^ A TfiEATISE ON TRILmEAE CO-ORDIIATES, INTEI^DED CHIEELY EOE THE USE OE JUNIOE STUDENTS. BY C. J. C. PRICE, M.A.,. FELLOW AND LECTIJEEE OF EXETER COLLEGE, OXFORD. BOSTON COLLEGE LIBRARY CHESTNUT HILL, MASS, OXrOED and LOmOl^: JOHISr HENEY aistd JAMES PAEKER. 1866. 151915 |nitt£b h) Itcssrs. "§vixhx, Contmnx-kt, ©xforb. PEEFACE. 'THE acknowledged want of a text -book on the subject of Trilinear Co-ordinates, adapted to the use of students for honours in the Mathematical Schools, has led to the publication of the present yolume. It has been found necessary, with a view to rendering the elementary portion of the work as complete as possible, to exclude the consideration of equations of an order higher than the second. For the same reason, problems 'relating to the focal properties of conic sections have not been dis- cussed ; but the Author does not regret that the limits of the work have compelled him to pass over investigations of a class in which little or nothing is gained by the em- ployment of the trilinear method. The introduction of matter belonging more properly to the department of Pure Greometry^ as also any reference to other systems of co-ordinates, has been as far as possible avoided; except, perhaps, in the fifth Chapter, where the importance of the subjects treated and the want of a succinct yet tolerably complete account of them seemed to warrant the digression. Besides an acquaintance with the principles of the Dif- ferential Calculus, such as a student who is about to enter upon this branch of Modern Geometry is sure to possess, the reader is supposed to have some knowledge of the Theory of Determinants. The Author feels it right to state that the papers on iv PREFACE. Trilinear Co-ordinates communicated by Mr. Allen Whit- worth, of St. John's College, Cambridge, in the first numbers of the "Messenger of Mathematics'' did not come under his notice until the earlier portion of this treatise was written. The results now published were arrived at independently, the perusal of the papers referred to having led only to the insertion of Art. 152. In writing Chap. Y. the Author has derived much assist- ance from Mr. Townsend's " Modern Geometry" and from a work by M. Housel entitled Introduction a la Geometrie Superieure. To books which are so well known as Dr. Salmon's " Conic Sections" and Mr. Ferrers' " Trilinear Co-ordinates" it is difficult to say to what extent he is indebted. In conclusion, the Author would take the present oppor- tunity of expressing his sincere thanks to Mr. J. D. Daven- port, Eellow of Brasenose College, for his kindness in re- vising a great portion of the manuscript for the press, as well as for many valuable suggestions, and to other friends for any assistance they may have rendered. ExETEE College, Oxeoed, - December 4, 1865. CONTENTS. CHAPTER I. EXPLANATION OF THE METHOD. THE STEAIGHT LINE. ARTS. PAGE 1. Teilineae, co-ordinates of a point ..... 1 2. Convention with regard to sign , . . . . ib. 3 — 5. Identical relation between the co-ordinates of a point . . 2 6. Interpretation of a few simple Trilinear Equations . . .3 7, 8. The general equation oe the stsaight line, in terms of the co-ordinates of a fixed point on it and the sines of the angles which it makes with the sides of the triangle of reference . . 4 9, 10. The equation of a straight line in terms of the co-ordinates of a fixed point on it and the direction-cosines of the line . . 6 11. To shew that the equation of a right line may be written in the form la + m^ + ny = . . . . .7 12. To shew, conversely, that every equation of this form represents a straight line ....... 8 13. Polar form of the equation of the second degree . . . ib. 14. Abbreviations explained ...... 9 15. To construct geometrically the straight line whose equation is given in the homogeneous form . . . . . . ib. 16. Equation of a straight line which passes through the intersection of two given lines . . . . . . .10 17. Condition that three straight lines should meet in a point . . ib. 18 et sqq. Examples . . . . . . .11 19. To find the equations of the bisectors of the angles of a triangle ABC, and to shew that they meet in a point . . . ib. 20. Co-ordinates of this point . . . . . . ib. 21. Equation of a straight line through one of the vertices . . ib. 22. To find the equations of the bisectors of sides of the triangle ABC, and to shew that they meet in a point . . . .12 23. Co-ordinates of this point ... . . . 13 24. To find the equations of the perpendiculars let fall from the vertices of the triangle ^.BCupon the opposite sides, and to shew that they meet in a point . . . . . . . ib. b VI CONTENTS. ARTS. PAGE 25. Co-ordinates of this point . . . . . .14 26. The proportional values of the co-ordinates of a point used instead of their actual values . . . . . . . ib. 27. Other methods of obtaining the results of Arts. 22, 24 . .15 CHAPTER II. THE STEAiaHT LINE CO]NrTIH"IJED. EELATIONS BETW^EElf THE CONSTANTS, 28 — 31. Fundamental relations between the constants in the tvro forms of equation of the straight line investigated in Arts. 7 — 10 . 16 32. Symmetrical equations of the straight line which passes through a given point and makes a given angle with a given straight line . 17 33. Symmetrical equations of a straight line perpendicular to a given straight line . . . . . . » 18 34 — 38. Relations between the direction-cosines of a given straight line - 19 35. Symmetrical expressions for the area of the triangle of reference in terms of the direction-cosines of any line . . . .20 39. To write down the homogeneous equation of a straight line whose symmetrical equations are given . . . . .22 40, 41. To express the direction-cosines of a straight line in terms of the coefficients of its homogeneous equation . . .23 42. To express the sines of the direction-angles in terms of the co- efficients of its homogeneous equation . . . .24 43, 44. Formulae relating to straight lines ... . . ih. CHAPTER III. THE STEAIGHT LINE CONTINUED. LINE AT INFINITY. 45, 46. Expression for the distance between two given points . . 26 47. Length of the perpendicular from a given point upon a given straight line . . . . . . .27 48, 49. Length of the line drawn from a given point to a given straight line so as to make with it a given angle . . . .28 50, 51. Expressions for the sine, cosine, and tangent of the angle be- tween two given straight lines . . . . . ih. 52, 53. Equation of the straight line at infinity . . . .30 CHAPTER IV. PEOBLEMS ON THE STKAIGHT LINE. 54. To find the co-ordinates of the point of intersection of the straight lines (Zu m^ ^i), (4, in^, %) ..... 31 CONTENTS. Vii ARTS. PAGE 55. Condition that the three straight lines (li, m^, n-^, (J^, m^, ^2), (Z3, w?3, %) may meet in a point . . . . .31 56. Condition that three given points may lie in the same right line . 32 57. Symmetrical equations of the straight line which joins two given points . . . . . . . . ih. 58. Homogeneous equation of the straight line which joins two given points ........ 33 59. 60. Homogeneous equation of a right line passing through a given point, and whose direction-cosines are given . . . ib. 61. Equation of the straight line which joins a given point to the inter- section of two given straight lines . . . . .34 62, 63. Equation of the straight line which bisects the angle between two given straight lines . . . . . . ib. 64. Condition that two straight lines whose homogeneous equations are given should be parallel to each other . . . .36 65, 66. Equation of the straight line which passes through a given point, and is parallel to a given straight line . . . . ib. 67. Two straight lines are parallel when their equations differ only by a constant term . . . . . . .37 68. Condition of parallelism of two straight lines whose equations are given in the homogeneous form . , . . . ib. 69 — 74. Condition of perpendicularity of two straight lines . . 38 75. Symmetrical equations of the perpendicular to a given straight line through a given point . . . . . .40 76. ^owo^eweot«5 equation of tlie same . . . . . ib. 77. Homogeneous equation of the straight line which passes through a given point and makes an angle co with a given straight line . 41 CHAPTER V. ANHAHMONIC EATIOS. HAEMONIC POINTS AND PENCILS. HOMOGEAPHIC SYSTEMS. INVOLTJTION. 78. 79. Definitions. — Pencil of rays, centre of pencil, transversal, range, axis ...... .43 80. Anharmonic ratio defined . . . . . ,44 81. A system of four points or rays gives six different anharmonic ratios, whereof three are the reciprocals of the other three . . ib. 82. Relations subsisting between these ratios . . . ,45 83. To shew that the anharmonic ratio of the four points in which a transversal meets a pencil is constant for any position of the transversal . . . . . . , . ib. Vlll CONTENTS. ARTS. PAGE 84, 85. Perspective pencils. — When two pencils are in perspective their anharmonic ratios are equal. — Axis of perspective; homology, axis of homology . . . . . ... 46 86. Definition of Harmonic section. Harmonic conjugates ; hariDonic range (or penciZ) ; equation of harmonicism . . .47 87. A segment whicli is harmonically divided is divided internally and externally in the same ratio ..... ih. 88. 89. Other forms of the condition of harmonicism . . .48 90. Anharmonic ratio of four concurrent lines whose direction-angles are given ........ ih. 91. Condition that four such lines should form a harmonic pencil . 49 92. Anharmonic ratio of a pencil of four rays passing through the intersection of two straight lines whose equations are given in the homogeneous form . . , . . .50 93. nomographic systems . . . . . .51 94. Condition that four straight lines which pass through the inter- section of a given pair should form a harmonic pencil . . ib. 95. Equations of a pair of straight lines which are harmonic conjugates with respect to a given pair . . . . .52 96. 97- The internal and external bisectors of the angle included between any pair of right lines are harmonic conjugates with respect to them . ...... ih. 98, 99, T^ prove the harmonic properties of a complete quadrilateral . 53 HOMOGEAPHIC SYSTEMS OF POINTS AND LINES. 100. Definition of homography . . . . , .55 101. Three pairs of corresponding points (or rays) are sufiicient to determine two homographic systems . . . .56 102. 103, Point on either axis which corresponds to the point at infinity on the other . . . . . . . ih. 104. If the points at infinity on the two axes correspond, the systems will be similar . . . . . . . ih. COAXAL (OE CONCENTEIC) HOMOaEAPHIC SYSTEMS. 105, 106. Double points (or lines). — In every system of two homographic coaxal rows (or concentric pencils) there are two points (or rays) P and Q, called double points (or lines) which are their own correspondents . . . , , . . 57 107. Any pair of correspondents divides PQ into segments whose an- harmonic ratio is constant . . . . .58 108, If one of the double points be at infinity, the systems will be similar ih. CONTENTS. iX ON INVOLUTIOlSr. ARTS. PAGE 109. Definition of involution — conjugate points (or lines) of an involution 59 110. Two pairs of conjugates are sufficient to determine a system in involution . . . . . . . ih. 111. Relation between three pairs of conjugates in an involution of points or lines . . . . . . . ih. 112. Any pair of conjugates are harmonic conjugates with respect to the double points of the involution . . . . . ih. 113. Centre of the involution. Geometrical construction . . 60 114 — 118. Definitions of involution . . . . . ih. 119. In a pencil of lines in involution there exist always two conjugate rays which are at right angles to each other . . .62 120. If more than one pair of conjugates of a pencil in involution be rectangular, every pair will be so , . . . . ih. 121. 122. Equations representing two systems of lines in involution . ih. CHAPTER VI. THE GENEEAL EQUATION OE THE SECOND DEGEEE. 123. The general equation of the second degree 124. To shew that it always represents a conic section 125. Conditions for an Ellipse, Parabola, or Hyperbola . 126. Equation of the chord . . . . . 127. 128, 129. Homogeneous equation of the tangent at any point 130. Symmetrical equations of the tangent 131. Condition that the straight line {I, m, n) should touch the conic 132. Symmetrical equations of the normal at any point 133. Homogeneous equation of the normal 134. 135. Equation of the polar of a given point 136. To find the co-ordinates of the pole of a given straight line with respect to the conic . . , . 137, 138. To find the co-ordinates of the centre 139. The centre is the pole of the straight line at infinity. The centre of a parabola lies on the straight line at infinity 140, 141. Equation of the pairs of tangents which may be drawn through a given point ...... 142, 143. Equation of the director of the conic, or locus of intersection of tangents which cut each other at right angles 144, 145. Equation of the asymptotes .... 64 ib. ih. 66 ih. 69 ih. 70 ih. 71 72 ih. 74 ih. 76 77 CONTENTS. ARTS. 146, 147. Condition that the equation of the second degree may repre sent a pair of right lines ..... 148. Condition for a parabola ..... 149. Conditions for a circle ..... 150. Condition for a rectangular hyperbola 151. Conditions that two conies whose equations are given should be similar and similarly placed .... 152. To find the direction of the principal diameter, when the locus i a parabola ...... 153. Equation of a circle whose radius and centre are given 154. Expression for the radius of a circle 155. Condition for a rectangular hyperbola 156. Expression for the area of a circle 157. Conjugate diameters ..... 158. Condition that two lines whose direction -cosines are given may be parallel to conjugate diameters of the conic 159. Condition that (^i, m\, n\), {l^, m^, n^ may be parallel to conjugate diameters ...... 160,161. Equation of the principal axes of the conic 162. Lengths of the semi-axes ..... 163. Expression for the area of the conic 78 80 ih. 81 82 ib. 83 ih. 84 ib. ih. 86 ih. 87 89 91 CHAPTER VII. INTEEPBETATION OF PAETICULAR FOEMS OF THE EQITATIOK OF THE SECOND DE&EEE. 164, 165. The equation ^^ - -^.S'^ = 166. Nature of the locus . . . 167, 168. Chords of intersection 169. The equation 8^ — ktu = 170, 171. The equations S^ - ku^ = 0, and S-^ - ku = 172. The equation S^^ - k^ ^ 173. . . vw - ktu = 174. . . vtv - ku^ = 175. . . vw - ku = 176, 177. . vw - k^ = 178. The equations S-^ - kafi = 0, S^ - kar ^0, S^ - ka = 179— 185. Equation of a conic described about the triangle of reference viz. i/37 + Mya + Na^ = . 186. The equation Uo? + lf-/8- + N'^y"- ±'iMN'fiy ± 2NLya± 2LMa^ 92 93 ih. 94 ih. 96 ih. 97 ih. 98 99 ih. 102 CONTENTS. XI ARTS, PAGE 187 — 193. Equation of a conic inscribed in the triangle of reference, viz. iV + j)[/2^2 + _z\r2^2 _ 2MN^y - 2.NLya - 2LMa& = .103 194 — 200. Equation of a conic with respect to which the triangle of reference is self-conjugate ..... 106 201, 202. The equation fiy - ka^ = , , , ^, . 109 203, 204. . , , fiy-ha =0 , . , . . ib. 205. . . . jSy - /c2 z= . , , , .110 206. . . . i62 - A-a = . . , . .111 CHAPTER VIIL EiQUATIONS OF THE SECOND OEDEK €ONTINUED. 207, 208. Principal forms of equations. 209. Equation of the chord 210. Equation of the tangent 211. Equation of the polar 212. Condition of tangency 213. Co-ordinates of the centre , 214. Condition for a parabola 216. Conditions for a circle EQUATION OE A CONIC EEFEEKED TO TWO TANGENTS AND THEIE CHOED OF CONTACT. 217— 220. The conic whose equation is of the form iil[f= ^2 ^ ^ I33 221, To find the equation of the chord when the equation is given in this form . . . . . . . . 134 222, 223. The equation of the tangent . . . . .135 224. The equation of the polar . . . . , . ib. 225. If from a given point Q two conies be drawn cutting a conic in the points Pj, and Q^, P^ and Q^, respectively; the two pairs of chords which join, directly and transversely, these four points, will intersect on the polar of O , . . . . 136 226. 227. Harmonic properties ...... 137 228. Every chord of a conic is harmonically divided by any point on it and the polar of that point . . , . . ib. 229. The poles, with reference to a given conic, of the straight lines which pass through a fixed point, lie on a fixed right line . 138 230. Geometrical theorems involved in the forms of some of the equa- tions discussed ....... 139 Their loci •. . , 112 . . . 113 . . . 118 ■• •• • . 121 . . , 123 . • . 128 . - . 130 •. . . 131 XU CONTENTS. CHAPTEK IX. THE CIECLE. ARTS. PAGE 231. To find the equation of the circle circumscribing tlie triangle ABC 140 232 — 234. To find the equations of the inscribed and escribed circles . 141 235. Notation explained ....... 142 236. The circumscribed circle ...... 143 237. Equations of chord and tangent. Condition of tangency . . 144 238. Co-ordinates of the centre . . . . . ' . ih. 239. The inscribed circle ....... 145 240. Equation of the tangent. Condition of tangency . . . ib. 241. Co-ordinates of the centre . . . . . . 146 242. Cenive^ oi the escribed circles . . . . . ih. 243. The circle with regard to tohich the triangle of reference is self- conjugate. Equation of tangent. Condition of tangency . ih. 244. Co-ordinates of the centre . . . . . . ih. THE NINE-POINT CIECLE. 245. M.Terquem's Theorems ...... 147 246. To shew that the equation of a circle may always be written in the form a^y + 57a + ca^ = {aa + J/8 + 07) (?a + mB + ny) .■ . ih. 247. All circles pass through the same two imaginary points at infinity . 148 248. The radical axis of a circle whose equation is given in the above form and the circumscribed circle ..... 149 249. 250. To find the radical axis of two circles whose equations are given . . . . . . . . ih. 251. Concentric circles touch each other in two imaginary points at infinity ........ 150 252. Equation of the inscribed circle . . . . . ih. 253. Equation of the circle through the middle points of the sides . 151 254. Equation of the circle through the feet of perpe7idiculars from the vertices . . . . . . . .152 255. 256. Proof of M. Terquem's Theorem . . . .153 257. Properties of these circles ...... 154 258. Co-ordinates of the centre of the nine-poiiit circle . . . ib. 259. Equation of the circle through the centres of the escribed circles . ib. 260. Equation of the circle through the centre of the inscribed and those of two escribed circles ...... 157 CONTENTS. Xlll ARTS. PAGE 261. Equations of the foregoing circles . . . . .158 262. Equations of their radical axes . . . . . ih. 263. Co-ordinates of the principal points referred to in Art. 235 . 160 CHAPTER X. G-ENEEAL THEOREMS AND PROBLEMS. 264. A conic is completely determined when five points upon it are given 161 265. A conic is completely determined when five tangents to it are given ih. 266. If from a point two chords OR^R^, OS ^8^ be drawn in given directions, to a curve of the second degree, the ratio of the rectangles under the segments of the chords is the same for every position of O . . . . . . ih. 267. 268. The anharmonic ratio of the pencil formed by joining four points on a conic to any fifth point is constant . . . 162 269. Four fixed tangents are cut by any variable tangent in points whose anharmonic ratios are constant .... 166 270. Pascal's Theorem. — The three pairs of opposite sides of a hexagon inscribed in a conic intersect in points which lie in the same straight line ....... 168 271- Brianclion' s Theorem. — The diagonals which connect the three ■ pairs of opposite angles of a hexagon described about a conic meet in a point ....... 169 272. To find the locus of the centre of a conic wliicli touches three given straight lines and passes through a given point . . . 171 273. To find the equation of the polar reciprocal of a given conic . 173 Examples ........ 