CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall CORNELL UNIVERSnV LIBRARY 3 1924 050 939 630 Cornell University Library The original of tinis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924050939630 MATHEMATICAL WOEKS. Mr ROUTH on THE DYNAMICS OF A SYSTEM OF EIGID BODIES. Crown 8vo. cloth, los. 6d. Mr SNOWBALL'S PLANE AND SPHEEICAL TRIGO- NOMETRY. Tenth Edition, crown 8vo. 7s. 6d. Mr LUND'S IMPROVED EDITION OF SNOWBALL'S COURSE OF MECHANICS AND HYDROSTATICS. Fifth Edition. Crown 8to. cloth, 5s. Mr DREW'S GEOMETRICAL TREATISE ON CONIC SECTIONS. SOLUTIONS. Crown 8vo. 4s. 6d. Third Edition, crown 8vo. cloth, 4s. 6d. Mr PUCKLE'S TREATISE ON CONIC SECTIONS AND ALGEBRAIC GEOMETRY. Second Edition, crown 8yo. cloth, 7s. 6d. Me PHEAR'S ELEMENTARY TREATISE ON HYDRO- STATICS. Third Edition, crown 8vo. cloth, 5s. 6rf. Mr TAYLOR ON GEOMETRICAL CONICS, INCLUDING ANHARMONIC RATIO AND PEOJECTION. Crown 8vo. cloth, 7s. 6d. MATHEMATICAL WORKS. Messrs FROST AND WOLSTENHOLME'S TREATISE ON SOLID GEOMETRY. 8vo. cloth, i?». Mr FROST'S NEWTON'S PRINCIPIA, Sect. I. II. III. With Notes and Problems. Second Edition, 8to. clotb, io«. 6d. Mr CHEYNE'S ELEMENTARY TREATISE ON THE PLANETARY THEORY. Crown 8vo. cloth, 6s. 6d. Mr MORGAN'S COLLECTION OF MATHEMATICAL PROBLEMS. With Answers. Crown 8vo. cloth, 6s. 6d. MACMILLAN AND CO. LONDON AND CAMBRIDGE. TRILINEAR CO-OEDINATES. aatnbrarge: rBINTED XT C. J. OLAT, Mjl. AT THE TJNrVEBEITY PEES8. AN ELEMENTAEY TREATISE ON TRILINEAR CO-ORDINATES, THE METHOD OF EECIPEOOAL POLARS, AND THE THEORY OF PROJECTIONS. BY THE REV. f. M. FEEREES, M.A. FELLOW AND TtTTOfi OF GONVILLE AND CAIUS COLLEGE. SECOND EDITION. HonlJOtt anS ©ambragE: MACMILLAN AND CO. 1866. In compliance with current copyright law, LBS Archival Products produced this replacement volume on paper that meets the ANSI Standard Z39.48-1984 to replace the irreparably deteriorated original. 1988 PREFACE. The prominence which the modern geometrical methods have recently acquired in the studies of the University of Cambridge, appears to justify the publication of a treatise devoted exclusively to these branches of Mathematics. This remark applies more especially to the method of Trilinear Co-ordinates, which forms the subject of the greater part of the following work. My object in writing on this subject has mainly been to place it on a basis altogether independent of the ordinary Cartesian system, instead of regarding it as only a special form of Abridged Notation. A desire not unduly to increase the size of the book has prevented me from proceeding beyond Curves of the Second Degree. I have introduced a short Chapter on' Determinants. The great utility of these expressions in investigations connected with curves of the second degree will, I hope, VI PREFACE. be a sufficient excuse for the employmeut of a notation which has hitherto been hardly admitted into Cambridge text-books. I have, however, confined myself rigorously to the demonstration of such elementary properties as are required in the course of this work. I should be glad if the very slight sketch contained in Chapter III. should be the means of inducing any of my readers to refer to the original memoirs on this and kindred subjects. In this Second Edition several new articles have been added, especially in the latter part of the work, and the chapter on Reciprocal Polars considerably enlarged. N. M. F. GOXVILLB AND CaIUS COLLEGE, August, 1866. CONTENTS. CHAPTER I. TBILINEAE CO- ORDISTATES. EQUATION OF A STRAIGHT LINE. ARTS. PAQB I. DEFlNrriON of TriliDear Co-ordinates i ■i. Identical relation between the TrUinear Co-ordinates of a Point »5. 3. Distance between two given Points 4 4 — 6. Investigation of Equations of certain Straight Lines ... 6 7. Every Straight Line may be represented by an Equation of the First Degree .......... 9 8. Every Equation of the First Degree represents a Straight Line . 11 9. Point of Intersection of Two Straight Lines . . . . la 10. Equation of a Straight Line passing through Two given Points . ti. 11. Equation of a Straight Line passing through the Point of Inter- section of Two Given Straight Lines . . . . .13 12. Condition that Three Points may lie in the same Straight Line . tj. 13. Condition that Three Straight Lines may intersect in a Point . 14 14. Condition that Two Straight Lines may be parallel to one another. Line at Infinity ......... 16 15. Equation of a Straight Line, drawn through a given Point, paral- lel to a given Straight Line 17 16. Inclination of a Straight Line to a side of the Triangle of Befer- ence 18 17. Condition of Perpendicularity 19 18. Distance from a Point to a Straight Line 20 Examples 11 19. Anhabmonio Eatio. Definitions 23 10. The Anharmonic Eatio of a Pencil is equal to that of the range in which it is cut by any Transversal ib. 21. Definition of an Harmonic Pencil 15 VIU CONTENTS. AKTS. PAOB 22. The Bisectors of any Angle form, with the Lines containing it, an Harmonic Pencil ......... 15 43. Anharmonic Ratio of a given Pencil ...... ^6 24. Fourth Harmonic to Three given Straight Lines zj. Harmonic Relation of Points and Lines 76. Ok Ikvolution. Definitions .... • 17 17 — 29. Anharmonic Properties of Points and Lines in Involution . it. CHAPTER II. SPECIAL FORMS OF THE EQUATION OF THE SECOND DEGEEK I. Every Equation of the Second Degree represents a, Conic Sec- ' tion • • • 33 2, 3. Equation of the Conic described about the Triangle of Refer- ence 34 4. Position of the Centre. Condition for a Parabola . . .35 5. Condition of Tangency. Every Parabola touches the Line at Infinity ........... 37 6. Equation of the Circumscribing Circle ... . ib. 7. Equation of the Conic touching the Three Sides of the Triangle of Reference ......... ib. 8. Position of the Centre. Condition for a Parabola . . .40 cj. Condition of Tangency ........ 42 10. Equations of the Four Circles which touch the Three Sides of the Triangle of Reference ....... 43 II — 15. Eqiiation involving the Squares only of the Variables . . 45 16. Condition of Tangency ........ 48 17. Condition for a Parabola 49 18. Co-ordinates of the Centre ....... ib. 19. Equation of the Circle, with respect to whidi the Triangle of Reference is self-conjugate 51 20. Equation of the Conic which touches two sides of the Triangle of Reference in the points where they meet the third . . 51 11. Any Chord of a Conic is divided harmonically by the Conic, any Point, and its Polar 53 23. Equation of a Line joining Two given Points .... ib. 24. Equation of the Tangent at a given Point . . . -54 15. Pole of a given Straight Line ib. 26. Condition of Tangency. Condition for a Parabola . . . ib. 21. Co-ordinates of the Centre 5."; Examples 56 CONTENTS. IX CHAPTER III. ELIMIMATION BETWEEN LINEAR EQUATIONS. AKT8. PAGE ■i. Definition of a Determinant 59 3 — 6. Law of Fonnation of Determinants ...... 60 7. SignB of the several Terms of a Determinant . . . .66 8. Sign changed by interchange of two Consecutive Lines or Co- lumns ........... ti. 9. Multiplication of a Determinant by a given Quantity . . 67 10. Minors of a Determinant ........ 68 11. Condition that a Quadratic Function may be resolvable into Two Factors ih. ,12, 13. PascaVs Theorem 69 Examples 71 CHAPTER IV. ON THE CONIC, REPRESENTED BY THE GENERAL EQUATION OF THE SECOND DEGREE. "i. To find the point in which a straight line, drawn in a given direction through a given point of the conic, meets the conic again 7^ 3. Equation of ^;he Tangent at a given Point . . . .74 4, 5. Condition that a given Straight Line may touch the Conic . 75 6. Condition for a Parabola 77 7. Condition that the Conic may break up into Two Straight Lines ib. 8. Equation of the Polar of a given Point ib. 9. Co-ordinates of the Pole of a given Straight Line . . -78 10. Equation of the pair of Tangents, drawn to the Conic from a given external Point 79 11. Co-ordinates of the Centre . . . . . . .81 11. Equation of the Asymptotes ,,,.... 82 13. Condition for a Rectangular Hyperbola 83 14. Conditions far a Circle 85 X CONTENTS. ARTS. PAGE 15. All Circles pa«s through the same two points at infinity . .87 16. All Conios, similar and similarly situated to each other, intersect in the same two points in the line at infinity . 17. Radical axis of two similar and similarly situated Conies . 18. Property of the nine-point Circle ..... 19. Equation of the nine-point Circle ..... 20. Locus of the intersection of two Tangents at right angles to one another. Directrix of a Parabola .... 21. To find the magnitudes of the axes of the Conic 22. To find the area of the Conic. Criterion to distinguish between an Ellipse and an Hyperbola . . . ■ . Examples 88 a. 90 91 92 93 OHAPTEE Y. TRIANGULAB CO-ORDINATES. 1. Definition of the Triangular Co-ordinates' of a Point . gg ■i. Formulae relating to Straight Lines 100 3. Formulse relating to Conies loi CHAPTER VI. RECIPROCAL POLARS. 3. Definition of a Polar Eeoiprocal 104 5. The degree of a curve is the same as the class of its reciprocal, and vice vend ......... 105 6, 7. The polar reciprocal of a conic is a conic . . . . .106 8. Equation of the Polar Reciprocal of one Conic with regard to . another ii. 10. Instances of Transformation .... . . 108 12. Brianchon't Theorem . . . . . . . .111 13. The anharmonic ratio of the Pencil, formed by four intersecting straight lines, is the same as that of the range formed by their poles . . . . . . . . . . .Ill 15. Any straight Hne drawn through a pven point A is divided har- monically by any Conic Section and the polar of A with re- spect to it • 114 CONTENTS. XL AETS. PAOB 17. If four straight lines form an harmonic pencil, either pair will be it? own polar reciprocal with respect to the other , -115 18.. Cpnditipu that two pairs of straight lines may form an harmonic pencil ib. 10. Keciprocation with respect to a Circle . . . . • 1 1 7 23. Eeciprocation with respect to a Point . . . . .118 24. The three circles described on the diagonals of a complete quad- rilateral as diameters have a common radical axis . . . ib. Foci of a quadrilateral ib. 25. Orthocentre of a triangle . . . . . . . .119 28. The director circles of all conies which touch four given straight lines have a common radical axis . . . . . . 1 20 49. Polar reciprocal of a Circle with regard to any point . . . ib. 30. Instances of Transformation of Theorems by Reciprocation with respect to a point . . . . . . . . .122 31. Cori'esponding Points and Lines. The angle between the radius vector and tangent in any curve is equal to the corresponding angle in the Reciprocal Curve . . . . . .124 33. Co-ordinates of the foci of a Conic . . . . . .125 34. Double application of the Method of Reciprocal Polars . . 127 ExAMrLES ib. CHAPTER VII. TANGENTIAL CO-OEDINATES. 1. Definition of the Tangential Co-ordinates of a Straight Line . 130 2. Interpretation of the Negative Sign. Equations of certain points 131 3. General Equation of a Point 132 4. Identical relation between the co-ordinates of any straight Line . 133 6. An equation of the »th degree represents a curve of the nth class . . . ■ ■ • • • • • • '35 7. Equation of a Conic, touching the three sides of the triangle of reference *"• 8. Equation of circumscribed Conic '37 9. Equation of the Pole of a given straight Une, and of the centre. Condition for a Parabola '3^ 10. Circular points at infinity ''■'>■ Conditions for a Circle '39 11. Self-conjugate Conic '4° Xll CONTENTS. AMS. PACK Examples 140 12. Tangential rectangular Co-ordinates 141 15. Tangential polar Co-ordinates 143 EXAUFLEB ' . . . . ib. CHAPTEE VIII. ON THE INTERSECTION OF CONICS, AND ON PROJECTIONS. I — 3, Any two conies intersect in four points, real or imaginary. Ver- tices of the quadrangle formed by these points . . . 145 4 — 7. If the four points of intersection be all real, or all imaginary, all the vertices are real. If two of the points of intersection be real, and two imaginary, one vertex only is real. If the four points of intersection be all real, all the common chords are real ; if not, one pair only is real 146 8. Invariants of two Conies 148 On Peojeotions. 9. Definition of Projections . , . . . . . ,149 10. Projection to infinity 150 13. Any quadrilateral may be projected, in an infinite number of ways, into a parallelogram of which the angles are of any magnitude 151 14. Projection of Tangents, of Poles, and Polars . . . . ib. 15. Any two conies may be projected into concentric carves . , 152 16. Also into similar and similarly situated curves .... iji 17. These projections may he effected in an infinite number of ways ib. 18. Any two intersecting conies may be projected into hyperbolas of any assigned eccentricity ib. 19. Any two conies may be prelected into conies of amy eccentricity, or into circlea 153 20. Projection of the foci and directrices of a Conic . . . ib. 21. The anharmonlc ratio of any pencil or range is unaltered by ' projection 154 22. Any two lines, which make an angle A with each other, form with the lines joining the circular points at infinity to their point of intersection, a pencil of which the anharmonic ratio ^ j(^-M V-i ISS CONTENTS. XUl ABTS. PAOB 23. The auharmonic ratio of any four points on, or any four tangents to, a conic, is constant ........ 156 25. Any system of points in involution projects into a system in involution, and the foci of one system project into the foci of the other .......,,. 157 16. A system of Conies, passing through four given points, cut any straight line in a system of points in involution . . . ib. 27. Orthogonal Projection ii. Examples 158 CHAPTER IX. MISCELLANEOUS PROPOSITIONS. ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN GEOMETRI- CAL CONDITIONS. ■i. If five points be given, one conic only can be drawn . . . 160 3. If four points and one tangent be given, two conies can be drawn . . ib. 4. If three points and two tangents be given, four conies can be drawn ........... ib. 5. If two points and three tangents be given, four conies can be drawn . t6i 6. If one point and four tangents be given, two conies can be drawn ...... ..... 162 7. If five tangents be given, one conic only can be drawn . . ii. 8. Reduction of certain other conditions to these . . . . i6. 9. Conjugate triad 163 ON THE LOCUS OF THE CENTRE OF A SYSTEM OF CONICS WHICH SATISFY FOUR CONDITIONS. II. Four points given 163 n. Three points and a tangent 165 13. Two points and two tangents 166 14. One point and three tangents 167 15. Four tangents 169 XIV CONTENTS. SUPPLEMENTAEY PEOBLfiMS. AKTS. TAOE i6. The product of any two determinants is a determinant . .170 17. Property of the co-ordinates of three points, forming a conjugate triad 171 18. Envelope of a side of an inscribed triangle whose other sides pass each through a fixed point 172 19. Locus of a vertex of a circumscribed triangle, whose other vertices move ea<;h along a fixed straight line . . . . .174 io. Trilinear co-ordinates of the Foci . . . . . • '75 Miscellaneous Examples 177 TEILINEAR CO-OEDINATES. CHAPTER I. TEILINEAR CO-OEDINATES. EQUATION OF A STRAIGHT LINE. ~ 1. In the system of co-ordinates ordinarily used, the position of a point in a plane is determined by means of its distances from two given straight lines. In the system of which we are about to treat, the position of a point in a plane will be determined by the ratios of its distances from three given straight lines in that plane, these straight lines not passing through the same point. ThejteiaD gl e^jbr med by these three^ straight lines is^alled the triangle of reference, its"?ides7iines of rraerence, and the distances of a point from its three sides wiT^jcalled the trilinear co-ordinates of that p^SJ;^ — Wr~shall usually "cfenote the angular points of the triangle of reference by the letters A, B, C, the lengths of the sides respectively opposite to them by a, I, c, and the dis- tances of any point from BC, CA, AB respectively by the letters a, /3, 7. When two points lie on opposite sides of a line of re- ference, the distance of one of these points from that line may be considered as positive, and that of the other as negative. We shall consider a, the distance of a point from the line BG, as positive if the point lie on the same side of that line as the point A does, negative if on the other side ; and similarly for /8 and 7. It thus appears that the trilinear co-ordinates of any point within the triangle of reference are all positive ; while no point has all its co-ordinates negative. 2. Between the trilinear co-ordinates of any point an important relation exists, which we proceed to investigate. If ^ denote the area of the triangle of rffer&nce, a, yS, 7, the trilinear co-01 dinates of any point, then — aa + 6/3 -1- C7 = 2A. Let P be the given point, and first suppose it to lie within F. /" 1 TEILINEAE CO-ORDINATES. the triangle of reference (fig. 1). Join PA, PB, PC, aiid draw PD perpendicular to BC. Then PD = a, and aa. = twice the area of the triangle PBC. Fig. I. Similarly J/3 = twice the area of POA, cy = twice the area of PAB. Adding these equations, we get aa + J/3 + C7 = 2A. Next, suppose Pto lie between AB, A C produced, and on the side oi BC remote from A (fig. 2). Then a will be Fig. 2. ■ '- AREA OF THE TEIANGLE OF REFERENCE. 3 negative, wMle /S, 7 are positive. Hence, twice tlie area PBG will be represented by — aa., and we shall therefore have as before aa + 5/3 + C7 = 2A. Thirdly, let P lie between AB, A G, produced backwards (fig. 3), so that /3, 7 are negative while a is positive. Twice Kg. 3. the areas of PEG, PGA, PAB, are now represented by aa., — &/3, — cy respectively, so that we still have aa + &jS + C7 = 2A. In all cases, therefore, aa + 5/3 + C7 = 2A. The importance of the above proposition arises from its enabling ns to express any equation in a form homogeneous with respect to the trilinear co-ordinates of any point to which it relates. Any locus may be represented, as in the ordinary system, by means of a relation between two co- ordinates, /8 and 7 for example, and this may be made homo- geneous in a, /8, 7 by multiplying each term by ^^r — - , 1—2 4 TEILINEAB CO-OKDINATES. raised to a suitable power. Thus, the equation /3'+ Ay+ Jt?=0 is equivalent to the homogeneous equation iA'/S" + 2 AAy {aoL + l^ + cri)+l^ {m'a—l'h)^+ [n'a—l'c) 7 = 0. • ' "jC:^^ »* " .^ And these straight lines must be at right angles to one an- other. If 6, 6' be the respective inclinations of these straight* lines to the internal bisector of the angle A, then, by the result of the last article, n (Zc — ncC) — (ma — Ih), A tan d = ;, ';——^ jjx tan — , (Zc — no) + [ma — lb) 2 tan tf — 777 ; 7 ; 7 -, YrTz tan — . {lc — na) + {ma — lb) 2 And, if these be at right angles to one another, l+tan5tan^' = 0. Hence {Ic — no) (Z'c — no) + {ma — lb) [m'a — I'l) + {(Zc — «a) [ma — I'h) + (»ia — W) (Z'c — n'a)] cos J. = ; 2-2 20 . . MODERN (JEOMETET. .•• W (&' + c" - 25c COS A) + mm'a'+ nn'a^ — {mn + m'n) a" cos A — (nl' + n'l) {ac — db cos A) — {Im' + I'm) {ah — ac cos A) = 0, which, since b" + c'— 2hc cos A=a', c — b cos A — a cobB, b — ccosA = a cos C, reduces to U+mm'+rin'- {mn'+m'n)coaA- {nl'+n'l) cosB- {Im'+l'm) cos C=0, the required condition. 18. To find the perpendicular distance from a given point to a given straight line. Let (/, g,h) be the given point,. (Z, m, w) the given straight line. Then, if g and r.be the distance from A, of the points where this straight line meets A 0, AB, respectively, we have shewn (Art. 7) that 1 _ 1 wa> "' ,-- I'b'lFc' ^^^"'-' 1" 1 ma r'^o'hc' C', ^ Now, let a' denote the distance from (/, g, h) to Q, m, n). Then ■ {^-\-'r'-'2qrcosAy'o: + qg-\-rh = ^{af+lg + ch), n 1- 2 COS A\^ , _c^+bg + ch_g_h \^ r^ qr ) be r q And from the values of q and r 1 cos J.- 1 ' cos^ , ..a = 1 — [n — m cos A) ^=- q r c ^ Ibc _ a {IcosB + mcosA — n) Wc • or Similarly DISTANCE FEOM A POINT TO A LINE. 21 1 COS A _a{l cos G + n cos A —m) , 1,1 2 cos J. • .2 "r s r £ Ibc ' g. ^. = {zW^^^ ~ "''^ (Zcos5+ m cos^ - m) + (Z5 — ma) {I cos C + w cos ^ — »w) [ = pp-2{f'(ccos^+5cos C) +ni'a + n^a — 2mn a cos A — nl(c + acosB — h cos A) — lni(b — c cos A + a cos C)}, which, by reduction, is equal to 2 TzTTs (Z''+ m' + w' — 2mKcos^ — 2»?cos5- 2?otcos C). Hence a' = + - Z/^+ ni^ + nh (f + m" + m' - 2»w« cos A - 2nl cosB-2lmcos Cf' the required expression. It will be observed, that the numerator of this expression vanishes if the point (/, g, h) lie upon the line (I, m, n), as manifestly ought to be the case. It will also be remarked, that the more nearly the ratios I : m : n approach to the ratios a : b : c, the less does the denominator of the above fraction become, and the greater, therefore, the distance from the point to the line ; which is in accordance with the remark made in Art. (14). Examples. 1. Find the equation of the straight line joining the middle points of two sides of the triangle of reference ; and thence prove that it is parallel to the third side. 22 MODERN GEOMETET. 2. Find the equations of the straight lines, drawn through the several angular points of the triangle of reference, respectively at right angles to f + Y = 0. I + ? = 0, -4 = Qi b c a a b and thence prove that they intersect in a point. 3. If 6 be the angle between the two straight lines {I, m, n) (X, /i, v), prove that l\+m ii,+nv—(mv+nfi) cos A— (n\+lv) cos^ - (Ifi + toX) cos C (mv — nil) sin J. + (»X — Iv) siu B + ili^. — wiA) sin G cote = 4. On the sides of the triangle ABO, as bases, are constructed three triangles A'BG, AFC, ABC, similar to each other, and so placed that the angle BA'C= B'AO = BAC, CB'A = CBA = GBA', AG'B = A'GB = AGE. Prove that the straight lines AAl, BB', GG' intersect in one point. ' 5. Prove that the straight line, joining the centre of the circle inscribed in the triangle ABG, with the middle point of the side BG, is parallel to the straight line joining A with the point of contact of the cii-cle touching BG externally and AB, AG produced. 6. On the sides BG, GA, AB of the triangle ABG, respectively, pairs of points are taken, B , G^; G , A^; A^, B^; such that the points of intersection of BC with B^G^, of CA with G^A^, and of AB with A^ lie in a straight line ; BG^ GB^ intersect in L ; GA,, AG^ in M; AB^, BA, in N. Prove that AL, BM, CiV inter- sect in one point. 