f LIBRARY ^ UNIVERSITY OP k CALIFORNIA ^ SAN DIEGO 3S SCIENCE AND ENGINEERING LIBRARY Un.vers.ty of California, San Diego Please Note: This item is subject to RECALL after one week. DATF niiF '^E 7 (Rev. 7/82} UCSD Libr. A TREATISE CONIC SECTIONS: COXTAISIXG AN ACCOUNT €F SOME OF THE MOST IMPORTANT MODERN ALGEBRAIC AND GEOMETRIC METHODS. GEORGE SALMON, D.D., D.C.L., F.R.S. REGIUS PROFESSOR OF DIVINITY IN THE UNIVERSITT OF DUBLIN. Jiftlj ^bithit. LONDON: LONGMANS, GEEEN, EEADEE, AND DYER. ... 1869. Q/ISTS, CAMBRIDGE: PRINTED BY WILLIAM METCALFE, GREEN STREET. CONTENTS. [Junior readers -will find all essential parts of the theory of Analytical Geometry included in Chapters i., ii., v., vi., x., xi., sii., omitting the articles marked with asterisks.] CHAPTER I. THE POINT. PAOE Des Cartes' Method of Co-ordinates . . . . .1 Distinction of Signs ...... 2 Distance between two Points . . . . . .8 Its sign ....... 4 Co-ordinates of Point cutting that Distance in a given Ratio . . .5 Transformation of Co-ordinates ..... 6 does not change Degree of an Equation , . . .9 Polar Co-ordinates ...... 10 CHAPTER II. THE RIGHT LINE. Two Equations represent Points . . . . . .11 A single Equation represents a Locus , , , . 12 Geometric representation of Equations . . . . .13 Equation of a Right Line parallel to an Axis .... 14 through the Origin . . . . . .15 in any Position ...... 16 Meaning of the Constants in Equation of a Right Line . . .17 Equation of a Right Line in terms of its Intercepts on the Axes . . 18 in tenns of the Perpendicular on it from Origin, and the Angles it makes with Axes . . . . . . .19 Expression for the Angles a Line makes with Axes ... 20 Angle between two Lines . . . . . .21 Equation of Line joining two given Points . ^ . . .23 Condition that three Points shall be on one Right Line . . .24 Co-ordinates of Intersection of two Right Lines ... 25 Middle Points of Diagonals of a Quadrilateral are in a Right Line (see also p. G'2) 26 Equation of Perpendicular on a given Line .... 26 of Pei-pendiculars of Triangle . . . . .27 of Perpendiculars at Middle Points of Sides , . , 27 of Line making a given angle ^vith a given Line . . .27 IV CONTENTS. PAOR Length of Peipendiculav from a Point on a Line . . . 2« Equations of Bisectors of Angles between two given Right Lines . . 29 -Area of Triangle in terms of Co-ordinates of its Vertices . . 30 Area of any Polygon . . . . . . . iU Condition that tlrrec Lines may meet in a Point (see also p. 34) . . 32 Area of Triangle formed by three given Lines . . . .32 Equation of Line through the Intersection of two given Lines . . 33 Test that three Equations may represent Right Lines meeting in a point . 31 Connexion between Ratios in which Sides of a Triangle are cut by any Transversal 35 by Lines through the Vertices wliicli meet in a point 3(5 Polar Equation of a Right Line . . . . , 3G CHAPTER III. EXAMPLES ON THE RIGHT LINE. Investigation of rectilinear Loci , . . . . .39 of Loci leadmg to Equations of Higher Degree ... 47 Problems where it is proved that a Moveable Line always passes through a Fixed point . . . . . . .47 Centre of Mean Position of a series of Points ... 50 Right Line passes through a Fixed Point if Constants in its Equation be connected by a Linear Relation . ' . . . .50 Loci solved by Polar Co-ordinates . . . . . 51 CHAPTER IV. THE RIGHT LINE. — ADRIDGED NOTATION. Meaning of Constant I- in Equation a = kji . , . .53 Bisectors of Angles, Bisectors of Sides, &c. of a Triangle meet In a point . 34, 54 Equations of a pair of Lines equally inclined to a, /3 . . . 55 Theorem of Anharmonic Section proved .... 55 Algebraic Expression for Anharmonic Ratio of a Pencil . . ,56 nomographic Systems of Lines . . . . .57 Expression of Equation of any Right Line in tei-ms of three given ones . 57 Hai-monic Properties of a Quadrilateral proved (see also p. 305) . . 57 Homologous triangles : Centre and Axis of Homology . . .59 Condition that two Lines should be mutually perj^jendicular . . 59 Length of Perpendicular on a Lure . . . . .GO Perpendiculars at middle Points of Sides meet in a Point . . 34, CO Angle between two Lines . . . . • .60 Trilinear Co-ordinates . . . . . .61 Trihnear Equation of Parallel to a given Line . . . .61 of Line joining two Points ..... 62 Proof that middle Points of Diagonals of Quadrilateral lie in a Right Line . 62 Intersections of Perpendiculars, of Bisectors of Sides, and of Perpendiculars at middle points of Sideis, lie in a Right Line (see also pp. 60, 127) . 63 Equation of Line at infinity . . . . . .64 Cartesian Equations a case of Trilinear .... 65 Tangential Co-ordinates . . . . . .65 Reciprocal Theorems ...... 66 CONTENTS. CHAPTER V, RIGHT LINES. Meaning of .in Equation resolvable into Factors of a Homogeneous Equation of the w"' Degree . . Imaginary Right Lines ..... Angle between two Lines given bj^ a single Equation Equation of Bisectors of Angles between these Lines Condition that Equation of second Degree should represent Right Lines (see ; pp. 144, 148, 150, 255) ..... Number of conditions tha,t higher Equations may represent Right Lines Number of. terms in Equation of «"' Degi'ee .... 07 68 C9 70 71 CHAPTER VI. THE CinCLE. Equation of Circle ..... Conditions that general Equation may represent a Circle Co-ordinates of Centre and Radius Condition that two Circles may be concentric . that a Curve should pass through the origin Co-ordinates of Points where a given Line meets a given Circle Imaginaiy Points ..... General definition of Tangents Condition that Circle should touch either Axis Equation of Tangent to a Circle at a given Point Condition that a Line should touch a Circle Equation of Polar of a Point with regard to a Circle or Conic Length of Tangent to a Circle .... Line cut harmonically by a Circle, Point, and its Polar . Equation of pair of Tangents frnm a given Point to a Circle Circle through three Points (see also p. 128) Condition that four Points should lie on a Circle, and its Geometrical meanin Polar Equation of a Circle .... 75 75 76 77 77 79 80, 81 81 83 84 85 85 86 86 87 CHAPTER VII. EXAMrLES ON CIRCLK. Circular Loci . . . . , . .88 Condition that intercept by Circle on a given Line may subtend a right Angle at a given Point ....'. 90 If a Point A lie on the polar of B, B lies on the polar of -1 . . . 91 Conjugate and self-conjugate Triangles . ' . . , 91 Conjugate Triangles Homologous . . , . .92 If two Chords meet in a Point, Lines joining their extremities transversely meet on its Polar ...... 92 Distances of two Points from the centre, proportional to the distance of each from Polar of other . . , . . .93 Expression of Co-ordinates of Point on Circle by auxiliary Angle . 94 Problems where a variable Line always touches a Circle . . .95 Examples on Circle solved by Polar Co-ordinates ' . . ,96 VI CONTENTS, CHArTEll VIII. ruoriiUTiEs of two or moue ciucles. PAQE Equation of radical Axis of two Circles . . . . .98 Locus of Points wlience Tangents to two Circles have a given Ratio . 99 Radical Centre of three Circles . . . . ,99 Properties of system of Circles having common radical Axis . . 100 The limiting Points of the system , , . . . 101 Properties of Circles cutting two Circles at light Angles, or at constant Angles 102 Common Tangent to two Circles . . . , .103 Centres of Similitude , , . , , . .105 Axis of Similitude , . . . . .108 Locus of centre of Circle cutting three given Circles at equal Angles , . 108 To describe a Circle touching three given Circles (see also pp, 115, 130, 279) 110 Mr. Casey's Solution of this Problem , . . . .113 Relation connecting common Tangents of four Circles touched by same fifth , 113 Method of Inversion of Curves , , , . ,114 Quantities michanged by Inversion . . . ■* . . 114 Equation of Circle cutting three at right Angles (see pp. 128, 347) . 115 CHAPTER IX. THE CIUCLE — ABRIDGED NOTATION. Equation of Circle circumscribing a Quadrilateral , . . . 116 Equation of Circle circumscribing Triangle a, (3, y . . . 118 Geometrical meaning of the Equation , . . . .118 Locus of Point such that Area of Triangle formed by feet of Perpendiculars from it on sides of Triangle may be given . , . 119 Equation of Tangent to circumscribing Cu-cle at any vertex . . .119 Tangential Equation of circumscribing Circle . . . 120 Conditions that general Equation should represent a Circle . , . 121 Radical Axis of two Circles in Trilinear Co-ordinates . . . 122 Equation of Circle inscribed in a Triangle , • . . . 122 Its Tangential Equation ...... 124 Equation of inscribed Circle derived from that of circumscribing . , 1 25 Length of Tangent to a Circle in Trilinear Co-ordinates . . 127 Feuerbach's theorem, that the four Circles which touch the sides of a Triangle are touched by the same Circle , . . 120, 128, 301, 345 Tangential equation to Circles whose Centre and Radius are given . , 128 Distance between two Points expressed in Trilinear Co-ordinates , 128 Determinant Notation ...... 128 Determinant expressions for Equations of Circles through three Points, or cutting three at right Angles .... 128 Relation connecting mutual distances of four points in a plane . .129 Proof of Mr. Casey's theorems ..... 130 CHAPTER X. GENERAL EQUATION QF SECOND DEGREE. Number of conditions which determine a Conic Transformation to Parallel Axes of Equation of second Degree 131 132 CUN TENTS. VU Discussion of Quadratic which determines Points where Line meets a Oonic Equation of Lines which meet Conic at infinity Distinction of Ellipse, Hypei-bola, and Parabola Co-ordinates of centre of Conic . Equations of Diameters Diameters of parabola meet curve at infinity Conjugate diameters . , . . Equation of a Tangent Equation of a Polar .... Class of a Curve, defined Harmonic Property of Polars (see also p. 28i) Polar propeities of mscribed Quadrilateral (see also p. 307) Equation of pair of Tangents from given Point to a Conic (see also p. 257) Rectangles under segments of parallel chords in constant ratio to each other Case where one of the Unes meets the Curve at mfinity Condition that a given Line should touch a Conic (see also pp. 255, 328) Locus of centre of Conic through four Points (see also pp. 243, 260, 290, 308) PACE 133 131 135 138 139 140 141 141 142 142 143 143 144 145 146 147 148 CHAPTER XI, CENTRAL EQUATIONS. Transformation of general Equation to the centre Condition that it should represent i-ight Lines Centre, the Pole of the Line at infinity (see also p. 284) , Asymptotes of Ciu-ve . ... Equation of the Axes, how found .... Pimctions of the Coefficients which are unaltered by transformation Sum of Squares of Reciprocals of Semi-diameters at right Angles is constant Sum of Squares of conjugate Semi-diameters is constant . Polar Equation of Ellipse, centre being Pole Figure of Ellipse investigated .... Geometrical construction for the Axes (see also p. 1C8) Ordinates of Ellipse in given ratio to those of concentric Circle Figure of Hyperbola ..... Conjugate Hyperbola . . . . * . Asymptotes defined ..... Eccentricity of a Conic given by general Equation . Equations of Tangents and Polars Expression for Angle between two Tangents to a Conic . Locus of intersection of tangents at fixed angle Conjugate Diameters : their Properties (see also p. 154) Equilateral Hyperbola : its Properties Length of central Peii^endicular on Tangent . .' Angle between Conjugate Diameters Locus of intersection of Tangents which cut at right Angles (see also pp 258, 339) . . . . Supplemental Chords ..... To construct a pair of Conjugate Diameters inclined at a given Anglo Relation between intercepts made by variable tangent on two parallel tan (see also pp. 275, 287) .... Or on two conjugate diameters .... Given two conjugatadiametens to find the axes 149 150 150 150 151 152 b 154 154 155 156 156 157 158 159 150 159 159 ICO 161 101 , 161 162 , 163 164 , 164 ). 161, 166 . 166 166 igents 107 167 168 \ Vin CONTENTS. PACE Normal : its Properties ...... 168 To draw a Normal through a given Point (see also p. 323) . . . 169 Chord subtending a right Angle at any Point on Conic passes througli a fixed Point on Normal (see also pp. 259, 273) . . . 170 Co-ordinates of intersection of two Normals . . . .170 Properties of Foci . . , . . ,171 Sum or difference of Focal Radii constant . ■ . • . . 172 Property of Focus and Directrix ..... 173 Rectangle under Focal Perpendiculars on Tangent is constant . . 174 Focal Radii equally inclined to Tangent .... 174 Confocal Conies cut at right Angles ..... 175 Tangents at any Point equally inclined to Tangent to Confocal conic through the Point ....... 176 Locus of foot of Focal PeiiDendicular on Tangent .... 176 Angle subtended at the Focus by a Chord, bisected by Line joining Focus to its Pole (see also pp. 24-1, 272) ..... 177 Line joining Focus to Pole of a Focal Chord is perpendicular to that Chord . 177 Polar Equation, Focus being Pole (see p. 309) .... 178 Segments of Focal Chord have constant Harmonic Mean . . . 179 Origin of names Parabola, Hyperbola, and Ellipse (see also p. 31G) . 180 Asymptotes : how foimd ...... 180 Intercepts on Chord between Curve and Asymptotes are equal . . 181 Lines joining two fixed to variable Pomt make constant Intercept on Asymptote 182 Constant area cut off by tangent ..... 182 Mechanical method of constracting Ellipse and Hyperbola . 172, 183, 207 CHAPTER XII. THE PARABOLA. Transformation of the Equation to the form y- = 73a; . . .184 Expression for Parameter of Parabola given by general Equation . . 186 ditto, given lengths of two Tangents and contained Angle (see also p. 203) . . . . . . .188 Parabola the limit of the Ellipse when one Focus passes to infinity . 189 Intercept on Axis by two Lines, equal to projection of distance between their Poles . . . . . . ,190 Subnormal Constant ...... 191 Locus of foot of Pei-pendicular from Focus on Tangent . . . 193 Locus of intei-section of Tangents which cut at right Angles (see also 273, 339) 194 Angle between two Tangents half that between coiTCsponding Focal Radii . 194 Circle circumscribing Triangle formed by three Tangents passes through Focus (see also pp. 203, 263, 273, 308) . . . . ,196 Polar Equation of Parabola , . . . . .196 CHAPTER XIII. EXAMPLES ON CONICS. Loci ..,,.... 197 Focal Properties . . . , , . .198 Locus of Pole with respect to a series of Confocal Conies . . .198 If a Chord of a Conic pass through a fixed. Point 0, then tan^PFO. tan \r'FO is constant (see also p. 319) . , . . . 199 CONTENTS. IX PAOR Locus of intersection of Normals at extremities of a Focal Chord (see also p. 323) 200 Expression for angle between tangents to ellipse from any point , .201 Eadii Vectores through Foci have equal diiference of Reciprocals , 201 Examples on Parabola . . .... 201 Three Perpendiculars of Triangle formed by three Tangents intersect on Directrix (see also pp. 236, 263, 278, 329) . . .201 Area of Triangle formed by thi-ee Tangents , . . ,201 Radius of Circle circumscribing an inscribed or circumscribing Triangle 202 Locus of intersection of Tangents which cut at a given Angle (see also pp. 245, 273) ...... 202 Locus of foot of Perpendicular from Focus on Nonnal . . . 202 Co-ordinates of intersection of two Normals . . , 203 Locus of intersection of Normals at the extremities of Chords passing through a given Point (see also p. 326) . . . . , 203 Given three Points on Equilateral Hyperbola, a fourth is given (see also pp. 278, 329) . . . . . . .204 Circle cu'cumscribing any self-conjugate Triangle with respect to an Equilateral Hyperbola passes through centre (see also p. 310) . . . 204 Locus of intersection of Tangents which make a given Intercept on a given Tangent , . . . . . 204 Locus of centre, given four Tangents (see also pp. 243, 256, 327) . , 205 ditto, given three tangents and sum of squares of axes . . 205 Locus of Foci given four Points ..... 206 Eccentric Angle ...... 206 Construction for Conjugate Diameters .... 207 Radius of Circle circumscribing an inscribed Triangle (see also p. 321) . 209 Area of Triangle formed by thi-ee Tangents, or three Normals . . 209 Similar Conic Sections ..... 211 Condition that Conies should be similar, and similarly placed . , 211 Properties of similar Conies . . , . .212 Condition that Conies should be similar, but not similarly placed . . 213 Contact of Conics ...... 214 Conies having double contact . . . . .215 Osculating Circle defined , . . . .216 Expressions and construction for Radius of Curvature (see also pp. 223, 231, 356) . . . . . . .217 If a Circle intersect a Conic, chords of intersection are equally incUned to the Axes (see also p. 223) . . . . .217 Condition that four Points of a Conic should lie on a Circle . . 218 Relation between three Points whose osculating Circles meet Conic again in the same Point . . . . , 218 Co-ordinates of centre of Curvatm-e . . .i , .219 Evolutes of Conics (see also pp. 325, 326) . . . .220 CHAPTER XIV. abridged notation. Meaning of the Equation (S' = A;S' . . . . . 221 Three values of k for which it represents right lines . . . 222 Equation of Conic passing through five given Points . . . 222 Equation of osculating Circle ..... 223 Equations of Conics ha^ong double contact with each other . . . 223 Every Parabola has a Tangent at infinity (see also p. 317) , . 224 X CONTENTS. PAOV Similar and similarly placed Conies have common Points at infinity . . 225 if concentric, touch at infinity . . , . . 226 All Circles have imaginary common Points at infinity (see also p. 313) . 227 Form of Equation referred to a self -conjugate Triangle (see also p. 242) . 227 Conies having same Focus have two imaginary common tangents (see also pp. 308, 3-10) . . . . . . .228 Method of finding Co-ordinates of Foci of given Conic (see also p. 340) . 228 Kelation between Perpendiculars from any point of Conic on sides of inscribed quadrilateral ...... 228 Anharmorric Property of Conies proved (see also pp. 240, 276, 306) . 229 Extension of Property of Focus and Directrix .... 229 Result of substituting the Co-ordinates of any Pbint in the Equation of a Conic 230 Diameter of Circle circumscribing Triangle formed by two Tangents and therr ' Chord ....... 231 Property of Chords of Intersection of two Conies, each having double contact with a third . . . . . . .231 Diagonals of inscribed and cireumseribed Quadrilateral pass through the same Point ....... 231 If three Conies have each double contact with a fourth, their Chords of Intersection intersect in threes ..... 232 Brianchon's Theorem (see also pp. 268, 304) .... 233 If three Conies have a common Chord, their other Chords intersect in a Point , 233 Pascal's Theorem (see also pp. 268, 289, 304, 307) . . .234 Steiner's Supplement to Pascal's Theorem (see also p. 360) . . . 235 Circles circumscribing the Triangles formed by four lines meet in a Point , 235 When five Lines are given, the five Points so found lie on a Circle . . 235 Given five Tangents to find their Points of Contact . . . 236 MacLaurin's Method of generating Conies (see also p. 287) , . . 236 Given five Points on a Conic to construct it, find its centre, and draw Tangent at any of the points ..... 236 Equation referred to two Tangents and their Chord . . . 237 Corresponding Chords of two Conies intersect on one of their Chords of Intersec- tion (see also pp. 232, 234) ..... 238 Locus of Vertex of Triangle whose sides touch a given Conic, and base Angles move on fixed Lines (see also pp. 307, 336) . . . 239 To inscribe in a Conic a Triangle whose sides pass through fixed Points (see also pp. 261, 269, 289) ., ... 239 Generalizations of MacLaurin's Method of generating Conies (see also p. 288) . 240 Anharmonie properties of Points and Tangents of a Conic (see pp. 229, 276, 306) 240 Anharmonic r^tio of four Points on a Conic {abccT} = {a'b'c'd'], if the Lines aa', &c. meet in a Point ; or if they touch a Conic having double contact with the given one . , . . .241 Envelope of Chord joining corresponding Points of two homographic systems on a Conic (see also p. 291) ..... 242 Equation referred to sides of a self-conjugate Triangle . . . 242 Locus of Pole of a given Line with regard to a Conic passing through four fixed points (see also pp. 148, 256, 260, 290) . . . .243 of touching four right Lines (^ee also pp. 256, 265, 269, 309, 327) , 243 Focal properties of Conies ...... 244 Locus of Intersections of Tangents to a Parabola which cut at a given Angle (see also pp. 207, 273) . . . . .245 Self-conjugate Triangle common to two Conies (see also pp. 335, 347) , . 245 when real, when imaginaiy . . . . • 245 CONTENTS. XI Locus of Vertex of a Triangle inscribed in one Oonic, and whose sides touch one another (see also p. 336) ..... 246 IEnvslopes, how found ...... 247 Examples of Envelopes ,..,.. 248 iPormation of Trilinear Equation of a Conic from Tangential, and vice versa 249 iCriterion whether a Point be within or with«ut a Cenic , . 250 Discriminant of Tangential Equation ..... 250 ■Given two Points of a Conie having double contact with a thii'd, its Chord of contact passes through one or two fixed Points . . .251 Equation of a Conic having double contact with two given Conies . 251 touching four Lines . ..... 251 Locus of a Point whence sum or difference of Tangents to two Circles is constant 252 Malfatti's problem ...... 252 Tangent and Polar of a Poiut with regard to a Oonic given.by the general Equation 254 Discriminants, defined ; discriminant of a Conic found (see also pp. 72, 144, 148, 150) . , . , , , .255 ■Co-ordinates of Pole of a given Line .... 255 Condition that two Lines should be conjugate , . . . 256 'Condition that a Line should touch a Conic (see aiso ^p. 147, 328) . 255 Seam's Method of finding Locus of Centre of a Conic, four conditions beieg given 256 Equation of pan* of Tangents through a given Point (see also p. 144) . . 257 Property of Angles of a circumscribing Hexagon (see also/p. 277} . 258 Test whether three paii-s of Lines touch the same Conic . . . 258 Equations of Lines joining to a given point intersections of two Curves . 259 Chord which subtends a right Angle at a -fixed Poiint on Conic, passes through a fixed Point . . . , .. . .259 Locus of the latter Point when Point on Curve varies . . , 259 Envelope of Chord subtending constant An^e, or subtending right Angle at Point not on Curve ...... 259 Given four Points, Polar of a fixed Point passes through fixed Point , 259 Locus of intersection of coiTCsponding Lines of two homographic pencils . 260 Envelope of Pole of a given Point with regard to a .Conic having double contact with two given ones ..... 260 Anharmonic Ratio of four Points the same as that of their Polars , . 260 Equation of AsjTnptotes of a Conic given 'by general Eqr.arion (see also p. 328) 260 Given thi-ee Points on Conic, and Point on one Asymptote, Envelope of other 261 Locus of Vertex of a Triangle whose sides pass through fixed Points, and base Angles move along Conies . . . .261 To inscribe in a Conic a Triangle whose sides pass throxigh fixed Points (see also pp. 239, 269, 289) . . , . . .261 Equation of Conic touching five Lines . . '. . 262 Co-ordinates of Focus of a Conic given three tangents (see also pp. 228, 340) 263 Directrix of Parabola passes thi-ough intersection of Pei-pendiculars of circum- scribing Triangle (see pp. 201, 236, 278, 329) . . .263 Locus of Focus given four Tangents (see also p. 265) ... 263 CHAPTER XV, RECIPROCAL rOLARS. Principle of Duality ....... 264 Locus of Centre of Conic toucliing four Lines . . . 265 Locus of Focus of Conic touching foui- Lines .... 265 Xll CONTEXTS. PAGR Director Circles of Conies toucliing four Lines have a common radical axis 265 Degree of Polar Reciprocal in general .... 2()7 Pascal's Theorem and Brianchon's mutuallj' reciprocal . . . 268 Radical Axes and Centres of Similitude of Conies having double contact with a given one ...... 270 Polar of one Circle w-ith regard to another . . . .271 Reciprocation of Theorems concerning Angles at Focus . . 272 Envelope of Asymptotes of Hj^erbolas having same Focus and Directrix . 273 Reciprocals of equal Circles have same Parameter . . . 27-1 Relation between Pei-pendiculars on Tangent from Vertices of circumscribing Quadrilateral ...... 275 Tangential Equation of Reciprocal Conic .... 374 Trilinear Equation given Focus and either three Points or three Tangents . 276 Reciprocation of Anharmonic Properties .... 275 Camot's Theorem respecting Triangle cut by Conic (see also p. 307) . . 277 Reciprocal, when Elhpse, Hyperbola or Parabola ; when Equilateral Hyperbola 278 Axes of Reciprocal, how foimd ..... 279 Reciprocal of Properties of Confocal Conies .... 279 To describe a Circle touching three given Circles . . . 279 How to form Equation of Reciprocal ..... 280 Reciprocal transformed from one origin to another . . . 280 Recii^rocals with regard to a Parabola . . . . .281 CHAPTER XVI. HARMONIC AND ANHARMONIC PROPERTIES. Anharmonic Ratio, when one Point infinitely distant . . . 283 Centre the Pole of the Line at infinity ..... 284 Asymptotes together with two Conjugate Diameters form Harmonic Pencil 284 Lines from two fixed Points to a variable Point, how cut any Parallel to Asymptote ...... 285 Parallels to Asymptotes through any Point on Curve, how cut any Diameter . 286 Anhaitnonic property of Tangents to Parabola . . . 287 How any Tangent cuts two Parallel Tangents .... 287 Proof, by Anharmonic Properties, of Mac Laurin's Method of Generating Conies, of Newton's mode of Generation .... 287 Chasles's extension of these Theorems ..... 288 To inscribe in a Conic a Polygon whose sides pass through fixed Points . 289 To describe a Conic touching three Lines and having double contact with a given Conic (see also p. 345) ..... 289 Anharmonic proof of Pascal's Theorem .... 289 of Locus of Centre, when four Points are given . . . 290 Envelope of Line joining con-esponding Points of two nomographic Systems 290 Criterion whether two Systems of Points be nomographic . . 292 Analytic condition that foiu: Points should form a Hai-monic System . . 293 Locus of Point whence Tangents to two Conies form a Harmonic Pencil (see also p. 333) ...... 294 Condition that Line should be cut Hannonically by two Conies . . 294 Involution ....... 295 Property of Centre . . . . . . .296 of Foci ....... 297 Foci, how found when two Pairs of coiTesponding Points are given . . 298 CONTENTS. Xlll PAOR Condition that six Points or Lines sliould form a system in Involution . 298 Sj'Stem of Conies through four Points cut any Transversal in Involution . 299 System of Conies toucliing four Lines, when cut a Transversal in Involution 301 Proof by Involution of Feuerbach's Theorem concerning the Ckcle through middle Points of Sides of Triangle . . . .301 CHAPTER XVII. THE METHOD OF PROJECTION'. All Points at Infinity may be regarded as lying in a right Line . . 303 Projective Properties of a Quadrilateral .... 305, 306 Any two Conies may be projected into Cii'cles .... 306 Projective proof of Camot's Theorem (see also p. 277) . . . 30G of Pascal's Theorem ...... 307 Projections of Properties concerning Foci .... 308 The six Vertices of two Triangles circimiscribing a Conic, lie on the same Conic, (see also p. 331) . . . . . .308 Projections of Properties concerning Right Angles . . . 309 Locus of Pole of a Line with regard to a System of Confocal Conies . .310 The six Vertices of two self -con jugate Triangles lie on same Conic (see also p. 320) 310 Chord of a Conic passes through a fixed Point, if the Angle it subtends at a fixed Point in Curve, has fixed Bisector . . . 311 Projections of Theorems concerning Angles in general . . .311 Locus of Point cutting in given ratio intercept of variable Tangent between two fixed Tangents . . . . . ,312 Analytic basis of Method of Projection . . . . .312 Sections of a Cone ...... 314 Every Section is EUipse, Hyperbola, or Parabola . . . .315 Origin of these Names ...... 316 Every Parabola has a Tangent at an infinite Distance . . .317 Proof that any Conic may be projected so as to become a Circle, while a given Line passes to Infinity . , . . .318 Determination of Focus of Section of a right Cone . . .319 Locus of Vertices of right Cones from which a given Conic may be cut . 319 Method of deducing properties of Plane Cm-ves from Spherical . .319 Orthogonal Projection ...... 320 Radius of Cu-cle circumscribing inscribed Triangle . . .321 CHAPTER XVIII. INVARIANTS AND COVARIANTS. Equation of Chords of Intersection of two Conies . , . 322 Locus of Intersection of Normals to a Conic at the extremities of Chords passmg through a given Point ..... 323 Condition that two Conies should touch . . • . 324 Criterion whether Conies intersect in two real and two imaginary Points or not 325 Equation of Curve 7)ff)'o//e/ to a Conic ..... 325 Equation of Evolute of a Conic . , , . . 326 Meaning of the Invariants when one Conic is a pair of Lines . . 326 Criterion whether six lines touch the same conic . . . 327 XIV CONTENTS. PAOP. Equation of pair of Tangents whose Chord is a given Line . . 327 Equation of Asymptotes of Conic given by Trilinear Equation . . 328 Condition that a Triangle self -con jugate with regard to one Conic, should be inscribed or circumscribed about another .... 328 Six vertices of two self -con jugate Triangles lie on a Conic , , 329 Circle circumscribing self -con jugate Triangle cuts the director circle orthogonally 329 Centre of Circle inscribed in self -con jugate Triangle of equilateral Hyperbola lies on Curve ...... 329 Locus of intersection of Perpendiculars of Triangle inscribed in one Conic and circumscribed about another .... 329 Condition that such a Triangle shoidd be possible . . . 330 Tangential equation of four Points common to two Conies . . 331 Equation of four common Tangents ..... 332 Their eight Points of Contact lie on a Conic .... 332 Covariants and Coutravariants defined ..... 333 Discriminant of Covariant F, when vanishes . . . 335 How to find equations of Sides of self -conjugate Triangle common to two Conies (see also p. 347) ...... 335 Envelope of Base of Triangle inscribed in one Conic, two of whose sides touch another ...... 336 Locus of free Veiiex of a Polygon all whose sides touch one Conic, and all whose Vertices but one move on another .... 337 Condition that Lines joining to opposite vertices, Points where Conic meets Triangle of reference should form two sets of three meeting in a Point . 337 General Tangential Equation of two Circular Points at infinity . . 337 Condition for Equilateral Hyperbola and for Parabola in Trilinear Co-ordinates 338 JEvery line through an imaginary Circular Point, perpendicular to itself . 338 General Equation of Dh-ector Circle ..... 339 Equation of Directiix of Parabola given by Trilinear Equation . . 339 Co-ordinates of Foci of Curve given by general Equation . . . 340 Extension of relation between perpendicular Lines . . . 341 Equation of reciprocal of two Conies having double contact . . 342 Condition that they should touch each other . . . 343 To di-aw a Conic having double contact with a given one, and touching three other such Conies ...... 343 Four Conies having double contact with S, and passing through three Points, or touching three Lines, are touched by the same Conies . . 345 Condition that three Conies should have double contact with the same conic . 345 Jacobian of a system of three Conies .... 345 Corresponding points on Jacobian ..... 346 Lines joining corresponding Points cut in involution by the Conies , 346 General equation of Jacobian ...... 346 To draw a Conic through four Points to touch a given Conic . . 346 To form the equation of the sides of self-conjugate Triangle common to two Conies . . . . . . .347 Jacobian of three Conies having two Points common : or one of which reduces to two coincident Lines ..... 347 Equation of Circle cutting three Circles orthogonally . . . 347 Condition that a line should be cut in involution by three Conies . . 347 Invariants of a sj-stem of three Conies . . . . . 348 Condition that they should have a common Point . . . 348 Condition that Xf/ -I- yuF-f- vir can in any case be a perfect Square . . 349 Three Conies derived from a single Cubic : method of forming its Equation . 349 CONTENTS. XV CHAPTER XIX. THE METHOD OP INFINITESIMALS. PAGE Direction of Tangents of Conies ..... 352 Determination of Areas of Conies ..... 353 Tangent to any Conic cuts off constant Area from similar and concentric Conic . SOI Line which cuts off from a Curve constant Arc, or which is of a constant length where met by its Envelope ..... 355 Determination of Radii of Curvatxire . . . ■ . . 355 Excess of sum of two Tangents over included Arc, constant when Vertex moves on Confocal Elhpse ..... 357 Difference of Arc and Tangent, constant from any Point on Confocal Hyperbola 358 Fagnani's Theorem ....... 358 Locus of Vertex of Polygon circumscribing a Conic, when other Vertices move on Confocal Conies ..... 359 NOTES. Theorems on complete Figure formed by six Points on a Conic . . 3C0 On systems of Tangential Co-ordinates .... 363 On Mr. Casey's form of the Equation of a Conic having double contact with a given one and touching three others .... 366 On the Problem to describe a Conic under five conditions . . 367 On systems of Conies satisfying four Conditions .... 368 ERRATA. p. 129, last line. The determinants should be connected by the sign of multiplica- tion, not addition. p. 130, lines 1, 23, the words "rows" and "columns" should be interchanged. p. 384, at end of Art. 389, add " Mr. Bumside has expressed this invariant T in terms of the elementary invariants : T= 6^,3 - 4 (0,226133 + e.„e233 + ©3,16322) + 12* ; where 4> is the invariant, corresponding to Gjjj, formed with the coefficients of the reciprocal systems." ANALYTIC GEOMETRY. CHAPTER I. THE POINT. 1. The following method of determining the position of any point on a plane was introduced by Des Cartes in his Geometric^ 1637 -; and has been generally used by succeeding geometers. We are supposed to be given the position of two fixed right lines XX', YY' intersecting in the point 0. Now, if through any point P we draw PM^ PN parallel to YY' and XX', it is plain that, if we knew the position of the point P^ we should know the lengths of the par- allels PM, PN\ or, vice versd^ that if we knew the lengths of PM^ PN^ we should know the position of the point P. Suppose, for example, that we are given PN=a^ PM= h, we need only measure OM the a and ON=h^ and draw parallels PM^ PAT which will intersect in the point required. It is usual to denote PM parallel to OY by the letter ?/, and PiV parallel to OX by the letter cc, and the point Pis said to be determined by the two equations x = a^y = h. 2. The parallels PM^ PN are called the co-ordinates of the point P. Pilf is often called the ordinate of the point P; while PN^ which is equal to OM the intercept cut of by the ordinate, is cfilled the ahscissa. 1 THE POINT. The fixed lines XX' and YY' are termed the axes of co- ordinates ^ and the point 0, in which they intersect, is called the origin. The axes are said to be rectangular or oblique, ac- cording as the angle at which they intersect is a right angle or oblique. It will readily be seen that the co-ordinates of the point M on the preceding figure are ic = a, y = ; that those of the point N are x = 0, y — b] and of the origin itself are a; = 0, ?/ = 0. 3. In order that the equations x = a^ y = ^ should only be satisfied by one point, it is necessary to pay attention, not only to the magnitudes^ but also to the signs of the co- ordinates. If we paid no attention to the signs of the co-ordinates, we might measure OM=a and 0N= b, on either side of the origin, and any of the four points P, P^, P„ P3 would satisfy the equations x = a, 2/ = ^• It Is possible, however, to distinguish algebraically between the lines OMj OM' (which are equal In magnitude, but opposite in direction) by giving them different signs. We lay down a rule, that if lines measured in one direction be considered as positive, lines measured in the oppo- site direction must be con- sidered as negative. It IS, arbitrary in which it is customary to of course, direction we measure positive lines, but consider OM (measured to the right hand) and ON (measured npivards) as positive, and OM', ON' (measured In the oppo- site directions) as negative lines. Introducing these conventions, the four points P, Pj, J!^, Pg are easily distinguished. Their co-ordinates are, respectively, y = + b x = -{-a] x = — a[ ' y = + ^\' x = + a) x = — a g = -b THE POINT. These distinctions of sign can present no difficulty to the learner, who is supposed to be already acquainted with tri- gonometry. N.B. — The points whose co-ordinates are x = a, y = h^ or x = x\ y = y\ are generally briefly designated as the point (a, J), or the point xy . It appears from what has been said, that the points (+ a, -f 5), (—a, — IS) lie on a right line passing through the origin ; that they are equidistant from the origin, and on opposite sides of it. 4. To express the distance between two points xy\ x"y'\ the axes of co-ordinates heing supposed rectangular. By Euclid i. 47, PQ" = PS^ + 8Q\ but PS = PM- QM' = y' - y'\ and Q8== OM- OM' = x'-x"; hence 8' = PQ' = {x - x'J + [y' - y")\ To express the distance of any point from the origin, we must make x" = 0, y" = in the above, and we find ^' = x"' + y'\ 5. In the following pages we shall but seldom have occasion to make use of oblique co- ordinates, since formulse are, in general, much simplified by the use of rectangular axes ; as, however, oblique co-ordinates may sometimes be employed with advantage, we shall give the principal formulse in their most gene- ral form. Suppose, in the last figure, the angle YOX oblique and = w, then PSQ= 180° -to, and PQ' = PS'' 4- QS' - 2PS. QS.cosPSQ, or, PQ' = Q/' - y'y + [x - x")' + 2 (y' - y") [x - x") cos o). Similarly, the square of the distance of a point, x'y\ from the origin = x'^ + y'^ + 2x'y' cos w. Y' Q s O M' M 4 'J'llE ruiNT. In applying these formulae, attention must be paid to the signs of the co-ordinates. If the point Q^ for example, were in the angle XO Y\ the sign of y" would be changed, and the line PS would be the sum and not the difference of y and y". The learner will find no difficulty, if, having written the co- ordinates with their proper signs, he is careful to take for PS and QS the algebraic difference of the corresponding pair of co-ordinates. Ex. 1. rind the lengths of the sides of a triangle, the co-ordinates of whose vertices are x' -2, y' = 3; x" = 4, y" = - b ; x'" = - 3, y'" = - 6, the axes being rectangular. Ans, ^68, .J50, ^106. Ex. 2. Find the lengths of the sides of a triangle, the co-ordinates of whose vertices are the same as in the last example, the axes being inclined at an angle of 60°. Ans. J52, J57, .J151. Ex. 3. Express that the distance of the point xy from the point (2, 3) is equal to 4. Ans. {x-2f+{i/- 3)^ = 16. Ex. 4. Express that the point xy is equidistant from the points (2, 3), (4, 5). Ans. {x-2f+{y- 3)"- = (a; - 4)^ -f- Q/ - 5)^ ; or x + y = 7. Ex. 5. Find the point equidistant fi-om the points (2, 8), (4, 5), (6, 1). Here we have two equations to determine the two unknown quantities x, y. , .. . J(50) Ans, X— ^, y = I, and the common distance is — —. 