye that there will be one term involving a", 
 two involving a""', three involving a""'', four involving a""', and 
 so on, and (w + 1) not involving any power of a. 
 
 Hence the whole number of terms is 
 
 (n + 1) (w + 2) 
 2 
 
 There are, therefore, — ~ coeflScients involving 
 
 ^« ^jn ; — 1, or — ^-— — - ratios, and therefore by reason- 
 
 w. 
 
 29 
 
450 THE GENERAL EQUATION OF THE n"^ DEGREE. 
 
 ing analogous to that of Art. 147, a curve of the n*'^ order 
 can generally he found to pass through — ^— — '- given points, 
 or to fulfil this number of simple conditions. 
 
 ,m(»i+3)-2 
 532. All curves of the re* order which pass through 
 
 fixed points pass also through — y~ other faced points. 
 
 Let ^ (a, j8, 7) = and ^ {a, /3, 7) = he the equations to 
 
 two curves of the w"" order passing through -^ ^ given 
 
 points. And let any other curve of the w*'' order be determined 
 l3y passing through these points and the point (a', /3', 7'), thus 
 
 making the requisite number of points altogether. 
 
 The equation to this curve will be 
 
 ^ (a, P, 7) ^ ^ (a, A 7) f,s 
 
 4>{a:,P',i) t(«'./3'.7') ^'' 
 
 for this equation is satisfied at the point (a', /?, 7') and at any 
 point of intersection of the given curves. 
 
 But the two given curves being each of the w"" order intersect 
 in r^ points, and the locus of the equation (1) passes through all 
 
 these points, i.e. through the — ^^ — -^ given points, and the 
 
 remaining ^ -^ points of intersection. Hence all 
 
 w fw 4- 3) 2 
 
 curves of the n^^ order which pass through — ^ — — -^ fixed 
 
 points pass also through -^ ^ '- other fixed points. 
 
 Q. E. D. 
 
 533. For a more extensive discussion of the properties of 
 the locus of the general equation of the «* degree, the reader 
 is referred to Dr Salmon's Higher Plane Curves. 
 
EXERCISES ON CHAPTEE XXVII. 451 
 
 Exercises on Chapter XXVII. 
 
 (296) Shew that the curve 
 
 wV= [u—vYyz 
 
 has a point of osculation at the intersection of the straight lines 
 M = and v = 0, the tangent being u = v. 
 
 (297) Shew that the curve 
 
 (/i-j/)a"+ (v-X)/3'+ (\-m) 7" = 
 touches the straight line 
 
 (/*- i;) a + (i; - X) yS + (\- /a) 7 = 
 at the point a=^ = y. 
 
 (298) Shew that the curve 
 
 touches the conic 
 
 at the four points 
 
 ±a = ±/3=±7, 
 
 and that the common tangents are represented by the equations 
 + (ji-v)a±{v-X)^± (\-/i)7 = 0. 
 
 (299) Find the asymptotes of the curve whose trilinear 
 equation is 
 
 a" (aa. + J/S + 07)' =k{b0 + cyf (a/37 + h^ + ca^)> 
 
 and shew that the curve passes through the circular points at 
 infinity. 
 
 29—2 
 
452 EXERCISES ON CHAPTER XXVII. 
 
 (300) K a conic be inscribed in the triangle of reference so 
 that one focus lies on the conic 
 
 then the other focus will lie on the curve 
 
 VZa + Vot/S + Vw7 = 0. 
 
 (301) The general equation to a curve of the fourth order 
 having double points at the points of reference is 
 
 \ /i V ?g + OT/8 + ny 
 
 (302) The tangents at the double points in the last exercise 
 are given by the equations 
 
 — +- + -^ = 0, ^ + -+— i- = 0, - + ^ + -7-^-0. 
 
 fi V fiV V \ V\. \ /J, \fJ. 
 
 (303) The general equation to a curve of the fourth order 
 having cusps at the points of reference is 
 
 f 7m' n' 2mn '2nl 2lm _ 
 
 and the equations to the tangents at the cusps are 
 /3_7 7_a a_/3 
 m n' n I' I m' 
 
 (304) If a Curve of the fourth order have three cusps their 
 tangents are concurrent. 
 
 (305) The tangents drawn from the double point (/3 = 0, 
 7 = 0) to meet the curve 
 
 X fi, V I m w_ 
 
 "a T "^ T 2 + ^ 1 — -p, — 
 
 a p 7 py 7a ap 
 
 are represented by the equation 
 
 /S' {m? - 4Xv) + 2/87 {mn - 27J) + 7' (n» - 4 V) = 0. 
 
EXERCISES ON CHAPTER XXVII. 453 
 
 (306) If the equation 
 
 represent a real curve, it has a conjugate point and two double 
 points at the points of reference. 
 
 (307) The curve whose equation is 
 
 1 1 1 
 
 passes through all the points of intersection of the three conies 
 m n _ I n I _m I m_n 
 
 (308) The tangent to the same curve at the point {x, /3', 7') 
 is represented by the equation 
 
 = 0. 
 
 (309) The equation 
 
 a' {m'^ + n<f + llM) + ^ ("V" + '"*' + 2m7a) 
 
 + 7' (Pa' + m"/3' + 2mai8) + a/87 (X187 + /i7a + i/a/3) = 
 
 is the general equation to a curve of the fifth order referred to 
 a triangle formed by joining three double points. 
 
 (310) The tangents at the points of reference in the last 
 exercise are given by the equations 
 
 9tt'y3= + »i'V + 2?/37 = 0, w^" + V'(k + 27»7a = 0, 
 
 ZV+»»"/3' + 2nay3 = 0. 
 
 (311) The general equation to a curve of the fifth order 
 referred to the triangle formed by joining three cusps is 
 
 a' (»n/3 + M'7)' + /S" («7 + ''«)" + 7' (^a + '«'/3)' 
 
 + a/37 (\/37 + /ii7a + j/a/S) = 0. 
 
 
 a 
 
 
 
 /3 
 
 
 
 7 
 
 
 
 a' 
 
 
 1 
 
 /8' 
 
 
 1, 
 
 7' 
 
 
 r 
 
 Wl 
 
 w 
 
 Til 
 
 n 
 
 i 
 
 ' n 
 
 I 
 
 m 
 
 -7 " 
 
 a 
 
 "/8'' 
 
 — 7 
 7 
 
 /S' 
 
 7' 
 
 a' 
 
 i' 
 
 a 
 
 /3' 
 
454 EXERCISES ON CHAPTER XXVII. 
 
 (312) The general equation to a curve of the m"" order 
 having double points at the points of reference is 
 
 a)87./(a, A 7) +/3V. </> (^,7) +7'a°-t (7, «) +a''/3=.x(«,/8) =0, 
 
 where / (a, /3, 7) is any function whatever of the {n — 3)"" degree 
 of the three coordinates, and <j) {/3, 7), 1^ (7, a), 'x^ (a, j8) any func- 
 tions whatever of the {n — 4)'" degree of two coordinates each. 
 
 in 
 
 (313) The two tangents at the double point (/3 = 0, 7 = 0) 
 the last exercise are represented by the equation 
 
 ^. X (1, 0) + /S7 ./(I, 0, 0) + 7» t (0, 1) = 0. 
 
MISCELLANEOUS EXEKCISES. 
 
 (314) Shew that 
 
 U, W , V 
 
 — w', V, u 
 
 — v', — u, w 
 
 = uvw + uu' + vv" + ww'\ 
 
 (315) Shew that 
 
 ■ah, —ca I + 2 
 
 — ah, V, — he 
 
 — ca. 
 
 ■he, 
 
 0, c\ V 
 c^ 0, a^ 
 h\ a\ 
 
 = 0. 
 
 (316) Shew that 
 
 ;8+7-o, 7+a-/3, a+y9_^ =4 
 /3'+7'-a'> i +0: -pi, a! +^ -i 
 ^" + 7" -a", 7"+a"-;S", a" + /3"-7" 
 
 (317) Prove that if w> 2, the determinant 
 
 = 0. 
 
 ^i-°'i' ajj-a,,. 
 
 ■ .x^-a„ 
 
 «2-«i. asj-a^,. 
 
 ..x^-a„ 
 
 253- «!, a;, — o„. 
 
 ..a;, -a,. 
 
 «„-«„ a:,.-CTj, . 
 
 ..a;„-a„ 
 
 a, iS, 7 
 a, P, 7 
 a , P , 7 
 
456 MISCELLANEOUS EXERCISES. 
 
 (318) Prove that 
 
 0, 1, 1, 1 
 
 1, 0, z\ f 
 1, z\ 0, x' 
 
 1, y", a;^ 
 = {x+y-{-s){x-y-z)iy-z-x){z-x-y). 
 
 (319) Shew that if 
 H = 
 
 tlien will 
 
 \Ha^+Ku, Hal + Kw, Hca + Kv' =0. 
 
 Hah + Kw', Hb' + Kv, Hbc + Ku 
 
 Hca + Kv', Hbc + Ku, JI& + Kw 
 
 (320) Prove that the determinant of the (« + 1)"" order 
 
 0, 1, t, 1, 
 
 1, 0, a + b, a-Vc, 
 1, t + a, 0, 5 + c, 
 1, c + a, c + 5, 0, 
 
 M> 
 
 «;', 
 
 «;' 
 
 and K= 
 
 u, 
 
 to, 
 
 V, 
 
 a 
 
 W, 
 
 *', 
 
 m' 
 
 
 w 
 
 ■0, 
 
 11, 
 
 b 
 
 *>', 
 
 
 w 
 
 
 V, 
 
 u, 
 
 w, 
 
 c 
 
 
 
 
 
 a, 
 
 h, 
 
 c. 
 
 
 
 = _(_2)"-^aJc. 
 
 : : &c. 
 
 /Ill 
 
 (- + 7+- + . 
 \a c 
 
 •) 
 
 (321) Prove that 
 b + c a 
 
 a ' 
 
 b+c' 
 
 b + c 
 
 b 
 
 c + a 
 
 b 
 
 c + a' 
 
 b ' 
 
 c + a 
 
 c 
 
 c 
 
 a + h 
 
 a + J' a + b' c 
 
 _ 2(a + b + cy 
 ~ {b + c) {c + a) {a + b) 
 
MISCELLANEOUS EXERCISES. 
 
 457 
 
 (322) If («„ ^,, 7^, (a„ ^,, 7^, (a,, ^„ 7,) be the trilinear 
 coordinates of three points, the ratios of the sides of the triangle 
 of reference are given by the equations 
 
 1 1, 1, 1 
 
 
 1, 1, 1 
 
 
 1, 1, 1 
 
 /3., /3„ /9s 
 
 
 7i. %, % 
 
 
 «!, Oj, a. 
 
 7i. 7,. 73 
 
 
 a„ a,, a. 
 
 
 ^1. iSa, /S3 
 
 (323) If (a, /3, 7), («', /3', 7'), (a", /3", 7") be the coordinates 
 of three points, and 
 
 (/ea, Ka', ««"), (k'^, k'0, «'/3"), (k'V, k'V, k'V) 
 
 the coordinates of three other points, shew that 
 
 ah c , , , 
 
 -+-,+ — = a + 6 + c. 
 
 K K K 
 
 Also prove that if the first three points are collinear, so also 
 are the other three. 
 
 (324) If in a homogeneous equation in trilinear coordinates 
 the sum of the coefficients on each side of the equation be the 
 same, then the equation will be satisfied by the coordinates of 
 the centre of the circle inscribed in the triangle of reference. 
 
 (325) ABC is a triangle, right-angled at G: draw AE, BF 
 perpendicular and equal to AG, BC respectively; join AF, BF, 
 and draw CD perpendicular to AB. Then the three lines AF, 
 BE, CD will be concurrent. 
 
 (326) The straight line whose equation is 
 
 h + myS + ny = 
 
 meets the lines of reference BC, GA, AB in the points A', B', C 
 respectively; and AO, BO, GO meet the same lines in the 
 points P, Q, E respectively, being given by the equations 
 
 Find the equations to the straight lines A'Q, BP. 
 
458 
 
 MISCELLANEOUS EXEECISES. 
 
 (327) If, with the construction of Ex, (326), the straight 
 lines BR, C'Q intersect in X, the straight lines C'F, A'R in Y, 
 and the straight lines A' Q, B'P in Z, find the equations to the 
 sides of the triangle XYZ. 
 
 (328) If a = 0, /8 = 0, 7 = he the equations to the sides 
 of a triangle, find the equations to the straight lines joining 
 the centre of the circumscribed circle with the centres of the 
 inscribed and escribed circles. 
 