174 PKELIMIMRY CHAPTER. A. To shew that if - =: - = -^ ; then X y z X y % Ix -\- my -\- n% Let each of the given fractions = X ; we shall have y — \%', and, multiplying these equations by I, m, n, respectively, and adding, we get la + WjS -\- ny = Xlx -\- Xmy -\- \nz = \{lx + my + n%) ; , a iS 7 . la-^ m^-\- ny whence, - = -=-=:A=r: '-. X y % Ix -{- my -\- nz Examples of the theorem here proved occur in Arts. 20, 23, 25, 31, 34, 54, etc. B. To shew that if — = - z= — • fjigji X y z a /3 y fifty -\- my a + TlaftYs X y z \lyz -\- mzx + nxyj XVI PRELIMINARY CHAPTER. Multiplying equations (i) two and two together, we get ya = \hx, (ii) a/3 := X^,ry; and multiplying these by I, m, 7i, respectively, and adding, we have = ^^{ly% + mzx + nxy). a_^_7 _ (l^y + mya + na^\k Therefore X y % \lyz + mzx ■\- nxyj The reader will find instances of the application of this theorem in Arts. 40, 45, etc. 0. To shew that if - = ^^z=z 1 - then X y z a ^ ^ _ y _ ^ ^/a^ + m0' + 7Zy-^i X y z " \lx^ + my^ -\- n^ , Squaring equations (i), we have a^ = X2^^ /3"' = X'/, (iii) / = XV; and multiplying these respectively by /, m, n, and adding, we get Ja^ 4- m^'' + ny'' = \^Jx" + \-my^ + X2^^s2 PHELIMINARY CHAPTER. XVll Hence, a X y \ = _ /la^ -\- m0^ 4- ny^\^ This theorem is employed in Art. 45. D. The discriminant of the quadric [denoted hereafter by ^(a, /3, y),] or the condition that it should be resolvable into linear factors, is obtained (Art. 147) by the elimination of a, /3, y between the equations fj = «' d^ 0, ^^- dy 0: i.e. : between Aa-{-F^-\-i:y= 0, Fa->rB^ + Dy= 0, Fa + J)^+ Cy= 0. It is, therefore, A, F,F F,B,JD F, B, C = 0. (iv) The determinant which forms the left-hand member of equation (iv) occurs frequently in the course of the present work. It is convenient, therefore^ to have some abbreviation for it. We shall denote it by A, and wherever this symbol is employed (see Arts. 138, 144, 146, 162, etc.) the above meaning is to be attached to it. The abbreviation {A^B, C) in which only the diagonal consti- XVIU PRELIMINARY CHAPTER. tuents are given is sometimes (Art. 167) used to represent tlie same determinant. The^rs^ minors of this determinant [viz. the determinants, with proper signs, which we obtain by cancelling successively the column and row to which the several constituents A, B, C, D, E, F are common], are also of frequent occurrence : these, therefore, it will be convenient to denote by A',£', C, D', JE\ F', respectively ; so that B. D D. C A, F ^, C A, E F, B = BC-I)^ = A\ - = CA-F^ = B', = AB-F^= C\ - ^, E F, B F. B E, B F. B F, C = EF-AB = B', = FB-BE = E\ = BE- CF = F'. The value of A may be expressed in terms of these minors as follows : — A = AA' + FF' + EE\ A=FF' -{- BB' + BB, A = EE' + BB' + CC. An instance of the use of this notation will be seen in Art. 188. If a determinant be of the form «i + k, h + 'tn^, ^1 + ^1 ^3 + k, h + ^^h, Ci + n^ or [a^ -f- \^ ^2 + ?^^2j ^3 + %)j ^ach constituent being the sum of two others, the determinant is equal to the sum of all the deter- minants which can be formed by taking for each column one of the partial columns of the corresponding column of the original determinant ; that is to say, PRELIMINARY CHAPTER. xix (, m ^, -Z>, c, n I, m. n. l\^n ^(?, m, n)'. r. If (^(a, /3, y) represent the general quadric, as in section [D) of this chapter ; then 0-.+ (!>.+(?)" For, da «i gi-+(S)"{|- = {Aa,^F^,-\-:Ey,)a ^{Fa,^B^,^Dy,)^ J^[i:a^D^,-\-Cy{)y. TRILIMAR CO-ORMMTES. CHAPTER I. EXPLANATION OF THE METHOD. THE STRAIGHT LINE. 1. The position of a point in Trilinear Co-ordinates is given by means of its perpendicular distances from three fixed straight lines which do not meet in a point. In accordance with the usual Tri- gonometrical notation, A, B, C are employed to denote the angles of this triangle of reference, and a, h, c the sides which respectively subtend them ; the area of the triangle being represented by S. The perpendicular distances of any point from the three sides are called the co-ordinates of the point, and are denoted by the Gieek letters a, iS, 7 ; and the point itself, for the sake of brevity^ by (a, ^, y). Thus (^, 0, OJ, [Q, ^, 0^, [o, 0, ^j, and U, ^sin C^ %in^J, (-sin C, 0, -sin^J, 1 -sin^, -sin^, O), repre- sent respectively the three vertices of the triangle of reference, and the middle points of the opposite sides. 2. It is necessary, as in the Cartesian system, to lay down some rule for the interpretation of signs ; the position of a point depend- ing upon the sign, no less than upon the numerical value of its co-ordinates. The direction of a perpendicular is indicated by the one, as its absolute length is by the other. Thus, a and —a refer to points v/hich are equidistant from the side BC, but which lie on opposite sides of that line. It is customary to regard the a- co-ordinate of any point P as positive or negative, according as the vertex A and the point P lie B 2 TRILINEAR CO-ORDINATES. on the same or on opposite sides of £C; the sign of the /3- and y- co-ordinates being determined in a similar manner. If the Tig. 1. P^ (^-^,A>Yz> -"^3 (^J.y,)' point lie on the side £C, it is obvious that its a- co-ordinate will be = 0. And similarly for points which lie on either of the other two sides. Thus then the co-ordinates (oi, /3i, yi) of any point Pi, within the triangle of reference, are all positive ; whilst in the figure Pj has ^ne (viz. the /3- co-ordinate), and P3 two (viz. the ^3- and y- co-ordinates) negative. 3. Bearing in mind this convention with regard to signs, the student will have no difficulty in proving for himself that the co- ordinates of any point (a, /3, y) satisfy the relation «a -f JiS -f cy = 28. (1) For if the point (a, j8, y), whatever its position, be joined to the vertices of the triangle of reference, it will be seen at once that the algebraical sum of the triangular areas represented by 1 1-1 ~aa, -0/3, -cy, IS always equal to the area of the triangle ABC. 4. By means of the relation (1) any trilinear expression may be THE STRAIGHT LINE. O rendered homogeneous, or raised to any required order. For, since , . aa -\- h^ -\- cy . . . . , ,^ the quantity —z is equal to unity, we can, without alter- ing the value of the expression, multiply any term by such a power of it as may be necessary to raise it to the given dimensions. Thus the equation may be written in the homogeneous form 48^ a"" + 2Sh{aa + 5/3 + Cy){^ + r) 4- P {aa + 5^ + CyY = 0, and the linear expression la -\- m^ -\- fly replaced by the equivalent quadric ^ {aa + 5/3 4- cy){la + mjB -\- ny\ or ^ j aW-\-lm^^ -f my^-\- {hn-\-cm)^y-\- {cI-\-an)ya-\- {am-\-U)a^ 5. The relation of Art. 3 is also useful if it is required to deter- mine the co-ordinates of a point, when their ratios only are given. Instances of its being so employed will be met with in the present chapter (Arts. 20, 23, 25). 6. The foregoing account of the method will suffice for the in- terpretation of the following simple forms of trilinear equations. (i.) a = represents the locus of points the perpendicular distances of which from BC are = 0. This being true only of points which lie on that line of reference, a = must be the equation of BC. Similarly j3 = 0, y — 0, are the equations of the sides CA and AB respectively. Hence the three lines of reference are sometimes conveniently denoted by the letters a, jS, y ; and their points of intersection by (/3y), (ya), (a/3). (ii.) Again a=zjc (a constant) is the equation of a straight line parallel to B (7, at a distance Jc from it ; for it is true only 4 TRILTNEAR CO-ORDINATES. of points which lie on such a line. In the same way ^= a con- stant, 7= a constant, represent straight lines parallel to CA,AB respectively. (iii.) The equation j3— y = 0, or ^ = -y, is true only for points which are equidistant from CA, AB, the perpendiculars upon those sides being either both positive or both negative ; whereas iS + y = is the equation of the locus of points whose distances from the same two sides are equal but of opposite sign. Hence iS— y=:0, /3 + y = are respectively the equations of the in- ternal and external bisectors of the angle at A ; and the equa- tion /3^— y^ = represents this pair of bisectors. Similarly y^ — a^ = 0, c? — /3^ = represent the internal and ex- ternal bisectors of the angles at JB and C. (iv.) I3y—a^=. is the equation of the locus of points such that the squares of their perpendicular distances from one of the sides of the triangle of reference are equal to the pro- ducts of their distances from the other two sides. The nature of this locus will be determined hereafter (Art. 216. (^).). 7. On the general equation of a straight line. Let PQR be any straight line, passing through the fixed point Fig. 2. -'^(aoj /3oj yo)> and making with the sides of the triangle of reference the acute angles 6, ^, ■^. THE STRAIGHT LINE. Take F{a, ^, y) any other point on the line and let the perpen- diculars a, uq be drawn. Then, if r be the distance between the two points, we shall have and, therefore, In the same way, we get Hence, Oq — a =: JEG = r sin ^, a — oq =: — r sin 6. ^ — ^0 == ^ sin (^, y — yo = r sin ^. « — «o _ ^ — ^0 _ 7 — To — sin 6 sin sin -^ + ^ (2) is the equation of the straight line in terms of the co-ordinates of a fixed point on it, and the acute angles which it maJces with the sides of the triangle of reference. The lower sign must be taken with r if (a, /3, y) be on the other side of E, as at F'. 8. It will be observed that the denominators of (2) have not all the same sign : the reader will do well to remember the following rule with respect to them. One of the segments QR, RP, FQ of the line always subtends (as does QR in the above figure) the supplement of the correspond- ing vertical angle, instead of the angle itself; and it will be found that, in all cases those members of the equation have their deno- minators of the same sign which correspond to segments that are in this respect similar. Thus the lines marked (1), (2) in the annexed figure have their equations respectively of the forms oo _ ^ — /3o _ 7 — 7o sin 6 — sin sin -v//- and "~ "0 — ^~^o _ 7 — 70 ^ sin^ sin (^ — siux//" 6 TRILINEAR CO-ORDINATES. Fig. 3. We have given this form of the equation of the right line, be- cause, owing to the simplicity of the angles involved, it is more easily written down in particular cases than the form which we now proceed to give, and which, on account of its greater sym- metry, we shall hereafter employ. 9. Let JS, F, as before, be the points (oq, /3o, yo), (a, jS, y) respec- tively, and UF=: r. Kg. 4. C Let <^o, 5o> ^0 be the angles which the line FF makes with the perpendiculars from any internal point, as (oq, /Sq, yo) ; the angles THE STRAIGHT LINE. 7 being measured in one and the same direction (namely in tliat in which we pass from oq to /3o, from /Sq to y^, and from y^ to oq), as indicated by the dotted lines in the figure. IVe shall then have sin ^ = cos % sin<^ = cos(3i— tt) = — cos^i, sin\/r = — cos^i; ' and the equation of the last Article becomes, on changing the sign of the whole g — gp /3 — ^0 y — yo — J— — = + r, • (3 ! cos aQ cos Oq cos Cq — ■ \ ^ which is therefore tJie equation of the straight line in terms of the oo-ordinates of a fixed point on it and what we shall in future call the direction-cosines of the line. These direction-cosines we shall occasionally denote bv X, u, v, and the straight line itself by (X, /x, v), or by (cos a^, cos Iq, cos ^o)- 10. The student would do well, by employing a variety of figures, to satisfy himself that the forms of the equation given in the preceding Articles hold in all cases, whatever be the position chosen for the line itself, or for the points (go, /3o, yo), (g, iS, y). 11. To shew that the equation of the right line may he written in the form la + M/3 -^ ny=zO. Let the direction-cosines of the right line be X, ft, i/; and its equations « — Qq _ /3 — ^0 _ y — yo X /Lt l* From these we get fxy — vfi = yofi — ^o'', va — Xy =r oqV — yoX, which give (yoX — aov)(fxy — v^) = (yofi — /3o'')(^y — va) ; whence, multiplying out, dividing by u, and arranging the terms, we get {^qV — yo/^) a + (yoX — a^u) ^ -\- {aofx — ,3o\) y = 0, 8 TRILINEAK CO-ORDINATES. or /3o» 7o [X, V a + 7o' oo 3 + X, /x 0, (4) a? /3, r ao5 /^o, 7o >^, A*' r/ 0. (5) an equation of the required form. It may be written in tlie form of a determinant, thus : 12. To shew conversely that every equation of the form la + m^ -\- ny =■ represents a straight line. Take any point (oo, /3o, yo) ^^ po^e, and let the equations of the radius vector from it to any point (a, ^, y) on the locus of the equation la-\- m^ -^ ny = a — Op _ /3 — /3o _ 7 — yo _ be r. (6) (7) We have from (7) a = aQ-]-Xr, ^ = ^0 + f^r, y = 70 + vr; and, substituting these values of a, jS, y in (6), we get {l\ 4- mfi. + ?^I')r -j- {la^ + m/3o + ^7o) = 0, which can give only one value of r for each separate value of (oo, ^05 7o): 01' of (^5 ^? O- Hence, no straight line can meet the locus in more than one point; the locus of (6) is therefore a straight line^. 13. Polar form of the general equation of the second degree. The method of the preceding Article may be applied to the general equation of the second degree, (/)(a, /3, y) = Aa^ + £^^ + Cy^ + 2Dj3y + 2^ya + 2Fa^ = 0. * It will be observe^, moreover, that if (cq, /3q, Jq) be taken on the locus, since in that case laQ + 'm^Q + nyQ=0, r will nlways = 0, except when ZA + m^ + nv = also, (which, as will shortly be seen, is the condition that the radius vector should coincide in direction with the locus), and then it is indeter- minate. THE STRAIGHT LINE. For putting, as before, a ■= ao -\- Xr, 7 = To + »'^> we get, by Taylor's Theorem, ♦(•••I'- »•)+!(*) '+{«;) '+(*)• or, since VUq ^d^QdyJ KdyQdaQ/ \dyQ aanJ (8) which is the polar form of the general equation of the second degree ; (oq, /Soj To) being the pole, and X, [i, v, the direction-cosines of the radius vector. 14. As it will be necessary sometimes to distinguish between the two forms of the equation of the straight line which have been investigated in Arts. 8 — 11, we propose to call the former the symmetrical and the latter the homogeneous form. The latter form will be denoted by (/, m, n) = 0, or simply by {I,m,n) ; the former, as has been already stated, by (X, p., v) or (cos ^, cos 5, cose). Examples will be found at the end of the work which will fami- liarize the reader with the method of forming the symmetrical equations of a right line in particular cases. 15. To construct geometrically the straight line whose eqiiation is given in the homogeneous form. Suppose the given equation to be la 4" ?w/3 "i" wy = 0, 10 TRILINEAR CO-ORDINATES. and, in order to find the co-ordinates of the points P, Q, J2, (see fig. Art. 7) in which the locus intersects the sides of the triangle of reference, make a r= 0, ^ == 0, y — 0, successively. "We get (Prelim, chap. {A).), for P, Q and E respectively n a ^ —n a ^ m -I = r —m = 2S hn—cm = 7 I = 2S cl—an y 28 am — 1)1 Hence these points may be obtained geometrically by drawing parallels to the sides of the triangle of reference at distances determined by the preceding equations. And, when any two of the three points P, Q, R, are found, the position of the straight line is known. 16. If a straight line pass through the intersection of two other straight lines whose equations are \a -f »ii/3 -f- ^ly = 0, l^a -\- M2/3 + ^27 = 0» lis equation ivill he of the form {l^a 4- mi/3 + n^y) — kij^a + m^^ 4- n^y) = 0, Jc leing an arbitrary constant. For the equation (Z^, m^, n^ — h'f^^, m.2, %) = (Art. 12) must re- present so7ne straight line ; also, it is satisfied by those values of a, /3, y, which satisfy the equations (/i, ?%, n-^ = 0, (4, 'i^^^ n^ = 0, simultaneously. It therefore represents a straight line which passes through the point of intersection of the lines {l^,mi,ni), {U.nh.n^). 17. If the equations of three straight lines {l^, ^i, n^), (4, 'W?2. ^2), (4, ^^3, %) he such that the sum of the left-hand memhers, tvhen multi- plied each ly some constant, vanishes identically, these three straight lines meet in a point. For suppose X^, I.2, L^ to be certain multipliers, such that Xi(?ia+mi/3+%y)-f Z2(4a+W2/3+M27)+^3(4a+^%/3+%y) = 0. Then it is manifest that the co-ordinates of the intersection of any EXAMPLES. 11 two of the three vstraight lines (since they satisfy simultaneonsly the equations of that pah- of straight lines,) will, by virtue of this relation, also satisfy the equation of the third. Hence any one of the three given straight lines passes through the intersection of the other two. 18. In the following examples the given triangle is taken as the triangle of reference, so that the equations of its sides are a = 0, 19. To find the equations of the bisectors of the angles of a triangle j and to shew that they meet in a point. The straight line Avhich bisects the /_A of the triangle ABC may be regarded as the locus of points which are equidistant from the two sides CA (/3 = 0), AB{y = 0) ; its equation, therefore, will be /3 m y, or ^ - 7 = 0. Similarly, y — a = 0, (9) a — jS = 0, are the equations of the bisectors of the angles B and C respect- ively. Also, since the left-hand members of (9), when added together, are identically equal to zero, the condition of Art. 17 is satisfied, and the three bisectors meet in a point. 20. If (co, /3o, yo) ^^ t^® co-ordinates of this point, we shall have, by equations (9) 2>S' ao = /3o = yo = ^_^^ , ^ . (Prelim, chap. {A).), S — f s (10) 2 (22) sin (^i — Jj ) = sin (7 j cos (% — 5i) = — cos C ) between the direction-angles of any right line and the angles of the triangle of refer ence^ follow at once from the equations FUNDAMENTAL RELATIONS. 17 180°-^= ^1-^1, 180° -5= Cy-a„ (23) 180«+ C ——[a^-h), the truth of which appears from an inspection of the figure given in Art. 9. 31. To prove the relations « COS «i + J COS ^1 + -\- Jj, /3--iSi V — n (28) cos{co-{-ai) cos{ i^ij Ti); their equations are of the form = j- = = r, (30) COS^Tj COSC'i cos^i ^ ^ ^-^^^i^z;^^^,. (31) sm^i sm^i smCi ^ ^ In other words, the direction-cosines of two perpendicular right lines are of the form (cos a^ cos $i, cos i-\- c cos c^ =. 0. The relation a sin ai-\- h sin hi + c sin S^= JVsin^^=5Vcos^5i+5Vcos^Ci+25Vcos3iCos^iCos^; FUNDAMENTAL RELATIONS. 21 or, since l cos ^i = —{c cos c^-\-a cos «i), c cos S^ a cos A\'^-\-b cos B n"^ -\- c cos Cv ^2 — ^2 — ^2 —^2 ^2 ^2— ^ ^^p also, = abc 38. By adding the two expressions involved in abc = a cos A)^ + b cos ^/4^ + c cos Ci/^, since, sin V + cos V = 1? etc. = etc.; or those involved in since (Art. 30. (22).), cos {b^—c-^ = —cos A, etc. = etc. ; we get in each case the expression 8 S^ -^ =: a cos A -\- b cos B -^ c cos C, (37) which is sometimes useful, and may assist the student in remem- bering the more important formulae from which it is derived^. 39. To write down the homogeneous equation of a straight line whose symmetrical equations are given. Let the given equations be — -— = = , (6i!>) A fl fl and suppose la + m/3 + ?^y = (39) to be the homogeneous form of the equation of the same straight line. Since (oq, 3oj To) is ^ point on the line, we shall have lao + m/So -\-nyo = 0; ^ This formula is easily proved independently; for tlie right-hand member 2J2c2 + 2c2a2 + 2aH^ —a^ — h^ —c^ 8s(s — a)(s—b){s—c) _ SS^ ~~ 2abc ~ abc abc VALUES OF THE DIRECTION-COSINES. also, (Art. 31. (26).) l\ •\- m^x -\- nv =0; whence, I m n /3o, 7o To, «o oo, /3o /X, V v,X X, /x and (39) becomes /3o» To /X, J/ a + 7oi Qo /3 + X, /x = 0. 23 (40) (41) 40. To find the direction-cosines of a straight line whose equation is given in the homogeneous form. Let 4" + ^i/3 + n^y = be the given equation: then, since (Art. 31. (24), (26).) a cos a^-\-h cos hi-\- c cos Cj = and Ix cos a^ -\- m^ cos h^ -\- n^ cos Ci = 0, we have, by cross-multiplication, cos«^ cos^i cose b, c ca cos 5 — 2a5 cos C). will be represented by ^a, h, c?. j- 24 TRILINEAU CO-ORDINATES. 