6. The distance between two points, being expressed in the form of a square root, is necessarily susceptible of a double sign. If the distance PQ^ measured from P to Q, be con- sidered positive, then the distance QP, measured from Q to P, is considered negative. If indeed we are only concerned with the single distance between two points, it would be un- meaning to affix any sign to it, since by prefixing a sign we in fact direct that this distance shall be added to, or subtracted from, some other distance. But suppose we are given three points P, ^, P in a right line, and know the distances PQ^ QB, we may infer, PB = PQ + QR. And with the explana- tion now given, this equation remains true, even though the point P lie between P and Q. For, in that case, PQ and QR are measured in opposite directions, and PR which is their arithmetical difference is still their algebraical sum. Except in the case of lines parallel to one of the axes, no convention has been established as to which shall be considered the positive direction. THE POINT. 7. To find the co-ordinates of the point cutting in a given ratio m : w, the line joining two given points x'y'^ ^"y"- Let ic, y be the co-ordinates of the point R which we seek to determine, then m:n::FE:EQ::MS: SNj P r'l/ m: n:: x —X : X — x'\ or mx—mx" = nx'-nXj hence mx" + nx M m + n In like manner my" + ny' " m + n If the line were to be cut externally in the given ratio, we should have m : n :: x — x : X — x\ T , „ mx — nx my" — mi and therefore x = , ?/ = -^ — . m — n m — n It will be observed that the formulae for external section are obtained from those for internal section by changing the sign of the ratio : that is, by changing m '. -\-n into m : — n. In fact, in the case of internal section, PR and RQ are measured in the same direction, and their ratio (Art. 6) is to be counted as positive. But in the case of external section PR and RQ are measured in opposite directions, and their ratio is negative. find the co-ordinates of the middle point of the linejoiniag the points x' + x" y' + y" Ans. X = — 5 — , y = — - — . Ex. 1. To Ex. 2. To find the co-ordinates of the middle points of the sides of the triiuigls, the co-ordinates of whose vertices are (2, 3), (4, — 5), (— 3, — 6). Ans. (i, - V), (- h - f), (3, - !)• Ex. 3. The line joining the points (2, 3), (4, -5), is ti-isected; to find the co- ordinates of the point of trisection nearest the fonner point. Ans. a; = f , y — \- Ex. 4. The co-ordinates of the vertices of a triangle being xy', x"y", x"'y"', to find the co-ordinates of the point of trisection (remote from the vertex) of the hne joining any vertex to the middle point of the opposite side, Ans. x = ^ {x' + x" + x'"). y = i{!/' + y" + !>'")- TRANSFORMATION OF CO-ORDINATES. Ex. 5. To find the co-ordinates of the intersection of the bisectors of sides of the triangle, the co-ordinates of whose vertices are given in Ex. 2. Ans. x = 1, ^ = — f . Ex. 6. Any side of a triangle is cut in the ratio m : n, and the line joining this to the opposite vertex is cut in the ratio m + n : I; to find the co-ordinates of the point of section. , Ix' + mx" + nx'" ly' + mi/" + mi'" Ans. X ■ I + m + n 2/- I + m + 71 TRANSFORMATION OF CO-ORDINATES.* 8. When we know the co-ordinates of a point referred to one pair of axes, it is frequently necessary to find its co- ordinates referred to another pair of axes. This operation is called the transformation of co-ordinates. We shall consider three cases separately: first, we shall suppose the origin changed, but the new axes parallel to the old; secondly, we shall suppose the directions of the axes changed, but the origin to remain unaltered; and thirdly, we shall suppose both origin and directions of axes to be altered. First. Let the new axes be parallel to the old. Let Ox^ Oy be the old axes, (9'X, y ^ O'Y the new axes. Let the co-ordinates of the new origin referred to the old be a?', ?/', or 0'S=x\ 0'B = 7/'. Let the old co-ordinates be X, ?/, the new X, Y^ then we have 6>J/= OB + BM, and FM= FN-\- NM, that is, x = x +X, and y = y' + Y. These formulae are, evidently, equally true, whether the axes be oblique or rectangular. 9. Secondly, let the directions of the axes be changed, while the origin is unaltered. X * The beginner may postpone the rest of this chapter till he has read to the end of Art. 41. TRANSFOKMATION OF CO-ORDINATES. Let the original axes be Ox, Oy^ so that we have OQ = x^ PQ = y. Let the new axes be OX, Oy, so that we have ON=X, PN=Y. Let OX, OY make angles respectively a, yS, with the old axis of a?, and angles a', fi' with the old axis of y' and if the angle xOy between the old axes be 0), we have obviously a4-a'=a), since XOx + XOy = xOy\ and in like manner /3 + /3' = «. The formulae! of transformation are most easily obtained by expressing the perpendiculars from P on the original axes, in terms of the new co-ordinates and the old. Since PM=PQ ^mPQM we have PM = y sin to. But also PM= NR + PS= ON smNOR + PN smPNS. Hence y sin &> = X sin a + Y sin /3. In like manner x &in(o = X sin a' + Y sin/3' ; or X sin w = X sin (w — a) + 1" sin (&> — /9) . In the figure the angles a, y8, (o are all measured on the same side of Ox, and a', /3', &>, all on the same side of Oy. If any of these angles lie on the opposite side it must be given a negative sign. Thus, if OY lie to the left of Oy, the angle /3 is greater than co, and yS' (= &> — /3) is negative, and therefore the coefficient of Y in the expression for x sinw is negative. This occurs in the following special case, to which, as the one which most frequently occurs in practice, we give a sepa- rate figure. To transform from a system of rectangular axes to a neio rectangular system making an angle 6 ivith the old. Here we have a = d, yS-90 + ^, a' = 90-^, 13' = -0- and the general formulae become y = X smO + Ycosd, £c = X cos ^ — F sin ^ : 8 TRANSFORMATION OF CO-ORDINATES. the truth of which may also be seen directly, since y = PS-\- NR^ x=OB- SNjwhWe PS = FN cos 0, NP = ON sin 0-, OB = ON cos 0, SN= PN sin 0. There is only one other case of transformation which often occufo in practice. To transform from oblique co-ordinates to rectangular ^ retain- ing the old axis of x. We may use the general for- mulae making a = 0, y8 = 90, a' = w, /S' = a)-90. But it is more simple to inves- tigate the formulae directly. We have 0<3 and P(3 for the old a; and ^ ^ y, OM and PM for the new ; and since PQM= &>, we have 1^=3/ since), X=x-{-y co^w. while from these equations we get the expressions for the old co-ordinates in terms of the new y sin w = Y^ x sin w = X sin a> — Y cos w. 10. Thirdly, by combining the transformations of the two preceding articles, we can find the co-ordinates of a point re- ferred to two new axes in any position whatever. We first find the co-ordinates (by Art. 8) referred to a pair of axes through the new origin parallel to the old axes, and then (by Art. 9) we can find the co-ordinates referred to the required axes. The general expressions are obviously obtained by adding x' and y' to the values for x and y given in the last article. Ex. 1. The co-ordinates of a point satisfy the relation a;2 + ^2 — 4a; — 6y = 18 : what will this become if the origin be transformed to the point (2, 3) ? Ans. Z2 + F2 = 31. Ex, 2. The co-ordinates of a point to one set of rectangular axes satisfy the relation y^ — x^ — Q: what will this become if transformed to axes bisecting the angles between the given axes ? Ans. XY = 3. Ex. 3. Transform the equation 2^^ - hxy + 2^/^ = 4^ from axes inclined to each other at an angle of 60°, to the right lines which bisect the angles between the given axes, Ans. X"^ - 271'^ + 12 = 0, Ex. 4, Transform the same equation to rectangular axes, retaining the old axis of .r, Ans. 3X2 + ior2 - 7XY J3 = R, rOLAR CO-ORDINATES. 9 Ex. 5. It is evident that when we change from one set of rectangular axes to another, 3? + y^ must ~ X^ + Y^, since both express the square of the distance of a point from the origin. Verify this by squaring and addmg the expressions for X and Y hi Art. 9. Ex. 6. Verify in like mamier in general that x^ + i/ + 2xij cos xOy = X^ + Y^ + 2 AT cos A'OF. If we write A' sin a + F sin /3 = L, X cosa + Y cos (i = M, the expressions in Art. 9 may be written y smo3 — L, x sin to — M sin to — L cos a> ; whence sLn^o) [x"^ + y^ + 2xy coato) = {L- + 31 -) sin-w. But U- + J)/2 = A'2 + 3'2 + 2A'r cos (a - /3) ; and « - /3 = A'OF. 11. The degree of any equation between tJie co-ordinates is not altered by transformation of co-ordinates. Transformation cannot increase the degree of the equation: for if the highest terms in the given equation be x\ y"\ &c., those in the transforaied equation will be {x sinw+a; sm[(o- a)-{y s\n[(o—^)y'\ [y' sincu+a? sina+^ sin^)'"', &c., which evidently cannot contain powers of x or y above the tn^ degree. Neither can transformation diminish the degree of an equation, since by transforming the transformed equation back again to the old axes, we must fall back on the original equation, and if the first transformation had diminished the degree of the equation, the second should increase it, contrary to what has just been proved. POLAR CO-ORDINATES. 12. Another method of expressing the position of a point is often employed. If we were given a fixed point (9, and a fixed line through it OB^ it is evident that we should p know the position of any point P, if we knew the length OP^ and also the angle FOB. The line OP is called the radius vector ; the fixed point is called ^ the pole ; and this method is called the method of polar co- ordinates. It is very easy, being given the x and y co-ordinates of a point, to find its polar ones, or vice versa. 10 First, let the fixed line coincide with the axis of x^ then we have OP: PM:: smPMOismPOM; denoting OP by p, POM by 6^ and YOX by eo ; then ^,, p sin^ PM or y = ~ ; POLAR CO-ORDINATES. and similarly, OM=x = p sin (o) — 0) sinca X For the more ordinary case of rectangular co-ordinates, (o = 90°, and we have simply x — p cos^ and ?/ = p sin^. Secondly. Let the fixed line OB not coincide with the axis of Xj but make with it an angle = a, then P0£= e and POM= d-a, and we have only to substitute 6 — a for 6 in the preceding formula. For rectangular co-ordinates we have x = p cos (^ - a) and y = p sin {0 - a). Ex. 1, Change to polar co-ordinates the following equations in rectangular co- x'^ + y" = 5mx. Ans. p = 5ot cos0. X- — y- = a-. Ans. p- cos26 — a^. Ex. 2. Change to rectangular co-ordinates the following equations in polar co- p-sin 26 = 2«-. Ans. xy — a-, p- = a- cos20. Ans. {x- + y"Y — ci?- {x- — y^). p' cos^0 = a". Ans. x'^ + y- — (2a — x)^. p^ - a^ cos^e. Ans. {2x- + 2y" - ax^ = a^ {x^ + y-). 13, To express tlie distance hetween two jJOints^ in terms of their jjolar co-ordinates. ^ Q Let P and Q be the two points, OP=p, POB^d'] OQ = p", QOB^e"; o then PQ' = OP' + 0Cf-20P. OQ.cosPOQ, or S' = p" + p"' - 2p'p" cos {6" - &). ordinates : ordinates : ( 11 ) CHAPTER 11. THE EIGHT LINE, 14. Any two equations between the co-ordinates represent fjeometrically one or more points. If the equations be both of the first degree (see Ex. 5, p. 4), they denote a single point. For solving the equations for X and ?/, we obtain a result of the form x = a^ y — ^i which, as was proved in the last chapter, represents a point. If the equations be of higher degree, they represent more points than one. For, eliminating y between the equations, we obtain an equation containing x only 5 let its roots be a,, a^, ag, &c. Now, if we substitute any of these values (a,) for X in the original equations, we get two equations in y, which must have a common root (since the result of elimination be- tween the equations is rendered =0 by the supposition x = a^. Let this common root be ?/ = /S^. Then the values a; = a,, ?/ = ySj, at once satisfy both the given equations, and denote a point which is represented by these equations. So, in like manner, is the point whose co-ordinates are x=a^^y = yS^, &c. Ex. 1. "What point is denoted by the equations 3a; + 5y = 13, 4j- — ?/ = 2 ? Ans. X — \, y = 2. Ex. 2. What points are represented by the two equations a;- + y- = 5, xy — '2? EUminating y between the equations, we get x* — ox^ + 4 = 0. The roots of this equation are x- =\ and x^ = 4, and, therefore, the four vahies of x are x = +\, x = -\, a:-+2, x--2. Substituting these successively in the second equation, we obtain the coiTespondmg values of y, y = + 2, y = -2, »/ = +!, y = -l. The two given equations, therefore, represent the foiu" points (+ 1, + 2), (- 1, - 2), (+ 2, + 1), (- 2, - 1). Ex. 3. What points are denoted by the equations x-y=\, x'^ + y--2c>? Ans. (4, 3), (- 3, - 4) . Ex. 4. What points are denoted by the equations f 3? — hx -f // + 3 = 0, x" + y- — bx — oy + 6 = ? Ans. (1, 1), (2, 3;, (3, 3,', vJ. 1). 12 THE EIGHT LINE. 35. -4 single equation between the co-ordinates denotes a geometrical locus. One equation evidently does not afford us conditions enough to determine the two unknown quantities a;, y ; and an Inde- finite number of systems of values of x and y can be found which will satisfy the given equation. And yet the co-ordinates of any point tahen at random will not satisfy it. The assemblage then of points, whose co-ordinates do satisfy the equation, forms a locus^ which is considered the geometrical signification of the given equation. Thus, for example, we saw (Ex." 3, p. 4), that the equation {x-2Y+{y-?,y=\% expresses that the distance of the point xy from the point (2, 3) = 4. This equation then is satisfied by the co-ordinates of any point on the circle whose centre is the point (2, 3), and whose radius is 4 ; and by the co-ordinates of no other point. This circle then is the locus which the equation is said to represent. We can illustrate by a still simpler example, that a single equation between the co-ordinates signifies a locus. Let us recall the construction by which (p. 1) we determined the position of a point from the two equations x — a^ y = h. We took (9J/=a. we drew MK parallel to OY] and then, measuring MP=h^ we found P, the point required. Had we been given a different value of y, x = «, y = h'j we should proceed as before, and we should find a point F' still situated on the line il/Zi, but at a diff"erent distance from M. Lastly, if the value of y were left wholly Indeterminate, and we were merely given the single equation x = a, we should know that the point F was situated somewhere on the line MK, but Its position in that line would not be determined. Hence the line MK Is the locus of all the points represented by the equation a; = a, THE RIGHT LINE. 13 since, whatever point we take on the line il//v, the x of that point will always = a. 16. In general, if we are given an equation of any degree between the co-ordinates, let us assume for x any value we please {x = ci)^ and the equation will enable us to determine a finite number of values of y answering to this particular value of x ; and consequently, the equation will be satisfied for each of the points (^?, 7, r, &c.), whose x is the assumed value, and whose y is that found from the equation. Again, assume for X any other value {x = d)^ and we find, in like manner, ano- ther series of points, P) 4i *'? whose co- ordinates satisfy the equation. So again, if we assume x = a" or a; = a'", &c. Now, if x be supposed to take successively all possible values, the assemblage of points found as above will form a locus^ every point of which satisfies the conditions of the equation, and which is, therefore, its geometrical signification. We can find In the manner just explained as many points of this locus as we please, until we have enough to represent its figure to the eye. Ex. 1. Represent in a figure* a series of points which satisfy the equation «/ = 2a; + 3. Ans. Giving x the values — 2, — 1, 0, 1, 2, &c., we find for y, — 1, 1, 3, 5, 7, &c., and the coiTesponding pouits will be seen all to lie on a right line. Ex. 2. Represent the locus denoted by the equation y = x- — ix — 2. Ans. To the values for x, -1, - ^, 0, ^, 1, f, 2, f, 3, f, 4; coiTespoud for y, 2, - ^, - 2, - 1^, - 4, - V, - 4, - V, - 2, - i, 2. If the points thus denoted be laid down on paper, they will sufiiciently exhibit the form of the curve, which may be continued indefinitely by giving x greater positive or negative values. Ex. 3. Represent the cm-ve y = 3 + J(20 — x — x-). Here to each value of x con-espond two values of y. No part "of the cui-ve lies to the right of the line a; = 4, or to the left of the line cc = - 5, since by giving greater positive or negative values to x, the value of y becomes imaginaiy. * The learner is recommended to use paper ruled into little squares, which is sold tmder the name of logarithm paper. 14 THE RIGII'f LINE. 17. The whole science of Analytic Geometry is founded on the connexion which has been thus proved to exist between an equation and a locus. If a curve be defined by any geo- metrical property, it will be our business to deduce from that property an equation which must be satisfied by the co-ordinates of every point on the curve. Thus, if a circle be defined as the locus of a point (a:, ?/), whose distance from a fixed point (rt, h) is constant, and equal to r ; then the equation of the circle to rectangular co-ordinates, is (Art. 4), On the other hand, it will be our business when an equation is given, to find the figure of the curve represented, and to deduce its geometrical properties. In order to do this systematically, we make a classification of equations according to their degrees, and beginning with the simplest, examine the form and pro- perties of the locus represented by the equation. The degree of an equation is estimated by the highest value of the sum of the indices of x and y in any terra. Thus the equation a;^/ + 2£c 4 3?/ = 4 is of the second degree, because it contains the term xy. If this term were absent, it would be of the first degree. A curve is said to be of the n^ degree when the equation which represents it is of that degree. We commence with the equation of the first degree, and we shall prove that this always represents a right line^ and, con- versely, that the equation of a right line is always of the first degree. 18. We have already (Art. 15) interpreted the simplest case of an equation of the first degree, namely, the equation x = a. In like manner, the equation y — h represents a line PA^ parallel to the axis OX^ and meeting the axis OF at a distance from the origin ON=b. If we suppose h to be equal to nothing, we see that the equation y — denotes the axis OX] and in like manner that x = Q denotes the axis Y. Let us now proceed to the case next in order of simplicity, and let us examine what relation subsists between the co- ordinates of points situated on a right Hue passing through the origin. THE RIGHT LINE. 15 - X If we take any point P on such a line, we see that hoth the co-ordinates Pil/, OM^ will vary in length, but that the ratio PM: OM will be constant, being = to the ratio sinPOJ/: sinJ/PO. Hence we see, that the equation _ sin P6> J/ y~ ^mMPO^ will be satisfied for every point of the line OP, and therefore, this equation is said to be the equation of the line OP. Conversely, if we were asked what locus was represented by the equation y = mx^ write the equation In the form - = in. and the question is, " to X find the locus of a point P, such that, if we draw Pil/, PN parallel to two fixed lines, the ratio PM'.PN may be constant." Now this locus evidently Is a right line OP, passing through 0, the point of intersection of the two fixed lines, and dividing the angle between them in such a manner that sin POJ/=w sin PO.V. If the axes be rectangular, sinPOA^^eosPOJ/; therefore, ?*< = tan PO J/, and the equation y^vix represents a right line passing through the origin, and making an angle with the axis of x^ whose tangent is m. 19. An equation of the form y — + 7nx will denote a line OP^ situated in the angles FOX, Y'OX'. For it appears, from the equation y = -\- nix, that whenever x is positive y will be positive, and whenever x is negative y will be negative. Points, therefore, represented by this equation, must have their co-ordinates either both -positive or both negative, and such points we saw (Art. 3) He only In the angles FOA", F'OA". 16 THE RIGHT LINE. On the contraiy, in order to satisfy the equation ?/ = — mx^ if X be positive y must be negative, and if x be negative y must be positive. Points, therefore, satisfying this equation, will have their co-ordinates of different signs; and the line represented by the equation, must, therefore (Art. 3), He in the angles T OX, YOX'. 20. Let us now examine how to represent a right line PQ, situated in any manner with regard to the axes. Draw OR through the origin parallel to PQ, and let the ordinate PM meet OR in R. Now it is plain (as in Art. 18), that the ratio RM : OM will be always constant [RM always equal, sup- pose, to m.OM)] but the ordinate PM differs from RM by the constant length PR = OQ, which we shall call h. Hence we may write down the equation PM= RM-\- PR, or PM= m . OM + PR, that is, y = rax + h. The equation, therefore, y = mx -\- b, being satisfied by every point of the line PQ, is said to be the equation of that line. It appears from the last Article, that m will be positive or negative according as OR, parallel to the right line PQ, lies in the angle YOX, or Y' OX. And, again, b will be positive or negative according as the point Q, in which the line meets OY, lies above or below the origin. Conversely, the equation y = mx + b will always denote a right line ; for the equation can be put into the form y — b •^ =m. x Now, since if we draw the line QT parallel to OM, TM will be = b, and PT therefore =y — b, the question becomes : " To find the locus of a point, such that,- if we draw PT parallel to OF to meet the fixed line QT, PT may be to QT in a THE RIGHT LINE. 17 constant ratio ;" and this locus evidently is the right line PQ passing through Q. The most general equation of the first degree, Ax->!- By + C=Oj can obviously be reduced to the forra y = mx + hj since it is equivalent to _ ^ _C ^~ B^ B' this equation therefore always represents a right line. 21. From the last Articles we are able to ascertain the geometrical meaning of the constants in the equation of a right line. If the right line represented by the equation y = mx + b make an angle = a with the axis of Xj and = ^ with the axis of y, then (Art. 18) sin a smp and if the axes be rectangular, m = tan a. We saw (Art 20) that h is the intercept which the line cuts off on the axis of ?/. If the equation be given in the general forra Ax-{-By + C=0, we can reduce it, as in the last Article, to the form y = mx + hj and we find that A _ sin a (J or If the axes be rectangular =tana; and that — y, is the length of the intercept made by the line on the axis of y. Cor. The lines y = mx-\-h^ y — in'x + h' will be paraHel to each other if m = m\ since then they will both make tlie same angle with the axis. Similarly the lines Ax + By + C=o, A'x + B'y+C' = 0, will be parallel if A_A B" B" Beside the forms Ax + By + C=0 and y = mx + hj there are two other forms in which the equation of a right line is frequently used ; these we next proceed to lay befoi-c the reader. D 18 THE RIGHT LINE. 22. To express the equation of a line MN in terms of the intercepts OM=a, ON=h xohich it cuts off on the axes. We can derive this from the form already considered A B Ax + Bi/ + C=0, or _ cc + -^?/+l = 0. This equation must be satisfied by the co-ordinates of ever?/ point on J/iV, and there- fore by those of J/, which (see Art. 2) arc x = aj y = 0. Hence we have -a+l = 0,-^ = --. In like manner, since the equation is satisfied by the co-ordinates of N,{x=0^7/=b)j we have B__l C~ h' Substituting w^hich values in the general form, it becomes x y a This equation holds whether the axes be oblique or rect- angular. It is plain that the position of the line will vary with the signs of the quantities a and h. For example, the equation X y . ... - + ^ = 1, which cuts ofi" positive mtercepts on both axes, re- presents the line MN on the preceding figure ; f = 1) cutting off a positive intercept on the axis of x^ and a negative in- tercept on the axis of y, represents MN'. Similarly, •" f ~ ^ represents NM' and y — J = I represents MN'. By dividing by the constant term, any equation of the first degree can evidently be reduced to some one of these four forms. THE RIGHT LINE, 19 Ex. 1. Examine the position of the followmg lines, and find the intercepts they make on the axes : 2x-3// = 7; Sx + % + 9 = ; 3x + 2i/ = G; 47/-5x = 20. Ex. 2. The sides of a triangle being taken for axes, form the equation of the line joining the pomts which cut off the wi'" pai-t of each, and show, by Art 21, tliat it is parallel to the base. yi«,?j.y — i. " « b m' 23. To express the equation of a right line in terms of the length of the perpendicular on it from the origin^ and of the angles which this perpendicular makes with the axes. Let the length of the perpendicular OP=p^ the angle POM which it makes with the axis of cc = a, FON=^, 031= a, ON=h. We saw (Art. 22) that the equa- tion of the right line MN was X y , - + f = l. a Multiply this equation by p, and we have P V V \x^^^y=p. P - o m\ But - =cosa, 4 = cos)S; therefore the equation of the line Is a ^ a; cosa + y cos/3=p. In rectangular co-ordinates, which we shall generally use, wc have yS = 90° - a ; and the equation becomes x cos a -\y sina =^. This equation will include the four cases of Art. 22, if we suppose that a may take any value from to 360". Thus, for the position iVJ/', a is between 90° and 180°, and the coefficient of a; is negative. For the position il/W, a is between 180° and 270°, and has both sine and cosine negative. For JAY', a is between 270° and 360°, and has a negative sine and positive cosine. In the last two cases, however, it is more convenient to write the formula a; cosa + y sina=:— ^?, and consider a to denote the angle, ranging between and 180°, made with the positive direction of the axis of a?, by the perpendicular pro- duced. In using then the formula a* cosa + ?/ sina =^', wc suppose^) to be capable of a double sign, and a to denote the 20 THE RIGHT LINE. angle, not exceeding 180°, made with the axis of x either by the pcrpcndlcidar or Its production. The general form Ax-\- Bij +C=0^ can easily he reduced to the form x cosa-f ?/ sin a =2^; for, dividing it by ^J[A^ -\- B'^)^ we have A. B C _ ^f[A' + £^) "" + ^J{A' + J5--') y "^ ^/{A' + B') ^' But we may take A , B . cos a, and -77-ri d2\ =sina, V(^' + B') ' — V(^' + £") since the sum of squares of these two quantities = 1. A B Hence we learn, that ,, ■■■ — =^7- and ,. ,.. — 77:77 are re- ' ^[A' + B-] sl[A'-vB-) spectlvely the cosine and sine of the angle which the per- pendicular from the origin on the line [Ax + By -\- C = 0) makes G with the axis of x. and tliat ,, ,■■ — ^^^r is the length of this ' ^'[A' + B-) ° perpendicular. *24. To reduce tlie equation Ax+By + C^O [referred to oblique co-ordinates) to the form £c cosa-f ?/ cosy3=^. Let us suppose that the given equation when multiplied by a certain factor R is reduced to the required form, then i?^l = cosa, ii5=cos/?. But it can easily be proved that. If a and /3 be any two angles whose sum is &), we shall have cos'a + cos^/3 — 2 cosa cos/3 cosca = sin" &>. Hence E^ [A' + B'-2 AB cos a) = sin' «, and the equation reduced to the required form Is A sin 6) B slnw ^{A + B' - 2AB coso)) ^ "^ ^/{A + B' - 2AB costo) ^ C sin (o _ ■*" V(^' + -S'-2^5cosa>) ~ And we learn that A slnw ^ slnw V'U' + B' - 2AB cos w) ' V (^i' + i^' - 2^i? cos «) ' * Articles and Chapters marked with an asterisk may be omitted on a first reading. THE KIGHT LINE. 21 are respectively the cosines of the angles that the perpendicular from the origin on the line Ax-]- Bji/+ C=Oj makes with the axes of a; and y. and that ,. .■■ ^., ^ . ^^ c Is the length ^ ' \/(^ +B'- 2AB cos ft)) ^ of this perpendicular. This length may be more easily cal- culated by dividing the double area of the triangle NOJT^ [ON.OM smco) by the length of il/iV, expressions for which are easily found. The square root In the denominators Is, of course, susceptible of a double sign ; since the equation may be reduced to either of the forms a;cosa4 1/ cos^-p = 0, x cos(a+ 180°) +y cos(/3+ 180°)+7> = 0. 25. To find the angle between two lines wJiose equations loith regard to rectangidar co-ordinates are given. The angle between the lines is manifestly equal to the angle between the perpendiculars on the lines from the origin ; If therefore these perpendiculars make with the axis of x the angles a, a', we have (Art. 23) A . B cosa = ,, ... — —^f^\ sma = ^J{A' -^ B')' — ^{A' + B') A' . B' cosa' = -77-775 — TTT^ ; sin a = BA' - AB' Hence sin (a - a') = ^^j;^ ^ b'^) ^J[A'^.^ B-) ' AA'-^BB' cos (a a j - ^^^, ^ ^,^ ^^^„ ^ ^,,^ ; and therefore tan (a — a) = -r-r, tttt, • ^ ^ AA + BB COK. 1. The two lines are parallel to each other when BA'-AB' = (Art. 21), since then the angle between them vanishes. Cor. 2. The two lines are pei'pendlcular to each other when AA' + BB' = 0^ since then the tangent of the angle between them becomes infinite. 22 THE RIGHT LINE. If the equations of the lines had been given in the form y = mx + h^ y = m'x + ?»' ; since the angle between the lines is the difference of the angles they make with the axis of x^ and since (Art. 21) the tangents of these angles are in and m\ it follows that the tangent of the required angle is ; : that the lines are parallel if wi = m ; ^ 14- riim ^ and perpendicular to each other if mm +1=0. *26. To find the angle betioeen two lines^ the co-ordinates being oblique. We proceed as in the last article, using the expressions of Art. 24, A sin 6) consequently, Hence sin (a — a') = cos a = cos a = sin a = sin a = A' sin ft) V(^"H5''-2^'5'cosft))' B — A cos© V(^' + ^'-2^5cosa))' B' - A' cosw '^J{A^TB'^^^^2AW'^^)' {BA'-AB') sin« cos a — a = V(^' + B' - 2AB cosw) ^f{A'^ + B"' - 2A'B' cosco) ' ■ BB' + A A' - [AB' + A'B) cos co V(^' + B^ - 2AB cos w) V(^" + -B" - 2 A'B' cosw) ' r _ {BA'-AB ') &ma> ^"^* ^^~ AA' + BB' - {AW + BA') cos CO ' Cor. 1. The lines are parallel {fBA' = AB'. Cor. 2. The lines are perpendicular to each other if AA' + BB' = {AB' + BA') cos w. 27. A right line can be found to satisfy any two conditions. Each of the forms that we have given of the general equa- tion of a right line includes two constants. Thus the forms y = inx + by ic cosa + y sina=2> involve the constants m and Z*, 2> and a. The only form which appears to contain more con- THE RIGHT LINE. 23 stants is Ax + By + C= ; but in this case we arc concerned not with the absolute magnitudes, but only with the mutual ratios of the quantities A, B, C. For if we multiply or divide the equation by any constant it will still represent the same line : we may divide therefore by (7, when the equation will only A B contain the two constants 7^ , 77 • Choosing then any of these forms, such as ?/ = rax + h, to represent a line in general, we may consider m and b as two unknown quantities to be deter- mined. And when any two conditions are given we are able to find the values of m and Z», corresponding to the particular line which satisfies these conditions. This is sufficiently illus- ti-ated by the examples in Arts. 28, 29, 32, 33. 28. To find the equation of a right line parallel to a given one^ and passing through a given i^oint x'y. If the line y = mx + h be parallel to a given one, the con- stant m is known (Cor., Art. 21). And if it pass through a fixed point, the equation, being true for every point on the line, is true for the point x'y\ and therefore we have y' — mx + Z*, which determines h. The required equation then is y = mx +y' — mx'^ or y — y — m [x — x). If in this equation we consider m as indeterminate, we have the general equation of a right line passing through the point x'y'. 29. To find the equation of a right line passing through two fixed 23oints xy\ x'y" . "VYe found, in the last article, that the general equation of a right line passing through xy is one which may be written in the form y - y = w. where m is indeterminate. But since the line must also pass through the point x'y" ^ this equation must be satisfied when the co-ordinates x", ?/", are substituted for x and y ; hence X — x -^. = in. 24 THE RIGHT LINE. Substituting this value of ?«, the equation of the line becomes y-y _ y"-y' t If t * VO ~~ JC tX/ **~ iC In this form the equation can be easily remembered, but, clearing it of fractions, we obtain it in a form which is some- times more convenient, {^y -.y)x-[x -x)y + xy -yx =0. The equation may also be written in the form (^-«^')(y-/) = (^-^")(y-y). For this is the equation of a right line, since the terms xy^ which appear on both sides, destroy each other; and it is satisfied either by making x = x\ y =y' ] or x = x\ y = y" , Expanding it, we find the same result as before. Cor. The equation of the line joining the point xy to the origin is y'x = xy. Ex. 1. Form the equations of the sides of a triangle, the co-ordinates of whose vertices are (2, 1), (3, - 2), (- 4,-1). An$, x + 7y + 11 = 0, 3^ - a; = 1, 'ix + y-1. Ex, 2. Form the equations of the sides of the triangle formed by (2, 3), (4, - 5), (- 3, - 6). Ans, a; - 7y = 39, 9a; - 5^ = 3, 4a; + 2/ = 11. Ex. 3. Form the equation of the line joining the jDoints , , , mx' + nx" my' + ny" xri and , -^ ^ . 711 + 11 m + n Ans. {jj' — y") X — {x' — x") y + x'y" — y'x" — 0. Ex. 4. Form the equation of the Hne joining , , , x" + x"' y" + y'" xy and 2 ' 2 ' Ans. {i/" + y'" — 2y') x — {x" + x'" — 1x') y + x"y' — y"x' + x"'y' — y"'x' — 0. Ex. 5. Form the equations of the bisectors of the sides of the triangle described In Ex. 2. Ans. 17a; - 3^ = 25 ; 7a; + 9*/ + 17 = j 5a; - 6^ = 21. Ex. 6. Foi-m the equation of the line joining /x' — mx" ly' — my" Ix' — nx'" ly' — ny'" l — m ' l—m l—n ' l — n Ans. X {l{in-n) y' + m {n — T)y" + n {l—m) y'"} —y{l{m-n) x' + m{ii—r)x"+n (?-»?) x"'} = Im {y'x" — x'y") + mn [y"x"' — x"y"') + n? (^"'x' — y'x'"). 30. To find the condition that three points shall lie on one right line. We found (in Art. 29) the equation of the line joining two of them, and we have only to see if the co-ordinates of the third will satisfy this equation. The condition, therefore, is (y. - 3/ J ^. - (^, - »^J y, + (a'j. - ^.y.) = % THE RIGHT LINE. 25 which can be put into the more symmetrical form 31. To find the co-ordinates of the point of intersection of two right lines whose equations are given. Each equation expresses a relation which must be satisfied by the co-ordinates of the point required ; we find its co-ordinates, therefore, by solving for the two unknown quantities x and ?/, from the two given equations. We said (Art. 14) tliat the position of a point was deter- mined, being given two equations between its co-ordinates. The reader will now perceive that each equation represents a locus on which the point must lie, and that the point is the intersection of the two loci represented by the equations. Even the simplest equations to represent a point, viz. x = a^ ^ = ^j ^^^ t^^ equa- tions of two parallels to the axes of co-ordinates, the intersection of which is the required point. When the equations are both of the first degree they denote but one point ; for each equation represents a right line, and two right lines can only intersect in one point. In the more general case, the loci represented by the equations are curves of higher dimensions, which avIU inter- sect each other in more points than one. Ex. 1. To find the co-orcliuates of the veitices of the triangle the equationi5 of whose sides are a; + y = 2 ; a; — 3y = 4 ; Sx + 5^ + 7 = 0. ^"^. (-T^. -if). (V- -¥% (1- -^). Ex. 2. To find the co-ordinates of the intersections of ox + y -2 = 0; .t + 2 // = 5 ; 2a; - 3^ + 7 = 0. Ans. (}, V). (- ^\, H), (- I V)- Ex. 3. Find the co-ordinates of the intersections of 2a.- + 3^ = 13 ; ox-y = l; x- 4ij -f 10 = 0. .'Ins. They meet in the point (2. 3\ Ex.4. Find the co-ordinates of the vertices, and the equations of the diagonals, of the quadrilateral the equations of whose sides are 2y-Zx= 10, 2y + x = 6, 16a; - lOy = 33, 12a; + Uy + 29 = 0. Ans. (-1, I), (3, f), a, -t), (-3, i); Gy-x = 6, 8x + 2y+l = 0. * In using this and other similar fonnulre, which we shall afterwai-ds have occa- sion to employ, the learner must be careful to take the co-ordmates in a fixed order (see engi-aving). For instance, in the second member of the formula just given, ?/2 takes the place of ?/„ 3:3 of x.,, and .r, of X3. Then, in the third member, we advance fi-om ?/._, to //j, ft-om \^^v__^/,,^ Xj to a-,, indicated. Xj to a-,, and from .r, to .r._,. always proceeding in the order just ^- ■» E 26 THE RIGHT LIXE. Ex. 5. Find tlie iiitcisectioiis of opposite sides of the same quadrilateral, and the equation of the line joining them. Ann. [Hd, ^% (- Vj %')? 