 (329) If two straight lines be given by equations of the 
 form 
 
 lot. + m/3 + W7 = 0, 
 
 what are the equations to the lines which pass through their 
 intersection and bisect the angles between them ? 
 
 (330) If upon the sides of a triangle as diagonals, paral- 
 lelograms be described having their sides parallel to two given 
 straight lines, the other diagonals of the parallelograms will 
 meet in a point. 
 
 (331) If sti-aight lines be drawn bisecting the interior angles 
 of a quadrilateral, shew that they will form another quadri- 
 lateral whose diagonals pass through the intersections of the 
 opposite sides of the first. 
 
 Shew further, that if three of the straight lines pass through 
 a point, the fourth will also pass through that point. 
 
 (332) If p, q, r be the distances of a variable straight line 
 from the vertices of the triangle ABC, shew that the value 
 of the determinant 
 
 0, 
 
 0, 
 
 1, 
 
 1, 
 
 1 
 
 0, 
 
 0, 
 
 p. 
 
 ?. 
 
 r 
 
 1, 
 
 i'. 
 
 0, 
 
 c\ 
 
 V 
 
 1, 
 
 ?. 
 
 c\ 
 
 0, 
 
 a 
 
 1, 
 
 r, 
 
 h\ 
 
 a\ 
 
 
 
 is constant. 
 
MISCELLANEOUS EXERCISES. 459 
 
 (333) If the anharmonic ratio of four collinear points 
 A, P, B, Q be fi, shew that 
 
 /* 1 _ijk — l 
 AP~AQ~^AB' 
 
 (334) Through each angle of a triangle let two straight 
 lines be drawn, equally inclined to the bisectors of those angles, 
 but the inclination not necessarily the same for each of the 
 three ; then the straight lines joining the intersections of these 
 lines will meet the corresponding sides of the triangle in three 
 collinear points. 
 
 (335) Three straight lines AD, AE, AF&r& drawn through 
 a fixed point A, and fixed points B, C, D are taken in AD. Any 
 straight line through G intersects AE and AF in E and F; and 
 BE, DF intersect in P ; DE, BF in Q. Shew that the loci of 
 P and Q are straight lines passing through A, and if AD be 
 harmonically divided in the points B, C, D, the loci of P and 
 Q coincide and form with the lines AD, AE, AF a harmonic 
 pencil. 
 
 (336) Find the locus of a point, the sum of the perpendicu- 
 lars from which on a series of given straight lines shall be 
 equal to a given line. 
 
 (337) If the two sides BC, B C of any hexagon ABCA'B C 
 intersect on the diagonal AA' produced, and the two sides 
 CA!, C'A on the diagonal BB" produced ; then will the remain- 
 ing sides AB, A'B intersect on the diagonal CO' produced. 
 
 (338) The two conies which have B, C for foci, and pass 
 through A are represented in tangential coordinates by the 
 equations 
 
 [ap, hq, crY+ibcqr cos"— = 0, 
 
 A 
 and {ap, hq, crj" — ihcqr sin' ^ = 0. 
 
 (339) Find the equation to the straight line joining the 
 middle points of the diagonals of the quadrilateral formed by the 
 triangle of reference and the polar of the point (a', ^ , 7') with 
 respect to the circle circumscribing that triangle. 
 
460 MISCELLANEOUS EXERCISES. 
 
 (340) If M = be a tangent to a conic S=0, the two conies 
 8 + Atf' = 0, S+ k'u" = have four-pointic contact. 
 
 (341) If a series of conies have four-pointic contact at a 
 fixed point, and if from any point on the common tangent other 
 tangents he drawn to the conies, their points of contact are col- 
 linear. 
 
 (342) Two conies have four-pointic contact at a fixed point 
 P, and through P a variable straight line is drawn cutting the 
 conies in Q and B : find the locus of the intersection of the tan- 
 gents at Q and B. 
 
 (343) The triangle whose sides are 
 
 ni^ + n'Y= 0, la — 2ny = 0, h — 2m/3 = 
 is self-conjugate with respect to the conic 
 VZa + V»w/S + Vwy = 0. 
 
 (344) A conic section is described round a triangle ABO; 
 lines bisecting the angles of this triangle meet the conic in the 
 points A', B , C respectively ; express the equations to 
 
 A'B, A'C, A'B'. 
 
 (345) The triangle whose sides are 
 
 q r ' r p ' p q ' 
 will be self-conjugate with respect to the conic 
 
 VS + Vw/S + Vm7 = 0, 
 provided p + q + r = 0. 
 
 (346) The straight lines which bisect the angle of a triangle 
 meet the opposite sides in the points P, Q, B respectively ; find 
 the equation to an ellipse described so as to touch the sides of 
 the triangle in these points. 
 
 (347) If a conic section be described about any triangle, 
 and the points where the lines bisecting the angles of the tri- 
 
MISCELLANEOUS EXERCISES. 461 
 
 angle meet the conic be joined, the intersection of the sides of 
 the triangle so formed with the corresponding sides of the original 
 triangle lie in a straight line. 
 
 (348) If from any point perpendiculars be drawn on the 
 three sides of any triangle, the area of the triangle formed by 
 joining the feet of the perpendiculars bears a constant ratio to 
 the rectangle under the segments of a chord of the circle circum- 
 scribing the triangle, drawn through the point. 
 
 (349) If conies pass through two fixed points and touch at 
 another fixed point, the common tangents to any pair of them 
 intersect on a straight line passing through the point of contact. 
 
 (350) If two conies have four-pointic contact at A, and if 
 any straight line touch one conic in A' and cut the other in B, C; 
 and if AB, A G cut the former conic in C, B, then AA', BB', 
 CC are concurrent. 
 
 (351) If a conic circumscribe a triangle and if three conies 
 be described having four-pointic contact with the first at the 
 angnlar points, and touching the opposite sides, the straight 
 lines joining the points of contact to the opposite angular points 
 are concurrent. 
 
 (352) If Vte + "Jmy + Vwa = 
 
 be the equation in triangular coordinates to a parabola, the 
 equations 
 
 x—x _y —y' _z —z' 
 I m n 
 
 will represent a straight line meeting it in only one finite point. 
 
 (353) Find the equation to the hyperbola conjugate to the 
 hyperbola represented by the equation 
 
 ua? + v^ + wf + 2m'/37 + 2«'7a + 2w'aP = 0. 
 
 (354) If through the extremities of one side of a triangle 
 any circle be described, cutting the other sides in two points, and 
 these points be joined ; shew that the locus of the intersection of 
 
462 MISCELLANEOUS EXEECISES. 
 
 the diagonals of the quadrilateral thus formed will be a conic 
 passing through the angular points of the triangle. 
 
 (355) From the middle points of the sides of a triangle 
 draw perpendiculars, proportional in length to those sides, and 
 join the ends of the pei-pendiculars with the opposite angles of 
 the triangle ; then the locus of the point of intersection of the 
 joining lines will be a conic described about the triangle. 
 
 (356) The locus of a point from which the two tangents to 
 a given conic are at right angles is a circle. 
 
 (357) Two conies have double contact, shew that the locus 
 of the poles with respect to the first, of tangents to the second, is 
 another conic having double contact with both at the same 
 points. 
 
 (358) If a conic be inscribed in the triangle of reference and 
 also touch the straight line whose trilinear equation is 
 
 ^ + ^ + ^ = 0, 
 
 its centre will lie on the chord of contact of tangents from the 
 point (\ : /i : v) to the circle circumscribing the triangle of 
 reference. 
 
 (359) If a series of conies be inscribed in a quadrilateral, 
 their centres will lie upon the straight line joining the middle 
 points of the diagonals. 
 
 (360) AA'B'B is a quadrilateral inscribed in a conic. 
 Two tangents PF, QQ' meet the diagonals AB', A'B in the 
 points P, P, Q, Q' respectively. Shew that a conic can be 
 described so as to touch AA', BB', and also to pass through the 
 four points P, P', Q, Q'. 
 
 (361) If three conies circumscribe the same quadrilateral 
 the common tangent to any two is cut harmonically by the 
 third. 
 
 (362) If a series of parabolas are inscribed in a triangle the 
 poles of any fixed straight line lie on another straight line. 
 
MISCELLANEOUS EXERCISES. 463 
 
 (363) If ABOD be a quadrilateral circumscribing a conic, 
 and P, Q be any two points on the conic, then a conic can be 
 described touching AP, A Q, GP, CQ, and passing through the 
 points B and D. Also the pole of PQ with respect to the first 
 conic will be the pole of BD with respect to the second, 
 
 (364) Given four tangents to a conic, find the locus of the 
 foci. 
 
 (365) If P and Q be any points on a conic, S and H the 
 foci, a circle can be inscribed in the quadrilateral formed by 
 SP, SQ, HP, HQ having its centre at the pole of PQ with 
 respect to the conic. 
 
 (366) 8S', HH', O are the diagonals of a quadrilateral cir- 
 cumscribing a conic which cuts SS' m. A, A'; HH' in B, B'; 
 00' in I, T. Shew that if SS' , HH' intersect in (7, CI, CI are 
 the tangents at /, /'. Also if D, B' be the points of contact of the 
 sides SH, and E, E' the points of contact of the sides SH, S'H', 
 the tangent at any other point P will meet DE in a point Z 
 such that [S . POZO'] is harmonic. Prove also that if the tan- 
 gents to the conic from the point Q divide the angle OQO, 
 harmonically, the locus of Q will be a conic passing through 
 O, 0' and having its centre at C. 
 
 (367) Suppose 0, 0' in the last exercise are the circular 
 points at infinity, and consider what focal properties will be 
 derived. 
 
 (368) Apply the property proved in Art. 417 to shew that 
 if a parabola be inscribed in a triangle its focus lies on the 
 circle circumscribing the triangle. 
 
 (369) Two conies intersect in a point and touch the sides 
 of a quadrilateral whose diagonals are AA', BB', CO'. If OP, 
 OQ be the tangents at O.then pencils [O.APA'Q], [O.BPB'Q], 
 [ . CPC Q] are harmonic. Hence prove that two confocal conies 
 intersect at right angles. 
 
 (370) Given a focus and two tangents to a conic section, 
 shew that the chord of contact passes through a fixed point. 
 
464 MISCELLANEOUS EXERCISES. 
 
 (371) Shew that \i fl+gm + hn = (i, the locus of the foci 
 of the conies represented by the equation 
 
 la mp 717 
 is a cubic curve : and find its equation. 
 
 (372) Two imaginary parabolas can be drawn having their 
 foci at the circular points at infinity, and intersecting in the 
 centres of the inscribed and escribed circles of a given triangle. 
 
 (373) If a variable conic to which a fixed triangle is always 
 self-conjugate always passes through the centre of the circle 
 inscribed in the triangle, the locus of its centre will be the circle 
 circumscribing the triangle. 
 
 (374) B, C are the foci of a conic P; C; A those of a conic 
 Q; A, B those of a conic B; and common tangents to Q, B 
 intersect in A!, common tangents to B, P in B' , and common 
 tangents to P, Q in C. Shew that the systems of points 
 A', B', C; a; B, C; A, B', C; A, B, C are collinear. 
 
 (375) If /(a, (8, 7) =0 be the trllinear equation to a conic, 
 its directrices will be represented by the equation 
 
 where k must be so detei-mined that the first member may 
 resolve into two linear factors. 
 
 (376) Trace the cubic whose trilinear equation is 
 F^ {^ + 7^^ + 2/S7 cos u4) = 7 (aa. + 5/3 + 07)', 
 
 (377) Shew that the six points in which the cubic 
 (/3 + 7 cos A) (7 + a cos B){a + ^ cos C) 
 
 = (7 + j8coSu4) (a + 7Cos5) (fi + acosO) 
 is cut by any circle concentric with that which circumscribes 
 
MISCELLANEOUS EXERCISES. 465 
 
 the triangle of reference lie two and two at the extremities of 
 three diagonals of the circle. 
 
 Shew also that the centre of the circle is a point of inflexion 
 of the cubic. 
 
 (378) If two cubics intersect in six points on a conic, 
 their other three points of intersection are coUinear. 
 
 (379) Through six points on a conic there are drawn three 
 cubics. Shew that their other points of intersection lie by threes 
 on three concurrent straight lines. 
 
 (380) Find the conic of five-pointic contact at any point 
 of the cuspidal cubic -tf = 3?z. 
 
 The next eightt Exercises are selected from 
 Cambridge College Examination Papers. 
 
 (381) If the inscribed circle of a triangle ABG pass through 
 the centre of the circumscribed, then 
 
 cos A + cos 5+ cos G= V2. 
 
 (382) Determine the value of Te that the equation a — Z;/S = 
 may represent a tangent to the circle described about the triangle 
 of reference. 
 