41. The direction-cosines of {li, m^, n^) may obviously be written in the somewhat more convenient form cos a. sin B, sin C Vh cos hi j sin C, sin A I COSCi sin^, sin 5 {^1,^1,%} (43) the same abbreviation being made. 42. To find the sines of the direction-angles of a straight line, vjhen its equation is given in the homogeneous form. By equation (43) of the last Article, we have li,mi,ni ^— {n-y sin B—m^ sin C)^ {/i, m^, n^Y 1 Ul + miCOS^C + n\ cos^5— 2»^lWl(cos^ — sin ^ sin Cj — 2%?i cos jB—2lxmi cos C] sin% = {/l,»^l,7^l}■ {l^—mi cos C— % cos ^)^ . {k,mi,niY Therefore sin«, ?i — m^ cos C— ^1 cos B the values of sin bi, sin c^ following by symmetry. Hence sin a^ sin h^ sin c^ /j — miCosC — ^iCos^ ^1— ?^lCos^ — ^iCOsC % — /iCos5— micos^ = L__. (44) 43. To shew that li sin ^1 + Ml sin J^ -f ?^l sin Cj = {li, m^^ n^]. Since Zi cos a^ + ^i cos&i + n^ coscj = (Art. 31. (24).), we have (J I sin a^ -\- mi sin hi -t- w^ sin Ci)^ = (/^ sin with it, its Kg. 11. (s^.^;r) (^C^o/c) {li,mijni;) length p is known: since, from the geometry of the figure, P = smco Thus ~ {^1,^1, %} sin ca (52) 49. But this result may be arrived at independently; for the equation of the line which meets (Ji, nii, n^) at the given angle will be (Art. 32) a — Oo y— yo cos (co + tti) cos (co + Ji) cos (o) + ^i) and, proceeding as in Art. 46, we shall get finally I^ao + mi/3o + ^lyo IP, z=z + P - li COS (a>+ : ^2 cos <^i + w?.2 cos 5i + ^2 cos , respectively, ^2, % sin^+ ^1, h 11=} , ^9 sin^+ sin (7 1^2 + w^iWi2+ ^1^2 ~~ 0^1% + m2ni)cosA — (wi4+ %^i) cos£— (^l»^2+ ?2^Wi)cos C (56) ^2, % sin^-f ^1, /i ^2» '2 sin54~ ^^'^^ I sin (7 - (57) {Zi, mi,Wi}. {4, ^2, %} Zl4+^^lW^2 + %^2— (^1^2+^2^l)C0S^— (^^l?2+%^l)C0S^— (^1^2+ /2^Zi)cos C' {/i, mi, n^}. {4, ^2, %} (58) 52. jTo investigate the meaning of the equation aa -\- h^ + cy = 0. The given equation, being linear, must (Art. 12) represent some straight line. Also the equation («x 4" V + ^^y + (^«o + i^o + ^70) = . - gives, as in Art. 12, the distance of the required locus from a point (ao, /3o, yo) ^^^ ^^ the locus, in any direction (X, ji, v). Therefore, since (Art. 31. (24).) a\ -{• hn -{■ cv=^Q always, the radius vector is in every direction infinite. Hence the equation aa •]- b^ -^ cy = 0, or sin Aa -}- sin B^ + sin Cy = 0, or, (Art. 3. (1).), a constant = 0, must he interpreted as representing a straight line which lies altogether at an infinite distance. 53. It will be observed that, as {a, 6, c} = identically, the direction-cosines of this line are (Art. 41) indeterminate. 31 CHAPTER IV. PROBLEMS ON THE STRAIGHT LINE. 54. To find the co-ordinates of the ^oint of intersection of the straight lines (/i, mi, n^, (/g, Wg, %). Let (ao, i3o5 7o) be the co-ordinates of the point of intersection of the given straight lines, then we shall have ^ Zioo + ml^o + ^lyo = 0, and ^oo + wiSq + ^270 = ; whence, by cross-multiplication. a© _ /3o _ To _ 2^ mi, nx m2,n2 %, ^1 ^2, 4 Zi, mi 4' ^^2 a mi, % + ^ ^2, 4 4, mi kiin2 (59) 55. To find the condition that the three straight lines (l^, m^^ n^ (4, ^2» ^2)? (4> ^3j %) ^^y ^ ^3 0. (60) 32 TRILINEAR CO-OEDINATES. 56. To find the condition that the three points {a^, ^i, yi), (ag, /So. ^2)? (03, jSg, 73) may lie in the same straight line. Suppose the three given points to lie on the right line whose equation is la •{- m^ -\- ny =■ : we shall then have hi + mjSi + nji = 0, la^ + mjSa + Wy2 = 0, and Zas + m/33 + ^73 = : and these equations cannot exist together, unless = 0, (61) which is therefore the required condition. ^3, ^3» 73 57. To find the symmetrical equations of the straight line which passes through the two given points (oi, /3i, yi), (02, 32, 72). The equations of a straight line through (02, /S,, 72) must be of the form a — (h _ ^ — ^2 _ y — yi \ jJ, V Also since (ai, jSj, 71) is a point upon the line, we have «! — g 2 _ ^1 — 1^2 _ yi ~ 72 Hence, eliminating \, fi,v between (62) and (63) we get a — a2 _ ^ — ^2 _ 7 — 72 «i — 02 ^1 — /32 71 — 72 which are the equations required. (62) (63) (64) PROBLEMS ON THE STRAIGHT LINE. 33 58. To find the homogeneous equation of the straight line which passes through the two given points (a^, ^x, yj), (02, /32, 72)- Suppose the equation of the joining line to be Za + m/S -1- ?^y = : (65). then, since the given points lie on the line, we shall have, to determine I, m, n, lax + vn^i + nyx = 0, (66) and la^ + m^2 + ^Yn = 0. (67) From these, by cross-multiplication, we get I m n /3i, 71 7i' «i ai, /3i ^2, 72 72' «2 ^2, ^2 whence, substituting in (65), we have for the required equation ^]. 7i i32, 72 a + or, more briefly. 7i> '^i I «. I I "i' ^1 72, 02 1 I 02, /^a = 0. = a? /3, 7 Ol' /^i, 71 Og, /:^r 72 (68) (69) In the latter form the equation might have been written down at once as the eliminant of the three equations (65), (66) and (67). 59. To find the homogeneous equation of the straight line which passes through the point {oq, ^q, 7^), and whose direction-cosines are Let the equation be la -f- w^^ -j- ^7 = ; then, since (oq, ^q, 7^) is a point on the line, lao + m^Q 4- ?^7o = : 34 TRILINEAR CO-ORDINATES. also, (Art. 31. (26).), IX ■\- mix •\- nv — 0. And, eliminating /, m, n, between these tliree equations, we have for the required equation This equation may also be written thus : ^o» 7o II, V a + 705 "0 V, X /3 + ■J X, /x = 0. (70) (71) 60. If the equations of the two preceding Articles be compared, it will be seen that the homogeneous equation of a right line may be formed with equal ease, whether the co-ordinates of two points on the line, or the co-ordinates of one such point and the direction- cosines of the line, be given. 61. To find tlie equation of the straight line which joins the point (co, iSo, yo) to the point of intersection of the straight lines (/i, m^t %), (4 ^2, n^. The required equation must (Art. 16) be of the form l^a + m^^ + n^y + lilc^a -f m^^ + W27) = 0, (72) because it passes through the intersection of the given straight lines. Also, since it passes through (ao, jSq? 70) j we have, to determine Tc, kaQ + ^1^0 + ^i7o + ^'(4ao + mSa + ^270) = J and (72) becomes, when this value is substituted for h, (4ao+m2/3o+%yo)(^ia + ^i/3+%7)=:(?iao+mii3o+%7o)(4a+*^^2^+%7)- (73) 62. To find the equation of the straight line which passes through the point of intersection of the straight lines (/i, ^i, %), (4? '^h'> %); and bisects the angle between them. PROBLEMS ON THE STRAIGHT LINE. 35 The bisector is the locus of points the perpendicular distances of Fig. 12. '£,m^,rt^) which from the two given straight lines are equal ; its equation therefore (Art. 47. (51).) will be Zja + Wild -f ^i7 /2a 4- 'in^S + n^y T {^1, ^h, ^h} {4. '^^2, %} (74) the upper sign belonging to the internal, and the lower to the external bisector. 63. If the direction-angles of the tw^o lines be given, and (oo, /3o, 7o) be their point of intersection, the bisectors will be re- spectively represented (Art. 33) by the symmetrical equations a — flo cos %-f ^2 cos 7—70 cos Ci-\-c^ (75) and ^-^0 7—70 sm a^ -\- «2 sm -!— ! — -^ sm Gi + ^2 (76) For the direction-angles of the internal bisector are the Arithmetic means between a^ and «2» \ and l^, c^ and c^, respectively ; and the external bisector cuts the former at right angles. 36 TRILINEAR CO-ORDINATES. 64. To find the condition that two given straight lines (J^, m^, n^), (Zg, m^. n^) should be parallel to each other. The two given lines must, if they are j^arallel, have the same direction-cosines; hence (Art. 31), we must have, not only a cos ai-\- h cos ii-{- c cos Ci = 0, and li cos % + mi cos hi + nj cos Ci = 0, but also 4 cos ai -\- m^ cos hi -\- % cos Ci = 0. whence, eliminating cos ai, cos hi, cos Cj, we get for the required condition a, h, c \ 0. (77) a, h, e /i. nil. ni \ 4? ^2, % 1 65. To find the equation of the straight line which passes through the point (qq, iSo' 7o). <^^^ is parallel to the straight line {Ji, nii, %). This straight line may be regarded as the locus of points the perpendicular distances of which from {li, »^l, w^) are equal to that of (oo, /3o' To) fi'om the same right line. This (Art. 47) Avill be expressed as follows : lia + mil3 -\- niy /i«o H- ^i/3o + ^i7o or {li, Ml, %} {li, nil, ni} lia -\- Mi^ + Uij = liao + W2i^o + ^l7o- (78) 66. Or we may proceed as follows : Since the straight line passes through the point (oq, /3o, 70) and is parallel to (li, nii, n-^, its direction-cosines (Art. 40) are respec- tively proportional to J, C c, a a, h mi, Ui ? Ui, li ' li, mi and its equation (Art. 59. (70).) is /3, 7 /3o» 70 hni — cmi, cli — a%, atni — hli = 0. (79) CONDITION OF PARALLELISM. 37 This equation is in the homogeneous form, and, being identical with («ao+^/3o+<^7o)(^ia+^i^+^i7) = (^iao+^i^o+^i7o)(«a+^i3+Cy),(£0) is equivalent to (78) of the last Article, since (laQ-{-h^Q-\-CyQ =. aa-\'h^-\-Cy, (Art. 3). 67. Prom (78) it appears that the general equation of a straight line parallel to (Z^, m^, n{) is l^a -\- mi/3 + n^y =z k (a constant), (81) and, conversely, two straight lines are parallel when their equations differ only hy a constant term. Equation (81) becomes (Art. 4), when rendered homogeneous, h ha + ^i^ '^^^'y~2s ^^" + ^^ + ^y)> and is therefore of the form la^ -f m/3i + ny^ = k' {aa + i^ + Cy), (82) the meaning of which (Arts. 16. 52) is that every straight line which is parallel to {li,mi,ni) passes through the intersection of{li,mi,ni) with the straight line at infinity. This follows also from the condition of parallelism (Art. 64. (77).), which expresses (Art. b^. (60).) that the given straight lines must intersect on the straight line at infinity. Indeed we might have assumed this property of parallel straight lines, and deduced from it the condition of parallelism by the method of Art. bb. For the convenience of the student this form of proof is given in the next Article. 68. To find the condition that the straight lines (?i, m^, Wi), (^2, »^25 ^2) ^<^y he parallel. If the given lines be parallel they will intersect on the right line at infinity. Let (ao, /3o> 70) be their point of intersection ; we shall have «ao 4- J/3o -j- cy^ = 0, ?i«o + »»i^o + ^170 = 0, 38 and TRILINEAR CO-ORDINATES. and, eliminating a^, ^o, yo from these equations, we get for the re- quired condition a, i, c ?1, Ml, % = 0. '2? ^2» ^2 69. It was shewn in Art. 33 that the straight lines (cosfl^i, cos ii, cos^i), (cos«2» cos ^2? C0SC2) «^*^^ intersect at right angles, if cos a^ = sin a^f COS h.j = sin 5i, cos C2 ^ sin ^1. Hence also, since (Art. 31) 4 cos a2 -\- m^ cos h.2 + ^2 cos C2 = 0, ^A^ condition that (4, ^^^21 ^' 2) should he perpendicular to (cos %)» when the direction-cosines satisfy the equation (X, /x, r) = 0, (86) or ?i4X^+Wim2/A^-|-WlW2''^+(^lW2+^2%)/iJ'4-(w/2 + ^2^1'^ -r (4«?2+^l^l)^/A = 0. But (Art. 33) the values X, fx, v, w^hich correspond to a pair of perpendicular lines, must be of the form (cos «i, cos li, cos c^ and (sin ai, sin bi, sin c{), respectively. 40 TRILINEAR CO-ORDINATES. We have therefore, by (86), both (cos %, cos ^1, cos Cj) = 0, and

with the straight line (Ji, Ml, Wi). Assuming 4a + WgS + ffzY — 0, (94) for the equation of this straight line, we have, since (ao, ^q, yo) lies upon it, 4ao -h W2/3o -f ^2yo = 0. (95) Also, since its direction-cosines (Art. 32) are cos (co -\- a^y cos (o) + 5i), cos (o> + ^i), the relation (26) of Art. 31 becomes 4 cos (co + ^0 + ^2 cos (o) 4- l^ + % cos ( ^1 sinw — ^1 sin (o— ^)— Wisin(co-|-^) (98) 43 CHAPTER V. ANHARMONIC RATIOS. HARMONIC POINTS AND PENCILS. HOMOGRAPHIC SYSTEMS. INVOLUTION. 78. In the present Chapter a short account will be given of the theory of Anharmonic and Harmonic section, so far as it relates to the properties of the point and line. For a more complete in- vestigation of the subject the reader is referred to geometrical treatises ^. 79. When a pencil of rays^ originating at a centre 0, is cut by a transversal in points P^,P^,P^, . . . ., P^P^,P^F^^F^P\ .... are Fig, 13. conveniently employed to represent not only the corresponding segments of the transversal, but likewise the angles which those segments subtend at the centre of the pencil. In the same way we shall occasionally use Pi, Pg? Pz^ .... to denote the rays OPi, OP^, OP...... With reference to the range of points Pi, Pg, P3, .... the trans- versal is sometimes called the axis of the system. * I may mention Townsend's " Modern Geometry," Hodges, Smith, and Co., Dublin, 1863—5. 44 TRILINEAR CO-ORDI^^ATES. 80. Definition of Anharmonic ratio. Let the segment (or angle) P1P3 be divided, as in the figure of the last Article, bj the two P P P P f points (or lines) P^, P4, ; and let the two ratios -^-^> J j^ [ o^^ sin p. P ' sin P P / <^enoted by ps, Pi (or 0-2, 0-4), respectively. Then the ratio p2 * P4 (or 0-2 I o-J and its reciprocal p^ ', p^ (or 0-4 : 0-2) are the two anharmonic ratios which are due to the division of the segment (or angle) P1P3. Written at full length this pair of Anharmonic ratios will be ^ ^ : ^ ^ , and its reciprocal, P3 P2 P3 P4 f sin Pi P2 sin P^ P4 , ., . i^ or — - : —^, and its reciprocal . : V sinP3P2 sinP3P4' ^ J but the following abbreviations are often used ; viz. {P1P3, P2P4} and {P1P3, P4P2}, (99) to express the pair of ratios for the system of points, and { O.P1P3, P2P4} and { O.P1P3, P4P2} (100) to represent the corresponding ratios for the pencil. 81. But when four points on a common axis (or rays through a common vertex), P^, P2, P3, P4, are given, any one out of the six segments (or angles) P1P2, P1P3, P1P4, ^2-^3? ^2^4j AA^ i^ay be re- garded as the divided segment (or angle), and we shall have ac- cordingly twelve anharmonic ratios ; viz. the following six, {PlP2,P3P4}, {PlPs^P^P,}, {PlP,,P2P3h etc., and their six reciprocals {P,P„ P,Ps}, {P,Ps, P,P,}, {P,P„PP,}, etc. : (or the corresponding ratios of sines). Their number, however, is only half of that stated above ; for it will be found that the ratios of the segments (or sines of the segments) of P1P3, made by P2P4, ANHARMONIC RATIOS OF FOUR POINTS OR LINES. 45 are the same in sign and magnitude as those, of the segments (or sines of the segments) of P^P^^ made by Pi,P^; so that {P1P3, P^P^ =: {P^P^^- P\P'i}^ and so on. Hence for a system of four points {or lines) there are hut six different Anharmonic ratios, whereof three are the reciprocals of the remaining three : viz, {P,P,. P,F,}, {P,P„ P,P,}, {P,P,, P,P,}. {P,P,.P,Ps}, {P,P,.P,P.}, {P,P,,P^P,}. 82. These six Anharmonic ratios may be represented briefly by X, T, Z, and their reciprocals ; and from an inspection of their actual values it appears that XF^=-1, and^-^.^=: - 1. (101) The student will have no difficulty in proving that they are also connected by the following relations ; X^~ = l, ¥-{.^=1, Z-\-^=\. (102) 83. If a pencil of four concurre7tt lines he cut hy a transversal, the anharmonic ratios of the four points of intersection are equal to the corresponding ratios of the pencil, and are therefore the same for any position of the transversal. Let a pencil, having its centre at 0, be cut by a transversal in the points Pj, P^, P^. P^. Fig. 14. O 46 TRILTNEAE. CO-ORDINATES. Using the notation of Art. 80, we have -L ^ J- C) J. oJ- I 3^ 2 -^3-^4 OP,. OF^. sin F,P, OPi. OPs.smPgPs sin P1P2 , sin P1P4 , (Euc. Ti. 1,) OPi- OP4. sinPiP4 OP,. OP,, sin P.P, sin P3P2 sin P3P4 = {0.P,P3, P2P4}. (103) And the same may be proved with respect to the remaining five anharmonic ratios of the range. 84. It follows from the theorem of the preceding Article that if two pencils originating at and 0', are such that their rays intersect, Fig. 15. o tivo andj two, in points which lie in the same straight line, they will have the same anharmonic ratios. For, if Pi, P2, P3, P4 be the points of intersection of the two pencils, then, since their anharmonic ratios are equal to the cor- responding ratios of each of the pencils, we have HARMONIC POINTS AND PENCILS. 47 = {0'.P,P,,P,P,} (104) etc. =: etc., and the pencils are therefore equi-anharmonic'^ . 85. Any two pencils which are so mutually related that their corresponding rays intersect in pairs collinearly are said to be in perspective, or in homology with each other ; and the line on which their common points lie is called the axis of perspective, or the axis of homology, 86. Definition of harmonic section. Any segment (or angle) is said to be harmonically divided by two points (or lines) when each of its two anharmonic ratios (Art. 80) =r — 1. Thus the line (or angle) Pi Pg will be divided harmonically by the points (or lines) P^, P^, ., , , P1P2 PiPJ sin P,P, sin PiPA provided that ^^ : ^^ ^or _^ : -_^_j, or its recipro- cal, = — 1 ; and then the points (or lines) Pg, P4, are said to be harmonic conjugates with respect to P^, P3 ; and Pj, Pg, Pg, P^ form a harmonic range (or pencil). 87. The equation of harmonicism P3 P2 P3 P4 = -1 (105) may be written in the form P,P,___P,P, (106) Hence we conclude that a segment which is harmonically divided is divided internally and externally in the same ratio. This is some- times taken as a definition of harmonic section. ^ Equi-anharmonic systems of points or lines are said to be JiomograpMc. The two pencils in fig, 15 are both homogra'pliic and 'perspective, (Art. 85). 48 TRILINEAR CO-ORDINATES. 88. Anotlier definition is afforded by the following property, which is easily deduced from (105). If the segment Pi, P3 le har- monically divided ly P^^ P45 ^^^ C' le the point which bisects P1P3 ; then CF^ = CPi = CP^ . CP^. (107) 89. The condition of harmonic section may also be written in the form {P,P3, P2P4} = {P1P3, P4P2}, (1 08) (or {O.P,P,,P,P,} = {O.P,P,,P,P,})', (109) for the only case in which the two reciprocal ratios of a divided segment (or angle) are equal to each other is that in which the dividing points (or rays) are harmonic conjugates of the two points (or rays) by which the divided segment (or angle) is terminated «. 90. To find the anharmonic ratios of a pencil of four concurrent lines whose direction- angles are given. Fig. 16. PI Let the direction-angles of the four rays OP^, OP^, OP^, OP^, be ( ^o? %) in Tj. Then, (Art. 83. (103).), { O.P1P3, P2P4} == tei^3,i?2i?4} _ PlP2 . PiPa. ~ PzP^ ' P3Pa A 3 /(/^ A/3 Ixi^ and the other five ratios may be formed in a similar manner. ^ YoT p^Lyp^L, etc. are proportional to the lengths of the perpendiculars CONDITION OF HARMONICISM. ^ 51 93. From (113) it appears that the anharmonic ratios of the system of rays are independent of the constants in the equations of the given pair of lines {l^, mo, %), {I, m, n). Hence, if there be another pencil of lines having for their equations (Z'o, m'o, n'o) — Jci {!', m, n) = 0, il\, m'o, ^'o) — h (}', ^', n) = 0, (/'o, m'o, ?^'o) — h Q'y ^', n) = 0, {l\, m\, n'o) — \ (?, m, n') = 0, this pencil will have the same anharmonic ratios as that of the last Article; in other words, the equations (112) and (114) represent homographic systems. 94. To find the condition that four straight Jines which pass through the intersection of the pair of lines (4, ni^, %), {I, m, n) should form a harmonic pencil. Let us employ the same notation as in Art. 92. The equation of harmonicism ■i^^-t^i -t^s-t A -1, 3-1-2 -t 3-t 4 (Art. 87. (105).), becomes, by reason of (113), K^ A/2 f^\ f^^ iCo A^o ko ica = -1; (115) and, if this condition hold, the second and fourth lines of (114) will be harmonic conjugates with respect to the first and third. from_pj, j^2> stc. •upon (Zq, ^Q, Wq), and therefore, as will now be shewn, to k^, h^, etc. respectively. Suppose 5 to be the length of the perpendicular upon (I, m, n) from any point on the transversal. Then (Art. 46) the perpendicular from ^ ^ [(a, )8, 7) say] upon (Z„ ^„ .,) = -Jl-;^,^ = Jl^;^-^^^ [(112) first eq.] 1^0, Wq, ?^o| |Z, m, n^ |Zo, m^, n^^ C is a constant for all points on the transversal. 52 TRILINEAR CO-ORDINATES. 95. To find the form of the equations of any pair of straight lines which a/re harmonic conjugates with respect to the given pair {Iq, Mq, Uq), {I, m,n). Let {lo, mo, n^—h^ {I, m,n)=:0 (l'^6) {lo,mo,no)—Jc^{I,m,n) = (117) be the equations of the pair of conjugates. Applying the condition (115) to the equations (?o, mo, no) = {Iq, Mq, no)—^ {h m, n) = {I, m, n) =10 {Iq, mo, no)—h {h m, n)=:0, we find that (116) and (117) will represent harmonic conjugates with respect to the given lines provided that or ^2 ^= — ^'4 = K- (suppose). Hence four straight lines whose equations are of the form {h,^o.no) = {I, m, n) = ^ ^ {lo. Mo, no) + k{1 m, n) = {lo, mo, no) — k{1, m, n)=zO form a harmonic pencil, (118) and (119) being conjugate pairs^ 96. CoE. Since the equations (?i, m^, n^) = (4, m.2. n^) ~ {li,m^,n^\ - {4, ^^'2,%} = 0, HARMONIC PROPERTIES OF THE QUADRILATERAL. 