162;y - 199x = 1402. Ex. 6. Find the diagonals of the parallelogi'am fonned by ,T = rt, X — a', y — l>, y ~h' . Ans. (h — V) X— {a — a') y — a'h — ah' ; (6 — 6') a; + (« — «') y — ah — ah' . Ex. 7. The axes of co-ordinates being the base of a triangle and the bisector of the base, form the equations of the two bisectors of sides, and find the co-ordmates of their intersection. Let the co-orduiates of the vertex be 0, y', those of the base angles x', ; and — x, 0. Ans. Sx'y — y'x — x'y' = ; Sx'y + y'x — x'y' = ; ( 0, ^ j . Ex. 8. Two opposite sides of a quadrilateral are taken for axes, and the other two are X y . X y ^ 2a^2^» ' 2«' 26' ' find the co-ordinates of the middle points of diagonals. Ans. («, V), («', h). Ex. 9. In the same case find the co-ordinates of the middle point of the line joining the intersections of opposite sides. . a'b . a — ah', a' u'b . V — ah'. 6 , , . Ans. jj J-, — , yi 77 — : and the fonn of the result shows (Art. 7) ao — ab ab — ah ^ that this point divides externally, in the ratio a'b : ab', the line joming the two middle points {a, b'), («', b). 32. To find the equation to rectangular axes of a right line passing through a given point., and perpendicular to a given line^ y = mx -f h. The condition that two luies should be perpendicular, being mm= — \ (Art. 25), we have at once for the equation of tlie required perpendicular I , ,. y-y =-- [x-x). It Is easy, from the above, to see that the equation of the per- pendicular from the point x'g' on the line Ax + Bg-\-C=0 is ^{y-y') = -^{^-^')^ that is to say, we interchange the coefficients of x and ?/, and alter the sign of one of them. Ex. 1. To find the equations of the peri:)endiculars from each vertex on the opposite side of the triangle (2, 1), (3, — 2), (— 4, — 1). The equations of the sides are (Art. 29, Ex. 1) .T + 7;/ + 11 = 0, 3y — .T = 1, 3j; + ^ = 7 ; and the equations of the pei-pendiculars Ix — y— 13, 3.r + y ~ 7, 3y — .r = 1. The triangle is consequently right-angled. Ex. 2. To find the equations of the perpendiculars at the middle points of tlie sides of the same triangle. The co-ordinates of the middle points being (-1- -f). (-1. 0). (f. -i). THE lilUlIT LINE. 27 The peipendiculars are 7a; - ^ + 2 = 0, 3x + y + 3 = 0, 3^ - a; + 4 = 0, intersecting in (- ^, - J). Ex. 3. Find the equations of the peipendiculars from the vertices of the triangle (2, 3), (4, - 5), (- 3, - 6) (see Art. 29, Ex. 2). A}is. Ix + y-ll, 5a; + % + 25 = 0, a; - 4»/ = 21 ; intersectmg in (|g, - u,")- Ex. 4. Eind the equations of the perpendiculars at the middle poLats of the sides of the same triangle. Ans. Ix + y + 2-Q, 5a; + 9_y + 16 := 0, a; - 4^ = 7 ; intersecting in (- ^l^ _ ^). Ex. 5. To find in general the equations of the pei-pendiculars fi-om the vertices on the o]3posite sides of a triangle, the co-ordinates of whose vertices are given. Ans. {x" - x'") x + {jf" - y'") y + {x'x'" + y'y'" ) - {x'x" + y'y" ) = 0, [x'" -x' ) a; + Q/'" -y' ) y + {x''x' + y"y' ) - {x"x"' + y'y'") = 0, {x' - x" )x+{t)' -y")y + {x"'x" + y"'y") - {x"'x' + y'"^ ) = 0, Ex. 6. Pind the equations of the perj)endiculars at the middle points of the sides. Ans. {x" - x'") x+(j/" - y'") y-l {x"^ - x""^) + ^ (^"^ - ^'"2), W" -x')x+ (y'" -y' )y = l ix'"-' -x'n + i il/'"^ - y'^ ), {x' -x")x + {?/' -y")y = i (a;'2 - x"^ ) + i(jj"^ - y'"- ). Ex. 7. Taking for axes the base of a triangle and the perpendicular on it from the vertex, find the equations of the other two perpendiculars, and the co-ordinates of theu- intersection. The co-ordinates of the vertex are now (0, y'), and of the base angles (x", 0), (- x"', 0). Ans. x'" (x — x") + y'y = 0, x" {x + x'") — y'y = 0, (0, — j- \ . Ex. 8. Using the same axes, find the equations of the perpendiculars at the middle points of sides, and the co-ordinates of their intersection. Ans. 2{x"'x+y'y)=y'^-x"'^, 2{x"x-y'y)=x"--y'"-, 2x=x"-x"', (— ^ , ^^) • Ex. 9. Foi-m the equation of the peipendicular fi-om x'y' on the line x cosa +yshia =p ; and find the co-ordinates of the intersection of this peipentliculai- with the given line. Ans. {x' + cosa {}) — x' cosa — y' sina), y' -I- sina {p — x' coso — y' sina)}. Ex. 10, Find the distance between the latter point and x'y'. Ans. ± (p — x' cosa — y' sina). 33. To find the equation of a line passing through a given 2)oint and making a given angle </>, ivith a given line g = mx + b {the axes of co-ordinates being rectangular). Let the equation of the required line be 7/-y' = Vl{x-x'), and the formula of Art. 25, m - m tan© = 7 , , enables us to determine in — tan (^ m = 1 4 m tan (f) 28 THE HIGHT LINE. 34. To fad lilt length of the ijerpendicular from any point xy\ on the line whose equation is x cosa-ry cos^- p = 0. We liave already indicated (Ex. 9 and 10, Art. 32) one way of solving this question, and we wish now to show how the same result may be obtained geometrically. From the given point Q draw QR parallel to the given line, and QS perpen- dicular. Then OK—x', and OT will be =x' cos a. Again, since SQK=^^ and QK—y\ BT=Q8 = y' cos^', hence x cos cc+y' cos ^ = OB. Subtract OP, the perpendicular from the origin, and x eosa + y cos^-j) = PE= the perpendicular QV. But if in the figure the point Q had been taken on the side of the line next the origin, OE would have been less than OF, and we should have obtained for the perpendicular the expression p—x cosa — ?/' cos/3; and we see that the perpendicular changes sign as we pass from one side of the line to the other. If we were only concerned with one perpendicular, we should only look to its absolute magnitude, and it would be unmeaning to prefix any sign. But if we were comparing the perpendiculars from two points, such as Q and 8, it is evident (Art. 6) that the distances QV, SV, being measured in opposite directions must be taken with opposite signs. We may then at pleasure choose for the expression for the length of the perpendicular either + {p — x cosa- y' cos/S). If we choose that form in which the absolute term is positive, this is equivalent to saying that the perpendiculars which fall on the side of the line next the origin are to be regarded as positive, and those on the other side as negative ; and vice versa if we choose the other form. If the equation of the line had been given in the form Ax -\-By + C = 0, we have only (Art. 24) to reduce it to the form X cos a + // cos /3 —p — 0, THE RIGHT LINE. 29 and the length of the perpendicular from any point xy\ - ^^' + By' + G [Ax' + By' + C] sino) ~ V(^' + B") ' ^^ Vi^' + B' - 2AB cos 0)) ' according as the axes are rectangular or oblique. By comparing the expression for the perpendicular from x'y' with that for the perpendicular from the origin, we see that x'y' lies on the same side of the line as the origin when Ax + By + G has the same sign as C, and vice versa. The condition that any point x'y' should be on the right line Ax + By + C—Oj is, of course, that the co-ordinates ic'?/' should satisfy the given equation, or Ax' + By' + C=0. And the present Article shows that this condition is merely the algebraical statement of the fact, that the perpendicular from the point x'y' on the given line is = 0. Ex. 1. Find the length of the perpendicular from the origin on the line 3x + 4ij + 20 = 0, the axes being rectangular. Ang, 4, Ex. 2. Find the length of the pei-pendicular from the point (2, 3) on 2x + y — 4 = 0. 3 A7is. -p : and the given j^roint is on the side remote from the origin. Ex, 3. Find ihe lengths of the peipendiculars fi-om each vertex on the opposite side of the triangle (2, 1), (3, - 2), (- 4,-1). A71S. 2 J(2), J(10), 2 4(10), and the origin is %vithin the triangle. Ex. 4. Find the length of the perpendicular from (3, — 4) on -ix + 2ij -1, the angle between the axes being 60°. Alls. I : and the point is on the side next the origin. Ex. 5. Find the length of the pei-pendicular from the origin on a (a; - a) + i 0/ - J) = 0. Ans. ^{(i" + b"). 35. To find the equation of a line bisecting the angle hetween two lines, x cosa -f y sina — ^> = 0, a; cos/3 + y sln^ —p = 0. We find the equation of this line most simply by expressing algebraically the property that the perpendiculars let fall troni any point xy of the bisector on the two lines are equal. This immediately gives us the equation x cosa + y sina— 2> = ± {x cos/S-f y sin/3— y), since each side of this equation denotes the length of one of those perpendiculars (Art. 34). 30 THE RIGHT LINE. If the equations had been given in the form Ax+Bi/ + C=Oj A'x + B'y 4-0' = 0, the equation of a bisector would be A x + By + G _ A'x- ]- ^ + C' ^/{A' + B') ~- V(-^"-+i?'') ' It is evident from the double sign that there are two bisectors : one such that the perpendicular on what we agree to consider the positive side of one line is equal to the perpendicular on the negative side of the other: the otlier such that the equal perpendiculars are either both positive or both negative. If we choose that sign which will make the two constant terms of the same sign, it follows from Art. 34 that we shall have the bisector of that angle in which the origin lies ; and if we give the constant terms opposite signs, we shall have the equation of the bisector of the supplemental angle. Ex. 1. Eeduce the equations of the bisectors of the angles between two lines, to the form x cos a + 1/ sin a = p. Ans. X cos [I {u + li) + 90°} + y Bm{i {a + (3) + 90°} = ^^-^|^^i^ ; , cos. (a + ^) + 2/ sina („ + /J) = ^_|±Z_ . Ex. 2. Find the equations of the bisectors of the angles between 3.r + 4y - 9 = 0, 12x + 5^ - 3 = 0. Ans. 7x-9i/ + 3i = 0, Ox + 7//= 12. 36. To Jind the area of the triangle formed hy three ]^o{nts. If we multiply the length of the line joining two of the points, by the perpendicular on that line from the third point, we shall have double the area. Now the length of the perpen- dicular from x^y^ on the line joining x^y^.^ ^-iV-ii ^^^ axes being rectangular, is (Arts. 29, 3-4) and the denominator of this fraction is the length of the line joining cc,y„ x^j^^ hence represents double the area formed by the three points. If the axes be oblique. It will be found on repeating the in- vestigation with the formulae for oblicjue axes, that the only change that will occur Is tliat the expression just given Is to be multij)Hcd by slnw. Strictly speaking, we ought to prefix to THE RIGHT LINE. 31 these expressions the double sign implicitly involved in tlic square root used in finding them. If we arc concerned with a single area we look only to its absolute magnitude without regard to sign. But if, for example, we are comparing two triangles whose vertices x^y^^^ a?^^, are on opposite sides of the line joining the base angles x^y^^ ^-iU'ii ^^^ must give their areas different signs ; and the quadrilateral space included by the four points is the sura instead of the difference of the two triangles. Cor. 1. Double the area of the triangle formed by the lines joining the points x,?/^, x^j^ to the origin, Is y^x,^—y„x^ as appears by making x^ = 0, y^ = 0, in the preceding formula. Cor. 2. The condition that three points should be on one right line, when interpreted geometrically, asserts that the area of the triangle formed by the three points becomes = (Art. 30). 37. To express the area of a polygon in terms of the co-ordi- nates of its angular points. Take any point xy within the polygon, and connect it with all the vertices x^y^^ x^y^...x^^y^'^ then evidently the area of the polygon is the sum of the areas of all the triangles into which the figure Is thus divided. But by the last Article double these areas are respectively ^ iyz -y^-y i^^ - ^ J + ^jj, - ^,y.^ ^ iyn-i -y„) - y (^„-, - ^J + ^„-,y„ - «'»y»-ij ^(yn -yr)-yi^n -^,)+^^n y.-a^.y,,- When we add these together, the parts which multiply .7^ and y vanish, as they evidently ought to do, since the value of tlic total area must be independent of the manner In which we divide it into triangles ; and we have for double the area (^,2/2 - 3^2^.) + (^22/3 - '^33/2) + (-^3^4 - ^4^3) + • • • (-^..yi - ^d/J- This may be otherwise written, a^t (y, - yn) + ■■^'■z (y, - y.) + ^. iy, - y) +• • ••^'. (//- - y,.-X or else y. (^„ - a-J + y, (•^■, - ■'-:■) + y. i-'^ - ^"4) ^- • • -y,. (•*•»-, - ■*•,)• 32 THE RIGHT LINE. Ex. 1. Find the area of the triangle (2, 1), (3, - 2), (- 4, - 1). Ans. 10. Ex. 2. Find the area of tlic triangle (2, 3), (4, - 5), (- 3, - 6). Am. 29. Ex. 3. Find the area of the quadrilateral (1, 1), [2, 3), (3, 3), (4, 1). Ans. 4. 38. To find the condition that three right lines shall meet in a point. Let their equations be Ax + By+C=Q, A'x + B'y + G' = 0, A"x ■\- B"y + C" = 0. If they intersect, the co-ordinates of the intersection of two of them must satisfy the third equation. But the co-ordinates of the intersection ot the nrst two are —r^FVi 77^ , —r-pr, ttt^ . AB -AB^ AB -AB Substituting in the third, we get, for the required condition, A" {BC'-B'C) + B" ( CA' - C'A) + C" {AB' - A'B) = 0, which may be also written in either of the forms A[B'C"-B"C')^B {C'A"-C"A')+C [A'B" -A"B') =0, A \b' C" - B" C) + A [B" G - BG") + A" [BG' - B'G)= 0. *39. To find the area of the triangle formed hy the three lines yla; + % + (7=0, A'x-^B'y + G'=^0^ A"x + B"y + G" = 0. By solving for x and y from each pair of equations in turn we obtain the co-ordinates of the vertices, and substituting them in the formula of Art. 36, we obtain for the double area the expression BG'-B'G (A'G"-G'A" A"G-G"A] AB'-BA' {B'A"-A'B" B'A-A"B B'C"-B"G' \ A"G-G"A _ AG'- GA' ] ■^ A'B" - B'A" \B"A - A"B BA' - AB'] B"C-BG" (AG'-GA' A'G"-G'A" A"B-B"A [BA'-AB' B'A" -A'B"} But if we reduce to a common denominator, and observe that the numerator of the fraction between the first brackets is {A"{BG'-B'G)-\-A {B'G"-B"G') + A' {B"G-G"B]} multiplied by A" ; and that the numerators of the fractions be- tween the second and third brackets are the same quantity multiplied respectively by A and A', we get for the double area the expression {A{B'G"~B"G'] + A'[B"G-BG") + A"iBG'-B'G)]' {AB' - BA') {A'B" ~ B'A") {A"B - B"A) THE RIGHT LINE. 33 If the three lines meet In a point, this expression for the area vanishes (Art. 38) ; if any two of them are parallel, It becomes infinite (Art. 25). 40. Given the equations of two right lines, to find the equation of a third through their point of intersection. The method of solving this question, which will first occur to the reader, is to obtain the co-ordinates of the point of inter- section by Art. 31, and then to substitute these values for x'y in the equation of Art. 28, viz., y-y z=in {x — x). The question, however, admits of an easier solution by the help of the following important principle : If 8=0, S' = Q,le the equations of any two loci, then the locus represented hy the equation S+kS' = {where k is any constant) passes through every point common to the two given loci. For it is plain that any co-ordinates which satisfy the equation S=0, and also satisfy the equation S' = 0, must likewise satisfy the equation S+kS' = 0. Thus, then, the equation {Ax + By + C)-]-Jc{A'x-{B'y + C')=0, which is obviously the equation of a right line, denotes one passing through the intersection of the right lines Ax-^By + G=0, A'x + B'y-]-C' = 0, for if the co-ordinates of the point common to them both be sub- stituted in the equation [Ax + By + C) + k {A'x + B'y + C') = 0, they will satisfy it, since they make each member of the equa- tion separately = 0. Ex. 1. To find the equation of the line joining to the origin the intersection of Ax + By + C-0, A'x + B'y+C = 0. Multiply the first by C, the second by C, and subtract, and the equation of the required line is (AC — A'C) x + {BC — CB') y = 0; for it passes through the origin (Art. 18), and by the present article it passes through the interaection of the given lines. Ex. 2. To find the equation of the line drawn through the intersection of the same lines, parallel to the axis of .i*. Ans. (BA' — AB') y + CA' — AC — 0. Ex. 3. To find the equation of the line joinmg the intersection of the same lines to the point x'y'. "Writing do^-u by this article the general equation of a Une through the intersection of the given lines, we determine k from the consideration that it must be satisfied by the co-ordinates x'y', and find for the requu'ed equation <^Ax + By + C) {A'x' + B'y' + C) = {Ax' + By' + C) {A'x + B'y + C). Ex. 4. Find the equation of the line joining the point (2, 3) to the intersection of 2j: + 3y + 1 = 0, ox-Ay = b. Ans. 11 (2,(; + 3// + 1) + U (,3x- - 4y - 5) = : or CA.r - 23./ = .M'. 34 THE RIGHT LINE. 41. The principle established in the last article gives us a test for three lines intersecting in the same point, often more convenient in practice than that given in Art. 38. Tliree riglit lines loill jjass through the sai^ie point if their equations heing multijylied each hy any constant quantity^ and added together^ the sum is identically = ; that is to say, if the following relation be true, no matter what x and y are : I [Ax + By + C) + m [A'x + B'y + C) + n {A"x + B"y + C") = 0. For then those values of the co-ordinates which make the first two members severally = must also make the third = 0. Ex. 1. The three bisectors of the sides of a triangle meet in a point. Their equations are (Art. 29, Ex. 4) {y" + ij'" — 2/y' ) X — {x" + x'" - 2x' ) y + {x"y' — y"x' ) + {x"'y' — y"'x' ) = 0, (.'/'" + .'/' - 2^" ) « - (■«'" + -c' - 2j.-" ) y + {x"'y" - y"'x") + {x'y" - y'x" ) - 0, (.'/' + ll" - 2^'") x-ix' + x" - 2x"') y + {x'y'" -y'x'" ) + (/'«/'" - y"x"') = 0. And since the three equations when added togeth-er vanish identically, the hnes represented by them meet in a point. Its co-ordinates are found by solving between any two, to be i (x' + x" + x'"), i {y' + y" + y'"). Ex. 2. Prove the same thmg, taking for axes two sides of the triangle whose lengths are a and h. , 2x y _ „ a- 2// , r. ^ V r. (to a a Ex. 3. The three pevpendicidars of a triangle, and the three pei-}Dendiculars at middle points of sides respectively meet in a point. For the equations of Ex. 5 and G, Art. 32, when added together, vanish identically. Ex. 4. The three bisectors of the angles of a triangle meet in a point. For their equations are {x cosa + »/ sina — p ) — (.r cos/3 + y sin/3 — jj' ) = 0, (x cos/3 + y sinfi —j)' ) — {x cosy + y smy —p") = 0- (.r cos y + // sin y — 2>") — {x cos a + y sina — p ) = 0. *42. To find the co-ordinates of the intersection of the line joining the points xy\ x"y'.\ with the right line Ax + By + C=0. We give this example in order to illustrate a method (which we shall frequently have occasion to employ) of determining the point in which the line joining two given points is met by a given locus. We know (Art. 7) that the co-ordinates of any point on the line joining the given points must be of the form mx" -f nx mi/" + ny x = , y=-— ^; m + n m + n and wc take as our unknown quantity — , the ratio, namely, in THE KIGHT LINE. which the luie johihig the points is cut by the given locus ; and we determine this unknown quantity from the condition, that the co-ordinates just written shall satisfy the equation of the locus. Thus, in the present example we have .rnx' + nod ^my"-^ny' ,^ m + n m-\- n hence m n x = Ax +B^' iC ^ Ax" + Bi/"+cf' and consequently the co-ordinates of the required point are _ {Ax + %' + C) x" - [Ax" + By" + G) x ^ {Ax' -h By' + C)- {Ax" + By" + C) ' with a similar expression for y. This value for the ratio m : n might also have been deduced geometrically from the considera- tion that the ratio in which the line joining x'y'^ x"y" is cut, is equal to the ratio of the perpendiculars from these points upon the given line ; but (Art. 34) these perpendiculars are Ax' + By' + C , Ax" + By" 4 C -p,^^,^ and _-p---^^. The negative sign in the preceding value arises from the fact that in the case of internal section to which the positive sign of m : n corresponds (Art. 7), the perpendiculars fall on opposite sides of the given line, and must, therefore, be understood as having different signs (Art. 34). If a right line cut the sides of a triangle BC, CA^ AB, in the points LMN^ then BL.CM.AN _ LC.MA,NB~ Let the co-ordinates of the vertices be x'y'^ ^'y" ■> ^"y" i ^''cn BL _ _Ax" +By" -+0 M rc " GM__ MA~ AN __ NB ~ Ax"' + By'"-\-C' Ax'" + By'"+C ^ Ax +By' 4 C ' Ax' +By' +G ^ Ax -^By" -VC' and the truth of the theo- rem is manifest. 36 THE KIGllT LINE. *43. To find the ratio in which the line joining two points aJ,y„ ic.^25 is cut hy the line joining two other points x,^^^ ^^i- The equation of this latter line is (Art. 29) Therefore, by the last article, !!.^ = _ (3/3 - 3/ 4 ) ^1 - (^3 - ^4) Vx + ^^Jf, - ^43/3 ■^ (3/3 - y^ ^. - K - a'4) 3/2 + ^33/4 - ^4.y3 * It is plain (by Art. 36) that this is the ratio of the two tri- angles whose vertices are a;,?/^, x^y^^ x^^^ and x^^^ x^^^ x^y^^ as also is geometrically evident. If the lines connecting any assumed point with the vertices of a triangle meet the opposite sides BC, CA^ AB respectively^ in X>, Ej F^ then BD.CE.AF _ DC.EA.FB'^ Let the assumed point be xjj^^ and the vertices aj^y,, x.^^^ cc3^3, then BD ^ x^ {y.^ - yj + x^ [y^ - .y,) + x^ {y^ - yj I^G x^ {y,-y,) + x^ (^3 - 3/J + a?3 (y, - yj ' ^ ^ ^2 {ys - y,) + ^3 (3/4 - 3/ J + ^4 (3^. - 3^3) ^^ ^1 (3/2 - 3/4) + ^2 (3^4 - yx) + «^4 (yi - 3^2) ' ^ ^ ^1 (3/4 - 3/ 3) + ^4 (3/3 - 3/t) + ^3 (3/1 - 3/4) -^-^ ^2 (3/3 - 3/4) + ^3 (3/4 - 3/2) + «^4 (3/2 - 3^3) ' and the truth of the theorem is evident. 44. To find the polar equation of a right line (see Art. 12). Suppose we take, as our fixed axis, OP the perpendicular on the given line, then let OR be any radius vector drawn from the pole to the given line OE = p, ROP^d', but, plainly, OR cosd= OP, hence, the equation is THE EIGHT LINE. 37 If the fixed axis be OA making an angle a with the perpen- dicular, then BOA = 6, and the equation 13 p cos {6 - a) =p. This equation may also be obtained by transforming the equation with regard to rectangular co-ordinates, X cosa + ^ sin a = p. Rectangular co-ordinates are transformed to polar by writing for a;, p cos^, and for y^ p sin^ (see Art. 12) ; hence the equa- tion becomes p (cos Q cos a + sin ^ sin a) =^ ; or, as we got before, p cos(^ — a) =p. An equation of the form p{A cose + Bsine)=C can be (as in Art. 23) reduced to the form p cos(^ — a)=2', by dividing by \/{A!^ + B'^) ; we shall then have A . B C ''''^'^~^J[A' + B•')' '''^''~>^[A'^By P~^/[A' + B')' Ex. 1. Reduce to rectangular co-ordinates the equation p-ta sec (^ + ^) • Ex. 2, Find the polar co-ordinates of the intersection of the following lines, and also the angle between them : p cosf 6 - - j = 2a, p cos( — -J = «. Ans. p-2a, 6=2'' °-^S^^ = 5" • Ex. 3. Find the polar equation of the line passing through the points whose polar co-ordinates are p', 6'; p", 6". Ans. p'p" sin ((J' - 6") + p"p sin (6" - ti) + pp' sin (6 - 6') = 0. ( 38 ) CHAPTER III. EXAMPLES ON THE RIGHT LINE. 45. Having In the last chapter laid down principles by which we are able to express algebraically the position of any point or right line, we proceed to give some further examples of the application of this method to the solution of geometrical problems. The learner should diligently exercise himself In working out such questions until he has acquired quickness and readiness in the use of this method. In working such examples our equations may generally be much simplified by a judicious choice of axes of co-ordinates : since, by choosing for axes two of the most remarkable lines on the figure, several of our expressions will often be much shortened. On the other hand, it will sometimes happen that by choosing axes uncon- nected with the figure, the equations will gain in symmetry more than an equivalent for what they lose in simplicity. The reader may compare the two solutions of the same question, given Ex. 1 and 2, Art. 41, where, though the first solution is the longest, it has the advantage that the equation of one bisector being formed, those of the others can be written down without further calculation. Since expressions containing angles become more complicated by the use of oblique co-ordinates, it will be generally advisable to use rectangular axes in any question in which the considera- tion of angles is involved. 46. Loci. — Analytical geometry adapts itself with peculiar readiness to the investigation of loci. We have only to find what relation the conditions of the question assign between the co-ordinates of the point whose locus we seek, and then the statement of this relation in algebraical language gives us at once the equation of the required locus. EXAMPLES ON THE RIGHT LINE. 39 Ex. 1. Given base and difEerence of squares of sides of a triangle, to find the locus of vertex. Let us take for axes the base and a perpendicular thi-ough its middle point. Let the half base = c, and let the co-ordinates of the vertex be X, y. Then AC^ - BC- - 4:cx, and the equation of the locus is 4ea; = m-. The locus is therefore a line perpendicular to the base at a dis- tance from the middle point x = j- . It is easy to see that the difference of sqiiares of segments of base = difference of squares of sides, Ex. 2. Find locus of vertex, given base and cot^ + m cot B. It is evident, from the figure, that ^ , AR c + X cot B — • ; y ' y and the required equation is c + j; + m (c — x) = pij ; the equation of a right line. Ex. 3. Given base and sum of sides of a triangle, if the peipendicular be pro- duced beyond the vertex until its whole length is equal to one of the sides, to find the locus of the extremity of the pei-pendicular. Take the same axes, and let us inquire what relation exists between the co-ordi- nates of the point whose lociis we are seeking. The x of this point plainly is MR, and the y is, by hyiDothesis, =i AC ; and if m be the given sum of sides, BC=m-y. Now (EucUd II. 13), BC' = AB^- + AC- - 2AB. AR ; or {jii — y)- = 4c- H- y- — 4c (c -I- x). Reducing this equation, we get 2 my — icx = m", the equation of a right line. Ex. 4. Given two fixed lines, OA and OB, if any line AB be drawTi to intersect them parallel to a third fixed line OC, to find the locus of the point P where AB is cut in a given ratio ; viz. PA = nAB. Let us take the Unes OA, OC for axes, and let the equation of OB he y = mx. Then since the point B hes on the latter line, its ordinate is m times its abscissa ; or AB = mOA. Therefore PA = mnOA; but PA and OA are the co-ordinates of the iDoint P, whose locus is there- fore a right line through the origin, having for its equation, y — mnx. * This is a particular case of Art. 4, and c -f a; is the algebraic diffei'ence of the abscissae of the points A and C (see remarks at top of p. 4). Beginnei-s often reason that since the line AR consists of the parts AM = ~ c, and MR — x, its length is — c + X, and not c -\- x, and therefore that A C^ - y" + {x — c)-. It is to be observed that the sign given to a lino depends not on the side of the origin on which it lies, but on the direction in which it is measm-ed. We go from A to R by proceeding in the positive direction AM = c, and stiU further in the same dii-ection MR = x, therefore the length AR = c + x: but we may proceed from R to B by rii-st going in the negative direction R^f= — x. and then in tlie opposite direction .!//? = <•, hence the length NB is r r. 40 EXAMPLES ON THE RIGHT LINE. Ex. 6. PA drawn parallel to OC, as before, meets any number of fixed lines in points B, B', E", &c., and PA is taken proportional to the sum of all the ordiuatea BA, B'A, &c., find the locus of P. Ans. If the equations of the lines be y = mx, y — m'x + n', y = m"x + n", &c., the equation of the locus is l-y — mx + [m'x + n') + {m"x + n") + &c. Ex, 6. Given bases and sum of areas of any number of triangles having a common vertex, to find its locus. Let the equations of the bases be X cosa + y sina — p = 0, a; cos/3 + y 8in/3 — Pi — 0, itc, and their lengths, a. b, c, &c. ; and let the given sum =: m^ ; then, since (Art. 34) X cos a + y sin a — p denotes the perpendicular from the point xy on the first line, a {x cosa + y sina—p) vrill be double the area of the first triangle, &c., and the equation of the locus will be a{x cosa + y sin a —p) + b{x cosft + y sin/3 — ^,) + c{x cos y + ysmy —p^) + &C. — 2m'', which, since it contains x and y only in the first degree, will represent a right line. Ex. 7. Given vertical angle and sum of sides of a triangle, find the locus of the point where the base is cut in a given ratio. JL The sides of the triangle are taken for axes; N/^\P and the ratio PK : PL is given = n : m by similar triangles, {7n + n)x ^r„(wi + n)y Then 0K=: OL and the locus is a right line whose equation is — I- - = — ^^ — . m n m + n Ex. 8, Find the locus of P, if when perpendiculars PM, PN are let fall on two fixed lines, OM + ON is given. Taking the fixed lines for axes, it is evident that OM = X + y cos to, ON = y + x cos to, and the locus is x + y — constant, Ex. 9. Find the locus if MN be parallel to a fixed line. Ans. y + X cos to = m {x + y cosoj), Ex. 10. If J/.Vbe bisected [or cut in a given ratio] by a given line y — mx + n. The co-ordinates of the middle point ex- pressed in terms of the co-ordinates of P are ^{x + y cosoj), ^{y -¥ x cosw) ; and since these satisfy the equation of the given Une, the co-ordinates of P satisfy the equation y + X cos to = m {x + y cos ui) + 2n. Ex. 11. P moves along a given line y = mx + n, find the locus of the middle point of JIN. If the co-ordinates of P be a, /3, and those of the middle point x, y, it has just been proved that 2x = a + (3 cosw, 2y = (3 + a cosco. Whence solving for «, /3, a sin^ a> = 2a; — 2y cos w, /3 sin^ w = 2y — 2x cos to. But a, /3 are connected by the relation fi — ma. + n, hence 2y — 2x cosu) — m {2x — 2y cosw) + n sin^w. EXAMPLES ON THE RIGHT LINE. 41 47. It is customary to denote by x and y the co-ordinates of a variable point which describes a locus, and the co-ordinates of fixed points by accented letters. Accordingly in the preceding examples we have from the first denoted by x and y the co- ordinates of the point whose locus we seek. But frequently in finding a locus it is necessary to form the equations of lines connected wuth the figure ; and there is danger of confusion between the x and y^ which are the running co-ordinates of a point on one of these lines, and the x and y of the point whose locus we seek. In such cases it is convenient at first to denote the co-ordinates of the latter point by other letters such as a, /3, until we have succeeded in obtaining a relation connecting these co-ordinates. Having thus found the equation of the locus, we may if we please replace a, /S by a? and y^ so as to w^rite the equation in the ordinary form in which the letters x and y are used to denote the co-ordinates of the point which describes the locus. Ex. 1. Find the locus of the vertex of a triangle, given the base CD, and the ratio AM : NB of the parts into which the sides divide a fixed Une AB parallel to the base. Take AB and a perpendicnlar to it through A for axes, and it is necessary to express AM, NB in tenns of the co-ordinates of P. Let these co-ordi- nates be a/3, and let the co-oi'dinates of C, D be x'y', x"y', the y' of both being the same since CD is pamllel to AB. Then the equation of PC joining the points a/3, x'y' is (Art. 29) (/3 -y') x-{a- x') y = ^x' - ay'. This equation being satisfied by the x and y of every point on the line PC, is satisfied by the point J/, whose ?/ = and whose x = AM. Making then ?/ = in this equa- tion, we get jQ.r'' — ay' AM = In like manner, AN== §x" — ay' and if ^B = c, the relation AM = kBN gives (3x' - ay' _ ^ f^ fix" /5 - .'/' V (i-y' We have now expressed the conditions of the problem in terms of the co-ordinates of the point P ; and now that there is no fui-ther danger of confusion, we may replace a, /3 by X, y ; when the equation of the locus, cleared of fractions, becomes yx' - a-//' = Jc {(• ijj-y')- {>jx" - xy')]. Ex. 2. Two vertices of a triangle ABC move on fixed right lines LM, LX, and the thi-ee sides pass through three fixed points 0, P, Q which lie on a right line; find the locus of the third vertex. G 42 EXAMPLES ON THE RIGHT LINE. Take for axis of x the right line OP, containing the three fixed points, and for axis of y the line OL joining the inter- section of the two fixed lines to the point through which the base passes. Let the co-ordinates of C be a, /3, and let OL = b, OM=(t, OX=a', OP = c, OQ-c'. Then obviously the equations of LM, LN are + .'/. •f=l and ^-|--f=l. b a b The equation of CP through a/9 and P (1/ = 0, a; = c) is {a ~ c) II -I3x + (ic = 0. The co-oixlinates of -1, the intersection of this line with y - 4- - — 1 ab (a — c) + acft y\-- h {a - c) j3 b {a-c) + aft ' "' b{a-c) + ali' The co-ordinates of B are found by simjily accentuating the letters in the preceding ; b («' - c') /3 _ n'b (a — c') + a'c'ft ^2" b{a-c') + a'ft ''■ Vi 6 (a - c') -I- «'/3 Now the condition that two points a;,yi, x^^ shall lie on a right line passing through the origin, is (Art. 30) — = — . Applying this condition we have b {a - c) /3 b («' - c') /3 ab {a — c) ■\- acfi a'b (o — c') + a'c'ft We have now derived fi-om the conditions of the problem a relation which must be satisfied by a/3 the co-ordinates of C: and if we replace a, /3 by x, y we have the equation of the locus written in its ordinary form. Clearing of fractions, we have (a — c) \a'b {x — c') + n'c'y] = (a' — c') [ab (x — c) + acy], {ac' — a'c) X + 1=1, cc' (« — a') — aa' (c — c') the equation of a right line through the point L. Ex. 3. If in the last example the points P, Q lie on a right line passing not through but through L, find the locus of vertex. We shall first solve the general problem in which the points P, Q have any position. We take the fixed lines L3f, LN for axes. Let the co-ordinates of P, Q. 0, C be respectively x'y', x"y", x"'y"', a/3; and the condition which we want to express is that if we join CP, CQ and then join the points A, B, in whicJi these lines meet the axes, the line AB shall pass through 0. The equation of CP is {ft -y') x-[a- x') y = ftx'- ay'. And the intercept which it makes on the axis of x is ftx' - ay' In like manner the intercept which CQ makes on the axis of // is ay" - ftx" LA = ' LB = The equation of AB is LA^m='^ T (0 - y') y (a - x") ftx' — ay' ' ay" — ftx" EXAMPLES ON THE EIGHT LINE. 43 And the condition of the problem is that this equation shall be satisfied by the co-ordinates x"'y"'. In order then that the point C may fulfil the conditions of the problem, its co-ordinates a/3 must be connected by the relation ^"' O - ?/) y'" (a - x") _ J fix' — ay' ay ' — /3x" When this equation is cleared of fractions, it in general involves the co-ordinates a/3 in the second degi-ee. But suppose that the points x'y'^ x"y" lie on the same line passing through the origin y — mx, so that we have y' — mx', y" — mx", the equation may be wiitten x") (fi-?/')_^y"' ("■_: = 1. x' (fi — am) x" {am — /3) Clearing of fractions and replacing a, /3 by a; and y, the locus is a light line, viz., x"'x" (y — y') — y"'x' {x — a:") = x'x" {mx — y). 48. It is often convenient, instead of expressing the condi- tions of the problem directly in terras of the co-ordinates of the point whose locus we are seeking, to express them in the first instance in terms of some other lines of the figure ; we must then obtain as many relations as are necessary in order to eliminate the indeterminate quantities thus introduced, so as to have remaining a relation between the co-ordinates of the point whose locus is sought. The following Examples will sufficiently illustrate this method. Ex. 1. To find the locus of the middle points of rectangles inscribed in a given triangle. Let us take for axes CR and AB; let CR =]>, RB = .«, AR = s'. The equations of .4 C and -BC are -7= 1 and ^ + - = 1. p s p Now if we di-aw any line FS parallel to the base at a distance FK = k, we can find the absciss£e of the points F and S, in which the line FS meets AC and BC, by substituting in the equations of AC and BC the value y = I: Tlius we get from the first equation, 7=1; .'. a; or RK = — s' 2) s and from the second equation - + - = 1 ; .•. a; or RL = s { 1 - - ) . P s \ pj the abscissa of F and S, we have (by Art. 7) the abscissa of the middle Havin, I)oint of FS, viz., x This is evidently the abscissa of the middle point of the rectangle. But its ordinate is ?