7. Prom the vertices of a triangle ABG, three straight lines AP, BQ, GR are drawn to pass through one point, and three straight lines AF, BQ', GR to pass through another point, the points P, P lying on BG, Q, Q' on GA, R, R on AB ; BQ, GR meet AP in .D„ D, ; GR,.AP meet BQ' va.E^,E^; AP, BQ meet GR in F , F-, GD^, BD^ intersect in L ; AE^, GE^ in M ; BF^, AF^ in iv. Prove that AL, BM, GN intersect in a point. ANHAEMONIC EATIO. 23 Anhabmonic Katio. 19. We shall introduce, in this place, a short account of harmonic and anharmonic section, as a familiarity with this conception is useful in the higher geometrical investiga- tions. Def. 1. If OP, OQ, OR, 05' befour_fitraight lines in- tersecting in a point, the ratio ■J? ts< svaPOQ.smROS c[. sin FOS. sin QOB is called the anharmonic ratio of the pencil OP, Q, OR, OS, and is expressed by the notation {O.P QR8}*. Dep. 2. If P, Q, R, 8 be four points in a straight line, PO Jf^ the ratio p„' „ is caUed the anharmonic ratio of the range P, Q, R, 8, and may be expressed thus \PQRS]. In using these definitions, attention must be paid to the order in which the lines or points follow one another. Thus, the an- harmonic ratio of the pencil OP, OR, OQ, OS, is different from that of the pencil OP, OQ, OR, OS, the former being equal to sin POR . sin QOS ,, , , , , Bin POQ . sin ROS ain POS. sin QOR' ^'^^ ^""^^^ *° sin POS . sin QOR ' Def. 3. If any number of straight lines, intersecting in a point, be cut by another straight line, the straight line which cuts the other is called a transversal. 20. Peop. If four given straight lines, intersecting in a point O, he cut by a transversal in the points P, Q, R, S, the anharmonic ratio of the pencil OP, 0(3, OR, OS, will be equal to that of the range P, Q, R, S. * This notation is due, I believe, to Dr. Salmon. See his Conic Sections, p. 273 (third edition). 24 ilODEEN GEOMETRY, Fig. 9. For And sin POQ _ sin POQ sin OPS sin P08~ sin OPQ ' sin P08 _PQ 08^ ~OQ-PS' s in BOS _ smROS sin OEQ sin ^0-B "sin OB 8' sin QOE _R8 OQ ~ 08- QR' sinPOQ. sin B08 PQ.R8 ;'■ sin P08 . sin Q0B~P8.QB' Thus the proposition is proved. CoE. 1. It appears, from the above proposition, that if a pencil be cut by two distinct transversals in P, Q, R, 8 and P, Q, B, 8' respectively, the anharmonic ratio of the range P, Q, R, 8 will be equal to that of the range P, Q, E, 8', since each is equal to that of the pencil OP, OQ, OB, 08. COE. 2. It appears also that, if four points P, Q, B, 8, lying in a straight line, be joined with each of two other points 0, 0', the anharmonic ratios of the pencils OP, OQ, HA.EMONIC PENCILS AND RANGES. 25 OR, OS; O'F, O'Q, O'R, O'S, will be equal to one another, since each is equal to that of the range P, Q, B, 8. 21. Dep. a pencil, of which the anharmonic ratio is"] unity , is called an harmonic pencil. J A range, of which the anharmonic ratio is unity, is called an harmonic range, and the straight line, on which the range lies, is said to be divided harmonically. From what has been said above, it will be seen that, if an harmonic pencil be cut by a transversal, the four points of section will form an harmonic range. And if four points, forming an harmonic range, be joined with a fifth point, the four joining lines will form an harmonic pencil. The line 08 is said to be a fourth harmonic to the pencil OP, Q, OB ; and the point >S to be a fourth harmonic to the range P, Q, B. The term harmonic is employed on account of the cir- cumstance, that if the points P, Q, B, 8 form what is above defined as an harmonic range, PB will be an harmonic mean between PQ and PS. For PQ.BS=PS.QB; 3 - .: PQ{P8-PB)=P8{PB-PQ); "^^^ • '^' .-. PQ : PS :: PB-PQ : PS -PR, P ^ whence PQ, PB, PS are in harmonical progression. From the above proportion it appears that if PQ = QB, PS=oa. Hence, if PB be bisected in Q, the fourth har- monic to the range P, Q, B is infinitely distant. Or, as it may otherwise be stated, if PB be bisected in Q, and P, Q, R be joined with any point 0, not in the line PR, the fourth harmonic to the pencil OP, Q, OR, will be parallel to the transversal FQR. 22. Prop. The external and internal bisectors of any angle farm, with the lines containing the angle, an harmonic 26 MODERN GEOMETEY. Fig. 10. Let the angle POR be bisected internally hj OQ, let PO be produced to any point P', and let the angle P'OE be bisected by OS, then sin P0^ = sin (30^, Bin P0^= sin P'05f = am BOS; am POQ. sin BOS •"• Bin POS. Bin QOB~ Hence the truth of the proposition. 23. Prop. If ABC he the triangle of reference, and AD, AE straight lines respectively represented hy the equations ^-J• which determines the centre. 27. Pbop. The anharmonic ratio of four points is equal to that of their four conjugates. FoT,iiOP=p, OQ = q, OB = r, 08 = s, L ■* J {s-p){r-q}' and[PW^T=i|^ik^ \s p) \r q) ^{ p-q){r- s) {p -s){q- r) = \.PQB8], ' which proves the proposition. 32 MODERN GEOMETRY. COE. It is evident that [PQBF'] = [P'^BT]. 28. Prop. Ani/ two conjugate joints form, with the two foci, an harmonic range. Let -^1^2 ^^ *^^ ^^^h t^en K,P=p-lc, K,P=p + }c, then E,P.K,F={p-h)(^ + k)=^{f-h% and K,P'.K,P^{h-^{h+p)=^-{f-¥); .-. K,P.K^ = K^F.K^P, or the four points in question form a harmonic range. Conversely, if there be a system of pairs of points in a straight line, such that each pair forms, with two given points, an harmonic range, the aggregate of the pairs of points will form a system in involution, of which the two given points are the foci. 29. A system of straight lines, intersecting in a point, may he treated in the same manner as a system of points lying in a straight line, the sine of the angle between any two lines taking the place of the mutual distance of two points. From the "proposition, proved in Art. 20, it will follow that, if a system of straight lines in involution be cut by a transversal, the points of section will also be in invo- lution. 33 CHAPTEE II. V SPECIAL FOEMS OP THE EQUATION OF THE SECOND DEGREE. 1. We now proceed to the discussion of the curve repre- sented by the equation of the second degree. We shall first prove that every curve, represented hy such an equation, is what is commonly called a conic section ; and then, before proceeding fui'ther with the consideration of the general equa- tion, shall investigate the nature of the curve corresponding to certain special forms of the equation. , PeoP. Every curve represented hy an equation of the second degree is cut hy a straight line in two points, real, coincident, or imaginary. The general equation of the second degree is represented by u^ + v^ + wrf + 2m' ySv + 2«' 7a + 2to' a/3 = 0, To find where the curve, of which this is the equation, is cut by the straight line h. + JW/S + W7 = 0, we may eliminate a between the two equations. This will give us a quadratic for the determination of — , to each of the two values of this ratio, real, equal, or imaginary, one value of a will correspond ; whence it appears that the straight line and the curve cut one another in two real, coincident, or imaginary points. Hence, the curve is of the same nature as that represented by the equation of the second degree in Cartesian co-ordinates, and is, therefore, a conic section. F. 3 34 MODERN GEOMETBT. 2. We shall now inquire what are the relations of the conic section to the triangle of reference, when certain rela- tions exist among the coefficients of the equation. First, suppose u, v, w, all = 0. The equation then assumes the form 'u'0y + v'rya + w'al3 = O, which we shall write X/37 + ftya. + vayS = 0. Now, if in this equation we put a = 0, it reduces itself to which requires either that /3 = 0, or that 7 = 0. It hence appears that the curve passes through two of the angular points {B, C) of the triangle of reference. It may similarly be shewn to pass through the third. Hence the equation \^y + /xrya + va^ = 0, or, as it may also be written, a /3 7 represents a conic, described about the triangle of reference. 3. Let us now inquire how the line is related to this conic. If in the equation of the conic we put - +"^=0, or, which is the same thing, /i7 + 1//3 = 0, it reduces to \yS7 = 0. Hence the line - + -i = meets the conic in the points in which it meets the lines /3 = 0, 7 = 0; but these two points coincide, since the line in question evidently passes through the point of intersection of ^ = and 7=0. Hence the straight line and the conic meet one another in coincident points, that is, they touch one another at the point A. CENTEE OF THE CONIC. 35 Similarly, the equations of the tangents at B and C are - + i = 0' A, jJ, 4. To determine the position of the centre of the conic. Through the angular points A, B, C of the triangle of reference draw the tangents EAF, FBD, DCE. Bisect Fig. 14. AC, AB respectively in H, I, join EH, FI, and produce them to intersect in 0. Then, since every straight line drawn through the intersection of two tangents so as to bisect their chord of contact passes also through the centre, will be the centre of the conic. 3—2 36 MODERN GEOMETRY. Now, at the poin1 \,E, we have a. I at the point H /3 = 0, cy = aa. Hence the equation of UK is vc \A, (ij \a Xji vj Similarly that of FI is \a \jji v) jih \v xj ' Hence, at the point 0, Xa \/i v) fib \v Xj vc [^ fij ' a ^ 7 X fi V — Xa + iJ^ + vc Xa — fji,b+ vc Xa + fib — vc' These equations determine the position of the centre. Cob. We may hence deduce the relation which must hold between X, fi, v, in order that the conic may be a para- bola. For, since the centie of a parabola is at an infinite distance, its co-ordinates will satisfy the equation aa + b^ + cy = 0. We hence obtain the following equation : W + fiV + A' - 2fJLvbc - 2vXca - 2X/jiab = 0, CONDITION FOE A PARABOLA. 37 which is equivalent to ± 0^y± (jJ'hy± {vc)*=0, as the necessary and sufficient condition that the conic should be a parabola. 5. To determine the condition that a given straight line may touch the conic. If the conic be touched by the straight line (Z, m, n) , the two values of the ratio yS : 7, obtained by eliminating a between the equations \^j + fiya. + va^ = 0, la, + JWyS + W7 = 0, must be coincident. The equation which determines these is - Xl^y +(jj/y + v^) (jw/3 + ny) = 0, and the condition that the two values of /3 : 7 be equal, is 4/in . vm — {iMin + vn — Xl)' = 0, or X'l' + fi^mJ' + fV — 2fiv . mn — 2v\ . nl — iXfi ,lm = 0, which is equivalent to ± (X,Z)* ± [iimf ± {vnf = 0. If this be compared with the condition investigated in Art. (4) that the conic may be a parabola, it will be observed that the parabola satisfies the analytical condition of touching the straight line aa + J/S + 07 = 0. This is generally ex- pressed by saying that every parabola touches the line at infinity. 6. To investigate the equation of the circle, circumscribing the triangle of reference. This may be deduced from the consideration that the co-ordinates of the centre of the circumscribing circle are respectively proportional to cos A, cos £, cos (see p. 4). Or 38 MODERN GEOMETEY. it may be independently investigated as follows. Draw EAF, FAD, DAE (fig. 2), tangents to the circle, then the angle EAC is equal to ABG, and FAB to ACB (Euc. ill. 32). Hence the equation of the tangent EAF must he /3 sin B sin C = 0. ^ig- IS- or 1+^ = 0. c Similarly the equations of the other tangents FBD, DCE are c a a , /3 a - + ^=0; CONIC TOUCHING THREE SIDES. 39 and, comparing these with the forms of equations of the tangents given in Art. (3), we see that the equation of the circumscribing circle is a b c . „ or, as it may also be written, sin A sin 5 sin (7 +-— — + =0. a /O 7 7. Having thus discussed the equation of the conic, cir- cumscribing the triangle of reference, we may proceed to in- vestigate that of the conic which touches its three sides. The condition that the conic wa' + v^' + wf + 2m' jS7 + 2v' 7a + 2w' a/8 = 0, may touch the line a = is, that the left-hand member of the equation obtained by writing a = in the above may be a perfect square. This requires that m" = vw, or u = ± (vioy. Similarly, v = ± (wm)^ w' = ± {uv)^, are necessary conditions that the conic should touch the lines /3 = 0, 7 = 0. We must observe, however, that if the conic touch all three of the sides of the triangle of reference, the three double signs in the above equations must be taken all negatively, or two positively and one negatively. For, if they be taken otherwise, the left-hand member of the equation of the conic will become a perfect square, as may be ascertained by sub- stitution, and the conic will degenerate into a straight line, or rather into two coincident straight lines. 40 MODERN GEOMETET. Taking then the double signs all negatively, and writing for convenience, i", if, N'', instead of u, v, w, the equation of the conic which touches the three sides of the triangle of reference becomes iV + Jir/3' + N'rf - HMN^y - 2NLyoi - 2iJfa/3 = 0, which is equivalent to It may be remarked, that the condition that the point (l, m, n) should lie in the above conic, is the same as the con- dition that the straight line (Z, m, n) should touch the cir- cumscribing conic L^y + Mya. + Na^ = 0. See Art. 5. This we shall return to hereafter. 8. To find the centre of the conic. Let D, E, F be the points of contact of the sides BC, CA, AB respectively. Join EF, FD, BE, bisect FB, BE in H, I, join BH, CI, and produce them to meet in 0. Then will be the centre of the conic (see p. 32). We have then to fipd the equations of BH, CI, which, by their in- tersection, determine 0. Kg. 1 6. CO-OEDINATES OF THE CENTRE. 41 Let f^ , _^j, \ be the co-ordinates of D. Then j^ = ; and 5fj, Ti^ will be the values of ;8, 7, which satisfy the equations W^ + iVy - 2MZV37 = 0, or Jf/3-iV7 = 0, and l^ + cy = 2A. Hence, g',=Trri — jt- 2 A J_ M M^ N 2A. In like manner it may be proved that, if/j, g^, h^ be the co-ordinates of E, _1_ Now, for 7, and therefore for every point in the line CI, Therefore the equation of GI is or iV6 -h Ifc Lc + Na' Similarly that of BE is 7 _. ■ Ma + Lb Nb + Mc' 42 . MODERN GEOMETEY. Therefore at the point we have a ^ _ 7 m + Mc Lc + Na Ma + Lb' These equations with aa + 5/3 + C7 = 2A determine the co-ordinates of the centre. COE. Hence may be obtained the condition , that the conic may be a parabola. For the centre of a parabola is infinitely distant, its co-ordinates must therefore satisfy the algebraical relation aa + 6/3 + C7 = 0, whence we get Lie + Mm + Nab = 0, LMN . or — + T" "• = 0' a c as the required condition. This will be observed by reference to Art. 9, to be iden- tical with the condition that the conic should touch the straight line, aa+b^ + cy = 0, and thus we are again led to the conclusion noticed in Art. 7, that every parabola touches the line at infinity. 9. To find the condition that the conic should touch a given straight line. If the straight line (Z, m, n) be a tangent to the conic, the values of the ratio /3 : 7, obtained by eliminating a between the equation of the conic and the equation ' la + »»/3 + M7 = 0, must be equal to one another. For this purpose, it is most convenient to take the equation of the conic in the form CONDITION OP TOUCHING A GIVEN STRAIGHT LINE. 43 Eliminating a, we then get L {m^ + ny) + l {(il//S)* + (iV?/)^}' = 0, or {Lm +MI)I3+ {Ln + JV?) 7 ± 2l (ilfiV/S^)* = 0, and, if the roots of this, considered as a quadratic in ( — j be equal, we have (Lm + Ml) {Ln + Nl) - VMN= 0, or Lmn + Mnl + Nhn = 0, which may also be written L M N ^ T- + - + - = 0- lmn It hence appears that the condition, that the line (Z, m, n) should touch the conic (ia)i + (ilf/3)^+(iV"7)* = 0, is identical with the condition that the point (Z, m, n) should lie in the conic L M N ^ -+^ + - = 0; a P 7 a result analogous to that obtained in Art. 13, chap. i. 10. To find the equations of the four circles which touch the three sides of the triangle of reference. These may be obtained most readily by the employment of the equations for the determination of the centre, obtained in Art. 8. Thus, let it be required to find the ratios of L, M, N in order that the conic may become the inscribed circle. At the centre of this circle we have, as we know, a = ^ = 7. 44 MODERN GEOMETEY. This gives, by the result of Art. 8, Nb + Mc = Lc + Na = Ma + Lb. To solve these equations, put each member equal to r, we then get M N_r^ h'^ c~hc' c a ca' L M _ r a b ab' Adding together the last two of these equations, and sub- tracting the first, we get r (b + c — a) L = 2 be Similar expressions being obtained for M and N, we see that b+c—a c+a—b a+b—c L:M:N:: be ' ca ' ab O :: cos — : cos — : cos 2 2 2 Hence the inscribed circle is represented by the equation cos -— , a^ + cos — . y8^ + cos — . 7^ = 0. 2^2 It may similarly be proved that the escribed circles, of which the centres are respectively given by _a = /3 = 7, a = -)S = 7, a = ^ = -y, will be represented by the equations A , ,i , . B ^, . . G cos- : (- a)* + sin - , /3^ + sin - . 7^ = 0, ESCEIBED CIRCLES. 45 A. H C sin — . a* + cos — (— yS)* + sin — . 7^ = 0, . A 1 . B ^i C , ^1 ^ sin -- . a^ + sm — . /3^ + cos — (— 7)* = 0. We may remark that, at every point in the circle which touches BG externally, a. is essentially negative, so that the form (— a)^ represents a real quantity. Similarly the appear- ance of (— yS)*, (—7)^ in the equations of the other two escribed circles may be accounted for*. 11. The next form of the general equation of the second degree which we propose to consider is that in which u , v, w, the respeictive coeflacients of 2/87, 27a, 2a/3, are all = 0. The equation then assumes the form We observe in the first place, that if this equation represent a real conic, the coefficients of 0^, /3^, 7", cannot be all of the same sign. Suppose the coefficient of a" to be of a different sign from the other two, then writing, for convenience of future investigations, L", — M", — JV" for u, v, w respectively, our equation assumes the form iV - M'/B" - Ny = 0. 12. We have now to enquire how this conic is related to the triangle of reference. Putting /3 = 0, we get La=± Ny. ABC • If these equations be rationalised, and the sines and cosines of — , — , — ^ ^ ^ be expressed in terms of the sides, they assume the following forms : a''(a-aya' + l^{s-h)^P^ + (?{s-cf'f-2ic{s-h){a-c)py-2ca(s-c){s-a)ya -2ab{s-a) (s-6)oj3=0, oV a" + 5» (3 - c)» |3« + c= (3 - 6) V - 25c (s - J) (« - c) /37 + 2cos (8 - J) 7a + 2abs{s-c)ap=0, 46 MODERN GEOMETRY. The interpretation of this equation is, that the two straight lines drawn from B to the points in which the conic is cut by GA, form, with BC, BA, an harmonic pencil. It may similarly be proved that the two straight lines drawn from G to the points in which the conic is cut by AB, form, with GA, CB, an harmonic pencil. If we put a = 0, we get M^=± V(- 1) Ny, shewing that BG cuts the conic in two imaginary points. The analytical condition of harmonic section is, however, satisfied here also. 13. We may next investigate the equations of the tan- gents drawn through the points A, B, C. If in the equation of the conic we put La — Ny, we get /8 = 0, shewing that the straight line La — Ny = meets the conic in two coincident points, and, therefore, touches it. Similarly La + Ny = 0, La-M^ = 0, La + M^ = 0, are tangents to the conic. The tangents to the conic drawn through A would be analytically represented by the equations Jlf/3 = V(- 1) Ny, M^ = - V(- 1) Ny, which shew that these tangents are imaginary, or that the point A lies within the concavity of the conic. 14. Since the two tangents drawn through B meet the conic in points situated in the line GA, it follows that GA is the chord of contact of tangents to the conic drawn through B, or that GA is the polar of B, and B the pole of GA with respect to the conic. Similarly, G, AB, stand to one another in the relation of pole and polar. Again, since the pole of AB is the point G, and the pole of A C is the point B, it follows that the line joining B and C is the polar of the point of intersection of AB, A C, i. e. that A is the pole of BG, and BC the polar of A. CONDITION OF TANGENCT. 47 We come then to this conclusion, that when an equation of the second degree does not involve the terms /37, 7a, a/3, the conic represented by it is so related to the triangle of reference, that each side of the triangle is the polar, with respect to the conic, of the opposite angular point*. This is expressed by saying that the triangle is self-con- jugate with respect to the conic ; or that the three angular points of the triangle form a conjugate triad. The geometrical properties of the conic having been thus established, we shall, in future investigations, write for the sake of symmetry of form, — L^ instead of U, so that the equation of the conic will be written It must here be borne in mind that one of the three quan- tities L, M, N\s, essentially imaginary. 15. Any two conic sections represented by such equa- tions as have important relations to one another, which we proceed to consider. They will of course intersect in four points, which may be real or imaginary. We will first suppose them real, and represent them by the letters P, Q, R, S. Now the locus of the equation {L'M" - L"M') ^' + {L'N" - L'^N") 7^ = passes through all the points F, Q, R, 8; and, since it may be resolved into linear factors, represents two straight lines. • If the coeflBoients of /3^ and y be equal, and the triangle of reference be right-angled at A, the form of the equation shews that A will be a focus of the conic, and BC the corresponding directrix. 48 MODERN GEOMETBY. Suppose them to be PQ and BS. The intersection of these two straight lines is given by the equations (iW= - L'^W)^ ^ = - (i' W - iW") i 7, which evidently give yS = 0, 7 = 0. Hence PQ, MS intersect in A. Similarly, PE, Q8 intersect in B, and PS, QR intersect in C. Hence, the angular points of the triangle of reference coincide with the intersections of the line joining each pair of points of intersection of the conies with the line join- ing the other pair. Hence also, if any number of conic sections be described about the same quadrangle*, and the diagonals of that quadrangle intersect in A, while the sides produced intersect in B and G, then A, B, G form, with respect to each of the circumscribing conies, a conjugate triad. The points A, B, G may themselves be called vertices of the quadrangle, or of the system of circumscribing conies. It will be seen, from the preceding investigation, that ani/ two conies which intersect in four real points can be reduced, by a proper choice of the triangle of reference, to the form The same reduction may also be effected in every case with the reservation that if two of the points of intersection of the conies be real and two imaginary, then two of the angular points of the triangle of reference (or vertices) will be imaginary and the remaining one real. If all the points of intersection be imaginary, the vertices of the conies wiU be all real. This we shall prove hereafter. 