 (383) Shew that the trilinear coordinates of the centre of 
 the conic section 4ay3 — Xt" = are 
 
 XaJc sin 5 \a6c sin ^ oJc sin (7 
 
 2(c'-\a6) ' 2(c»-\aJ)' \ab-(^ ' 
 
 (384) Prove that if m = 0, v = 0, lo = be the equations of 
 the sides of a triangle, the equation of a conic section circum- 
 scribed about the triangle will be 
 
 I in , n „ 
 - + -+-=0, 
 U V to 
 
 w. 
 
 30 
 
466 MISCELLANEOUS EXERCISES. 
 
 and that the equations of the tangents of the conic section at the 
 three vertices of the triangle will be 
 
 m n „ n I ^ I m ^ 
 
 - + - = 0, - + - = 0, - + - = 0. 
 
 V W W U U V 
 
 (385) The equation to the self-conjugate rectangular hyper- 
 bola passing through (/, ff, h) is 
 
 (386) If ABC be a triangle such that the angular points 
 are the poles of the opposite sides with respect to a conic, and 
 abc be another triangle possessing the same properties with 
 respect to the same conic, then that one conic will circumscribe 
 the two triangles. 
 
 (387) lf-+o + - = 0, -+-75 + - = 0be two comes, find 
 ' a p y a p y 
 
 the equations of the several lines joining the centre of the circle 
 
 inscribed in the triangle of reference with the four points of 
 
 intersection of the two conies. 
 
 (388) If A', B'; C be the middle points of the sides of a 
 triangle ABC, and a parabola drawn through A', B', C meet 
 the sides again in A", B", C"— then will the lines AA", BB", 
 CC" be parallel to each other. 
 
 (389) Conies circumscribing a triangle have a common 
 tangent at the vertex; through this point a straight line is 
 drawn: shew that the tangents at the various points where it 
 cuts the curves all intersect on the base. 
 
 (390) OA, OB are tangents to a conic section at the points 
 A, B; and C is any point on the curve. It AC, BC be joined 
 and OPQ be drawn to intersect A C, BC (or these lines produced) 
 in P and Q, prove that BP, A Q intersect on the curve. 
 
 (391) AP, BP, CP are drawn to meet a conic circumscribing 
 ABC in DUF. The tangents at BEF meet BC, AC, AB in 
 A'B'C. Prove that A'B'C lie on a straight line. 
 
 (392) A conic is described about a triangle so that the 
 normals at the angular points bisect the angles. Shew that the 
 
MISCELLANEOUS EXESCISES. 467 
 
 distances of the centre from the sides are inversely proportional 
 to the radii of the escribed circles. 
 
 (393) Find the equation to the conic section circumscribing 
 the triangle of reference and bisecting the exterior angles of the 
 triangle. 
 
 (394) . The diameter of a conic circumscribing ABC which 
 bisects the chords parallel to AP, BP, GP where P is a given 
 point, meet the tangents to the conic at A, B,C in DEF, prove 
 that DEF lie on the polar of P. 
 
 (395) The tangents to a conic at ABC meet the opposite 
 sides of the triangle produced in PQR. The other tangents 
 from Q and R being drawn meet AB and CA respectively in 
 q, r; prove that Pqr lie on a straight line. 
 
 (396) A conic section is inscribed in the triangle ABC and 
 touches the sides opposite to A, B, C in A', B', C respectively, 
 any point P is taken in B'C and CP, BP meet AB, AC in c, b 
 respectively; prove that be is a tangent to the inscribed conic. 
 
 (397) If perpendiculars be drawn from the angular points 
 of a triangle on the opposite sides, an ellipse can be drawn 
 touching the sides at the feet of the perpendiculars ; construct it. 
 
 (398) If a conic touch a triangle at the feet of the perpen- 
 diculars from the angular points, the distance of the centre from 
 the feet is proportional to the length of the sides. 
 
 (399) Two conies touch each other in two points A, B. If 
 be any point in the straight line AB and if OPP' QQhe 
 any chord cutting the two conies in P, Q and P, Q' respectively, 
 prove that 
 
 1.+ 1 = 1 +.1 
 
 OP^ 0Q~ OF^ OQ" 
 
 (400) The four common tangents to two conies intersect two 
 and two on the sides of their common conjugate triad. 
 
 30—2 
 
468 MISCELLANEOUS EXERCISES. 
 
 (401) Shew that the general equation to a circle in trilinear 
 coordinates is 
 
 S= {aoL + h^ + cy) {la + »n/3 + ny) - (o/Sy + hya + ca^) = 0, 
 and that the square of the tangents drawn to it from a point 
 whose trilinear coordinates are a', /S', y is j^ 8': where a, h, c 
 are the sides and A the area of the triangle of reference. 
 
 (402) The self conjugate, the nine-points', and the circum- 
 scribing circle of a triangle have a common radical axis, which 
 is the polar of the centre of gravity with respect to the self-con- 
 jugate circle. 
 
 (403) The radical axes of the circles (areal) 
 
 (m, V, w, u, v, w'){a/3yY = 0, 
 
 {p,q,r,p',q',r){a0yy = O, 
 
 will he represented by 
 
 ua + v^ + wy poL + q0 + ry 
 
 u-^v+ w — v! —v —w p + q-i-r —p' — q' — r ' 
 
 (404) Three circles described on the chords of a complete 
 quadrilateral as diameter have a common radical axis. 
 
 (405) Shew that the equation to any circle that passes 
 through the points B, C of the triangle of reference, may he 
 expressed in the form 
 
 /37 sin A + yaamB+ a/S sin C 
 
 + ia(asin.4 +)Ssin5 + 7sin (7) = 0; 
 
 and determine the value of the constant i in order that the circle 
 may Umch the side AB, 
 
 (406) Shew that the equation of the fourth tangent common 
 to the circle inscribed in the triangle of reference, and to the 
 escribed circle that touches BG externally is 
 
 A^,^ . . B-G . 
 acos -^ + \p-y) sin— — =0. 
 
MISCELLANEOUS EXERCISES. 46D 
 
 (407) The two points at which the escribed circles of a 
 triangle subtend equal angles, lie on the straight line whose 
 equation in trilinear coordinates referred to the triangle is 
 
 a cos -4 {b- c)+^coaB{c-a)+yco3 C{a-b) = 0. 
 
 (408) If T be the intersection of perpendiculars from 
 A, B, C, on the opposite sides of the triangle ABO and L 
 the middle point of BG, and if TL be produced to meet the 
 circle circumscribing ABC in A' ; shew that AA' is a diameter 
 of the circle. 
 
 (409) Prove that four fixed points on a conic subtend at 
 any other point on the curve a pencil of constant anharmonic 
 ratio, which is harmonic if the line joining two of the points 
 which are conjugate passes through the pole of the line joining 
 the other two. 
 
 (410) The anharmonic ratio of the pencil formed by joining 
 a point on one of two conies to their four points of intersection is 
 equal to the anharmonic range formed on a tangent to the other 
 by their four common tangents. 
 
 (411) Pp, Qq, Br, Ss are four chords of a conic passing 
 through the same point, shew that a conic can be drawn touching 
 
 SB, BQ, PQ, sr, rq, qp. 
 
 (412) Having given five tangents to a conic, shew how to 
 determine their points of contact. 
 
 (413) The equation of the line passing through the feet of 
 the perpendiculars from a point (aj, ^i, %) of the circle 
 
 on the sides of the fundamental triangle, may be put in the form 
 ^,cos(7-7,co3if 7,cosul-aiCOsa '^ ^ a^cos B - ^^cos C 
 
470 MISCELLANEOUS EXERCISES. 
 
 (414) AP, BP, CP&i-e drawn to meet a conic circumscrib- 
 ing the triangle ABC in Z>, E, F; EF, FO, DE meet BG, 
 CA, AB in A^, 5,, C, respectively. Shew that these three 
 points are in a straight line, which is the polar of P with regard 
 to the conic. 
 
 (415) One conic touches OA, OB in A and B, and a 
 second conic touches OB, 0(7 in Sand G: prove that the other 
 common tangents to the two conies intersect on A G. 
 
 (416) Two conies touch each other, and through the point 
 of contact any chord is drawn : if the tangents to the conies at 
 the other extremities of the chord meet on the common tangent, 
 the common chord of the conies will pass through their iuter- 
 eection. 
 
 (417) Two rectangular hyperbolas intersect in four points, 
 shew that each point is in the intersection of perpendiculars 
 from the angles on the sides of the triangles formed by joining 
 the other three. 
 
 (418) If three conies be drawn each touching two sides of a 
 triangle and having the third for their chord of contact, shew 
 that the three chords of intersection pass through a point. 
 
 (419) If three parabolas are drawn having two of the sides 
 of a triangle for tangents and the third for their chord of con- 
 tact, shew that their other three points of intersection form a 
 triangle similar to the original one and of one-ninth its area. 
 
 (420) If a triangle is self-conjugate with respect to each of 
 a series of parabolas, the lines joining the middle points of its 
 sides will be tangents: all the directrices will pass through 
 the centre of the circumscribing circle : and the focal chords, 
 which are the polars of 0, will envelope an ellipse inscribed in 
 the given triangle which has the nine-points' circle for its auxili- 
 ary circle. 
 
MISCELLANEOUS EXERCISES. 471 
 
 (421) Shew that there are two points P, Q in the polar of 
 with respect to a conic, such that PO is perpendicular to the 
 polar of P, and QO to the polar of Q and that then POQ is a 
 right angle. 
 
 (422) Through a point P within the triangle ABC a line is 
 drawn parallel to each side. Prove that the sum of the rect- 
 angles contained by the segments into which each of these lines 
 is divided by the point P is equal to E^— OP', R being the 
 radius of the circumscribed circle, its centre. 
 
 (423) The diameter of the circumscribing circle of the tri- 
 angle ABC 
 
 sin 2^ sin 2.B sin 2 C" 
 
 where a', /S', 7' are the perpendiculars on any tangent from 
 A, B, C. 
 
 (424) Similar circular arcs are described on the sides of a 
 triangle ABC, their convexities being towards the interior of 
 the triangle; shew that the locus of the radical centres of these 
 tliree circles is the rectangular hyperbola 
 
 sin(^-C) sin(g-^) sin(^-^) _^ 
 a /3 7 
 
 a, ^, 7 being the trilinear coordinates of a point with respect to 
 the sides of the triangle. 
 
 (425) The pole of a tangent to a fixed circle with re- 
 spect to another fixed circle will have a conic section for its 
 locus. 
 
 (426) A conic circumscribes a triangle ABC, the tangents 
 at the angular points meeting the opposite sides on a straight 
 line DEF. The lines joining any point P to A, B, and C meet 
 the conic again in A', B , C: shew that the triangle A'B'C 
 envelopes a fixed conic inscribed in ABC, and having double 
 contact with the given conic at the points where they are met 
 
472 MISCELLANEOUS EXERCISES. 
 
 by DEF. Also the tangents at A, B', C to the original conic 
 meet B'C, C'A', AS in points lying on BEF. 
 
 (427) If straight lines be drawn from the angular points of 
 a triangle ABC, through a point P, to meet the opposite sides 
 in a, ;8, 7, shew that if P moves on a conic, the intersection of 
 BA and /Sy traces out a conic, and that tangents to corre- 
 sponding points of the conies intersect on BG. 
 
 (428) A tangent to a conic cuts two fixed tangents in 
 Tand T', B and B are fixed points, shew that the locus of 
 intersection of TB and T'B' is a conic. 
 
 (429) A, B, C are three fixed points, and G such that 
 tan 5^ G^ varies as t&nBGG; prove that the locus of (? is a 
 conic passing through A, B, and C. 
 
 (430) The locus of the centre of rectangular hyperbolas 
 inscribed in the triangle of reference of trilinear coordinates is 
 the circle 
 
 a' sin 2A+^ sin iB+r sin 2C7= 0. 
 
 (431) A rectangular hyperbola circumscribes a triangle; 
 shew that the loci of the poles of its sides are three straight lines 
 forming another triangle, whose angular points lie on the sides 
 of the first, where they are met by perpendiculars from the 
 opposite angular points. 
 
 (432) Ellipses are described on AB as diameter, and touch- 
 ing BC; if tangents be drawn to them from C, shew that the 
 locus of the points of contact is a straight line. 
 
 (433) The locus of the centres of conies inscribed in a 
 triangle and such that the centres of the escribed circles form 
 a conjugate triad with respect to them is a straight line parallel 
 to aa + J;8 + C7 = in triangular coordinates. 
 
 (434) If P move so that a tangent to its path is always 
 parallel to its polar with respect to an ellipse, then P traces out 
 an ellipse similar and similarly situated to the former. 
 