/ = ^k. Now we want to find a relation wliich will subsist between this ordinate and abscissa whatever k be. We Lave only then to eliminate k between these equations, by substituting in the fii-st the value of k (= 2y), derived from the second, when we have 2x=(5-o(l-^0- 44 EXAMPLES ON THE RIGHT LINE. This is the equation of the locus which we seek. It obviously represents a right Une, and Lf we examine the intercepts which it cuts off on the axes, we shall find it to be the line joining the middle point of the perpendicular CR to the middle point of the base. Ex. 2. A line is drawn parallel to the base of a triangle, and the points where it meets the sides joined to any two fixed points on the base ; to find the locus of the point of intersection of the joining lines. We shall presei-ve the same axes, &c., as in Ex. 1, and let the co-ordinates of the fixed points, Tand V, on the base, be for T {m, 0), and for V [n, 0). The equation of FT will be found to be Y il--\+viVy + kx- km = 0, and that of SV to be is f 1 j — n\y —'kx + hn - 0. Now since the point whose locus we are seeking lies on both the lines FT, SV, each of the eqviations just written expresses a relation which must be satisfied by its co- ordinates, StUl, since these equations involve h, they express relations which are only true for that particular point of the locus wliich corresponds to the case where the parallel FS is di-awn at a height k above the base. If, however, between the equa- tions, we eUminate the indetenninate k, we shall obtain a relation involving only the co-ordinates and known quantities, and which, since it must be satisfied whatever be the position of the parallel FS, wall be the required equation of the locus. In order, then, to eliminate k between the equations, put them into the form FT (a-' + m) y - k {^- y - X + 'ni\ -Q, P and SV is - n) y - ki- y + x- n\-Q ; and, eliminating k, we get for the equation of the locus i.s-n)\^^y-x + mj = (s' + m) ^% 4- a; - wj . But this 13 the equation of a right Une, since x and y are only in the first degree. Ex. 3. A line is drawn parallel to the base of a triangle, and its extremities joined transversely to those of the base ; to find the locus of the point of intersection of the joming lines. Tliis Is a particular case of the foregoing, but admits of a simple solution by choosing for axes the sides of the triangle AC and CB. Let the lengths of those lines be a, b, and let the lengths of the proportional intercepts made by the parallel be fxa, fib. Then the equations of the transversals will be -4-^ = 1 and — -f | = 1. a fxo fia Subtract one fi-om the other ; divide by the constant 1 , and we get for the equation of the locus ^ y ^ -—0 a b~ ' which we have elsewhere foimd (see p. 34) to be the equation of the bisector of the base of the triangle. Ex. 4. Given two fixed points A and B, one on each of the axes ; if A' and B' be taken on the axes so that OA' + OB' — OA + OB ; find the locus of the intersection Qt AB', A'B. EXAMPLES ON THE RIGHT LINE. 45 Let OA = a, OB — b, OA' = a + k, then from the conditions of the problem, OB' = b — k. The equations of AB', A'B are respectively - H — = 1, a b — k X y 1- i — 1 or bx + ay — ab + k {a — x) — 0, bx + (11/ — ab + k {y — b) — 0. Subtracting, we ehminate k, and find for the equation of the locus X + y = a + h. Ex. 5. If on the base of a triangle we take any poi-tion A T, and on the other side of the base another portion BS, in a fixed ratio to AT, and draw JtJT and FS parallel to a fixed line CR ; to find the locus of 0, the point of intersection of JJB and FA. Take AB and CB for axes ; let ^r= k, BE = s, AB = s', CR — p, let the fixed ratio be m, then BS will = mk ; the co-ordinates of S will be (s — mk, 0), and of T {- (s' - k), 0}. The ordinates of E and F will be found by sub- stituting these values of x in the equations ol AC and BC. We get for E, x--{s' - k), y = and for F, x = s — mk, y = - Now form the equations of the transverse Unes, and the equation of EB is {s + s'-k)y+'-rx-'-r=0, mpk mpks' and the equation of ^F is {s + s' — mk) y ■ To eliminate k, subtract one equation fi-om the other, and the result, divided by k, will be which is the equation of a right line. Ex. 6. PP' and QQ' are any two parallels to the sides of a pai-allelogi-am ; to find the locus of the intersection of the Unes PQ and P'Q'. Let us take two of the sides for om- axes, and let the lengths of the sides be n D and b, and let AQ' = m, AP = n. Then the equa- tion of PQ, joining P (0, ?i) to Q (m, b) is {b — ») X — my + mil = 0, and the equation of P'Q' joining P' {a, n) to Q' (?«, 0) is nx — {a — m) y — mil = 0. There being ttoo indeterminates m and n, we should at first suiDpose that it would not be pos- sible to eliminate them fi-om two equations. However, if we add the above equations, it will be found that both vanish together, and we get for our locus bx — ay = 0, the equation of the diagonal of the parallelogi-am. Ex, 7. Given a point and two fixed lines : di-aw any two lines through the fixed point, and join transversely the points where they meet the fixed lines ; to find the locus of intersection of the transverse Unes. A Q' 46 EXAMPLES ON THE RIGHT LINE. Take the fixed lines for axes, and let the equations of the lines through the fixed point be — + - = 1, and —, + —,= 1. The condition that these Unes should pass through the fixed point x'y' gives us or, subtracting, Now the equations of the transverse lines clearly are or, subtracting, + ^ = 1, and — + ^ = 1 ; \m m ] \n n J X y ^ , X y ^ - + -S = 1, and -; + ^ = 1 ; \m m J ^ \n n J Now fi'om this and the equation just found we can eUminate (i_i,)and(i-ll \m m J \n n J ' and we have x'l/ + y'x — 0, the equation of a right line through the origin. Ex. 8. At any point of the base of a tiiangle is drawn a line of given length, parallel to a given one, and so as to be cut in a given ratio by the base : find the locus of the intersection of the lines joining its extremities to those of the base. 49. The fundamental idea of Analytic Geometry is that every geometrical condition to be fulfilled by a point leads to an equation which must be satisfied by its co-ordinates. It is important that the beginner should quickly make himself expert in applying this idea, so as to be able to express by an equation any given geometrical condition. We add, therefore, for his further exercise some examples of loci which lead to equations of degrees higher than the first. The interpretation of such equations will be the subject of future chapters, but the method of arriving at the equations, which is all with which we are here concerned, is precisely the same as when the locus is a right line. In fact until the problem has been solved, we do not know what will be the degree of the resulting equation. The examples that follow are purposely chosen so as to admit of treatment similar to that pursued in former examples, ac- cording to the order of which they are arranged. In each of the answers given it is supposed that the same axes are chosen, and that the letters have the same meaning as in the corre- sponding previous example. EXAMPLES ON THE RIGHT LINE. 47 Ex. I. Find tlie locus of vertex of a triangle, given base and sura of squares of sides. Ans. x^ + y"^ = ^m'^ — c-, Ex. 2. Given base and m squares of one side + ii squares of the other. Ans. [m ± ft) [x'' + y") + 2 {m + w) ex + [m ± n) c^ -J'-. Ex. 3. Given base and ratio of sides. Ex. 4. Given base and product of tangents of base angles. In this and the Examples next following, the learner will use the values of the tangents of the base angles given Ex. 2, Art. 46. Ans. y^ + w^x* = m-c-. Ex. 5. Given base and vertical angle, or, in other words, base and sum of base angles. Ans. x'^ + y^ - 2cy cotC = c-. Ex. 6. Given base and difference of base angles. Ans. x^ — y^ + 2xy cotZ) = c-, Ex. 7. Given base, and that one base angle is double the other. Ans. 3x- — y"- + 2cx = c^. Ex. 8. Given base, and tan C = m tan B. A ns. m (x- + y" — c^) = 2c (c — a-). Ex. 9. PA is drawn parallel to OC, as in Ex. 4, p. 39, meeting two fixed lines in points B, B' ; and PA- is taken = PB.PB', find the locus of P. Ans. mx {in'x + n') — y {mx + ni'x + n'). Ex. 10. PA is taken the harmonic mean between AB and AB'. Ans. 2mx {m'x + n') = y {mx + m'x + w'). Ex, 11. Given vertical angle of a triangle, find the locus of the point where the base is cut in a given ratio, if the area also is given. Ans. xy = constant. Ex.12. If the base is given. . x"^ y- 2xy cobm b- Ans. — ;, + '—„— ~ Ex. 13. If the base pass through a fixed point. . mx' ny' _ m? II? mn {in + n)- ' y Ex. 14. Find the locus of P [Ex. 8, p. 40] if MN\& constant. Ans. X- + y- + 2xy coso) = constant. Ex.15. If J/.V pass through a fixed point. x' y' _. Ans. ~ H — 1. X + y cos u) y + x cos ai Ex. IG. If MN pass through a fixed point, find tlie locus of the intei-section of parallels to the axes through M and X. , ^ ^ , -V' _ i ' "' X y~ ' Ex. 17. Find the locus of P [Ex. 1, p. 41] if the line CD be not parallel to AB. Ex. 18. Given base CD of a triangle, find the locus of vertex, if the intercept AB on a given line is constant. Ans. {x'y - y'x) {y - y") - {x"y - y"x) (// - y') = c {y - y') {y - y"). 50. Prohlems whe7'e it is required to prove that a moveable right line passes through a fixed point. We have seen (Art. 40) that the line Ax^-By + C-\h{A'x + B'y + C') = <); or, what Is the same thing, [A + hA') :v + (7? + l-B') 7j + C-\-l-C' = 0, 48 EXAMPLES ON THE EIGHT LINE. where h is indeterminate, always passes through a fixed point, namely, the intersection of the lines Ax-\-Bij + C=i)^ and A'x + B'y^C = Q. Hence, if the equation of a riglit line contain an indeterminate quantity in the first degree^ the right line will always ^ass through a fixed point. Ex. 1. Given vertical angle of a triangle and the sum of the reciprocals of the sides ; the base will always pass through a fixed point. Take the sides for axes ; the equation of the base is - + 7 = 1, and we are given j-v. j-i- > ^ a b ' " the condition 111 111 therefore, equation of base is X y y , - + - - - = 1. where m is constant and a indetenninate, that is, - (^ - 2/) + - - 1 = 0, •where - is indeterminate. Hence the base must always pass through the intersection . of the two lines x — ?/ = 0, and y = 'm. Ex. 2. Given three fixed lines OA, OB, OC, meeting in a point, if the thi-ee vertices of a triangle move one on each of these Unes, and two sides of the tiiangle pass through fixed points, to prove that the remaining side passes through a fixed point. Take for axes the fixed lines OA, OB, on which the base angles move, then the line OC on which the vertex moves will have ' ' an equation of the form y = mx, and let the j\ fixed points be x'y', x"y". Now, in any position of the vertex, let its co-ordinates be x = a, and, consequently, y — ma ; then the equation of vlC is (x' — a) y — (]/ — ma) x + a (;/' — mx') = 0. Similarly, the equation of BC is {x" — a) y — {y" — ma) x + a (y" — mx") = 0. O B Now the length of the intercept OA is found by making a: = in equation AC, or a («' — mx') X — a Similarly, OB is found by making ^ = in BC, or a {y" — mx") ~ y" — ma Hence, from these intercepts, equation of AB is y" — ma x' — a X ^7> r, - y — , - a. y —mx y — mx But since a is indetenninate, and only in the first degree, this line always passes through a fixed point. The particular point is found by arranging the equation in the form y" ^' ( '"a; y ,\ „ -yr^ x ; ; y - a[ -r, r - • . + 1=0. y — mx y — mx " \y" - mx' y — mx' J EXAMPLES ON THE KIGIIT LIxNE. 4[) Hence the line passes through the intersection of the two lines 1/ — 7nx y = 0, and -;; ' - ,-- — ; +1 = 0. y — mx y — mx Ex. 3. If in the last example the line on which the vertex C moves do not pass thi-ough 0, to determine whether in any case the base will pass through a fixed point. "We retain the same axes and notation as before, with the only diiference that the equation of the line on which C moves will be ?/ = mx + n, and the co-ordinates of the vertex in any position will be ci, and ma + n. Then the equation of AC is {x' ~ a) y ~ iy' — ma — n) x + a {//' — mx') — nx' — 0. The equation of BC is [xJ' — a) y — (jj" — ma — n) x + a (y" — 7nx") — nx" = 0, ^ I _ « (/ - ^^') - "■■p' . Q^ _■ « Q/" - ^^") - "•^' _ x' — a ' y" ~ "** ~ ** The equation of AB is therefore y" — m.a — n x' — a , y'—f ~\ ^=^' * '^ W ~ '»^x") ~ «2;" « 0/' — mx') — nx' Now when this is cleared of fractions, it wiU in general contain a in the second degree, and therefore, the base will in general not pass through a fixed point ; if, however, thepobits x'y', x"y", lie in a right line (i/ = kx) passiny through 0, we may substitute in the denominators y" = hx", and y' — kx', and the equation becomes ii" — ma — n x' — a X . - y . — ,— = a (b- 77)) — 71, x X which only contains a in the first degree, and, therefore, denotes a right line passing through a fixed point. Ex. 4. If a line be such that the sum of the perpendiculars let fall on it from a nmnber of fixed points, each multiplied by a constant, may = 0, it will pass through a fixed point. Let the equation of the line be X cos a ->t y ^ma — p — 0, then the perpendicidar on it fi-om x'y' is x' cos « + y' sin « — p, and the conditions of the problem give us m' {x' cos a + y' sin a —pi) + m" {x" cos a + y" sin a — p) + m'" {x"' cos a + 7j"' sinrt — p) + ic = 0. Or, using the abbreviations S (/nx') for the sum* of the wi.r, that is, 7n'x' + 7n"x" + 7)i"'x"' + itc, and in Uke manner 2 {mij') for m'y' + m"y" + m"'y"' + ic, and 2 {in) for the sum of the ?«'s or m' + ni" + ?»'" + >fcc.. * By sum we mean the algebraic sum, for any of the (luantities w', m", A-c. may be negative. II 50 EXAMPLES ON THE RIGHT LINE. we may write the preceding equation 2 (ma-') cosrt + S ('»/) sin« — p'2 {m) = 0. Substituting in the original equation the value of j) hence obtained, we get for the equation of the moveable line icS (to) cos a + yS (m) sina — S (ma;') cos« — S (m?/') sin a = 0, or xS (m) — S {mx') + {y'S (m) — 2 {my')} tana = 0. Now as this equation involves the indeterminate tan a in the first degree, the line passes through the fixed point detennined by the equations a-2 (m) - 2 {mx') = 0, and ?/2 (?») - 2 {my') ~ 0, or, writing at full length, _ m'x' + m"x" + m"'x"' + &c. _ m'y' + m"y" + m"'y"' + &c. ~ m' + m" + m"' + &c. ' m' + m" + m'" + &c. This point has sometimes been called the centre of mean position of the given points. 51. If the equation of any line involve the co-ordinates of a certain point x'y' in the first degree, thus, {Ax' + By' + C)x+ {A'x' + B'y' + C')y+ [A"x' + B"y' + G") = ; then if the point x'y' move along a right line, the line whose equation has just been written will always pass through a fixed point. For, suppose the point always to lie on the line Lx' + Mtj+N=0, then if, by the help of this relation, we eliminate x from the given equation, the indeterminate y' will remain in it of the first degree, therefore the line will pass through a fixed point. Or, again, if the coefficients in the equation Ax -^ By + C=Oj he connected by the relation aA + bB+ cC=0 [ichere a, 5, c are constant and A^ Bj C may vary) the line represented hy this equa- tion will always pass through a fixed point. For by the help of the given relation we can eliminate G and write the equation {ex -a) A-^[cy-b)B=0, a right line passing through the point ( a; = - , 3/ = - ) . 52. Polar co-ordinates. — It is, in general, convenient to use this method, if the question be to find the locus of the extremities of lines drawn through a fixed point according to any given law. Ex. 1. A and B are two fixed points; draw through £ any hne, and let fall on it a perpendicular from A, AP ; produce AP so that the rectangle AP.AQ may be constant ; to find the locus of the point Q. EXAMPLES ON THE RIGHT LINE. 51 Take A for the pole, and AB for the fixed axis, then AQ is our radius vector, designated by p, and the angle QAB = 0, and our object is to find the relation existing between p and 0. Let us call the constant length AB — c, and from the right-angled triangle APB we have AP=c cos 0, but AP. A Q — const. = k- ; therefore pc cos 6 = k-, or p cos Q — — ; but we have seen (Art. 44) that this is the equation of a right line perpendicular to AB, and at a distance fi-om .1 = — . Ex. 2. Given the angles of a triangle ; one vei*tex A is fixed, another along a fixed right line : to find the locus of the third. Take the fixed vertex A for pole, and AP perpendicular to the fijied line for axis, then AC = p, CAP — 6. Now since the angles of ABC are given, AB is in a fixed ratio to A C (= mA C) and BAP -Q-a; hvA AP-AB cos BAP i therefore, if we call AP, a, we have mp cos [Q — a) — a, which (Art. 44) is the equation of a right Une, making an angle a with the given line, and at a distance from B moves Ex. 3. Given base and sum of sides of a triangle, if at either extremity of the base B a perpendicular be erected to the conterminous side BC; to find the locus of P the point where it meets the external bisector of vertical angle CP. Let us take the point B for oui- pole, then BP will be our radius vector p ; and let us take the base produced for om- fixed axis, then PBD — 0, and our object is to express p in terms of 0. Let us designate the sides and opposite angles of the —S^^ triangle «, i, c. A, B, C, then it is easy to see, that the angle BCP = 9Q° -^C, and fi-om the triangle / \ /I* PCB, that a — p tan-^C. Hence it is evident, that if we could express a and tan^C in terms of 6, we could A. B express p in terms of 0. Now from the triangle ABC we have 62=a* + c2-2a<;cos^, but if the given sum of sides be in, we may substitute for b, m — a ; and cos^ plauily = sin ; hence ni^ — 2am + a- = a- + c- — 2ac sin 0, and 2 (ot — c sin 0) Thus we have expressed a in tenus of and constants, and it only remains to find an expression for tan JC. isow tanJ^C = But Hence 6 (1 + cosC) " b sinC = c sin^ = c cos0 ; and b cosC= a — c coaB -a c sin 0, taniC=' c COS0 m — c sin ' 52 EXAMPLES ON THE liiailT LINE. We are now able to express p in terms of 6, for, substitute in the equation n = p tan^C the values we have found for a and tan^C, and we get pc cos I or p cos V — — — , 2 {m — c sin 0) (m — c sin 0) ' ' 2c Hence the locus is a line peipendicular to the base of the triangle at a distance from £ = 2c The student naay exercise himself with the con-esponding locus, if CF had been the internal bisector, and if the difference of sides had been given. Ex. 4. Given n fixed right lines and a fixed point ; if through this point any radius vector be drawn meeting the right lines in the points Vi, r^, 7'3...»'n, and on this a point Ji be taken such that — — = \ \ +... , to find the locus of i?. ^^ ^'-1 ^"^ ^'--^ ^'■" Let the equations of the right lines be p cos (6 — a) =iJi ; p cos (6 — /?) = po, &c. Thenit is easy to see that the equation of the locus is n cos(0 — a) cos (d — B) P Ih Pi the equation of a right line (Art. 44). This theorem is only a particular case of a general one which we shall prove aftenvards. We add, as in Art. 49, a few examples leading to equations of higher degi-ee. Ex. 5. BP is a fixed line whose equation is p cos 6 = in, and on each radius vector is taken a constant length PQ, to find the locus of Q [see fig., Ex. 1], AP is by hypothesis = x : therefore AQ = p = r, + d, which transfoimed •' cos e ' ^ f cos a ' to rectangular co-ordinates is {x — in)" {x^ + y") = cPa;-. Ex. 6. Find the locus of Q, if P describe any locus whose polar equation is given, P = (p (6). We are by hypothesis given AP in tei-ms of 6, but AP is the p of the locus — d ; we have therefore only to substitute in the given equation p — d for p. Ans. p — d= (j) (0). Ex. 7. If AQ be produced so that ylQ may be double AP. Then AP is half the p of the locus, and we must substitute half p for p in the given equation. Ex. 8. K the angle PAP were bisected and on the bisector a portion AP' be taken so that AP'^ = mAP, find the locus of P', when P describes the right line in p COS0 = m. PAB is now twice the of the locus, and therefore AP= - — zr,, and cos2o the equation of the locus is p^ cos 26 = in'. ( 53 ) ^CHAPTER IV. APPLICATION OF ABRIDGED NOTATION TO THE EQUATION OF THE EIGHT LINE. 53. We heave seen (Art. 10) that the Tine [x cosa + 7/ suia —jj) — k [x cos/3 + y sin/? — ^/) = deuotes a line passing through the intersection of the lines X cosa 4 y sina — p = 0, a; cos/3 + y siny3 — p' = 0. We shall often find it convenient to use abbreviations for these quantities. Let us call X cosa + y sina— p, a; x cos/S + y sin/3— p', /3. Then the theorem just stated may be more briefly expressed, the equation a — Jl/3 = denotes a line passing through the intersec- tion of the two lines denoted by a = 0, ^ = 0. AYe shall for brevity call these the lines a, /3, and their point of intersection the point a^. We shall, too, have occasion often to use abbre- viations for the equations of lines in the form Ax + By -\-C=0. We shall in these cases make use of Roman letters, reserving the letters of the Greek alphabet to intimate that the equation is in the form X cos oi-\- y sin a —p = 0. 54. We proceed to examine the meaning of the coefficient h in the equation a — /i'/3 = 0. We saw (Art. o\) that the quantity a (that is, a; cosa + ?/ sina— ^>) denotes the length of the perpendicular PA let fall from any point xy^ on the line OA (which Ave suppose represented by a). Similarly, that yS is the length of the perpendicular PB from the point .ry, on the line OB^ represented by /S. Hence the equation a - /v/3 = asserts, that if from any point of the locus represented by it, perpen- diculars be let fall on the lines OA^ OB^ the ratio of these per- pendiculars (that is, PA : PB) will be constant, and = k. Ilcncc 54 THE RIGHT LINE — ABRIDGED NOTATION. the locus represented by a-k/3^0 is a right line through 0, and J _PA _ s'mFOA ''~m' ""'' ~BmPOB' It follows from the conventions concerning signs (Art. 34) that a + ^•/3 = denotes a right line dividing externaUy the angle A OB into parts such that . — ti^7^ = A-. It is, of course, as- {iinJrOii ' sumed in what we have said that the perpendiculars PA^ PB are those which we agree to consider positive; those on the opposite sides of a, /3 being regarded as negative. Ex. 1. To express in this notation the proof that the three bisectors of the angles of a triangle meet in a point. The equations of the three bisectors are obviously (see Arts. 35, 54) a — fi~Q, /3 — y = 0, y — a = 0, which, added together, vanish identically. Ex. 2. Any two of the external bisectors of the angles of a triangle meet on the third internal bisector. Attending to the convention about signs, it is easy to see that the equations of two external bisectors are a + /3 = 0, a + y = 0, and subtractmg one from the other we get /3 — y = 0, the equation of the third internal bisector. Ex. 3, The thi-ee perjjendiculars of a triangle meet in a point. Let the angles opposite to the sides a, /3, y be A, B, C respectively. Then since the peiiJendicular divides any angle of the triangle into parts, which are the com- plements of the remaining two angles, therefore (by Art. 54) the equations of the pei-pendiculars are aCOBA — (i cosB — 0, j8 cosi? — y cosC= 0, y cosC— a cos^ = 0, which obviously meet in a point. Ex. 4. The three bisectors of the sides of a triangle meet in a point. The ratio of the perpendiculars on the sides from the point where the bisector meets the base plainly is sin A : sin B. Hence the equations of the three bisectors are asinA ~ (i smB = Q, /3 sin ^ — y sinC = 0, y sinC — a sin^ = 0. Ex. 5. The lengths of the sides of a quadrilateral are a, b, c, d; find the equation of the line joining middle points of diagonals. Ans. aa — bfS + cy — dS — ; for this line evidently passes through the inter- section of «a — 1(3, and cy — dS; but, by the last example, these are the bisectors of the base of two triangles having one diagonal for their common base. In like manner aa — do, i/3 — cy intersect in the middle point of the other diagonal. Ex. 6. To foi-m the equation of a perpendicvilar to the base of a triangle at its extremity. A7is. a + y cosB = 0. Ex. 7. If there be two triangles such that the peiTpendiculars from the vertices of one on the sides of the other meet in a point, then, vice versa, the perpendiculars from the vertices of the second on the sides of the first will meet in a point. Let the sides be a, (i, y, a', /3', y', and let us denote by {ajS) the angle between a and /J. Then the equation of the perpendicular from uj3 on y' is a cos (fiy') — (i cos (ay') = 0, from (iy on a' is ft cos {ya!) — y cos {fia') = 0, from yn on ft is y cos {aft') — a cos (y/3') = 0. THE RIGHT LINE — ABRIDGED NOTATION. 55 The condition that these should meet in a point is found by eUminating ft between the first two, and examining whether the resulting equation coincides with the thii'd. It is COs(a^') cosd^y') C03(7a') = cos(a'/3) cos(/3'7) cos(7'a). But the symmetiy of this equation shows that this is also the condition that the perpendiculars fi-om the vertices of the second triangle on the sides of the first should meet in a point. 55. The lines a — l-^ = 0, and ha- (3 — 0, are plainly such that one makes the same angle with the line a which the other makes with the line /3, and are therefore equally inclined to the bisector a- ^. Ex. If through the vertices of a triangle there be drawn any three lines meeting in a point, the thi-ee lines drawn through the same angles, equally iaclined to the bisectors of the angles, wUl also meet in a point. Let the sides of the triangle be a, ft, y, and let the equations of the first three lines be la — mft = 0, mft — My = 0, ny — la = 0, which, by the piinciple of Art. 41, are the equations of thi-ee lines meeting in a point, and which obviously pass through the points aft, fty, and ya, Now, from this Article, the equations of the second three lines will be « /^ = 0, ^-51=0 and5:-" = 0, which (by Art. 41) must also meet in a point, 5Q. The reader is probably already acquainted Mnth the fol- lowing fundamental geometrical theorem: — '"'' If a pencil of four right lines meeting in a i^oint he intersected hy any transverse right line in the four points A, P, P', P, tJieii . AP.FB . the ratio —t^ft, — ^:ri=: ts constant, no matter how AP .PB ' the transverse line he drawn.'''' This ratio is called the anharmonic ratio of the pencil. In O fact, let the perpendicular from on the transverse line =]) : then ;p.AP= OA. OP.s'niA OP (both being double the area of the triangle AOP); p.P'B=OP'.OBsmP'OB; p.AP'=OA.OF smAOF ] 2).PB = OP. OB. sin POB', hence /. AP. P'B = OA . OP. OF. OB. sin A OP. sinP' OB ; /. AP' .PB = OA. OP' . OP. OB. sin A OP' . sin POB ; A P.FB _ sin^ OP. sm FOB . AP\PB ~ sin A OF . sin POP ' but the latter is a constant quantity, independent of the position of the transverse line. 56 THE RICUiT LINE — ABRIDGED NOTATION. 57. If a — A;/3 = 0, a — 7//3 = 0, be the equations of two lines, h then p will be the anliarmonic ratio of the pencil formed by the four lines a, /3, a - /i'/3, a — h'^^ for (Art. 54) sinylOP sin^OP' therefore smFOB' s'mF'OB^ k &'mAOP.smP'OB Ic sin ^ OP', sin POP' but this is the anharmouie ratio of the pencil. The pencil is a harmonic pencil when y-, = — 1, for then the angle A OB is divided internally and externally into parts whose sines are in the same ratio. Hence we have the important theo- rem, two lines whose equations are a — ^•yS = 0, a + k^ = Oj form with a, /3 a harmonic pencil. 58. In general the anharmonic ratio of four lines a — ^/3, 7 1 r> ^ . (n— T) (m — k) T^ 1 , ., , a — tp, a— riiB, a — nB is ; — - , , , — ^ . h or let the pencil be ' [n — m) [I — k) cut by any parallel to /3 in the four points iT, P, M^ N^ and the . . NL.MK ^ . ratio IS ^^-,_— _^^. Jiut since p has the same value for each of these four points, the perpen- diculars from these points on a are (by virtue of the equations of the O ^ lines) proportional to h^ /, «?, w ; and AK^ AL^ AM^ AN are evi- dently proportional to these perpendiculars ; hence NL is propor- tional to w — ? ; MK to m - k ; NM to n — m ; and LK to ^ - h. 59. The theorems of the last two articles are true of lines represented in the form P—kP\ P —IP\ &c., where P, P' de- note ax + hy ^ c, a'x + h'y + c', &c. For we can bring P to the form X cosa+3/ sina— ^^ by dividing by a certain factor. The equations therefore P-kP' = 0, P— ?P' = 0, &c. are equivalent to equations of the form a — kpl3 = 0^ a — ?p/3 = 0, &c., where p is the ratio of the factors by which P and P' must be divided in order to bring them to the forms a, /?. But the expressions THE KIGHT LINE — ABRIDGED NOTATION. 57 for anharmonic ratio are unaltered when we substitute for /i, /, ?/?, 7i ; kp^ ?/?, mp^ np. It is worthy of remark, that since ihe expressions for an- harmonic ratio only involve the coefficients /c, Z, ?«, n, it follows that if we have a system of any number of lines passing through a point, F—kP'j F—IP\ &c. ; and a second system of lines passing through another point, Q-lcQ\ Q — IQ\ &c., the line P—kP' being said to correspond to the line Q — kQ\ &c. ; then the anharmonic ratio of any four lines of the one system is equal to that of the four corresponding lines of the other system. We shall hereafter often have occasion to speak of such systems of lines, which are called homographic systems. 60. Given three lines a, /3, y^ forming a triangle;* the equatvm of any right lii^e^ ax + ht/ -\- c = 0^ can he thrown into the form la + mfi + «7 = 0. Write at full length for a, /S, 7 the quantities which they represent, and leu + m^ + n<y becomes [l cosa + m cos/3 + n cosy) x-\- [l sina-f m sin/3 + M sin7)y — [lp + mp'-\-np") = 0. This will be identical with the equation of the given line, if we have / cosa + ?H cos/8-f n cos7 = a, ? sina4- »i sin/3+ n sln7 = />, Ip) -f wp + np' = — <5, and we can evidently detei'mine /, w, ??, so as to satisfy these three equations. The following examples will illustrate the principle that it is possible to express the equations of all the lines of any figure in terms of any three a = 0, /8 = 0, 7 = 0. Ex. 1. To deduce analj'ticaily the haraionic properties of a complete qiuid- rilateral. (See figure, next page). Let the equation of AC be a = 0; of AB, /3 = ; of BD, y = 0; of AD. la — mfi = ; and of BC, m^ — ny = 0. Then we are able to express in temis of these quantities the equations of all the other lines of the figure. * We say " forming a triangle," for if the lines a, /3, y meet in a point, la + mfi+ny ■must always denote a line passing through the same point, since any values of the co-ordinates which make a, fi, y separately = 0, must make la + mfi + ny = 0. I 58 THE RIGHT LINE — ABRIDGED NOTATION. For instance, the equation of CD is la — W(/3 + ny = 0, for it is the equation of a right line passing through the intersection of la — mfi and y, that is, the point D, and of a and ?«/} — ny, that is, the point C. Again, la — ny = Q is the equa- tion of OE, for it passes through ay or E, and it also passes through the intersection of AD and BC, since it is = {la — wi/3) + (»«/3 — ny). EF joins the point ay to the point [la — wjS + ny, /3), and its equation wnll be foimd to be la + ny = 0. From Art. 57 it appears, that the four lines EA, EO, EB, and EF foi-m a harmonic pencil, for their equations have been shown to be a = 0, y = 0, and la + ny = 0. Again, the equation of FO, which joins the points {la + ny, /3) and {la—mft, m/3-My) is la — 2?»/J + 7iy — 0. Hence (Art. 57) the four lines FE, FC, FO, and FB are a harmonic pencil, for their equations are la — 7)i(3 + ny = 0, /3 = 0, and la — to/3 + ny ± ni^ = 0. Again, OC, OE, OD, OF are a harmonic pencil, for their equations are la — mp = 0, TO/3 — wy = 0, and la — mfi ± (/h/3 — ny) = 0. Ex. 2. To discuss the properties of the system of lines fonned by di-awing through the angles of a triangle three lines meeting in a point. Let the equation of A B he y = ; of ^C, /3 = ; ol BC, a - ; and let the lines OA, OB, OC, meeting in a point, be 7«|3 — ny, ny — la, la — ?»/3 (see Art. 55). Now we can form the equa- tions of all the other lines in the figure. For example, the equation of EF \^ m-P + ny — la = 0, since it passes through the points {ft, ny — la) or E, and (y, 7nft — la) or F. In like manner, the equation of DF is la — mft + ?iy = 0, and of DE la + mji — ny = 0. Now we can prove, that the three points L, M, N are all in one right line, whose equation is la + mft + ny = 0, for this line passes through the points (7a + mft — ny, y) or N ; {la — mft + wy, ft) or M; and {mft + ny — la, a) or L. The equation of CN is la + mft - 0, for this is evidently a line through {a, ft) or C, and it also passes through N, since it = {la + nift + ny) — ny. THE RIGHT LINE — ABRIDGED NOTATION. 59 Hence BN is cut liannomcally, for the equations of the four lines CN, CA, CF, CB are a = 0, /3 = 0, la- ?«/3 = 0, la + m(i = 0. The equations of this example can be applied to many particular cases of fre- quent occurrence. Thus (see Ex. 3, p. 54) the equation of the line joining the feet of two pei-pendiculars of a triangle is a cos^l + /3 cos£ — y cosC= 0; while a cos,4 4- /3 cos5 + y cosC passes through the intersections with the opposite sides of the triangle, of the lines joining the feet of the perpendiculars. In like manner a sin^ + /3 sin5 — y sinC represents the line joining the middle points of two sides, &c. Ex. 3. Two triangles are said to be homohgous, when the intersections of the coiTesponding sides lie on the same right line called the axis of homology: prove that the lines joining the corresponding vertices meet in a point [called the centre of hmnologg] . Let the sides of the fii-st triangle be a, (3, y: and let the line on wliich the coiTe- sponding sides meet be la + m(i + ny : then the equation of a line through the intersection of this with a must be of the form I'a + mji + ny — 0, and similarly those of the other two sides of the second tiiangle ai'e la + 7«'/3 + ny = 0, la + mft + n'y — 0. But subtracting successively each of the last three equations from another, we get for the equations of the lines joining corresponding vertices {I —V) a — {m — m') (i, (jii — m') (3 — {n — n'] y, (« — «') y =: (^ — l') a, which obviously meet in a potut, 61. To find the condition that two lines la -f 7»/3 4 ny, I'a + m'^ + n'<y may he mutually 'perpendicular. Write the equations at full length as in Art. 60, and apply the criterion of Art. 25, Cor. 2 [A A + BB' = 0), when we find W + 7nm' + nn + [mn -\- m'n) cos(/3 - 7) + {nV -f n'l) cos (7 - a) + {hn + I'm) cos (a - yS) = 0. Now since ^ and 7 are the angles made with the axis of x by the perpendiculars on the lines yS, 7 ; /3 — <y is the angle between those pei-pendiculars, which again is equal or supplemental to the angle between the lines themselves. If w^e suppose the origin to be within the triangle, and A, B, C to be the angles of the triangle, /3 - 7 is the supplement of A. The condition for perpendicularity therefore is H'+mm'+nn'-{mn'+m'n) cosA-{nl'+n'l) cosB-{hn'+Tm) cos (7=0. As a particular case of the above, the condition that la+m^ + ny may be perpendicular to 7 is n = m cos A + I cosB. In like manner wc find the length of the perpendicular from .r'l/' 60 THE RIGHT LINE — AB1UD«ED NOTATION. on lcf.-\-m^-\-n<y. Write the equation at full length and apply the formula of Art. 34, when, if we write x cosa + y sina — p = a', &c., the result is la 4- m^' + n'y' '^{P + m^ + n^ — 2mn cos A — 2nl cosB— 2lm cos G) ' Kx. 1. To find the equaiioo of a perpendicular to y through its extremity. The equation is of the farm la + wy — 0. And the condition of this article gives n = l cosB, as in Ex. 6, p. 54. Ex. 2. To find the equation of a perpendicular to y through its middle point. The middle point being the intersection of y with a sia ^4 — /3 sin B, the ecjuation of any Hne through it is of the form a sin .4. — /3 sin B + ny = 0, and tlie condition of this article gives n = shi{A — B). Ex. 3. The tlu-ee perpendiculaxs at middle points of sides meet in a point. For eliminating a, /3, y in turn, between a smA - (3 smB + y am{A - B) - 0, (SsmB - y sinC + a sin (B - C) = 0, we get for the lines joining to the three vertices the intersection of two perpen- diculars = = — -- • and the symmetry of the equations proves that the cos.4 cosii cosC J J 1 f tliird perpendicular passes thi-ough the same point. The equations of the perpen- diculars vanish when multiplied by sin^C, sin^J, s'm-B, and added together, Ex. 4, Find, by Art. 25, expressions for the sine, cosine, and tangent of the angle between la + 7«/3 + «y, I'a + m'^ + n'y. Ex. 5. Prove that a cos A + /3 cosB + y cos C is perpendicular to a sin.4 cos.4 sin(B — C) + /3 sinjS cosB sin(C— A) + y sinCcosCsin(4 — B), Ex. 6. Find the equation of a line through the point a'/3'y' perpendicular to the line y. Alls, a (Ji' + y' cos A) — /3 (a' + y' coaB) + y ifi' cosB — a cos J). 62. We hare seen that we can express the equation of any- right line in the form la + m^ + ^7 = 0, and so solve any problem by a set of equations expressed in terms of a, yS, 7, without any direct mention of x and y. This suggests a new way of looking at the principle laid down in Art. 60. Instead of regarding a as a mere abbreviation for the quantity x cosa + ?/ sina— p, we may look upon it as simply denoting the length of the pei'pen- dicular from a point on the line a. We may imagine a system of trilmear co-ordinates in which the position of a point is de- fined by its distances from three fixed lines, and in Avhich the position of any right line is defined by a homogeneous equation between these distances of the form la + wz/3 + ^7 = 0. The advantage of trilinear co-ordinates is, that whereas in THE RIGHT LINE — ABRIDGED NOTATION. 61 Cartesian (or x and ?/) co-ordinates the utmost simplification we can introduce is by choosing two of the most remarkable lines in the figure for axes of co-ordinates, we can in trilinear co-ordi- nates obtain still more simple expressions by choosing three of the most remarkable lines for the lines of reference a, /3, 7. The reader will compare the brevity of the expressions in Art. 54 with those corresponding In Chap. 11. 63. The perpendiculars from any point on a, yS, 7 are connected by the relation ao^ + h^ -\- cy = il/, where a, h, c are the sides, and M double the area, of the triangle of reference. For evidently oa, h^, cy are respectively double the areas of the triangles OBC, OCA^ GAB. The reader may suppose that this is only true if the point be taken ivifhin the triangle ; but he is to remember that if the point were on the other side of any of the lines of reference (a), we must give a negative sign to that perpendicular, and the quantity aa + h^ + cy would then be double OCA +OAB-OBC, that is, still = double the area of the triangle. Since sin^ is proportional to «, it is plain that a sin^ 4 /3 sin i?+ 7 sin (7 is also constant, a theorem which may otherwise be proved by writing a, /3, 7 at full length as in Art. 60, multiplying by sin(/5-7), sin (7 -a), sin(a-/8), re- spectively, and adding, when the coefficients of x and 1/ vanish, and the sum is therefore constant. The theorem of this article enables us always to use homo- geneous equations in a, /3, 7, for if we arc given such an equa- tion as a = 3, we can throw it into the homogeneous form J/a = 3(«a+ J/3 + C7). 64. To exjjress in trilinear co-ordinates the equation of the parallel to a given line la + m/B + ny. In Cartesian co-ordinates two lines Ax+Bi/+Cj Ax+Bi/+C' are parallel If their equations differ only by a constant. It follows then that la + w/3 + ny + k (a slnvl + /3 s'mB+y sin C) = denotes a line parallel to la 4 rn/3 + ny, since the two equations differ only by a quantity which has been just proved to be constant. 