16. To find, the condition that a given straight line may touch the conic. Let the equation of the straight line be la. + m/S + W7 = 0. * I employ the term quadrangle in preference to qtrndrilateral, coneidering a quadrangle as a figure primarily determined by four point/, a quadrilateral by four indefiDite straight Una. CONDITION OF TANGENCT. 49 Where this meets the conic, we have and, making the two values of yS : 7 equal, we get (LW + M'r) (iV + IPl'^ = i'mV, whence M^N^l^ + N'LW + L'M^' = 0, the required condition. 17. 2b ^raa' ' , , 4A.i'c Similarly ''^^^iPa' + LV Hence the equation of 5 § is 7 _ a or N'y _ i'^a This gives one straight line on which the centre lies. It may be similarly proved to lie on the straight line X°a_ilf'j8 a b Therefore the co-ordinates of the centre are given by the equations a b c ' Combining therewith aa + J/S + C7 = 2A, we get for the co-ordinates of the centre a b c T? „^ W „. IT* ^^T^ 55i TT, 2A-3 75 y, 2A- SELF-CONJUGATE CIRCLE. 51 Each of these becomes infinite when the conic is a para- bola, as manifestly ought to be the case. 19. To find the equation of the circle with respect to which the triangle of reference is self-conjugate. It is a distinguishing property of the circle that the line joining the centre with any other point is perpendicular to the polar of that point. Hence the line -^ - - b c ~"' which joins the centre with the point A, must be perpendicu- lar to a = 0. This gives (see Art. 5, p. 8) b cosB ccos C Similarly, since 'the lines joining the centre with B, C are respectively perpendicular to j8=0, 7 = 0, we shall have c cos G a cos A ' a cos A b cos B ' Hence the equation of the required circle is a cos -4. a' + 5 cos 5. /3'' + ccos G.'f = 0, or sin2^.a'' + sin25./3' + sin2(7.y=0. It will be remarked that this circle will be imaginary/, unless one of the quantities sin 2 A, sin 25, sin 2 (7 be nega- tive, that is, unless one of the angles 2 A, 2B, 2(7 be greater than two right angles, or unless the triangle of reference be obtuse-angled. COK. By referring to the expressions for the co-ordinates of the centre of the conic, given in Art. 18, we see that at the centre of the circle we have a cos ^ = jS cos 5 = 7 cos G. 4—2 52 MODEBN GEOMETRY. Or, the centre of the circle, with respect to which the tri- angle of reference is self-conjugate, coincides with the intersec- tion of the perpendiculars drawn from the angular points to the opposite sides. This is otherwise evident from geometri- cal considerations. 20. To find the equation of tJie conic which touches two sides of the triangle of reference in the points where they meet the third. Let AB, AC \)& the two sides which the required conic touches in the points B, C. We then require that the con- stants in the equation ia" + M^ +Nrf + 2XjS7 + Z/iya -t- 2va/3 = should he so related to one another, that when /8 = we have the two values of a = 0, and also when 7=0 the two values of a may each = 0. Hence the two equations L^" + Nrf + 2fi'ya = 0, La^ + M^'-i-2pa^=0, must both be identically satisfied when a = 0, and hy no other value. This requires that JV = 0, 11 = 0, M=0, v = 0. Hence -the -equation reduces to or, writing — P for — , Fa' = ^7. This equa,tion, it will be observed, involves only one arbi- trary constant, as ought to be the case, since when a tangent and its point of contact are given, the conic is thus subjected CONIC TOUCHING TWO SIDES. 53 to two conditions, and, therefore, when two tangents and their points of contact are given, to four, 21. If any straight line whatever be drawn through A, and meet the conic in P, Q, and be represented by the equation then BF, BQ will be represented by the equations ka = ny, ka. = — ny, from the form of which it is apparent that BA, BP, BC, BQ form an harmonic pencil. Or ani/ chord of a come is divided harmonically/ by the conic itself, any point on the chord, and the polar of the point with respect to the conic. 22. We may observe, that the two straight lines repre- sented by the equations ka. = ft)/3, ka = -y, €0 intersect on this conic whatever be the value of ta. Hence any point on the conic may be expressed by giving the value of the ratio ka. 7 If (o be the value of this ratio at any point, that point •may be denoted by the letter a*. The line joining the two ppints a>, a>' may be called the line axo'. 23. To find the equation of the line coco'. Let the required equation be ka. + jw/3 + W7 = 0, we have then to determine m and n. Since, when ka. = aB, ka = — j, we get 1 H \- no} = 0. ° CO • This mode of expression is given by Salmon in his Conic Sections. 54 MODERN GEOMETRY. 971 Similarly 1 + -; + noa' = 0. Hence m = to O) and n = ; . (0+ CO Hence the line wta' is represented by the equation ww'/S + 7 = (&) + «') kx. 24. To find the equation of the tangent at co. This is obtained at once, from the result of the preced- ing article, by simply putting w' = to. It will then be seen to be w'/S H- 7 = 2'/3 -7 = 0, 2a)'.A:a-ft)"/3-7 = 0, whence ^a /8 7 6)" — 0)'' 2 (ft) — ft)') 2&)ft)' (o) — co') ' 2ia „ 7 or -——,=^=-±. ft) + ft) &)&) 26. ^0 _^w K ^^ *^s co-ordinates of the points in which the straight line meets the conic, those of the centre will be A+A 9^+9, K + h 2 ' 2 ' 2 " N'ow^,/^, ff„, g^, \, \ are the respective values of a, ^, 7, obtained from the equations iV - /37 = 0, J/3-C7 = 0, aa + &/3 -H C7 = 2A. 56 MODERN QEOMETRT. Eliminating 7, a between these, we get a'5^ = 4Fc(5/3-A)', whence 9^+9.^ 4&°cA „. ., , K + \ 4FSA Similarly __^ = ^^^,^_^. These are the values /8, 7 at the centre. The correspond- ing value of a may be ascertained by substitution in the equation aa + J/3 + C7 = 2 A * , -2aA to be 4F&C - a' ' These values all become infinite when 4FSc = o", as mani- festly ought to be the case, since, as has been shewn in Art. 24, the conic is then a parabola. Examples. 1. A triangle is inscribed in a conic ; prove that the points, in. which, each side intersects the tangent at the opposite angle, lie in a straight line. 2. A triangle is described about a conic ; prove that the straight lines, joining each angular point with the point of contact of the opposite side, intersect in a point. + 3. Find the equations of the normals to the conic X^y+fiya . voyS = 0, drawn at the angular points of the triangle of reference ; and prove that they will intersect in a point if 4. If the normals to a circumscribing conic, at the angular points of the triangle of reference, meet in a point, prove that the EXAMPLES. 57 locus of tte centre of the conic is made up of the curve repre- sented by the equation and of the three straight lines -which join the middle points of the sides of the triangle of reference. 5. Three conies are drawn, touching respectively each pair of the sides of a triangle at the angular points where they meet the third side, and all intersecting in a point. Prove that the three tangents at their common point meet the sides of the triangle ■which intersect their respective conies in three points lying iu a straight line ; and that the other common tangents to each pair of conies intersect the sides of the triangle which touch the several pairs of conies in the same three points. 6. Prove that the points of intersection of the opposite sides of any quadrangle, and the point of intersection of the diagonals, form a conjugate triad with respect to any conic described about the quadrangle. 7. If i? be the radius of the circle described about the triangle of reference, p that of the circle with respect to which the triangle of reference is self-conjugate, prove that p' + 45^ cos A cos B cos C = 0. 8. If BC, GA, AB be three given tangents to a conic, P, Q, R three points on the curve, and if the areas of the triangles PBG, PCA, PAB be denoted by p,, p^, p^, respectively, and those of the triangles obtained by successively writing Q and R in place of P by q^, q^, q^ r„ r^, r^, prove that 9. Prove that the diagonals of any quadrilateral described about a conic, and the lines joining the points of contact of .opposite sides, all intersect in a point. 10. A system of conies is described touching three straight lines ; prove that, if one of the foci move along a given straight line, the other will describe a conic about the triangle. Hence prove that the circle, which passes through the points of intersection of thi-ee tangents to a parabola, passes also through the focus. 58 EXAMPLES. 11. ProTe that the equation of the line passing through the feet of the perpendiculars from a point a,, yS,, y,, of the circle ajSy + hya + caji = 0, on the sides of the triangle of reference, may be put in the form, P^cosG—y^co&B Yj cos j1 — a, cos (7 a^ cos 5 — /3j cos il cy = 0. 12. Shew that the axis of the parabola, whose equation is a'a' = 46cj8y, is given by the equation (c + 6cosjl)/3-(6 + ccos^)y = ^f — j j aa. 13. The equation of the directrix of the parabola, which touches the sides of the triangle of reference, and also the straight line la + mfi + i/y = is a cos .4 ( ) + S cos 5 ( — 7 ) + y cos C ( -=• ) . \m n/ '^ \n IJ ' \{ mj 14. If the equation (Za)i + {m^f + {ny)^ = represent a parabola, the equation of its axis is ( 59 ) CHAPTER III. ON ELIMINATION BETWEEN LINEAR EQUATIONS. 1. Before entering upon the discussion of the conic re- presented by the general equation of the second degree, it will be necessary to devote a few pages to the subject of elimina- tion between homogeneous linear equations, and to explain some of the terms recently introduced in connection with this branch of analysis. We shall, however, only state and prove such elementary theorems as will be necessary in our future investigations; referring the reader who may be desirous of fuller informa- tion to Salmon's Lessons on the Higher Algebra ; Spottis- woode, On Determinants (the second edition of which will be found in Crelle's Journal, t. 51, pp. 209, 328), and to the ori- ginal memoirs communicated to various scientific Journals by Messrs Boole, Sylvester, Cayley, and others. 2. If we have given n homogeneous linear equations, con- necting n unknown quantities x^, a;^ ... aj„, such as a^T^ + a^^ 4- . . . + a^x„ = 0, h^x^ + b^^+... + h„x„ = 0, h'^l + ^^2 + • • • + ^n^Jn = 0> the quantities x^,x^...x„ can be eliminated between them, and the result of the elimination may be expressed by omitting x^,x^...Xn, and writing the coefficients only in the order in which they appear in the given equations, thus 60 MODERN GEOMETRY. .a .1 K> h • • ■ h = 0. The left-hand member of this equation is what is called the determinant of the given system of equations. We proceed to investigate the law of its formation. 3. First, suppose we have two equations, a^x^ + a^x^ = 0, \x^ + \x^ = 0. Multiply the first by h^, the second by a^, and subtract, and we get Hence K h = «A-«2^i- «1. h = «1. «. «2. K h, K We may remark in passing that we shall obtain the same result by eliminating \, \ between the equations «i\ + K\ = 0, Hence _ A like theorem will be proved to be true for all deter- minants. 4. Next, suppose we have the three equations ^A + a^i + %x^ = 0, l^x^ + 62*2 + 1^^ = 0, C,X, + C^j + CjCBj = 0. DETERMINANTS OF THREE EOWS. 61 .(A). lyiultiply these equations in order by the arbitrary mtilti- pliers \j, Xj, \g, and add them together. Let the two ratios \j : \ : \g be determined by the conditions that the coeffici- ents ot' aSj and x^ in the resulting equation shall each be zero, i.e. let o A + K\ + "A = 0" «3\ + K\ + "s^ = 0, The resulting equation is then reduced to (aA + ^A + cA) ^1 = o> which requires that aA+ 5A+ <'A = •, •(B). Multiply the first of equations (A) by a^, the second by flj, and subtract, we then get or CA-^a^a «2*a-«8^i! h^B-K'^i by symmetry. ..(C). Hence, dividing each term of (B) by the corresponding member of (C) we get «i (Vs - ^sCa) + h (''A - "3^2) + Ci {a A - a^^, or h h , «3 7, = -a,{l. Ca-^a" 2) + ^(C2«3- C3«2) "i> "a^ "3 + Ci(a2&8-«s^2) = «i + h S. «a + C. «2. «3 62,5a = «i J,,/ -"a -\ «2> «3 C2, Ca + C, ^2, «3 62 MODERN GEOMETRY. It will be seen that the ahove process is really equivalent to that of eliminating Xj, Xj, Xj between the equations (A) and (B). Hence «1> «2> ''s a^, \, Cj ^1. K h = «2. K % c„ %, c. a,. ^8. C3 5. Next, let us have the four equations ttjAjj + a^cCj + a^x^ + ap^ = 0, JjiTj + h^^ + \x^ + &,ar, = 0, c^x^ + CjXj + 0,333 + c^aJi = 0> tZ^aj^ + rf^2 + ^%^% + <^A = 0. To effect the elimination, multiply the equations in order by Xj, Xj, Xj, X^, add them, and equate the coefficients of ajj, ojj, o;^ severally to zero. "We shall then have a^X, + JjX^ + c,X, + <;,X, =: 1 a,Xj + JjXj + CjX, + Ca, C?s «2> Cj, C^j 6a> Cs, d^ «8. C3. <^8 K C4. < Oa> *s, ^3 «4. h, ^4 Oj, K Cj «8. ^8. Ca «4> ^4. C4 These equations may be more conveniently written in the following equivalent forms : K K h «a, «a. «4 a„ «3: «4 «S, Oa. «4 <=,, C3> C4 . - c., "a. C4 K> K ^4 — &., h, *4 d,, ^a, d. ^8) <^4 -h «2> «8' "4 + c. «a> <*8> «4 ia. 58. K di, d^, d. -d. «2> «S> <*4 6a, ^8. \ d^, <^4 And since the above process is equivalent to tlie elimi- nation of X.J, \, Xj, X^ between the equations (A') and (B'), we see that «i, «2. o„ a^ h' K> K \ ^1, "2, ''s) "4 t?i, *8, C8...A;3 Aj, Kj, k^...K„ ««, 5„, c^'-K 5a> J8-S» «2, «8 — <*« =«1 Ca, C,...C„ *2» ""8 ■••""« -5. Cj, C3...C„ • ■ • ■ • • + ... + (-1)-^^;, «2, «8 I =2, ^8 ...A„ DETEKMINANTS OP FOUB BOWS. 65 It may also be proved that if we have m — 1 equations connecting n quantities X„ X,... X„, sucli as flj \ + Jj\ + Cj\ + ... + k,\„ = 0, «3 \ + h\ + c^\ + .-+ K\ = 0, «4 \ + ^A + cA + — + ^A = 0. a„\ + h„\ + c„\+ ... +k„\ = 0, we shall obtain the following ratios between \,\,X^,...\„, h,c,. ■K K^s- ■ K — K,Cn- ■K a„ c,. ..*, «3. c,. ../fcs «B, c„. •4 O3' *S— ^3 On, K---K i-ir «2. *2- «8. h- By reference to the expanded values of the determinants «1. «2. «3 h, KK Cx> C.. C3 it will be seen that the former contains 1 . 2 or two terms, the latter 1.2.3 or six. It may also be proved that, if w quanti- ties be eliminated from n linear homogeneous equations, the resulting determinant will contain 1.2.3...n terms. For, referring to the relation between determinants of n and « — 1 rows, given in Arts. (4), (5), (6), it will be seen that this theo- rem is true for a determinant of n rows, if it be true for one of n — 1. But it is true for three rows, therefore it is universally true. 7. The horizontal rows of a determinant are commonly spoken of as "lines," the vertical ones as "columnsi" It F. 5 a,, ttj, «8 K h, J, C,, Cj, C3 66 MODERN GEOMETRY. will be observed, moreover, that each term is the product of n factors, one taken from each line and from each column, and that the coefEcieijts of one half of the terms are + 1, of the other— 1. To determine the sign of any particular term we proceed as follows. Considering for simplicity the case of three rows, we have Here we observe, first, that (the factors of each term being arranged in alphabetical order, that is, in the order of the columns) the term ajb,c, (in which the suffixes follow the arithmetical order, that is, the order of the lines) has a positive coefficient. Now every other term may be formed from this by making each suffix change places with either of its adjacent suffixes a sufficient number of times. Thus the term afi^c^ is produced by simply making the suffixes 2 and 3 exchange places. The term ajb^c^ is produced by making the suffix 3 change places, first with 2, and next with 1, which is then adjacent to it. If this process of interchanging the suffixes of two consecutive letters be called a " permutation," we may enunciate the following law, which by inspection will be seen to hold. " Every term derived from the first by an odd number of permutations has a negative sign. Every term formed by an even number of permutations has a positive sign." Thus, it will be observed that the terms OjJjC, ; aj>^c„ each of which is derived from afi^c^ by one permutation, have negative signs. The terms ajb^c^; aj>c„ each formed by two permutations, have positive signs. The term a^b^c,, formed by three permutations, has a negative sign. In like manner, in the case of a determinant of four rows, if apffi^ have a positive sign, such a term as aj>^cji^, derived by two permutations, will nave a positive sign, while ap^cji^, derived by three, has a negative sign. 8. The sign of a determinant is changed by interchanging any two consecutive lines or columns. CHANGE OF SIGN. In the first place, we observe that = aj}^ - aj>, = - {J.a. - 5,a.) = - 67 Again, C« 1 Co » Co h>K = a. h, K -I. = — a. h, \ + c. "2> "8 5., i. -J, by what has been shewn above, The theorem enunciated is thus proved for determinants of two and of three rows, and may by successive inductions be extended to any number. Cor. It hence follows that, if any two lines or columns of a determinant be identical, the determinant will vanish. For we see, by the theorem, that and therefore = 0. • 9. We see that OTC, "!> '<) «8 ^3 = WlflSj KK — ?w5, a.. «3 3 Cj. c. "j. c. Cs + mCj «.. «3 S., *3 = m «1. a„ «8 ^, K, ^3 Cl, c„ Cj 5—2 68 MODERN GEOMETBY. Hence, if all the terms in any line or column of a determi- nant he multiplied by any given quantity, the determinant itself will he multiplied by the same quantity. 10. Def. From any given determinant, other determin- ants may be formed, by omitting an equal number of lines and columns of the given determinants. These are termed Minors of the given determinant, and are called first, second, &c. minors, according as one, two, &c. lines and columns have been omitted. Thus h,h are first minors of a, "■>» "8 a. ■l> "2' 11. To investigate the relation which must hold amoruj the coefficients L, M, N, X, /t, v, in order that the quadratic function La' + M/?" + Ny" + 2\/37 + 2/t7a + 2i'a/9 muy be the product of two factors of the first degree in a, P, 7- The given expression is identical with (ia+ vyS + py) a+ (i/a +M/3+-\/y) ^ + Qioi + \^ + Ny) 7. Now, if the relation between L, M, N, X, fi, v be such that, for all values of a, /3, 7, the three linear functions La + v^+fj/y, va + M^ + Xry, fioi+\0 + 2fy may bear to one another constant ratios {p : q : r, suppose), then the given expression will be the product of two factors,' respectively proportional to La + v0 + /j^, pa + q^+ry. DISCRIMINANT. 69 The necessary condition is then that La + v^ + fiy _ va + M^ + 1^ _ fi.a+\^ + Ny p ~ q ~ r for all values of a, /8, 7, and therefore for those which make the numerators of any two of the above fractions = 0. That is, values of a, ^S, 7 exist, which simultaneously satisfy the equations ia + v^ + fiy =0, pa+M^ + \y= 0, /ia + X^ +Ny = 0. Hence, eliminating a, /3, 7, we get, as the condition that the given expression may be the product of two factors, the equation L, V, fi V, M, X =0, /*, X, N or LMN+ 2Xfiv - LX^ - Mfi" -Nv' = 0. This expression L, V, fi V, M, X y-, \ N the evanescence of which is the necessary condition that the given quadratic function may break up into two factors, is termed the Discriminant of that function. 12. Pascal's Theorem. From the analytical result stated in Art. 6 of the present chapter, that the value of a determinant is not altered by changing its lines into columns and its columns into lines, we obtain a proof of Pascal's theorem, which asserts that If a hexagon be inscribed in a conic, and the pairs of opposite sides be produced to intersect, the points of intersection lie in the same straight line. 70 MODERN GEOMETET. Let AFBDGE be the conic ; take ABC as the triangle of reference, and let the equation of the conic be a P 7 (!)• Let the equation of AE be /S = n^, of AFhe, 7 = mfi, BF ...y=l^a, oiBD ...a= n^y, CD ... a = «t.;S, oiCE...^^l^a. Then, since D lies in the conic (1), we have \ + fim^+ vw, = 0, E X?j + /*+TO, = 0, F X?5 + /l,7W3 + l/=0, whence 1, JW, «I k, 1, «2 4, OTj, 1 = 0j. .(2) is the necessary condition that the six points A, F, B, D, G, E may lie in a conic. Again, if the pairs of opposite sides intersect in points lying in a straight line, let the equation of that straight line be jja + j/S + ry = 0. Then, since BF and CE intersect in this line, we. have p + qli + vl^ = 0, CPand AF pm^ + q + vm^=0, AE&nA. BD jWi + 2«j + V = 0, whence 1, I. I »Mj, 1, Wis «1. «2 1 = 0, ...(3) is the condition that these points of intersection nnay lie in the same straight line. But (2) and (3) are identical. Hence the proposition is proved. 13. From Pascal's Theorem many interesting conse- quences may be deduced. Thus, if the point F coincide with^ PASCAL S THEOREM. 11 A, D with B, E with C, then AF, BD, CE become the tangents at A, B, C respectively, and we obtain the theorem enunciated in Ex. 1, Chap. ii. Again, by supposing J) to coincide with B, and E with C, we readily obtain the follow- ing theorem : " If the opposite sides of a quadrilateral, in- scribed in a conic, be produced to meet, and likewise the pairs of tangents at opposite angles of the quadrilateral, the four points of intersection will lie in the same straight line." And, by supposing F to coincide with A, we obtain a geometrical construction, by which, having given five points of a conic, we can draw a tangent at any one of them. For, since AF then becomes the tangent at A, we see that, if AE, DB be produced to meet in Q, AB, EC in H, and GH in- tersect UD in /, then AI will be the tangent at A. EXAMPLES. 1. Prove that o, h, c, d i. «. d, c c, d, a, b d, e, b, a = {a + b + e + d){a — b + c — d) {a — b—c+d){a+b-c—d). 2. If c , b = j1, ..., prove that and that A„A„A 8 <^X,«2,«8 3 = K K K a <^i' ^a) ^3 72 3. If b, c and = A, b', c y, « MODERN GEOMETET, a, b c, a z, X = B, = A', = ■8-, = C, a',b' = C7', prove that 4. Prove tliat 5. Prove that 6. Prove that B, G B,G' 8 + G,A C',A' + A, B A',B' a = a, b, c a'.b',c' X, y,z 0, 1, 1. 1,... 1, 0, a+b, a+c... 1, b+a, 0, b+c... 1, c+a, c+b, ... = a5c. {x'+f+z'). /Ill \ \a 6 c / m + n-y+z, —y + z — l, —y + z-l — z + x-m, n+l — z + x, —z+x-m — x + y — n, —x + y — n,l+m—x + y = 0. b + c a b + c' b c + a' a + b a a a ' c + a b + c c + a b c 2{a + b + cY b ' a + b (6 + c){c + a) {a + b) c ' a + b ( 73 ) CHAPTER IV. ox THE CONIC EEPKESENTED BY THE GENERAL EQUATION OP THE SECOND DEGREE. 1. We may now proceed to the discussion of the gene- ral equation of the second degree, which we shall express under the form, uo? + v0' + wi' + 2u'^y + Iv'r^d + 2w'ay3 = 0. This we may write, for shortness, («, /3, 7) = 0. This equation, as we have shewn (Art. 1, Chap. 11.), represents a conic section. 2. To find ike point in which a straight line, drawn in a given direction through a given point of the conic, meets the conic again. Let /, g, h be the co-ordinates of the given point, a, /3, 7 those of any other point whatever. Then, for all points of the straight line joining these two, the quantities a-f, fi-ff, f-K will bear constant ratios to one another. Let these ratios be denoted hy p : q : r, so that we have a — f li — q y — h — - = - — - = = s, suppose. p q y ' rr 74 MODERN GEOMETRY. To find where the line again meets the conic, we must substitute in the equation of the conic f+ps for a, g + qs for yS, h + rs for 7. We thus get, arranging the result according to ascending powers of s, ^ (/) 5'. ^) + 2 { {up + w'q + v'r) /+ {w'p +vq-\- u'r) g + {v'p + u'q + wr) Ji]s+ ^ [p, q, r) s' = 0. The two roots of this equation, considered as a quadratic in s, determine the two points where the line meets the conic. Now, since {/, g, h) is, hy supposition, a point on the conic, it follows that