MISCELLANEOUS EXERCISES. 473 
 
 (435) The square of the distance of a point from the base 
 of a triangle is equal to the sum of the squares of its distances 
 from the sides. Prove that its locus is a conic, and will be 
 an ellipse, parabola, or hyperbola, according as the vertical 
 angle of the triangle is obtuse, right or acute. 
 
 In the case of the parabola find its focus and directrix. 
 
 (436) The section of a cone cannot be an equilateral hyper- 
 bola unless the angle of the cone is at least a right angle. 
 
 (437) The equation of the asymptotes of the conic 
 
 (lo?+ m0'+ n-f) (j + ^+ ^) = («a + h^ +c yf. 
 
 (438) A conic is described about a given quadrilateral, 
 prove that its centre always lies on a conic which passes through 
 the middle points of the sides and diagonals of the quadri- 
 lateral and also through the three points of intersection of the 
 diagonals. 
 
 (439) Shew that a conic section can be described passing 
 through the middle points of the four sides and of the two 
 diagonals of any quadrilateral, and also through the intersections 
 of the diagonals and of the two pairs of opposite sides, its centre 
 being the centre of gravity of four equal particles placed at the 
 angular points of the quadrilateral. Prove also that this conic is 
 similar to each of the four conies which have their centres 
 respectively at the four angular points of the quadrilateral and 
 to which the triangle formed by joining the other three points is 
 self-conjugate. 
 
 (440) Shew that the conic which touches the sides of a 
 triangle and has its centre at the centre of the circle passing 
 through the middle points of the sides, has one focus at the 
 intersection of the perpendiculars from the angles on the oppo- 
 site sides, and the other at the centre of the circle circumscribing 
 the triangle. 
 
474 MISCELLANEOUS EXERCISES. 
 
 (441) One focus of a conic inscribed in the triangle ABO 
 lies in a conic touching AB, AC aX B, C respectively; prove 
 that the other focus lies on another conic touching as before. 
 
 If these two conies coincide, the major axis of the conic 
 inscribed in the triangle passes through a fixed point. 
 
 (442) If a conic be inscribed in a triangle and its focus 
 move along a given straight line, the locus of the other focus 
 is a conic circumscribing the triangle. 
 
 (443) If an ellipse inscribed in a triangle has for one focus 
 the point of intersection of the perpendiculars from the angular 
 points of the triangle on the opposite sides, shew that (i) the 
 other focus is the centre of the circle circumscribing the triangle, 
 and (ii) the major axis of the elliptic is equal to the radius of 
 this circle. 
 
 (444) If a = 0, /9 = 0, 7 = 0, S = be the equations to four 
 straight lines all expressed in the form 
 
 X cos a + y sin a —p = 0, 
 
 and if aa + 6/3 + C7 + (Z8 = for all values of x and y, then the foci 
 of all the conic sections which touch the four straight lines lie 
 on the curve 
 
 a h c d . 
 
 0/070 
 
 (445) If a hyperbola be described touching the four sides 
 of a quadrilateral which is inscribed in a circle, and one focus 
 lie on the circle, the other focus will also lie on the circle. 
 
 (446) The poles of any fixed straight line with respect to a 
 series of confocal conies lie on another straight line. 
 
 (447) Tangents are drawn from any point on an ellipse, to 
 an interior confocal ellipse, and with the points of contact as 
 foci a third ellipse is described passing through the given point 
 on the first : prove that its latua rectum is constant. 
 
 (448) If a series of parabolas touch three straight lines their 
 foci lie on a circle and their directrices are concurrent. 
 
MISCELLANEOUS EXERCISES. 475 
 
 (449) Three parabolas are drawn touching the three sides 
 of a triangle ABC. If D, E, Fhe the foci, prove that 
 
 ABC _ a.b.c 
 DEF'f.g.h' 
 where /, g, h are the sides of the triangle DEF. 
 
 (450) ABC is any triangle and P any point: four conic 
 sections are described with a given focus touching the sides of 
 the triangles ABC, PBC, PC A, PAB respectively, shew that 
 they all have a common tangent. 
 
 (451) Tangents are drawn at two points P, P' on an 
 ellipse. If any tangent be drawn meeting those at P, F in 
 R, R', shew tliat the line bisecting the angle RSR intersects 
 RR' on a fixed tangent to the ellipse. Find the point of contact 
 of this tangent. 
 
 (452) Four conies can be described about a triangle having 
 a given point as focus. If the sides of the triangle subtend 
 equal angles at the given point, one of the conies will touch the 
 other three, 
 
 (453) Circles are described on a system of parallel chords 
 to an ellipse as diameters, shew that they will have double 
 contact with an ellipse, having the extremities of the diameter 
 of the chords as foci, and itself having double contact with the 
 original ellipse. 
 
 (454) With the centre of the circumscribed circle as focus 
 three hyperbolas can be described passing through ABC with 
 excentricities cosec 5 cosec C, cosec cosec -4, cosec^ co-sec^, 
 their directrices being the lines joining the middle points of the 
 sides. The fourth point of intersection of any two lies on the 
 line joining one of the angles to the middle point of the oppo- 
 site side. 
 
 (455) An ellipse is described round a triangle, and one 
 focus is the intersection of perpendiculars from the angular 
 points on the opposite sides. Shew that the latus rectum 
 
 2.B cos A cos B cos G 
 
 ~ . A . B . C ' 
 
 sm-sm-sin- 
 
 where R is the radius of the circumscribing circle. 
 
476 MISCELLAKEOUS EXERCISES. 
 
 (456) ABO is a triangle, 8 is any point, 8A, SB, 8C are 
 joined, 8a, 8b, 8c are drawn perpendicular to 8A,8B, 80, meet- 
 ing the sides in abc. Straight lines Aa, Bb, Cc are drawn, 
 forming a triangle PQR. If two conies with 8 as focus be 
 inscribed in the two triangles ABO, PQR, shew that the latus 
 rectum of one is half that of the other. 
 
 (457) The equation to the directrix of a parabola which 
 touches the sides of the fundamental triangle and the straight 
 line la + m^ + M7 = may be expressed 
 
 ocot^('i-il+jScot5fi-i)+7Cotcf7-i') = 0. 
 
 (458) Three points ABG are taken on an ellipse. The 
 circle about ABC meets the ellipse again in P, and PP^ is a 
 diameter. Prove that of all the ellipses passing through ABCP', 
 the giyen ellipse is the one of minimum excentricity. 
 
 (459) Shew that the reciprocal of a given conic A with 
 respect to another conic B will be a rectangular hyperbola, if the 
 centre of B lies on a certain circle. 
 
 (460) A series of equal similar and equally eccentric ellipses 
 are reciprocated with respect to a circle, shew that, if one of the 
 reciprocals be a rectangular hyperbola, they will all be so, and 
 have double contact with a hyperbola whose eccentricity e' is 
 given by 
 
 e being the eccentricity of the ellipses. 
 
 The foett Exercises which follow are selected from 
 
 THE papers of THE CAMBRIDGE MATHEMATICAL TeIPOS 
 
 Examinations. 
 
 (461) Let four points be. taken at random in a plane, join 
 them two and two in every possible way, the joining lines being 
 produced, if necessary, to intersect. Join these points of inter- 
 
MISCELLANEOUS EXERCISES. 477 
 
 section two and two, in every possible way, producing as before 
 the joining lines. Every line in the figure so formed is divided 
 harmonically. 
 
 (462) Prove the following method of drawing a tangent to 
 any curve of the second order from a given point P without it. 
 From P draw any two lines, each cutting the curve in two 
 points. Join the points of intersection two and two, and let 
 the points in which the joining lines (produced if necessary) 
 cross each other be joined by a line which will, in general, cut 
 the curve in two points A, B. PA, PB are tangents at A and B. 
 
 (463) Two points are taken within a triangle : each is joined 
 with an angular point, and the line produced to intersect the 
 opposite side. Prove that the six points of intersection so 
 formed will lie on the same conic section, and find its equation. 
 
 (464) An ellipse is described so as to touch the three sides 
 of a triangle ; prove that if one of its foci move along the cir- 
 cumference of a circle passing through two of the angular points 
 of the triangle, the other will move along the circumference of 
 another circle, passing through the same two angular points. 
 Prove also that if one of these circles pass through the centre of 
 the circle inscribed in the tiiangle, the two circles will coincide. 
 
 (465) Shew that all conic sections, which have the same 
 focus, have two imaginary common tangents passing through 
 that focus ; and hence derive a general definition of foci. 
 
 (466) Prove that the locus of the centre of a conic section; 
 passing through four given points, is a conic section ; and shew, 
 (1) that when the straight line joining each pair of the given 
 points is perpendicular to the straight line joining the other pair, 
 this locus will be a circle, (2) that when the four given points lie 
 in the circumference of a circle, this locus will be a rectangular 
 hyperbola. 
 
 (467) Find a point the distances of which from three given 
 points, not in the same straight line, are proportional to p, q, 
 and r respectively, the four points being in the same plane. 
 
478 MISCELLANEOUS EXERCISES. 
 
 (468) OA, OB are common tangents to two conies having 
 a common focus S; CA, CB are tangents at one of their points 
 of intersection ; BD, AE tangents intersecting CA, CB in D, K 
 Prove that SBE is a straight line. 
 
 (469) Two tangents OA, OB are drawn to a conic, and are 
 cut in P and Qhy a variable tangent ; prove that the locus of 
 the centres of all circles described about the triangle OPQ is a 
 hyperbola. 
 
 (470) The circles wliich touch the sides AC,BCoia. triangle 
 at G, and pass through B, A respectively, intersect AB in E and 
 F. Lines drawn from the centres of the circles inscribed in the 
 triangles ACF, 5 (IE parallel to CE, CF respectively, meet AC, 
 BC in P, Q. Prove that CP is equal to CQ. 
 
 (471) If ABC be a triangle whose sides touch a parabola, 
 and^, q, r be the perpendiculars from A, B, C on the directrix, 
 prove that 
 
 p tan A +q tan B+r tan C= 0. 
 
 (472) A, Pand B, Q are points taken respectively in two 
 parallel straight lines, A, B being fixed and P, Q variable. 
 Prove that if the rectangle AP, BQ he constant, the line PQ 
 will always touch a fixed ellipse or a fixed hyperbola according 
 as P and Q are on the same or opposite sides of AB, 
 
 (473) Three hyperbolas are drawn whose asymptotes are 
 the sides of a triangle ABC taken two and two, prove that the 
 directions of their three common chords pass through the angu- 
 lar points A, B, C and meet in a point, — which will be the centre 
 of gravity of the triangle, if the hyperbolas touch one another. 
 
 (474) Prove that the straight lines represented by the 
 equation 
 
 (a^-^) sin C+k (a sin A+^sin B) (/3 cos P- a cos A) = 0, 
 
 are parallel to the axes of the conic section 2a/S = h^. 
 
MISCELLANEOUS EXERCISES. 479 
 
 (475) If the lines which bisect the angles between pairs of 
 tangents to an ellipse be parallel to a fixed straight line, prove 
 that the locus of the points of intersection of the tangents will be 
 a rectangular hyperbola. 
 
 (476) Tangents to an ellipse are drawn from any point in a 
 circle through the foci, prove that the lines bisecting the angle 
 between the tangents all pass through a fixed point. 
 
 (477) P is a point within a triangle ABC, and AF, BP, CP 
 meet the opposite sides in A', E , C respectively ; if Pa, Pb, Pc 
 be measured along PA , PB, PC so that these last are harmonic 
 means between PA', Pa ; PB, Pb ; PC, Pc respectively, prove 
 that a, b, c lie on a straight line. 
 
 (478) Prove that the envelope of the polar of a given point, 
 with respect to a system of confocal conies, is a parabola the 
 directrix of which passes through the given point. 
 
 (479) ABC is a given triangle, P any point on the circum- 
 scribing circle, through P are drawn PA', PB', PC at right 
 angles to PA, PB, PC to meet BC, CA, AB respectively; shew 
 that A', B", C lie on one straight line that passes through the 
 centre of the circumscribing circle. 
 
 (480) If tangents be drawn to the circle bisecting the sides 
 of a triangle, at the points where it has contact with the four 
 circles which touch the sides, these tangents will form a quadri- 
 lateral whose diagonals pass one through each angular point 
 of the triangle. 
 
 ^o^ 
 
 (481) If POP', QOQ, ROE, SOS' be four chords of an 
 ellipse, the conic sections passing through 0, P, Q, R, S and 
 0, F, Q', E, S' will have a common tangent at 0. 
 
 (482) Four circles are described, each self-conjugate with 
 respect to one of the triangles formed by four straight lines in 
 the same plane; prove that the four circles have a common 
 chord. 
 