62 THE RIGHT LINE — ABRIDGED NOTATION. In tlic same case Ax -^ Bi/ -{ C + {Ax -{■ By + C) denotes a line also parallel to the two given lines and half way between them: hence if two equations i^=0, P' = are so connected that P— P' = constant, then P-1- P' denotes a parallel to P and P' half way between them. Ex. 1, To find the equation of a parallel to the base of a triangle drawn through the vertex. Ans. a sin A + /3 sin B — 0. For this, obviously, is a line through a/8 ; and writing the equation m the form y sinC — (a sin^ + /3 sinS + y sinC) = 0, it appears that it differs only bj' a constant from y = 0. We see, also, that the parallel aainA + ^ sinU, and the bisector of the base a sin A — /3 sini? form a harmonic pencil with a, /3 (Art. 57). Ex. 2. The line joining the middle points of sides of a triangle is parallel to the base. Its equation (see Ex. 2, p. 58) is a smA + (3 sinB — y sinC = 0, or 2y sinC = a siaA + /3 sin^ + y sinC. Ex. 3. The line «a — 6/3 + cy — dS (see Ex. 5. Art. 54) passes through the middle ix>int of the line joining ay, /3o. For {aa + cy) + (i/3 + dd) is constant, being t^vice the area of the quadrilateral ; hence «a + cy, b^ + do are parallel, and {aa + cy) — {bji + dd) is also parallel and half-way between them. It therefore bisects the line joining (ay) which is a point on the first line, to (fio) which is a j)oint on the second. 65. To write in the form la -f on/S + 7iy = the equation of the line joining two given jyoints xy\ x"y". Let a', as before, denote the quantity a;' cos a + 3/' sin a— ^?. Then the condition that the co-ordinates xy' shall satisfy the equation la. + m^ + «7 = may be written la! +mfi' +')iy' =0. Similarly we have la" + 9»/3" + ny" = 0. Solving for - , — , from these two equations, and substituting in the given form, we obtain for the equation of the line joining the two points a (/Sy - 7'/3") + /3 (7'a" - 7'V) + 7 (a'yS" - a"/3') = 0. It is to be observed that the equations in trilinear co-ordi- nates being homogeneous, we are not concerned with the actual lengths of the perpendiculars from any point on the lines of reference, but only with their mutual ratios. Thus the preceding equation is not altered if we write pa', p/S', ^7', for a', /8', 7'. Accordingly if a point be given as ^liC intersection of the lines •^ = — = - , we may take L m, n as the trilinear co-ordinates I m n^ -^ ' ' THE EIGHT LINE — ABRIDGED NOTATION. 63 of that point. For let p be the common value of these fractions, and the actual lengths of the perpendiculars on a, )8, 7 are Ip^ 7np, np where p is given by the equation alp + bmp + cnp = J/, but, as has been just proved, we do not need to determine p. Thus, in applying the equation of this article, we may take for the co-ordinates of intersection of bisectors of sides, sin5sin(7, sin C sin ^, sin^ sin^; of intersection of perpendiculars, cos^ cosC, cos (7 cos^, cos^ cos 5; of centre of inscribed circle 1, 1, 1 ; of centre of circumscribing circle cos-4, cos-B, cosO, &c. Ex. 1. Find the equation of the Ime joining intersections of perpendiculars, and of bisectors of sides (see Art. 61, Ex. 5). Ans. a siaA cos A sin (5 -C) + /3 sLn^cosi? sin(C-^) + y sinC cosCsin(.l-i?) = 0. Ex. 2. Find eqixation of line joining centres of inscribed and circumscribing circles. Ans. a (cos5 - cosC) + (i (cosC - cos.l) + y (cos.-l - cos^) = 0. 66. It is proved, as In Art. 7, that the length of the per- pendicular on a from the point which divides in the ratio I : m the line joining two points whose perpendiculars are a', a" is — j ■ . Consequently the co-ordinates of the point dividing in the ratio I : m the line joining a'ySy, a"(3"'y" are la + ?«a", ?/3' + m/3", ly 4 my". It is otherwise evident that this point lies on the line joining the given points, for if a'/SY, a"^"y" both satisfy the equation of a line Aa + B^ + Cy = 0, so will also la' + ma'\ &c. It follows hence without difficulty that la! - ma", &c. is the fourth harmonic to la + ma\ a', a" : that the anharmonic ratio of a — ha!\ a! — la\ a — wa", a — no!' is ; ~ \ , ~ J : and also that, given two s^^stems of points on [n - m) [l-lc]^ ' ^ "^ two right lines a! — ha!', a — la!\ &c., a' — ha!"', a!" — la", &c., these systems are homograpMc, the anharmonic ratio of any four points on one line being equal to that of the four corresponding points on the other. Ex. The intersection of peiioendiculai-s, of bisectors of sides, and the centre of circumscribing circle lie on a right line. For the co-ordinates of these points arc cos^ cosC, kc, sini? sinC, <S:c., and cos J, &c. But the last set of co-oi-dmates may be written sini? siuC — cos^ cos (7, iSrc. The point whose co-ordinates are cos(i? - C), cos(C- .1), cos(.l - B) e\-identlj' lies on the same right Hue and is a fourth harmonic to the tlirce preceding. It wUl be found hereafter that this is the centre of the circle thi-ough the middle points of the sides. 64 THE RIGHT LINE — ABRIDGED NOTATION. 67. To examine lohat line is denoted by the equation a sin^ + /3 sini? + 7 sin (7=0. This equation is included in the general form of an equation of a right line, but we have seen (Art. 63) that the left-hand member is constant, and never =0. Let us return, however, to the general equation of the right line Ax + By + C = 0. We saw that the intercepts cut off on the axes are — -7 j ~" "S 5 consequently, the smaller A and B become, the greater will be the intercepts on the axes, and, therefore, the more remote the line re- presented. Let A and B be both = 0, then the intercepts become Infinite, and the line is altogether situated at an infinite distance from the origin. Now it was proved (Art. 63) that the equation under consideration is equivalent to Oic -j- Oj/ + C = 0, and though it cannot be satisfied by any finite values of the co-ordinates, it may by infinite values, since the product of nothing by infinity may be finite. It appears then that a sinJ. + /8 sini?+7 sin (7 denotes a right line situated altogether at an infinite distance from the origin ; and that the equation of an infinitely distant right line, in Cartesian co-ordinates, is O.cc-f 0.y + C=0. We shall, for shortness, commonly cite the latter equation in the less accurate form (7=0. 68. We saw (Art. 64) that a line parallel to the line a = has an equation of the form a + (7=0. Now the last Article shows that this is only an additional illustration of the principle of Art. 40. For, a parallel to a may be considered as intersecting it at an infinite distance, but (Art. 40) an equation of the form a + (7=0 represents a line through the intersection of the lines a = 0, (7=0, or (Art. 67) through the intersection of the line a with the line at infinity. 69. We have to add that Cartesian co-ordinates are only a particular case of trilinear. There appears, at first sight, to be an essential difference between them, since trilinear equations are always homogeneous, while we are accustomed to speak of Cartesian equations as containing an absolute term, terms of the first degree, terms of the second degree, &c. A little reflection, however, will show that this difference is only apparent, and THE rJGHT LINE — ABRIDGED NOTATION. 05 that Cartesian equations must be equally homogeneous in reality, though not in form. The equation a? = 3, for example, must mean that the line x is equal to three feet or three inches, or, in short, to three times some linear unit ; the equation x7/ = d must mean that the rectangle x?/ is equal to nine square feet or square inches, or to nine squares of some linear unit ; and so on. If we wish to have our equation homogeneous in form as well as in reality, we may denote our linear unit by 2, and write the equation of the right line Ax-]- B7/ + Cz = 0. Comparing this with the equation and remembering (Art. 67) that when a line is at an infinite dis- tance its equation takes the form s = 0, we learn that equations in Cartesian co-ordinates are only the particular form assumed hy trilinear equatio?is when two of the lines of reference are tohat are called the co-ordinate axes^ while the third is at an infinite distance. 70. We wish in conclusion to give a brief account of what is meant by systems of tangential co-ordinates^ in which the position of a right line is expressed by co-ordinates, and that of a point by an equation. In this volume we limit ourselves to what is not so much a new system of co-ordinates as a new way of speaking of the equations already in use. If the equation (Cartesian or trilinear) of any line be \x + fiy ■\- vz = 0, then evidently, if X, /A, V be known, the position of the line is known : and wc may call these three quantities (or rather their mutual ratios with which only we are concerned) the co-ordinates of the right line. If the line pass through a fixed point x'y'z\ the relation must be fulfilled x'\-\ y'iJb + z'v = 0^ if therefore wc are given any equation connecting the co-ordinates of a line, of the form a\ ■\-hiJt,-\-cv = 0, this denotes that the line passes througli tlio fixed point (a, J, c), (see Art. 51), and the given equation may be called the equation of that point. Further, we may use abbreviations for the equations of points, and may denote by a, /3 the quantities x'X + y'fju + sV, x'\ + ^'V + ^"^ '■> then it is evident that h + m^ = is the equation of a point dividing in K 66 THE RIGHT LINE — ABRIDGED NOTATION. a given ratio the line joining the points a, /3j that loL = m^j m,^ = W7, W7 = la are the equations of three points which lie on a right line ; that a + /•"yS, a — h^ denote two points harmonically conjugate with regard to a, /S, &c. We content ourselves here with indicating analogies which we shall hereafter develope more fully; for we shall have occasion to show that theorems concerning points are so connected with theorems concerning lines, that when either is known the other can be inferred, and often that the same equations differently interpreted will prove either theorem. Theorems so connected are called reciprocal theorems. Ex. Interpret in tangential co-ordinates the equations used Art. 60, Ex. 2. Let a, /3, y denote the points A, B, C; m/3 — ny, ny — la, la — mp, the points L, M, N ; then ??i/3 + ny — la, ny + la — m/3, la + 7n/3 — ny denote the vertices of the triangle formed by LA, 3IB, NC; and la + »?/3 + ny denotes a point in wliich meet the lines joining the vertices of this new triangle to the corresponding vertices of the original : »i/3 + ny, ny + la, la + to/3 denote D, E, F. It is easy hence to see the points in the figure which are harmonically conjugate. ( 67 ) CHAPTER V. EQUATIONS ABOVE THE FIRST DEGREE REPRESENTING RIGHT LINES. 71. Before proceeding to speak of the curves represented by equations above the first degree, we shall examine some cases where these equations represent rigid lines. If we take any number of equations X = 0, M= 0, N= 0, &c. and multiply them together, the compound equation L3IN &c.=0 will represent the aggregate of all the lines represented by its factors ; for it will be satisfied by the values of the co-ordinates which make any of its factors = 0. Conversely, if an equation of any degree can he resolved into others of lower degrees^ it will repre- sent the aggregate of all the loci represented hy its different factors. If, then, an equation of the n^ degree can be resolved into n factors of the first degree, it will represent n right lines. 72. A homogeneous equation of the n " degree in x and y denotes n right lines passing through the origin. Let the equation be X - px y + qx y —occ. ...+ ty =0. Divide by y\ and we get (i) -4) -^(i) ;*^-=;- Let a, Z», c &c. be the n roots of this equation, then it is re- solvable into the factors and the original equation is therefore resolvable into the factors {x — ay) {x — by) [x - cy) &c. = 0. It accordingly represents the 7i right lines x — ay — 0^ &c., all of which pass through the origin. Thus, then, in particular, the homogeneous equation x^ —pxy -\- qy'^ = ^ 68 EQUATIONS REPRESENTING RIGHT LINES. represents the two right lines x — ay = 0^ x — hy = 0^ where a and 1) are the two roots of tlie quadratic (iy-©--- It is proved, in like manner, that the equation [x - ay -p [x - ay-' [y-h) + q[x- ay'" {y - Z»)\ . .+ t [y - ly = denotes n right lines passing through the point (a, h). Ex. 1. What locus is representee! by the equation xi/ — 0? Ans. The two axes ; since the equation is satisfied by either of the suppositions a; = 0, 2/ = 0. Ex. 2. What locus is represented by a;^ — ^/^ = ? Ans. The bisectors of the angles between the axes, x ±y = (see Art. 35). Ex. 3. What locus is represented by x- — 5xy + 6i/- = 0? Atis. x—2y = 0, x — 3i/ = 0. Ex. 4. What locus -is represented by x- — Ixy sec 6 + «/^ = ? Ans. X = ?/ tan (45° + J;^). Ex. 5. What lines are represented by x- — 2xy tan 6 — y- = 0? Ex. G. What lines are represented by x^ — 6x-y + llxy"^ — 6^' = ? 73. Let us examine more minutely the three cases of the solution of the equation x^ —pxy + qif = 0, according as its roots are real and unequal, real and equal, or both imaginary. The first case presents no difficulty : a and h are the tangents of the angles which the lines make with the axis of y (the axes being supposed rectangular), ^j is therefore the sum of those tangents, and q their product. In the second case, when a = h., it was once usual among geometers to say tluit the equation represented but one right line [x — ay = 0). We shall find, however, many advantages in making the language of geometry correspond exactly to that of algebra, and as we do not say that the equation above has only one root, but that it has two equal roots, so we shall not say that it represents only one line, but that it represents two coin- cident right lines. Thirdly, let the roots be both imaginary. In this case no real co-ordinates can be found to satisfy the equation, except the co- ordinates of the origin a; = 0, y = Oj hence it was usual to say that in this case the equation did not represent right lines, but was the equation of the origin. Now this language appears to us very objectionable, for we saw (Art. 14) that iivo equations EQUATIONS KEPRESENTINU RIGHT LINES. 69 are required to determine any point, lience wc are unwilling to acknowledge any single equation as the equation of a point. Moreover, we have been hitherto accustomed to find that two dif event equations always had different geometrical significations, but here we should have innumerable equations, all purporting to be the equation of the same point ; for it Is obviously Immaterial what the values of ^:> and q are, provided only that they give Ima- ginary values for the roots, that is to say, provided that p' be less than 4(7. We think It, therefore, much preferable to make our language correspond exactly to the language of algebra; and as we do not say that the equation above has no roots when p'' Is less than 4^', but that It has two imaginary roots, so we shall not say that, In this case, it represents no right lines, but that It represents two imaginary right lines. In short the equa- tion x^ — jyxy -{ qy"^ = Q being ahvays reducible to the form [x - ay) [x — hy) — 0, we shall always say that It represents two right lines draw^n through the origin ; but when a and h are real, we shall say that these lines are real ; when a and h are equal, that the lines coincide ; and when a and h are imaginary, that the lines are imaginary. It may seem to the student a matter of In- difference which mode of speaking we adopt ; we shall find, how- ever, as we proceed, that we should lose sight of many Important analogies by refusing to adopt the language here recommended. Similar remarks apply to the equation Ax' + Bxy -}- Cy' = 0, which can be reduced to the form x' —pxy + qy" = 0, by dividing by the coefficient of x\ This equation will always represent two right Hues through the origin ; these lines will be real if B' — iA C be positive, as at once appears from solving the equa- tion ; they will coincide if B' — ^AC-()] and they will be ima- ginary if B' — ^AC be negative. So, again, the same language is used if we meet with equal or Imaginary roots In the solution of the general homogeneous equation of the ;<"' degree. 74. To find the angle contained hy the lines rejjresented hy the equation x^ —pxy 4- qy'' = 0. Let this equation be equivalent to {x~ ay) {x — hy) = 0, then a — h the tangent of the angle between the lines is (Art. 25) , , 70 EQUATIONS REPRESENTING RIGHT LINES. bat the product of the roots of the given equation = q^ and their difference =\/(p^ — 42'). Hence ^ l+q If the equation had been given in the form Ax' + Bxy + Of = 0, it will be found that , , >J{B'-iAC) '^"■^^ A + C ■ CoR. The lines will cut at right angles, or tan^ will become Infinite, if ^^ = — 1 in the first case, or if A-}-C=0 in the second. Ex. Find the angle between the liaes X- + xi/ — &y" = 0. Ans. 45°. x^ — 2xy sec Q + i/^d. Ans. Q. *If the axes be oblique, we should find, in like manner, tan = — -. — ~ — ^ . ^ A+C- B cosoi 75. To find the equation ivMcIi will represent the lines bisecting the angles between the lines represented hy the equation Ax^ + Bxy + Cy'^^. Let these lines be a; - «?/ = 0, a; — Z*?/ = ; let the equation of the bisector \>q, x — [xy = 0, and we seek to determine //.. Now (Art. 18) y[A is the tangent of the angle made by this bisector with the axis of ?/, and it is plain that this angle is half the sum of the angles made with this axis by the lines themselves. Equating, therefore, tangent of twice this angle to tangent of sum, we get 2/A a + h ^ l-fju' ^ 1-ab^ but, from the theory of equations, B , G therefore ^ J 1-fi' A-C A —C or /a' - 2 — ^ /^ - 1 = 0. EQUATIONS REPRESENTING RIGHT LINES. 71 This gives us a quadratic to determine /u, one of whose roots will be the tangent of the angle made with the axis of y by the mternal bisector of the angle between the lines, and the other the tangent of the angle made by the external bisector. We can find the combined equation of both lines by substituting in the last quadratic for yw. its value = - , and we get and the form of this equation shows that the bisectors cut each other at right angles (Art. 74). The student may also obtain this equation by forming (Art. 35) the equations of the internal and external bisectors of the angle between the lines x — ay = Oj x — hy = 0, and multi- plying them together, when he will have (pc-^yf _ [x-byY l + a' ~ 1 + b' ' and then clearing of fractions, and substituting for a + b, and ab their values in terms of -4, B, C, the equation akeady found is obtained. 76. We have seen that an equation of the second degree may represent two right lines ; but such an equation in general cannot be resolved into the product of two factors of the first degree, unless its coefficients fulfil a certain relation, which can be most easily found as follows. Let the general equation of the second degree be written ax'^ + 2hxy + by- -f 2qx + 2fy + c = 0,t or ax^ + 2{hy+g]x + by- + 2fy-^c = 0. * It is remarkable that the roots of this last equation will always be real, even if the roots of the equation Ax- + Bxy + Cy- = be imaginaiy, wliich leads to the curious result, that a pair of imaginary lines may have a pair of real lines bisecting the angle between them. It is the existence of such relations between real and imaginary lines which makes the consideration of the latter profitable. •f- It might seem more natural to wi-ite tliis equation ax- + bxy + cy- + dx + ey +/= 0, but as it is desirable that the equation should be written with the same lettci-s all through the book, I have decided on using, from the fii-st, the form which will hereafter be found most convenient and symmetrical. It will appear hereafter 72 EQUATIONS REPRESENTING RIGHT LINES. Solving this equation for a?, wc get ax = - (Jiy+fj) ± ^J[[]l' - ah) if + 2 ilig - of) y + [cf - «c)|. In order that this may be capable of being reduced to the form x — my + n^ it is necessary that the quantity under the radical should be a perfect square, in which case the equation would denote two right lines according to the different signs we give the radical. But the condition that the radical should be a perfect square is (A' - ah) (/ - ac) = [hg - afY- Expanding, and dividing by «, we obtain the required condition, v'z. ahc + 2^/i - af - hg'' - cW = 0.* Ex. 1. Verify that the following equation represents right lines, and find the lines : X- - bxy + 4?/2 + a; + 2y - 2 = 0. Ans, Solvuig for x as in the text, the lines are found to be a;-y-l = 0, a;-4^ + 2 = 0. Ex. 2. Verify that the following equation represents right lines : {ax + fy~ r-y = {a- + ^- r-) {x"- + / - r^). Ex. 3. What lines are represented by the equation x- ~ xy + y^ — X — y + 1 — d'} Ans. The imaginaiy lines x + dy + 6- = 0, x + 6-^ + 6 = 0, where is one of the imaginary cube roots of 1 . Ex. 4. Determine li, so that the following equation may represent right Unes : x^ + 2hxy + y- — bx — ly + & —Q. A71S. Substituting these values of the coefficients in the general condition, we get for /* the quadratic, 12/j- — 3ok + 25 = 0, whose roots are f and |. *77. The method used in the preceding Article, though the most simple in the case of the equation of the second degree, is not applicable to equations of higher degrees ; we therefore give another solution of the same problem. It is required to ascertain that this equation is intimately connected with the homogeneous equation in three variables, which may be most symmetrically written ax- + by- + C22 + 2fyz + 2ffzx + 2hxy = 0. The form in the text is derived from this by making 2=1. The coeflacient 2 is aifixed to certain terms, because formulas connected with the equation which we shall have occasion to use, thus become simpler and more easy to be remembered. * If the coefficients /, g, h in the equation had been written without numerical coefficients, this condition would have been 4.abc +fgh - af- - hg"- - cli^ - 0. EQUATIONS REPRESENTINa RIGHT LINES. 73 whether the given equation of the second degree can be Identical with the product of the equations of two right lines [ax + ^i/-l) [ax + ^'y - 1) = 0. Multiply out this product, and equate the coefficient of each term to the corresponding coefficient in the general equation of the second degree, having previously divided the latter by c, so as to make the absolute term in each equation = 1. We thus obtain five equations, viz. aa' = -, a + a' = -^, /9)S' = -, yS + /3' = -^, a;9' + a';3 = -; c c c c c from which eliminating the four unknown quantities a, a', ^, /3' we obtain the required condition. The first four of the equa- tions at once give us two quadratics for determining a, a ; /3, ;Q' ; which indeed might have been also obtained from the considera- tion that these quantities are the reciprocals of the intercepts made by the lines on the axes ; and that the intercepts made by the locus on the axes are found (by making alternately a; = 0, y = 0, in the general equation) from the equations ax"" H- 2^a; + c = 0, hif + 2fij + c =^ 0. We can now complete the elimination by solving the quadratics, substituting in the fifth equation and clearing of radicals ; or we may proceed more simply as follows: Since nothing shows whether the root a of the first quadratic is to be combined with the root /3 or /S' of the second, it is plain that — may have either of the values a/3' + a'/3 or a/3 4 a'/3'. This is also evident geometrically, since if the locus meet the axes in the points iy, L' ; M^ M' ; it is plain that if it represent right lines at all, these must be either the pair LM^ IJM\ or else LM\ i'J/, whose equations are [<xx\-^y-\){a:x^^'xj-\)=^^ or (a.c+/3'^-l)(a'a:-Fy8^-l) = 0. The sum then of the two quantities a.^' + a'/3, ayS + a'/3' = (a + a')(/3 + ^') = ^^ and their product 74 EQUATIONS EEPIIESENTING EIGHT LINES. Hence - is given by the quadratic T^_fy 2A af + Iq' - ale _ which, cleared of fractions, is the condition abeady obtained. Ex. To determine A so that x- + 2hxy + y'' — 5x — 7i/ + 6 = may represent right lines (see Ex. 4, p. 72). The intercepts on the axes are given by the equations x--5x + 6 = 0, y--7ij + Q-0, whose roots are x — 2, a; = 3; «/=l, «/ = 6. Forming, then, the equation of the Hnes joinmg the points so foimd, we see that if the equation represent right lines, it must be of one or other of the forms {x + 2y- 2) (2j; + y - 6) = 0, (a; + 3y - 3) (3.c + y - 6) = 0, whence, miUtiplying out, h is determined. *78. To find how many conditions must he satisfied in order that the general equation of the n^ degree may represent right lines. We proceed as in the last Article ; we compare the general equation, having first by division made the absolute term = 1, with the product of the 7i right lines {ax + ^y-l) {ax + I3'y-1) {ax + ^"y _ 1) &c. = 0. Let the number of terms in the general equation be N ; then from a comparison of coefficients we obtain N— 1 equations (the absolute term being already the same in both) ; 2w of these equations are employed in determining the In unknown quan- tities a, a', &c., whose values being substituted in the remaining equations afford N— 1 — 2?i conditions. Now if we write the general equation A + Bx + Cy + Dx' + Exy 4- Fy' + Gx^ + Hx'y + Kxtf + Ly^ + &c. = 0, it is plain that the number of terms is the sum of the arithmetic SGI*1GS i^=l + 2 + 3+...(«+l)=i?i±ii|i±^; hence N- I = -^ — — ' ; N- I - 2w = \ ^ ' . 1.2 ' 1.2 ( 75 ) CHAPTER VI. THE CIRCLE, 79. Before proceeding to the discussion of the general equa- tion of the second degree, it seems desirable that we should show in the simple case of the circle, how all the properties of a curve may be deduced from its equation, without assuming any previous acquaintance with the geometrical theory. The equation, to rectangular axes, of the circle whose centre is the point (a^) and radius is ?•, has already (Art. 17) been, found to be Two particular cases of this equation deserve attention, as occurring frequently in practice. Let the centre be the origin, then a = 0, /3 = 0, and the equation is x' + y' = r\ Let the axis of a? be a diameter, and the axis of y a per- pendicular at its extremity, then a = r, /S = 0, and the equation becomes x'^ + ^^ = 2r.r. 80. It will be observed that the equation of the circle, to rectangular axes, does not contain the term xy^ and that the co- efficients of x^ and y^ are equal. The general equation therefore ax" + 2hxy + ly' + 2()x + 2/j/ + c = 0, cannot represent a circle, unless we have h = 0, and a = h. Any equation of the second degree wliich fulfils these two conditions may be reduced to the form [x - of +{y — /3)" = r^, by a process corresponding to that used in the solution of quadratic equations. If the common coefficient of x^ and y- be not ab-eady unity, by division make it so ; then having put the terms containing x and y on the left-hand side of the equation, and the constant term on the right, complete the squares by adding to both sides the sum of the squares of half the coefficients of x and y. 76 THE CIRCLE. Ex. Reduce to the form {x — a)- + [y — /3)^ = »-^, the equations x^ + y'-2x-4:y = 20; Zx^ + 3/ -bx-ly+\=0. Ans. {x - 1)2 + (y - 2)2 = 25 ; {x - 1)- + {y - lY - f i ; a^d the co-ordinates of the centre and the radius are (1, 2) and 5 in the first case; (f, f) and ^4(62) in the second. If we treat in like manner the equation a{x^ + f) + 27a; + 2/7/ + c = 0, we get f a; + ^) + f^/ + - ) = — ^ ~ ^^ * a \ tt a -q - f and the co-ordinates of the centre arc — ^ , -- , and the radius a ^ a I£ g'^ -i-f'^ is less than ac, the radius of the circle is imaginary, and the equation being equivalent to {x — a)^ + {]/ — ^Y + *"^ = ^j cannot be satisfied by any real values of x and y. If g^ +f'^ = aCj the radius is nothing, and the equation being equivalent to (a?— a)^+ (?/- /9)^ = 0, can be satisfied by no co- ordinates save those of the point (a/3). In this case then the equation used to be called the equation of that point, but for the reason stated (Art. 73) we prefer to call it the equation of an injimtely small circle having that point for centre. We have seen (Art. 73) that it may also be considered as the equation of the two imaginary lines {x — d)±[y — IB) \J[ — 1) passing through the point (a/3). So in like manner the equation a;'' + ?/'' = may be regarded as the equation of an infinitely small circle having the origin for centre, or else of the two imaginary lines x±y\j{—\). 81. The equation of the circle to oblique axes Is not often used. It is found by expressing (Art. 5), that the distance of any point from the centre is equal to the radius ; and is (a; - ay + 2 (a.^ - a) [y - /3) cos w + (?/ - /S)' = r'. If we compare this with the general equation, we see that the latter cannot represent a circle unless a = 5, and ]i = a costw. When these conditions are fulfilled, we find by comparison of coefficients that the co-ordinates of the centre and the radius are given by the equations Q 'f C a + /8cosa> = --, /S -f- a cos w = --'- , a'''+/3^+ 2a/9 cosw - ?'^= - . THE CIRCLE. 77 Since a, yS are determined from the first two equations wliicli do not contain c, we learn that two circles will he concentric if their equations differ only in the constant term. Again, if c = 0, the origin is on the curve. For then the equation is satisfied by the ccT-ordinatcs of the origin a? = 0, ?/ = 0. The same argument proves that if an equation of any degree loant the ahsolute term^ the curve represented j)asses through the origin. 82. To find the co-ordinates of the points in which a given right line x cos a 4 ?/ sina =7;, meets a given circle x^ + ?/ = r\ Equating to each other the values of y found from the two equations, we get for determining a;, the equation p — x cosa ,, ., ,, ^—. ■=V(*-"-a;'), sma or, reducing x^ — 2])x cos a -\-p^ — 7-^ sin'"'a — ; hence, x =]) cos a + sin a s/[r^ - p^), and, in like manner, y=p sina + cosa \/{r^—p'). (The reader may satisfy himself, by substituting these values in the given equations, that the — in the value of y corresponds to the + in the value of cc, and vice versa.) Since we obtained a quadratic to determine a?, and since every quadratic has two roots, real or imaginary, we must, in order to make our language conform to the language of algebra, assert that every line meets a circle in two points, real or imaginary. Thus, when^:> is greater than r, that is to say, when the distance of the line from the centre is greater than the radius, the line, geometrically considered, does not meet the circle ; yet we have seen that analysis furnishes definite imaginary values for the co-ordinates of intersection. Instead then of saying that the line meets the circle in no points, we shall say that it meets it in two imaginary points, just as we do not say that the corre- sponding quadratic has no roots, but that it has two imaginary roots. By an imaginary point we mean nothing more than a point, one or both of whose co-ordinates are imaginary. It Is a purely analytical conception, which we do not attempt to repre- sent geometrically ; just as when we find imaginary values for roots of an equation, we do not try to attach an arithmetical 78 THE CIRCLE. meaning to our result. And attention to these imaginary points is necessary to preserve generality in our reasonings, for we sLall presently meet with many cases in which the line joining two imaginary points Is real, and enjoys all the geome- trical properties of the corresponding line in the case where the points are real. 83. When p = r, it is evident geometrically that the line touches the circle, and our analysis points to the same conclu- sion, since the two values of x in this case become equal^ as do likewise the two values of y. Consequently the points answer- ing to these two values, which are in general diiferent, will in this case coincide. We shall therefore, not say that the tangent meets the circle in only one point, but rather that it meets it in two coincident points; just as we do not say that the corre- sponding quadratic has only one root, but rather that it has two equal roots. And in general we define the tangent to any curve as the line joinijrg two indefinitely near joints on that curve. We can in like manner find a quadratic to determine the points where the line Ax + By + C meets a circle given by the general equation. When this quadratic has equal roots, the line is a tangent. Ex. 1. Find the co-ordinates of intersection of x- + y" = G5 ; 3x + y = 25-. Ans. (7, 4) and (8, 1). Ex. 2. Find intersections of (x - c)- + (y - 2c)2 = 25c2 ; ix + 3i/ = 35c. Ans. The Une touches at the point (5c, 5c). Ex. 3. When wUl y - mx + b touch x"^ + y" = r^ ? Ans. When V^ = r"^ (1 + vi^). Ex. 4. When will a line through the origin, y — mx, touch « (a;- + 2xy cos w + y'^) + 2gx + 2fy + c? The points of meeting are given by the equation a (1 + 2;?i cos w + m-) x- + 2 {g +fiii) x + c = 0, wHch will have eqixal roots when (g +/m)" = ac (1 + 2m cosw + m-). We have thus a q^uadratic for determining m, Ex. 5. Fmd the tangents from the origin to x'' + y^ - Gx — 2y + 8 = 0. Ans. X — y = 0, x + 7y = 0. 84. AVhen seeking to determine the position of a circle re- presented by a given equation, it is often as convenient to do so by finding the intercepts which it makes on the axes, as by finding its centre and radius. For a circle is known when THE CIRCLE. 79 three points on it arc known ; the determination, therefore, of the four points where the circle meets the axes serves com- pletely to fix its position. By making alternately y = 0, x = in the general equation of the circle, we find that the points in which it meets the axes are determined by the quadratics ax^ + 2(jix + = 0, af + 2fi/ + c = 0. The axis of x will be a tangent when the first quadratic has equal roots, that is, when g~ = ac ; and the axis of ?/ when /"'' = ac. Conversely, if it be required to find the equation of a circle making intercepts \, X' on the axis of x, we may take a = 1 , and we must have 2g = — {X + \')^ c = XV. If it make intercepts /A, fjb' on the axis of y, we must have 2/=— (//. + /i,'), c = /j,/i'. Thus we see that we must have W — fjLfi' (Euc. iii. 36). Ex, 1. Find the points where the axes are cut hj x- + y^ — ox — 7y + 6 = 0. Ans, a; = 3, a;=:2; y = 6, y = 1. Ex. 2. What is the equation of the circle -which touches the axes at distances from the origin =a? Ans. x- + y- — '2ax — 2ay + o^ =: 0. Ex. 3. Find the equation of a circle, the axes being a tangent, and any line through the point of contact. Here we have \, X', fj. all = ; and it is easy to see from the figm-e that /u' = 2r sin w, the equation therefore is X- + 2xy cos (1} + y- — 2ry sia cd = 0. 85. To find the equation of the tangent ai the j^oint x'y to a given circle. The tangent having been defined (Art. 83) as the line joining two indefinitely near points on the curve, its equation will be found by first forming the equation of the line joining any two points {;x'y\ x'y") on the curve, and then making x—x and y —y" in that equation. To apply this to the circle : first, let the centre be the origin, and, therefore, the equation of the circle x^ + y^ = r'. The equation of the line joining any two points {x'y') and {x"y") is (Art. 29) ^ ^^ y-y' ^ y -y^ . x — x x — x" ' now if we were to make In this equation y =y" and x =-x"^ the right-hand member would become indeterminate. The cause of this is, that we have not yet introduced the condition, that the two points (a;'?/', xi'y") are on the circle. By the help of this condition we shall be able to write the equation in a form which 80 THE CIRCLE. will not become Indeterminate when the two points are made to coincide. For, since r' = x'^ + y"" = X" + y"'\ we have a?" - x'"^ = y""" - y""^ I ft t . tr and, therefore, ^, — ^ = ; 7, • X —X 3/ + 2/ Hence the equation of the chord becomes y-y' ^ x' + x" ^ x — x y' + y" ' And if we iioio make x = x" and y' —y'\ we find for the equation of the tangent, y-y ^ _ ^' x — x ?/' * or, reducing, and remembering that x^ + y'' = r\ we get finally XX + yy = r''. Otherwise thus :* The equation of the chord joining two points on a circle may be written, [x - x') [x - x") + [y- y') {y ~ y") = a;^ + / - r\ For this is the equation of a right line, since the terms a? + y^ on each side destroy each other 5 and if we make x = x\ y = ?/', the left-hand side vanishes identically, and the right-hand side vanishes, since the point x'y is on the circle. In like manner the equation is satisfied by the co-ordinates x"y". This then is the equation of a chord ; and the equation of the tangent got by making x = x\ y = y'\ is ^a,-x'y + {y-y'y = x'' + f-7'\ which reduced, gives, as before, xx +yy' = r\ If we were now to transform the equations to a new origin, so that the co-ordinates of the centre should become a, y8, we must substitute (Art. 8) a; — a, a;' — a, ?/ — /3, ?/'— /S, for x^ ic', ?/, y'j respectively : the equation of the circle would become {x-aY + {y-^y = r% and that of the tangent [x-a){x-a) + {y-^){y'-^) = r'', a form easily remembered, from its similarity to the equation of the circle. * This method is due to Mr, Bumside. THE CIKCLli:, 81 Cor. The tangent is perpendicular to the radius, for the equation of the radius, the centre being origin, is easily seen to be XT/ —2/'x = 0] but this (Art. 32) is perpendicular to xx-^yy'—r\ 86. The method used in the last article may be applied to the'general equation* ax^ + "llixij + Inf + Igx + 2/?/ + c = 0. The equation of the chord joining two points on the curve may be written a [x - X) ix - x) + 2A [x - x) [y - y") ■^- b {y - y) [y - y") = ax^ 4- 2Jixy + hy^ + 'igx + 2fy-\-c. For the equation represents a right line, the terms above the first degree destroying each other ; and, as before, it is evidently satisfied by the two points on the curve x'y\ x"tj". Putting x' = x\ y" = ?