53 (Art. 62. (74).), are of the required form, we see that the internal and external bisectors of the angle included ly any pair of right lines are harmonic conjugates with respect to them. 97. In the preceding Articles we have taken the most general case. It will be at once seen that the harmonic pencil formed by the lines /3 = 0, y r= 0, jS + y = 0, jS— y =■ 0, affords an instance of the theorem of Art. 96 : also that the pencil /3 — i?;iy = ^-% = o /3 — hy = is homographic with (112) and (114). (120) 98. To prove the harmonic properties of a complete quadrilateral. Let BiCi, C1B2, JB^C^, C^B^, be the sides of the tetragram; and Kg. 18. -.. \ 54 TRILINEAE, CO-OEDINATES. let Ai, Ac^\ B^, B^; C\, C^ be the pairs of intersections of opposite sides ; also, let X, Y, Z be the points of intersection of the three diagonals. Take A^B^ C^ for the triangle of reference, and let the equations of its sides, taken in order, be a = 0, ^3 = 0, y = 0, respectively. Assume for the equations of the two diagonals XF, ZX. la —m^ = . . . {XY) ny- la =0 . . . (ZX) respectively. The equation of B2 C^, the fourth side of the quadri- lateral, will be of the form oty — la -\- Jc^ =1 ; also of the form la — m/3 + Ivy = 0, (Art. 16) : and, identifying these two equations, we get X- = w, k' =: —n; from either of which we find la — m^ — ny = 0... {B2C2) to be the equation required. Also, since the line AiX passes through the intersections of /3 = with 7 = 0, and of XY{la—m^ = 0) with ZX{ny— la ■=.()), its equation will be of the form /3 -j- X'y =r 0, or la — m/3 + h'{ny—la) = ; and identifying, as be- nJc' n . ' fore, we sret k :=^\. fc=^ r=— - ; therefore the equation of ^iXis m,S — wy = . . . {A^X). Again, since YZ passes through the intersections of ^ = with 7 = 0, and of a = with B^C^, {la — m3 — ny = 0), we obtain, by a similar process, for its equation, m/3 -f ?^y =: . . . ( YZ). Hence A^C^{^ — 0\ A,C2 {y = 0), A^X {m,S — ny= 0), A^ Y (m/3 4- ny = 0), (Art. 95,) form a harmonic pencil, and (7i, (72, X, Y are four har- monic points. HOMOGRAPHIC SYSTEMS. 55 In the same way it may be sliewn that {B, . A^A„ TZ}, { C, . B,B„ XY} are harmonic pencils, and, consequently, {A,A,, YZ}, {B,B,, XY} harmonic ranges. The triangle XYZ is sometimes called the harmonic triangle, since its sides are harmonically divided by the pairs of points ^1, A^ ; Bi, B2 ; Ci, C2, respectively. 99. The reader will observe that the harmonic properties of the quadrangle (or tetrastigm) are also established by the proof just given in the case of the quadrilateral. We have, therefore, as the result of Art. 98, the two following reciprocal theorems. (a) In every tetragram the three pairs of opposite intersections divide harmonically the three sides of the triangle determined ly their three lines of connection. (j3) In every tetrastigm the three pairs of opposite connectors divide harmonically the three angles of the triangle determined hy their three points of intersection, (Townsend's "Modern Geo- metry," Art. 236.) HOMOGEAPHIC SYSTEMS OP POIISTTS AI^D LINES. 100. Definition of homography . — Two rows of points on any axes (or pencils of rays from any centres), Px^p^.Pz^ • • • and q^, ^o, q^, . . ., are said to be homographic when they correspond in such a manner that the anharmonic ratios of any four points (or rays) of the one are equal to those of their four correspondents in the other. (Art. 84, note.) The homography of the two systems is expressed by the equation {PlP-2Pz " '} = {Mi^i " •} (or { .pxp^pz ...} = { . q^q^q^ ...}), but in the following Articles, for the sake of brevity, the former 56 TRILINEAR CO-ORDINATES. notation only will be used ; the student will have no difficulty in seeing where the latter is implied, and the ratios of segments may be replaced by the corresponding ratios of sines. 101. Three pairs of corresponding points {or rays) are sufficient to determine two homographic systems; for if we suppose the three pairs ^1, qi ; p^, q^^ ; p^, q^, to be given, the correspondent q^ of any fourth point (or ray) p^ (taken in the first system) is known from the homographic relation {PxP2PzP^} = {m^M^} • (121) 102. Point on either axis which corresponds to the point at infinity on the other. Let /, e/be those points of two homographic systems {Pi^fi^Pz ' ' •}? {^i, $'2,^3 • • '}, which correspond respectively to q a:), p oc,} the points at infinity on the two axes. We have, by the condition of homography (Art. 100), that is, piPs . Pil _ q^qs . q^q^ P2Pz P2l~ q.2(i% q^q.^' (122) (123) ~~ ^2 $'3 Similarly, q^q^ . qxJ' PiPz q^qz q^J p^pz 103. From (122) and (123) it appears that if the correspondent of the point at infinity on either axis, and two pairs of corresponding points, he given, the correspondent of any third point, taken on either axis, is known, 104. If the points at infinity on the two axes correspond, the systems will he similar : for then {PiP^PzPcc } = {qi q2 ^3 qoo h whence p^ _ 5^3 P2Ps ~ q2 qz DOUBLE POINTS AND LINES. 57 Similarly, ^3 ^ q^ ^^^ ^^^^^^^^^ V^ ^ m\ PzPx q^qi P1P2 qiq2 Therefore p^p^ : p^p^ : p^P^ = ?2 ?3 : ?3 ?i : S'l ?2. And, conversely, tlie points at infinity on the axes of two simila/r homographic systems correspond. COAXAL (OE CONCENTEIC) HOMOGEAPHIC SYSTEMS. 105. Double points and lines. — In every system of two Jiomographic roivs on tlie same axis {or pencils of rays from a common centre) there exist two points {or rays), called double points {or rays), which are their own correspondents. For suppose the two coaxal systems to be {PiP^Pz • • •}? {qi^'i^z' ' '}' Since they are homographic, we have {PiP^Pz . . .} = {$'i^2^3 • . .}. (124) Let /, t7'(Art. 102) be the points which correspond to the points at infinity p^ , q^ , and suppose ^ to be a point which belongs to both systems and is its own correspondent ; we have by (124) {p,p^ Ix} = {q^Jq^ x}, or, Pil . PiX _ g,q^ . q^ . .p-^ Pco^~ Jq^ ' ^^ ' therefore piX q^x Pil Jx Pi^ — Pi gi ~~ PiX—p^J (by introducing an origin ^^1), whence we get p^xf — {piI+PiJ)pix\ +pj.p,qi = 0, (125) a quadratic which determines two positions of x, its distance being measured from pi. 58 TRILINEAE, CO-ORDINATES. If C be the middle point of the segment //, we have Pi^ — o ' and (125) becomes Pix\^ — 2piC .pix\ -\- pil.piqi = 0. (126) There are therefore two double points (which we shall call P and Q), real or imaginary. They will manifestly be on opposite sides of the origin if piC = 0, that is, if the origin be at C the middle point of U; since their position will then (126) be given by the equation 6'^P+ CI. CC'=0, (127) in which 0' is the correspondent of C. 106. Hence the segments IJ and FQ, are concentric. Also from (127) it appears that if CC ^ 0, P and Q coincide in the point C. 107. Any pair of correspondents {say Pi,qi) divide PQ into seg- ments whose anharmonic ratios are constant, For since, by their definition, P and Q are their own correspondents, we have (124), from the homography of the systems, p2i q-2 being any other pair of correspondents, {p,p,PQ} = {q^q.PQ}, PiP PiQ qiP 9iQ or . ::^ • . ' p^P ' P2Q q^P ' q^Q' in other words, the ratio PiP , PiQ q.P ' qi Q is constant. 108. From the theorem of the last Article it follows that, if one PiP of the double points (Q, suppose) he at infinity, the ratio — 5, for any qi^ HOMOGRAPHIC SYSTEMS IN INVOLUTION. 59 pair of correspondents jK»i, q^, is constant: that is to say, the systems are similar. O's iisrYOLUTioisr. 109. Definition of involution. — When two homographic coaxal rows {or concentric pencils) are such that every point (or ray) has the same correspondent, to whichever system it he regarded as helonging, the two rows {or pencils) are said to form a system in involution, and the cor- responding constituents are called conjugates, 110. Two pairs of conjugates are sufficient to determine a system in involution. Por, if two points pi, p^., and their conjugates §'i, g'2, be given, the conjugate of any fifth point ^3, is known (121) from the equation {Pi^i;^2y3} = {§'ii^i?2i?3}- (128) 111. It follows at once from (128) and the other similar equa- tions that any three pairs of conjugates of two homographic roivs {or pencils) in involution are connected ly the following relation^ Ml . M2 . Piq^ _ J (129) smp^qi sinp3^2 smp^gs \ or -■ • -: • ^ = 1 ; sm^^g^-i smpi^2 sinj?2?3 J and conversely, when two homographic rows {or concentric pencils) a/re such that these relations hold between any three pairs of corresponding constituents, the rows {or pencils) are in involution. 112. Any pair of conjugates are harmonic conjugates with respect to the double points of the involution. For, since P, Q (Art. 105) are self- conjugates, we have, supposing p-^, qi to be any pair of conju- gate points, {p,q,PQ} = {q,p,PQ}, which shews (Art. 89. (108).) that pi, qi are harmonic conjugates with respect to P and Q. 60 TRILINEAR CO-ORDINATES. 113. Hence (Art. 88. (107).), if Che the point of bisection of PQ, we have CP' = CQ' = Cp,.Cq,= Cp,.Cq,= ...=:.... C is called the centre of the involution, and evidently, if circles be described on the segments piqi, p^qi, Piq^, ... as diameters, they will Kg. 19. all pass through the same two points, and C will be that point on the line of centres through which the radical axis of the system passes. 114. The results of Arts. 112, 113, afford definitions of involu- tion which are more convenient than the one given in Art. 109, since they admit of a simple geometrical interpretation by refer- ence to the well-known properties (i.) of harmonic points, (ii.) of a system of circles having a common radical axis. Thus any one of the following definitions might have been given. 115. Wlien three or more pairs of collinear points {or concurrent DEFINITIONS OF INVOLUTION. 61 lines) are harmonic conjugates with respect to a fixed pair, they are said to form a row {or pencil) in involution, and the fixed pair are called the double points {or lines) of the system. 116. Or, when three or more pairs of collinear points, p^, q^ ; ^^^ q^ ; j^g, ^3*, . . ., are so situated with reference to a fixed point C on their common axis that Opi . Cqi = Cp2 . Cq2 = Cp^. Cq3 = . . . = a constant = + B{say), (130) the points are said to he in involution, and the point C is called the centre of the system, the involution being said to be positive or nega- tive according as h^ is affected by tbe upper or lower sign. The double points P, Q, since they are given by the equation (130), are equidistant from C and on opposite sides of it, and will be real or imaginary according as the involution is of the positive or negative kind. 117. Or again ; if a system of three or more pairs of collinear points Pu 9i ; P2^ Q2 '■) Ps) 93 'y • ' • ^^ '^^^^^ i^^i t^^^ circles described about the segments p^qi^p^q^-^p^qz, • • - as diameters have a common radical axis, the system is said to be in involution. The point C, in which the radical axis meets the axis of the collinear system, is called the centre of the involution, and the involution is said to be positive or negative according as the points Pi, p2,P3, ■ • • ^^^ their conjugates q^, q^, q^, • - - lie on the same or on opposite sides of this central point. In the former case there are two points, P and Q, — viz. the limiting points of the system of circles (being, in fact, circles v^^ith infinitesimal radii) — equidistant from C and on opposite sides of it, which, when con- sidered with reference to the collinear system, are their own conjugates, and are therefore called the double points of the involution. They are the points in which the axis of the involution is met by the circle, of radius h (131), which cuts the above system of circles orthogonally. In the latter case such double points do not exist, there being no points 62 TRILINEATl CO-ORDINATES. belonging to the system which possess the property of being self- conjugate. In hoth cases the centre C is the conjugate of the point at infinity. 118. It will be observed that the definitions of Arts. 116, 117, do not apply to a system of lines in involution, which may be defined as in Arts. 109, 115. It is obvious, however, that any pencil of rays through an involution of points forms a system of lines in involution, of which the corresponding rays are those which pass through conjugate points. Also the rays which pass through the double points will be the double lines of the involu- tion. Similarly, a pencil of lines in involution is cut by any transversal in points which are also in involution. 119. In a pencil of lines in involution there exist always two con- jugate rays ivhich are at right angles to each other : for suppose p^, q^-, jt?2, q^ ; Pi-, qzi to be pairs of conjugates of the involution in which the pencil is cut by any transversal, and let circles be described on the segments />! g-i, p2 5'25 i^s S's ; these circles (Art. 113) will have a common radical axis. Let now that circle of the system be described which passes through the vertex of the pencil, and let j9^ be its intercept on the transversal : j9, q are evidently conjugates of the involution of points, and Op, Oq, consequently, conjugate rays of the given pencil. Also Op, Oq are rectangular, since pq is a diameter. 120. If more than one pair of conjugates of a pencil in involution he rectangular, every pair will he so ; for then the vertex of the pencil is evidently one of the two common points of the system of circles described as explained in the last Article. 121. From the definition given of involution in Art. 115 it ap- pears that straight lilies whose equations are of the form (4, ^0, ^o) + ^^1 {h m, n) = (4, mo, n^) + h {I, m,n) = (/o, mo, no) -f X-g {I, m, n) = = INVOLUTIONS OF POINTS AND LINES. 63 form a pencil which is in involution with the concentric homographic pencil {Art, 93) represented hj the following equations, (4, ^0, ^o) — h {h ^n, n) — 0, (4, m^, no) — h {h ^^h ^) = 0, {lo, Mq, Wo) — h (J, m, n) = 0, = 0; (133) the straight lines (134) {lo, Mo, %) = {I, m, n) =^ being the doulle lines of the system. For we have seen (Art. 95) '' that (132) and (133), taken in pairs, are harmonic conjugates with respect to (134). 122. Similarly the straight lines i3+ y = - '^ (135) i3 + yl-2y=0 . . . . = form a system in involution of which )3 = (136) y = are the double lines, since each pair of (135) forms with (136) a harmonic pencil. 64 CHAPTEE YI. THE GENERAL EQUATION OF THE SECOND DEGREE. 123. We now proceed to discuss the trilinear equation of the second degree, the most general form of which (Art. 13) is (^(a,/3,y) = ^a^ + ^/32+ Cy' + 2l)^y -{• 2Uya + 2Fa^ = ; (137) and whenever, in this chapter, the equation 0(a, /S, y) = 0, or the curve cf){a, /3, y) = 0, is spoken of, the reader will understand that the complete equation (137), or its locus, is intended. An investi- gation of various modified forms of the equation of the second degree will be found in the next chapter. 124. Tg shew that the general equation of the second; degree alivmjs represents a conic section. It was shewn (Art. 13) that the general equation of the second degree, ^(a, /3, y) = 0, may be thrown into the polar form ^(X, „ .) .^ + { gj X + (IJ. + ( J)^ j - 4- 0(ao, /3o, yo) = 0. (138) Now, since (ao, /Sq, yo) may be amj point and the direction-cosines (X, /x, v) may have any values whatever, this quadratic shews that if a straight line be drawn through a point (oq, ^o, To), '^^ ^'^V direc- tion, it will meet the locus of 0(a, ^, y) = in two points, which are either real, coincident, or imaginary. Hence the locus of the general equation is a curve of the second degree, that is to say, a conic section. 125. To find the conditions that tlie general equation of the second degree should represent a Kyperlola^ a Paralola, ^ or an I]llipse. LOCUS OF AN EQUATION OF THE SECOND DEGREE. 65 Prom tlie equation (138) it appears that the directions of the asymptotes are given by the equation cjy{X,fji,v) = 0, (139) or AX^ -f ^/x' + Cv'' + 2DfjiV + 2Uv\ + 2FXix = ; whence, eliminating v by means of the relation aX -{- bfji -^ ci/ = 0, we get, to determine the ratio X : /x, which, on arranging the terms, becomes [Ae'+ Cd'~2ma)X^^2{Fe'-Elc-I)ca-\- Cal)X}i^{Ch''-\-Bc'-2mGy=0. (140) Now the roots of this equation are real and unequal, coincident, or imaginary, according as {Fc'-Ehc-Bca-yr Cahf = {A^-ir Ca^-2Eca) ( Cy^Bc'-2I)lc). This condition may be put into another form, for, multiplying out and dividing by c^, we get, after arranging the terms, {IP -B 0)0" + (^2- CA)l'' + {F^-ABy + 2{AB-EF)hG + 2{BF-FB)ca + 2{CF-BF)ah = 0, < that is (Prelim, chap, {B).) -{A' a" + B'V + C'c" + 2B'hG + 2F' ca + 2F'al) = ; or, according to the notation explained in Preliminary chap. {F), -(t){a,h,cy=0 (141) 66 TRILINEAR CO-ORDINATES. or, "A' = 0. (142) Hence the equation ^(a, I3,y) = represents a Syperlola^ Parabola, or Ellipse, according as c})[a, h, c)' is negative, zero, or positive, or as "d. is positive, zero, or negative. 126. Equation of the chord joining the two joints (ai, jS^, yi), (a2> i32> 72) (^'^ t^^^ conic (p{a, /3, y) = 0. The equation A{a-a,) (a-a2) + ^(/3-/30 {^-^2)+ C{y-y,) (y-y^) + i){(/3-^0(7-y2)+(y-7i)(/3-/3,)}+^{(y-yi)(a-a2) + («-«i)(7-72)}+^{(a-aO(/3-/32) + (/3-^0(a-a2)}=^a^+^^2 + Cy^+2D^y+2Eya^2Fa^, (143) is evidently satisfied by the co-ordinates of both the given points : and is linear, since the higher terms disappear on expansion. It therefore represents the straight line on which the given points lie, and is the required equation of the chord of the conic. To find the equation of the tangent to the conic ({){a, /3, y) = at the point (ai, j3i, yi). 127. First method. — "Writing oi, ^i, yi for 03, ^2; 72 respectively in (143) (since this is the same as making the two points on the curve, through which the chord passes, coincident), we get for the equation of the tangent at (ai, j3i, y^) . ^(a-aO^ + ^(/3-/30^+ (7(y-yO^+2i)(/3-/30(y-yi) + 2^(y-yO (a-aO + 2F{a-a,) (/3-/30 = Aa' + W' + Cy' -f 2i)/3y + 2^ya + 2Fa^ J or, ^a,^+m'+ Cy-'+2J)^,y,^2Fy,a,^2Fa,^, = 2{{Aa,+F^,+Fy,)a ^{Fa,+£^,-\-Dy,)^-^{Fa,-^m,+ CyM. (144) EQUATION OF THE TANGENT. 67 But the left-hand member of this equation vanishes, since (ai,/3i,yi) is on the curve ; we have therefore (Aa, + F^, + ^yi)a + {Fa, + ^ + i>7i)/3 + {Fa, + D^, + Cy{)y =z 0, which may be written in the abbreviated form da, I or again (Prelim, chap. {F).), thus, the expressions which form the left-hand members of these two last equations being identically the same. Other methods of find- ing the equation of the tangent will now be given. 128. Second method. — Suppose /a + w/3 + wy =0 (147) to be the equation of the tangent. Since it passes through the points (ai, iS^, yi), {ai-\-da,,^i-\-d^i, yi+^i)> "^® have the two equations la, + m^x + nji = 0, Ida, + md^x ■\- ndy, =z 0, which give, by cross multiplication, I m n (148) /3i, 7i Tl, «! ai, /3i ^/3i, dy. dyi, da. da,^ dj3i Also, by Euler's Theorem of homogeneous functions. ''■V + (S>^ + (?>=0- ai and, from the equation of the conic, 68 TRILINEAR CO-ORDINATES. whence, as before, dcji da\ /3i, yi d^i, dyi d(f)^ fdcf) d^J _ V^i dyi^ dui I daii d^i From (148) and (149) it follows that I m fd(j) \~ fd(li\ \daj \d^J n \dy. and (147) becomes e)-(S)»- fdcf>\ which is the required equation. 7 = 0, (149) (150) 129. Third method. — Taking the point (ai, /3i, yi) as pole, we have (Art. 124) to determine r, But, since the point (a^, /3i, yi) is on the curve, <^(ai, ^\, yi) = ; and, if the radius vector be a tangent, both values of r will = at the point ; we must have, therefore, ©'+(!)-+ (|)-=»- <-> Hence, since (Art. 12. (7).) X, /x, v are proportional to a— oi, ^—^i, y— yi» respectively, dcf) dcj)' g) ("--) + (I) c^-^-) +©(--)=«- (-) 69 (153) CONDITION OF TAXGENCY. or, by Euler's Theorem of homogeneous functions, daj ^ Vw wiy which is the equation of the tangent at (ai, ]3i, yi). 