480 MISCELLANEOUS EXERCISES. 
 
 (483) A conic always touches four given straight lines; 
 prove that the chord of intersection of the circle, described about 
 any one of the triangles formed by three of these straight lines, 
 with the circle which is the locus of the intersection of two tan- 
 gents to the conic at right angles to each other, always passes 
 through a fixed point. 
 
 (484) A triangle is circumscribed about a given conic, and 
 two of its angular points lie on another given conic ; prove that 
 the locus of the third angular point is another conic, and that the 
 three conies have a common conjugate triad. 
 
 (485) Two triangles, ABG, A'B'C, are described about an 
 ellipse, the side BO being parallel to B'C, CA to C'A', ABia 
 A'B'. If B'C, C'A', A'B' be cut by any tangent in P, Q, B 
 respectively, prove that AP, BQ, OR will be parallel to one 
 another. 
 
 (486) If a point be taken, such that each of the three dia- 
 gonals of a given quadrilateral subtends a right angle at it, prove 
 that the director circle of every conic which touches the four 
 sides of the quadrilateral will pass through this point. 
 
 Prove also that the polars of this point with respect to all the 
 conies will touch a conic of which the point is a focus. 
 
 (487) If the perpendiculars Aa, Bb, Cc be let fall from 
 A, B, C the angular points of a triangle upon the opposite sides, 
 prove that the intersections of BC and he, of OA and ca, of AB 
 and ah will lie on the radical axis of the circles circumscribing 
 the triangles ABC, and ahc. 
 
 (488) A series of conies are circumscribed about a triangle 
 ABC, having a common tangent at A. Prove that the locus of 
 the intersection of the normals at B and C is a conic passing 
 through B and C, — and also through A if the given tangent form 
 a harmonic pencil with AB, A C and the diameter of the circum- 
 scribing circle through A. 
 
 If in this case the corresponding locus be found for B and C, 
 prove that the three conic? fIW have a fourth point in common. 
 
MISCELLANEOUS EXERCISES. 481 
 
 (489) Prove that if a rectangular hyperbola be reciprocated 
 with respect to a circle, the tangents drawn to the reciprocal 
 conic from the centre of the circle will be at right angles to one 
 another. 
 
 (490) If /(«, ^, 7) =0 be the trilinear equation to a plane 
 curve, and <j> (I, m,n)=0 the condition that the line 
 
 la + 7»y3 + ny = 
 
 may be a tangent to it; prove that/(?, m,n)=0 is the condi- 
 tion that the straight line h + mfi + ny = may be a tangent to 
 the curve ^ (a, /3, 7) = 0. 
 
 (491) If a triangle circumscribe a circle, and p^, p^, p^ be 
 the algebraical perpendiculars let fall from any point in the 
 plane of the triangle upon the line joining its angular points to 
 the centre of the circle, prove that 
 
 A B G ^ 
 
 p, cos 2" + P2 cos - + p, cos 2=0. 
 
 A, B, C being the angles of the triangle. 
 
 (492) If an ellipse of given area be circumscribed about a 
 given triangle, the locus of the centre, referred to the same tri- 
 angle, will be represented by the equation 
 
 (bfi + cj- ay) (cry + aa- J/3) (aa + 6y3 - 07) = Ca^^-y', 
 
 C being a constant depending on the length of the sides at the 
 triangle. 
 
 (493) A rectangular hyperbola passes through the angular 
 points, and a parabola touches the sides of a given triangle : 
 shew that the tangents drawn to the parabola, from one of the 
 points where the hyperbola cuts the directrix of the parabola, are 
 parallel to the asymptotes of the hyperbola. Which of the two 
 points on the directrix is to be taken? When the two points 
 coinqide, shew that one curve is the polar reciprocal of the other 
 with regard to the coincident points. 
 
 w. .^1 
 
482 MISCELLANEOUS EXERCISES. 
 
 (494) Five straight lines are drawn in a plane thus forming 
 five quadrilaterals: shew that the straight lines joining the 
 middle points at the diagonals of these quadrilaterals meet in a 
 point. 
 
 (495) A parabola is drawn so as to touch three given 
 straight lines, shew that the chords of contact pass each through 
 a fixed point. 
 
 (496) With any one of four given points as centre, a conic 
 is described self-conjugate with regard to the other three ; prove 
 that its asymptotes are parallel to the axes of the two parabolas 
 which pass through the four given points. 
 
 (497) If a triangle be self-^conjugate with respect to a 
 parabola, shew that its nine-points' circle passes through the 
 focus. 
 
 (498) A triangle is described about the conic 
 
 two of its vertices moving along the lines 
 
 la. + jn/3 + wf, I'a + ?n'/3 + n'<y = : 
 prove that the locus of the third vertex will be the conic 
 
 I, m, n ' 
 {ir + mm' + nny{u' + 0' + r/)+ V , m' , n =0. 
 
 a, &, 7 
 
 (499) On the sides BG, CA, AB of a triangle ABC are 
 taken points A' , E, C respectively, each of the angles C'A'B', 
 A'B' C, B' C'A' being of given magnitude ; prove that, if the 
 area of the triangle A'B' C be a minimum subject to these con- 
 ditions, and K be the area of the triangle ABC, R the radius 
 of the circle ^'-B'C, 
 
 (sin J/sin(^jK42 , sin B' sin {B + B') 
 ( Sin A sm B 
 
 sin C sin (0-1- 0')! j^ 
 
 ■^ nST? 1 = ^- 
 
MISCELLANEOUS EXERCISES. 483 
 
 (500) If /(ot, )8, 7) =0 be the equation of any curve refer- 
 red to trilinear coordinates, and if 
 
 /a(«,A7), Moi,^,i), /y(a,)S,7) 
 
 be the partial differential coeflElcients o{f{a, ^, 7) with respect to 
 a, /8, 7 respectively, shew that the line 
 
 a/a («»i. c, - ax, -b) + fif^ {cx^ , c,-ax^-b) 
 
 + lfyip^i> c, - ax, - 5) = 
 
 is in general an asymptote of the curve, x, being a root of the 
 equation 
 
 f{cx, c,—ax — h)= 0, 
 
 and a, h, c being the sides of the triangle of reference. 
 
 Hence find the asymptotes of 
 
 a(/3-7)' + ^(7-ar + 7(a-/Sr = 0, 
 and trace the curve, the triangle of reference being equilateraL 
 
 31—2 
 
NOTES ON THE EXERCISES. 
 
 Ebsults and Occasional Hints. 
 
 (2) They all lie on the straight line joining (a, /8) to the 
 origin. 
 
 (3) a + y3cos(7=0. (4) oa = J/3. Area = a6 sin C. 
 (5) p' sin" C = (a - a')' + {/8 - /S')'' + 2 (a - a') (fi - 0) cos C. 
 
 (6) 
 
 i(a;8'~a'/3)cosec(7. (7) \^d. 
 
 (8) 
 
 ^'. ^^- (9) ia-a/3=(aa-&/3)co3a 
 
 (10) 
 
 The points of trisection of BG are 
 
 
 / 4A 2A'\ , / 2A 4A\ 
 [''3b- 3c)^°n«'36' 3c)- 
 
 (") 
 
 2A 2A 2A 
 3a' 3b' 3c' 
 
 ('1t)^ 
 
 a cos B COS C bcoaCcosA c cob A cob B 
 
 siaA ' sin 5 ' sin G 
 
 (13) The centre of the escribed circle opposite to A is 
 
 given by 
 
 « 2A 
 
 -a =^ = 7=1— . 
 
 b + c — a 
 
 (14) {mb" + nc^) a" + Z (S/S + 07)= = (mc" + U?)^ + m{py + aa)' 
 
 = {W + mff) -/+ n (aa + b^f. 
 
EESULTS AND OCCASIONAL HINTS. 485 
 
 (15) J. (16) 2Acos^cos£cosa 
 
 (17) — a cos ^ + y3 cos 5 + 7 cos C = 0, 
 
 o cos -4 — /3 cos 5 + 7 cos (7 = 0, 
 o cos ^ + yS cos 5 — 7 cos (7 = 0. 
 
 (18) ^«^c 
 
 {bn + cm) {cl + an) {am + bl) ' 
 
 (19) They all lie on the line la. + m^ + m7 = 0. 
 
 (20) See (10). (21) aa - 2J/3 - 2c7 = 0. 
 
 (22) The equation to PQR will be aa + 4J/8 - 2c7 = 0. 
 
 (23) aa - (»re - 1) */3 - (n - 1) 07 = 0. The coordinates are 
 
 ~ "~1 ?^ »t-l 
 b ' n — m' c ' m—n' 
 
 (24) The straight line is a cos ^ = /3 cos B. 
 
 (25) [In each of the given equations for — read +]. 
 
 The equation to AP is m^-n'^ = 0. To A Q, m^+3ny= 0. 
 The other equations may be written down by symmetry. 
 
 (26) They are respectively parallel to the lines a + /3 = 0, 
 o — /3 = 0, which are known to be at right angles. 
 
 (29) ^ = ^. (30) 1=^. 
 
 ^ ' b c ^ ' a b+c 
 
 /o«\ i -1 2 sin G 
 
 (32) tan ' ^ = ^ . 
 
 *• ' 3 cos ^ cos ^ — cos C 
 
 (34) m^+wy— la = 0, (35) ma. + mn^ + 7 = 0, 
 
 W7 + Za — wiyS =0, «/8 + wJy + a = 0, 
 
 la+m^— ny = 0. ly+ lma+0 = O. 
 
 (36) If J5C, B'C intersect in P; C4, C'A' in g; ^5, 
 ^'5' in B; P, Q, R will be found to lie on the fourth straight 
 line required. 
 
 (37) A particular case of (40), when AP, BQ, CR are per- 
 pendicular to the sides of the triangle of reference. See (40). 
 
486 RESULTS AND OCCASIONAL HINTS. 
 
 (38) A particular case of (40), when AF, BQ, CR are the 
 bisectors of the angles of the triangle. 
 
 (39) A particular case of (40), when P, Q, B are the 
 middle points of the sides of the triangle of reference. In this 
 case the fourth straight line is the line at infinity. 
 
 (40) To construct the lines, let be the point given by 
 la. = 'm^ = ny, join OA, OB, 00 and produce them to meet 
 BO, OA, AB respectively in P, Q, R; also let BO, QR intersect 
 in F; OA, RP in Q'; AB, PQ in E; then three of the straight 
 lines required will be the sides of the triangle PQR, and the 
 fourth will pass through F, Q, R'. The coordinates of the 
 middle points of PF will be 
 
 m 
 
 m n n m 
 
 and the coordinates of the middle points of QQ', RR can be 
 
 written down by symmetry. These three points lie on the 
 
 straight line 
 
 la m^ ny _ 
 I 7~ ~r — — U. 
 
 a o c 
 
 (42) The equations to the straight lines joining the point 
 of reference A to the two given points at infinity can be written 
 down, and the condition that they should be at right angles can 
 be reduced to the given form. 
 
 (43) The line through A will have the equation 
 
 6y3 {q -p) +cy{r -p) = 0. 
 
 (44) aa. {2p-q-r) + 6/3 (2q-r-p) + cy (2»- -p-q) = 0. 
 
 (45) [aa — b^) coaA=y{h + a cos 0) . 
 
 (46) 2Aa + «?(aa + &/3+C7)=0. (47) 4A. 
 
 ,^g, 4APmV A 
 
 {nl + lm- mn) {Im + mn — nl) (mn + nl—lm)' ^ •' I ' 
 
 (53) The straight line is the perpendicular from on BA. 
 
RESULTS AND OCCASIONAL HINTS. 
 
 487 
 
 (55) [For r = 0, read r = d.] Two straight lines will 
 satisfy the conditions, and their equations are 
 
 d{aa+i0+cy)== + {aa + b0)bsmA. 
 
 (56) Let s be the altitude required. Then the equation to 
 BQ isaaip + s) + JyS {q+s) +cy {r + s)=0. Hence at B the 
 
 value of /S is -^ .^ — - . But at P the value ot B is -,-. — ^ , 
 o p—q b r—q 
 
 and the middle point of the diagonal PB lies on the locus of 
 
 ;S = 0, therefore ^ 1 = 0, which gives 
 
 p—q r—q ° 
 
 ^^2prj-pq-qr 
 q — r 
 
 (58) The two paragraphs must be read as separate ques- 
 tions. In the second paragraph, for ' this point,' read 
 
 'the point ? = ^ = ^.' 
 
 (59) 
 
 a b c 
 
 la + m^ + nr/ I'd + »n'/3' + n'y 
 IX + mfi + nv 
 
 l'\+ m'fi +n'v 
 
 (60) The coordinates of P arc a' + V> ^' + W> t' + "P- ^^ 
 for the other points. 
 