/', we get the equation of the tangent a{x-x'Y+2h {x-x')[y-y') +h {y-y'Y=ax;'+2Jixy+hy'-{2r/x-^2fy+c ; or, expanding, 2ax'x + 2h [x'y + y'x) + 2hy'y + 2gx + 2fi/ + c = ax"' + 2hx'y' + hy'\ Add to both sides 2gx' + 2fy' + c, and the right-hand side will vanish, because x'y' satisfies the equation of the curve. Thus the equation of the tangent becomes ax'x + h {x'y + y'x) + hy'y +g {x + x) +f{y +y') + c = 0. This equation will be more easily remembered if we compare it with the equation of the curve, when we see that it is derived from it by writing x'x and y'y for x' and y\ x'y + y'x for 2xy, and x +x^ y' -{■ y for 2x and 2y. Ex. 1. Find the equations of the tangents to the curves xi/ - c-, and y- = ;w. Ans. x'y + y'x - 2c- and 2^/ =2> (^ + x'). Ex. 2. Find the tangent at the point (5, 4) to (x - 2)- + (;/ - 3)= = 10. Alls. 3x + y = 10. Ex. 3. What is the equation of the chord joming the points x'y', x"y" on the circle a;^ + ,^2 = ,.2 ? Ans. {x' + x") x + (y' + y") y - »•- + x'x" + y'y". Ex. 4. Find the condition that Ax + By + C^d should touch [x-af + ijj- py- = r-. Ans. " "]" ' ^ = ,' ; since the periDcndicular on the line from a/3 is equal to r. 4{A^ + B^) * Of coiurse when this equation represents a circle we must have b = a, h = a cosoi ; but since the process is the same, whether or not b or h have these particular values, we prefer in this and one or two similar cases to obtain at once formulse which will afterwards be required in our discuspion of the srcneral equation of the second detrree. 82 THE CIKCLE. 87. To draw a tangent to the circle x^ + y'' = ?-^, from any point x'y. Let the point of contact be x"y"^ then since, by hypo- tliesis, the co-ordinates x'y' satisfy the equation of the tangent at a;"?/", we have the condition xx' -\-y'y" = r\ And since x"y" is on the circle, we have also x"'-{-y"'' = r\ These two conditions are sufficient to determine the co-ordinates a;", y". Solving the equations, we get „ rV' ± ry V [x^ -\- y'^ — f^) » f'^y' + '^x \J {x^ + y''^ — r^) x^ + y' ' "^ x^ + y- Hence, from every point may be drawn tico tangents to a circle. These tangents will be real when x'^ + y''^ is>r'"', or the point outside the circle ; they will be imaginary when x'^ + y''^ is < r'*, or the point inside the circle ; and they will coincide when x'^ + ^'^ = ?*■'} or the point on the circle. 88. We have seen that the co-ordinates of the points of contact are found by solving for x and y from the equations xx -1- yy' = r^'j x^ -{■ y^ = r\ Now the geometrical meaning of these equations evidently is, that these points are the intersections of the circle a? -\-y^ = r^ with the right line xx -I- yy = r^. This last then is the equation of the right line joining the points of contact of tangents from the point xy \ as may also be verified by forming the equation of the line joining the two points whose co-ordinates were found in the last article.* We see, then, that whether the tangents from xy be real or imaginary, the line joining their points of contact will be the real line XX -f yy = r^, which we shall call the polar of x'y' with regard to the circle. This line is evidently perpendicular to the * In general the equation of the tangent to any curve expresses a relation con- necting the co-ordinates of any point on the tangent, with the co-ordinates of the point of contact. If we are given a point on the tangent and required to find the point of contact, we have only to accentuate the co-ordinates of the point which is supposed to be known, and remove the accents from those of the point of contact, when we have the equation of a curve on which that point must he, and whose intersection with the given curve determines the point of contact. Thus if the equation of the tangent to a curve at any point x'y' be xx'^ + yy'^ = r^, the points of contact of tangents drawn from any point x'lj' must lie on the curve x'x"^ + y'y- — r'. It is only in the case of curves of the second degi-ee that the equation which deter- mines the points of contact is similar in form to the equation of the tangent. THE CIRCLE. 88 line {x'y—y'x = 0)^ which joins x'y' \.o the centre; and its dis- tance from the centre (Art. 23) is ,, ,„ ^, . Hence, the polar of ^ ,J[x^ + y') ' ' any point P is constructed geometrically by joining it to the centre" C, taking on the joining line a point J/, such that CM.CP=r'^^ and erecting a perpendicular to CP at M. AVc see, also, that the equation of the polar is similar in form to that of the tangent, only that in the former case the point x'y' is not supposed to be necessarily on the circle : if, however, x'y' be on the circle, then its polar is the tangent at that point. 89. To find the equation of the polar of x'y' xcith regard to the curve ^_^2 _j_ 2/^^^ ^ j^2 ^ 2^_^ + 2/?/ + c = 0. We have seen (Art. 86) that the equation of the tangent is axx -I- h {x'y + y'x) + hy'y + g [x + x) +f{y + y') + c = 0. This expresses a relation between the co-ordinates xy of any point on the tangent, and those of the point of contact x'y'. We indicate that the former co-ordinates are known and the latter unknown, by accentuating the former, and removing the accents from the latter co-ordinates. But the equation, being sym- metrical with respect to the co-ordinates xy, xy ^ is unchanged by this operation. The equation then written above, (which when xy is a point on the curve, represents the tangent at that point), when xy is not on the curve, represents a line on which lie the points of contact of tangents real or imaginary from x'y'. If we substitute xy for xy in the equation of the polar, we get the same result as if we made the same substitution in the equation of the curve. This result then vanishes when xy is on the curve. Hence the polar of a point passes through that point only when the point is on the curve, in which case the polar is the tangent. Cor. The polar of the origin is gx +fy + c = 0. Ex. 1, Findthepolarof (4,4) 'w-ith regard to (.r-l)2+Cy-2)=tl3. Ans. Zx+2y=-20. Ex.2. Find the polar of (4, 5) with regard to a;2+/-3x-4^=8. Ans. 5j; + 6y=48. Ex, 3. Find the pole of Ja; + % + C = vnth. regai-d to x- + f/- r^. Ans. ( -^ , -^ j , as appears from comparing the given equation with 31-' + yi)' = r-. Ex. 4. Find the pole of 3j; + 4y = 7 with regard to x- + y- = 14. Ans. (6, 8). Ex. 5. Find the pole of 2.r + 3^ = 6 wth regard to (.r - 1)- + (y - 2)- = 12. Ans. (-11. - 16). 84 THE CIRCLE. 90. To find the length of the tangent drawn from any j)oiy\t to the circle {x- af + [y - ^f - r' = 0. The square of the distance of any point from the centre = {a--af+{y-fiy- and since this square exceeds the square of the tangent by the square of the radius, the square of the tangent from any point is found by substituting the co-ordinates of that point for x and y in the first member of the equation of the circle {x~aY + {y-^y-r^ = (). Since the general equation to rectangular co-ordinates a {x^ + if) + 2gx + '2fy + c = 0, when divided by «, is (Art. 80) equivalent to one of the form {x-aY+{y-^y-r' = 0, we learn that the square of the tangent to a circle whose equa- tion is given in its most general form is found by dividing by \ the coefficient of x\ and then substituting in the equation the co-ordinates of the given point. The square of the tangent from the origin is found by making x and 3/ = 0, and is, therefore, = the absolute term in the equation of the circle, divided by a. The same reasoning is applicable if the axes be oblique. *91. To find the ratio in ivhicJi the line joining two given 2>oints x'y'^ ^v" 1 ^'-^ ^^^^ % ^ given circle. We proceed precisely as in Art. 42. The co-ordinates of any point on the line must (Art. 7) be of the form Ix" + tnx ly" + my l + m ' 1+ 7n Substituting these values in the equation of the circle x' + f - r' = 0, and arranging, we have, to determine the ratio Z:w, the quadratic V {x'"' + y"' - r') + 2lm {x'x" + y'y" - r') + m"" {x"' + y" - r'^) = 0. The values of l:m being determined from this equation, we have at once the co-ordinates of the points where the right line meets the circle. The symmetry of the equation makes this method sometimes more convenient than that used (Art. 82). THE CIRCLE. 85 If x"y" He on the polar of x'_y', we have xx ■\-y'ij" — r^ — () (Art. 88), and the factors of the preceding equation must be of the form l-\-[jbm^ l—fim ; the line joining x'y'j x'y" is therefore cut internally and externally in the same ratio, and we deduce the well-known theorem, any line drawn through a point is cut har- monically hy the pointy the circle^ and the p)olar of the point. *92. To find the equation of the tangents from a given point to a given circle. We have already (Art. 87) found the co-ordinates of the points of contact; substituting, therefore, these values in tlie equa- tion xx" + yy" - r' = 0, we have for the equation of one tangent r (xx + yy' — x^ — y''^) + [xy — yx) -sJix"^ + y"^ — r^) = 0, and for that of the other, r [xx + yy — x"^ — y''^) — [xy — yx) \J{x''^ + y"^ - r^) = 0. These two equations multiplied together give the equation of the pair of tangents in a form free from radicals. The preceding article enables us, however, to obtain this equation in a still more simple form. For the equation which determines I : m will have equal roots if the line joining x'y\ x'y" touch the given circle ; if then x'y" be any point on either of the tangents through xy ^ its co-ordinates must satisfy the condition [x'' + y"^ - r^) [s^ + f - r) = {xx + yy - r'')^ This, therefore, is the equation of the pair of tangents through the point xrj . It is not difficult to prove that this equation is identical with that obtained by the method first indicated. The process used in this and the preceding article is equally applicable to the general equation. We find in precisely the same way that I : m is determined from the quadratic f [ax'' + Ihx'y" + ly"" + "Igx + ify" + c) + ilm [axx + h [x'y" + x"y') + hy'y" -Vg («' + '■x^") +f{l/'+!/") + <^| + vi' {ax^ + 2hx'y' + hy'- + 2gx' + 2fy' + c) = ; from which we infer, as before, that when x"y" lies on the polar of x'y' the line joining these points is cut harmonically ; and also that the equation of the pair of tangents from x'y' is [ax"' + 2hx'y' + hy"' + 2gx -f 2fy + c) {ax^ + 2hxy + hy' -\- 2gx + 2fy + c) = [ax'x -\r h {.vy + xy') + hyy' + g [x + x!) +f{y + y') + <?]"• 86 THE CIRCLE. 93. To find tlie equation of a circle passing through three given points. We have only to write down the general equation «- + if + 2gx + 2/?/ + c = 0, and tlien substituting in it, successively, the co-ordinates of each of the given points, we have three equations to determine the three unknown quantities </, f c. We might also obtain the equation by determining the co-ordinates of the centre and the radius, as in Ex. 5, p. 4. Ex. l.'Find the circle through (2, 3), (4, 5). (6, 1). An.s. (a- - VT + Cy - §)- = ■¥' (see p. 4). Ex. 2. Find the circle through the origin and through (2, 3) and (3, 4). Here c = 0, and we have li + Ag + G/= 0, 25 + 6<7 + 8/= 0, whence 2y = - 23, 2/= 11. Ex. 3. Taking the same axes as in Art. 48, Ex. 1, find the equation of the circle through the origin and thi'ough the middle pouits of sides ; and show that it also passes through the middle j)oint of base. Ans. 1p {x" 4- y") —p (•< — «') x — (p- + .<■«') y = 0. *94. To exjyress the equation of the circle through three 2>oints xy\ x'y\ x"y" in terms of the co-ordinates of those iwints. We have to substitute in x' + y' + '2gx + 2/?/ + c = 0, the values of g^ f c derived from [x!^ +y'^)+ 'igx + 2fy' -I- c = 0, {x'"' 4- y'"' ) + 2gx!' -f- 2fy" ^- c = 0, [x""' + y""') + "igx" 4- 2/y ' + c = 0. The result of thus eliminating g^f c between these four equa- tions will be found to be ix" +/ ){x {y"-y"')-^x {y"-y')+x"'{y' - y" )] -{x" +y"')[x [y"'-y )^x"{y -y")-\-x {y" -y'")] + K'+/')K'(3/ -y') + x {y -u"') + x' {y"'-y )} -[:^'- + f-)[x iy' -y")^x {f - y )+x'{y -y )]=0, as may be seen by multiplying each of the four equations by the quantities which multiply (x^ + y^) &c. in the last written equa- tion, and adding them together, when the quantities multiply- ing g^f c will be found to vanish identically. If it were required to find the condition that four points should lie on a circle, we have only to write x^^ y^ for x and y THE CIHCLE. 87 ill the last equation. It is easy to see that the following is the geometrical interpretation of the resulting condition. If yl, 7i, C\ D be any four points on a circle, and any fifth point taken arbitrarily, and if we denote by BCD the area of the triangle BCD, &c., then OA'.BCD + C\ABD = 0B\ ACD + OD\ABC. 95. We shall conclude this chapter by showing how to find the polar equation of a circle. We may either obtain it by substituting for x^ p cos^, and for ?/, p s'md (Art. 12), in either of the equations of the circle already given, a {x' + tf) + Igx + 2/y + c = 0, or {x - a)' + (y - /S)' = r\ or else we may find it independently, from the definition of the circle, as follows : Let be the pole, C the centre of the circle, and OG the fixed axis; let the distance OC=d, and let OP be any radius vector, and, therefore, =/?, and the angle PO (7=^, then we have PC^=0F'+0C''-20P.0C co^POC, that is, r^ = p^ + d'^ - 2pd cos 6, or p^ - 2dp cos 6 + d''- r' = 0. This, therefore, is the polar equation of the circle. If the fixed axis did not coincide with OC, but made with it any angle a, the equation would be, as in Art. 44, p- - 2dp cos (^ - a) + d'' - r' = 0. If we suppose the pole on the circle, the equation will take a simpler form, for then ;• = d, and the equation will be reduced to p = 2r cos^, a result which we might have also obtained at once geometrically from the property that the angle in a semicircle is right ; or else by substituting for x and ?/ their polar values in the equation (Art. 79) x' + / = 2rx. ( 83 ) CHAPTER Vll. THEOKEMS AND EXAMPLES ON THE CIRCLE. 96. Having in the last chapter shown how to form the equations of the circle, and of the niost remarkable lines related to it, we proceed in this chapter to illustrate these equations by examples, and to apply them to the establishment of some of the principal properties of the circle. We recommend the reader first to refer to the answers to the examples of Art. 49, to examine in each case whether the equation represents a circle, and if so to determine its position either (Art. 80) by finding the co-ordinates of the centre and the radius, or (Art. 84) by finding the points where the circle meets the axes. We add a few more examples of circular loci. Ex. 1. Given base and vei-tical angle, find the locus of vertex, the axes having any position. Let the co-ordinates of the extremities of base be x'y', x"y" . Let the equation of one side be y-y' = »i (•» - «'). then the equation of the other side, making with this the angle C, will be (Art. 33) (1 + m tanC) (.(/ — y") = [in — tanC) {x — x"). Eliminatmg m, the equation of the locus is tanC [{y-y'){y- y") + (x - x') {x - x")} ■+ x{y' - y") -y{x' - x") + x'y" - y'x" = 0. If C be a light angle, the equations of the sides aie y-y' - 111 [x - x') ; m {y - y") + (a; - x") = 0, and that of the locus (// - y') (y - y") + {x- x') (x - x") = o. Ex. 2. Given base and vertical angle, find the locus of the intereection of perpen- diculars of the triangle. The equations of the perpendiculars to the sides are m (j/ — y") + (x — x") — 0, (to — tanC) (^ — y) + (1 + »» tanC) {x — x') = 0. Eliminating m, the equation of the locus is tanC {{y - y') (y - y") + {x - x'} {x - x")} -x{y' - y") - y {x' - x") + x'y" - y'x" ; an equation which only differs from that of the last article by the sign of tanC, and which is therefore the locus we should have found for the vertex had we been given the same base and a vertical angle equal to the supplement of the given one. Ex. 3. Given any number of points, to find locus of a point such that m' times square of its distance from the first -1- m" times square of its distance from the second + &c. = a constant ; or (adopting the notation used in Ex 4, p. 49) such that S {mr^) may be constant. THEOREMS AND EXAMl'l.ES ON THE CIRCLE. 89 The square of the distance of any point x)j from x'y' is (a; — x'Y + {y — y'f. Multiply this by »«', and add it to the corresponding terms found by expressing the distance of the pomt xy from the other points x"i/\ &c. If we adopt the notation of p. 49, we may write for the equation of the locus, 2 (m) a;2 + S (m) y" - 2S {mx') a; - 22 {my') y + 2 {vix'^) + 2 {jny'^) = C. Hence the locus will be a circle, the co-ordinates of whose centre wiU be 2(maO ^,_^{my') 2 \rn) ' -^ - 2 (m) ' that is to say, the centre will be the point which, in p. 50, was called the centre of mean position of the given points. If we investigate the value of the radius of this circle, we shall find BT-'S. {m) - 2 (m7-2) - 2 {mp% where 2 {mry = C= sum of in times square of distance of each of the given points from any point on the circle, and 2 {mp") — sum of m times square of distance of each point from the centre of mean position. Ex. 4. Find the locus of a point 0, such that if parallels be drawn through it to the three sides of a triangle, meeting them in points B, C; C, A'; A", B" ; the sum may be given of the three rectangles BO.OC+C'O.OA' + A"O.OB". T&ing two sides for axes, the equation of the locus is / a \ f, b \ c^xy „ x{a-x--y)+y[h-y--^x)^.— ^m-, or .T- + 2/^ + Ixy cosC — ax — by + m- = 0. This represents a circle, which, as is easily seen, is concentric with the circumscribing circle ; the co-ordinates of the centre in both cases being given by the equations 2 (a -f- /3 cosC) = «, 2 (/3 + o cosC) = b. These last two equations enable us to solve the problem to find the locus of the centre of circumscribing circle, when two sides of a triangle are given in position, and any relation connecting theu" lengths is given. Ex. 5. Find the locus of a point 0, if the line joining it to a fixed point makes the same intercept on the axis of x, as is made on the axis of ?/ by a perpendicular thixiugh to the joining line. Ex. 6. Fmd the locus of a point, such that if it be joined to the vertices of a triangle, and perpendiculars to the joining lines erected at the vertices, these pei-pen- diculars meet in a point. 97. We shall next give one or two examples Involving the problem of Art. 82, to find the co-ordinates of the points where a given line meets a given circle. Ex. 1. To find the locus of the middle points of chords of a given circle, drawn parallel to a given Une. Let the equation of any of the parallel chords be X cos a + y sin a — p = 0, where a is, by hypothesis, given, and p is indeterminate : the abscissa of the points where this line meets the circle are (Art. 82) found from the equation X- — 2px cos a + p- — r- sin- a — 0. Now. if the roots of this equation he .r' and .r". the .r of the middle point of the N 90 THEOREMS AND EXAMPLES ON THE CIRCLE. chord will (Art. 7) be ;} (x' + x"), or, from the thcoiy of equations, will = p cos a. In like manner, the t/ of the middle point will equal p sin a. Hence the equation of the locus is y = a- tan a, that is, a right line drawn through the centre pei-pendicular to the system of parallel chords ; since a is the angle made with the axis of x by a ix;rpendicidar to any of the chords. Ex. 2. To find the condition that the intercept made by the circle on the line X cosa + ?/ sina —p should subtend a light angle at the point x'/. We foimd (Art. 90, Ex. 2) the condition that the lines joining the points x"i/", x"'y"' to xy should be at right angles to each other ; viz. {x - x") {x - x'") + (y~ y") {y - y'") = 0. Let x"i/", x"'y"' be the points where the line meets the circle, then, by the last example, x" + x"' = 2p cosa, x"x"' = p^ — r^ sin!' a, y" + y"' = 2p sina, !/"//" =7^^ — »'2 cos- a. Putting in these values, the required condition is ^'2 ^ y'2 _ 2px' cos a — Ipy' sin a + Ip" — r" = 0. Ex. 3. To find the locus of the middle point of a chord which subtends a right angle at a given point. If X and y be the co-ordinates of the middle point, we have, by Ex. 1, p cos a = X, p sin a= y, ^^ — 3.2 ^ ^2^ and, substituting these values, the condition fomid in the last example becomes {x — x')- + {y — y'Y + x'^ + y- = r^. Ex. 4. Given a line and a circle, to find a point such that if any chord be drawn through it, and peiiiendiculars let fall fi-om its extremities on the given line, the rectangle imder these peiijeudicidars may be constant. Take the given line for axis of x, and let the axis of y be the perpendicular on it from the centre of the given circle, whose equation will then be a:2 + (y - /3)2 = r2. Let the co-ordinates of the sought point be x'y' ; then the equation of any line through it will be y — y' = m {x — x') . Ehminate x between these two equations and we get a quadratic for y, the product of whose roots will be found to be {y' — mx')'' + in' (/3- — r^) This will not be independent of m imless the numerator be divisible by 1 -}- m', and it will be found that this cannot be the case, unless x' — 0, y'- = /3- — »•-. Ex.5. To find the condition that the intercept made on a; cosa -f-^ sina —^;, by the circle x"- + y"- + 2yx + 2fy + c - may subtend a right angle at the origin. The equation of the pair of lines joining the extremities of the chord to the origin may be written down at once. For if we multiply the terms of the second degi-ee in the equation of the circle by 7^-, those of the first degi-ee by p {x cosa + y sina), and the absolute term by (x cosa + y sina)'^, we get an equation homogeneous in x and y, which therefore represents right lines drawn through the origin ; and it is satisfied by those points on the circle for which x cosa + y sin a —p. The equation expanded and arranged is (7^^ + '^UP cosa + C COS^a) .T^ + 2 {(J sina H-^f cosa -f- c sina cosa) xy + (/>^ -)- 2fp s\na + c sin- a) y'^ — 0, THEOPvEMS AND EXAMPLES ON THE CIRCLE. Vl These two lines cut at right angles (Art. 74) if 2p2 + 2p (ff cos a +J' sin a) + c — 0. Ex. 6. To find the locus of the foot of the perpendicular fi"om the origin on a chord which subtends a right angle at the origin. The polar co-ordinates of the locus are jj and a in the equation last found ; and the equation of the locus is therefore 2 (a;2 + if) + 2ffx + 2fy + c = 0. It will be found on examination that this is the same circle as in Ex. 3. Ex. 7. If any chord be dra\vn through a fixed point on a diameter of a circle and its extremities joined to either end of the diameter, the joining lines cut of? on the tangent at the other end, portions whose rectangle is constant. Find, as in Ex. 5, the equation of the lines joining to the origin the intersections of x'' + y- — 2rx with the chord y — m {x — x') which passes through the fixed point (x', 0). The intercepts on the tangent are found by putting x — 2/- in this equation and seeking the corresponding values of y. The product of these values wUl be x' — 2r found to be independent of m, viz. 4?'- ; — . 98. We shall next obtain from the equations (Art. 88) a few of the properties of poles and polars. If a point A lie on the polar of B^ then B lies on the polar of A. For the condition that x'y' should lie on the polar of x"y" is X X ■\- y y" = r^ \ but this is also the condition that the point x'y" should lie on the polar of xy . It is equally true if we use the general equation (Art. 89) that the result of substituting the co-ordinates x'y" in the equation of the polar of xy is the same as that of substituting the co-ordinates oi^y in the polar of x'y" . This theorem then, and those which follow, are true of all curves of the second degree. It may be otherwise stated thus ; if tlie polar of B pass through a fixed point A^ the locus of B is the polar of A. 99. Given a circle and a triangle ABC, if we take the polars with respect to the circle, of ^, i?, C; we form a new triangle A'B'C called the conjugate triangle. A' being the pole of BGj B' of CA^ and C of AB. In the particular case where the polars of Aj Bj C respectively are BC^ GA^ AB, the second triangle coincides with the first, and the triangle is called a self-conjugate triangle. The lines AA'^ BB'^ CC'^ joining the corresp)onding vertices of a triangle and of its conjugate^ meet in a point. The equation of the line joining the point x'y' to the inter- 92 THEOREMS AND EXAMPLES ON THE CHICLE. section of the two lines xx' + yy" — r" = and xx" + yy'" - r* = is (Art. 40, Ex. 3) AA\ {xx" + xjy" - ?•') [xx + yy" - r"") — [xx" + y'y" - /•"'') [xx"' + yy'" — z-^) = 0. In like manner BB^ {xx -\-yy -r){xx +yy -?■) — [xx" + 3/"?/'" - r') {xx + y»/' - r') = ; and 6'C", (jj"iK"' + y"y " — **'') («;•«' + ?/?/' — r'^) — (a; V + y'y"' — r") (cca;" + yy" — r*^) = ; and by Art. 41 these lines must pass through the same point. The following is a particular case of the theorem just proved : If a circle be inscribed in a tricmyJe^ and each vertex of the tri- angle joined to the point of contact of the circle loith the opposite side^ the three joining lines loill meet in a point. The proof just given applies equally if we use the general equation. If we write for shortness -Pi = for the equation of the polar of x'y'^ {ax'x + &c. = 0) ; and in like manner /!,, P^ for the polars of x'y''^ ^'"y" ] and if we write [1 , 2] for the result of substituting the co-ordinates x"y" In the polar of x'y'y {ax'x'-^&c.)^ then the equations are easily seen to be AA' [1,3]P,= [1,2]P„ BB' [1,2]P3=[2,3]P„ CG' [2,3]Pi = [l,3]i^,, which denote three lines meeting in a point. It follows (Art. 60, Ex. 3) that the Intersections of corresponding sides of a triangle and its conjugate He in one right line. 100. Given arvy point Oy and any two lines through it ; join both directly and transversely the points in which these lines meet a circle ; then^ if the direct lines intersect each other in P and the transverse in Q, the line PQ will be the p)olar of the point 0, with regard to the circle. Take the two fixed lines for axes, and let the intercepts made on them by the circle be A, and A,', [x and fi!. Then X 11 X 11 r + ~ - 1 = " », -, + -^-1=0 THEOREMfe'^ AND EXAMPLES ON THE CIRCLE. 93 will be the equations of the direct lines ; and - + !(_, = 0,^ + 1-1=0, the equations of the transverse lines. Now, the equation of the line PQ will be A, A, /A yw. for (see Art. 40) this line passes through the Intersection of X y ^ X y , K It, ' K fJb and also of ^ + ■^, - 1, ^, + ^ - 1. If the equation of the curve be ax' -\- 2hxy + hf + 2gx + 2/7/ + c = 0, \ and V are determined from the equation ax' + 2gx + c = (Art. 84), therefore, II 2rtr 1 1 2/ ^ + r-, = - ^ , and - + - = - f . A, A C fJU fl C Hence, equation of FQ is .9-^ +fy + c = ; but we saw (Art. 89) that this was the equation of the polar of the origin 0. Hence it appears that if the point were given, and the two lines through it were 720t fixed, the locus of the points P and Q would be the polar of the point 0. 101. Given any two points A and i?, and their polars with respect to a circle whose centre is : let fall a perpendicidar AP from A on the polar of B, and a perpendicular BQ from B on the J . , , OA OB polar of A ; then -j^ = -^ . The equation of the polar of A [x'y) is xx -f yy — r'^ = ; and BQ^ the perpendicular on this line ivoxa..B[x'y")^ is (Art, 34) x x" + y' y" — r'' ^{x' + if)-' Hence, since \^{x" + y"') =OAj we find OA.BQ = x'x" + y'y" - r' ; 94 THEOREMS AND EXAMPLES ON THE CirX'LE. and, for the same reason, OB.AP=xx" + y'y" -r\ ,T OA OB Hence AP^BQ- 102. In working out questions on the circle It is often con- venient, instead of denoting the position of a point on the curve by its tivo co-ordinates x'y\ to express both these in terms of a single Independent variable. Thus, let 6' be the angle which the radius to x'y makes with the axis of ic, then x = r cos 6'^ y = r sin 0\ and on substituting these values our formulae will generally become simplified. The equation of the tangent at the point xy will by this sub- stitution become X cos & ■\-y sin & = r\ and the equation of the chord joining xy\ x'y\ which (Art. 86, Ex. 3) is X [x + cc") +y {y + y") = r^ + x'x" + y'y"^ will, by a similar substitution, become X cosi [6' -f d") +y sini [& -f 6") = r cos J- {6' - 6"), & and 6" being the angles which radii drawn to the extremities of the chord make with the axis of x. This equation might also have been obtained directly from the general equation of a right line (Art. 23) x do^a.-^ y siaa=^, for the angle which the perpendicular on the chord makes with the axis is plainly half the sum of the angles made with the axis by radii to its extremities ; and the perpendicular on the chord = r cos^(f?'-r). Ex. 1. To find the co-ordinates of the intersection of tangents at two given points on the circle. Tlie tangents being X cos 6' + y'sin 6' = »•, x cos 6" + ?/ sin 0" = r, the co-ordmates of their intersection are _ cosj-(y + e") _ sin^(e' + e") "^ ~ '' cos^ {%' - 0") ' y -^ cos J- (f)' - a") ■ Ex. 2. To find the locus of the intersection of tangents at the extremities of a chord whose length is constant. Making the substitution of this article in {x — x"Y + {>/ — y")- — constant, it reduces to cos (6' - 6") = constant, or 6' - 6" = constant. If the given length of THEOREMS AND EXAMPLES ON THE CIRCLE. 95 the chord be 1r sin o, then W — 6" = 2o, The co-ordinates therefore found in the hxst example fulfil the condition (u?2 + 1/') COS^ S = ?'2. Ex. 3. "What is the locus of a point where a chord of a constant length Ls cut in a given ratio '? Writing down (Art. 7) the co-ordinates of the point where the chord is cut in a given ratio, it will be found that they satisfy the condition x- + y- = constant. 103. We have seen that the tangent to any circle x^ + y'' = r* has an equation of the form X cos 6 + 7/ shi ^ = r ; and It can be proved, In like manner, that the equation of the tangent to [x — a)'^ + (^ — /3)'' = r'^ may be written [x — a) cos 6 -\-[y — ^) sin ^ = r. Conversely, then. If the equation of any right line contain an indeterminate 6 in the form [x — a) cos 6 + [y — P) sin ^ = r, that line will touch the circle (x — «)'' + (^ — ^f = r\ Ex. 1, If a chord of a constant length be inscribed in a circle, it will always touch another ckcle. For, in the equation of the chord X cosi (6' + Q") + i/sm} (6' + 6") = r cosi (6' - 6") ; by the last article, 6' — 6" is known, and 6' + 6" indetemainate ; the chord, therefore, always touches the circle X- + y- — r- cos^ S, Ex. 2. Given any number of points, if a right Une be such that m' times the peipendicular on it from the first point, + m" times the perpendicular from the second, + A-c, be constant, the line will always touch a circle. This only differs fi"om Ex. 4, p. 49, in that the sum, in place of being = 0, is con- stant. Adopting then the notation of that Article, instead of the equation there found, {x'2. (m) - S {mx')] cos« -I- {;y2 (in) - 2 {jny')] sin a - 0, we have only to write [x'2,m — S {mx')] cos a + {yS {m) — S [my')] sin a = constant. Hence this line must always touch the cii-cle f 2 {:mx')]^ ( _ S {my') r S(«OJ F S(;«) whose centre is the centre of mean position of the given points. T+{^-¥?l?F='°°'''"'' 104. We shall conclude this Chapter with some examples of the use of polar co-ordinates. Ex. 1. If through a fixed point any chord of a circle be drawn, the i-ectangle under its segments will be constant (Euclid iii. 35, 36). Take the fixed point for the pole, and the polar equation is (Art. 95) p^ - 2p(l cos e + d--}--~0; 96 THEOREMS AND EXAMPLES ON THE CIRCLE. the roots of which equation in p are evidently OP, OP', the vahies of the radius vector answering to any given value of 6 or POC. Now, by the theory of equations!, OP.OP', the product of these roots will = (fz _ ■)•-, a quantity independe/it of 0, and therefore constant, whatever be the direction in which the line OP is drawn. If the point be outside the circle, it is plain that d^ _ ^2 must be — the square of the tangent. Ex. 2. If through a fixed point any chord of a cu-cle be drawn, and OQ taken an arithmetic mean between the segments OP, OP' ; to find the locus of Q. We have OP + OP', or the sum of the roots of the quadratic in the last example* = 2(1 cost) ; but OP + OP' = 20Q, therefore OQ = dcose. p^ Hence the polar equation of the locus is p = d cos 6, ^ ^ \ kj I Now it appears from the final equation (Art. 95) \ / that this is the equation of a circle described on the line OC as diameter. The question in this example might have been othei-wise stated : " To find the locus of the middle points of chords which all pass through a fixed point." Ex. 3. If the line OQ had been taken a harmonic mean between OP and OP', to find the locus of Q. tynp OP' That is to say, OQ = gp gp, , laut OP.OP' = d^ - r^, and 0P+ OP' = 2d cos0 ; therefore the polar equation of the locus is /3=-= . . or p cose = — T — . dcosd ■ d This is the equation of a right line (Art. 44) perpendicivlar to OC, and at a distance from = d , and, therefore, at a distance from C= -; . Hence (Art. 88) d d the locus is the polar of the point 0. We can, in like manner, solve this and similar questions when the equation is given in the form a {x^ + /) + 2gx + 2/1/ + c = 0, for, transforming to polar co-ordinates, the equation becomes o^ + 2 f^ cose +-^ sine") o + - = 0, ^ \a a J '^ a and, proceeding precisely as in this example, we find, for the locus of harmonic means, ^ (/ cose +/ sine' and, returning to rectangular co-ordinates, the equation of the locus is yx +fy -I- c = 0, the same as the equation of the polar obtained already (Art. 89). Ex. 4. Given a point and a right line or circle ; if on OP the radius vector to the line or circle a part OQ be taken inversely as OP, find the locus of Q. Ex. 5. Given vertex and vertical angle of a triangle and rectangle under sides ; if one extremity of the base describe a right line or a circle, find the locus described by the other extremity. Take the vertex for pole ; let the lengths of the sides be p and p, and the angles they make with the axis 6 and 0', then we have pp' = I? and e - 6' = C. THEOREMS AND EXAMPLES ON THE CIRCLE. 97 The student must wiite dowTi the polar equation of the locus which one base angle is said to describe ; this will give him a relation between p and 6 ; then, writing for p, — , and for 6, C + 6', he will find a relation between p' and 6', which will be the p polar equation of the locus described by the other base angle. This example might be solved in like manner, if the ratio of the sides, instead of their rectangle, had been given. Ex. 6. Through the mtersection of two circles a right line is dra'wn ; find the locus of the middle point of the portion intercepted between the circles. The equations of the circles will be of the form p = 2?* cos (0 — a) ; p = 2r' cos (9 — a') ; and the equation of the locus will be p = )' cos (G — a) + )•' cos (6 — a') ; which also represents a circle. Ex. 7. If through any point 0. on the circumference of a circle, any three chords be drawn, and on each, as diameter, a circle be described, these three circles (which, of course, aU pass through 0) will intersect in three other points, wliich Ue in one right line. (See Cambridge Jfathe/natical Journal, Vol. I., p. 1G9). Take the fixed point for pole, then if d be the diameter of the original cii'cle, its polar equation will be (Art. 95) p = fZ cos 0. In like manner, if the diameter of one of the other circles make an angle a with the fixed axis, its length will be = (Z cos a, and the equation of this cii'cle will be p = d cosa cos(0 — a). The equation of another cu'cle will, in like manner, be p — d cos/3 cos (0-/3). To find the polar co-ordinates of the point of intersection of these two, we should seek what value of woiild render cosa cos (6 — a) = cos/3 cos(0 — /3), and it is easy to find that must = a + /3, and the con'esponding value of p — d cosa cos/3. Similarly, the polar co-ordinates of the intersection of the fii-st and third circles arc = a -r y, and p = d cos a cos y. Now, to find the polar equation of the line joining these two points, take the general equation of a right Une, p cos {k — d) = j) (Art. 44) and substitute in it suc- cessively these values of d and p, and we shall get two equations to determine 7> and 1-. We shall get p = d cosa cos/3 cos{^- — (a + /3)} = d cos a cos y cos {I- — (a + y)}. Hence k = a + (3 + y, and j) = d cos a cos j8 cos y. The symmetiy of these values shows that it is the same right line which joins the intersections of the first and second, and of the second and third circles, and, therefore, that the thi-ee points are in a right line. ( '-^^ ) CHAPTER VIII. rnorERTiES of a system of two or more circles. 105. To find the equation of the chord of intersection of two circles. If S=0^ S' = be the equations of two circles, then any equation of the form >S'+^->S" = will be the equation of a figure passing through their points of intersection (Art. 40). Let us write down the equations S ={x-ay+i7/-^y-r' =0, S-={x-ar+{y-/3'Y-r'-^ = 0, and it is evident that the equation S+kS' = will in general represent a circle, since the coefficient of xt/ = 0, and that of x^ = that of 3/^. There is one case, however, where it will re- present a right line, namely, when k = — 1. The terms of the second degree then vanish, and the equation becomes >S- /S"= 2 (a - a) a; + 2 (/3'- /3) ?/ + r" - r' + a' - a' + /S'^ - /3'' = 0. This is, therefore, the equation of the right line passing through the points of intersection of the two circles. What has been proved in this article may be stated as In Art. 50. If the equation of a circle be of the form S+kS' = involving an indeterminate k In the first degree, the circle passes through two fixed points, namely the two points common to the circles S and S'. 106. The points common to the circles ;S^ and >S" are found by seeking, as in Art. 82, the points in which the line S—S' meets cither of the given circles. These points will be real, co- incident, or imaginary, according to the nature of the roots of the resulting equation ; but it is remarkable that, whether the circles meet in real or imaginary points, the equation of the chord of intersection, S—S' = 0^ always represents a real line, having important geometrical properties in relation to the two PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. 99 circles. This is in conformity with our assertion (Art. 82), that the line joining two points may preserve its existence and its properties when those points have become imaginary. In order to avoid the harshness of calling the line S— S' the chord of intersection in the case where the circles do not f/eo- metrically appear to intersect, it has been called* the radical axis of the two circles. 