d'^ d' I m = d/3d0L' d^' d^drf' d' d^ dr^da' d-/d^' drf' ^ I, m, n, as the necessary condition that the line (l, m, n) should touch the conic (j> {/, g, h) = 0. Expanding the determinant, this may be written {yw — m") V + (mm — «'') m' + {uv — w'^) n'+2 {v'w' — mm') mn + 2 (m)V - vv') nl + 2 {u'v' - ww) Im = 0. 5. The coefficients of P, m", n", 2mn, 2nl, 2lm, in the above equation, will be observed to be the several minors of the determinant u, 1 1 W , V w, V, u' v, 11, w They will frequently present themselves in subsequent investigations, and it will be convenient, therefore, to denote each by a single letter. We shall adopt the following notation : vw — m'" = TJ, wu — v' =V, uv — w'" = W, VW —UU — U , WU —VV = V , UV — WW = W . The condition of tangency investigated in Art. 4 may then be written, Ul^ + Vm"- + TFft" + 2 TJ'mn + 2 V'nl 4 2 W'lm = 0, CONDITIOX OF TANGENCT. 77 the same condition, it will be observed, as that which must hold, in order that the point {I, m, n) may lie on the conic Ua" + F/3= +Wrf+2 U'^y + 2 V'yx + 2 W'ajS = 0. 6. 1o find the condition that the conic may be a parabola. Since every parabola touches the line at infinity, the con- dition required will be obtained bv writing a, b, c respec- tively in place of I, m, n, in the condition of tangency. This gives, as the necessary and suflBcient relation among the coefficients, M, w', v, a =0 w', V, u', b v', u, w, c a, b, c, or Ua' + Vb^ +Wc^ + 2 U'bc + 2 V'ca + 2 W'ab == 0, 7. To find the condition that the conic may break up into two straight lines, real or imaginary. For this purpose it is necessary and sufficient that the expression «^ (a, /3, .7) should break up into two factors. The condition for this has been shewn in Art. 9, Chap. iii. to be M, w, v' =0 w, V, u v, u , w or wow + 2u'v'w' — uu'^ — vv'^ — ww" = 0. 8. To find the equation of the polar of a given point. If through a given point any straight line be drawn cutting a conic in two points, and at each point of section a tangent be drawn to the curve, the locus of the intersection of these tangents is the polar of the given point. We proceed to find the equation of the polar of (/, g, n). Let /, g^, \;f^, g^, \ be the co-ordinates of the points in which any straight line drawn through [f g, h) meets the 78 MODERN GEOMETRY. conic. Then, since (/, g, h), {f^, g^, \), (/„ g^, A,) lie in the same straight line, we have (see Art. 12, Chap. I.). Again, the equations of the tangents at ifv 9v K\ (/a. 9^, K respectively, are f^ {wx + w'^ + v'y)+g^ {w'a + v^ + u''y)+h^(v'a.+ m'/8 + wy) = 0, j^ (m + to'/3 + v'y) +g^ {w'a + v^ + u'y) + h^ {v'a+ m'/3 +wy) = 0. Where these intersect, we have Ma + w'/3 + v'y _ w'a + vl3 + u'y _ v'a + m'/3 + toy , , 9K-9h Kf\-Kf\ ~ f,92-f,9x Combining this with equation (I), we get / (wa + M)'/3 + v'y) + j (to'a + 1)/3 + u'y) +h{v'a. + m'/9 + wy) = 0, or (m/+ w'g + v'h) a + {w'f+ vg + u'h) /S + {v'f+ u'g + wh)y= 0, as a relation which holds at the intersection of the tangents ; and which, since it is independent of the values of _/„ g^, A, ; /«' 9>i ^>! ™ist be the equation of the locus of the point of intersection of the tangents drawn at the extremities of any chord passing through (/, g, h), that is, it is the equation of the polar of (f, g, h). It may also be written, It will be remarked that this equation is identical in form with that already investigated for the tangent at a point (f, g, h) of the curve. In fact, when the point (/, g, h) is on the curve, the polar and the tangent become identical. 9. To find the co-ordinates of the pole of a given straight line. Let the equation of the given straight line be la. + jwyS + Miy = 0. POLE AND POLAR. 79 If {f, g, h) be the co-ordinates of its pole, we must have, applying the equation just investigated for the polar of v,f+ w' g + v'h _ w'f+ vg + u'h _ v'f-\- u'g + wh I m n ' Putting each memher of these equations = — k,we get «/+ w'g + v'h +lk=0, w'f+ vg + u'h + mk = 0, v'f+ u'g + wh + nk = 0, whence / g h w v, I ' u, w' TO V, u, n V. u, m W, V, n u, w', I «', w, n V, u, I w , V, m These equations, together with af+ hg + ch = 2A, determine the co-ordinates of the pole. They may also be written / _ ff ^ Ul+Wm + V'n W'l+Vm+U'n V'l+U'm+Wn' 10. To find the equation of the pair of tangents drawn to the conic from a given external point. Consider the equation ^ (a, /3, 7) + i {(«/+ w'g + v'h) a + {w'f+ vg + u'h) /3 + {v'f+u'g+wh)yY = 0, where k is an arbitrary constant. This, being of the second degree in a, /8, y, represents a conic ; and meets the conic ^ (a, ^, 7) = in the two points in which that conic meets the line (m/+ w'g + v'h) a + (w'f+ vg .+ u'h) /3 + (r/+ u'g + wA) 7 = 0, so MODEEN GEOMETET. and in these points only. Hence since two conies in general intersect in four points, it follows that in this case the four points of intersection coincide two and two, that is, the conies touch one another at the two points where they meet the above-mentioned line, or have double cordact with each other. The arbitrary constant Jc may be determined by making the conic pass through any assigned point. Suppose now that the conic is required to pass through the point (/, ff, h), of which the line of contact is the polar. This gives, for the determination of h, the condition if, ff,h) + k {(«/+ w'g + v'h)f+ {w'f+ vg + u'h) g + {vf+u'g + w1i)JiY = Q, whence 1c = — -r-n — n • Hence the equation if> ff> ^) 'P («> A 7) - {(«/+ w> + v'h) a + {w'f+ vg + u'h) (S + {v'f+ u'g + wh) 7]' = 0, represents the curve of the second degree, passing through the point [f, g, K) and touching the conic <^ (a, /3, 7) = 0, at the points where the polar of this point intersects it. But this curve must evidently be coincident with the two tangents drawn from that point to the given conic (a, yS, 7) = 0. This equation may be put under another form, also under form like that of asymptotes, for the coefficients of a" will be found, by actual expansion, to be u («/' + vg^ + wh' + 2u'gh + 2v'hf+ iw'fg) - {uY + w'Y + v'V + iv'w'gh + iuv'hf+ luw'fg) = (m« - w"^) / + {vm - v") h' + 2 [uu' - v'w') gh = Wg'+Vh'-2U'gh. That of 2/87 is «' (mT + f/ + wh' + 2u'gh + 2v'hf+ Iw'fg) - {wf-\- vg + m'A) (v/+ vig + wh) = {uv!-v'w')f + (m"- vw) gh + («'«' - ww') hf+ {w'u'- vv')fg = -U'f-Ugh+Whf+Vfg. CO-OKDINATES OP THE CENTRE. 81 Similar expressions holding for the coefficients of ^, y", 27a, 2a^, we obtain the equation of the two tangents under the form ( Wg'+ Vh'-2 U'ghy+{ Uh'-\- Wf-2 rhf)0'+{Vf'+ Uff'-2 W'fg)rj' -2{Zrr+ Ugh - Whf- V'fg) 0y - 2 ( r/ + Vhf - U'fg - W'gh) 7a - 2 ( WV+ Wfg-V'gh- U'hf) a^^O. If the point (/, g, h) be within the conic, these two tan- gents will be imaginary. 11. To find the co-ordinates of the centre. Since the two tangents, drawn at the extremities of any chord passing through the centre, are parallel to each other, it follows that the polar of the centre is at an infinite distance, and may therefore be represented by the equation aa + b^ + cy=0. Hence, if a, /3, 7, be the co-ordinates of the centre, we obtain, by ah investigation similar to that of Art. 9, Ma + w'^ + v'y +ak=0, no a. + v/8 + uy -k-hk — ^S, VOL + u'0 + icy + ck = 0, (A). Hence, a /3 7 k w, v, a m', w', h v, u', c u, w, v V, u, b w, v', c u, w', a w', V, u' m', w, c v, u, a w', V, b v', u, w or ^ Ua+W'b+V'c W'a + Vb+U'c V'a+U'b + Wc ^ k uvw + 2u'v'w' — uu" — vv* — ww'* ' These equations determine the centre. F. 6 82 MODERN GEOMETET. 12. To find the equation of the asymptotes. Writing a, /3, 7, for /, g, h, in the investigation of Art. 10, and paying regard to equations (A) of Art. 11, the asymptotes will be found to be represented by the equation ^ («, A 7) (a, A 7) - {(«a + JyS + C7)'} ^ = 0, or <^ (a, A 7) ^ (a, A 7) - (2 A)= i' = 0. But, multiplying equations (A) in order by a, yS, 7, and adding, we get ^(a, A 7)+2A.A; = 0. Hence the asymptotes may be represented by the equa- tion .^(a,/3,7)-.^(a,A7)=0, or ^(a, /S, 7)+2A.i = 0, which may be put under the homogeneous form (aa + 5^ + C7) ^ (a, ^, 7) + i (aa + 6/8 + 07)' = 0. But, by the final result of Art. 11, it may be seen that gg + 5^+ g^ ^ Pffl" + Fy + PFc" + 2 C/^'ic + 2 F'ca + 2 W'ah k uvw + luv'w — uu^ — W* — ww'^ ' whence the equation of the asymptotes becomes ( Ua' + FS» + TFc" + 2 Wbc + 2 V'ca + 2 W'ah) (a, /3, 7) — (wrw + 2u'v'w' — uu'^ — vv'^ — ww") {aa. + 5/8 + 07)' = 0. This may also be written under the form u, t w, f a w i>, f u, b ■», I u, w, c a, i, c, ^ (ai P, 7) + «, to', v' w', V, u v , u, w ASYMPTOTES. 83 COE. It appears, from the preceding investigation, that if a, yS, 7 be the co-ordinates of the centre of the conic repre- sented by the equation ^ (a, /3, 7) = m' +v^ + w'f + 2u'^y + 2w'7a + 2to'a/3 = 0, then ^(a,/8, 7) = - M, w', v' W, V, u' V, f u, w 4A^ M, w, V, a w', «, u, h v'. u', w, c a. h, c, 13. To find the condition that the conic may he a rect- angular hyperbola. If the equations of the asymptotes be la. + OT/8 + n7 = 0, ra + «i';e + w'7=0, the condition of their perpendicularity is IZ + j»m' -1- nn — {mn' + m'n) cos A — (nl' + n'l) cos B — {Im + I'm) cos C = 0. Writing, for shortness, Ua^ + VV +Wc' + 2 U'hc +2V'ca + 2 W'ah = D, uvw + luvvD — mm'* — m'^ — ww'^ = K, we see, by reference to Art. 12, that W mm _ nn' Du-Ka^'Dv-KU' ~ Dw - Kc* ' _ ^ (mn' + m'n) _ ^ jnl' + n'l) _ ^ (bn' + I'm) Du'-Kbc ~ Dv'-Kca ~ Dw'-Kah 6-2 8i MODERN GEOMETRY. Hence the required condition is D{u + v + w — 2u'co3A — 2v' cos B — 2w' cos C) — K{a' + V + 'bc ^0^+ ra' -2q'ca «a* + mS" — 2w'ab ~~ qa^ + jab" — 2r'ab' (The identity of these three values of k is ensured by the condition of similarity already investigated.) k may also be written U V w u v w c^ ¥ c" bo ca ab a' b" c" be ca ab 90 MODEEN GEOMETET. Hence, the equation of the radical axis becomes ua v^ wy poi q^ ry a c a c u V w u w w p q r p q r a' 6* c° he ca ah a" J" c* he ca ah 18. As an example of the application of this formula we may take the following theorem. The nine-point cirele of a triangle [that is, the cirele which passes through the middle points of its sides) touches each of the four circles which touch the three sides of the triangle. Suppose that is the equation of the radical axis of the inscribed and nine- point circles. The equation of the nine-point circle will then be (see Chap. ii. Art. 10), o' (s-a)=a= + &= (s- 5f /3» + c' (s-c)V — 2hc (s — h) [s— c) /Sy — 2ca {s — c) [s— o) 7a -2o5(s-a) [s-h) a^+{Xa + ii^ + vy) (aa+h^ + cy) = 0. If this represent the nine-point circle, it must be satisfied when a = and b^ = cy. Hence (s-hf + {s-cy-2{s-h)is-c)+2(^^ + fj=0, ft, V (h — c)" or j: + -= „ ■. be 2 Similarly ^ + ^ = ^\ a 2 \ /* _ (g - hy ^ a'^b~ 2 ' 2\ _ (c-ay + (a-hy- (h-cY a 2 = {a-h){a-c); IKTEESECTION OF RECTANGULAR TANGENTS. 91 .". \ = ^a{a — h) {a — c). Similarly fi = ^h{h — c) {h — a), j' = ^c {c—a) (c— h). This gives, for the equation of the radical axis, h — c c — a a — b Now, to ascertain whether this touches the inscribed circle, we have, applying the condition of Chap. II. Art. 9, to in- vestigate the value of h — c ,A c—a ,B a — h „G cos" — + — r- cos" — + cos"— , a 2 b 2 c 2 or ^ {(5-c) (s-a) + (fi-a) {s-b) + {a-h) (s-c)}, which is 0. Hence, the radical axis touches the inscribed circle, and therefore the inscribed and nine-point circles touch one another. Similarly, it may be proved that the nine-point circle touches each of the escribed circles. 19. The equation of the nine-point circle may be de- duced by substituting the above values of \, fi, v, or (perhaps more neatly) by expressing the fact that the curve wa" + «/S' + vr/' + 2u'0y + 2v''^a. + 2w'a^ = passes through the middle points of the sides of the triangle, and combining the equations thus obtained with those inves- tigated in Art. 14. The former gives w° + wV + 2u'be = 0, wffl" + Mc" -1- 2v'ca = 0, uV + va^+2w'ah=^0. Hence, by Art. 14, u'hc = v'ca = w'ah. 92 MOJDEEN GEOMETRY. Supposing m' = — a, we get V w 2a with two similar equations, whence u _ y + c° — a' _ 2 cos ud d' aba a ' .: u = 2a COS A. Hence, the nine-point circle is represented by the equation acos A.cc' + h cos B.^ + ccosC.'f — aySy — hya. — ca/3 = 0, CoE. It hence appears that the nine-point circle passes through the points of intersection of the circumscribed and self-conjugate circles, or has a common radical axis with them. ■20. We have investigated, in Art. 10, the equation of the pair of tangents drawn to the conic from a given point (/> 9> ^)- If these two tangents be at right angles to one another, they may be regarded as the limiting form of a rectangular hyperbola, and must therefore satisfy the equa- tion investigated in Art. 13. This, therefore, gives as the locus of the intersection of two tangents at right angles to one another Wf + Vh'- 2 U'gh +Uh'+ Wf- 2 V'hf+ Vf+ Ug' - 2 W'fg + 2(U'f+ Ugh -V'fg- W'hf) cos A + 2 ( Vy + Vhf- W'gh - U'fg) cos B + 2 {WV+ Wfg- U'hf- V'gh) cos (7=0. This may be shewn (see Art. 15) to represent a circle, as we know ought to be the case. This equation may also be expressed in the following form fV+W+2U'eosA^ W+U+2V'coaB h ff /■ ^, 7, , 7^ f^+ TF-|-2f7'cos^ ,, (o/+ hg + ch) y f+ ■ U+V+2W'cobC. MAGNITUDES OF THE AXES OF THE CONIC. 93 - {^ ^^ )iagh+bhf+cfff)=0. If the conic be a parabola, then (see Art. 6) this breaks up into two factors, one of which is the line at infinity; and the other must represent the directrix, since that is the locus of the point of intersection of two tangents to a parabola at right angles to one another. The appearance of the line at infinity as a factor in the result in this case may be explained as follows : Every para- bola touches the line at infinity, and this line also satisfies the algebraical condition of being perpendicular to any line whatever, since, whatever I, m, n may be, aZ+J»i+cw— (Jw+cot) cos^— (cZ+an) cos,B—{am+hl) (a, ^, 7) + 2x {ua + w'J8+ v'y) + 2y {w'a + v^ + m't) + 2a {v'a + u'0 + toy) + {x,y, s). Now, by Art. 11 of the present chapter, ua + w'P + v'y _ w'a. + v0 + u'y _ v'a + u'^ + toy a b c Aiso,ax + hy + cz = a{a-a) + J (/3-y8) + c (7-7) = 0...(2) ; •■• ^(aJ, 2^, a)=-^(a, A 7), or, ux^ + vy^ + ws' + luyz + Ivzx + 2w'xy u, w, v W, V, u (2A)' v, m', w u, w, V , a w', V, u, h v, m', w, c a, b, c, .(3). (See Art. 12, Cor.) Now the semi-axes are the greatest and least values of the semi-diameter. We have then to make 4A^ abc r^ = a COS A.a? + b COS B .y' + ccoa C .s' •(4) a maximum or minimum, x, y, z being connected by the rela- tions (2) and (3). Multiply (2) by the indeterminate multiplier 2yii, (4) by \, adding them to (3), differentiating, and equating to zero the coefficients df each differential, we get AEEA OF THE CONIC. 95 ux + w'y + «'« + \a cos ^ . a; + /(40 = 1 w'x-V vy +UZ + 7d> COS B.y + fib = > v'x +u'y + wz+ Tim cos O.z+/ic = } Multiplying these equations in order by x, y, z, and add- ing, we get abc Substituting this value of \ in equations (5), and elimi- nating X, y, z from the equations combined with (2), we obtain the following quadratic for the determination of -j : u. w. v' w' V, t u r r u, w u, w V, a w' ^'. u', b V, «', w, c a, 6, c, fas cos A ■w, — w fls COS B -V, fbs cos B \ , /cscos G \ -M, = 0, where s is written for u, w', v' ale w\ V, v! V, u', w u, w , V , a w', V, u, h f V, u', w, c a, h, c, This equation determines the semi-axes. 96 MODERN GEOMETRY. 22. To find, the area of ike coreac. In the above equation, the coefficiejjt of -j is \ — cihcs' {a cos 5 cos (7 + & cos C cos ^\^ c cos A cos B), which is equal to ■ (sin A cos 5 cos C + sin B cos C cos A 2A 2A + sin C cos A cos 5) sin >4l sin 5 sin C = — AA''. s*. The term independent of r' is u, w, V, a w, v, 1 u; h t t u, w, c a, h, 0, Hence the product of the two values of r' is 4AV M, w', V, a w'. V, u', b v; < w, c a, h c, The area of the conic is, therefore, 27rAa&c M, w', v' w; V, u' v\ u', W u, w, Vf —a V), V, m'i, —h v', u', w, —c a, h, c, EXAMPLES. 97 From the above investigation may be obtained the crite- rion which determines whether the conic be an ellipse or hyperbola. For, in the hyperbola, the two values of r^ have opposite signs, hence the cui-ve will be an ellipse or hyper- bola according as t r M, w'. w, V, a 1>, u, b t u, w, c b, c, is negative or positive ; or according as Ua' +Vb'+Wc' + 2 Vbc + 2 V'ca + 2 PT'aJ is positive or negative. Examples. 1. Each angular point of a triangle is joined with each of two given points ; prove that the six points of intersection of the joining lines with the opposite sides of the triangle lie in a conic. 2. A conic is described, touching three given straight lines and passing through a given point ; prove that the locus of its centre is a conic. Express, in geometrical language, the position of the given point relatively to the straight lines, in order that the locus of the centre may be a circle. Also find the locus of the given point, in order that the locus of the centre may be a rectangular hyperbola. 3. Four circles are described, so that each of the four tri- angles, formed by each three of four given straight lines, is self- conjugate with respect to one of them ; prove that the four circles have a common radical axis. 4. If A, B, C, A\ B, C" be six points, such that the straight lines B'C, C'A', A!B! are the several polars of the points A, B, C, with respect to a given conic, prove that F. 7 98 MODERN GEOMETRY. The ttree straight lines AA', BB', GO', intersect in a point ; and that The points of intersection oi BC with B'C, GA with C'A', AB with A'B', lie in a straight line. 5. If two triangles circumscribe a conic, their angular points lie in another conic. 6. The equation of a conic circumscribing the triangle of reference, and having its semi-diameters parallel to the sides equal to r,, r^, r^ respectively, is ah c - j-i" '•//s d'4> d'4> da.' do?' dud^' doidfY df d'4> d' d' d^' d^doL' dff" d^dy dyjr d'(l> d' d' dy' dydoi' djd^' &f = 0. 108 MODERN GEOMETRY. 9. Since the tangents at the extremity of any diame- ter of a conic are parallel to one another, it follows that the polar of the centre is at an infinite distance, and con- versely, that the line at infinity reciprocates into the centre of the auxiliary conic. Hence it follows that parallel lines reciprocate into points lying on a straight line passing through the centre of the auxiliary conic; and that the • asymptotes of any curve, being the tangents drawn to it at the points where it meets the line at infinity, reciprocate into the points of contact of the tangents drawn to the reciprocal curve from the centre of the auxiliary conic. Since the asymptotes of an hyperbola are real, while those of an ellipse are imaginary, it follows that the tangents, drawn from the centre of the auxiliary conic (supposed real) to the reciprocal curve, will be real or imaginary, according as the original curve is an hyperbola or an ellipse. If it be a para- bola, the reciprocal curve will pass through the centre Of the conic, which is in accordance with what has already been stg,ted, that every parabola touches the line at infinity. Con- versely, if one conic be reciprocated with respect to another, the reciprocal curve will be an ellipse, parabola, or hyperbola, according as the centre of the auxiliary conic lies within, upon, or without, the original conic. 10. We have now sufficient materials for transforming any descriptive proposition, that is any proposition relating to the position of lines and points, without reference to consi- derations of magnitude, into another. Before proceeding fur- ther, we will give a few examples of this process. We will first take the following proposition. " If two of the angular points of a triangle move each along a fixed straight line, and each side pass through a fixed point, the three points lying in the same straight line, the third angular point will move along a straight line, passing through the intersection of the straight lines along which the other angu- lar points move." ^ The reciprocals of the three sides of the given triangle will be three points, which may be, considered as the angles of a triangle, which may be called the reciprocal ti-iangle. Those TRANSFORMATION OF THEOREMS. 