 (62) Apply the result of (59). 
 
 (63) 
 
 tan"\- 
 
 2A {{m-nY + {n-iy+{l-my} 
 
 {b'-c')(l-m){l-n)+{e'-a'){m-n) {m-l) +{a''-b'){n-l){n-m) " 
 (64) Apply Art. 27. All the straight lines are parallel to 
 X cos'a + y cos' /3 + » cos" 7=0. 
 
 (69) 
 
 (70) 
 (74) 
 
 l — m' 
 
 A {Imn + 1) 
 
 {l+l){m+l) [n+l) 
 
 = 0. 
 
 u, 
 
 V, 
 
 w 
 
 I, 
 
 m, 
 
 n 
 
 I', 
 
 m', 
 
 n 
 
438 
 
 RESULTS AND OCCASIONAL HINTS. 
 
 (75) -U, V, W =0. 
 
 a, b, c 
 
 b, a, c 
 (81) See Art. 331. 
 
 (85) Tlie straight line ] a, /3, 7 =0. 
 
 /, 3, h 
 f> 9, ^t 
 
 (86) The equation may be written 
 
 (a cos ^+/8 cos 5+ 7 008 0)"+ (asin^ + /3sin J9+ 73in Cy—0. 
 
 (87) If -^--e^B-c, l-4,= c-A, 
 
 then will 
 
 -y^=^A-B, 
 
 and the equation may be written 
 
 (m cos .4+ V cos B + wcos 0)'+ (m sin ^ + « sin i? + lo sin C)*= 0. 
 
 (88) The equation may be written 
 
 {a? + «/" + «"+ 'iyz) {a? + y' + z' + 2sj;) (a;» + y" + s'' + 2xy) = 0, 
 
 each factor of which, when equated to zero, represents a pair of 
 imaginary straight lines parallel to a line of reference. 
 
 (90) If la = m^ = ny be the point 0, the points of inter- 
 section lie on the straight line 
 
 la, + m0 + W7 = 0. 
 
 (91) See Art. 129. 
 
 (92) Use one of the equations of Art. 108. 
 (93), (94) Apply Art. 130. 
 
 (95), (96), (97) Apply Art. 125. 
 
 (100) One system of lines satisfying the required conditions 
 has the equations 
 
 
 '-^ + ^ + 1 = 0. 
 
 \ /J, V 
 
RESULTS AND OCCASIONAL HINTS. 489 
 
 (101) Fonn the determinant as in Art. 149. The sum of 
 three rows will be identically equal to the sum of the other three. 
 Hence the determinant vanishes and the condition is fulfilled. 
 
 (102) See Art. 97. 
 
 (103) 3{x-yy-Az{x + y)+z' = 0. 
 
 (104) Apply Art. 149. The conic is the circle of Art. 307. 
 
 (105) The straight lines 
 
 la + w/3 + W7 = 0, r a + m'0 + n'y = 0, 
 are asymptotes and their point of intersection the centre. 
 
 (106) Three parabolas. 
 
 (107) The four points in which the two straight lines 
 x= ±1/ cut the two straight lines a; + 3y= — 1. 
 
 (110) See Arts. 161 and 129. 
 
 (114) If (a, /8', y) be the coordinates of P, the three tan- 
 gents have the equations 
 
 ^ ^1-^' 'yj.fi-?^ ?,^_?5: 
 
 (116) With the notation of Art. 91, the three conies have 
 the equations 
 
 u' = v'+ w\ v^ = w' + u', w^ = u' + v\ See (1 19). 
 
 (117) A conic with respect to which two particular triangles 
 are self-conjugate. 
 
 (120) Two imaginary straight lines dividing the right 
 angle harmonically. 
 
 (123) The resulting equation should be 
 
 {Ikol + m/^' + {W + m) ny' = 0. 
 
 (124) (^ + ^V f ) {?«' + ^^^ + ''7') - (.A + ^/3 + ^7)' = 0. 
 (1 26) a^yz + Vzx + <?xy=^ (a;' cot ^ -I- / cot 5 -f- s" cot C). 
 
490 RESULTS AND OCCASIONAL HINTS. 
 
 (130) Either condition is V^ + Vwi/* + Vwi* = 0. 
 
 (132) If i +^+r?=o be the conic, 
 
 a /3 
 
 -tH 1-- = is the straight line. 
 
 (134) Take the triangle as triangle of reference, and let 
 {x, y, z) be the fixed point. The centre will lie on the conic 
 
 a'" V" s" „ 
 X y z 
 
 (147) The polar of the point (a', ^ ■, y) with respect to 
 
 {la +m0 + ny) {I'a + «i'/8 + n'y) = 0, 
 is given by 
 
 {loL + m^' + ny) {I' a + m'/S + n'y) 
 
 + {To! + m'^' + n'y) {la + m/S + ny) = 0, 
 
 which proves the proposition (Art. 88). 
 
 (148) Apply Arts. 46 and 146. 
 
 (149) Let a = be one of the straight lines : let the others 
 be parallel to y3 = 0, and let 7 = be the straight line at infinity. 
 Then the equation to the locus can be written 
 
 la' +771 {13 + kyf + m'{fi + k'yf +... = ny', 
 or , ua'' + v^+wy'' + 2u'^y=0, 
 
 which shews that a = is the chord of contact of tangents firom 
 13 = 0, 7 = 0, i. e. of tangents parallel to /3 = 0. Which proves 
 the proposition. 
 
 (150) Apply a method similar to that of (149). 
 
 (151) Let a = be the chord of contact, ^ = the tangent 
 to the first conic, 7 = the straight line joining the point of con- 
 tact of this tangent to one of the points of contact of the conies. 
 Then the equation to the first conic may be written 
 
 ly' + ma^ + n^y = 0, 
 
RESULTS AND OCCASIONAL HINTS. 491 
 
 and the equation to the other will be 
 
 ka? + I'f + ?wa/3 + w/Sy = 0, 
 which meets yS = in two points given by 
 
 which therefore divide harmonically the line joining the point 
 of contact of the tangent with the point where the tangent meets 
 the common chord. 
 
 (152) A conic circumscribing the triangle which is self- 
 conjugate with respect to both the conies. 
 
 (156) The equation to the locus is 
 
 (df if df\' 
 
 i/c>^-'))-=M^.'l'l 
 
 (1 57) If ZySy + wya + wa/S = be the first conic, the fixed 
 point is given by la. + m^ = 0, 7=0. 
 
 (161) See Art. 417 (first column). 
 
 (164) Apply Art. 65. (165) Apply Art. 65. 
 
 (166) If X, /*, V be the direction sines of the chords, and 
 A" the constant area, the equation to the locus is 
 
 /(a,A7) + /'y(X,/^, i')=0, 
 
 which may be rendered homogeneous. 
 
 (] 67) If the constant be ^ , the equation to the locus will be 
 
 /(..ft„ + *(.|+^|+,f).o. 
 
 which may be rendered homogeneous. 
 
 (168) Refer the conic to a self-conjugate triangle having 
 one vertex at the focus. 
 
 (1 72) Tan f = as : a. Apply Art. 285. 
 
 (174) To deduce Euclid in. 31. Apply Art. 159. 
 
492 RESULTS AND OCCASIONAL HINTS. 
 
 (177) Take the given centres as points of reference. 
 
 (181) In trilinear coordinates : 
 
 (i) a sin A{l3siaB+y sin C) = [/ScosB—y cos C), 
 
 (ii) 2a cos" 5- (/3 + 7) = (/3 - 7) (/8 cos 5 - 7 cos C), 
 
 (iii) a {c^ + 67) + a cos A (b/3 + cy) 
 
 = {b^ - cy) {13 cos 5 - 7 cos O), 
 and similar equations. 
 
 (182) If the area of the given rectangle he 2//, times that of 
 the triangle, the equation to the locus may be written 
 
 ^y sin A-\-ya sin B + a^ sin C 
 
 = fi{a.amA + ^s'mB + y sin Cy, 
 
 shewing a circle concentric with the circle circumscribing the 
 triangle. 
 
 (183) If the given constant be 2/j, times the area of the 
 triangle the locus will be represented by the equation 
 
 i^y sin A + yu sin B + a^ sin C) sin A sin B sin C 
 = /i (a sin -4 + /3 sin 5 + 7 sin Cy. 
 
 (184) Apply Arts. 210, 289, 314. 
 
 (185) In general the equation 
 
 [da ' d^' dy] ~ 
 
 represents the polars of the circular points with respect to the 
 conic /(o, /3, 7) = 0. If the latter be a circle, the polars are 
 tangents and pass through the circular points. Hence, Art. 318, 
 the equation represents an indefinitely small circle. But since 
 the polars of points at infinity intersect at the centre of the 
 curve, this indefinitely small circle is at the centre, or is concen- 
 tric with the given circle. 
 
 (186) Apply a method analogous to that of Art. 273. 
 
RESULTS AND OCCASIONAL HINTS. 493 
 
 (188) Any conic touching the sides a = 0, /3 = 0, 7=0 is 
 known to have the equation 
 
 f'a* + m^/S* + nV - 2jMn/37 - 2nlya - 2lma^ = 0. 
 
 The straight line 8 = 0ora + ^ + 7 = will be a tangent (Art. 
 210) provided l + m + n=0. Eliminate the terms involving 
 a", ^, rf (Art. 327) and we have 
 
 {m + rif ^7 + (n + If 7a + (Z + m)=' a/3 + PaS + vd^l + mV = 0, 
 
 or in virtue of 
 
 Z + »w + w = 0, P(/37 + aS) +»i'(7a + /SS) + m''(a/8+7S) =0. 
 
 And we may write 
 
 l = lju — v, ■m = v — \, n = \ — fi,. 
 
 (189) Apply the condition that the line at infinity should 
 be a tangent. A solution is given at length in Vol. \. of the 
 Messenger of Mathematics, p. 201. 
 
 (190) This may be deduced from (189) by writing 
 
 \ = 1, X' = i, fJL = fi = V = v = 0. 
 
 (191) The two roots of the quadratic in (190) cannot be 
 equal unless either A=B and C = D, or else A= C and B = D. 
 In either of these cases the parabola would be altogether at in- 
 finity. In any other case there can therefore be two parabolas 
 drawn through four fixed points, one of which will however 
 degenerate into two parallel straight lines if the points lie on 
 two such lines. 
 
 (192) In virtue of the relation 
 
 a + /3 + 7+S = 0, 
 the equation may be written 
 
 X(a + /3) (a + 7) + /* 03+ 7) (/3 + a) + " (7 + a) (7 + ^) = 0, 
 
 which shews that its locus circumscribes the triangle whose 
 sides are^ + 7 = 0, 7 + « = 0, a + /S = 0. 
 
494 RESULTS AND OCCASIONAL HINTS. 
 
 (195) Apply the result of (194). If (a', /3', 7', 8') be the 
 point 0, the two tangents are given by 
 
 a' /3'+7' 8' "' 
 
 a ^ 7 8 _(. 
 
 and ~+of '~V~> 
 
 a p y o 
 
 which form a harmonic pencil (Art. 124) with the lines 
 
 °^ ^ r, A ^ y n 
 
 - - -r, = and ^ - -^ = 0. 
 
 a P 7 
 
 (196) Let both conies circumscribe the quadrilateral of 
 reference, and let (a,, /3j, 7,, 8,), (a^, ^j, 7,, B^ be the two 
 points of contact of a common tangent. Then the equations 
 
 to the two conies are 7^— = — =r and 75 — = — s- , and the equa- 
 
 tions to the straight lines can be readily formed. 
 
 (197) Apply a method analogous to that of (196). 
 
 (200) One of the points is given by a = /8 = 7. The other 
 two are the points in which 6 = meets the fourteen-points' 
 conic. 
 
 (215) If p = 0, q = 0, r = 0, lp+mq + nr = are the equa- 
 tions to the angular points in order, the middle points of the 
 diagonals lie upon the straight line given by 
 
 {n—l)p = {l + 2m + n)q = (l — n)r. 
 
 (217) By Art. 393 the centre is at the point 
 
 p tan A + q tan B + r tan C = 0, 
 
 which by Ex. (211) is the centre of the circle with respect to 
 which the triangle is self-conjugate (Art. 179, Cor. 2). 
 
 (218), (219), (220) Apply a method similar to that of 
 Ex. (217). 
 
 (221) See Ex. (214) and apply Art. 380. The second form 
 of the equation shews that the circle circumscribes the triangle 
 whose angular points axQ q + r = 0, r +p = 0, p + q=0. 
 