107. We saw (Art. 90) that if the co-ordinates of any point xy be substituted in >S', it represents the square of the tangent drawn to the circle >S', from the point xy. 8o also S' is the square of the tangent drawn to the circle S' ; hence the equation S— S' = asserts, that if from any point on the radical axis tan- gents be drawn to the two circles, these tangents loill he equal. The line {S — S') possesses this property whether the circles meet in real points or not. When the circles do not meet in real points, the position of the radical axis is determined geome- trically by cutting the line joining their centres, so that the diiference of the squares of the parts may = the ditference of the squares of the radii, and erecting a perpendicular at this point ; as is evident, since the tangents from this point must be equal to each other. If it were required to find the locus of a point whence tan- gents to two circles have a given 7'atio, it appears, from Art. DO, that the equation of the locus will be S-~Jc^S'=0, which (Art. lOo) represents a circle passing through the real or imaginary points of intersection of S and S'. When the circles S and *S" do not intersect in real points, we may express the relation which they bear to the circle S—IS'S' by saying that the three circles have a common radical axis. Ex. Find the co-ordinates of the centre of S — L-S'. a-k'^a B-Jc-B' . . , . „ ^, . Ans. - — - , .;^— : that is to say, the une joimng the centres of S, S is divided externally in the ratio 1 : k". 108. Given any three circles^ if we take the radical axia of each pair of circles, these three lines loill meet in a jwint, irhich ■is called the radical centre of the three circles. * By M. Gaultier of Toiu's {Jotinial de TEcoIe Pohjtechui<iue, C.il'ier xvi., 18U?). 100 PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. For the equations of the three radical axes are S-S' = 0, S'-S" = 0, S"-S=0, which, by Art. 41, meet in a point. From this theorem we immediately derive the following : If several circles jpass through two fixed points^ their chords of intersection loitJi a fixed circle will pass through a fixed point. For, imagine one circle through the two given points to be fixed, then its chord of intersection with the given circle will be fixed ; and its chord of intersection with any variable circle drawn through the given points will plainly be the fixed line join- ing the two given points. These two lines determine, by their intersection, a fixed point through which the chord of intersection of the variable circle with the first given circle must pass. Ex. 1, Find the radical axis of x--\-y--Ax- by + l = Q; x" + ?/2 + 6,c + 8_y - 9 = 0. Alls. lOx + \oy - 16, Ex. 2. Find the radical centre of {x-\f+{ij-2r- = 7; {x-ZY + y- = b; (.; + 4)= + (z/ + 1)- = 9. Ans. (- J-, - \^. *109. A system of circles having a common radical axis pos- sesses many remarkable properties which are more easily inves- tigated by taking the radical axis for the axis of ?/, and the line joining the centres for the axis of x. Then the equation of any circle will be x' ^-y' - 2hx±h'' = 0., where h"^ Is the same for all the circles of the system, and the equations of the ditferent circles are obtained by giving different values to h. For it is evident (Art. 80) that the centre .Is on the axis of a;, at the variable distance h ; and if we make x = in the equation, we see that no matter what the value of k may be, the circle passes through the fixed points on the axis of y, y^±^ = 0. These points are Imaginary when we give 6^ the sign 4- 5 and real when avc give It the sign — . ''•110. The pohirs of a given pointy with regard to a system of circles having a common radical axis^ ahoays p)(tss through a fixed p)Oint. The equation of the polar of xy with regard to it? + if - 2i'x + a' = 0, PROPEETIES OF A SYSTEM OF TWO OR MORE CIRCLES. 101 is (Art. 89) xx + yy - ^< [x + x) + §- = (); therefore, since tbls involves the indeterminate h in the first degree, the line will always pass through the intersection of XX 4 yy + S" — 0, and x-\-x = 0. *111. There can always he found two ]jotnts^ lioicever^ such that their jjolars, ivith regard to any of the circles^ will not only pass through a fixed pointy hut ivill he altogether fixed. This will happen when xx -^-yy -\-^' = 0^ and x + x' = re- present the same right line, for this right line will then he the polar whatever the value of Ic. But that this should be the case we must have ?/' = and x~ — S^, or x = + S. The two points whose co-ordinates have been just found have many remarkable properties In the theory of these circles, and are such that the polar of either of them, with regard to any of the circles, is a line drawn through the other, perpendicular to the line of centres. These points are real when the circles of the system have common two imaginary points, and imaginary when they have real points common. The equation of the circle may be written in the form which evidently carmot represent a real circle if P be less than 6''; and if A.' =8', then the equation (Art. 80) will represent a circle of infinitely small radius, the co-ordinates of whose centre are y = 0, a? = + S. Hence the points just found may themselves be considered as circles of the system, and have, accoi'dingly, been termed by Poncelet* the Umiting points of the system of circles. *112. If from any point on the radical axis we draw tan- gents to all these circles, the locus of the points of contact must be a circle, since we proved (Art. 107) that all these tangents were equal. It is evident, also, that this circle cuts any of the given system at right angles, since its radii are tangents to the given system. The equation of this circle can be readily found. * Ti-diti} (/t'x Vrnprictvs rmjectlces. p. -11. 102 PROPERTIES OF A SYSTEM OF TWO OR IVUORE CIRCLES, The square of the tangent from any point [x = 0, >/ — h) to the circle x'' + f-2kx+8-^0, being found by substituting these co-ordinates in this equation, is /i^ + 8^; and the circle whose centre is the point [x = i), y = li)i and whose radius squared = /r' + S", must have for its equation a;H(?/-/0' = /^' + S'-', or x^ '^if ~ 2% = ^■^• Ilcnce, whatever be the point taken on the radical axis (^*.e. whatever the value of A may be), still this circle will always pass through the fixed points (3/ = 0, x = ±h) found in the last Article. And we infer that all circles which cut the given system at rUjlit angles pass through the limiting points of the system. Ex. 1, Find tlie condition tliat two circles a;2 + )/2 + 2(jx + 2/)/ + c = 0, x" + if + 1(j'x + ^tfy + c' = should cut at right angles. Expressing that the square of the distance between the centres is equal to the sum of the squares of the radii, we have (5' - 'JY + {f-f"f =■- ff- +/- -c + 3" +/'- - C, or, reducing. 2////' + 2/f' = c + c'. Ex. 2. Find the cu-cle cutting three circles orthogonally. "We have thi-ee equations of the first degree to determine the three miknown quantities r/, J', c ; and the problem is solved as in Art. 94. Or the problem may be solved otherwise ; since it is evident from this article that the centre of the required circle is the radical centre of the tlu'ee circles, and the length of its radius equal to that of the tangent from the radical centre to any of the circles. Ex. 3. Fmd the circle cutting orthogonally the three circles. Art. 108, Ex. 2. Ans. (x + ^\y + i>j + f^y = Vicf'. Ex. 4. If a circle cut orthogonally three circles S', S", S'", it cuts orthogonally any circle kS' + IS" + viS'" = 0. Writing down the condition 2i' U<-ff' + hj" + mg"') + y ijcf + If" + mj"') = {k +1 + m) c -\- {he' + h" + mc'"), we see that the coefficients of k, I, m vanish separately by hypothesis. Similarly, a circle cutting ,S', S" orthogonally, also cuts orthogonally kS' + IS". Ex. 5. A system of circles which cuts orthogonally two given circles S', S" has a common radical axis. This, which has been proved in Art. 112, may be proved otherwise as follows : The two conditions 2yi/' + 2ff = c + c', %jf' + yf" = c + c", enable us to determine ij and / linearly in terms of c. Substituting the values so foxmd in x- + y~ + 2r/x + 2f// + c ^ 0, the equation retains a single indeterminate c in the first degi-ee, and therefore (Art. 105) denotes a system having a common radical axis. Ex. G. If AB be a diameter of a circle, the polar of A with respect to any circle which cuts the first orthogonally, will pass through B. PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. 103 Ex. 7. The square of the tangent from any point of one circle to another is proportional to the pei-pendicnlar from that point upon theu- radical axis. Ex. 8. To find the angle (a) at which two circles intersect. Let the radii of the circles be R, r, and let D be the distance between their centres, then D"- = Er- + r''--2R,'cosa: since the angle at which the circles intersect is equal to that between the radii to the point of intersection. If *S = be the equation of the circle whose radius is r, the co-ordinates of the centre of the other cu'cle must fulfil the condition R- — 2/^;- cos a = 5, as is evident from Art. 90, since 1)- — r- is the square of the tangent to S from the centre of the other circle. Ex. 9. If we are given the angles a, (i at which a circle cuts two fixed circles ;S^, *S", the circle is not determined, since we have only two conditions : but we can determine the angle at which it cuts any cii'cle of the system kS + IS', For we have Rr - 2Rr cosa- S, R? - 2Rr' cos j8 = 6", 1 TV. o n ^'' cos a + Ir' cos fl kS + IS' whence Rr — 2R , -. ~ = -, — ,- , k+l k+l ' which is the condition that the moveable circle should cut kS + IS' at the constant angle y : where {k + I) r" cosy = kr cosa + Ir' cos jS, r" being the radius of the circle kS + IS' . Ex. 10. A cu'cle which cuts two fixed circles at constant angles will also touch two fixed circles. For we can determine the ratio ^- : ? so that y shall = 0, or cosy = 1. "Wilting S and <S" at fuU length and calculating the radius r" of kS+lS', we easily find {k + ly- r"^ = {k + J) {kr"- + 7?-'2) - kW\ where D is the distance between the centres of S and S', Substituting tliis value for r" in the equation of the last example, we get a quadratic to determine k : 1, 113. To draw a common tangent to two circles. Let their equations be and {x-dY-{[ij-^J = r^ [S'). We saw (Art. 85) that the equation of a tangent to {S) was {x-a){x'-o) + {jj-^)[y'-^)=r'; or, as in Art. 102, writing X -a y -^ = cos 6", = sin V. r r {x - a) cos 6 + [y — ^) sin 6 = r. In like manner, any tangent to [S') is {x -- a) cos 0' + {y — ^') sin 6' = r. Now, if we seek the conditions necessary that these two equations should represent the same right line ; first, from com- paring the ratio of the coefficients of x and y^ we get tan^=tan^', 104 rilOPERTIES OF A SYSTE^I OF TAVO OR MORE CIRCLES. whence 6' cither = 6^ or = 180" + 6. If cither of these conditions be fultilled, we must equate the absolute terms, and we find, in the first case, (a - a) cos (9 + (/3 - /3') sin (9 + r - r' = 0, and in the second case, (a - a') cos 9+{^- /3') sin d-\- r + 7-' = 0. Either of these equations would give us a quadratic to deter- mine 0. The two roots of the first equation would correspond to the direct or exterior common tangents, Aa, A' a ; the roots of the second equation would correspond to the transverse or interior tangents, Bh^ B'h'. If we wished to find the co-ordinates of the point of contact of the common tangent with the circle (/S), we must substitute, in the equation just found, for cos^, its value, , and for sin 9. , and we find r ' (a_a')(a;'_a) + (/3-/3')(//'-/3) + r(r-r') = 0; or else, (a - a) [x - a) + (^ - /3') {y - /3) -}- r (r + r') = 0. The first of these equations, combined with the equation (>S') of the circle, will give a quadratic, whose roots will be the co- ordinates of the points A and A'^ in which the direct common tangents touch the circle [S) ; and it will appear, as in Art. 88 j that (a'-a)(a^-a) + (^'-/S)(y-^)=r(r-/) is the equation of AA\ the chord of contact of direct common tangents. So, likewise, (a' -o)[x-a) + (/3' - /3) (2/ - /3) = r [r + r') is the equation of the chord of contact of transverse common PKOrERTIE.S OF A SYSTEM OF TWO OR MORE-CIKCLES. lO'j tangents. If the origin be the centre of the circle (>S;, then 7. and yt^ = ; and we find, for the equation of the choj'd of contact, <xx + ^'y — r[r + r) . Ex. Find the common tangents to the cii'cles x"- + y- - Ax - 2y + 4: - 0, x^ + ?/ + Ax + lij -4 = 0. The chords of contact of common tangents with the first circle are 2x + y — Q, 2x + y — o. The first chord meets the circle in the points (2, 2), {^, %), the tangents at which are y = 2, Ax-3y=l0, and the second chord meets the circle in the points (1, 1). (j, i), the tangents at which are .r = 1, 3x + Ay — 5, 114. The points and 0', in which the direct or transverse tangents intersect, are (for a reason explained in the next Article) called the centres of similitude of the two circles. Their co-ordinates are easily found, for is the pole, with regard to circle (>S'), of the chord AA\ whose equation is (a'-a)r. , iff — ^)r „, .^ r — r r — r Comparing this equation with the equation of the polar of the point xy\ [x-a) (oj- a) + (?/'-/3) (y -/3) = r, (a - a) r , a'r — ar' we sret tc — a = ,— , or a; = —- , ° r — r r — r So, likewise, the co-ordinates of 0' are found to be ar + ox , /3'r + |6r' X = r- , and y= , . r + r ' -^ r + r These values of the co-ordinates indicate (see Art. 7) that the centres of similitude are the points where the line joining the centres is cut externally and internally in the ratio of the radii. Ex. Find the common tangents to the circles x- + y- — 6.r — % = 0, x- + y- — Ax — (i^ = 3. The equation of the pair of tangents through x'y' to {x - a)"- + {y- l3y- = >-2 is found (Art. 92) to be {{x' - ay- + iy' - /3)2 - 7-"-} {(.r - ay + {y- fty - r"-] = {{x- a) (x' - a) + {y - /?) Cv' -ft^- >•-'■ P 106 PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. Now, the co-ordinates of the exterior centre of similitude are found to be (— 2, — 1), and hence the paii- of tangents through it is 2b {x^ + y- - 6x - 8i/) = {ox + 5>/ - lOy- ; or xy + x + 2y + 2-0 ; or (x + 2) {t/+\) = 0. As the given circles intersect in real points, the other pair of common tangents become imaginary ; but their equation is foimd, by calculating the pair of tangents through the other centre of simiUtude {-^, y), to be 40x2 + xy + 40/ - 199a; - 278y + 722 = 0. 115. Every right line drawn through the intersection of com- mon tangents is cut similarly hy the two circles. It is evident that if on the radius vector to any point P there be taken a point (>, such that OP=m times OQ^ then the x and y of the point P will be respectively m times the x and y of the point Q] and that, therefore, if P describe any curve, the locus of Q is found by substituting mx^ my for x and y in the equation of the curve described by P. Now, if the common tangents be taken for axes, and if we denote Oa by a, OA by «', the equations of the two circles are (Art. 84, Ex. 2) X' + ?/" + ^xy coso) — '^ax — lay + a^ =0, o? -\-y^ -\- 2xy cos w — 2a x - 2a! y + a!~ = 0. But the second equation is what we should have found if we Ct.CV fill had substituted -r , -v , for x^ ?/, in the first equation ; and it therefore represents the locus formed by producing each radius vector to the first circle in the ratio a : a, COR. Since the rectangle Op. Op is constant (see fig. next page), and since we have proved OR to be in a constant ratio to Op, it follows that the rectangle OR.Op =OR'.Op is constant, however the line be drawn through 0. 116. If through a centre of similitude we draw any two lines meeting the first circle in the points R, R', S, /S", and the second in the points p, p', cr, cr', then the chords RS, pa ; R'8\ p<r ; will he parallel, and the chords RS, p'cr' ; R'S', pa; will meet on the radical axis of the two circles. Take OR, 08 for axes, then we saw (Art. 115) that OR = rnOp, OS=mOa-, and that if the equation of the circle pap a' be a {x' + 2xy cos w + y^) -f 2gx + 2fy + o = 0, PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES, 107 that of the other will be a [x^ + 2xy cos tw + y") 4- 2m [gx +/>/) + m'c = 0, and, therefore, the equation of the radical axis will be (Art. 105) 2(5'-»+/i/) + ("«+l)c = 0. Now let the equations of pa and of pa be X y ^ X V , a a then the equations of ES and R' S' must be JL j^ y =.^ ^ , .V ma ml) 7 + y, = 1. ma onb It is evident, from the form of the equations, that RS is parallel to po-; and RS and pa must intersect on the line 1 1 - 4- - a a +y + r, ) = l-f »', or, as in Art. 100, on 2(,9'-«+/3/) + (m + l)c = 0, the radical axis of the two circles. A particular case of this theorem is, that the tangents at R and p are parallel, and that those at R and p meet on the radical axis. 117. Given three circles Sj S', S" ; the line joining a centre of similitude of S and S' to a centre of similitude of S and S" icill pass through a centre of similitude of S' and S". Form the equation of the line joining the points ^ra — ar' r/3' — ^r'\ ■ fra" — ar" r/B" — (3r" (Art. 114), and we get (see Ex. 6, p. 24), {•/• (/3' - /3") + r' (/3" - /3) + /' (^ - yS')l a' - [r [a — a") + r (a" - a) + r" (a - a')} y + r (yS'a" - /S"a') + r [13" a - /3a") + r" {/3u' - /S'a) = 0. 108 I'liDPEUTIES UF A SY8TE.M OF TWO OR MOKE CIRCLES. Now the symmetry of this equation sufficiently shows, that the line it represents must pass tlu-oug-h the third centre of similitude, r'0" - 7'" 13' r a — r a y This line is called an axis of slnuUtude of the three circles. Since for each pair of circles there are two cen- tres of similitude, there will be in all six for the three circles, and these will be distributed along four axes of similitude, as represented in the figure. The equations of the other three will be found by changing the signs of either r, or ?•', or r", in the equation / • just given. ' / Cor. If a circle (S) touch two others [S and S')^ the line join- ing tJie points of contact ivill pass through a centre of similitude of S and S\ For when two circles touch, one of their centres of similitude will coincide with the point of contact. If 2 touch S and /S", either both externally or both internally, the line joining the points of contact will pass through the exter- nal centre of similitude of 8 and S'. If 2 touch one externally and the other internally, the line joining the points of contact ■will pass through the internal centre of similitude. *118. To find the locus of the centre of a circle cutting three given circles at equal angles. If a circle whose radius is B cut at an angle a the three circles >S', S'. S", then (Art. 112, Ex. 8) the co-ordinates of its centre fulfil the three conditions S=E'- 2Rr cos a, S' = E' - 2Rr' cos a, S" = R'- 2Rr" cos a. From these conditions we can at once eliminate R^ and 11 cos a. Thus, by subtraction, >S'- S' = 2E {r - r) cosry, S- S" = 2E [r" - r) cos a, rROPEllTIES OF A SYSTEM OF TWO OK lAFOKE CIliCLES. I<j9 whence [S- S') (r — r") = [S - S") (r - >•'), the equation of a line on which the centre must lie. It obviously passes through the radical centre (Art. 108) ; and if we write for S—S', S- S", their values (Art. 105), the coefficient of x in the equation is found to be - 2 [a (>•' - r") + a' [r" - r) + a" [r - r')}, while that of ?/ is -2{^(r'-r") + /3'(r"-r)+/3"(r-/)}. Now If we compare these values with the coefficients in the equation of the axis of similitude (Art. 117), we Infer (Art. 82), that the locus is a perpendicular let fall from the radical centre on an axis of similitude. It is of course optional which of two supplemental angles we consider to be the angle at which two circles Intersect. The formula (Art. 112) which we have used, assumes that the angle at which two circles cut, Is measured by the angle which the distance between their centres subtends at the point of meeting : and with this convention the locus under consideration Is a per- pendicular on the external axis of similitude. If this limitation be removed, the formula we have used becomes S= R'^ ±2B7- cosa ; or, in other words, we may change the sign of either r, r', or r" in the preceding formulae, and therefore (Art. 117) the locus is a perpendicular on any of the four axes of similitude.* When two circles touch Internally, their angle of intersec- tion vanishes, since the radii to the point of meeting coincide. But If they touch externally, their angle of Intersection accord- ing to the preceding convention is 180°, one radius to the point of meeting being a continuation of the other. It follows, from * In fact all circles cutting three cii-cles at equal angles have one of the axes of simihtude for a common radical axis. Let 2, 2', 2" be three circles cutting the given circles at angles a, ft, y respectivelj'. Then the co-ordinates of the centre of S must fulfil the conditions 2 = ?•- - 2j-i? cos a, ^' = r--2rE' cosft, I." = r" - 2 rR" cosy; whence {R cosa - R" cosy) (2 - 2') = (R cosa - R' cos/3) (2 - 2"). Now this which appears to be the equation of a right line is satisfied by the co- ordinates of the centre of S, of S', and of S". three points which are not supposed to be on a right line. It denotes therefore an identical relation of the form 2 - X-2' + /2" shewinsr that the tliree circles have a common radical axis. 110 PROPERTIES OF A SYSTE:\I OF TAVO OR MORE CIRCLES. what has been just proved, that the ])erpcndicuLir on the external axis of shnllitude, contains the centre of a circle touching three given circles, either all externally or all internally. If we change the sign of >•, the equation of the locus which we found denotes a perpendicular on one of the other axes of similitude which will contain the centre of the circle touching ^S' externally, and the other two internally, or vice versa. Eight circles in all can be drawn to touch three given circles, and their centres lie, a pair on each of the perpendiculars let fall from the radical centre on the four axes of similitude. *119. To describe a circle touching three given circles. We have found one locus on which the centre must lie, and we could find another by eliminating R between the two conditions S=E'-^2Br, S' = B' + 2Rr'. The I'esult however would not represent a circle, and the solu- tion will therefore be more elementary, if instead of seeking the co-ordinates of the centre of the touching circle, we look for those of its point of contact with one of the given circles. We have already one relation connecting these co-ordinates, since the point lies on a given circle, therefore another relation be- tween them will suffice completely to determine the point.* Let us for simplicity take for origin the centre of the circle, the point of contact with which we are seeking, that is to say, let us take a = 0, y8 = 0, then if A and B be the co-ordinates of the centre of E, the sought circle, we have seen that they fulfil the relations S-8' = 2B (r - r'), S-S" = 2B {r - r"). But if X and y be the co-ordinates of the point of contact of S with >S', we have from similar triangles r ' r Now if in the equation of any right line we substitute onx^ my for X and ?/, the result will evidently be the same as if we multiply the whole equation by m and subtract [m — 1) times the absolute terra. Hence, remembering that the absolute term in ;S'— >S" is * This solution is hy M. Gergonne, Annnle.t ties Mathematifptes, Yol. vil.. p. 289. PROPEKTIES OF A SYSTEM OF TWO OR MORE CIRCLES. 1 1 1 (Art. 105) r'^ — r^ — a^ — ^'\ the result of making the above sub- stitutions for A and B in (>S'— S') = 2R (r — r) is ^^* {S- S') + ^ [a!' + /3'^ + r^ - r") = 2R [r - r'), or [R + r) {S- S') = R {{r - rj - a""' - (3"]. Similarly [R + r) [8- S") = R [[r - r'J - a!"' - /3"']. Eliminating i?, the point of contact is determined as one of the intersections of the circle S with the right line S~S' S-S" a" + yS'^ - (r - ry a'"' + yS"" - (r - r"f ' 120. To complete the geometrical solution of the problem, it is necessary to show how to construct the line whose equation has been just found. It obviously passes through the radical centre of the circles ; and a second point on it is found as follows. Write at full length for S— S' (Art. 105), and the equation is 2a.' X + 2, 8' 7/ + r'^-r'-a"'-^''' _ 2a"x +2 l3"i/ +r"' - r' - a"--/3"^ oc'=^ + /3'^-(r-r')'^ ~ ar' + ^"'-{r-r"f ' Add 1 to both sides of the equation, and we have a'x + /3'y + [7-' —r)r_ a"x + ^"y + [r" — r) r l^'T^"-{r-r'y'~ ^ a"' + ^"'-{r-r"Y ' showing that the above line passes through the intersection of a'x + ^'y + [r — r) r = 0, a"x + ^"y + (?•" — r) r — 0. But the first of these lines (Art. 113) is the chord of common tangents of the circles /Sand S' ] or, in other words (Art. 114), is the polar with regard to S of the centre of similitude of these circles. And in like manner the second line is the polar of the centre of similitude of /Sand >S"'; therefore (since the intersection of any two lines is the pole of the line joining their polos) the intersection of the lines ax + ^'y + (?•' — 7']r = 0, a"x + /3"y + (?•" — ?•) r = 0, is the pole of the axis of similitude of the three circles, with regard to the circle S. Hence we obtain the following construction : Drawing any of the four axes of similitude of the three circles, take its pole with respect to each circle, and join the 112 PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. points so found (P, P', P") with the radical centre ; then, if the joining Hues meet the circles in the points (a, h I a', h' ; a", h")^ the circle through a, a', a" will he one of the touching circles, and that through 5, />', Z'" will he another. Repeating this process with the other three axes of similitude, we can de- termine the other six touching circles. 121. It is useful to show liow the preceding results may he derived without algebraical calculations. (1) By Cor., Art. 117, the lines aJ, a'h\ ah" meet in a point, viz., the centre of similitude of the circles add\ hh'b". (2) In like manner dd\ h'h" intersect in ;S', the centre of similitude of C", C". (3) Hence (Art. 116) the transverse lines n'h\ ah" intersect on the radical axis of C, G" . So again a"5", ah intersect on the radical axis of 0", C. Therefore the point E (the centre of similitude of add\ hh'h") must be the radical centre of the circles (7, 0', C". (1) In like manner, since a'h\ d'h" pass through a centre of similitude of ««'«", hh'h" \ therefore (Art. 116) dd'^ h'h" meet on the radical axis of these two circles. So again the points S' and >S"' must lie on the same radical axis ; therefore SS'S", the axis of similitude of the circles C, 6", C", is the radical axis of the circles ada"^ hh'h". (5) Since d'h" passes through the centre of similitude of add', hh'h", therefore (Art. 116) the tangents to these circles where it meets them intersect on the radical axis SS'S". But this point of intersection must plainly be the pole of d'h" with regard to the circle C". Now since the pole of d'h" lies on SS'S", therefore (Art. 98) the pole of SS'S" with regard to G" lies on d'h". Hence d'h" is constructed by joining the radical centre to the pole of >S'>S"/S"' with regard to G" . PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. 113 (G) Since the centre of similitude of two circles is on the line joining their centres, and the radical axis is perpendicular to that line, we learn (as in Art. 118) that the line joining the centres of aa'a'\ hh'h" passes through i?, and is perpendicular to SS'S". 121 (a).* Mr. Casey has given a solution of the problem we are considering, depending on the following principle due to him : If four circles be all touched by the same fifth circle, the lengths of their common tangents are connected by the following relation T2 .34+14 .23 ±13.24 = 0, where 12 denotes the length of a common tangent to the first and second circles, &c. This may be proved by expressing each common tangent in terms of the length of the line joining the points where the circles touch the common touching circle. Let JR be the radius of the latter circle whose centre is 0, r and r' of the circles whose centres are A and B, then, from the isosceles triangle a Oh, we have ah = 2R sin^aO^. But from the triangle A OB, whose base is D, and sides B — r, B — r', we have o jy^ — ir- r'f s\VL^haOl> = —r^ — \ , „ ,v . Now the numerator of this frac- ^ 4:{B — r){B — r) tion is the square of the common tangent 12, hence B.U s'[R-r)[B-r)' But since the four points of contact form a quadrilateral in- scribed in a circle, its sides and diagonals are connected by the relation ah.cd-\-ad.'bc = ac.hd. Substitute in this equation the expression just given for each chord in terms of the corre- sponding common tangent, and suppress the numerator B' and the denominator sJ[B - r) {R - r') [B - r") {B - r'") which are common to every term, and there remains the relation which we are required to prove. 121 [h). Let now the fourth circle reduce itself to a point, this will be a point on the circle touching the other three, and * In order to avoid conf asion in the references, I retain the numbering of the articles in the last edition, and mark separately those articles which are added in this edition. Q ] 14 PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES. 41, 42, 43 will denote the lengths of the tangents from that point to these three circles. But the lengths of these tangents are (Art. 90) the square roots of the results of substituting the coordinates of that point in the equations of the circles. We see then that the co-ordinates of any point on the circle which touches three others must fulfil the relation 23 ^'{S)±3i ^/{S')±12 s/{S")=0. If this equation be cleared of radicals it will be found to be one of the fourth degree, and when 23, 31, 12 are the direct common tangents, it will be the product of the equations of the two circles (see fig., p. 112) which touch either all externally or all internally. 121 (c). The principle just used may also be established without assuming the relation connecting the sides and dia- gonals of an inscribed quadrilateral. If on each radius vector OP to a curve, we take, as in Ex. 4, p. 96, a part OQ in- versely proportional to OP, the locus of ^ is a curve which is called the inverse of the given curve. It is found with- out difficulty that the equation of the inverse of the circle X- + if + 2gx-\-2fy-\-c is c (a;' + if) + 2gx + 2/?/ + 1 = 0, which denotes a circle, except when c = 0, (that is to say, when the point is on the circle) in which case the inverse is a right line. Conversely, the inverse of a right line is a circle passing through the point 0. Now Mr. Casey has noticed that if we are given a pair of circles, and form the inverse pair with regard to any point, then the ratio of the square of a common tangent to the product of the radii is the same for each pair of circles.* For if in <f+f' — c-, which (Art. 80) is 7*^, we substitute for g-, f-, c] - , - , - , we find that the radius of the inverse circle is r divided by c ; and if we make a similar substitution in c -\- c — 2gg - 2ff which (Ex. 1, p. 102) is D'^ — r^ — r"\ we get the same quantity divided by cc. Hence the ratio of D^ — r' — r"^ to rr is the same for a pair of circles * This is equivalent (see Ex. 8. p. 103) to saying that the angle of intersection is the same for each pair, as may easily be proved geometrically. PROPERTIES OF A SYSTEM OF TWO OR i\IORE CIRCLES. 115 and for the inverse pair; and therefore, so is also the ratio to rr' of I)'' - [r f r')^ Consider now four circles touching the same right line in four points. Now the mutual distances of four points on a right line are connected by the relation 12.34+14.32 = 13.24; as may easily be proved by the identical equation [b -a)(d-c) + [d- a) [c-h) = {c- a) [d-h), where a, 5, c, d denote the distances of the points from any origin on the line. Thus then the common tangents of four circles which touch the same right line are connected by the relation which is to be proved. But if we take the inverse of the system with regard to any point, we get four circles touched by the same circle ; and the relation subsists still ; for if the equation be divided by the square root of the products of all the radii, it consists of members — - — j- . ,, „ ,„^ , &c., \/\rr) \/\i' r ] which are unchanged by the process of inversion. The relation between the common tangents being proved In this way,* we have only to suppose the four circles to become four points, when we deduce as a particular case the relation connecting the sides and diagonals of an inscribed quadrilateral. This method also shows that, in the case of two circles which touch the same side of the enveloping circle, we are to use the direct common tangent ; but the transverse common tangent when one touches the concavity and the other the convexity of that circle. Thus then we get the equation of the four pairs of circles which touch three given circles, 23 s/{S)±Jl s/{S')±\2 v'(/S"') = 0. When 12, 23, 31 denote the lengths of the direct common tan- gents, this equation represents the pair of circles having the given circles either all inside or all outside. If 23 denotes a direct common tangent, and 31, 12 transverse, we get a pair of circles each having the first circle on one side, and the other two on the other. And similarly we get the other pairs of circles by taking in turn 31, 12 as direct common tangents, and the other common tangents transverse. * Another proof will be given in the appendix to the next chapter. ( IIG ^CHAPTER IX. APPLICATION OF ABRIDGED NOTATION TO THE EQUATION OF THE CIRCLE. 122. If we have an equation of the second degree expressed in the abridged notation explained in Chap. IV., and If we desire to know whether it represents a circle, we have only to transform to X and y co-ordinates, by substituting for each abbreviation (a) its equivalent (aJCOsa + ?/ sina— p); and then to examine whether the coefficient of xy in the transformed equation vanishes, and whether the coefficients of j? and of y^ are equal. This is suffi- ciently illustrated in the examples which follow. When loiU the locus of a point he a circle if the product of 'perpendiculars from it on tivo op)posite sides of a quadrilateral he in a given 7'atio to the product of perpendiculars from it on the other tioo sides ? Let a, yS, 7, S be the four sides of the quadrilateral, then the equation of the locus is at once written down ay = A;/3S, which represents a curve of the second degree passing through the angles of the quadrilateral ; since It is satisfied by any of the four suppositions, a = 0, /3 = 0; a = 0, S = 0; /3 = 0, 7 = 0; 7 = 0, 8 = 0. Now, in order to ascertain whether this equation represents a circle, write It at full length {x cosa +y sina - p) [x C0S7 + ?/ sin7 -p") = k[x cos/3 + 3/ slnyS-^?') [x cosS +y sinS —p")- Multiplying out, equating the coefficient of ic'"' to that of ?/^, and putting that of xy = 0, we obtain the conditions cos (a + 7) = k cos(/3 + B] ; sin (a + 7] = ^ sin (yS + 8). Squaring these equations, and adding them, we find k=±l ; and if this condition be fulfilled, we must have a + 7 = /Q + 8, or else = 180° + /3 + 8 ; whence a- ^ = h-y^ or 180 + 8-7. Tin: C'IKCLE— ABRIDGED NOTATION. 117 Recollecting- (Art. 61) that a - /3 \a the sup]>lemcnt of that angle between a and /3, m which the origin lies, we see that this condition will be fulhlled if the quadrilateral formed by a/378 be inscribable in a circle (Euc. ill. 22). And it will be seen on examination that when the origin is within the quadrilateral we are to take ^ = — 1, and that the angle (in which the origin lies) between a and /3 is supplemental to that between 7 and 8 ; but that we are to take A; = + 1, w^ien the origin is without the quad- rilateral, and that the opposite angles are equal. 123. When ivill the locus of a point he a circle^ if the square of its distance from the base of a triancjle he in a constant ratio to the product of its distances from the sides ? Let the sides of the triangle be a, /8, 7, and the equation of the locus is a^ = ky^ If now v;e look for the points where the line a meets this locus, by making in it a = 0, we obtain the perfect square 7^ = 0. Hence a meets the locus in two coincident points, that is to say (Art. 83), it touches the locus at the point a7. Similarly, j3 touches the locus at the point /By. Hence a and 13 are both tangents, and 7 their chord of contact. Now, to ascertain whether the locus is a circle, "writing at full length as in the last article, and applying the tests of Art. 80, we obtain the conditions cos (a + /S) = k cos 27 ; sin (a + /S) = Z; sin 27 ; whence (as in the last article) we get k=l^ a — 7 = 7 — /3, or the triangle is isosceles. Hence we may infer that if from any point of a circle perpjendiculars he let fall on any two tangents and on their chord of contact^ the square of the last loill he equal to the rectangle under the other ttvo. Ex. When ^vill the lociis of a point be a cu-cle if the sura of the sciuares of the perpendiculars from it on the sides of any triangle be constant ? The locus is a^ + ^ + y"^ — c- : and the conditions that this should represent a chcle are cos 2a + cos 2^ + cos2y = ; sin2a + sin2j8 + sin2'y = 0. cos 2a = — 2 cos(^ + y) cos {(3 - y) ; sin 2a = - 2 sin(/3 + y) cos(/3 - y). Squaring and adding, 1 = 4 C0S2 03 - y) ; /3 - y = 60=^. And so, in like manner, each of tlie other two angles of the triangle is proved to be 60°, or the triangle must be equilateral. 118 THE CIRCLE — AHRIDGED NOTATION. ] 24. 2b obtain the equation of tlie circle circumscribing the triangle formed by the lines a = 0, /8 = 0, 7 = 0. Any equation of the form Wy 4- in'ya + ??a/3 = 0, denotes a curve of the second degree circumscribing the given triangle, since it is satisfied by any of the suppositions a = 0,/8 = 0; /3 = 0, 7 = 0; 7 = 0, a = 0. The conditions that it should represent a circle are found, by the same process as in Art. 122, to be I cos (/3 + 7) + m cos (7 + a) + w cos (a -f yS) = 0, I sin (/3 + 7) + m sin (7 + a) + n sin (a + /3) = 0. Now we have seen (Art. 65) that when we are given a pair of equations of the form la + ?n/3' + 927' = 0, la 4 m^" + ^7" = 0, ?, ?», n must be respectively proportional to /3'7"— yS'V, 7'a" — 7"a', a'/3" — a"y3'. In the present case then Z, «z, n must be pro- portional to sin(/3 — 7), sin (7 — a), sin(a-/3), or (Art. 61) to sin^, sini?, sInO. Hence the equation of the circle circura- sci'ibing a triangle is /87 sin A + 7a sin B+a^ sin (7= 0. 