109 of the angular points of tlie first triangle will be the sides of the reciprocal. Those of the fixed straight lines, along which two of the angular points of the first triangle move, will be fixed points through which two of the sides of the reciprocal triangle pass. Those of the three points, lying in the same straight line, through which the sides of the given triangle always pass, will be three straight lines, intersecting in a point, along which the angular points of the reciprocal triangle always move. Hence the data of the reciprocal proposition will be " Two of the sides of a triangle pass each through a fixed point, and each angular point moves along a fixed straight line, the three straight lines passing through the same point." In the given theorem, the thing to be proved relates to the motion of the third angular point. To this will correspond the third side of the reciprocal triangle. To the straight line, passing, through the intersection of the two given straight lines, along which the third angular point may be shewn to move, corresponds a point lying in the same straight line with the two given points, and through this the third side will always pass. Hence, under the cir- cumstances stated above as data of the reciprocal theorem, " the third side will pass through a fixed point lying in the straight line joining the two fixed points, through which the first sides pass*." * The giren tbeorem may be expressed, by the aid of letters, as follows : Let PQfi be the given triangle, and let ita angular point Q move along a fixed straight line OX, its angular point B along a fixed straight line OY. Also, let the straight line QiJ always pass through a fixed point F, BP through a fixed point G, PQ through a fixed point H, the three points F, G, S lying in the same straight line. Then the given theorem tells us that the point P will always move along a fixed straight line, passing through 0. Now let the whole figure be reciprocated with respect to any conic section. Let the line which is the polar of any point be denoted by accenting the same single letter by which the point is denoted in the original figure ; the polar of P, for example, being denoted by P'. Then the point of intersection of the lines P', Q! will be denoted by the two letters P' Q', and this will be the pole of the line PQ,. We have then a triangle of which the sides are P', Q!, B', the side Q! always passing through a fixed point 0' X', the side B' through a fixed point CT' Also the angular point Q'B' always moves along a fixed straight line F', the point B'P' along a fixed straight line G', the point P'Q; along a fixed straight line H', the three straight lines F', G', H' passing through the same point. Then the reciprocal theorem is that the side P" will always pass through a fixed point lying in the line Q'. The student will find the above mode of transformation, in which a 110 MODERN GEOMETRY. Again, turn to Example 4, on page 57, and let us inves- tigate the reciprocal theorem. The three conies touching respectively each pair of the sides of a triangle at the angu- lar points where they meet the third side, will reciprocate into " three conies passing respectively through each pair of the angular points of a triangle, and touching the lines joining them with the third angular point," that is, the sides of the triangle themselves. This condition, therefore, reciprocates into itself. The condition " all intersecting in a point" reciprocates into " all touching a straight line." Hence the data are, "Three conies are drawn, touching respectively each pair of the angular points of the sides of a triangle at the points where they meet the third side, and all touching a straight line." In the matter to be proved, we may first enquire what are the reciprocals of "the sides oi the triangle which intersect" (that is, which do not touch) "their respective conies." These will be " the angular points of the triangle not lying on their respective conies." The three tangents at their common point will reciprocate into "the three points of contact of their common tangents." And the meeting of the tangents with the sides will reciprocate into the fines joining the points of contact with the angular points. Hence the first thing to be proved is, " That the three straight lines joining the points of contact of the common tangent with the angu- lar points of the triangle not lying on the respective conies all pass through a point." Again, "the other common tangents to each pair of conies" reciprocate into "the other points of intersection of each pair of^ conies," and "the sides of the triangle which touch the several pairs of conies" into the angular points of the triangle " common to the several pairs of conies." Hence the latter part of the theorem will run : " And that the same three straight lines respectively join the other point of inter- section of each pair of conies with the angular point of the triangle common to each pair." straight line is denoted by a single letter, and a point by the pair of letters representing any two straight lines which Intersect in it, -• useftil mode of familiarizing hiinself with the method of reciprocal polars. TRANSFORMATION OP THEOREMS. Ill 11. After a little practice, the process of reciprocating a given theorem will be found to consist simply in writing " straight line" for "point," "join" for "intersect," "locus" for "envelope," &c., and vice versd. The word "conic" will of course remain unaltered. 12. Brianchon's Theorem. By reciprocating Pascal's Theorem (given in Art. 12, Chap. III.), we obtain Brianchon's Theorem, which asserts that "If a hexagon be described about a conic section, the three diagonals will intersect in a point*." • It may be well to append an independent proof of this important theorem. Take three sides of the hexagon as lines of reference, and let the equations of the other three be Let the equation of the conic be (io)* + (ilfi3)^ + (2VV)*=0. The conditions of tangency are whence i + — + — = 0, mil »i 1 1 i i 1 1 1. 1 = 0. The line passing through the intersections of /3=0 with (Z„ wij, 1) and of 7=0 with (Zj, 1, »j) is represented by the equation a+f+I=0. Similarly, the other two diagonals are represented by ffll TTIq - + ^+7=0;; ]12- MODEEN GEOMBTET. The student will find it useful to transform, by the method of reciprocal polars, the special cases of Pascal's Theorem, given in Art. 13, Chap. III.; and to obtain a geo- metrical construction by which when jive tangents to a conic are given, their points of contact may he found. 13. The anharmonic ratio of the pencil formed by four intersecting straight lines is the same as that of the range formed by their poles. This may be proved as follows. Let OP, OQ, OR, OS be the four straight lines, P', Q', E', S' their poles, which will lie in a straight line, the polar Fig. 1 8. of 0; let P, Q, R, 8 be the points in which the pencil is cut by the transversal P'Q'R'8 . Let this transversal cut the conic in jE'^, K^. Bisect 1 1 whence, if these intersect in a point, i 1 1 the same condition as that already investigated. = 0; ANHAEMONIC PEOPEETIES." 113' K^ K^ in V. Then, since PK^ PK^ ia divided harmonically in P, J^i, P', K^ (Art. 21, Chap, ii.), it follows that whence iVP- VK,) irK,+ VP) = {VP+ VK^ (JK, - VF), which since VK^= VK^, reduces to rP.VP'=VK^\ which, by similar reasoning, = VQ . VQ' = rB.rF= V8. VS'. Hence the eight points P, Q, E, S, P', Q', B', S' are in an involution, of which K^, K^ are the foci, and therefore (Art. 27, Chap, i.) {o.PQES] = [_p'gB:s]. 14. In Art. 13, Chap. i. we saw that the condition that the three points (Z,, ot„ m,), {l^, m^, n^),^ (Z,, m^, w,) shall lie in the same straight line is identical with the condition that the three straight lines {l^, m,, Wj) (Zj, m^, n^, {l^, m^, n^ shall intersect in the same point. Now these several points and lines are respectively the poles and polars of each other, with respect to the imaginary conic Thus the theory of reciprocal polars explains the fact that the condition for three points lying in a straight line is identical with that "for three straight lines intersecting in a point. It also explains the identity of conditions noticed in Chap. II. Arts. 7 and 9. For the reciprocial of the conic XV + z*"^ + vY - 2/*i'iS7 - 2i'X7a - 2XAta/3 = (1), with respect to o' + /3' + 7= = 0, will be found to be \0y + fJ^n + vi^ = O.. (2). F. 8 114 MODEEN GEOMETET. And the polar of {f, g, h) isfa+fffi + hy = 0. Hence if the line fa+ff^ + hy = touch (1), the point (/> fft ^) li^s i° (2)> giving for the condition of tangency f ff ^ And if the line fn + ff^ + hy = touch (2), the point (/ ff, h) lies in (2), giving for the condition of tangency in that case ^Y + A^y + v^h^ - 2fj,vffh - 2vXhf- 2Xfifff = 0. These conditions of tangency are identical with those already investigated. Again, every parabola touches the line at infinity. Now the co-ordinates of the pole of this line are proportional to a, h, c. Hence, if the conic, represented by the general equation of the second degree, be a parabola, the point (a, b, c) must lie in the reciprocal conic. This gives, as the condition for a parabola, Ua''+Vh^+ Wc'+2U'ho + 2V'ca+2W'ai = 0, the same as that already investigated. 15. Peop. Ani/ straight line drawn through a given point A is divided harmonically hy any conic section, and the polar of A with respect to it. This proposition may be proved as follows. Let the straight line cut the curve in P and Q, and the polar of A in B. Let C be the polar of the straight line, and let ABC be the triangle of reference. The conic will be self-conjugate with respect to ABC, and will therefore be represented by the equation ua^ + v^'' + wy' = 0. Hence the lines CP, CQ, which are tangents to the conic, are represented by the equation -^ and therefore fprpa ^ij harmonic pencil with CA, CB. ANHAEMONIC PKOPERTIES. 115 16. The straight line OB may be regarded as the polar of A with respect to the locus made up of the two straight lines CP, CQ. For the values of ^ and 7 at ^ being O, 0, and the equation of CP, CQ being ud'^ + v^ = 0, we get for the equation of its polar, a = 0, that is, the polar is the line AB. 0, ua, */S M, 0, 1 V^, 1, 17. If four straight lines form an harmonic pencil, either pair will be its own polar reciprocal with respect to the other. For, adapting the equation of Art. 8, to the case of two variables only, we get for the polar reciprocal of a^ = 0, with respect to wa" + v^ = 0, the following equation, = 0, or uva^ = 0. And, conversely, for that of ua^ + vl3^ = with respect to a/3 = 0, 0, or Ma° + v^ = 0, in either case reproducing the reciprocated curve. Hence the proposition is proved. 18. We may hence deduce the condition that two pairs of straight lines may form an harmonic pencil. First let them all intersect in A, and the equations of the two pairs be wa'' + «y3' + 2«j'a/3 = (1), pa.'+q^ + 2v'a^==0 (2). The polar reciprocal of (1) with respect to (2) is 0, pa + r'^, r'a-^-q^ poL + r'^, u, ra + ^^, w', 0, /3, a. y8, u, a> 0, V w V = 0, 8—2 116 MODERN GKOMETET. or u (/a + gy9)' + v(j>a + v'^f - 2m;' (r'a + q^) (pa + r'^) = 0. Suppose that uo? + v/8' + 210 OL^ = M (a + ^j/3) <« + i^) identically, i.e. that At the point of intersection of the line a + \^ = 0, with 7 = 0, we have «i • Taking the polar of this with respect to the curve (2) we get (pA;.-r')a + (r'i,-2))8 = 0. If this be identical with a + h^ = 0, we get or jikjc^ - 2v' {k^ + Jc^+q=0; .: pv — 2r'w' + qu = 0, the reqttired condition. The symmetry of this equation shews that (2) is also its own polar reciprocal with respect to (1), as ought to be the case. 19. If the point of intersection of the four straight lines do not coincide with one of the angular points of the triangle of reference, we have then only to express the condition that the range formed by their intersection with any one of its sides, 7=0, for instance, be an harmonic range. If this be the case, the pencil formed by joining these four points with C will be an harmonic pencil, and we shall have, as before, pv — 2r'w' + qu = 0. EECIPEOCATION WITH EESPECT TO A CIECLE. 117 20. We next proceed to consider the resuTts to be de- duced from the theory of reciprocal polars, when the auxiliary conic is a circle. It is here that the utility of the theory is most apparent, as we are thus enabled to transform metrical theorems, i. e. theorems relating to the magnitudes of lines and angles. We know that, if BQ be the polar of a point T with respect to a circle, of which the centre is 8 and radius Tc, then fiTwill be perpendicular to FO. Let BT co.i PQ in F. Then 8T.8V=Tc\ Hence the pole of any line is at a distance from the centre of the auxiliwry circle inversely proportional to the distance of the line. And conversely, the polar of any point is at a dis- tance from the centre of the auxiliary circle, inversely propor- tional to the distance of the point itse^. 21. If TX, TY be any two indefinite straight lines, JP, Q their poles, then, since 8P is perpendicular to TX, 8Q to TY, it follows that the angle P8Q is equal to the angle XTY or its supplement, as the case may be. Hence, the angle included between any two straight lines is' equal to the angle subtended at the centre of the auxiliary circle by the straight linejoirdng their poles, or to its supplement. 22. From what has been said in Art. 15, and the earlier articles of this chapter, it will appear that to find the polar reciprocal of a given curve with respect to a circle, we may proceed by either of the following two methods. First. Draw a tangent to the curve, and from 8, the centre of the auxiliary circle, draw 8Y perpendicular to the tangent, and on 8Y, produced if necessary, take a point Q, such that 8Q . SY= F. The locus of Q' will be the required polar reciprocal. Secondly. Take a point P on the curve, and join 8P; on 8P, produced if necessary, take a point Z, such that 8P.8Z=F. Through Z draw a straight line perpendicular to SP. The envelope of this line will be the required polar reciprocaL 118 MODERN GEOMETET. 23. It will be observed that the magnitude of the radius ■of the auxiliary circle affects the absolute, but not the relative, magnitudes or positions of the various lines in the reciprocal figure. As our theorems are, for the most part, independent of absolute magnitude, we may generally drop all considera- tion of the radius of the auxiliary circle, and consider its centre only. We may then speak of reciprocating " with re- spect to S" instead of " with respect to a circle of which 8 is the centre." 8 may be called the centre of reciprocation, h the constant of reciprocation. 24. As an example' of the power of this method we will reciprocate the following theorem, "The three perpendiculars from the angular points of a triangle intersect in a point." This may be expressed as follows : "If 0, A, B, (7 be four points, such that OB is perpendicular to GA, and 0(7 to AB, then will OA be perpendicular to 5C." Eeciprocate -this with respect to any point S, and the four points 0, A, B, give four straight lines, which we may call each by three letters cJ)c, ah'c', a'hc, a'b'c, respectively. Then, the fact that OB is perpendicular to GA is expressed hj h and V subtending a right angle at 8, or by ISb' being a right angle. Again, the fact that OJ is perpendicular to AB, shews that cSc' is a right angle. Then the reciprocal theorem tells us that a8a is also a right angle. We may express this more neatly as follows : aa, bb', cc', are the diagonals of the complete quadrilateral formed by the four straight lines, hence it appears that at any point at which two of the diagonals of a complete quadrilateral subtend a right angle, the third diagonal also subtends a right angle. Or, in other words. The three circles, described on the diagonals of a complete quad- rilateral as diameters, have a common radical axis. The extremities of this axis may be conveniently called the_/ooi of the quadrilateral.* 25. If the system formed by the four points 0, A, B, C be reciprocated with respect to any one of them, for in- stance, the triangle thus .obtained will be similar, and similarly situated, to that formed by the other three points A, B, C. *, This name is propos«d by Mr Clifford, in the Meitmger of Mathematics. KECIPEOCATION WITH RESPECT TO A CIECLE. 119 ' . For if on OA, OB, 00 respectively (produced if neces- sary), we take points A', £', 0', so that OA . OA' = OB . OB' = 00 . OC, and through A', B, C draw YZ, ZX, XY, severally at right angles to OA', OB, OC, then YZ, ZX, XFare respectively parallel to BC, GA, AB, or the triangle XYZ is similar and similarly situated to the triangle ABC. We may observe further, that the point X, since it is the intersection of the polars of B and C, is itself the pole of the line BC, and therefore OX is perpendicular to BC, that is to YZ. Similarly, OY, OZ, are respectively perpendicular to ZX, XY. Hence, is the intersection of the perpendicu- lars dropped from X, Y, Z on YZ, ZX, XY respectively. It may be convenient to call the point of intersection of the perpendiculars let fall from the angular points of a triangle on the opposite sides, the orthocentre of the triangle, or of its three angular points. Here we may say that " If a triangle be reciprocated with respect to its orthocentre, the reciprocal triangle will be similar and similarly situated to the given triangle, and will have the same orthocentre." It will be seen by Art. 19, that any three points and their orthocentre, reciprocated with respect to any point S, give a quadrilateral, of which S is a, focus. 26. If any conic be reciprocated with respect to an ex- ternal point S, the angle between the asymptotes of the re- ciprocal hyperbola will be the supplement of that between the tangents drawn from iS to the conic. (See Art. 9 of this chapter.) Conversely, if an hyperbola be reciprocated with respect to any point 8, we obtain a conic, which subtends at 8 an angle the supplement of that between the asymptotes of the hyperbola. 27. From the last article it follows that, if a parabola be reciprocated with respect to any point 8 on its directrix, we obtain a rectangular hyperbola, passing through 8. If a rectangular hyperbola be reciprocated with respect to 120 MODERN QEOMETET. any point S on its circumference, we obtain a parabola whose directrix passes through S. Again, if a conic be reciprocated with respect to any point on its director circle (i.e. the circle which is the locus of the intersection of two perpendicular tangents) we obtain a rect- angular hyperbola. If a, rectangular hyperbola be reciprocated with respect to any point 8 not on the curve, we obtain a conic, whose cU- rector circle passes through S. 28. It is known that the conies passing through the four points of intersection of any two rectangular hyperbolas, is itself a rectangular hyperbola ; and also that any one of these four points is the orthocentre of the other three. If, then, we reciprocate these theorems with respect to any one of the four points of intersection, we obtain the theorem that, " If a parabola touch the three common tangents of two given para- bolas, its directrix passes through the intersection of the di- rectrices of the two given parabolas, that is, through the orthocentre of the triangle formed by their common tangents." In other words, " If a system of parabolas be described, touch- ing three given straight lines, their directrices all pass through the orthocentre of the triangle formed by the three given straight lines." Again, reciprocating this system of rectangular hyperbolas with respect to any point S, we get, "All conies, which touch four given straight lines, subtend a right angle at either focus of th6 quadrilateral formed by these four straight lines." Or, in other words, "The director circles of all conies which touch four given straight lines, have a common radical axis, which is the directrix of the parabola which touches the four given straight lines." 29. To find the polar reciprocal of a circle with respect to any point. From what has already been shewn, we know that this win be a conic; we have now to investigate its form and position. Let 8 be the centre of reciprocation, Jc the constant of reciprocation, MPM' the circle to be reciprocated, its centre, RECIPEOCATION OP A CIECLE "WITH EESPECT TO A POINT. 121 MM' its diameter passing through ;S^, p its radius, and let OS=c Wig. 19. Through 8 draw any straight line cutting MPM' in P and Q. On 8PQ, produced if necessary, take two points Y and Z, such that SP.8Y=8Q.8Z=h\ The straight lines drawn through Y and Z perpendicular to 8P will be tangents to the reciprocal conic. Now 8Y. 8Z= SP.SQ'p'-c" which is constant. Hence, the reciprocal is a conic of such a nature that the rectangle under the distances from ;S^ of any two parallel tangents is constant. It is therefore a conic, of which >Si is a focus, and of which the axis-minor is r . It will be an ellipse, parabola, or hyperbola, according as p is greater than, equal to, or less than c, that is, according as 122 MODERN GEOMETET. the centre of reciprocation lies within, upon, or without, the circle to be reciprocated. This agrees with what has been already shewn, Art. 9. Let 2a, 2h, be the axes of the conic, 2l its latus-rectum, e its eccentricity. To determine their magnitudes, we proceed as follows. The axis-major will be in the direction SO. Let A, A' be its extremities. Then 2._ 1 1 __ SM+8M' _2p ^ l~ 8A^ iSA'~ k' ~ k"' k' Hence, l= — , or the latus-rectum is inversely propor- P tional to the radius of the circle. Again, a ~ l~ ¥p ■p'-c^' e' = 1- a' = !■ ¥ (p'-cT P'- c' ky or e c Thus the eccentricity yaries directly as the distance of the centre of the circle from the centre of reciprocation, and in- versely as the radius of the circle. If d be the distance from S of the corresponding directrix, d=-=- £=- e p ' c~ c ' or, the directrix is the polar of the centre of the circle. 30. We have now the means of obtaining, from any property of a circle, a focal property of a conic section. FOCAL PEOPERTIESi 123 Take, for example, Euc. iii. 21. This may be expressed as follows : " If three points be taken on the circumference of a circle, two fixed and the third moveable, the straight lines joining the moveable point with the two fixed points, make a constant angle with one another." This will be recipro- cated into " If three tangents be drawn to a conic section, two fixed and the third moveable, the portion of the move- able tangent intercepted between the two fixed ones, subtends a constant angle at the focus." This angle will be found, by reciprocating Euc- iii, 20, to be the complement of one- balf of the angle subtended at the focus by the portion of the corresponding directrix intercepted between the two fixed tangents. Again, it is easy to see that "if a circle be described touching two concentric circles, its radius will be equal to half the sum, or half the difference, of the radii of the given circles, and the locus of its centre will be a circle, concen- tric with the other two, and of which the radius is half the difference, or haK the sum, of the radii of the two given circles." Hence we deduce the following theorem. " If two conies have a common focus and directrix, and their latera-recta be 2l, 21', and another conic, having the same focus, be described ill' so as to touch both of them, its latus-rectum will be 7-7-7, , and the envelope of its directrix will be a conic, having the same focus and directrix as the given conies, and of which 11 " 4fcZ' ,, the latus-rectum is , _ „ . 