EESULTS AND OCCASIONAL HINTS. 495 
 
 (227) The angular points of a triangle co-polar with the 
 triangle of reference may be expressed by the equations 
 
 'Kp = mq + nr, /iq = nr+lp, vr=l2) + mgi; 
 the equation to the conic is then 
 
 \lp^ + fimgr^ + vnr^ + {jw + mn) gr 
 
 + {vX + nl)rp+ (\/i + Im) pq = 0. 
 
 (236) See Euclid iii. 32, 
 
 (237) See Euclid iii. 20 and 22. 
 
 (239) Any tangent to the interior of two concentric circles 
 is bisected at the point of contact. 
 
 (240) A circle can be described touching the escribed and 
 inscribed circle of a triangle. 
 
 (241) Reciprocate with respect to the focus. 
 
 (242) Beciprocate with respect to S. 
 
 (243) Tangents dra^vn from the point of reciprocation arc 
 at right angles. 
 
 (245) Eeciprocate 244. 
 
 (246), (247) A series of confocal conies may be recipro- 
 cated into a system of circles with the same radical axis. 
 
 (248) Eeciprocate with respect to 0. 
 
 (249) The locus of the intersection of tangents to a conic 
 which are at right angles is a circle. 
 
 (250) See Euclid in. 10. 
 
 (251) Eeciprocate with respect to S. 
 
 (252) Eeciprocate (110). 
 
 (253) Eeciprocate with respect to any point. 
 
 (254) Extend Exercise (132), applying Art. 95. Then 
 reciprocate. 
 
496 RESULTS AND OCCASIONAL HINTS. 
 
 (266) That the centre lies on a given straight line is equi- 
 valent to saying that the given straight line and the straight 
 line at infinity are conjugate. Hence this is a line condition. 
 
 (269). The double point (m = 0, u = 0). 
 
 (270) The double tangent (m = 0, v = 0). 
 
 (271) At the point (m = 0, v = 0). 
 
 (272) The six points (l : m : n), {m : n : I), {n : I : m), 
 {nx m : I), {I : n : m), {mil: n). 
 
 (275) If S=0 be the equation to one cubic, and m = the 
 straight, line joining the points of contact, tlie equation to the 
 cubic will be ;Sf + u'v = 0, where v = represents a straight line 
 on which the other points of intersection lie. 
 
 (276) Form the equations as in (275). 
 
 (277) See Art. 481. (279) Apply (275). 
 
 (280) The equation to the curve is 
 
 a (P/3= + Q0y + R^) = /Sy (AT/S + M-^). 
 
 (282) The equation to the cubic is 
 
 {Iji + JH/S + wy) [ma + n0 + lr/) (no. + l^ + my) 
 
 = Imn (\a + m^ + vy) (^ + " +A {ol + + y). 
 
 (283) The equation to the cubic is 
 (Pa-/8-7) {m'^-y-a.) {n'y-a-^) 
 
 = (a+/3+7) {h + m^ + ny)'. 
 
 (286) The six points 
 
 {a:h -.c), {b: c: a), {c : a : b), {c : b : a), [a: c:b), {b : a : c) 
 
 on a conic, and the three points in which the straight line 
 x + y + s= cuts the lines of reference. 
 
RESULTS AND OCCASIONAL HINTS. 497 
 
 (288) A particular case of the next exercise. 
 
 (289) If a = be the asymptote, the cubic will have the 
 equation a ./(a, /8, 7) «= {la + m^ + n<y) {a + fi + 7) in triangular 
 coordinates, where /{<*■, ff,'y)=0 is the equation to a conic. 
 The equation to the required locus will be 
 
 which represents a hyperbola. In the particular case when 
 l = ni = n, the locus reduces to the straight line 
 
 ^ = !^ 
 
 (290) If a; = 0, y = 0, s = be the tangents, and m = 
 the line of contact, the equation must take the form of Art. 
 483. 
 
 (292) Using the equation of Art. 472, the chords are 
 represented by 
 
 rijjS + m^y = 0, l^y + M,a = 0, m^a + 1^ = 0, 
 and the tangents by 
 
 »ii/3 + Tijy = 0, nj7 + Z,a = 0, l^a + mfi = 0. 
 The condition that either system should be concurrent is 
 
 (304) Apply the last result. 
 
 (306) The coeflScients \, /*, v cannot be all of one sign- 
 The real or imaginary tangents at the points of reference have 
 the equations 
 
 ^ 7« 7' a'' a' & 
 
 '± + 'L = o, ^ + -=0, 1 + ^ = 0. 
 
 w. 32 
 
498 RESULTS AND OCCASIONAL HINTS. 
 
 (307), (308) The equation may be written 
 
 1 + r+ > =0. 
 
 (311) Take the equation of (309) and apply the condition 
 that each equation of (310) may represent a pair of coincident 
 straight lines. 
 
INDEX. 
 
 [The numereJs refer to pages.] 
 
 Abbidged Notation. 
 
 The straight line, 104 
 
 The straight line in terms of the 
 equations to three other straight 
 lines, 105 
 
 Condition of concurrence of straight 
 lines, 113 
 
 Conic Sections, 165—171 
 
 Abridged notation for the circle, 288 
 
 Conic sections in tangential coordi- 
 nates, 361—363 
 
 Curves of the third degree and curves 
 of the third order, 410—413 
 
 Anqle 
 
 Between given line and line of refer- 
 ence, 49 
 Between two straight lines in trilinear 
 coordinates : 
 
 the tangent, 50, 85 
 the sine, 52, 64, 85 
 the cosine, 54, 85 
 In terms of perpendicular distances 
 of the lines from three points of 
 reference, 62 
 In terms of the direction sines of the 
 straight lines, 81, 82 
 
 Anhabmonio Eatio. 
 Definitions, 132 
 
 Anharmonic ratio of straight lines 
 whose equations are given, 135-137 
 
 Different ratios obtained by taking 
 
 range of points in different orders, 
 
 139 
 Anharmonic property of a conic, 321 
 Auharmonic ratio of range of points 
 
 whose tangential equations are 
 
 given, 341 
 Anharmonic ratio of a range of points 
 
 is the same as that of the pencil 
 
 formed by their pob.rs with respect 
 
 to any conic, 381 
 Anharmonic property of tangents to 
 
 a conic, 384 
 
 Area 
 
 Of a triangle in terms of two perpen- 
 dicular coordinates of each angular 
 point, 7 
 
 Of a triangle in terms of trilinear co- 
 ordinates uf each angular point, 
 21 
 
 Of a triangle wheu equations to its 
 sides are given, 68 
 
 Area of an ellipse whose equation is 
 given, 281 
 
 Asymptote. 
 Definition, 160 
 Equation to the asymptotes of given 
 
 conic, 247 
 Tangential coordinates of asymptotes, 
 
 359 
 Its polar reciprocal, 375 
 
500 
 
 INDEX. 
 
 Given asymptote of a conic equivalent 
 
 to two simple conditions, 397 
 Asymptotes of cubic curves, 416 
 Parabolic asymptote, 418 
 General equation to asymptote, 445 
 
 Axis OP A CONIO. 
 Definition, 269 
 
 Equation to the axes, 270, 272 
 Lengths of the axes, 279 
 Axis of a conic given in position 
 equivalent to one point- and one 
 line-condition, 398 
 
 Beianohon's Theorem 
 Enunciated and proved, 218 
 The reciprocal of Pascal's Theorem, 
 387 
 
 Centre of a Conic. 
 
 Its coordinates, 241 
 
 Its coordinates in terms of the dis- 
 criminants, 261 
 
 Equation to centre in tangential coor- 
 dinates, 352 
 
 Its polar reciprocal, 375 
 
 Polar reciprocal of concentric conies, 
 375 
 
 ClBCLE. 
 
 Equation to a circle I'eferred to a self- 
 conjugate triangle, 181 
 
 Equation to a circle referred to an 
 inscribed triangle, 198 
 
 Equation to a circle referred to a cir- 
 cumscribed triangle, 214 
 
 Conditions that the general equation 
 of the second degree should repre- 
 sent a circle, 243 
 
 Equation to circle whose centre and 
 radius are given, 287 
 
 Every circle passes through the cir- 
 cular points at infinity, 239 
 
 The intersection of circles (see Radi- 
 cal Axis), 290 
 
 The nine-points' circle, 296 
 
 Evanescent circles, 302 
 
 General equation in tangential coor- 
 dinates, 346 
 
 Condition that eqaation should repre- 
 sent a circle in tangential coordi- 
 nates, 353 
 
 Circles reciprocate into conies having 
 a focus at the centre of reciproca- 
 tion, 381 
 
 ClRCOLAR points AT INFINITY. 
 Definition, 126 
 Their coordinates, 127, 129 
 Every circle passes through them, 289 
 Every conic which passes through 
 
 them is a circle, 289 
 Tangential equation, 347 
 
 Class of a Coeve. 
 Definition, 364 
 Class of a curve the same as the order 
 
 of its reciprocal, 372 
 Curves of the third class, 401 — 413 
 
 CoAXiAX Triangles. 
 Definition, 109 
 Arc co-polar, ] 10 
 
 CoLLiNEAB Points. 
 Definition, 109 
 
 Condition in triliuear coordinates, 22 
 Condition in quadrilinear coordinates, 
 314 
 
 Common Chords. 
 
 Definition of pair of common chords, 
 168 
 
 Two conies have three pairs of com- 
 mon chord.s, 168 
 
 Equation to common chords of two 
 conies whose equations are given, 
 246 
 
 Common chord of circles (see Radi- 
 cal Axis), 290 
 
 Common Tangents. 
 
 To conic and great circle pass through 
 
 the foci, 351 
 Tangential equations, 361 
 
 CONCDRHENT STRAIGHT LiKES. 
 DeGnition, 109 
 
INDEX. 
 
 501 
 
 Condition in tiilineaT coordinates, 27 
 The perpendiculaia from vertices on 
 
 sides of any triangle are concui- 
 
 rent, 33 
 
 CONDITIOHS. 
 
 A conic will satisfy five simple condi- 
 tions, 158 
 
 Point-conditions and line-conditions 
 defined, 39i 
 
 One conic fulfilling five line-condi- 
 tions or five point-conditions, 394 
 
 Two conies fulfilling four line and 
 one point-coudition, or four point 
 and one line-condition, 394 
 
 Four conies fulfilling three line and 
 two point-conditions, or three point 
 and one line-condition, 394 
 
 Analysis of compound conditions, 
 396—398 
 
 Cubic curve will satisfy nine condi- 
 tions, 404 
 
 Curve of »th order will satisfy 
 
 !i*^ conditions, 449 
 
 CONFOCAL CONIOS 
 
 Are inscribed in the same imaginary 
 quadrilateral, 351 
 
 Conic Section. 
 
 Every conic section is a curve of the 
 second order, and conversely, 156 
 
 Equation to conic through five given 
 points, 158 
 
 Equation to conic referred to a self- 
 conjugate triangle, 1 73 
 
 Equation to conic referred to an in- 
 scribed triangle, 192 
 
 Equation to conic referred to a cir- 
 cumscribed triangle, 206 
 
 Tangential equation to conic with 
 given foci, 348 
 
 General tangential equation, 349 
 
 Tangential equation to conic referred 
 to a self- conjugate triangle, 360 
 
 Tangential equation to conic referred 
 to a circumscribed triangle, 359 
 
 Tangential equation to conic referred 
 
 to an inscribed triangle, 361 
 Polar reciprocal of conic section, 375 
 
 Conjugate Conios. 
 Definition, 273 
 Are similar and similarly situated, 
 
 274 
 Two conjugate conies cannot be both 
 
 real unless they be hyperbolas, 275 
 
 CONJDGATE Point on a Cuete. 
 
 Definition, 402 
 
 Equation to the imaginary tangents 
 
 at a conjugate point, 437 
 Cubic can have only one conjugate 
 
 point, 403 
 
 Conjdoatb Points, and Conjugate 
 
 Lines. 
 
 Definition, 391 
 
 Condition that two given points should 
 be conjugate (trilinear coordinates) 
 391 
 
 Condition that two given lines should 
 be conjugate (trilinear coordinates) 
 392 
 
 Condition that two given points should 
 be conjugate (tangential coordi- 
 nates) 392 
 
 Condition that two given lines should 
 be conjugate (tangential coordi- 
 nates) 393 
 
 Coordinates. 
 
 A system of perpendicular coordinates 
 
 referred to two axes, 1 
 Trilinear coordinates, 10 
 Their ratios often sufficient, 27 
 Areal and triangular coordinates, 94, 
 
 95, 
 Quadrilinear coordinates, 307 
 Tangential coordinates, 332 
 
 CO-POLAR TkIASGLES. 
 