130. To determine the direction-cosines of the tangent, we have (Art. 129. (151).) daj ^ \d^J ^ ^ \dyj and the relation whence aX -\- hjjL -\- en =: : X ^ V /d(p \ ld(p\ \d^J' \dyj h, e id(p\ (d^\ \dyj' \daj (d^\ (d<\>\ \daj' WJ a, h and the symmetrical equations of the tangent may be formed at once. 131. To find the condition that the straight line {l,m, n) should touch the conic \ dcf) dui ■+(l)'^-(l)'=° <■"' is the equation of the line on which the two points of contact lie, and therefore represents the polar of the point (aj, /3i, 71). 135. If the polar be defined as the locus of points whose dis- tances from (ci, /3i, 7i) are harmonic means between the radii vec- tores drawn from that point, in the same direction, to the curve, the following method may be employed. 72 TRILTNEAR CO-ORDINATES. The equation of the conic referred to the given point as pole is daj_^\d^ ^ \dyj_ = _x. (suppose): (161) li Ml ni that is, Aa-^ + i^/3i + Uy^ + IJc = 0, Fa, + £13, + Dy, + 7?^i/(; = 0, (162) Ua, 4-i)/3i 4- Cyi+ %^- = 0; these, with the equation aa, + 5/3i + cy, = 2>S', completely determine the values of {a,, ^„ y,). (See Art. 137.) 137. To find the co-ordinates of the centre- of the conic <^(a, /3,y) = 0. Let (a, ^, y) be the co-ordinates of the centre ; the polar equa- tion of the conic, referred to this point, will (Art. 13) be CO-ORDINATES OF THE CENTRE. 73 (163) Hence, as the two values of r must be, for any direction, equal in magnitude but of opposite sign, we must have fd^ x+f^V+f^V = o, (164) \da} \d^J \dy. where X, /x, v are not independent but connected by the relation a\-{-hix-{'Cv = 0. (165) Now (164) and (165) are true simultaneously for all values of X, fjL, v: they give therefore d^ dy da'' ^djS^ \d■^J ' dy a aa-\-l^^Cy (166) (Prelim, chap. {A), and Euler's Theorem). Equations (166) may be written Aa+F-^ + Fy-a'^^^^^O Fa + £^ + Dy-h'^^^^^=0, Fa^D^-\- Cy 2S (167) 2S = whence we get (Prelim, chap. {F).) a /3 y — <^(a, ^, y) 2^ F,F, a B,D,h D, C, G A,F, a F,B,h F, (7, G A,F, a F,B,i F,B,G — A,F,F F,B,B F,B, C (168) 74 TRILINEAK, CO-ORDINATES. 138. Equations (168) of the last Article may be put into the form <^(a, /3, y) a /3 y 2S a,F,I! A, a, JE A, F, a A,F,F I, B,B F, l,B F,B,h F,B,B c, B, C E, c, C F,B, c F,B, C (169) or, if we expand the determinants in terms of their first minors and write A for the denominator of the last member of (169), a ^ 7 _ (^(g, ^, y) aA'-\-bF'+cF' ~ aF'^hB-\-cB' ~~ aE'^lB-^cC ~ A.S (Prelim, chap. (^).) ; (170) also zzz A' a^-{-B'h^-\- C c^+2B'hc-\-2F'ca+2F' ah 28 — «A' «a6 whence (Prelim, chap. {F).)^ 0(a,^,y)=-4>S^^ A 6' c}){a, h, c)' (171) (172) a relation which will be useful hereafter. 139. A comparison of (166) with the equations (161) of Art. 136 shews that in every conic the centre is the pole of the straight line at infinity (Art. 52). And in finding the co-ordinates of the centre of the conic ^(a, /3, y) = 0, the student may, if he please, start with this property of the centre, and, deducing the equations (167) as in Art. 136, proceed as in the last Article. Again, since (Art. 125. (142).) "A^ ^ when the conic is a parabola, it appears from (171) that the centre of a parabola lies on the straight line at infinity. 140. To find the equation of the pair of tangents which may be drawn to the conic (f){a, jS, y) z= 0, through the point {qq, jSo, yo)* PAIR OF TANGENTS THROUGH A GIVEN POINT. 75 The equation *^^' "• "y + { © ^ + (I) " + ©^ ! ^ + *("»' ^»' ^») = ° gives the length of the radius vector drawn to the curve in any direction from the given point. It vrill have equal roots, and the radius vector, whose equation is « — «o _ /3 — /3o __ y — yo will touch the curve, provided that r. (173) n2 = 4(^(ao,/3o,yo)(^(X,M.^)- (^4) Hence, substituting for \,ii,v from (173), we get for the equa- tion of the pair of tangents through (oo, ^q, yo) [©("-«») +©(^-^») +©(-»! = 40(ao, ^0) yo) (f){a — ao, ^—^q, y— yo) = 4(^(ao, /3o, yo) (175) ^{a,fi,y)-[{jAa, ^0+ (z^jyo I + <^(«oj ^o> yo) ; which becomes, since, by Euler's Theorem, (^o)"" + fe)^» + (^„)^» = 2*^""' ^»'^»)' The form of (176) shews (Art. 170. ((7).) that its locus is a curve of the second order having double contact with the conic (jb(a, /3, y) = at the points where it is met by the polar of (ao,/3o,yo). 76 TRILINEAR CO-ORDINATES. 141. Equation (176) of the last Article may be written in the form ( (7'/3^-l-^Vo-2i)'/3oro)a^+ (^70+ C'al^2E'y,a,)^'-\- {B'al^A'0^-2F'aMy' -2{F'yl-\- C'ao^o-^'^oyo-Jyyoao)a^ = 0. (177) 142. To find the locus of the point of intersection of tangents to the conic \ fdcf)\ , fd(b\ T , r . _Wor^''^^ + VSor'''^^'^VV''''^J =4^(«o,^o,7o)[^cosX -\- B cos \ -f C cos ^^1 -\- 21) cos h^ cos c^ + &c. . .], and ~[d\ . (d(l>\ fd6\ 7 Ad^J'''' ''' + 1^0^'^ ^' + [chor^ '0 " ^ ^^"°' ^°' ^°^ ^^ '^ '' -\- B sin ^^ -\- C'sin^c^ -\- 21) sin h-^ sin c^ + &c. . .] ; adding (Art. 30. (22).), and omitting the suffixes, we get for the equation of the locus (&"-(i)'Hi)'-®)(fi— ;i)o~- -2(^) (^) cos 6^= 4(/)(a, ^, y) lA +B+C-2J)cosA -2FcosB-2Fcos C]. (178) 143. The equation of the last Article may also be written in the form DIRECTOR OF THE CONIC. THE ASYMPTOTES. 77 (5' + C" + 2i>' COS A)a' + ( C +A' + 2U' COS ^)/3^ + (.4'+^' +2i^' COS C)/-2(i)'-^' COS ^ 4-i<^' COS ^+^' cos C)^y-2{E —B'co^B-^-B'qos C+F' cos A)ya— 2(1^- C cos C-\-E' cos a -\-iy cos B)a^ = 0. (179) This, therefore, is the equation of the director of the conic (^(a, j3, y) = 0, and will be found to satisfy the condition for a circle. (See Art. 149.) 144. To find the equation of the asymptotes of the conic 0(a,/3,y)=rO. The asymptotes being a pair of tangents which have the line at infinity for their chord of contact, their equation must be of the form 0(a, jS, 7) = h{aa + 5/3 + CyY Since they pass through the centre of the curve, we have, <^(a, ^, y) = 4:hS^ ; and consequently their equation is ^(a, ^, y) = 0(a, ^, y). (180) This equation may be written in the homogeneous form 4:8'' <^(a, ^, y) = <^(a, ^, 7) («a + h^ + Cyf ', (181) or, again (Art. 138. (172).), ^A' (^(a, /3, y) + A {aa + h^ -\- Cyf = 0. (182) 145. Equation (180) of the last Article may of course be ob- tained directly from the polar equation of the conic. For, as in Art. 125, we have the directions of the asymptotes given by the equation (^(X, ^, I/) = ; 78 TRTLINEAK CO-ORDINATES. hence their equation is cji{a — a, 13 — ^, y— y) = 0, or t'^'^'^'y^ - 1 ©"+ (S)^+ (|» +"^("'^'^) =■•'• (183) But, since {a, /3, y) is the centre, we have (Art. 137. (164).) » (3)<-'HS)'»-«+(|)<'-''="' that is, S)-+(l)»+(|)'=©- (!)»+©' = 2 <^(a, jS, y), by Euler's Theorem ; and (183) becomes, 9{a, /3, y) = ^(a, ^, y). 146. To find the condition that the equation (f){a, ]3, y) = mai/ represent a pair of right lines. Let (a, j3, y) be the point of intersection of the two lines. This point being the centre of the conic, and therefore the pole of the line at infinity, we have, as in Art. 136. (161), dcl)\ fd(f)\ I d(p\ da^ __ ^dP __ ^dy^ _ (f){a, ^, y) ^ /^g^N a b c S But, in this case, the centre lies on the conic ; so that ^(a, ^, y) = 0, DISCRIMINANT OF THE EQUATION. and we must have simultaneously da I \d^J \dy that is to say, Aa-\- F^-^Uy^ 0, i^a + ^^ 4- Dy = 0, ^a + i)^ + CV = ; and the condition that these should co-exist is A, F,E 0; 79 (185) (186) or, (Prelim, chap. (-D),) E.B, C A = 0. = 0, (187) (188) 147. The condition of the last Article may be otherwise ob- tained, as follows. If (^(a, 3, y) = represents a pair of right lines, (^(a, jS, y) must be the product of two linear factors u and v (suppose) : so that ^(a, /3, y)-=.uv =^Q, and we shall have , 'd^\ fdv\ (du\ \da j \da J ' \da y dcf) dv <«i3y="^rf^' + ''l^ du' Q /' d(p dv dy)-'^\dyj^\dy)^ du whence it appears that any values of a, (3, y which satisfy u = 0, V =i simultaneously, will also make ^<^' dcf)^ '^J~WJ~[dyJ~^' 80 TRILINEAR CO-ORDINATES. and we obtain, as before, A = 0. 148. To find the condition that the equation <^(a, j3, y) =: should represent a parabola. If the conic be a parabola, the straight line at infinity aa^h^-{- cy = (189) is a tangent to it (Art. 131. Cor.). Let (a^, /S^, y^ be the point of contact. We shall have (Art. 131. (154).) /^\ fd^\ fd^\ VdaJ ^\^) ^V^l , , a h c whence Aa^ + F^^ + Fy^ -\- ah = (i, Fa^ -f ^/3i 4- i)yi+ hi = 0, Ea^ + D^,+ Cy, + Ck = 0, also by (189) aa^ + b^^ + cy^ =0: and eliminating a^, /3i, yi between these equations we have for the required condition "A* = 0, (190) which (Prelim, chap. {F).) may also be written in the form 0(«, h, c)' = 0. (191) 149. To find the conditions that the equation ^{a, /3, y) = may represent a circle. The polar equation of the conic, referred to the centre, since the co-efficient of r vanishes, gives (^(X,/x,v)r^=-(^(a,^,y) (192) = a constant. CONDITIONS FOR A PARABOLIC OR CIRCULAR LOCUS. 81 Hence, if p^, Pq, p be tlie lengths of the semi-diameters respec- tively parallel to the three sides of the triangle of reference, we shall have ^(0, —sinC, sm£)pl = (/)(sin(7, 0, —smA)p^ = (ji{—smB,s'mA,0)p-^. (193) But, in the case of the circle, Pa = P^ — Py' hence we have for the required conditions, ^(0, —sin C,smB) = ^(sin C, 0,— sin^^) = (j){—smB, sin^, 0). (194) Or, if we write them at full length, jBsm'C-\- Csm'JB-2DsmB&mC = Cdn''A+Asm^C-2EsmCs'mA = AsiR^B+Bsin^A-2FsmAs'mM. (:195) 150. To find the condition that the equation (f){a, /3, y) = should rejjresent a rectangular hyperhola. The directions of the asymptotes (Art. 125) are given by the equation ^(X, p., v) = 0. But if these are mutually perpendicular, the two sets of values for X, p., V, given by this equation, will be (Art. 33) of the form (cos a^, cos hi, cos c^) and (sin a^, sin l^, sin c^. Hence ^(cos a^, cos 5^, cos c^) =. 0, <^(sin «i, sin h^, sin c^) = ; adding and remembering the relations (22) of Art. 30, we get, as in Art. 72, A-\-B-\- C-2DGosA-2UeosB-2FcosC==0 (196) for the required condition. G 82 TRILINEAR CO-ORDINATES. 151. To find the conditions that two conies whose equations are <^(a, /3, y) rr: 0, /(a, /3, y) =: 0, should he similar and similarly situated. Let p^, p^, p^ and r^, r^, r^ be the central radii vectores of the two conies, drawn parallel to the sides of the triangle of reference : we have, as in (193) Art. 149, (/)(0, — sinC,sin^)p^ = (^(sinC,0, — sin^)p^ = 0( — sin5,sin^,0)p^; /(O, -sinC',sin^)r^ =/(sin6; 0, - sinA)rl =f{-smB, sinA, Oy^. But, if the two conies are similar and similarly situated, we must have Pa _ Pp __ P^. ^a ~ ^^ ^ * hence the required condition is 0(0, —sin C, sin^) (/)(sin C, 0, — sin^) _ (^(— sin 5, sin^, 0) , /(O, — sin C, duJB) ~/(sin C, 0, — sin^i) ~ / (— sin B, sin^, 0) ' (197) in other words, the quantities jgsin"(7+ Csin^^— 2i)sin^sinC, Csin2^ + ^sin2(7-2^sinCsin^, and A sin "^ + ^ sin "^ - 2i^ sin A sin B, must be to each other in a constant ratio. 152. To find the direction of the axis of the parabola whose equation is ^(a, ^, y) == 0. The equation {Ac^^ Ca'^-'lEca)^ ^2{F(?-JElG-J)ca- Cal)\ii^{Ch^^Bc'-2Bhcy=0^ (Art. 125. (140).) gives the value of the ratio X: /x for the direc- tions of the asymptotes. SIMILAR AND SIMILARLY SITUATED CONICS. 83 If (j){a, ]3, y) = be a parabola, the asymptotes will be co-incident and in the direction of the principal diameter, and the above equation will give (198; the last member following by symmetry. These equations, therefore, determine the direction of the axis of the parabola. 153. To find the equation of the circle, of radius p, ivhose centre is at the point (a, /3, y). Let (a, /3, y) be any point on the curve ; then, since its dis- tance from the centre (a, ]S, y) is constant and = p, we have (Art. 45. (48).) — — — — — _4S^n^ To) li^s on the right line whose equation is S)"+(g)^+(?)-«- (205) this therefore is the equation of the diameter or locus of middle points. ^ CoE. I. Hence is the equation of the diameter which is conjugate to a — a_ ^— /3 _ y — 7 _ H' (206) (207) CoE. II. Its direction-cosines being formed as in Art. 40, we may write the equation of the diameter which is conjugate to a — a jS — /3 y — y = r, (208) in the symmetrical form a — a /3-^ y — y h, c ld(p\ fdcj)\ \dfij' \dv J c, a /d(j)\ fdcjA \dv j' \dxj a, b fdxf\ fdcp\ \dxj' KdfiJ (209) CoE. III. Again, it follows from Art. 157, that two straight lines, whose equations are of the form la + m^ -{■ ny = (210) 86 TRILINEAR CO-ORDINATES. and h, c m, n c, a n, I dj)' d^ ''^ a, h I, m dy = 0, (211) are parallel to conjugate diameters. 158. To find the condition that the straight lines (X^, /xj, i/i), (Xg, M2J ^2) '^^(^y ^^ parallel to conjugate diameters of the conic 4>{a, /3, 7) = 0. It was shewn in the last Article that the straight line \dXJ ydfij"^ ' \dvj ' is conjugate to (A^, /^i, v^. Hence we have d\ d(j) d(f) dvi (Art. 29,) and the relation 4X2 + W2M2 + %^2 = (Art. 31. (26).) becomes /d(i>\ /d(h\ /d+(S-- CONJUGATE DIAMETERS. We must have, therefore, 87 d(f) d\i J, d(f) \ /^d(f) dfjii I \dvi m, n^ = — Jc (suppose). Hence, A\i + F^i^ + Ev-^^ + IJc = 0, F\^ + ^/xi + Bv^ + mjc —. 0, we also have l{k^ + miii^ + tiiv^ = 0, (Art. 31. (26).). And, eliminating X^, /l^i, v^ and Jc between these equations, we get for the required condition A, F, E, k F, B, B, ^2 E, B, C, W2 0. (214) If this determinant be expanded it will be found to be A' Ilk + B^miM^ + C'%^2 + ^' (^1% + ^2^0 + E' {tz^I^ -{- n^h) + FXI^m^ + 4%) = . (215) 160. To find the equation of the principal axes of the conic (p^a, jS, y) = 0. The principal axes may be regarded as the locus of points whose polars are perpendicular to the lines joining those points to the centre. Let (ai, ^1, yi) be a point on the axes. The polar of (ai, /3i, y{) has for its equation (Art. 134) /dd)\ /d(b \ / d(b\ and the equation of the right line joining (ai, /S^, y^) to the centre is S8 TRILINEAR CO-ORDINATES. ^,7 a + 7, a ^ + 7 = 0; and the condition that these should be perpendicular is (Art. 71. (85).) ((2)-(l,)»-(S)~- /3i, 7i + cosA -©'=°^^! -iS''=°^^ 7ij °i y? « a, ^ + ^, 7 d(fi ^1 cos_5 = 0; (216) whence, suppressing the suffixes and using the determinant form, we have for the equation of the locus d(fi da J)'=°'^-(^)'=°'-^' "' "' dy J \da \d^J 7' 0.(217) 161. The equation of the axes may be obtained directly in the form of a determinant, as follows. Let (?, m, n) be an axis and let (ai, ,8i, y^ be a point on it ; then, since the centre (a, ^, y) also lies upon it, we have and lax + ^i^i + nyi = 0, la + m^ + ny = 0. (218) (219) If (/, m, n) be perpendicular to the polar of (oi, /3i, yi), whose equation is / 6?^ \ /d(b \ /d(b\ EQUATION OF THE PRINCIPAL AXFIS. we must have (Arts. 69. 42) 'l©-(S)»<'-{|)»-!+»l(S)- 89 dcji COS A ^Ucl +. |f'^Vf^W-ftW^[=0; daiJ dyi/ \dai J^y (220) and, eliminating 1, m^ n between (218), (219), and (220), and sup- pressing the suffixes, we get for the equation of the locus g)-(S)eo--@)eo.. = 0. (221) 162. To find the lengths of the semi-axes of the conic ?7= 0={{A-Paco^A)\-\-Fii-\-Ev-\-aQ}d\-\-{{B-Phco^B)ix -\-Bv-\-F\-\-hQ}dix-^{{C-PcQo^C)v-\-E\^Bix^cQ}dv', and therefore, simultaneously, {A— Pa cosA)\ -\-FiJL-\- Ep-\-aQ = 0, FX+{B-Ph cos B)ix-\-Bv-\- hQ = 0, (225) EX + Bix-Jr{C-Pc cos C)v-{- cQ=0. Multiplying the equations (225) by X, fx, v respectively and adding, we get, by means of (223) and (224), -^ = 0(X, ^, .). (226) Hence, by (222), p^^.^V (227) c and., substituting this value for P and eliminating X, /x, v and Q from (225) and (224), we get A- A c ale ' 2 «cos A F, E, a F, B- A ahc c COS £. D. I E, I>, C- A ahe c c a, i, 0, = 0, (228) THE AXES AND AREA OF A CONIC. 91 a quadratic in -3 which gives the lengths of the principal semi- diameters. 163. To find the area of the conic (f)[a, j3, y) = 0. Mr. Ferrers has deduced the area of the conic from the equation (228) of the last Article. If we expand the determinant, and, for convenience, write {A), {B), ( C), for its diagonal elements, this quadratic becomes (^) {B\D, h F, F, a F, F, a F, F, a D, {C), c -F B, (C), c ^F {B), D, h — a {B), B, h h, c, h, c, h, c, B, {C),c = 0. or. (A){-{By-\'2Bhc-{C)i'}-F{-Fc^-\-Bca-^Fhc-{C)ah} + F{-Fhc-\-{B)ca + FP-Bai}-a{FBc-{C)Fh -i-{B){C)a-{B)Ec+BFi-IPa} = 0; or, if we collect only the constant terms and those which involve —J, and make the necessary reductions, -{A'a^+B'P-Jr C'c^+2B'hc+2E'ca+2F'ah) + etc ' ^ahc\2 16 S' a\b = 0; that is (Prelim, chap. (-E').), «A' + etc. ... - [2ahG8—.] -^ = 0- (229) Hence if —^^ -^ be the roots of this quadratic, we have ;- Area of the conic = irpip^. 2'KaalcS/!s. ["A'] 613 2 (230) 92 CHAPTEE YII. INTERPRETATION OF PARTICULAR FORMS OF THE EQUATION OF THE SECOND DEGREE. 164. The present chapter will be devoted to the consideration of particular forms of the trilinear equation of the second order. We commence with those which occur in the subjoined list : ^S'l - ^-^2 = 0, {A) S^ — Uu = 0, {£) S,-kw'=0, {C) Si - ku = 0, (D) s,- h^ = 0, {E) vw — Uu = 0, (F) vw — • Icu^ = 0, (6) vw — Jcu = 0, {S) VW - F = 0. {J) In these, h is any constant and /S'l = 0, /S^g = represent any two conies ; ^ =z 0, w := 0, v = 0, ti? =: are the abridged forms of the homogeneous equations of four straight lines (?i, m^, n-^^ (4, ^^2) %)» .... in which the co-efficients may have any possible values. 165. Of the above forms, which will be examined in order, the second, third, fourth and fifth are successively derivable from the first, and the remainder from these, in a manner which will be understood as we proceed. (A). SYSTEM OF CONICS THROUGH FOUR POINTS. Si — kS^ =:: 93 Fig. 20. is evidently satisfied by any values of a, ^, y which satisfy S^ =z and 8-2=^0 simultaneously. It therefore represents a conic (or, if k varies, a system of conies) passing through the four points of intersection, whether real, coincident, or imaginary, of Si and 82- Such a curve is indicated by the dotted line in the figure. 166. If the conditions of the last Chapter (Art. 125) be applied to this equation, it will appear that it represents a hyperbola, an ellipse, or a parabola, according as Ai — IcA^, Fi — JcF^, JEi — hF^, a Fi - IF^, Bi - kB^, Bi - X'A, ^ El - TcF^, Bi - hB^, Ci - IC^, c a, 5, c, is positive, zero, or negative. Its locus will be a pair of right lines (Art. 146) if Ai - hA., Fi — IF^, Fi -IF^ Fi — IcF^, Bi — hB^, Bi —IB^^ — 0. . (231) Fi-hF^.Bi-hB^^Ci-hC^ In like manner the condition for a circular locus (Art. 149) is easily applied. 167. The cubic (231), when the determinant is expanded (Prelim, chap. (-0),) becomes {AiBiCi)-{{A,BiCi)-\-{AiB,Ci)^-AiBiC,)}Jc-\-{{AiB,C,) -\-{A,BiC,)+{A,B,Ci)}k'-{A,B,C,)Jc' = 0. (232) ^ It is assumed that S^ = A^a^- + B^^"" + C^y^ + 2D^By + 2U^ya + 2F^aB = 0, and S^ = A^a^ + B^fi^ + C^y"- + 2I)^By + 2'E^ya + ^F^a^ = 0. 94 TRILINEAR CO-ORDIXATES. Let hi, hi, Ic^ be the roots of this equation ; then it is manifest that Si — hS^ — 0, (233) Si — /.'s^S's = 0, are the equations of the three pairs of chords of intersection of the conies >S'i = 0, S2 = 0, and, therefore, of the whole system represented by the equation Si — tcS^ ^= 0. 168. Now all this will be true if either Si or >S'2, or both, be resolvable into linear factors ; that is to say, if one or both of the conies degenerate into a pair of right lines. Should a factor be of the form cia -f J/3 + cy, or, which is the same thing (Art. 3. (1).). a constant quantity, one of the right lines will be at an infinite distance (Art. 52). If the factors be identical, the pair of lines will be coincident. 169. First suppose the conic >S'2 to consist of two right lines {Ji, Ml, ni), (4j ^2. %)> whose equations in their abridged forms are z^ = G, u^O; the above equation becomes {B). Si — htu =z 0, which, therefore, represents a system of conies passing through the four points of intersection of these two right lines with the conic Si (fig. 21). In this case t and u are two of the common chords ; one root, therefore, of the cubic (232) will be infinite, and, as we should expect (Art. 146), {A2B^C^)=0. 170. JS'ext let the two right lines, of which /S'2 is composed, coincide. Making t z= u, we have SIMILAR AND SIMILARLY PLACED CONICS. 