1+ I Again, take the ordinary definition of an ellipse, that it is the locus of a point, the sum of the distances of which from two fixed points is constant. This is equivalent to " the sum of the distances Irom either focus, of the points of contact of two parallel tangents, is constant." The reciprocal theorem will be, " If a system of chords be drawn to a circle, passing through a given point, and, at the extremities of any chord, a pair of tangents be drawn to the circle, the sum of the reciprocals of the distances of these tangents from the fixed point is constant." 124 MODERN GEOMETET. The known properly of a circle, that "two tangents make equal angles with their chord of contact" will be found, when transformed by the merthod now explained, to be equivalent to the theorem that "if two tangents be drawn to a conic from an external point, the portions of these tangents, inter- cepted between that point and their points of contact, subtend equal angles at the focus." From the fact that "all circles intersect in two imaginary points at infinity," we learn that "all conies, having a common focus, have a commo'n pair of imaginary tangents passing through that focus." And, more generally, we majr say that all similar and similarly situated conies reciprocate into a system of conic& having two common tangents. 31. Two points, on a curve and its reciprocal, are said to correspond to one another when the tangent at either point is the polar of the other point. Two tangents are said to correspond when the point of contact of either is the pole of the other. The angle between the radius vector of any point (drawn from the centre of reciprocation), and the tangent at that rig. 2o. point, is equal to the angle between the radius vector of, and tangent at, the corresponding point of the reciprocal curve. ^ For, if P be the given point, PF the iangent at P, and ST the centre of reciprocation, and /SFbe perpendicular to ~^ and if F be the pole of PF, and P'F*^ the polar of P FY- then FOCI OF A CONIC. 125 P will lie on SY, produced if necessary; and if 8Y' be perpendicular to P'Y', 8Y' will pass through P. Hence, since SP, PY, are respectively perpendicular to P'Y', 8P', it follows that the angle iSPY is equal to the angle 8P' Y'. 32. We have investigated (Art. 10, Chap, iv.) the equa- tion of the two tangents drawn to a conic from any given point (/, g, h). If in the right-hand member of that equa- tion we substitute for 0, «(aa + J^ + C7)', « being an arbi- trary constant, we shall obtain the general equation of all conies of which these lines are asymptotes. Now, since the asymptotes of the reciprocal conic with respect to {/, g, h), are respectively at right angles to the two tangents drawn from (/, g, h), it follows that the family of conies thus ob- tained will be similar in form to the reciprocal conic. 33. To find the co-ordinates of the foci of the conic repre- sented hy the general equation of the second d^ree. Since the reciprocal of a conic with respect to a focus is a circle, it will follow from Art. 32 that the family of conies obtained as above must, if [f, g, h) be a focus, be circles also. Applying the conditions for a circle investigated in Art. 14, Chap. IV,, it will be found that the terms involving m dis- appear of themselves, and our conditions assume the form {Uh^ + Wf'-2V'hf)c*+{Vf+Ug'-2Wyg)W + 2{U'f + Ugh - W'hf- V'fg) he = {Vf+ U^-iW'fg) a''+{Wg'+rh'-2U'gh) c' -H 2 ( F'/ + Vhf- TJ'fg - W'gh) ca = ( W + ^^° - 2 U'gh) V+{Uh'+Wf-2 Vhf) a= -1-2 ( W'h^ + Wfg- V'gh ^ U'hf) ab, or ( Vb'+ Wc'+ 2U'lo)f-2 ( V'c+ W'b)f{bg-\-ch) + U{bg+chy = {Wc'+Ua'+2 V'ca)g'-2{W'a+ U'c)g{ch+af) + V{ch+afy = {Ua'+ VV+2Wd>)¥-2{U'b+ra)h{af+bg) + W[af+hg)\ 126 MODERN GEOMETEt. equations which, since «/+ hg + ch- 2A, may also be written under the form . ( !7a= + Vb' +Wc' + 2 U'bc + 2 F'ca + 2 W'ab)/' - 4A ( V'c + W'b + Ua)f+ 4:U.A' = {Ua''+ Vh' +Wc' + 2 U'bc + 2V'ca+2 W'ah) / - 4 A ( F'a + i7'c + FJ) 5- + 4 F. A= = {Ua'+VV + Wc" + 2 U'bc + 2 V'ca + 2 W'ab) h" - 4A ( U'b + V'a + TFc) A + 4F. A'. The equations, together with af+ bg + ch = 2A, determine the co-ordinates of the foci. It will be seen that they give four values oi f, g, h, two of which are real, two imaginary. If the conic be a parabola, then, applying the condition of Art. 6, Chap. IV., these equations reduce to {V'c + W'b + Ua)f- C/A = {W'a+U'c+Vb)g-VA = (U'b + V'a + Wc)h-WA, which give the focus in that case. If the equation ua? + vy' + wz' + 2u'i/z + 2v'zx + 2w'xy = 0, be expressed in triangular co-ordinates, we get, for the co- ordinates of the foci, the equations (p-+F+PF+2P"+2F'+2TF')/°-2(F'+TF'+C0/+U- _ (Er+F+TF+2Z7'+2F+2TF')ff°-2(TF'+g7^+F).y+F b' ' {UJrV+W+2U'+2V'+2W')Ji^-2{U'+V'+W)h + W EXAMPLES. 127 or, if the conic be a parabola, 2 (F'+ W'+ U) f- U _ 2 iW'+ U'+ V)g-V a" ~ b" _ 2{U'+r'+W)h-W c» 34. Interesting results may sometimes be obtained by a double application of the riiethod of reciprocal polars. Thus, the theorem that " the angle in a semicircle is a right angle" may be expressed in the form that " every chord of a circle, which subtends a right angle at a given point of the curve, passes through the centre." Keciprocating this with respect to the given point, we get " The locus of the point of intersection of two tangents to a parabola at right angles to one another, is the directrix." Now, reciprocate this with respect to any point whatever, and we find that " Every chord of a conic which subtends a right angle at a given point on the curve, passes through a fixed point." Again, take Euc. in. 21. This may be expressed under the form " If a chord be drawn to a circle subtending a con- stant angle at a fixed point on its circumference, it always touches a concentric circle." Reciprocating this theorem with respect to 0, we get " If two tangents be drawn to a para- bola containing a constant angle, the locus of their point of intersection will be a conic, having a focus and directrix in common with the given parabola." Reciprocate this, with respect to any point whatever, and we get, " If a chord be drawn to a conic, subtending a constant angle at a given point on the curve, it always touches a conic havmg double constant with the given one." Examples. 1. Having given a focus and two points of a conic section, prove that the locus of the point of intei-section of the tangents at these points will be two straight lines, passing through the focus, and at right angles to each other. 2. Prove that four conies can be described with a given focus and passing through three given points, and that the latus-rectum. 128 MODEEK aEOMETEY. of one of these is equal to the sum of the latet&-recta of the other three. 3. On a fixed tangent to a conic are taken a fixed point A, and two moveable points F, Q, such that AF, A Q, subtend equal angles at a fixed point 0. From F, Q are drawn two other tan- gents to the conic, prove that the locus of their point of inter- section is a straight line. 4. Two variable tangents are drawn to a conic section so that the portion of a fixed tangent, intercepted between them, subtends a right angle at a fixed point. Prove that the locus of the point of intersection of the variable tangents is a straight line. If the fixed point be a focus, the locus will be the correspond- ing directrix. 5. Chords are drawn to a conic, subtending a right angle at a fixed point j prove that they all touch a conic, of which that point ia a focus, 6. Three given straight lines £C, GA, AB, are intersected by two other given straight lines iD.A^,A^; B^, B^; (7,, G^ respectively. Prove that a conic can be described touching the six straight lines AA^, AA^, BB^, BB„ GC„ GG,. 7. A, B, G, S, are four fixed points, SB is drawn perpendicular to SA, intersecting BG in D, SB perpendicular to SB, intersecting GA in B, SF perjiendicular to SG, intersecting AB in F. Prove that D, E, F lie in the same straight line. Prove also that the four conies which have S as & focus, and which touch the three sides of the several trian^es ABC, AEF, BFB, GDE, have their latera-recta equal. 8. Two conies are described with a common focus and, their corresponding directrices fixed ; prove that, if the sum of the re- ciprocals of their latera-recta be constant, their common tangents will touch a conic section. 9. A conic is described touching three given straight lines BG, GA, AB, so that the pair of tangents drawn to it from a given point 0, are at right angles to each other. Prove that it will always touch another fixed straight line ; and that, if this straight line cut BG, GA, AB in D, E, ^respectively, each of the angles AOD, BOE, GOF is a right angle. Prove also that the polar of with respect to this conic will alvrays touch a conic, of which is a focus. EXAMPLES. 129 ] 0. OA, OB, are the common tangents to two conies haying a common focus S, CA, CB are tangents at one of their points of in- tersection, BD, AE tangents intersecting GA, CB in D, E. Prove that S, D, E lie in the same straight line. 11. Any triangle is described, self-conjugate with regard to a given conic ; prove that, if a conic be described, touching the sides of this triangle, and having the centre of the given conic as a focus, its axis-minor will be constant. 12. Prove that two ellipses, which have a common focus, can- not intersect in more than two points. 13. If a system of conies be described, passing through four given points, four fixed straight lines may be found, such that the chord of each, intercepted by any conic of the system, sub- tends a right angle at one of the points. ( 130 ) CHAPTEK VII. TANGENTIAL CO-OEDINATES. 1. In the systems of co-ordinates 'with which we have hitherto been concerned, we have considered a point as deter- mined, directly or indirectly, by means of its distances from three given straight lines ; and we have regarded a curve as the aggregation of all points, the co-ordinates of which satisfy a certain equation. It is equally possible, however, to con- sider _a^traight line as determined by means of its distances from three points, which distances may be termed its co- ordinates ; and to regard a curve as the envelope of all straight lines, the co-ordinates of which satisfy a certain equa- tion. This system is closely connected -with the theory of reciprocal polars. In fact, it may be looked upon as a means of so inter- preting equations as at once to obtain the results which the method of reciprocal polars vrould deduce from the ordinary me- thod of interpretation. The equations are the reciprocals of those described in Chapter v. with respect to x' + y' + i!^ = 0. We may then define the co-ordinates of a straight line to be the perpendiculars let fall upon it from three given points A, B, C. The lengths of these perpendiculars we will denote by the letters p, q, r, respectively, the lengths BG, CA, AB being represented as before by the letters a, i, c, and the angles of the triangle of reference ABO being denoted by A, Bf C, and its area by A. 2. Any. two co-ordinates, j and r for example, will be considered to have contrary signs if the line of which they EQUATION OP A POINT. 131 are the co-ordinat es cuts the li ne ■B(7JiLa-fiQini;J3ging,.ltfitSfi£ii-- -B.-and C, otherwise to have'tte same sign. Thus, the in- ternal bisector of the angle A has its co-ordinates of contrary- signs, the external bisector of the same sign. The sign of p relatively to ,j and r will be determined in the same manner. If D be any point on the line BG, q, r, the co-ordinates of any line passing through it, and BD = a^, CD = a^, distances measured along the line BG from B to G being considered positive, and from G to B negative, it will readily be seen that Since this is a relation between the co-ordinates of any line passing through the point B, it may be considered as the equation of the point B. If I) be the middle point of BG, Oi = — Oj, hence it appears that the middle points of the sides of the triangle of reference are represented by the equations, q+r = 0, r+p = 0, j) + q = 0. It may also be proved that tlic points where the internal bisectors of the angles meet the opposite sides, are repre- sented by hq+cr = 0, cr + ap = 0, ap + hq = 0. The points where the external bisectors of the angles meet the opposite sides, by bq—cr = 0, cr — ap = 0, op — hq=Q. ^ The feet of the perpendiculars from the angular points on the opposite sides, by q tan B + r tan (7=0, r tan (?+ p tan -4 = 0, p tan ^ + g' tan B'= 0. The points of contact of the inscribed circle, by^^ ' -^, + ^ = 0, ^ + -^ = 0, ^+ ^,=0, 8—0 s—c s—c a— a s—a s—o where 2s = a + b + c. 9-2 132 MODERN GEOMETET. 3. We shall next prove the following proposition ; that if be any given point within the triangle ABC, then the co-ordinates p, q, r (their signs being taken in the manner already explained) of any straight line QPB, passing through it, will be connected by the following equation, l^BOG.p + ^COA.q + ^AOB.r^Q. Fig. 2 1. Let the triangular equation of QPB be Ix + my + na = 0, and the trilinear co-ordinates of 0, f , i;, f. Then f : t? : ? :: h.BOC : LCOA : AAOB. And, since lies on QPB, Again, since p is the distance from the point (1, 0, 0) to the line, (I, m, n), .-. (t2A)' = {(Z-5m) {1-71)0'+ {m-n) {m-l)h'+(n-l){n-m)o'}p'. Similar equations hold for Q ^nd B. hence m n or p^ + nu] + n^=0, ABOC.p + ACOA.q + AAOB.r = 0. This equation may be regarded as the equation of the point 0. IDENTICAL EELA.TIO^f. 133 A similar equation may be proved to hold for any point without the triangle, BOG being considered negative, if A and be on opposite sides of BC. The following are the equations of some important points connected with the triangle of reference : Centre of gravity, p+q + r=Q. Centre of circumscribing circle, ^ sin 2^ + g sin 2B+ r sin 2 C = 0. Centre of inscribed circle, op + Jg + c = 0. Centres of escribed circles, — ap + hq + cr = 0, op — bq + ;}{r-g;)c''y' 2A. COE. Hence, if p be the radius of a circle, lp+mq+nr=0 the equation of its centre, the circle, being the envelope of a line whose distance from the centre is constant, will be repre- sented by the equation (p _ 2) (jj - r) a' + (2 - >) {q -p) V +{r-p){r- q) c' -m(' — /'?4V ( ^P + '>nq + nr l + m + n 6. An equation of a degree, higher than the first, may be regarded as representing the curve which is touched by all the straight lines, the co-ordinates of which satisfy the equa- tion of the curve. Adopting this mode of interpretation, the values of the ratios p : q : r which simultaneously satisfy two given equations will be the co-ordinates of the common tangents to the two curves represented by these equations, and the values obtained by combining any given equation with an equation of the first degree, will represent all the straight lines which pass through the point represented by the equation of the first degree, and which touch the curve. From this it follows, that an equation of the w"' degree will represent a curve such that n tangents, real or imaginary, can be drawn to it from any point, that is, a curve of the n"" class. It will hence follow that every equation of the second degree represents a conic. We may proceed to consider some of its more interesting special forms. 7. To find the equation of a conic which touches the three sides of the triangle of reference. 136 MODERN QEOMETEY. The co-ordinates of the sides of the triangle of reference are 2 = 0, r = for BG, r = 0, p = for CA, p = 0, jrrO for AB. Hence, the equation of the required conic must be satis- fied whenever two out of the three co-ordinates j?, q, r are = 0. It must therefore be of the form Lqr + Mrp + Npq = 0. The equations of .the points of contact are These may be established as follows: If in the given equation we make Mr + Nq = 0, we obtain either q = 0, or r = 0. It hence appears that the tangents drawn through the point Mr + Nq = 0, pass either through the point j = 0, or through the point r = 0. But the three points Mr + Nq = 0, q = 0, r = 0, lie in the same straight line ; hence the tangents drawn from Mr + Nq = coincide, that is, it is the point of contact of the tangent for which j = r = 0. Similarly for the other two points of contact. It will hence appear, by reference to the equations of the points of contact of the inscribed circle, given in Art. 2, that that circle is represented by the equation (« — a)qr+{s — b)rp + {s — c) pq = 0. CIRCUMSCRIBING CONIC. 137 The escribed circles will be represented as follows : — aqr+(s — c)rp+{s — h) pq = 0, {s — c)qr — srp + (a — a)pq = 0, (s — J) jr + (s — a) rp — spq = 0. 8. To find the equation of a conic circumscribed ahout the triangle of reference. The equations of the angular points of the triangle of reference are /> = 0, q = 0, r = 0. Now, since _each of t hese.^ points lies on the curvej_the two tangejfl8"^ii^wnthrough any olie of fKem~m\ist coincide, hence when any one of these quantities is put = 0, the remaining equation must have two equal roots. The required equation will therefore be of the form Ly + MY + NV - 2MNqr - 2NLrp - 2LMpq = 0. The co-ordinates of the several tangents at the angular points will be given by the equations p = 0, Mq -Nr =0, q = 0, Nr—Lp = 0, r = 0, Lp-Mq=0. If the conic be a circle, the tangent at A will be deter- mined by the equations -ft g _ y -P" ' csinC bBiuB' frhich last is equivalent to b'q — c^r = 0. Similar equations holding for the other two tangents, the aquation for the circumscribing circle will be ay + by + c'r^ - 2bVqr - 2cVrp - 2a%^pq = 0, [vhich may be reduced to + op* + bq* + cr* = 0. 138 MODERN GEOMETRY. 9. By investigations similar to those in Chap. IV. Art. 8, it may be shewn that the equation of the pole of the line (/, g, h) with respect to the conic 4> (Fi ?> *■) = ^P' + ^2* + ^^^ + ^"'S''' + ^v'^j? + 210^2 = 0, is (m/+ w'g + v'h) p + (w/+ vg +u'h) q + {v'f + v!g 4-wh)r = 0. Now, the centre is the pole of the line at infinity, which is given by the equations p = q = r. The equation of the centre is therefore {u + V + w') p+{ii -irv+w')q + {u+v' + w)r = 0. If the conic be a parabola, it touches the line at infinity ; the condition that it should be a parabola is therefore u + v + w + 2u' + 2v' + 2ib' = 0. 10. The two points in which the conic is cut by the line {f, g, Ji) are represented by the equation (See Chap. IV. Art. 10.) Hence the two points in which it is cut by the line at infinity are given by — [{u+v Jrw) p-i- [u -^ v+w) q+{u' +v' +w)rY=0. Hence may be deduced the equation of the two points at infinity through which all circles pass. For these are the same for all circles. Now, for the inscribed circle they are obtained by putting u = v = w — 0, 2u'=s — a, 2v'=s — h, 2w' = s — c. The equation then becomes is {{s-a) qr+{s-b) rp+(8-c)pq} - {np + hq+ery = 0, or aY+i'Y+cV-2bcqrcoaA-2carp coaB-2abpqcoaC=0, CIECULAE POINTS AT INFINITY. 139 wWcli may also be written a^ {P -q) {p-r) +¥ iq-r) {q-p) + + 2M' + 2i>' + 2w)')a* k' Two other corresponding expressions may of course be obtained for k, and the required condition is therefore wu—v'''+uv—w"'—'2(v'w'—uu') uv—w''+vw—u"—2{w'u'—vv') _vw — m" + wu — v'" — 2 (u'v — ww') 6' ■ 11. To find the equation of the conic with respect to which the triangle of reference is self-conjugate. Since each angular point of the triangle of reference is the pole, with respect to this conic, of the opposite side, it follows that the equation of such a conic will be of the form up^ + v^ + wr' = 0. From the last article it will be seen that the equation of the self-conjugate cii'cle is 2 S S P + £ I ^ -0 h* + c^-d'^ F"' (3). Now let (0, g, K) be the co-ordinates of either vertex. Then, since it has the same polar with respect to both conies, the equations vg^ + wJiqw + rv-vw; therefore multiplying both sides by — 4«w, which is a positive quantity, —Avw {qr —p'^ > ^vw {vw — qw — rv) ; ,', {gw-'t-rvf— 4«M7 (sr—p") > {qw+rvf + 4«V — ivw (gw + rv) > (gw + rv — 2vwy >0; .'. {qw — rvf + ivwp'^ > 0. Hence, when the four points of intersection are imaginary, the vertices are in all cases real. 7. Suppose now that these vertices are taken as angular points of the triangle of reference. Let the conies be repre- sented by the equations " ua-' + v^ + wf^O, pce + q^ + rrf = Q. Then + "— = + ^=+ "' {gw —rv)^ (ru —pw) * [jpo — qu^ are the equations of the several pairs of common chords of the two conies. Since two of the expressions qw — rv, ru —pw, pv — qu, must necessarily have the same sign, it follows that one pair at least of common chords is always real. The other two pairs will, as may easily be seen, be real or imaginary, accord- ing as the four points of intersection are, or are not, all real. 8. Returning to the equation (4) given in Art. 3 of this COMMON CHORDS. 149^ chapter we see that it may be partially developed into the following form : U, ID, v' W, V, u v, u', w p, w', v u, r, V u, w, q T, V, U + w, q, u' + w, V, p i, u, w v', p', w V, U, V u, v, q p, W% q' p. r, v' w', q, p + r, V, p + r, q, u' v, p, r q, u', r q, p, w P, r', q' r, q, p' F = 0, i, P\ r This equation is generally written Here A and A' are known to be the discriminants of the two given conies. Now the roots of this equation are the values of k which will give the three pairs of straight lines drawn through the points of intersection of the two conies, and since these values must remain unaltered by any trans- formation of co-ordinates, it follows that the ratios of the four coefficients, A, ©, ©', A', also remain unaltered by any such transformation. On this account they are called the Invariants of the system. They possess numerous interesting proper- ties, but a detailed examination of them would lead us too far from the object of this work. They will be found fully dis- cussed in Dr Salmon's treatise. On Peojections. 