 Definition, 109 
 Are co-axial, 111 
 
502 
 
 INDEX. 
 
 Cubic Loci and Cubic Ehvelopes 
 Defined, 401 
 General equation, 405 
 Cubic loci through eight fixed points 
 pass through a ninth, 420 
 
 Cdsp. 
 Definition, 402 
 
 Cubic can have only one cusp, 403 
 Cusped curve of the third order is of 
 
 the third class, 407 
 General equation to a tangent at a 
 
 cusp, 440 
 At a cusp all first polars touch the 
 
 curve, 441 
 
 CUKVATDBE. 
 
 General expression for radius of cur- 
 vature at any point on a curve, 
 446 
 
 Coordinates of centre of curvature, 
 448 
 
 Diameter of a Coiric. 
 
 Definition, 240 
 
 Its equation, 240, 262 
 
 Conjugate diameters defined, 240 
 
 Direction of diameter of given para- 
 bola, 251 
 
 Equation to diameter of parabola, 252 
 
 Condition that equations should re- 
 present a pair of conjugate dia- 
 meters, 265 
 
 Tangential coordinates of conjugate 
 diameters, 358 
 
 Polar reciprocal of pair of conjugate 
 diameters, 375 
 
 Given diameter equivalent to one 
 line-condition, 396 
 
 Conjugate diameters given in position 
 equivalent to three simple condi- 
 tions, 397 
 
 DiBECTioN Sines. 
 
 The equations to a straight line, 73 
 Kelations connecting the direction 
 sines, 75 
 
 Symmetrical forms of these relations, 
 
 77 
 Direction sines of a straight line in 
 
 terms of the coefiScients of its ordi- 
 
 dinary equation, 86 
 Proportional to the coordinates of 
 
 tlie point where the straight line 
 
 meets the straight line at infinity, 
 
 229 
 
 DlSCBIMlNAMT. 
 
 Discriminant {H) of function of 
 
 second degree, 255 
 Definition of bordered discriminant 
 
 (K ), 256 
 Meaning of the conditions H=0, 
 
 ir= 0,261 
 
 DiBTAKCE BETWEEN POINTS. 
 
 Expression in trilinear coordinates, 45 
 
 Distance of Point feom Straight 
 Line. 
 
 In trilinear coordinates, 48, 61, 87 
 In tangential coordinates, 339 
 
 Double Point. 
 Definition, 402 
 
 Bcciprocates into double tangent, 374 
 Cubic can have only one double 
 
 point, 403 
 General criteria for double points, 436 
 Tangents at a double point, 437 
 
 Double Tangent. 
 Definition, 403 
 Bcciprocates into double point, 374 
 
 Duality. 
 
 Principle of duality, 368 
 
 Example of double interpretation, 369 
 
 Ellifse. 
 
 Definition, 160 
 
 Condition that general equation of 
 
 second degree should repres;;nt an 
 
 ellipse, 249 
 Polar reciprocal of ellipse, 376 
 
INDEX. 
 
 503 
 
 Equation. 
 
 Equation of first degree in trilinear 
 coordinates, 23 
 
 Equation of second degree in trilinear 
 coordinates, 157 
 
 Discussion of its general form in tri- 
 linear coordinates, 226 
 
 Equation of third degree in trilinear 
 coordinates, 401 
 
 Equation of nth. degree in trilinear 
 coordinates, 431 
 
 Equation of first degree in tangential 
 coordinates, 333 
 
 Equation of second degree in tan- 
 gential coordinates, 345, 349 
 
 Equi-anhaemokic Range 
 Defined, 329 
 
 Focus OP A CONIO. 
 
 Definition, 183, 216, 266 
 Coordinates of the foci of a conic, 266 
 Keal and imaginary foci, 350 
 Given focus of a conic equivalent to 
 two line-conditions, 398 
 
 Great Cibcle at Inpinttt. 
 
 Explained, 42 
 
 Its tangential equation, 347 
 
 Common tangents to great circle and 
 any conic intersect in the foci of 
 the conic, 351 
 
 Great circle reciprocates into evane- 
 scent conic at the centre of recipro- 
 cation, 380 
 
 Habmonic Ratio. 
 
 Definition, 137 
 
 Equations to straight lines forming 
 harmonic pencils, 139 
 
 Harmonic properties of a quadrilate- 
 ral, 142 
 
 Fourth harmonic to three given points 
 or lines, 144, 143 
 
 HoMOOBNBors. 
 How to render a trilinear equation, 
 homogeneous, 15 
 
 Properties of homogeneous functions 
 
 of the second degree, 223 
 General properties of homogeneous 
 
 functions, 426—430. 
 
 Htpekbola. 
 Definition, 160 
 
 Condition that general equation of 
 second degree should represent a 
 hyperbola, 249 
 Polar reciprocal of hyperbola, 376 
 
 Identical Equation. 
 
 Connecting the trilinear coordinates 
 of any point, 11 
 
 Connecting the triangular coordinates 
 of any point, 
 
 Connecting the distances of a straight 
 line from three given points, 58, 80 
 
 Connecting the direction sines of a 
 straight line, 77 
 
 Connecting the qnadrilinear coordi- 
 nates of any point, 308 
 
 luAOiNABY Points and Lines. 
 Definitions, 117, 119, 129 
 Every such line passes through one 
 
 real point, 120 
 Every such point lies on one real 
 
 straight line, 1 21 
 Imaginary point at infinity, 122 
 Imaginary branches of an evanescent 
 
 conic, 284 
 Imaginary tangents, 351 
 
 iNriNITT. 
 
 The straight line at infinity, its equa- 
 tion, 38 
 
 Parallel to any other straight line, 42 
 
 Its equation in qnadrilinear coordi- 
 nates, 310 
 
 Equations in tangential coordinates, 
 339 
 
 Infinity reciprocates into the centre 
 of reciprocation, 374 
 
 Infinite branches of cubic curves, 
 415 
 Inflexion. 
 
 Definition, 402 
 
504 
 
 INDEX. 
 
 Beciprocal of point of inflexion, 174 
 Beal points of inflexion on a cubic arc 
 
 collinear, 409 
 Point of inflexion at infinity, 420 
 General criteria for points of inflexion, 
 
 442 
 
 iNTKBBEOTIOlf. 
 
 Form of equation to straight line 
 tlirongh the point of intersection 
 of given straight lines, 28 
 
 Coordinates of point of intersection of 
 given straight lines, 36 
 
 Intersection of conies (four points), 
 167 
 
 Equation to conic through the points 
 of intersection of given conies, 170 
 
 Tangential equation to point of inter- 
 section of given straight lines, 337 
 
 Multiple Points 
 
 Defined, 402 
 
 General criteria for a multiple point, 
 
 443 
 Equation to tangents at a multiple 
 
 point, 444 
 
 NiNE-POINTB' ClBCLE. 
 
 Its properties, 296 
 
 Equation to nine-points' circle of the 
 
 triangle of reference, 294 
 It touches the inscribed and escribed 
 
 circles, 300 
 
 NOBMAL. 
 Equations to normal to a conic, 228 
 
 Obseb op a Cdbve. 
 Definition, 364 
 Order of a curve the same as the class 
 
 of its reciprocal, 372 
 Curves of the third order, 401 — 421 
 
 OSCULATIOlf. 
 Definition, 402 
 
 Point of osculation reciprocates into 
 a point of osculation, 874 
 
 Pababola. 
 Definition, 160 
 Condition that general equation of 
 
 second degree should represent a 
 
 parabola, 249 
 Diameter of parabola, 251 
 Condition in tangential coordinates, 
 
 354 
 Polar reciprocal of parabola, 376 
 Parabolic asymptote, 418 
 
 Pab-vllel Straight Lises. 
 
 Condition in trilinear coordinates, 37 
 Equation to straight line parallel to 
 
 given line, 43 
 Parallelism of imaginary straight 
 
 lines, 124 
 Polar reciprocal of parallel straight 
 
 lines, 375 
 
 Pascal's Theorem 
 
 Enunciated and proved, 201 
 The reciprocal of Brianchon's Theo- 
 rem, 387 
 
 Pebfensiculab Sibaioht Lines. 
 
 Condition in trilinear coordinates, 50 
 Eqaations to straight line perpendi- 
 cular to ^ven line, 87 
 
 Point. 
 
 Bepresented by trilinear coordinates, 9 
 
 Coordinates of point dividing given 
 straight line in given ratio, 19 
 
 Point represented by equation in 
 tangential coordinates, 333 
 
 Tangential equation to a point at in- 
 finity, 338 
 
 Tangential equation to point dividing 
 given straight line in a given ratio, 
 340 
 
 Polab. 
 
 Polar curve of any point with re- 
 spect to a given curve defined, 434 
 
 Any polar curve passes through all 
 the singular points, 436 
 
INUEX. 
 
 505 
 
 I'OL.VR Keoiprooals 
 Explained, 372 
 Table of reciprocal loci with respect 
 
 to a corlic, 373—376, 380 
 Equations to reciprocal conies, 376 
 Table of reciprocal loci with respect 
 
 to a circle, 382—383 
 Keciprocation of angular magnitude, 
 
 383 
 Reciprocation of distances, 384 
 Conies reciprocated with respect to a 
 
 focus, 381 
 
 Pole and Polab. 
 DeBnition, 233 
 Kquation to polar of given point with 
 
 respect to a given conic, 233 
 Coordinates of pole of given straight 
 
 line, 234 
 Reciprocal properties of poles and 
 
 polars, 235, 236 
 Tangential equation to pole of given 
 
 straight line, 355 
 Polar of a finite point with respect to 
 
 great circle is at infinity, 3^7 
 
 Quadrilateral. 
 
 For 'complete quadrilateral' see te- 
 tragram. 
 
 Harmonic properties of a quadrila- 
 teral, 313 
 
 Conies circumscribing a quadrilateral, 
 336, 325 
 
 Conies inscribed in a quadrilateral, 
 325, 326 
 
 Radical Axis of Two Circles. 
 Definitions, 290 
 
 Three radical axes of three circles con- 
 current, 290 
 
 Self-Conjugate Triangle. 
 
 Definition, 175 
 
 Kquation to a conic refen-ed to a self- 
 conjugate triangle, 173 
 
 Triangle self -conjugate with respect 
 to each of a series of conies having 
 four common points, 189 
 
 W. 
 
 A given self-conjugate triangle equiva- 
 lent to three simple conditions, 397 
 
 Similar and Similaelt Situated 
 
 CoNioa. 
 
 Definition, 272 
 
 Equation to conic similar and simi- 
 larly situated to a given conic, 273 
 
 Their bordered discriminants are equal, 
 283 
 
 Their discriminants are in the dupli- 
 cate ratio of their Hnear dimen- 
 sions, 283 
 
 Singular Points. 
 Defined, 403 
 
 Cubic having a singular point, 406 
 Tangents at singular point, 407, 437 
 Singular point at infinity, 41 9 
 
 .Straight Lines. 
 
 Straight line parallel to a line of re- 
 ference, 4 
 
 Straight line bisecting angle between 
 lines of reference, 5 
 
 Straight line dividing angle into two 
 parts whose sines are in a given 
 ratio, 5 
 
 Perpendicular of the triangle of refer- 
 ence, 31 
 
 Straight line joining two points, 23 
 
 Equation in terms of perpendicular 
 distances from paints of reference, 
 25,60 
 
 Equations in terms of direction sines, 
 73 
 
 Condition that general equation of 
 the second degree may represent 
 two straight lines, 245 
 
 Two straight lines may be regarded 
 as a limiting case of a conic section, 
 284 
 
 Tangent. 
 
 Equation to tangent to a conic, 227 
 Its direction, 230 
 
 Two tangents can be drawn from any 
 point to a conic, 2-31 
 
 33 
 
506 
 
 INDEX. 
 
 Equation to pair of tangents from 
 
 any point, 237, 238 
 Tangents at a singular point, 407, 437 
 General equation to a tangent to any 
 
 curve, 433 
 
 Tetbagbam. 
 
 Hai-monic properties of a tetragram, 
 313 
 
 The middle points of the diagonals 
 are colliuear, 315 
 
 Properties of the foiulieen-points' co- 
 nic, 329 
 
 Tbaksformation of Coobdin aibs 
 
 From an oblique Cartesian system to 
 a perpendicular system, 8 
 
 From trillnear to triangular coordi- 
 nates, and vice versa, 96 
 
 From one trilinear or triangular sys- 
 tem to another, 146 — 152 
 
 From trilinear to tangential coordinates 
 for points and straight lines, 334 
 
 "Vektejc 
 
 Definition, 269 
 
 Equations to determine coordinates 
 of vprtices of a conic, 269 
 
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