95 {C). Si — hu^ =■ 0, Fig. 22. whose locus is, therefore, a system of conies passing the two pairs of coincident points in which u meets the conic Si, and, there- fore, having double contact with Si at the extremities of the chord m = (fig. 22). 171. It is obvious, from an inspection of the figure, that two out of the three pairs of chords common to >S^i, S2 (Art. 167. (233).) have now come into coincidence with the chord of contact u, and the remaining pair become tangents at its extremities. The cubic (232) will have two infinite roots, and therefore both (A^B^C^) = and {AiB.^C^) + {A^BiC^) + {A.B.^Ci) = : the latter being true independently of the values of Ai,Bi, Ci, €tc, . . ., we must have the first minors of the determinant {A2B2C2) all = ; that is B^C^-Bl = 0, C^A.^-I^l = 0, A^B^-Fj = 0, E2F2—A2B2 = 0, etc. . . . (whereof the three latter relations are involved in the three former), or B\=^ B^C^,, El=^ C^A^, Fl=:i A2B2, conditions which are evidently satisfied when S^ is the square of a linear expression. {D). Si — ku =i is of a form such as we should derive from {B) if we were to Fig. 23. 96 TRILINEATl CO-ORDINATES. replace ^ by a constant quantity. Its locus, therefore, (fig. 23) is a system of conies which meet S^ where it is cut by u and the line at infinity. This system will be similar and similarly situated with respect to Si, and the asymptotes of the conies of the system, whether real or imaginary, will be parallel. If the curves be parabolas, since the two points at infinity in this case coincide, they will have contact with each other at infinity. 172. The equation (JEJ). Si-k^ = 0, ao:ain, is obviously a particular case of ((7), from which it is de- rivable by the substitution of a constant for u. Hence it denotes a conic (or system of conies) having double contact with Si, where that curve is met by the straight line at infinity. This system of conies will not only pass through two common points at infinity, but will have common tangents at those points. All the curves of the system, therefore, have the same asymptotes, and are not only Fig. 24. similar and similarly situated, but likewise concentric (fig. 24). If the curves be parabolas, they will be equal and have with each other a contact of the third order at infinity. 173. Now let us suppose the conic S-^ also to degenerate into a pair of straight lines ; {B) will become of the form SIMILAR, SIMILARLY SITUATED AND CONCENTRIC CONICS. 97 (F). vw Mu =■ 0, Fig. 25. which therefore represents a system of conies circumscribing the quadrilateral of which ^ =: 0, «^ = 0, and «; == 0, ^i? = 0, are the pairs of opposite sides. The truth of this, however, may be seen without reference to the preceding equations, since (jP) is evidently satisfied by any one of the suppositions, V = 0, and ^ = ; II -= 0, and u =^ ; w =^ 0, and ^ = : or, w = 0, and ti = 0. 174c Similarly, from a comparison of the next equation with (C), it appears that (G). vw — hu^ = represents a system of conies which have contact at two fixed points ; w = (fig. 26) being their chord of contact, and -y = 0, «^ = the tangents at its extremities. Fig. 26. 175. Again, by a reference to Art. 171. (i)), it will be seen that {H). VW — hu := has for its locus a system of coiiics having that portion of u which is intercepted between v and iv for a common chord, and passing H 98 TRILINEAR CO-ORDINATES. through the two fixed points at infinity in which the line at infinity is met by v = 0, ^ = 0« These conies are therefore similar and similarly situated hyperbolas having their asymptotes parallel to V = 0, w = (fig. 27). 176. The locus of (/). ^^ — F =: is a system of concentric, similar, and similarly situated hyper- Fig. 28. bolas, having v = 0, «^ = for their common asymptotes and the intersection of those lines for a common centre, (fig. 28). 177. It is obvious that in the four last cases v = 0, tv = 0, taken together, form one of the family of curves, just as Si = did in the preceding instances. SPECIAL FORMS OF THE GENERAL EQUATION. 99 178. The meaning of the equations {K), Si + ^a/3 = 0. (Z), Si + ka' = 0, (M), Si + Jca = 0, follows at once from the interpretation just given of equations (B), {C), and [D) ; the two sides, a = 0, /3 = 0, of the fundamental triangle taking the place of the two straight lines ^ = and u^=.0, or l^a -\- Mi^ -f n-iy = and l^a -j- '^2^ + %7 = 0. 179. We now proceed to consider some of the more special forms of the equation {a, I, c)' =z M'N'a^ - N'Z'P - L'M'c\ Wherefore, ( Q) will represent an ellipse, a parabola, or hyperdola, according as a' y c^ V M^ m is positive, zero, or negative. CONIC REFERRED TO TWO TANGENTS AND THEIR CHORD. 109 201. (i2). I3y — ka" -=z 0. By reference to (Art. 174. ((9).) it will be seen that the eqtcation {R) represents a conic section to ivMch CA (/3 = 0) and AB{y ■= 0) are tangents, while BC (a=::0) is their chord of contact. Pig. 33. 202. Here ^(a, ^, y) =: 'iW + ^0- + 0/ _ 2/3y + Oya + Oa/3 = ; <^[a, b, cy=^ —a^-\- 4:Jcic : and the locus of {H) will he an ellipse, a parabola, or hyperbola, ac- cording as h > a"- < 45^ 203. (>S). /3y — X'a=0. The locus of this equation (Art. 175. (.ff).) is a system of similar Fig. 34. 110 TRILINEAK, CO-ORDINATES. and similarly -placed hyperlolas (fig. 34), having the side BC of the triangle of reference for a common chord, and passing through the points at infinity on the other two sides. Each curve of the system, therefore, has its asymptotes parallel to CA, AB, 204. Equation (/S), as we should expect, satisfies the condition for ■ a hyperlola ; for we have (^(a, /3, y) = 2aa^ + 0/3^ + Oy' - 2( y )/3y -f 2cya + 2ha^ = 0, and therefore 4„2 2 = Tg— J essentially a negative quantity. 205. Again the equation gives us (^(a,/3, y)= «V+ hY+cy-\-2(bc - ~]^y-{-2caya + 2aha^ = ; whence we shall find that cji{a,d,cy= --^, and that (Arts. 176. 125) {T) represents a system of concentric. Fig. 35. SYSTEMS OF SIMILAR CONICS. Ill similar, and similarly situated Jiyperholas, having the two sides CA, AB, of the triangle of reference for their common asymptotes (fig. 35). 206. Lastly, the equation lias for its locus a system of conies (Art. 175. {M).) to which a=i and the line at infinity are tangents at the extremities of the common chord /3 = 0. We shall now have <^(a, iS, y) = 2«a' - ^ j^^ ^ Qy^ + 0/3y + 2cya + 25a/3 =r ; ^{a, I, cy=z - c^' - (^ + p)c' 4- 2{hc)ic + 2(^)m, = 0; and (W) represents a system of similar and similarly placed paraholas Kg. 36. of which /3 =: is a diameter, and a=: the tangent at its extremity (fig. 36). 112 CHAPTER YIII. EQUATIONS OF THE SECOND ORDER CONTINUED. 207. Instances will now be set before the reader of the mode of application to particular equations of the general results of Chapter VI ; the proof being, however, given in an independent form where the case appears to deserve a separate investigation. For the convenience of the student some of the most important forms of the equation of the second order are collected in the next Article. 208. {N). L^y + 3Iya + iVa/S = 0, or _ 4_ 4. _ = 0, a P 7 which represents a conic circumscribing the triangle of reference (Art. 180). ( O5). Va- + M^/S^ + N'-y^-2MN^y-2NLya-2LMa^ = 0, or {La)^ + {M^)i + {Nyf = 0, tvhich represents a conic inscribed in the triangle of reference (Art. 187). (P). xv + M^^s^ + ivy = 0, which is the equation of a conic ivith respect to which the triangle of reference is self- conjugate (see Art. 194. (P). (Q).)- LOCI REPRESENTED BY CERTAIN EQUATIONS. 113 (E), /3y - U = 0, which is the equation of a conic touching the sides CA, AB, of the triangle of reference at the points C and B respectively (Art. 201). (>S). ^y - la = 0, which represents a hyperlola passing through B and C and having its asymptotes parallel to CA, AB (Art. 203). which is the equation of a hylerhola whose centre is at A, and to which the sides CA, AB are asymptotes (Art. 205). ( JI). 0" - ha — 0, which represents a parabola of which CA is a diameter and B C the tangent at its extremity (Art. 206). To these may be added the equation (F). ay-^i3S = 0, which represents a conic circumscrihing a quadrilateral whose sides are a = 0, j3 i=: 0, y = 0, 6 = 0, and, though not strictly speaking trilinear, may be regarded as a particular case of (Art. 173. {F).). 209. Equation of the chord joining the two points (a^, jS^, yi), (02' ^2, 72)' L , M ^ N ^ {N). — + — + — = 0. ^ ' , a p y 114 TKILINEAR CO-ORDINATES. The equation of the straight line joining the two given points (Art. 58. (68).) is (/3ir2 - ^27i)a 4- (riaa - y^a,)^ + {a^l3, - a^,)y z=z ; (247) But, since in this case the points lie on the conic {JSF), we have L , M , N — + ~ + — =0 oi Pi 7i and L M N "2 ^2 72 = 0; L M N 1 1 "" 1 1 - 1 I ' /3l72 ^27l 7l«2 72«1 «i^2 a2^l /3l72 — ^27l 7x02 ~ 7201 L ~ M ai^2 — a2/3i ~" N a^a^ /3i/32 7i72 whence, or H172 — H271 _ yi"2 — 72"i _ "iP2 ~ "2 Pi . (248). and (247) may be written Z , M N «+^/3+— -7 = 0, (249) oitta ^1^2 7i72 which is the required equation. ( O5). Zh^ + M^/3* + N^yi = 0. Since {ai, ^i, yi), (03, /Sa, 72) are points on the curve, we have and Z^4 + M^^l + N^yl = ; whence, Z^ __ M^ _ N^ ^hl - l^hl 7f 4 - yl4 °M - aM ' EQUATION OF A CHORD. 115 or, 18172 — /32yi _ yi«2 — 72"! _ «i^2 — Q2/3l . i>* (^M + ^lyh J-^* (yf «l + 72V ) ^' {a!^l + a|/3*) ' (250) and (247) becomes L^i^hl + ^l7f )a + MHy"i4 + 7l«n/3 + A'*(aM+ 4/3f)7 = 0, (251) the equation of tlie chord. (P). XV + Jf^/S^+iV^y^O- Since the given points are on the curve, we have and L'al^M'^l-\-Ny,= 0, which give i'^ M^ N^ o2 2 o2 2 2 2 2 2 2,;j2 2o2 P172 — P2yi 7l«2 — 72«1 "1P2 — «2Pi j3i72 ^27l 7l«2 72«1 «1^2 + «2''^1 Hence, = — — = — — „ ; ('252') I^ M^ N' ^ ^ /3l72 + ^27l 7l«2 + 72«1 "1^2 + «2^1 . and we have for the equation of the chord, /3iy2 + /327i 7i«2 Hr 72"i "1^2 + «2/3i (i?). iSy - Jca" — 0. We have, since the given points are on the curve, A7i = ^-4 (254) 116 TRILINEAR CO-ORDINATES. /3272 = hal (255) Hence, multiplying (254) and (255) by y| and y\ respectively, sub- tracting, and dividing by {y^a^ + y^ai), we get ^172 — %i _ 71^2 — 72^1 /256^ — hiy^a^ -j- 72ai) 7172 Similarly, if we multiply (254) and (255) by 0i and ^l and sub- tract, we get and (256) and (257) give ^]72 — /327l _ 7l°2 — 72°! _ «l/^2 — <^2&\ — k 7i72 /3i32 (258) 7i«2 -f 72ai ai32 + a2/3i whence substituting in (247), we have for the required equation yi«2 + 72°! "1P2 + "2P1 Second method. — The equation of the chord may, however, be more easily formed, in the present instance, after the manner of Art. 126, as follows. The equation 0-/30(7-72) + (/3-/32)(7-70-2X'(a-aO(a-a2) =2^7-2X-a^ (260) must represent a right line, since it is linear ; also it is satisfied by the co-ordinates of each of the given points ; it must therefore be the equation of the chord. Equation (260) may be written 2^-(ai-t-a2)a-(7i-f72)/3-Oi + ^2)7-2X-aia2-}-to + to = 0; (261) EQUATION OF A CHORD. 117 which may be readily thrown into the homogeneous form (Art. 4). {S), /3y — Jca =z 0. Here the chord will be represented by the equation 0-^0(7-72) + (^-/32)(7-7i) = 2^y-2ka, (262) or, 2Jca-{y, + y2)/3-(^i + ^-^y + (/S^ys + ^,y,) = 0, (263) for (262) is evidently linear and is satisfied by (a^, /3i, y^) or by (02,^2572)- The equation just found may be rendered homogeneous by the method of Art. 4. As in the last section, the equation of the chord of {T) is easily seen to be (/3-/3i)(7-72) + (^-/32)(7-7i) = 2/3y-2F, (264) which is also not in the homogeneous form. In this case the linear equation (/3 - /3i)(/3 - ^2) = iS^ - lea (265) is satisfied by the co-ordinates of the given points on the conic ( 17), and therefore represents the chord which joins them. It may be written -^a - (/3i + ^2)^ + /3i^2 = 0. (266) 118 TRILTNEAR CO-ORDINATES. 210. Equation of the tangent at the point (ai,/3i, yi). The equation of the tangent may either be deduced from that of the chord (Art. 209) by making og ■= oi, ^2 = /^i, 72 = yi ; oi" he formed after eqq. (152), (153). If the latter method be adopted, it must be remembered that the form (153) may only be used when the equation of the conic is homogeneous, and that in other cases (152), the general equation of the tangent, must be employed. (iV). X/3y + Mya -\- Na^ = 0. Putting 02 = ai, ^2 = /3ij 72 = 7i in (249) the equation of the chord, we have 4a + ^^ + Ty"=0, (267) «! Pi 7i which is the equation of the tangent at (ai,/3i, yi). Second method. — The equation of the tangent is (Art. 129. (153).) ©.+{l)»+©'=«. (-> and, in the present case, (|3 = ^. + L,., (269) hence, (Myi + m,)a + {Na, + Ly,)^ + {L^i + Ma,)y = (270) is the required equation. EQUATION OF THE TANGENT AT ANY POINT. 119 It is obvious that (267) and (270) may be derived the one from the other by means of (N). Making a^ = ai, /Sj = /3i, 72 = 71, in (251) we get for the equation of the tangent at (ai, /3i, 71) Z^ M^ N^ — a + -r^4-— 7 = 0. (271) The second method will give the same result, since daj aj- ' \d^x) ~ ^j ' W71/ ~ y2 (272) If the equation of the conic be taken in the form iV + M^^^ -f NY-2MN^y-2NLya-2LMa^ = (153) gives the equation of the tangent in the form (273) (p). iv + M'^' + ivy =: 0. The equation of the tangent, obtained by either method, is Z\a + M%^ + N'y.y =0. (2 74) (i2). ^7-X^a2 = 0. The equation of the tangent, derived from (259) Art. (209), is 2Jcaia — 71/3 — /3i7 = 0. (275) 120 TRILINEAR CO-ORDINATES. Also, since the equation (268) gives exactly the same result. If the equation (261) be employed, the equation of the tangent will be obtained in the non-homogeneous form 2^aia — yi/3 — ^,y — ha\ + ^^y^ — 0, which, however, reduces to the form (275) by reason of {R). {S). ^y-Jca = 0. The equation of the chord (Art. 209. (263).) gives for the equa- tion of the tangent at {ai, /3i, y^) Tea — yi^ — /3i7 -{- ^^y^ = ; or, since by {8) ^^y^ = Jca^, (277) k{a + a,) - y,^ - ^,y = 0, (278) which (Art. 4) may be rendered homogeneous, if necessary. Second method. — The equation of the tangent at (a^, /3i, yi), (Art. 129. (152).) is 'iy-"^ + iw}^-^^^ + (S)(^-^')= °' (2^^) dcf) di therefore, since the required equation is Jc{a - a,) - y,{^ - ^,) - ^,{y - y,) = 0, (281) EQUATION OF THE POLAR OF A GIVEN POINT. 121 which, when jd^y^ is replaced by Jca^ (277), is identical with (278). In this case (g) = 0, (|J = ,„ (gj = /3, and the equation of the tangent (Art. 129. (152).) is "7i(/3-/30 + /3i(7-yi) = 0, (282) or, since, by (T), jS.y^ = ]c\ yi/3 + Ay = 2R (283) and (152) becomes - lc{a-a,) + 2^,(^-/30 = 0, (284) or, by ( U), l{a^ a,) - 2/3,/3 = 0. (285) 211. Equation of the polar of the point {aj, ^i,yi). From the results of Arts. 134 and 135 it appears that the equa- tion of the polar of the curve (/)(a, ^, y) = 0, when the equation is homogeneous atid of the second degree^ is ^ ©•+(l)"©'=»' « identical with the equation of the tangent at the point (ai, /3i, yi) ; and is 122 TRILINEAR CO-ORDINATES. (287) when the equation of the curve, though not homogeneous, is of the second degree. Hence tlie equation of the polar may be readily formed in the case of any of the curves we are considering. It may also be deduced from the equation of the tangent by the method which was applied in the general case (Art, 134). Examples of each method are subjoined. {N). L^y + Mya + Na^ = 0. c. n . . . , s , , o l^4>\ (^4>\ (^^\ Substituting m (286) the values of I y- U I T5- I' \-i~ I given m (269), we have, for the equation of the polar, (Jiyi + iV/3i)a + (iVai + Lji)^ + {L^^ + Ma^)y = 0. (288) Second method. — Suppose (03, /Sg, yg), (og, /Sg, yg) to be the points of contact of tangents drawn through (oi, /3i, yi) to the conic. The equations of the tangents (Art. 210. (270).), since (ai, /3i, yi) lies on each, give {My. + m,)a + (iVas + Ly,)^ + {L^, + Ma^)y = 0, and (Jfyg + Nl3,)a + {Na, + iyg)^ + (Z/Sg + Ma3)y = 0. Hence the points of contact both lie on the line {My +iV^)ai+(iVa + Xy )^i + (i^ + Ma )yi = or {My, + m,)a 4- {Na, -\- Ly,)^ + {L^, + Ma,)y = 0, (289) and (289) is, therefore, the equation of the polar. It may be shewn, in a similar manner, that the equations of the EQUATION OF THE POLAR OF A GIVEN POINT. 123 polars of the given point with respect to (O5), (P) and (it) are identical with (273), (274) and (275), respectively. Again, if we take the non-homogeneous equation we have, by (287), -k{a-a{) +71O-/3O + /3i(7-7i) + 2^,y,-2ka, = ; or, Ka + a,)-y,^-^,y=0; (290) the same as (278) the equation of the tangent at (ai, j3i, -yi). Second method. — If (02, /Ss, 72)? ("s^ j^sj 73) be the points of contact of tangents through the given point, we have (Art. 210. (278).), since (a^, i3i,7i) lies on each tangent, ^("2 + ai)— 72^1— /327i = 0, and ^(03 -|- ai) — 73^1— /3371 = 0. Hence, k{a +aO -71/3 -/3i7 =0 is the equation of a straight line on which both points of contact lie, and therefore represents the polar of (ai, /3i, 71). In the same way the equations of the polars of the point with respect to (T) and {U) may be shewn to be (283) and (285) respectively. 212. Condition that the straight line {I, m, n) should touch the conic. The required condition may be obtained either, as in Art. 131, by comparing the equation of the given line with that of the tangent (Art. 210), or by the direct application of the results of Art. 131 (Eqq. 155, 157); or, again, by combining the equations 124 TRILINEAR CO-ORDINATES. of the straight line and curve (the latter in the homogeneous form,) and expressing the condition that their points of intersection should be coincident. The following are examples. (iY). L^y + Mya + iVa/3 =: 0. Eliminating a between (N) and the equation /a + W/3 + Wy = 0, (291) we get Nm^^—{ZI—M7n—Nn)^y + Mny^ = ; . , . . 8 . which will give coincident values for -, if y -{n—Mm—Nnf—iMNmn = 0. This, therefore, is the required condition ; it may be written in either of the forms, ZH' -f MV + N^n^-2MNmn-2NLnl-2LMlm = 0, (292) or {Ziy + {Mmy + {Nnf = 0. (293) Second method. — Identifying (291) with the equation of the tan- gent at (ai, /3i, 7i) (Art. 210. (270).), we get My, + iV/3i JVa, -f Zyi Zl3, + Ma, ^ . . — - — -^ = = =r — X (say) : therefore, Nl3, + My, J^ IX =0, Na, + Zy, + m\ = 0, Ma, + Z^, Jf.nX = 0. Also, la, -\- mjS, + ny, = ; CONDITION OF TANGENCY. 125 and from these, by the elimination of ai, ^i, yi, and X, we get for the condition of tangency, 0, iV, M, I N, 0, L, m M, L, 0, n /, m, n, = 0. (294) If the determinant be expanded and the sign of the whole changed, it may be seen that this result agrees with that before obtained (Eqq. (292), (293).), and might have been written down at once as 'A*" = 0, or (/>(/, m, ^)' = (Art. 131. (155), (157).). (^5). X^a^ + M^^^ 4- N^y^ — 0. A comparison of the equation of the tangent (Art. 210. (271).) with (291) gives IP X 5 I m n and substituting these values for oi, /3i, y^ in the equation lai + m^^ + Wyi =r 0, we get for the required condition, L M N -7- + — + — I m n 0. (295) (i^). XV + M^/s' + ivy = 0. The condition of tangency may be at once written down from (155) Art. 131 in the form i:\ 0, 0, I 0, IP, 0, m 0, 0, IP, n 1, m, n, = 0; (296) 126 TRILINEAR CO-ORDINATES. or obtained, from (274), as in the last case, in the form (297) (i2). /3y - ha^ = 0. Identifying (291) with the equation of the tangent Art. 210. (275), we get I m n whence, since /Siyi = 7^1, we have for the condition sought V - 4Jcmn = 0. (298) {S). Py — la = 0. The equation of the tangent to (>S) at the point (ai, jS^, y^), written in the homogeneous form (Art. 210. (278).) is la^ - -y 7ij/^ + (^«i - -[. ^ijy = 0- (299) Eeplacing 2>S' in the coefficient of a by aa^ -f J/3i + cy^ (Art. 3. (1).), and comparing (299) with (291), we have 2aai + h^i + Cyi I , 2>S^ 2S m ^— X (suppose). Whence, eliminating oj, /3i, y^, X, we get, as in (iV) of this Article, 2a, h, c, I J. n 2>S^ 0, 0, — y, m 28 c, — 7-, 0, n I, m. 71, = 0. (300) CONDITION OF TANGENCY. 127 If we make use of (157), and remember that, in this case, ^(a, /3, y) = aa' + 0^^ + 0/-2|/3y + 2^ya + 2|a/3 =r 0, IC Ji Ji we shall have (301) so that the condition is __ + (..._ J,)^_ __ ^_ _ __ _ _j ^ ; (302) a result which will be found to agree with (300). {T). /3y-F = 0. Here ^{a, ^, y) = C?c? + W + ^V + ^(^^ - ^ )^^ + ^^"^^ + ^'^^"^' and the condition of tangency (157) becomes (f + 4^ + «/»«(2-^-^) = 0. (303) In this case *(^,»,»)'= _|^,_(M_« + *')„.