9. Dep. The surface generated by a straight line of indefinite length, which always passes through a given fixed 150' MODEEN QEOMETBT. point, and always meets a given curve, the curve and point not lying in the same plane, is called a cone. The fixed point is called the vertex, and will be denoted in this chapter by the letter V. If a cone be cut by any two planes, either of the curves of section is said to be a, projection of the other. Also the two points in which any generating line is cut by two planes are said to be the projections, the one of the other. The straight line in which the two planes intersect is called the Urtprojected, It may easily be seen that the projection of any curve on a given plane coincides with the shadow of the curve which would be cast upon the plane by a luminous point coinciding with the vertex of the cone. The projection of a point of intersection of any two curves will be a point of intersection of their projections. The projection of any straight line will be a straight line ; and that of any curve of the nth degree will be a curve of the nth degree. For siijce any straight line and curve of the nth degree intersect in n points, their projections will also intersect in n points. 10. If AB be any given straight line, and a cone be cut by any plane parallel to VAB, the projection of the line AB will be infinitely distant. Hence it is always possible so to project a figure, that the projection of any given straight line shall be removed to an infinite distance. This is called pro- jecting the straight line to infinity. 11. Any quadrilateral may be projected into a parallelo- ^am. For, if ABCD be any quadrilateral, and the sides AB, CD be produced to meet in E, AD, BO in F, and the line EF projected to infinity, then, since the projections of AB, CD intersect at an infinite distance, they will be parallel to one another, as also those oi AD, BO, whence it follows that liie quadrilateral ABCD is projected into a parallelogram. PEOJECTION OP CONICS. 151 12. The angle UVF -will be the angle between the pro- jections of the sides AB, BG. For if the plane of projec- tion cut the lines VA, VB, VO, VD in A\ B', C, D' re- spectively, then the points A!, B, C, B' are respectively the projections of A, B, C, B. Now the plane ABAsB contains the points V, JE, and, since the plane of projection, in which the points A', B lie, is parallel to VEF, and therefore to VE, it follows that AB is parallel to VE. Similarly B C is parallel to VF, and therefore the angle A!B C is equal to the angle EVF. 13. Since the angle EVF may be made of any magni- tude, by taking the point V anywhere on any segment of a circle of which EF is the base and which contains an angle of the required magnitude, it follows that any quadrilateral may be projected, in an infinite number of ways, into a parallelo- gram of which the angles are of any assigned magnitude. 14. We may now proceed to detail the application of the theory of projections to curves of the second degree. It will easily be seen that the projection of any tangent to any curve will be a tangent to the projection of the curve. Again, if any point and straight line be the pole and polar of one another with respect to a given conic, their projections vill be the pole and polar of one another with respect to the projection of the conic. For, let be any given point, XY its polar with respect to any given conic. On XY take any point T, external to Fig. -44. 152 . MODERN GEOMETRY. . the conic, and from T draw two tangents TP, TQ, then PQ will pass through 0. Now project the whole system, and let 0', P', Q', T, X', Y' be the respective projections of 0, P, Q, T, X, Y. Then T'P, TQ will be tangents to the pro- jected conic, and P' Q will pass through 0'. Hence since T is any point on X' Y', X' Y' will be the polar of 0'. 15. From the proposition just proved, it will follow that any two conies may be projected into concentric curves. For it is always possible (Arts. 5 and 7) to find one real point at least, the polar of which with respect to two given conies is the same straight line. Let then this straight line be pro- jected to infinity, and its common pole, with respect to the two conies, will become the centre of the curves of projection. 16. It may also be proved that any two conies may be projected into similar and similarly situated curves. For it is always possible (Arts. 5 and 7) to find two straight lines which meet two given conies in the same two points, real or imaginary. Project either of these straight lines to infinity, and the conies will then be projected into curves, two of the points of intersection of which are infinitely distant, that is, into, similar and similarly situated conies. These will be ellipses or hyperbolas, according as the points, in which the line projected to infinity meets the conies, are imaginary or real, if the two conies have double contact with one another, their projections will also be concentric. 17. The projections, spoken of in the last two articles, may be effected in an infinite number of ways. For any point whatever may be taken as the vertex of the cone, and if the cpne be cut by a plane, parallel to that which passes through the vertex and the line which it is required to project to in- finity, the required projection will be effected. 18. It hence follows that it is possible to project any two intersecting conies into hyperbolas of any assigned eccen- tricity. Suppose, for example, that it is required to project two conies, intersecting in points A, B, into two similar and similarly situated hyperbolas, the angle between the asymp- totes of each being a. Take any point F, such that the angle ANT CONIC MAT BE PROJECTED INTO A CIRCLE. l53 A VB = a, and describe two cones, of which V is the common vertex, passing through the two given conies. The sections of these cones made by any plane parallel to the plane VAB will be hyperbolas, of which the asymptotes are parallel to VA, VB respectively, and will therefore be similar and simi- larly situated to one another, and of the required form. 19. We now come to the most important and most diffi- cult point of the theory of projections, the process by which, from the properties of the circle, those of conic sections in general may be deduced. We have just seen that any two conies may be projected into hyperbolas of any assigned ec- centricity. Now this process, the possibility of which we have shewn by a geometrical method, of course admits of algebraical proof. And the algebraical investigation, on ac- count of the continuity of the symbols employed, would not take any account of the restrictions introduced into the geo- metrical investigation, either as to the conies intersecting in real points, or as to the eccentricity of the conies into which they are projected being greater than unity. It is therefore possible, by an algebraical process, to transform the equations of any two conies whatever into those of conies of any eccen- tricity, and therefore into those of circles. The points and tangents common to the two given conies will be transformed into points and tangents common to their projections, and the relations of poles and polars will remain unaltered. Since all circles pass through the same two points on the line at infinity, it follows that all circles are transformed by projection into a system of conies passing through the same two points, or having a common chord. Again, since every parabola touches the line at infinity, it follows that all para- bolas will project into a system of conies touching the same straight line. A system of parabolas and circles will project into a system in which all the circles will become conies pass- ing through the same two points, and all the parabolas will become conies, having the straight line joining those two points for a common tangent. 20. We have seen, in the investigation of the co-ordinates of the real and imaginary foci, given in Chap, vi., that the pair 154 MODEBN QEOMETBY. of imaginary iangents, drawn to a conic from ahy one of its four foci, satisfy the analytical conditions of being asymptotes to a circle. !mnce these tangents must themselves meet the line at infinity in the two circular points. Conversely, if from the two circular points at infinity two pairs of tangents be drawn to any conic, these will form an imaginary quadrilateral, cir- cumscribing the conic, the four angular points of which are the four foci of the curve. Hence all conies having the same focus project into conies having a pair of common tangents; and all confocal conics into conics inscribed in the same quadrilateral. The directrix is the polar of the focus, hence, if two conics have the same focus and directrix, they project into two conics having a common chord of contact for their common tangents, that is, having double contact with one another. 21. The anharmomc ratio of any pencil or range is un- altered by projection. Let the transversal PQR8 cut the four Straight lines OP, OQ, OB, OS. Take any point V, not lying in the plane through these straight lines, join VO, VP, VQ, VB, VS, and let these lines be cut by any other plaiie in (X, P", Q, i?, /ST* Then {a.pqB8'\ = \_pqB!8'-\ P'Q.B'S' '^P'E',Q'8' BinPVQf.smR'VS' "" sin F VB. sin gV8' _ Bin PVQ. sin BV8 sin PVB. Bin QV8 PQ.R8 ^ PB.QB *^{O.PQBB]. PEOJECTION OF ANGLES. 155 Hence the anharmonic ratio of the given pencil and range is the same as that of their projection. 22. The following proposition is usefal in the projection of theorems relating to the magnitude of angles. Any two lines which make an angle A with each other, Jbrm with the lines joining the circular points at infinity to their point of intersection^ a pencil of which the anharmonic ratio is e*"^'^"^. It will be nnderstood that the two given lines are taken as the first and third legs of the pencil. Take the two lines as two sides of the triangle of reference, and let them be denoted by /3 = 0, 7 = 0. The lines joining their point of intersection to the circular points at infinity- are given by eliminating a between the equation of the line at infinity and that of the circumscribing circle, that is, between oa + J/3+07 = 0, - + | + - = 0, a p 7 This gives ^ + 20y cos A + 'f = Q. Now the two lines represented by the equation (/3-A;7)(/3 + A'7) = form with /S = and 7 = a pencil of which the anharmonic k' ratio is -r (Art. 23, Chap. I.). In the present case. Hence the anharmonic ratio is g-^V-i _ _ jMV-tX _. g(ir-34)V-l CoE. In the case in which the lines are at right angles to one another, A^- , and the anharmonic ratio becomes unity, that is, the four lines form an harmonic pencil. 156 - HODEBN GEOMETRY; 23. The known property of a circle, that " the angles in the same segment are eqnal to one another," gives rise to an important anharmonic property of conic sections. The pro- perty of the circle may be expressed thus, that " if .4, 5 be any two fixed points on the circumterence of a circle, any moving point on it, the angle AOB is constant." Project the circle into any conic, and let A\ B, 0' be the projections of A,B,0; n,K those of the circular points at infinity. Then, from the result of the last article, it follows that {O'.A'B'HK} is constant. Or, the anharmonic ratio of the pencil, formed by joining any point of a conic to four fixed points on the curve, is con- stant. Reciprocating this theorem, in accordance with Art. 13, Chap, VII., we see that if any tangent to a conic he cut iy four fixed tangents, the anharmonic ratio of the range, form,ed by the points of section, is constant. 24. If P, Q, R be three points in a straight line, and p, q, r be their projections, and s the projection of the point at infinity on the line PQB, then ^ r , pq.rs PQ.R8 T'or lpgrs1=.P^^^^-^, where 8 denotes the point at infinity on the line BQR. Also RS : PS in a ratio of equality, hence • !>?"] = 1^. 25. If P, P, C ^, .B, JB'... be a system 6f points in involution, and p, p', g,q\ r, r'... their projections, then since by Art. 27, Chap. i. \FQB8] = [rQ'R;8'], and by Art. 21 of this Chapter {PQRS] = [pgr8], [P'g'iZ'^S'] = [p'gVV], ORTHOGONAL PBOJKCTION. 157 it follows that [pqrs] = [p'q'r's'], or p, p', q, q', r, r'... are a system of points in involution. Hence, any system of pointy in involution projects into a system in involution. If P coincide with P', p will coincide with p', or the foci of one system project into the foci of the other. We may observe that the centre of one system will not, in general, project into the centre of the other. 26. Let a system of circles be described through two given points A, A', and let any circle of the system cut a given straight line in P, P", Produce AA' to meet the given straight line in 0. Then OP.OP'=OA.OA', or OP. OP' is constant for all circles passing through A, A'. Hence, the system of points in which a system of circles, passing through two given points, cut a given straight line, are in involution. Project the system of circles into a system of conies, passing through four given points, and we learn that " a system of conies, passing through four given points, cut any straight line in a system of points in involution." Of this system of conies, one can be drawn so that one of its points of intersection with the given straight line shall be at an infinite distance, — in other words, so that one of its asymptotes shall be parallel to the given straight line. The other point, in which this conic cuts the given straight line, will be the centre of the system. Again (see Art. 3, Chap. IX., infra), two conies can be described, passing through the four given pqints, and touching the given straight line. The two points of contact of these eonics will be the foci of the system of points in involution. By reciprocating these propositions, we obtain analogous properties of the system of conies, inscribed in a given quad- rilateral, whence, by projection, may be obtained those of a system of confocal conies. 27. When the vertex of the cone, used for purposes of projection, is infinitely distant, so that the cone itself becomes a cylinder, the projection is said to be orthogonal. In this 158 MODERN GEOMETRY. mode of projection, the line at infinity remains at an infinite distance, and any two parallel lines will therefore project into parallel lines. Also any area will bear to its projection a constant ratio ; and the mutual distances of any three points in the same straight line will bear to one another the same ratios as the mutnal distances of their projections. Two per- pendicular diameters of a circle will, since each is parallel to the tangent at the extremity of the other, project into two conjugate diameters of an ellipse. By this method, many properties of conic sections, more especially those relating to conjugate diameters, may be readily deduced from those of the circle. EXAUFLES. 1. If XYZ be a triangle which moves in such a manner that its side YZ always passes through a fixed point F, ZX through Q, XY through B,, and if the locus of F be a fixed conic passing through R and P, that oi Z a. fixed conic passing through P and Q, prove that the locus of X will be a fixed conic passing through Q, R, and through the other three points of intersection of the two given conies. 2. If two tangents be drawn to a conic so that the points in which they cut a given straight line fi>rm, with two fixed points on the straight line, a harmonic range, prove that the locus of their point of intersection will be a conic passing through the two given points. 3. A system of conies is described touching four given straight lines ; prove that the locus of the pole of any fifth given straight line with respect to any conic of the system is a straight line. If the fifth straight line be projected to infinity io that the points where it intersects two of the other given straight lines be projected into the circular points, what does this theorem become) 4. A system of conies is described about a given quadrangle ; prove that the locus of the pole of any given straight line, with respect to any conic of the system, is a conic passing through the vertices of the quadrangle. EXAMPLES. 159 5. A system of conies is described touching the sides of a given triangle, and from a given point a pair of tangents is drawn to each conic of the system. Prove that, if the locus of one of the points of contact be -a straight line, that of the other will be a conic circumscribed about the given triangle. 6. The tangent at any point P of a conic, of which S and H are the foci, is cut by two conjugate diameters in ^, «; prove that the triangles iSPT, HPt are similar to one another. 160 CHAPTER IX. MISCELLANEOUS PROPOSITIONS. ON THE DETERMINATION OP A CONIC FROM FIVE GIVEN GEOMETRICAL CONDITIONS, 1. If any five independent conditions be given, to which a conic is to be subject, each of these, expressed in algebraical language, will give an equation for the determination of the five arbitrary constants which the equation of the conic in- volves. Hence, five conditions suffice for the determination of the conic. It may, however, happen that some of the equations for the determination of the constants rise to a degree higher than the first, in such a case, the constants will have more than one value, and more than one conic may therefore be described, satisfying the required conditions, although the number will still be finite. The geometrical conditions of most frequent occurrence are those of passing through given points and touching given straight lines, with such others as may be reduced to these. We proceed to consider how many conies may be described in each individual case. 2. Let fiv^ points he given. In this case we have merely to substitute in the equation of the conic the co-ordinates of the several points for a, y8, 7 ; we shall thus obtain five simple equations for the determina- tion of the constants, and one conic only will satisfy the given conditions. 3. Let four points and one tangent he given. Take three of the points as angular points of the triangle of reference. Let / ^, A be the co-ordinates of the fourth DETERMINATION QP A CONIC FEOM FIVE CONDITIONS. 161 given point, la + myS-f ny = 0, the equation of the given tan- gent. Let the equation of the conic be Then for the determination of the ratios \ : /i : v, we have the equations X"? + ii'm' + i^n* - 2/ivmn — 2v\nl — 2Xfj,lm = 0. These equations will give two values for the ratios, and prove therefore that two conies can be described satisfying the required conditions. 4. JJet three points and two tangents he given. Take the three points as angular points of the triangle of reference. Let the two given tangents be represented by the equations la + 7»jS + ny = 0, ?a + m'y3 + re'7 = 0. If then the conic be represented by the equation a P y we have, for the determination of \ : /i : v, the equations XT + /iW + j^w* — 2fivnin — 2vXnl — 2\/ilm = 0, XY' + /i'm" + vV - 2nvm'n' - 2vXn'l' - 2\/tZ'm' = 0, which, being both quadratics, give four values for each of the ratios, shewing that four conies may be described satisfying the given conditions. 5. Let two points and three tangents be given. Take the three tangents as lines of reference, and let f,g,h; /', g, K, be the co-ordinates of the two given pointg. F. 11 4^ JiOblBBlN !^)XK)UE4Xt. Then, if -fhe 'eqatttion of the conic be XV + At'yS" + kV - iftvfiy - ^vXr/a. -^\im$ = 0, •we shall get, writing/, ^ h^, f,g', ft', successively for a, ^, 7, two quadratics for the determination of the ratios X : /( : v, giving therefore four conies. 6. Let one point and four tangents he given. Taking three of the tangents as lines of reference, the condition of touching the fourth given line gives a simple equation for the deterininatidii of the coefficients, and that of passing through the given point a quadratic. Hence, two conies may he described, satisfying the given conditions. 7. Let five tangents he given. Taking three ^ the tangents ss lines of r^el-ence, the condition of tou<^ing each of the others : gives a simple equation for the determination of the constants, shewing that one conic only can be described satisfying these conditions. The results of Arts. "5, ^, 7, 'inay of course be deduced by the method of reciprocal polars, £i-om those of Arts. 4, 3, 2. 8. Several other forms under which the data may be given, are reducible to a certain number of lines and points. Thus to have given a tangent fend its point of contact is equivalent to having two points given, the points being indefinitely close together. Or, it may be regarded as equi- valent to having ttpo tangents given, these tangents being in- definitely nearly coincident. To have given that a conic is a parabola is equivalent to having a tangent given, since every parabola touches the line at infinity. To have given that it is a circle is equivalent to having two points given, since all circles intersect the line at infinity in the same two points. And this explains the reason why four circles can be described touching the sides of a given triangle, but only one circumscribed «,bout it. So, to have given that a conic is similar and similarly situated to a given one is equivalent to having two points given. To have given an asymptote is equivalent to having two points given, for an asymptote may LOCUS OF THE CE^Il^E. 163 be regarded as a itangent, the -point of contact -of -which is given (at an infimte distance). To have giyen the direction of an asymptote is equivalent to hg,ving one point given, for this virtually determines the point in which me conic meets the line At. infinity. 9. If it be given -that 'three given points form a ^son- jugate triad, this is equivalent to three conditions, as the equation of the conic, when these are taken as angular points of the triangle of reference, is of the form Two more conditions wiU therefore completely determine the conic. If these conditions be that the conic shall pass through two given points, or touch two given straight lines, or pass through .one given point .and touch one given straight line, one conic only can be drawn to satisfy these .condi- tions. We may observe that, if the above conic pass through the point (/, g, h) it also passes through the three points (-/, 9, A), (/, -g, h),.{/, g. -h), and that, if it touch the line (Z, m, n), it also touches the lines (— I, m, n), {I, — TO, n), (I, m, —n). ON THE LOCUS OF THE CENTEE OF A SYSTEM OF CONICS WHICH SATISFY FOUR CONDITIONS, EXPRESSED BY PASSING THROUGH POINTS AND TOUCHING STRAIGHT LINES. 10. The locus of the centre of a conic, which passes through TO points, and touches n straight lines, m+n being equal to four, will be a conic, in every case except two. We will cpnsider the several cases in order. 