+2(^)„„-2(f)«/. (304) and the condition is that this quantity should equal zero. 128 TRILINEAR CO-ORDINATES. 213. To find the co-ordinates of the centre. Let, a, jS, y be the co-ordinates of the centre. This point (Art. 139) is the pole of the straight line at infinity whose equation is aa -\- l^ -{■ cy =^ 0. (305) W- Z/3y + Mya -f iVajS = 0. The equation of the polar of (a, /3, 7) with respect to this conic (Art. 211. (289).) is {My-\- W^)a + (iVa + Zy)(3 + (Z^-f Ma)y = 0. Identifying this equation with (305), we get My + ^'^^ Na-^Ly L^ -\- 3Ia ^ r s whence, proceeding as in Art. 137, we find a _ /3 y N,M,a 0, if, a 0, iv; a 0. iv;jz 0, Z, I — N,L, I ]sr,L,'b — iV, 0, z Z, 0, c M, 0, c M, Z, c M,Z, that is, a T M jsr (306) — Zflj -f Jf^ + iVc Z^ — if^ + iVc La-^Mh — Nc -2S (307) ZV + M'b^ + A^V - 2 Jf A^Jc - 2]VZca - 2ZJfo5 (Prelim, chap. (^).)' equations which completely determine the centre. CO-ORDINATES OF THE CENTRE. 129 ( 0,). m + if2/32 + iVy - 2Mmy - 2NLya - 2LMa^ = 0. Comparing the equation of the polar (Art. 210. (273).) with (305), we get Z(-Za-\-M^ -^Ny) _ Jf(Za- Ji"/3"+ Ny) _ ■2\^(Za + i/^-iVy) ^ a b c ' whence, proceeding as in the last example, we have finally a _ "^ _ 7 _ S Mc -\- Nb ]Sfa-\- Lc Lb + Ma Lbc + Mca + Nah (308) (p). zv + ip^^ + Ny = gives, in a similar way, a __ /3 _ 7 _ 2S a b G ^^ _i_ ^^ i_ '^^ -ji ^ jj^2 ;^ + if2 + jv^ (309) {R), and /3y — ha^ — 0,; a _ ^ _ y _ >S^ (310) 2/^ 4^ ~ ^"^ (/S). iSy - ha — 0. Here, comparing , - - - (- 2S-\ f - 2S-\ (2«a + 5^ + Cy)a + [ba- -j y j ^ ^ i^ca- ^ ^ jy = 0, the homogeneous equation of the polar, with (305), we find a /3 7 k 2Sa — ca — ab a' K (311) 130 TRILINEAK CO-ORDINATES. Similarly, in the two next examples, the subjoined results are obtained. a A— —1— ^ T ~ — (* — ;3i3) 214. Condition that the conic should he a parabola. Since (Art. 131. Cor.) the straight line at infinity is a tangent to every parabola; and, again, (Art. 139) the centre of every parabola lies on this line ; the required conditions may be deduced from the results of either Art. 212 or Art. 213. They are seen below and will be found to agree with those given in Arts. 185—206. {N). L^a" + JIf' 5' + Nh''-2Mmc-2NLca-2LMah = 0. (^)- '"-4^. (S). Impossible {the curve being always hyperbolic). {T). Impossible {the locus being always a hyperbola). L a + M h + N _ c 0. + b^ h = + a^ A 1 0^ _ JV2 :0. CONDITION FOR A PARABOLIC OR CIRCULAR LOCUS. 131 ( U), The condition is satisfied for every value of h [and the curve is always a parahola). 215. It appears also from Art. 213 that the vertex A of the triangle of reference is the centre of the conic {T) (see Art. 205). 216. Conditions that the conic should he a circle. Applying the conditions (194) of Art. 149 we obtain the fol- lowing results. a p r The conditions are M N N L L M sin G %vsiB ^vnA sin C sin jB sin A whence we easily deduce L M' N sinA s\nB sin C Hence (314) sm^ sm^ s\nC , . a /3 7 -^ + 1 + -1 =0. (316) a /3 7 is the equation of the circle descriled alout the triangle of reference. ( Og). XV+ir'/32H-iVy-2JfiV^7-2iVZya-2Xifa^ = 0. In this case the condition becomes if sin C-fi\^sin B = i\^sin ^+Z sin C = Z sin ^+ if sin A ; 132 TRILINEAR CO-ORDINATES. whence we shall get cos^— cos'— cos^— and iJie equation of the inscriled circle will he (317) cos a^ + cos — /3 +COS — y =0. (318) (P). iv + jf 2^^ + ivy = 0. The conditions of Art. 149 become, in this case, if 2 sin 2 C^N^ sin ^B=N^ sin ^A^L"" sin ^ (7 = Z^ sin ^^-f if ^ sin ^.4, and give Jf2 ^S (319) sin 2^ sin 2^ sin2C and the equation of the self- conjugate circle is sin 2 A c? + sin 2^/3^ + sin 2 Cy^ = ; (320) or, « cos ^a^ + J cos B^^ + c cos Cy' = ; (321) which will not, however, represent a real locus (Art. 194) unless one of the coefficients be negative, that is, unless the triangle of reference le ohtuse-angled, (i2). Py — ka" = 0. Here, the conditions (Art. 149) are sin Z sin C = ksin^C =z k sin ^B ; whence, k = 1, and the triangle of reference is isosceles. (322) 133 EQUATION OF A CONIC REFEHRED TO TWO TANGENTS AND THEIR CHORD OF CONTACT. 217. Let the equation of a conic Fig. 37. referred to a pair of tangents and their chord of contact (Art. 174. (G).) be Then, if X M 1 ■■' be the equation of any chord PQ, through the intersection of the fixed tangents, we get, by combining (323) and (324), the equations L R M to determir^ its points of intersection with the conic ; the upper and lower signs referring respectively to the points F and Q, which lie on opposite sides of P = 0. 218. The chord PQ may be denoted briefly by y?, and the points P and Q (after Dr. Salmon's notation) by -f- /a and — jtx respectively. 219. The reader will observe that the equations represent the pairs of lines which join {ZR) and (MR) respectively to P and Q. Similarly, any pair of straight lines through (LR) and (MR) whose equations are of the form 134 TRILINEATl CO-ORDINATES. It = IcL, M = hR, will intersect on the curve. 220. Any point whicti does not lie on tlie curve may be denoted by a pair of equations of the form {R — hL,M=lcR). 221. To find the equation of the chord which Joins two given points on the conic LM-=z. W. Let /xi, /Li2 (Art. 218) be the two points. At these points, as in Art. 217. (325), we have, respectively, and L 1 _ ^ Ml L _ ^ 1 M2 2 Ml 2 ' M2 (326) (327) Suppose the equation of the chord to be IL\ rR + mMr= 0, (328) Since /nj, ^^ both lie on this line, we have, by (326) and (327), ? + ^>i + '^A = 0, (329) ^ + ^M2 + ^¥2 = 0. (330) Hence, eliminating I, m and r between (328), (329) and (330), we get for the equation of the chord fiifi^ Z, R, M Ij Mi» m! 1> M2> Mi = 0; (331) which, when expanded and divided through by /xj — fji2, becomes H^fji^L — {fxj^ + n.)R + if = 0. (332) CONIC REFERRED TO TWO TANGENTS AND THEIR CHORD. 135 222. To find the equation of the tangent at any point fi^ on the conic LM=^ R^. Making /uo := /xj, in the result of the last Article, we see that the equation of the tangent at /^i is lx\L — 2fx,E + M= 0. (333) 223. Similarly the equation of the tangent at — fti (Art. 218), the other extremity of the chord fil, is fjilL + 2fXiE + M=iO. (334) 224. To find the polar of a given point with respect to the conic LMzrz R^. Suppose the given point (Art. 220) to be the intersection of the pair of lines {R — hL,R — hM), (335) and let ^i^ be the point of contact of one of the tangents through this point. The equation of the tangent (Art. 222. (333).) is lxlL-2fXiR-\-M=0. Since therefore the point (335) lies on this line, we have, by substitution Icfxl — 2hkfii + h—0. (336) But at the point of contact jxi we have (Art. 221. (326).) ^ = ^ = ^; (337) 1 Ml K and, eliminating fi^ between (336) and (337), we find for the equa- tion of the polar at the given point hZ-2hJcR'\-JcM=iO. (338) 136 TRILINEAR CO-ORDINATES. 225. If from a given point two straight lines he drawn cutting a conic in the points P^ and Q^, P^ and Q^, respectivehj ; the tivo pairs of chords which join^ directly and transversely, these four points, will intersect on the polar of 0. Let i= 0, il/= be the equations of the pair of tangents which can be drawn to the conic through the given point : and Fig. 38. let jS = represent the polar of 0. The equation of the conic referred to these three lines will be LM-R^ 0; (339) and, if the straight lines OP^Q^, OP^Q'^he represented by n\, til. respectively (Art. 218), the points P^, Q^ will be denoted by + ni, and P.2, Q2 by ± fx^, respectively. Hence (Art. 221. (332).) the chords P1P2, Qi Q2 will be represented by and the transverse pair, P1Q2, i^sQi^ by -fi,fi,L±{fx, - fx,)P ^M=0. (340) (341) HARMONIC PROPERTY OF THE POLAR. 137 From the form of these equations it follows that the two pairs have their points of intersection on ^ =: 0, the polar of 0, For (340) shews (Art. 16) that F^Q^, F2Q2, both pass through S, the point of intersection of ^ = with M-\- ni^^L = ; and from (341) it appears that F1Q2, P2Q1 both pass through T, the point of inter- section of It=zO and M—ii-^n^L = 0. 226. Hence also it follows that the equations of OS, OT are respectively if + /xi/iaL = 0, (342) M— fiifjL2L = 0. (343) 227. Therefore also OS, OT form with OPi, OF^ a harmonic pencil (Art. 95) ; for the equations of these four lines are L = 0, M=0, M+^^fx^L — O. 228. Again, from the form of the equations (340) it appears that the four lines SO, ST, SF^, SQi form a harmonic pencil. There- fore the chords FiQi, F2Q2 are cut harmonically by the point and its polar ; and we have the well-known Theorem ; — FJvery chord of a conic is harmonically divided ly any point on it and the polar of that point. This is also evident from the form of the equations F = ± f^iL, (344) F = ± JU2L, (345) representing (Art. 219) the pairs of lines which join the point of contact of one of the fixed tangents to the extremities of the chords FiQi, F2Q2 respectively. For (344) and (345) each form a harmonic pencil wdth Z =: 0, ^ := 0, and, consequently, the transversals OFiQi, OF2Q2 are harmonically divided. 138 TRILTNEAR CO-ORDINATES. 229. The poles, with reference to a given conic, of straight lines which pass through a fixed point, lie on a fixed right line. Let the conic be referred to any pair of tangents and their chord of contact, and let its equation be LM=R\ (346) Also let the straight line the locus of whose pole with respect to (346) is to be found be represented by the equation IL — 2pR + mil — 0, (347) in which p is indeterminate (Art. 16). Suppose {R = hZ, R = IM) (348) (Art. 224. (335).), to be the pole of (347) ; then (Art. 224. (338).) the equation hL — 2hhR + hM= (349) must be identical with (347). Comparing them, we get 4- = - = -. (350) I p m But at the pole we have, by (348), L R M , . hence, substituting for h, Ic, from (350) in (351), we get m p / ' and the locus of the pole of (348) is a straight line whose equa- tion is IL - mM= 0. (352) GEOMETRICAL THEOREMS. 139 230. We shall conclude this Chapter by calling the attention of the reader to a few geometrical theorems, which are involved in the forms of equations investigated in the preceding Articles. Thus from the forms of (^), (/S^), {T) and {JJ) respectively we deduce the following; — The product of the distances of any point on a conic from a pair of tangents hears a constant ratio to the square of its distance from their chord of contact. If at the extremities of any chord of a hyperhola parallels he draivn to the asymptotes, the distance of any point on the curve from this chord is in a constant ratio to the product of its distances from these two straight lines. The rectangle under the distances of any point on a hyperhola from the asymptotes is constant. The square of the distance of any point on a parahola from a diameter is proportional to its distance from the tangent at its extremity. Again, (316) may be written in the form a^y + hya -\- CajS = ; and, if P be any point on the circumscribed circle and Q, H, S the feet of the perpendiculars from P upon the sides of the triangle, this equation asserts that the algebraical sum of the triangular areas FUS, PSQ and PQR is r= 0. Hence Q, R, S are in the same straight line ; and we have the following theorem ; — If from a point on the circumference of a circle perpendiculars he drawn to the sides of any inscrihed triangle, the feet of these perpen- diculars will lie in one and the same straight line. Also from (322) it appears that if a circle he descrihed touching the sides of an isosceles triangle at the extremities of the hase, the rectangle under the distances of any point on the circumference from the two sides is equal to the square of its distance from the hase. 140 CHAPTER IX. THE CIRCLE. 231. To find the equation of the circle described about the triangle ABC. We may assume the equation Fig. 39. (Art. 180. (iY).) to be of the form L^y + Mya + Na^ = 0. (353) The equation of the tangent at C (Art. 181. (234).) gives for any point P upon it M Therefore (Euc. iii. 32), when (353) represents a circle, L a PR PC s'm A sin A M~~ ^~ PQ~ PCsinP' ~ sin^ we have, by symmetry, L M N sin A sin P sin C Hence (353) becomes sinA^y -\- sin Pya-{- sin Ca^ = 0, or, a^y + hya + Ca^= 0. (354) (355) THE ClilCUMSCRlBED AND INSCRIBED CIRCLES. 141 232. To find the equations of tlie inscriled and escribed circles. The equation must (Art. 187. (^5)-) be of the form {Laf + {M^f + {Nyf = ; (356) and the equation of the straight line which joins C to the point of Fig. 40. contact of the opposite side (Art. 187. (238).) gives, for an^/ point upon it and therefore for the point of contact D, L _^_I)R LA^inA Hence, by symmetry A A OB cot — sin A cos^-- ^ 2 On cot — sin B cos^— - ^ 2 M N .A B cos^— cos^— C cos and the required equation is, by (356), A ^ B ^ (7 X cos —a^ -i- cos -^ /3f_ + cos -^y2 _ Q. (357) 233. Similarly the equations of the escribed circles may be shewn to be COS y(-«)' + smY/3' + sm Yy=^ = 0, 142 TRILINEAR CO-ORDINATES. Bm—a-" + COSy(-/3)2 + 8111^7' =0, (358; sm 7? n 234. It will be seen tliat the equations just obtained for the circumscribed and inscribed circles are the same as those given in Art. 216. A full discussion of them here would be superfluous, since all that was said with regard to the conies (iV) and ( 0-^ in the two preceding Chapters applies, mutatis mutandis^ to these circles. A few, however, of the more important results there ob- tained will be given in the present Chapter under their modified forms. For the sake of clearness we commence by explaining the notation employed. 235. R, 0^; A, B, C r, 0^ ; A„ Br, C,. ^a? ^a '■) Aa, Ba, 6^ Ti, Ob ; Ai, Bi, Cj, ^CJ Oc } Ac, Be, Cc rs, Or, r^, O9; Ai, Bi, 61 -^2? -^2? ^2 <5j -Si !>. * A ■^i -f ■^ =0" !* ■^ ?§ M^ 5^ • <<> (?i <%i ^S s ^ 5g ";i < ^ ;-i ^ ^ ^ g '^ ' -^^ ■^i t^~i -f«i !»-;> <;:i ^ QJ "^ i^ v" 00 .^s V-i <«s -S C ■^i S 0) ^ -Si fn =0 "^ -*-J -Si. S^,. =: 0, . . hiBcribed /S'a :=0, . . escrihed{onBC) . Sb =:0, . . escribed (on CA) . Sc =■0, . . escribed {on AB) . Ss =0, . . self -conjugate /S^9 = 0, . . nine-point Similarly Sabc ^^rbc ^arc ^abr are the circles described through 0«, Ob, Oe ; Or, Ob, 0, ; Oa, Or, 0, ; Oa, Ob, Or, respectively. Also, A,, B„ c; -^2i B^, C/2 are the middle points of the sides, feet of ±J' from the vertices ; THE CIRCUMSCRIBED CIRCLE. 143 and Pi ^1 G,. is the iiitersec Ga tion of G, G. Gahc 236. i^^)' a^y the _L''* at the middle points, the ±J'^ from the vertices. A A,, BB,, CC,. AA,, BB,, CC,. AA„ BB„ CC,. AA,, BB,, CC,. AA,, BB„ CC,. a^y -\- lya -f- Ca^ =: 0. Suppose a triangle A'B' C to be formed by drawing tangents to the circumscribed circle at the points A, B, C. Then it will ap- pear, as in Art. 181, that its sides, whose equations are and h + G — = 0, y c + a a = 0, a a M- b = 0, (359) meet the opposite sides of the original triangle in points which lie on the straight line JL + 4 + JL=0; a c (360) also Art. 182. (236).) AA', BB', CC are represented by the equations 144 TRILINEAR CO-ORDINATES. ^ - i = 0. b -^-- = 0, (361) c a ^ and JL _ 4. = 0, a and (Art. 183. (237).) meet in a point (ao, jSq, 70) such that ^^ ^ — yi— ^^ (362) ^ b ~ c a' -[-P+ C' ^ 237. For the equation of the chord joining the points (a^, /Si, y{), (as, ^2, 72), we shall have (Art. 209. (249).) ^« + _M. + _fZ_ = 0, • (363) aifls ^1/^2 7172 and the equation of the tangent at (a^, ^Si, y^) will be - + ^|+fV^O; (364) «i Pi 7i while (Art. 212. (293).), for the condition of tangency of {I, m, n), we have {alf + {hnf + (c;i)* =r 0. (365) 238. The centre Oji of the circumscribed circle is given (Art. 213, (307).) by the equations a ^ y 2S cos A cosi^ cos C acosA -\- b cosi/ + c cos C ahe = R, (366) and therefore, as may easily be shewn, coincides with P^. THE INSCRIBED CIRCLE. 145 239. \^*)' COS — a2 4- cos— /3^ + cos-^7^ == 0. It may be shewn, as in Art. 187, that the equations of AA^., BBr, CCr are -/3-cos^- COS^ — /3 — COS^ — y = 0, a A ■ cos^-^y — cos^-^a = 0, (367) cos -^a — C0S'-^/3 ^ 0, respectively, and that G^, their point of intersection, is given by the equations ABC cos^Y^o^cos^Y^o^cos^Yyo* (368) or, a{s — a)aQ = h{s — h)^Q = c(5 — 6?)7o. (369) 240. The equation of the tangent at any point (oi, ^xj yO (Art. 210. (271).) is I A IB 1 C ^ ,^^^, — cos — a + — COS — ^ + — cos^y = 0, (370) and the condition of tangencij for {I, m, n) (Art. 212. (295).) is i_cos2-:^ + i-cos2^ + -icos2^=r0; (371) I 2 ^ m 2 ^ n 2 ' ^ ^ or ^(^ - ^) + ^^' - ^^ + ^(^ - ^) == 0. (372) ' I 7n n k 146 TRILINEAR CO-ORDINATES. 241. Again, 0^ the centre of the inscribed circle (Art, 213. (308).) J is given hy the equations a = ^=y=:- = r, (373) s and therefore (Art. 20. (10).) coincides, as we know from the geometry to be the case, with the point of intersection of the bisectors of the angles of the triangle. 242. ITie centre 0^ of the circle escribed on BC (since it is the in- tersection of the lines j3 — 7 = 0, y-|-a=rO, a-l-^ = 0)is given by the equations -^ c — a S S-- a 243. = ^«. (374) {S,). a cos Aa + b cos B^'^ -j- c cos (7/ = 0. The equation of the tangent at any point (a^, /Sj, y^) on the self- conjugate circle (Art. 210. (274).) is a cos Aa^a + I cos ^/3i/3 + C COS Cy^y = 0, (375) and the condition oftangency for (/, w, ^) (Art. 212. (297).) is P +_^ + ^!l^=.0. (376) a cos A b cos B c cos C 244. Also (Art. 213. (309).) 0„ the centre of the self- conjugate circle, is such that — — — 2 S cos ^ a = cos ^/3 = cos (7y = a sec A -\- b sec B -^ c sec C = 2i2cos^cos^cos C, (377) and therefore (Art. 25. (18).) coincides with P^- 147 THE NINE-POINT CIRCLE. 245. Theorem. — In any triangle AB C, the feet of the perpendiculars from the vertices {which meet in P^, the bisections of the segments Ap2, BP2-> GP%-, cf'i^d the middle points of the sides, are nine points ivhicli lie on the same circle ; and this circle touches the inscribed and the three escribed circles of the triangle. (!N^ouvelles Annales de Matliematiques, 1842). The following general form of this Theorem is given by M. Terquem in the same paper. If through the vertices of any triangle AB C, inscribed in a conic, three straight lines be draivn conjugate to the opposite sides % these three lines intersect in a point P ; and the three points in which they meet the opposite sides of the triangle, the bisections of the segments AP, BP, CP, and the middle points of the sides, are nine points which lie on a second similar and similo/rly situated conic : also this conic touches an inscribed conic which is similar to the given one and similarly placed. The equation of the nine-point circle will be found hereafter (Arts. 253, 254). 246. To shew that the equation of a circle may always be written in the form a^y -l" bya -f Ca^ = {aa -\- b^ -\- Cy) {la -\- m^ -\- Uy) J I, m, n being arbitrary constants. The general equation of the second degree (^(a, iS, y) =z Aa' + ^^3^ + Cy' + 2i>/3y + 2Uya -f 2Fa^ = may, since aa-\- b^ -\- cy = 2S, be written in the form Abc{b^ -\- cy-2S)a + Bca{cy + aa-2S)^ -\- Cab{aa-{-b^-2S)y — 2abc{P^y -\- Eya + Fa^) — 0, ^ straight lines are said to he conjugates with respect to a conic when they are parallel to conjugate diameters. 148 TUILTNEAR CO-ORDINATES, or, a^Bc"^ CP-2mc)^y-\-h{ Ca^-\-A(P-2i:ea)ya+c{AP+£a^-2Fai)a^ -2.,.s{f + f + ^)=0. (378) But, if i/3y + 2U,ya + 2F,a^ = 0, A^a' + B,^' -f C^y' + 2D^y + 2^oya + 2F,a^ = 0, (388) 150 TRILINEATl CO-ORDINATES. the equation of their radical axis (Art. 246. (381).) will be the values of Jc^, h^ being similar to that which h has in the Article referred to. 251. Suppose the given equations to differ only by a constant term, or, which is the same thing, by a multiple of aa ■\- l^ -{• cy. The circles, in this case, are concentric^, and we get on sub- traction {aa + J/3 + cyf = ; which shews (Art. 249. (386).) that both chords of intersection are imaginary and coincide with the straight line at infinity. Hence it is to be inferred that concentric circles touch each other in two imaginary points at infinity. 252. To find the equation of the circle inscribed in the triangle AB C. Let it be assumed (Art. 247. (383).) to be a^y + lya + Ca^ — {aa -f J/3 + Cy) {la + W/3 -\-ny). (390) ^ For the centre of the conic , \da j + 2ka \d0 J fd^, \dy \ + / c 2ke a ~ Therefore, \da J \dy / ale and the conic is concentric with the conic ^^(a, y3j 7) = 0. EQUATION OF THE INSCRIBED CIRCLE. I5l Since the circle passes through A^, (0, (s — S^^, Sa-, Si and So successively. 257. Fig. 41 exhibits the relations in which these circles stand to one another, and the position with respect to them of certain important points connected with the triangle of reference. Thus Oji, Gi, Oq and Pg ^i"© collinear, O9 being the bisection of P^Oji, and GiP2 = 2GiOji. Again 0^, G^ and G^u are collinear, and G^G^^c^^^ 2GiO,.. Hence also PiG^u is parallel to 0,.Ojj, and P^Gau = 20,0^. 258. The centre of the nine-point circle is manifestly the intersec- tion of perpendiculars to the segments A1A2, B^B^, Cx C2, at their middle points. If therefore a, /3, y be the co-ordinates of O9, we shall have (Art. 238. (366).) and (Art. 244. (377).), — 1 7? a = :^ (i2 cos ^ + 2i^ cos B cos C) = -^cos [B— C), and therefore, by symmetry, 'l_ _ ^^ y P^ cos{B-C) ~ cos{C-A) ~ cos{A-B) ~ 2 (408) 259. To find the equation of the circle which passes through 0„ Oi, 0, (Art. 235). Let the equation be assumed to be «/3y -f hya + Ca^ — {aa + J/3 + Cy) (la + m^ -{- ny) = 0. (409) CIRCLE THROUGH THE CENTRES OF THE ESCRIBED CIRCLES. 157 At Oa we have — a = ^ = y = r«. Therefore a-h-c-{a—h-c){l-m-n) = 0, by (409) ; whence, I — m — n =i 1. Similarly, — I ^ m — n = 1, and — I — m -{• n =: 1. These equations give 1= m z=zn= — 1, (410) and, substituting these values for I, m, n in (409), we get «/3y + lya + Ca^ -f {aa + ^iS -f Cy) (a + jS + y) = (41 1) for the equation of S^hc 260. To find the equation of the circle which passes through 0,., 0^, Oc. Proceeding as in the last example, and substituting in (409) the values of a, jS, y at the points 0,., Oj,, 0^, successively, we get to determine I, m, n, I -\- m -\- n =: 1, — I -\- m — n z=z I, and — I — m -\- n = 1 : whence, — I =: m = n = I, (412) and we have for the equation of S^ic ■ a^y -\- bya -{-Ca^-\-{aa + h^-\-Cy) {a— ^—y)=0. (413) I 158 TRILINEAR CO-ORDINATES. 261. The equations of some of tlie more important circles are collected in the subjoined list. For the sake of shortness 2>S' has been substituted for aa-\- hj3 -\- cy. For an explanation of the notation employed the reader is again referred to Art. 235. 28 (Sr). a^y-\-hya+Ca^ - ^^{S^ {S,). a^y+hya+Ca^ -—{as'a-\- h{s - cf^ + c{s - hf y} = 0. (Sj,). a^y-\-hya-\-ca^ r {(i{s — cf a -f- S-s'^/S + c{8 — afy\ = 0. 2 S {8^. «/3y+ Jya-f ^a/3 - -^ {«(s - IJ a + l{8 - of^ + cs'y') = 0. {Sq). a^y-\-hya-\-Ca^ — S {cos^ a + COS^/3 + COsCy} = 0. (S^). a^y-\-hya-\-Ca^ — 28 {cOS^ a + COS^/3 + COsCy} - = 0. (>S,0. a^y+hya + Ca^ ^ 28 {a -\- ^ -^ y) = 0. (8rbc)' a^y-{-hya-\-Ca^ -{- 28 {a ~ ^ — y) = 0. {8arc)- al3y-\-hya-\-Ca^ ^ 28 {^ - y - a) = 0. {Sabr)' a^y-\-hya-]-Ca^ + 2>S (y - a - /3) = 0. 262. The radical axes of these circles taken in pairs, are repre- sented (Art. 249) bj the following equations : — {8^) and {8r). a^s-afa^ i{s-hy^ + c{s-cyy=z 0. {Sm) . . . {8;). as'a+h(is-hy^-\-c{s-cyy = 0. (>S'jt) ...(89). cos^a-f cos -5/3+ cos(7y = 0. (>Sjt) ...{8,). cos^a4- cos^i3+ cos (77 = 0. EQUATIONS OF RADICAL AXES. 159 h-\- e T aa-\- 0^ — cy=. 0. w . . . {8,). (^«) . . . (-S,). {S,) . . . (So). (^c) •. . . (S,). (>^0 . ■ (8,). {^abc) • . . (Sric)- (>S'a6c) . • i^arc)- i^abc) - ' {Sabr)- i^arc) • • {^abr)- (>^a6r) • . . {S,hc)- i^'^rbe) ' ' ' i^arc)' da + + ^ = 0. b — c c — a Oj — h aa J/3 + cy G c -\- a a-\-}) = 0. ^aa . h^ cy _ ^ b -\- c — a a -\- b da + — —. = 0. b -{- e c -\- a a — b (Sg). COS Aa-}- COS ^^ -\- COS Cy:=0. y -\- a':=: 0, a 4-/3 = 0. ^-y=0. y — a =^ 0. a — ^=0. From tlie above it appears that Sji, Sg^ and Sq are coaxal circles, a conclusion to which we are also led by a comparison of the equation of Sq, which may be written in the form a^sin2^+^2sin2^-f y2sin2(7— 2(/3y sin^+yasin^+a^sin C) = 0, (414) with those of Sjt and S^ (Eqq. 315, 320). 160 TRILIKEAK CO-ORDINATES. 263. The co-ordinates of some of the principal points referred to in Art. 235 will be found below :^ (Or). (ft)- (ft)- COS.^ _ y COS 5 COS C — XD = ^ = y = r. = ^ = y = ra = -^ = y = n — ^ = —y = r,. ^ y _ ^ cos(^-C') cos(6'-^) cos(^-i^) 2 (P2) 01' (^0- cos^a = cosP/3 = COS 6'y = 2i2cos^cosPcos(7. - - - 2^ {Gr)' a{s—a)a=z'b{s—h)j3^=c{s—e)y. an hj3 Cy V abcj ^_^ ^_^ ^_^ 161 CHAPTEE X. GENERAL THEOREMS AND PROBLEMS. 264. A conic is completely determined ivJien five points upon it are given. For the most general form of the equation of a conic, viz. : Aa^ + ^/32 + Cy" + 2D/3y + 2Eya + 2Fa^ — 0, (415) involves six constants, and the five equations which we obtain by successively substituting in (415) the co-ordinates of the given points are sufficient to determine their proportional values. 265. A conic is completely determined when five tangents to it are given. The condition that any line {I, m, n) should touch the conic (415) is (Art. 131. (157).) AT + B'm'' + C'n^ + 2iymn -f "^E'nl + 2F'lm — ; (416) and, if five tangents (/i, m^, Wi), (4, 'nh^ %),... be given, we shall have five such equations, whence the proportional values of A', B', C, . . . may be determined. These being known, the pro- portional values of the coefficients in (416) — the constituents of the determinant A, are also known. This proves the proposition. 266. If from a point two chords ORiR^, OS1S2 he drawn in given directions^ to a curve of the second degree, the ratio of the rectangles under the segments of the chords is the same for every position of 0. IS. 162 TRILTNEAR CO-ORDINATES. For (Art. 13. (8).) tlie distance of (oq, jSo, y^) from the curve, in any direction, is given by the equation (^(X, ju, v)r 4- . . •. . + (^(ao, ^0^ 7o) = 0. Hence, if (Xj, /^i, i/J, (As, /X2j ^2) ^i'© the two fixed directions, Ave have Oi?i . 6)i?2 _ (^(qp, /3o, 70) ^ 0(^0, ^o> yo) 0>S'i . OS2 0(Xi, /^i, J/i) 0(X2, /X2, J/2) _ (^(Xg, /X2> '^2) an expression which is independent of oq, ^q, jq, and is, therefore, constant for every position of 0. 267. To 2Jt'ove that the anharmonic ratio of the pencil formed ly joining four points on a conic to any fifth point on it is constatii. Let JPi, F2, -P3, -P4 be the four given points, and A, B, C any other points on the conic. Take the inscribed triangle ABC for the triangle of refer- ence; then the equation of the conic (Art. 180. {N).) will be of the form L , M , N , , — + — + — - 0. 417 a /^ 7 . ANHARMONIC PROPERTY OF FIVE POINTS ON A CONIC. 163 Let us assume y — 7^1 a = 0, y — Jha = 0, y — 7^^ a := 0, (418) (419) and /3 — 7;i a = 0, /3 — 7)2 a = 0, ^- Jc,a= 0, ^ — 7;^ a = 0, for the equations of the successive rays of the pencils {JB. F,F,F,F,} and { C. F,F,F,F,}, respectively. The point of intersection of the first pair of corresponding rays in (418) and (419) is given by the equations _a_ _ _^ _ y I ~ ki ~ hi' therefore, since Pj is on the conic (417), ^ if jsr ^ Hence /h = Lk, + M' and we should get similar values for h2, \, A4, in terms of 7^2, Ic^, h^, respectively. 164 TRILINEAR CO-ORDINATES. Taking, therefore, one of the anharmonic ratios (Art. 92. (113).) of the first pencil, since hi — ^2 ^^ MN etc. = etc.. ( A/ J "~~ A/o j we get 1\ — ^2 ^1 — ^4 K ^2 ^1 — ^4 ^3 — /i2 h,^ — '^4 "'3 — "'2 ^3 ^4 The equality of the other ratios may be shewn in a similar manner. "We have therefore {B,P,F,P,P,} = {C,P,P,P,P,}, and the truth of the proposition is established. 268. If the centres of the two pencils be made the opposite vertices of an inscribed quadrilateral, the proof of the Theorem of the last Article will be somewhat simplified. Fig. 43. Thus; let A,B' be the centres of the two pencils, ABC the triangle of reference, and S = the equation of the fourth side B'B" of the quadrilateral. The conic will be represented (Art. 208. {V).) by an equation of the form ya = h^b. (420) ANHARMONIC PROPERTY OF FIVE POINTS ON A CONIC. 165 Let US assume for the equations of the corresponding rays of the two pencils ^- l,y = 0, j3- l,y = 0, ^ - hy = 0, (421) and a — ^1 S = 0, a — m^d = 0, a — m^d = 0, a — m^d = 0; (422) then, since they intersect on the conic, we get by substituting for a and /3 in (420), my = kli, m^ = hk, (423) etc. = etc. ; and the second pencil is represented by the eqifetions a — Jcl^d = 0, a — kl^^ = 0, a — ^4S = 0, a — ^^48 = 0, and is therefore (Art. 93) homographic with (421). 166 TRILINEAR CO-ORDINATES. 269. Four fixed, tangents to a conic are cut hy any variable tangent in points ivhose anJiarmonic ratios are constant. Let ABC, the triangle formed by any three of the fixed tan- gents, be taken as the triangle of reference. Eig.. 44. The equation of the conic (Art. 187. (O5).) will be of the form {Za)^ + {3I^y + {JVyf = 0. (424) Suppose the equation of the fourth fixed tangent to be la + mjS + ny = 0, (425) and let Xa "l" /xj3 -}- py =z (426) represent any variable tangent meeting the four fixed ones in Pi, P2, Pi, Pi' The condition of tangency (Art. 212. (295).) gives Z If JV -7-+ — + — =0, (427) t m n ^ ^ ANHARMONIC TROPERTY OF FIVE TANGENTS. 167 and — -^ 1 ^ = 0. (428) Now the successive rays of tlie pencil {A -^^iPzPsPi} i^^^y be readily shewn to have for their equations /3 = 0, {mX — ?/x) /3 + {n\ — Iv)y = 0, and H^-\- Vy = 0,, respectively. Hence (Art. 92. (113).), the ratio P1P2 Pi Pi _ -^'i — h . h — h PsP2 ' PsPi h — h h — h when ki = 0, 1^2 = n\ — Iv V n\ — Iv 2 m\ — lij. V lji{n\ — Iv) n 1 1 Iv nk 1 _ I Ifj, mXj z=. — T^— , a constant qnantitv ; JVm the last step following by reason of (427) and (428), wdiich give by cross-multiplication 3£ N 1 1 1 1 fifjL mv Iv n\ m\ Ifi 168 TRILTNEAR CO-ORDINATES. 270. Pascal's Theorem. — TJie three ijairs of op2^osite sides of a hexagon inscribed in a conic intersect in points which lie in the same straight line. Let AJECDBF be any inscribed hexagon, and let ABC be taken as the triangle of reference. Fig. 45. The equation of the conic will be (Art. 180. (iV^).) of the form (429) Suppose L M N — + -^ + — = 0. a ^ 7 {BIT), a = X'ly ^ to be the equations of the two {CB). a = X'sjS ) sides which intersect in {CE). ^=^ha\ {AE). &z=h,y] {AF). y=zk^ {BF), y = 4« i F. Since B, F,F lie on the conic, we have from (429) L 4- Mh. -f Nh^ — 0, X^3 + Jf + Wk, = 0, zh,-\-Mk-\-Jsr =0, and, therefore, = 0. 1, k^j h I^Zi 1, h H} ^5; 1 (430) PASCAL S THEOREM. 169 But the condition that the opposite sides should intersect on the same straight line, IS la -\~ m^ -\- ny=iO (suppose), = 0. 1, ^'3> "'6 "'2? 1, h 1, "'37 /Cq (431) (432) For the intersection on (431) of C^ and jBF gives I + mk^ -f nke = ; that of AF . . . CD . . . U^ -j- m +nk5 = 0; ... £D . . . AF . . .Ih -^ mh + n =:0. But the conditions (430) and (432) are identical ; hence the theorem is proved *. 271. BriancJion's Theorem. — The diagonals which connect the three pairs of opposite angles of a hexagon descriled alout a conic meet in a point. Fig. 46. Let the circumscribed triangle ABC he taken as the triangle of * This proof of Pascal's Theorem is given by Mr. Ferrers in his treatise on Trilinear Co-ordinates. 170 TRILTNEAH CO-ORDINATES. reference ; then the equation of the conic (Art. 187. (O5).) will be of the form {Laf + (if/3)^ + {^yf ^ 0- (433) ■ Let A^A^, B1B2, C^C^, the other three sides of the hexagon, have for their equations /ja + W?i^ + Wi7 = 0, ha + m^^ + W27 = 0, (434) and \a + mS + ^^37 — 0? respectively. It may be proved by the usual method (e.g. see Art. 98) that the equations of the three diagonals B.C^., A^C^, A^B^ are, re- spectively, a + y/3 4-^7 = 0, 1 Ql — a+ /3 + — 7 = 0, (435) — a + — /3+ 7 = 0. Also, since (434) are tangents to the conic (433), we have L M N ^ -7- + — + — = 0. L M N ^ -7- + — + — = 0, Z if A^ ^ k m. no BRIANCHON S THEOREM. 171 and, therefore, 1 1 1 V nil n-i 1 1 1 V fth ^2 1 1 1 m. = (436) but this may be written in the form or, 1 1 1 h' 4' 4 1 1 1 Wl ^^2 ^3 1 1 1 %' n.2 ^3 1, mo 1/ 4 9 1, rh Ml W3 m^ ^2 1 = 0, = 0, which (Art. 55. (60).) is the condition that (435) should meet in a point. This proves the theorem. 272. To find the locus of the centre of a conic which touches three given straight lines and passes through the given point (a^, /Sij-yi). Let the given tangents be taken as the triangle of reference, and suppose the equation of the conic to be {LaY 4- [M^y + [NyY = 0. 172 TRTLINEAR CO-ORDINATES. The centre (Art. 213. (308).) is given by tlie equations ^ Nh -\- Mc Lc -\- Na Ma + Lb y 1 = - IT (say) ; k whence, and which give Ze + ]Va-^^k= 0, Lh-\-Ma + 7^ = 0, z M i\^ C, i, a 0, h, ~a 0, C, a 0, a, "^ — c, a, J c, 0, ^ a, 0, 7 b, 0, 7 b, a, y or, If N a{ — aa + J^ + Cy) h(aa — b^ + C7) ^(fl^a + b^ — Cy) But, since (oj, ^i, 71) always lies on the curve, (ZaO^+(ifA)*+(iV7i)^ = 0; . (437) (438) and, substituting in (438) for Z, M, N from (437) and writing a, ^, 7 for a, /3, 7, we get for the equation of the locus s/{aa,{- «a + J/3 + ^7)} + ^{b^,{aa - J/3 + cy)} + s/{cyi {cia + J/3 - C7)} = 0. (439) ITow, «a + ^j3 + ^7= 0, and THE POLAR RECIPROCAL OF A GIVEN CONIC. «a — J/3 + Cy = 0, aa + h^ — Cy =: 0, 173 represent the sides of the triangle A^ B^ C^ whose vertices are the middle points of the sides of the triangle of reference. The re- quired locus, therefore, is a conic section which touches the three sides of the triangle A^B^Ci. 273. To find the equation of .the polar reciprocal of the conic cjy,(a, ^, y) = A,a' + B.^" + C,y' -f 2B,^y + 2E,ya + 2F,a^ = 0, with respect to the conic Ha, /3, y) = A,a' + B,^' + C,y' + 2A^y + 2B,ya + 2F,a^ =0. Let {a, 13', y) be a point on the reciprocal curve. Its polar with regard to <^2(a> /3, y) = has for its equation (Art. 134) @)"+(t)^Ht)-«' and the condition that it should be a tangent to ^i(a, /3, y) = is, therefore (Art. 131. (155),), dcf)^ A,, Fi, El, F,. B„ B„ -^1, A, c„ #2\ da' r fd^A fd,\ \di3' J' \dy y da d(f)2 d^ d(f)2 = 0. Hence, suppressing the accents, we have for the equation of the polar reciprocal 174' ^1, #2 da TRILINEAE- CO-ORDINATES ^^' -^^' ^^' \da Fly By, Dl, d(f)2 d(f)2 dy d^ y' Uy y' or, to use the notation hitherto employed, _\da J \d^ J \dy = 0; rr 0. (440) (441) EXAMPLES. Throughout the following examples, except ivhen the contrary is stated, AB C is used to denote the triangle of reference. I. EiND the proportional values (Art. 26) of the co-orclhiates of the follow- ing points ; ABC being the triangle of reference. 1. The middle point of the angle-bisector AA(i. If I be the length of AA^, the actual values of the co-ordinates will be -^^ sin — , — sin — , — sin — , and their proportional values {h + c, a, a). 2ici 2i 2i 2i 2i 2i 2. The middle point of the side-bisector AAi. The actual co-ordinates are — , 7- sin C, — sin 5, and the point may be repre- sentedby ( -, t-j - )• \a e J 3. The middle point of the perpendicular AA2. Ans. (1, cos C, cos B). 4. The point of intersection of the side-bisector BBi with the angle- bisector CCq. The equations of these lines are, respectively, cy — aa = and a — ^ = 0. Hence the actual values of the co-ordinates of their intersection are aa = a^ = C7 = ;, and their proportional values ( — , — , — ]. ' 2a + h \a a c j 5. The intersection of the external bisectors of the angles B and C. Ans. (- 1-, 1, 1). 176 TRILINEAR CO-ORDINATES. n. 6. Shew that the three pevpendiculars drawn to the sides of an equi- lateral triangle^ from any point within it, are together equal to the altitude of the triangle. , 7. Prove that A\ sin A -v B\%m.B -V Ci sin (7 = ; where A\, B\, Ci are the areas of the triangles formed by joining the ver- tices of the triangle ABC to the centres of the inscribed and escribed circles. 8. Shew that the sum of the reciprocals of the distances of Gi (Art. 263) from the sides of the triangle of reference is equal to three times the radius of the inscribed circle. For the notation employed in the following examples the reader is referred to Art. 235 and fig. 41. 9. Prove that Og is the middle point of F2OR (Art. 257). Let Cg, 02, ajt be, respectively, the a- co-ordinates of the three given points. Then, by Art. 263, Oil + o-% = HcosA + 2iR cos B cos C = Bcos{B - C) and the same is true, by symmetry, for the a - and $ - co-ordinates. 10. Prove that G1P2 = '^G^Or (Art. 257). 11. Shew that G^Gahc = ^GiOr. 12. The sides of the triangle of reference being 5, 12, 13 ; construct (Art. 15) the line whose equation is 2a - 3j8 -I- 47 = 0. III. Write down, in their symmetrical forms, the equations of the following right lines (Arts. 7, 8, 9) :— EXAMPLES. 177 13. The straight line through ^ which bisects the l,BAC of the triangle of reference. AAA The Q, ((> and i// of this line are, respectively, — — t- C, -—, and — ; its ii ^ ^ direction-cosines, therefore, (since, by Art. 8 % they are equal to — sin 6, sin /{-S(w7 + la — m^)fij + sy[C{la + m&- ny)y} = 0. 69. Obtain the equation of the chord which joins the two points («i, -3i, 7i), (as, 1^2, 72) on the conic ,8y — ka = in the form 1 1 \ U + — + 1 1 «l • ■ + — M 1 1 + — Jcc I 1 1 1 ITcc a I \ , + /3, iS^y ^^&^_ 1 \\ 2.U + I a\ — + — + 7x 7a/ 7i72 ha EXAMPLES. 193 70, If a quadrilateral be inscribed in a circle, the product of the per- pendicular distances of any point on the circumference from two opposite sides will be equal to tlie product of its perpendicular distances from the other two sides. Let 7a = Tc^{la + m^ + ny) [Art. 208. (v).] be the equation of a conic circumscribed about a quadrilateral whose sides are a = 0, ;8 = 0, 7 = 0, and la + mfi + ny = 0. If it be a circle, we shall have (Art. 149) c(mo — nh) = -— = a(ma — Ih) inc — nh 1 hi — ma h{an — cl) a Tc —c cP' ~ a? 1 a{an — cl) {hi — am) + h(hl — am) (cm — hn) + c{cm — hn) {an — cl) ^' -^{c^- c^) - ale + ^(c2 - «2) = P + 171^ + n^ — 2mn cos A — 2ln cos -S — 2lm cos C; and- = |Z, m, nj^ , (la + m$ + ny)0. Hence the equation of the cn^cumscribed circle is ya = Tt i ^ io, m, n j- and (Art. 47) the proposition is proved. 71. If three conies have each double contact with a fourth, their six chords of intersection will pass, three by three, through the same points. We may assume the equations of the three conies to be 8 — 11^ = 0, >Sf — -jjS = 0, /S — w^ = (Art. 169), Their chords of intersection will be repre- sented by the equations «2 _ -^(,2 = 0, w2 _ ^2 = 0, ?*2 _ ^2 ^ 0^ Thus we have four groups of three equations each representing chords which meet in a point : viz. — ?; + w = 0, w — « = 0, M + V = 0, ■y — «[7 = 0, «C; + M = 0, U Ar V = 0, 194 TRILINEATl CO-ORDINATES. V + IV = 0, w + u = 0, u — V = 0, 1) — w — 0, w — u = 0, u — V = 0. The student will observe that the above theorem includes that of Brianchon, of which an independent proof was given in Art. 271. 72. In any triangle the bisector of any angle and the straight line which is perpendicular to the opposite side at its middle point, intersect on the circumference of the circumscribing circle. For the point of intersection of a — /8 = and sin^a - sin^iS + sin(^ - B)y = 0, is given by the equations sm A — sni B whence we may easily shew that sin A sin JB sin C + — -— + = 0. a /3 7 73. Shew that if the tangents to a circumscribed ellipse at the vertices of the triangle of reference are parallel to the opposite sides the equation of the conic is j^ + 2^ + ^ = 0. a b c 74. The two pairs of tangents at the extremities of the diagonals of a quadrilateral inscribed in a conic, and the pairs of opposite sides intersect on the same straight line. This Theorem is easily deduced from that of Pascal which was proved in Art. 270. 75. Shew that if a conic cut two sides of a triangle in points equidistant from their middle points it will cut the third side in the same way. Let the sides BC, CA of the triangle of reference be cut in this manner; then if (p{a,^,y) = be the equation of the conic, (a^, /Sq, 7o) the middle point of BC, and A, /j., v the direction-cosines of that side, w^e shall have by equation (8) of Art. 13, since the roots are equal and of opposite sign d