11. Let the system pass through four-points. This is best treated by Cartesian co-ordinates. Of the conies which can be described passing through four Soints, two are. parabolas. Take, as co-ordinate axes, that iameter of each of these parabolas, the tapgent at the ex- 11—2 164 MODERN GEOMETET. tremity of which is parallel to the axis of the other. Then the two parabolas wifl be represented by the equations ai' + 2/^ + A = (1). f + 2g'x + K=0 (2). The system of conies is represented by the equation af + \f + 2\g'x Jrlify ■{■h + SK = (i (3), \ being an arbitrary multiplier. The centre is given by the equations Eliminating X, we get for the locus of the centres ^=fg (4), a conic, whose asymptotes are parallel to the axes of the parabolas (1) and (2). If the four points form a convex quadrangle, the parabolas will be real, and the locus (4) an hyperbola. If the quad- rangle be concave, the parabolas will be imaginary, and the locus of centres an ellipse. The curve (4) bisects the distance between each pair of the four points, and passes through the vertices of the quadr rangle. This may be seen from geometrical considerations, for of the three pairs of straight lines which belong to this system of conies, the vertices are respectively the centres. From the form of the equation (3) we see that every conic of the system has a pair of conjugate diameters parallel to the axes of the parabola (1) (2) ; in other words to the asymptotes of (4); The conic of minimum eccentricity is obtained by making \ = 1. In this case, these are the equal conjugate diameters. If the axes of the parabolas be at right angles to, one another, the four points lie on the circumference of a circle. The axes of every conic in (3) are then parallel to the co-ordi- nate axes, and (4) is a rectangular hyperbola. If each of the four points be the orthocentre of the othe* GIVEN, THPEE POINTS AND A- TANGENT. 165 tliriee, tKe syBtem of conies is a system of rectangular hyper- bolas, and (4) is the nine-point circle of the given points. 12. JJet three pbints and a tangent he given. In this case we may see, A jpriori, that the locus- will he a curve of the fourth degree, for we can describe four para- bolas satisfying the given conditions, and the locus will have four asymptotes, parallel to the axes of these parabolas. Take the triangle formed by the three points for the tri- angle of reference, and use triangular co-ordinates. Let the tangent be represented by ic + my + m = 0. Then, if the system of conies be represented by \ya + /izx + vxi/ = 0, the condition of tilngency gives ZV + mV + wV ^ 2mniiv — 2nlv\ — ilmX/J, = 0. The centre is given by the equation fiz + vy = vx+Xs = \y + /tta;. If each member of this be put for the moment = p, we have y+z—x z+x—y x+y—z therefore the equation of the locus becomes Z V (y + z-xf + my {z + x-yY + n^z^{x+y- z^ — 2mnyz {z + x — y) (x+y — z) —2nllx{x + y — z) (if + z — x) — 2lmxy (jy+z — x) (z + x — y) — 0, a curve of the fourth degree. Writing 1 - 2a;, 1 - 2y, 1 - 2H, for y + z — x, z + x—y, x + y-s, respectively, the terms of the fourth order become T'a? + 9»y + wV - 2nmy's' - 2nh*a? - 2lma?y'. X 16i5 • iiotet^ Gn&ia^tRY. Hence tlie asymptotes; aaid ttereft«fe the aies of ihe font parabolae, are paridlel to t'he icna Imes ± l*x ± m^y ± j^z = 0. 13. Lei twd points and tioo tangmtd he gvseii. In this case, aeaiti, four pai^abolas can be described satisfy- ing the given conditions, and we might therefore expect that the locus would be a curve of the fourth degree. It will be found, however, that ft bifeaks up into two factors of the second degree. Taking the line joining the two points as a = 0, and the other two as )3 = 0, 7 * 0, the eqtiatioQ of the system may be written 2yS7+ (Xa + wi)8 + 717)'= 0, Here X is a tradable parameter. For the' determination of m and n, we may proceed as follows. Let the values of — , corresponding to a = 0, be called l^ V. We have then W» tut ire Hence -= + (ZZ')i. For the centre, we have the equations X (Xa + m^ + wy) ^ m (X« + j»/S + *J7) + 7 -»(Xa + »i/8 + n7)+j8; .-. Xa + }n/3+B7=§^; GIVEN, ONE POINT AND THEEE TANGENTS. 1®? (m + n) (7m + mfi + ny) + jft-f ry m + n m-rnfi + y ^Q^ + mfi + nr^l ~ 2 ■*" 2 /S^7 _ m^ — m/ Hence, tbe locus becomes p— 7 ' ' m—n ox {m — n) a {m^ — ?^y]^ -|- (tot ») (^— j^ ^^.+ nv) = O - 7)'; .•. («i — m) [m^—n'/+ (ra S5 jw) J87 — 717a + ma/9} = (/8 — 7)'. This may be written (m* - win — 1) iS" + (n* — nm - 1) 7* — (»»* — 3mra + »' - 2) ^87 + (w*— mn) ya + {m' -^ mn) a^ = 0, or^ dividing by m^, and substituting the values of - already obtained, + U'r^q, + oyS + (ZZ')4 (7a + a^) = 0, giving, as stated above, two conic sections for the required locus. 14. Let 0|ie paint, and t^pee tangtnts he givent Here t^e required iocijs wU be fi coniQ, §ince ©nly t^o p^rabol^ can be described pati^Mng the given conditions. Take the three straigjit lines ^s Jine§ of rciference: and let /, ff, h, be the triangular co-ordinates of the given ppmt We have then, as the equation of the system, i V + m^y" + nV — 2mny9 — 2nl fas -r ^^ xj( = p, subject to the con4itio^ P/*+ my + n'h* - 2mn gh - 2reZ hf- 2hifg = (1). I6d MODEBN GEOMETBT.. For the centre, we have l(lx — my — m)=m{—lx + my — nz)= n (— h; —my + ns) ; Ix'^my — nz _ — lx + my + nz _ — h; — my + ma ^ Ix my m or '^(m + n) Jw(n+Q" n(Z + wi)' X V . e y__: m + n n + l i+m ' I m n '' y + z—x z + x—y x + y — z' giving, for the locus of the centres, f^y + z-xY+g'{z + x-yy + h''{x + y-zy -2gh{z+x-y) (x + y-z) -2hf{x + y-z) {z + x-y) -2fff{y + z-A){z + x-y) = 6, a conic touching the three straight .lines which join the middle points of the sides of the triatigle formed by the three given tangents. Its asymptotes are parallel to those of the curve fa?-\-^f + Vz^-2gTiyz-2Tifzx-2fgxy = 0. It will therefore be a rectangular hyperbola, if /»a*+^J' + AV+ (6» + c'- a») gh + {,? + a*-V) hf + {a^+V- 170 vooms etsQWBTK^. The •prodwct of any two determinants is a determinant. Fiist, take the case of two row» and eolucuu. Let And let iS.S» + )8^,-0l These equations lead to the following : A ("la'i + a.a's) +^2 (*!«! + *if».) - ; or 03,0,+ /3A) a;, + 03,a.+ ^,&^ «c, ^ oj " Now, if (2) lie satisfied, (3) will be. Aad (2) are satiafied, if either = 0. 0v ^t or if fi and ^ are each = 0. In the latter case, we have by (1), Mther h\ .(1). .(2). .(3). ■W, .(5), or a!, and aj, = 0. But if a, and x^ be not =0, then (3) gives /8.a. + /8A, /8,<^ + A6. 1 ^ ^" Hence (6) is satisfied whenever either (4) or (5) are, and therefore its left-hand member most involve, rs factors, the left-hand members of (4) and (5). The only other factor is nnmerical, and this will be seen, by oompariag the ooeffioieuts of any term, as for instance afi^fi^, to be unity. «1. «». «. K h \ Cl. <^> 5' > * For, generally. M, to', v' 1 w'.v, u' v% «', u> «, to', r' w', t>, «' «', m', w /, i;> h /'. /, h' /" 9", r „/ + w'g + v% uf + w'g + »T, uf + w Y'+ »T' wf+vg +u%, wf'+vg' +«'A', w'f'+vg" +u'h" vf + u'g + wh, v'f + u'g' + wh\ v'f + ug" + wK' ^/» ^#» ^A ^/"» ^fl"> ^i" wtete ^/ is written for g^j &t' 172 MODERN GEOMETET. M, W', V' f, 9, h w', V, m' f,9\t^ v', «', w f',9",K' Hence f^r + ^^tf" + Hh", f'^r + /^ff" + A>A", f'^r + 9"'i>^' + *"^a" Bnt/^^+5'«/>j + A<^fc = ^(/, ^, A) whatever/,^, h, maybe. And f^+g'g + h'<^H = 0. Similarly, all terms of similar form = 0. Hence tie theorem is proved. 18. A triangle is inscribed in the conic, nx" + vy° + wz* + 2u'yz + 2v'zx + 2w'xy = 0, two of its sides passing through the fixed points (f, g, h), (f ', g , h'), to find the envelope of the third. Call the fixed points K, K', and the angular points of the triangle PQR, BP passing through K, PQ through K'. Then, by projecting the conic into a cirde and the line KK' to infinity, the lines JBP, PQ will'project into two lines always parallel to 'themselves, and therefore containing a constant angle, hence QB will project into a line always touching a circle concentric with tne given one. Therefore, in the given problem, the envelope of QR will be a conic, having double contact with the given one along the line KK', and will there- fore be represented by the equation >^ (». V, ») + / 9> A, fr9'. K X, y. z = 0. .(1). j) (^, y, z) being written, for shortness, instead of iwc* + vy' + we' + 2u'yz + 2v'zx + 2w'xy, and \ being a constant to be determined. Now we observe, in the first place, that \ must be of two dimensions in/, g, h, of two in/', g"^, h'i aud of - 1 ip w, v, Wf u, V, w. SOPPLEMENTAET PROBLEMS. 173 Next, let V be the point of intersection of two consecutive positions of QB. Then, if a triangle inscribed in the conic so that two of its sides always pass through K', V, the en- velope of the third side will pass through K. Hence (1) must be satisfied when we exchange x, y, z, with^ g, h. Therefore we have \4>U',9,'h) + X, y^ z /, 9^ K f, 9^ h = 0, \ being what \ becomes when x, y, z are written for/, g, h. Hence M> {^^ Vi «) = \^ (/, g, h) idmtically, res ^ (/, g, h) as a factor. li) as a factor. Hence we ) ^^. (.f,ff,h)^(f',g;h') whence, X involves ^ (/, g, h) as a factor. Similarly it in- volves ^ (/', g', V) as a factor. Hence we may write fL being a function of u, v, w, «', v', w', of three dimensions, since \ is of — 1. The equation then becomes ^ (/, 9, *) if, 9', *') ^ («, /3, 7) + 1* f, 9> h f 9' K a, P, 7 = 0, To determine fi, we may suppose, since it is independent of the co-ordinates of K, K', that each of these points fies on the polar of the other. Then, the envelope of QB must pass through the pole oi KK', as nyiy be seen by projecting KK' to infinity, for then QB will always'pass through the centre of the conic. Hence, if (/", g", h") be the polar of KK' {f.9,^)U">9',h')if",9",^")+/^ f, 9'. ^' = 0. whence by Art. 17, A* = — w', V, u' v, u, w 174 MDDESN GEOMETSY. t t ' /, 57, * w', V, u . /'>y/^'. t)' m', w ' a;, ^, s Therefore llae required enveloipe is 19. A triangle is described about a conic, x' + y* +,»' = 0, two of its vertices move on fired straight lines, Ix + my + nz = 0, I'x + m'y + n'z = ; to prove 'that the locus of the third vertex will he given hy the equation (U' + mm' + nn7-(x= +,y' + z=) + 1'. m'. n' , = 0. It may be shewn, sby reciprocating the theorem in the last article, that the locus will have double contact with the given conic along the pole of the intersection of the two given straight lines; hence its equation will be of the form I, m, n 1, m, n 1', K n'; X, y, z \{x'+f + z') + I , m', = 0. (1), and it remains to determine X. For this purpose let the straight line Ix + my ■\-nz-=Q cut the given conic va. P, P'. Let T be the pole oi PP'. Now, suppose oneisideof the triangle to become the tan- gent at P, then the other tangent through P will coincide with it, hence the required locus passes through the point of intersection of Tx + my+n'e = 0, with the tangent at P, and also with the tangent at P'. Now, the co-ordinates of T are J, m, n, hence these two tangents are represented by the equation {r + m' + n') {a? + y'^z'')-{lx + my + nzy = (2). Hence (1) must be satisfied by the values of x, y, z, which satisfy (2), and also make I x-\- vny + riz = 0. I, m, n Now l,m'y7ib TEILINEAS JC<>ORDmATES OF FOCI. 175 P + m' + m', ir + mm' + «w', Ix + my + ne IV + mm + W, Z" + to" + n'", Z'a! + m'y + w'a £B+4»y +w», I'n + m'y +n'z, a? +y' +z^ which if I'x + m'jf + w'» *= 0, becomes r +n? +7?, IV -^mm' +mC, h)+-my + nz ZZ' + mm' + Mn', r+j»'» +m", Ix + my +nz, 0, x^ + y'^ +2* = (Z' + m' + n=) ;(Z" + m!^ +;«") (a;' +/ + »^) - {W + mm' + niiY {pi?+f + z") - (Z" + m** + m") (fe +-m^ + my = - "(Zr + mm' + nny. {a? +f-\- i"), if (2) 136 satisfied. Hence, by (1) X — ijl' + mm' + ««')' = 0, identically, therefore, -the eqtiation of the required locus becomes {IV + mm' + nn'Y {a? + / + »') + I, m, n , m, n x>,y, s TEILINEAR CO-OEDINATES OF "THE FOCI OF A CONIC. 20. The following investigation of the trilinear co-ordi- nates of the foci does not introduce the conception of the imaginary circular points at infinity, or of imaginary tan- gents. The trilinear co-ordinates of the focus of the conic wa' + v^' + wr/" + 2m'j87 + 2^^ + 2w'a/3 = 0, may be investigated in the following manner. Draw two tangents to the conic parallel to a= 0, and let_yj,^ be their respective distances feom that line. Then, ii f, g,h be the co-ordinates of a focus, we iave (f—fi) (j^ — /) = the square on the semi-axia minor. 176 MODEEN GEOMETET. If the equation of either tangent be ■o'(5/3 + c7) = (2A-aa')a, which represents a line parallel to, and at a distance a' from a = 0, the two values of a! obtained by introducing the con- dition of tangencj, will be /„ f^. Now, the condition of tangency is 0, a'a-2A, ha.', ca! = 0. oa'-2A, M, w', v' ha', w', V, u' ca,', v', u', w or J7(aa' -2A)* + F(Ja7 + Tr(ca')' ' +2C/''Ja'.ca' + 2r'ca'K - 2A) + 2W'(aa.'-2A) Ja' = 0, which may be written ( Ua' +Vh' + Wc' + 2 Who + 2 7'ca + 2 W'al) a" - 4 A { Z7a + W'h + V'c) a' + 4 A' U= 0. Hence, the left-hand member of this equation is identi- cally equal to (Ua*+ Yb' + We' + 2Vhc + 2V'ca + 2W'ah) (a' -/,) (a' -/J, and therefore the square on the semi-axis minor {Ua+W'h + V'c)f- A £7- = -/= + 4A Ua" + Vb* +Wd' + 2 U'bo + 2 V'ca + 2 W'ab ' Two similar expressions being obtained, we get for the determination of the foci, the equations {Ua'+Vb^ + W(? + 2 U'bc + 2 V'ca + 2 W'ab)f -4A{Ua+W'h+ V'c)f+ i^^U = {Uc?-i- 76' +Wc' + 2 U'bc + 2V'ca + 2 W'ab) g^ - 4A (76 + ITc + W'a) ^ + 4A''7 = ( Ca' + 7S' + Fb' + 2 Whc H- 2 V'ca + 2 W'ah) A' -4A (T7c + V'a -t- Z/'J) A + 4A*Tr, the same as those obtained in Chap. vi. Art. 33. EXAMPLES. 177 MISCELLANEOUS EXAMPLES. 1. Prove that the centre of the conic aa op cy coincides -with the centre of gravity of the triangle of reference. 2. Prove that 0, 1, 1, 1 1, 0, z', f 1, «^ 0, x' = (x+y+z) {x-y-z) (y-z-x) {z-x~y). 3. Prove that the square on the radius of the circle, de- scribed about the triangle of which the angular points are a, b, c, is 0, ab', ac' 1 ba', 0, bo^ •J ca\ cb\ 0, 1, 1, 1 1, 0, ab', ac' 1, ba', 0, be' 1 , ca% cb\ 1 Investigate a similar expression for the square on the radius of the sphere, described about the tetrahedron of which the angular points are a, b, c, d. 4. S is a. focus of a conic, PQ a chord subtending a constant angle at S; SB, ST are drawn meeting the tangents at P and Q in P, T respectively, so that the angles PSR, QST are constant ; prove that RT always touches a conic having S for a focus, and a directrix in common with the given conic. F. 12 178 MODERN GEOMETRY. 5. Prove that, if the conic {laf + {mISf + («y)* = be a para- bola, its focus and directrix are given by the equations la _ mfS _ ny tan A tan B tan C Hence prove that, if a parabola touch three straight lines, its directrix always passes through a fixed point. State, in geometri- cal language, the position of this point relatively to the three straight lines. 6. A system of parabolas is described so that a given triangle is self-conjugate with respect to each curve of the system; prove that the locus of the focus is a circle, that the directrix always passes through the centre of the circle described about the tri- angle, and that every parabola of the system touches the three straight lines which bisect each pair of sides of the triangle. 7. If P be any point on the circumference of a circle, any fixed point, prove that the locus of the point, in which the tangent at P intersects the line which bisects OP at right angles, is a straight line. 8. A rectangular hyperbola circumscribes a triangle; shew that the loci of the poles of its sides are three straight lines form- ing another triangle, whose angular points lie on the sides of the first, where they are met by perpendiculars from the opposite an- gular points. 9. If ABC, A'EG' be two triangles, each of which is self- conjugate with regard to the same given conic, shew that another conic can be described about both. 10. If a, ^, y, 8 be the distances of a point from four given straight lines, so connected that la. + m^ + ny + p8 = 0, prove that, if a conic be described, touching these four straight lines, the locus of either of its foci will be the curve of the third degree repre- sented by the equation I m n p ^ a p y 6 EXAMPLES. 179 11. Prove that the polar reciprocal of a rectangular hyperbdla ■with respect to any point »S', is a conic, the sum of the squares on the semi-axes of which is equal to the square on the distance of its centre from S. 12,. Two given conies are so related that each of their common tangents subtends a right angle at a given point. Prove that, if any two points be taken, one on each conic, so that the line join- ing them also subtends a right angle at that point, the envelope of this line will be a conic, of which that point is a focus. 13. In Example 2, p. 116, prove that if any conic (A) be drawn touching the directrices of the four conies, the polar of the given point with respect to it will be a tangent to a conic, having the given point as focus and touching the sides of the tri- angle ; and that the tangents from the given point to A are at right angles to each other. 14. If, through a fixed point 0, a straight line be drav/n cut- ting the sides A£, AC of a triangle ABC in F, Q respectively, and BQ, GP be joined, prove that the locus of their j)oint of intersec- tion is a conic circumscribing the triangle ABC. 15. If Pa, Phj Pc lie the semi-diameters of a conic, respectively parallel to the sides of the triangle of reference, prove that the area of the conic is sin il sm if sin \ Pa J\ pt /\ Pc J , „^ sin A sin B sin C where 22 = 1 1 • Pa Ph Pc 16. PQ is the chord of a conic, having its, pole on the chord AB or AB produced ; Qg is drawn parallel to AB meeting the conic in q ; shew that Pq bisects the chord AB. 17. Similar circular arcs are described on the sides of a tri- angle ABC, their convexities being towards the interior of the tri- angle ; shew that the locus of the radical centre of the three circles is the rectangular hyperbola si n(^-C ) sjn(C-A) sin(A-B) ^ a p y ' 180 MODERN GEOMETKT. 18. Prove that, if r be either semi-axis of the curve repre- sented by the equation Ma" + 1)/?' + wf + 2M'y37 + 2»'ya + 2w'o^ = 0, the values of r wiU be the roots of the equation a" V \ u+j-{au'—hv—cw') \r'—as cos A iv+ — ipv—cw'—au') \r'—bscosB = 0, ■j w + -7 {cvD — au - bo) (• r — cs cos ( ahc where s- M, w', V W ■v, u v', < w u, w, ■», a W, V, u, b v', u, w, c a, b, c, 19. If a triangle is self-conjugate with respect to each of a series of parabolas, the lines joining the middle points of its sides will be tangents: all the directrices will pass through 0, the centre of the circumscribing circle: and the focal chords, which are the polars of 0, will envelope an ellipse inscribed in the given triangle which has the nine point cu-cle for its aioxiliary circle. 20. A conic circumscribes a triangle ABC, the tangents at the angular points meeting the opposite sides on the straight line BEF. The lines joining any point P on DBF to A, £ and G meet the conic again in A', B' , C' : shew that the triangle ABC envelopes a fixed conic inscribed in ABG, and having double contact with the given conic at the points where they are met by BEF. Also the tangents at £, B', G' to the original conic meet B'C', G'A', A'B' in points lying on DEF. 21. The anharmonic ratio of the pencil formed by joining a point on one of two conies to their four points of intersection is equal to the anharmonic range formed on a tangent to the other by their four common tangents. EXAMPLES, 181 22. The four common tangents to two conies intersect two and two on the sides of their common conjugate triad. 23. The locus of the centres of conies inscribed in a triangle and such that the centres of the escribed circles form a self-con- jugate triad with respect to them is a straight line parallel to aa + bj3 + cy = in areal co-ordinates. 24. A triangle ABC, right-angled at A, is inscribed in a rectangular hyperbola; tangents at £ and C meet in P: prove that AB, AP, ilCand the tangent at A form a harmonic pencil. 25. AB, CD are two fixed chords of a conic. A straight line AFQ meets CD in P and the curve in Q, and on CQ a point if is taken so that PH subtends a constant angle at B : the locus of B will bs a conic passing, through B and C. 26. Conies circumscribing a triangle have a common tangent at the vertex ; through this point a straight line is drawn : shew that the tangents at the various points where it cuts the curves all intersect on the base. 27. One conic touches OA, OB in A and B, and a second ■conic touches OB, OC in B and C: prove that the other common tangents to the two conies intersect on AC. 28. With any one of four given points as centre, a conic is described, self- conjugate with regard to the other three; prove that its asymptotes are parallel to the axes of the two parabolas which pass through the four given points. 29. A rectangular hyperbola passes through the angular points, and a parabola touches the sides, of a given triangle; shew that the tangents drawn to the parabola, from one of the points where the hyperbola cuts the directrix of the parabola, are parallel to the asymptotes of the hyperbola. Which of the two points on the directrix is to be taken? When they coincide, shew that either curve is the polar reciprocal of the other with respect to the coincident points. 30. The triangular coordinates of the two circular points at infinity are given by the equations X _ y z 182 . EXAMPLES. 31. If each of two conies be reciprocated ■with respect to the other, prove that the four points of intersection of any two of the conies thus obtained, and the four points of intersection of the other two, lie on a conic. 32. With any one of four given points as centre, a conic is described, self-conjugate with regard to the other three; prove that its asymptotes are parallel to the axes of the two parabolas which pass through the four given points. 33. With each of four given straight lines as directrix, two conies are described, self-conjugate with regard to the other three; prove that the four pairs of conies thus obtained, will have the same pair of points as foci corresponding to the given directrix. 34. If a triangle be self-conjugate to a rectangular hyperbola, and any conic be described, touching the sides of the triangle, each focus of this conic will lie on the polar of the other with respect to the rectangular hyperbola. CAMBMDGE; PliltJTJJD AT THE UKIVEDSITr PKESS. 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