125. The geometrical interpretation of the equation just found deserves attention. If from any point we let fall per- pendiculars (9P, OQ^ on the lines a, /S, then (Art. 54) a, ^ are the lengths of these perpendiculars; and since the angle be- tween them is the supplement of C, the quantity a^ sin is double the area of the triangle OPQ. In like manner, 7a sin B and y37 sin A are double the triangles OPR^OQR. Hence the quantity ^7 sin^ + 7a sin 5 4- a/3 sinC is double the area of the triangle PQR^ and the equation found in the last article asserts, that if the point be taken on the circumference of the circumscribing circle, the area PQB will vanish, that is to say (Art. 36, Cor. 2), the three points P, Q^ R will lie on one right line. THE CIRCLE — ABRIDGED KOTATION. 119 If it were required to find the locus of a point from wliicli, if we let fall perpendiculars on the sides of a triangle, and join their feet, the triangle PQR so formed should have a constant magnitude, the equation of the locus would be yS7 sin^ +7a sin ^ + a/3 sin (7= constant, and, since this only differs from the equation of the circum- scribing circle in the constant part, it Is (Art. 81) the equation of a circle concentric with the circumscribing circle. 126. The following inferences may be drawn from the equa- tion l^'y + m<yiy. + na^ = 0, whether or not ?, «<, 7i have the values sin^, sin5, sinO; and therefore lead to theorems true not only of the circle but of any curve of the second degree circum- scribing the triangle. Write the equation In the form 7(?/3 + wa) + wa/3 = 0; and we saw in Art. 124 that 7 meets the curve in the two points where it meets the lines a and ^ ; since if we make 7 = in the equation, It reduces to a/3 = 0. Now, for the same reason, the two points in which Z/3 4- ma meets the curve, are the two points where it meets the lines a and /3. But these two points coin- cide, since ?/3 + ma passes through the point a^. Hence the line ?/3 -1- ?«a, which meets the curve in two coincident points, is (Art. 83) the tangent at the point a/9. In the case of the circle the tangent is a sin2?-f y8 sinu.4. Now we saw (Art. 64) that a sin^ + ^S sin^ denotes a parallel to the base 7 drawn through the vertex. Hence (Art. 55) the tangent makes the same angle with one side that the base makes with the other (Euc. iii. 32). Writing the equations of the tangents at the three vertices In the form -+-=0, -+7=0, - + ^=0, m n ' n I L m we see that the three points In which each intersects the opposite side are In one right line, whose equation Is ";+^ + ^-=0. . t m n Subtracting, one from another, the equations of the three tangents, we get the equations of the lines joining the vertices 120 THE CIKCLE — ABRIDGED NOTATION. of tlic original triangle to the corresponding vertices of the triangle formed by the three tangents : viz., ^-1-0 ^---0 ^-^=0 VI n ^ n I ' I m three lines which meet in a point (Art. 40).* 127. If a'ySVj a"/3"7" be the co-ordinates of any two points on the cnrvc, the equation of the line joining them is _L — _ 4- — '— = • a'a" ^ /3'/3" ^ 77" ' for if we substitute in this equation a'/3'7' for a^7, the equation is satisfied, since a'yS'V satisfy the equation of the curve which may be written I m n ^ - + -^ +-=0. a /3 7 In like manner the equation is satisfied by the co-ordinates a"yS"7". It follows that the equation of the tangent at any point a'/3'7' may be written and conversely, that if Xa -}- jx^ + 1/7 = is the equation of a tangent, the co-ordinates of the point of contact a^'^' are given by the equations I m n Solving for a', yS', 7' from these equations, and substituting in the equation of the curve, which must be satisfied by the point a'/3'y\ we get V(^^) + \/(«i/*) + ^{nv) = 0. This is the condition that the line \a + yu.y8 + vy may touch l^y + mya. + na^ '^ or it may be called (see Art. 70) the tan- gential equation of the curve. The tangential equation might also be obtained by eliminating 7 between the equation of the line and that of the curve ; and forming the condition that the resulting equation in a : /3 may have equal roots. * The theorems of this article are by M. BobiOier (Annales des Mathematiques, Yol. xviii., p. 320). The first eqimtion of the next article is by M. Hermes. THE CIRCLE — ABRIDGED NOTATION. 121 128. To find the conditions that the general equation of the second degree in a, /3, 7, aa' + 1)^' + cf -i- 2//37 + 29-7^^ 4- Ih'x^ = 0, may represent a circle. [Dublin Exam. Papers, Jan. 1857]. It is convenient to avail ourselves of the result of Art. 124. Since the terms of the second degree, x^ + y\ are the same in the equations of all circles, the equations of two circles can only differ in the linear part; and if S represent a circle, an equation of the form S-\- lx-\- my-\- ?« = may represent any circle what- ever. In like manner, in trilinear co-ordinates, if we have found one equation which represents a circle, we have only to add to it terms la + m^ + 727, (which in order that the equation may be homogeneous we multiply by the constant a sin^l+/3sin5+7sinC) and we shall have an equation which may represent any circle whatever. Thus then (Art. 124) the equation of any circle may be thrown Into the form [la + m^ -f 727) (a sin ^ + /3 sin 5 + 7 sin C ) + k {^y sin A+<ya smB+ a/3 sin C ) = 0. If now we compare the coefficients of a^, ^'\ <y'^ in this form with those in the general equation, we see that, if the latter represent a circle, it must be reducible to the form i^-, «+ . -d)S+ -^^7) (a sin^ + /3 sini?+7 sin (7) \sm^4 smij sm G / + k {/3y sin A + ya s'mB + a^ sin C) = 0, and a comparison of the remaining coefficients, gives 2/ smB sin C = c siifB+h sin'"' C + h sin A smB sin C, 2g sin C sin^ = a sin"C + c sin^yl + A; sin -4 sin 5 sin C, 2h smA sinJ5 = h sxvl'A + a sm^B + h sin .4 sin^ sin (7, whence eliminating h^ we have the required conditions, viz. h sin' C+c sin^jB-2/sin5 sin C= c sin'^ + « sin" C- 2g sin C sin^ = a sm^B-^h &\vl^A - 2A sin^ sini?. If we have the equations of two circles written In the form [la + mfi + ny) (a sin ^ + yS sin B + y sin G ) 4- h [^y smA + 7a sln5 + a/3 sin C) = 0, [la + m^ + n'y) [a sin A + ^ s\\\B ^ y sin C) + h [^y sln^l + 7a sin 5+ a^ sin C) = 0, R 122 THE CIRCLE— ARRIDGED NOTATION. it is evident that tlieir radical axis is la + ?«yS + ^7 — [To. + m'^ + 7/7), and that la + m^ + ny is the radical axis of the first with the circumscribing circle. Ex. 1. Verify that a/3 — y- represents a circle H A = B (Ai-t. 123). Tlie equation may be written a/3 sinC+ /3y sin^ + yo sini? - 7 (a sin^l + /3 sin5 + y sinC) = 0. Ex. 2. The three middle points of sides, and the three feet of perpendiculars lie on a circle. The equation a? sin /I cos.l + /3- &iuB cosB + y- sinC cosC— {(3y sin A + ya sini> + a/3 sinC) = 0, represents a cui-ve of the second degiee passing through the points in question. For if we make y = 0, we get a- sin^ cos^ + /3- sinS cosjB — a/3 (sin^ cos5 + sini? cos^) = 0, the factors of which are a sin ^ — /3 sin 5 and a cos ^4 — /3 cos 5. Now the curve is a circle, for it may be written (a 003^4 + /3 cosiJ + y cosC) (a sin^ + /3 sin5 + y sinC) - 2 03y sin .4 + ya sini? + a/3 sinC) = 0. Thus the radical axis of the circumscribmg cu'cle and of the cu'cle through the middle points of sides is a cos^ + /3 cosi? + y cosC, that is, the axis of homology of the given triangle with the triangle formed by joining the feet of perpendiculars. 129. We shall next show how to form the equations of the circles which touch the three sides of the triangle a, yS, 7. The general equation of a curve of the second degree touching the three sides, is Z"V + wi'yS' + wV - 2?n??/37 - 2»?7a - ilma.^ = 0.* . Thus 7 is a tangent, or meets the curve in two coincident points, since if we make 7 = in the equation, we get the per- fect square l^d^-\- m^l3^ — 2hna^ = 0. The equation may also be written in a convenient form V(/a) + V(wyS) + V(«7) = ; for if we clear this equation of radicals, we shall find it to be identical with that just written. * Strictly speaking, the double rectangles in this equation ought to be written with the ambiguous sign +, and the argument in the text would apply equally. If however we give all the rectangles positive signs ; or if we give one of them a positive sign, and the other two negative, the equation does not denote a proper curve of the second degree, but the square of some one of the lines la + m(i + ny. And the form in the text may be considered to include the case where one of the rectangles is negative and the other two positive, if we suppose that I, m, or n may denote a negative as well as a positive quantity. THE CIRCLE — ABRIDGED NOTATION. 123 Before determining tlic values of /, w«, «, for which the equa- tion represents a circle, we shall draw from it some inferences which apply to all curves of the second degree inscribed in the triangle. Writing the equation In the form 727 ('^7 - 2?a - 2wiyS) + {hx - m^f = 0, we see that the line {la.- r/2/3), which obviously passes through the point a/3, passes also through the point where 7 meets the curve. The three lines, then, which join the points of contact of the sides with the opposite angles of the circumscribing triangle are la — m/3 = 0) ?»/3 — ny = 0, ny — /a = 0, and these obviously meet in a point. The very same proof which showed that 7 touches the curve shows also that ny — 2/a — 2ml3 touches the curve, for when this quantity is put =0, we have the perfect square [la— in^Y = 0', hence this line meets the curve in two coincident points, that is, touches the curve, and la — m^ passes through the point of con- tact. Hence, if the vertices of the triangle be joined to the points of contact of opposite sides, and at the points where the joining lines meet the circle again, tangents be drawn, their equations are 2?a + 2»iy3 - ny = 0, 2w/3 + 2n7 - Za = 0, 2ny + 2la - m/3 = 0. Hence we infer that the three points, where each of these tan- gents meets the opposite side, lie In one right line, la + ?n/S + ny = 0, for this line passes through the intersection of the first line with 7, of the second with a, and of the third Avith /3. 130. The equation of the chord joining two points a'/3'y', a"y3"7", on the curve is a V(0 W{l3'y") + V(/3"7')} + ^ V(m) W{y'a") + VlV'a')} + 7V(n){V(a'/3") + V(a"iS')l=0.* For substitute a', )8', 7' for a, /3, 7, and it will be found that the quantity on the left-hand side may be written {V(a'/3'7") + V(/3'7'a") + V(7'a'/3")} {V(^a') + V(m/3') + x/{ny']] - V(a'/3'7') y{la") + ^{m^")j^^^/(^iylh * This equation is Dr. Hart's. 124 THE CIKCLE — ABRIDGED NOTATION. which vanishes, since the ])oints are on the curve. The equation of the tangent is found by putting a", yS", 7" = a', /8', 7' in the above. Dividing by 2 V(a'/3'7'j, it becomes Conversely, if Xa + /i/3 + V7 is a tangent, the co-ordinates of tlic point of contact are given by the equations ►Solving for a'^'y\ and substituting in the equation of the curve, ^veget / ,n n r + - + - = 0, A /Ji V which is the condition that \a + /j,/3 + vy may be a tangent ; that is to say, is the tangential equation of the curve. The reciprocity of tangential and ordinary equations will be better seen, if we solve the converse problem, viz. to find the equation of the curve, the tangents to which fulfil the condition I m n -+- + _ = 0. K jJi V We follow the steps of Art. 127. Let X'a + /t'/S + v'7, 'X,"a + fi"/3 + v"<y be any two lines, such that V/aV, X'ijl'v" satisfy the above condition, and which therefore are tangents to the curve whose equation we are seeking ; then ZA- mil nv VT77 -f ^ ,, + -7-77 = 0, A, A, fjb fJi V V is the tangential equation of their point of intersection. For (Art. 70) any equation of the form AX + B^l^Cv = 0^ is the condition that the line Xa -f- yttyS + vy should pass through a certain point, or, in other words, is the tangential equation of a point ; and the equation we have written being satisfied by the tangential co-ordinates of the two lines is the equation of their point of intersection. Making X', yu,', v' = X", /*", v" we learn that if there be two consecutive tangents to the curve, the equation of their point of intersection, or, in other words, of their point of contact, is ZX mu, nv cr^ + -^ + -75 = 0. THE Cri:CLE — ABRIDGED NOTATION. 12.5 The co-ordinates then of the point of contact are I in n Solving for X,', /a', v' from these equations, and substituting in tlic relation, which by hypothesis Xf^'v' satisfy, we get the required equation of the curve V(/a) + ^/{mj3) + v/(n7) = 0. 131. The conditions that the equation of Art. 129 shouhl represent a circle are (x\rt. 128) 7«^ fi'nfC+7i^ s'm'^B+2mn shxB sm C=n^ sia^A + P sin'C + 2nl sin A sin (7= P sm^ B+m^ sin^A + 2lm sin A sin/?, or m sinC+ w sin5=± [n smA + I sinC) = ±{l s'mB + m sinyl). Four circles then raay be described to touch the sides of the given triangle, since by varying the sign, these equations may be written in four different ways. If we choose in both cases the + sign, the equations are I sin C — m sin G+ti (sin A — sin B) = 0', I s\nB + m [s\nA — sinC) —n sin5=0. The solution of which gives (see Art. 121), /=sin^ (sin^+sinC — sin^), ?« = sin5(sin(7+ sin^l — sini?), n = sin C (sin ^ + sin 5— sin C). But since in a plane triangle sini?+ sinO- sin^ = 4 cos^A sin ^5 sin ^6', these values for ?, yn, n are respectively proportional to cos" ^--4, cos'^^i?, cos'' J (7, and the equation of the corresponding circle, which is the inscribed circle, is cosi^ ^(a) + cos^B V(yS) + cos |(7 \/{y) = 0,* * Dr. Hart derives this equation from that of the circumscribing circle as follows : Let the equations of the sides of the triangle formed by joining the points of contact of the inscribed cu-cle be a' = 0, /3' = 0, y' = ; and let its angles be A', B', C ; then (Art. 124) the equation of the circle is (i'y' sin A' + y'a' sinB' + aft' sinC" = 0. But (Art. 123) for every point of the cu-cle wo hare a'- = fty, ft'- = ya, y"' = aft, and it is easy to see that 4' = 90 - 4-4, &c. Substituting these values, the equation of the circle becomes, as before, cos^.4 4{a) + cos IB lift) + cos.jC J(y) = 0. 126 THE CIRCLE — ABRIDGED NOTATIOX. or a' cos'i^l + yS" co&'},B + y' cos'},C- 2a;3 cos'^A cos'^B - 2/37 cos- ^5 cos' I C- 27a cos' t C cos'^A = 0. We may verify that this equation represents a circle by writing it in the form fa cos* I A ^ cos* IB 'ycos*},C\. . 4 . o • n . • n\ — , — i 1- ^-. n - -+ ^ - A (a si»^ + p sin5+ 7 sinC V sm^ sin5 sinC / (/37 sin^ + 7a sin^ -I- a/3 sin 6') = 0. 4 cos"M cos'^J5 cos^\C svnA sini? sin (7 In the same way, the equation of one of the exscribed circles is found to be a: cos" }yA + ff' sin" l^B + 7' sin*|- G - 2l3y sin' -|5 sin' J- G + 27a sin* ^ G cos' ^A + 2a^ sin""' ^B cos' \A = 0, or cos hA V (- a) + sin ^B V(/3) + sin ^ 6' V (7) = 0. The negative sign given to a is in accordance with the fact, that this circle and the inscribed circle lie on opposite sides of the line a. Ex. Find the radical axis of the inscribed circle and the circle through the middle points of sides. The equation formed by the method of Art. 128, is 2 cos^^^ cos- ^5 cos-^C [a cosA + (3 cosB + y cosC} . , . „ . ^/ cosH,4 ^cos*J-5 cos*JC\ \ sm^ smB ' sm(7 / Divide by 2 cos^^ cos^B cos^C, and the coefficient of a in this equation is cos^.,-1 {2 cos^^A sinJ^5 sin J^C— cos^ cos;}£ cos^C}, or cos^^ sin J- {A — B) sin^ {A — C). The equation of the radical axis then may be written a cos^^ /ScosiJS 7 cos^C _ 8in'i(5-C) "*" sinJ(C-u4) "^ sui^{A-B) ~ ' and it appears from the condition of Art. 130, that this line touches the inscribed circle the co-ordinates of the point of contact being sin-J(JK— C), sin-^(C— --1), sin-|(^— 5), These values shew (Art. 66) that the point of contact lies on the line joining the two centres whose co-ordinates are 1, 1, 1, and cos {B — C), cos (C — A), cos {A — B). In the same way it can be proved that the circle through the middle points of sides touches all the circles which touch the sides. This theorem is due to Feuerbach.* * Mr. Casey has given a proof of Feuerbach's theorem, which will equally prove Dr. Hart's extension of it, viz. that the circles which touch thi-ee given circles can be distributed into sets of four, all touched by the same circle. The signs in the follow- ing correspond to a triangle whose sides are, in order of magnitude «, b, c. The exscribed circles are numbered 1, 2, 3, and the inscribed 4 ; the lengths of the direct and transverse common tangents to the first two circles are written (12), (12)'. Then THE CIRCLE— ABRIDGED NOTATION. 127 132. If the equation of a circle in trilinear co-ordinates is equivalent to an equation in rectangular co-ordinates, in which the coefficient of x^+}f is wi, then the result of substituting in the equation the co-ordinates of any point is m times the square of the tangent from that point. This constant m is easily deter- mined in practice if there be any point, the square of the tangent from which is known by geometrical considerations ; and then the length of the tangent from any other point may be inferred. Also, if we have determined this constant m for two circles, and if we subtract, one from the other, the equations divided respec- tively by m and ?«', the difference which must represent the ra- dical axis, will always be divisible by a &\nA 4 /3 sin -6 + 7 sin (7. Ex. 1.- Find the value of the constant m for the cu'cle through the middle points of the sides, a^ sin^ cos^l + /3- smB cosB + y- sinC cosC — (iy svaA — ya sin^ — a/3 sinC= 0. Since the circle cuts any side y at poiats whose distances from the vertex A, are Jc and b cosA, the square of the tangent fi-om A is ^bc cosA. But since for A we have fi = 0, y = 0, the ra'?ult of substituting in the equation the co-ordinates of -4 is a- sin A cos^, (where a' is the perpendicular from A on the opposite side), or is be sin ^ sin 5 sin C cos J. It follows that the constant m is 2 sin ^4 sinB sinC Ex. 2. Find the constant m for the circle /3y sin^ + ya sinjB + aj3 sinC. If from the preceding equation we subtract the linear terms (a cos^ + /3 cosB + y cosC) (a sin^ + /3 smB + y sinC), the coefficient of x- + y"^ is imaltered. The constant therefore for /3y sin^, ic. is — sin^i sinjB siuC. Ex. 3. To find the distance between the centres of the inscribed and circumscribing circle. We find Z*- — Ji^, the square of the tangent from the centre of the inscribed to ^.l, ■ -u- • 1 u V ^v *• « ^1 r2(sin^ + sinB + sinC) the circumscnbmg cu-cle, by substitutmg a = /3 = y = r, to be . — ~ — ^ — r~^^7^ — ' or, by a well-kno\vn formula, = — 2i?r, Hence D- = JR^ — 2Er. Ex. 4. Find the distance between the centres of the inscribed circle and of that through the middle points of sides. If the radius of the latter bep, making use of the formula, sinA cos A + sinjB cosB + sinCcosC= 2 sin A sinB smC, we have D"^ — p" — 7'- — i-R. because the side a is touched by the circle 1 on one side, and by the other three circles on the other, we have (see p. 115) (13)' (24) =: (12)' (34) + (14)' (23). Similarly (12)' (34) + (24)' (13) = (23)' (14), (23)' (14) = (13)' (24) + (34)' (12), whence, adding, we have, (24)' (13) = (14)' (23) + (34)' (12) : shovraig that the foiu- circles are also touched by a circle, having the circle 4 on one side, and the other three on the other. 128 DET li 1{ M I X A N T N ( )T AT ION. Assuming then that wc otherwise know R = 2/), we have D = r — p ; or the circles touch. Ex. o. Find the constant m for the equation of the inscribed circle given above. Ans. 4 cos- ^.4 cos-^B cos-^C. Ex. G. Find the tangential equation of a circle whose centre is a'^'y' and radius r. This is investigated as in Art. 8(j, Ex. 4 ; attending to the formida of Art. 61 ; and is found to be {Xa + fifi' + vy')- - r- {X- + /u- + j/- - 2ixv cos .4 - 2i/X cos 5 - 2/V cosC). The corresponding equation in a, (i, y is deduced from this by the method afterwards explained, Art. 285, and is r- (a sin^ + /3 sin5 + y sinC)- = {(iy' - fi'yY + {ya' - y'a)- + {afi' - a'(i)- -2 {ya'-y'a)[ali'-a'ft) cosA-2{afi'- a'ji){fiy'-p'y) C0sIi-2(Jiy'~li'y){ya'-y'a) COsC. This equation also gives an expression for the distance between any two points. Ex. 7. The feet of the peiToendicixlars on the sides of the triangle of reference from the points o', /3', y' ; — , -;, , — ; (see Art. 55) lie on the same circle. By the help of Ex. 6, p. 60, its equation is found to be (/3y sin^+ya smB+ajB sin C)(o' sin 44-/3' sin ^+y' sin C){fi'y' sin -4+y'a' sin B+a'(i' sinC) = sin-4 sin^ sin C {a sin A + (3 sin^ + y sin C) fa«'(/j'+y'cos.4) (y'+/3'cos4) /3/3'(y'+«'cos^') (a'+y 'cos5) yy'(a'+/3'cos C) {(i'+a'cosC)] \ sinA Binli sin 6' J DETERMINANT NOTATION. 132(a). In the earlier editions of this book I did not venture to Introduce the determinant notation, and In the preceding pages I have not supposed the reader to be acquainted with it. But the knowledge of determinants has become so much more common now than It was, that there seems now no reason for excluding the notation, at least from the less elementary chapters of the book. Thus the double area of a triangle (Art. 36), and the condition (Art. 38) that three lines should meet in a point, may be wi'Itten respectively 1, 1, 1 A B, C ^1) ^25 ^3 ^', B, C S'li 3/.J % = 0, A" B'\ C" = 0. The equations of the circle through three points (Art 94), and of the circle cutting three at right angles (Ex. 2, p. 102), may be written respectively x' +/, X, Vi 1 x^-^y\ X, y-, 1 ^"' +f\ X, y\ 1 <^'j -9'i -/, 1 x"'^y"-% x'\ y\ 1 c", -.9"i -/', 1 x"-^^y"'\x" J y" 1 = 0, c , -9" -/", 1 0. DETEKM IN ANT NOTATION. 129 The equation of the latter circle may also be formed by the help of the principle (Ex. G, Art. 102), as the locus of the point whose polars with respect to three given circles meet in a point, in the form ^•^f)\ y+f\ ij'-^ +fy +c' «+/? 3/+/\ fj"'-^ +f"y +^'" ^ -^ 9"\ y +/'"? !j"'^ +f"y + c" = 0. The corresponding equation for any three curves of the second degree will be discussed hereafter. Ex. 1. To find the condition for the co-existence of the equations ax + hy + c = u'x + Uy + c' = a"x + h"y + c" = a"'x + h"'y + c'". Let the common value of these quantities be A. ; then eliminating x, y, X fi'om the four equations of the form ax + by + c =\, we have the result in the form of a determinant I 1, 1, 1, 1 «, «', a", a'" b, b', b", b'" c, c', c", c'" = 0, or A+C=B + I), where .-1, B, C, D are the four minors got by erasing in tuni each column, and the top row in this determinant. To find the condition that four lines should touch the same circle, is the same as to find the condition for the co-existence of the equations a — ^ — y = h. In this case the determinants .4, B, C, D geometrically represent the product of each side of the quadrilateral formed by the fom- lines, by the sines of the two adjacent angles. Ex. 2. To find the relation connecting the mutual distances of four points on a circle. The investigation is Mr. Cay ley's (see Lessons on Higher Algebra, p. 21). Multiply together according to the ordinaiy rule the determinants xi' + yi-, -2x„ -2^1, 1 1, J"i, ^1, Xi- -t- yr «2- + y^-, -2x„, -2.y,, 1 1, X,, 1/21 x.^ + y^ ^^ + 2/3-, -2ar3, -2^/3, 1 X 1, a-3, Vzi ^z^ + 2/3- x^ + V^, - 2a-4, -2^4, 1 1, »•■.» 2/4, a-1- + y^- which are only different ways of writing the condition of Art. 94 ; and ^v^e get the required relation 0, (12)^ (13)S (14)^ (12)2, 0, (23)^ (24)2 (13)2, (23)2, 0, (34)2 (14)2, (24)2, (34)2, =0, where (12)2 ig tjig square of the distance between two points. This detenninant ex- panded is equivalent to (12) (84) + (13) (24) + (14) (23) = 0, Ex. 3. To find the relation connecting the mutual distances of any four points in a plane. This investigation is also Mr. Cayley's {Lessons on Higher Algebra, p. 22). Prefix a unit and cyphers to each of the determinants in the last example ; thus 1, 0, &c. 0, • 22/1, 1 0, 0, 0, 1 1, .r,, ?/,, .r,2 + y,' 130 DETERMINANT NOTATION. We have then four rows and five column?, the determmant formed fi-om which accord- ing to the rules of multiplication, must vanish identicallJ^ But this is n, 1, 1, 1, 1 1, 0, (12)S (13)^ (14)^ 1, (12)2, 0, (23)2, (24)2 1, (13)2, (23)2, 0, (34)2 1, (14)2, (24)2, (34)2^ =0; which expanded, is (12)2 (34)2 1(12)2 ^ (34)2 _ (13)2 _ (14)2 _ (93)2 _ (24)2} + (13)2 (24)2 {(13)2 ^ (24)2 _ (12)2 _ (14)2 _ (23)2 _ (34)2} + (14)2 (23)2 {(14)2 + (23)2 _ (12)2 _ (13)2 _ (24)2 _ (34)2| + (23)2 (34)2 (42)2 + (14)2 (43)2 (31)2 + (12)2 (24)2 (41)2 + (12)2 (23)2 (31)2 _ q. If we wiite in the above a, b, c for 23, 31, 12 ; and R + r, E + ?•', li + r" for 14, 24, 34, we get a quadratic in Ji, whose roots are the lengths of the radii of the circles touching either all externally or internally three circles, whose radii are r, r', r", and whose centres form a triangle whose sides are a, b, c. Ex. 4, A relation connecting the lengths of the common tangents of any five circles, may be obtained precisely as in the last example. Write down the two matrices 1, 0, 0, 0, 0, 0, 0, 0, 1 '2+/2 _,.'2, -2x', -2/, 2'-', 1 1, «', y', ?•', x'2 + I/'- — 1 ,'2 •"- + j/"2 - r"-, - 2.v" &c. -2/' 2r" 1 1, «"i y"> r", x""- + 2/"2 - J &c. . - where there are five rows and six columns, and the determinant formed according to the rules of multiplication must vanish. But tliis is 0, 1, 1, 1, 1, 1 1, 0, (12)2, (13)2, (14)2^ (15)2 1, (12)2, 0, (23)2, (24)2^ (25)2 1, (13)2, (23)2, 0, (34)2, (35)2 1, (14)2, (24)2^ (34)2^ 0, (45)2 1, (15)2, (25)2, (35)2, (45)2, = 0, where (12), &c, denote the lengths of the common tangents to each pair of cu-cles. If we suppose the circle 5 to touch all the others, then (15), (25), (35), (45), all vanish, and we get as a particular case of the above, Mr. Casey's relation between the common tangents of four circles touched by a fifth, in the form 0, (12)2, (13)2^ (14)2 (12)2, 0, (23)2, (24)2 (13)2, (23)2, 0, (34)2 (14)2, (24)2, (34)2, ( 131 CHAPTER X. PROPERTIES COMMON TO ALL CURVES OF THE SECOND DEGREE. DEDUCED FROM THE GENERAL EQUATION. 133. The most general form of the equation of the second degree Is ax' + 2hxy + hf + 2gx H- 2fy + c = 0, where a, 5, c, /, g^ li are all constants. It Is our object In this chapter to classify the different curves which can be represented by equations of the general form just written, and to obtain some of the properties which are common to them all.* Five relations between the coefficients are sufficient to deter- mine a curve of the second degree. For though the general equation contains six constants, the nature of the curve depends not on the absolute magnitude^ but on the mutual ratios of these coefficients; since. If we multiply or divide the equation by any constant, it will still represent the same curve. We may, therefore, divide the equation by c, so as to make the absolute term =1, and there will then remain but five constants to be determined. Thus, for example, a conic section can be described through five 2Joints. Substituting in the equation (as In Art. 93) the co-ordinates of each point {x'y') through which the curve must pass, we obtain five relations between the coefficients, which will enable us to determine the five quantities, - , &c. 134. We shall in this chapter often have occasion to use the method of transformation of co-ordinates ; and It will be useful * We shall prove liereafter, that the section made by any plane in a cone standing on a circular base is a curve of the second degree, and, conversely, that there is no curve of the second degree which may not be considered as a conic section. It was in this point of view that these curves were firet examined by geometers. We mention the property here, because we shall often find it convenient to use the terms '' conic section," or " conic," instead of the longer appellation. " curve of the second degree."' 132 GEXEKAL EQUATION UF THE SECOND DEGREE. to find what the general equation becomes when transformed to parallel axes through a new origin [x'y'). We form the new equation by substituting x + x for a;, and y -\^ y' for ?/ (Art. 8), and we get a[x^x'Y+2h{x^-x'){j/+y')+h[y+yJ-^2g {x-^x) -\-2f{y-^y') -^6=0. Arranging this equation according to the powers of the vari- ables, we find that the coefficients of x\ xy^ and y\ will be, as before, «, 2/^, h ; that the new g^ cj = ax + Jiy -f g ; the new/, /' = hx + l>y 4-/; the new c, c' = ax^ + llixy + hy"^ -f 2^-«' + 2_/j/' + e. Hence, «/ the equation of a curve of the second degree he trans- formed to imrallel axes through a neio origin^ the coefficients of the hifjhest 'powers of the variables ivill remain uncJianged^ ichiJe the nevi absolute term ivill be the result of substituting in the original equation the co-ordinates of the new origin.^ 135. Every right line meets a curve of the second degree in two real, coincident, or imaginary points. This is inferred, as in Art. 82, from the fact that we get a quadratic equation to determine the points where any line y = mx-\- n meets the curve. Thus, substituting this value of y in the equation of the second degree, we get a quadratic to determine the x of the points of intersection. In particular (see Art. 84) the points where the curve meets the axes, are determined by the quadratics ax' -f 2^:c + c = 0, Inf -1- 2// + c = 0. An apparent exception however may arise which does not present itself in the case of the circle. The quadratic may re- duce to a simple equation in consequence of the vanishing of the coefficient which multiplies the square of the variable. Thus a;^/ + 2?/''' + a; + 5?/ + 3 = is an equation of the second degree ; but if we make y—^, we get only a simple equation to determine the points of meeting of the axis of x with the locus represented. Suppose, however, that in any quadratic Ax' + 2Bx + C=0, the coefficient C * This is equally tnie for equations of any degree, as can be proved in like manner. GENERAL EQUATION OF THE SECOND DEGREE. 133 vanishes, we do not say that the quadratic reduces to a simple equation ; but we regard it still as a quadratic, one of whose roots is a; = 0, and the other x = j . Now this quadratic may be also written and we see by parity of reasoning, that if A vanishes, we ought to regard this still as a quadratic equation, one of whose roots • 1 ^ T 1 , 1 2J5 C' „„ IS - = 0, or a? = GO : and tbc other - = — — , or a: = j, . ibe X X U 2B same thing follows from the general solution of the quadratic, which may be written in either of the forms _ -B±s/{B''-AC) C ""' A ~ -B^ sJ{B^-ACy the latter being the form got by solving the equation for the reciprocal of o", and the equivalence of the two forms being easily verified by multiplying across. Now the smaller A is, the more nearly does the radical become =±B', and therefore the last form of the solution shows, that the smaller A is, the larger is one of the roots of the equation ; and that when A vanishes we are to regard one of the roots as infinite. When therefore we apparently get a simple equation to determine the points in which any line meets the curve, we are to regard it as the limiting case of a quadratic of the form O.x^ + 2Bx + C= 0, one of whose roots is infinite ; and we are to regard this as indi- cating, that one of the points where the line meets the curve is infinitely distant. Thus the equation, selected as an example, which may be written (?/ -f- 1) (a; + 2?/ + 3) = 0, represents two right lines, one of which meets the axis of x in a finite point, and the other being parallel to It meets it in an infinitely distant point. In like manner, if in the equation Ax' + 2Bx +(7=0, both B and C vanish, we say that it is a quadratic equation, both of whose roots are a; = ; so if both B and A vanish we are to say that it is a quadratic equation, both of whose roots are a; = oo . With the explanation here given, and taking account of infinitely distant, as well as of imaginary, points, we can assert that ever)/ right line meets a curve of the second degree in two points. 134 (JENER.VL EQUATION' OF THE SECOND DEGREE. 13G. The equation of the second degree transformed to polar co-ordinates* is [a cos'^ + 2h cos d smO + h sin" (9) p" + 2 (rj cos 6 +/siu 6) p + c=^0; and the roots of this quadratic are the two values of the length of the radius vector corresponding to any assigned value of 0. Now we have seen in the last article that one of these values will be infinite, (tliat is to say, the radius vector will meet the curve in an infinitely distant point,) when the coefficient of p^ vanishes. But this condition will be satisfied for two values of 6 J namely those given by the quadratic a + 2h tan + h tan'6' = 0. Hence, fhe^'e can be draicn through the origin tico real^ coin- cident^ or imaginary lines, which will meet the curve at an infinite distance ; each of which Hues also meets the curve in one finite point whose distance is given by the equation 2[g cos^-f/siu^)p + o = 0. If we multiply by p^ the equation a cos^ 6 + 2h cos 6 sin 6 -\-h sin" ^ = 0, and substitute for p cos^, p sln^ their values x and y, we obtain for the equation of the two lines ax" + 2hxy + hy"^ = 0. There are two directions in which lines can be drawn through any point to meet the curve at infinity ; for by transformation of co-ordinates we can make that point the origin, and the preceding pi'oof applies. Now it was proved (Art. 134) that «, 7i, h are unchanged by such a transformation ; the directions are therefore always determined by the same quadratic a cos'^ + 2/i cos^ sin ^ + Z* sin^^ = 0. Hence, if through any point two real lines can he drawn to meet the curve at infinity, parallel Jines through any other point will meet the curve at infinity. '\ * The follo\\-ing processes apply equally if the original equation had been in oblique co-ordinates. "We then substitute mp for x, and no for ?/, where m is -: and n is ' : ' (Art. 12) ; and proceed as in the text. sm 0) •f This indeed is evident geometrically, since parallel lines may be considered as passing through the same point at infinity. GENERAL EQUATION OF THE SECOND DEGREE. 135 137. The most important question we can ask, concerning Xheform of the curve represented by any equation, is, whether it be limited in every direction, or whether it extend in any direction to infinity. We have seen, in the case of the circle, that an equation of the second degree may represent a limited curve, while the case where it represents right lines shows us that it may also represent loci extending to infinity. It is necessary, therefore, to find a test whereby we may distinguish which class of locus is represented by any particular equation of the second degree. With such a test we are furnished by the last article. For if the curve be limited in every direction, no radius vector drawn from the origin to the curve can have an infinite value ; but we found in the last article, that when the radius vector becomes infinite, we have a + 2/i tan 6 + h tan''' ^ = 0. (1) If now we suppose h^ — ah to be negative, the roots of this equation will be imaginary, and no real value of 6 can be found which will render a cos"'' 6 + 2A cos 6 &m6^-h sin' ^ = 0. In this case, therefore, no real line can be drawn to meet the curve at infinity, and the curve will he limited in every direction. We shall show, in the next chapter, that its form is that represented in the figure. A curve of tills class is called an Ellipse. (2) If W - ah hejyositive, the roots of the equation rt+27i tan^ + & tan'^ = wiU be real ; consequently, there are two real values of 6 which will render infinite the radius vector to the curve. Hence, two real lines [ax^ -f 2kxt/ -f hy'^ = 0) can, in this case, be drawn through the origin to meet the curve at infinity. A cm've of this class is called an Hyperhola^ and we shall show, in the next chapter, that its form is that represented in the figure. 136 GENERAL EQUATION OF THE SECOND DEGREE. (3) If K' — ah = 0, the roots of the equation a + 2h tan^+5 tan'^ = will then be equal, and, therefore, the two directions in which a right line can be drawn to meet the curve at infinity will in this case coincide. A curve of this class is called a Parabola^ and we shall (Chap. XII.) show that its form is that here represented. The condition here found may be other- wise expressed, by saying that the curve is a parabola when the first three terms of the equation form a perfect square. 138. We find it convenient to postpone the deducing the figure of the curve from the equation, until we have first by transformation of co-ordinates, reduced the equation to its simplest form. The general truth however of the statements in the preceding article may be seen if we attempt to construct the figure represented by the equation, in the manner explained (Art. 16). Solving for y in terms of a?, we find (Art. 76) hy = - {Jix +/) ± ^/[[K' - ah) cc' + 2 [hf- hg) x + {f - he)]. Now, since by the theory of quadratic equations, any quantity of the form x^ + px + q is equivalent to the product of two real or imaginary factors {x — a.){x — ^)j the quantity under the radical may be written [h^ - ah) {x — a) [x — ^). If then h^ — ah be negative, the quantity under the radical Is negative, (and therefore y imaginary), when the factors a; — a, cc — /3 are either both positive, or both negative. Real values for y are only found Avhen x is intermediate between a and /3, and therefore the curve only exists in the space Included between the lines a? = a, x = 13 (see Ex. 3, p. 1 3). The case is the reverse when h^ - ah is positive. Then we get real values of y for any values of a?, which make the factors a? - a, x- ^ either both positive or both negative ; but not so If one Is positive and the other negative. The curve then consists of two branches stretching to infinity both in the positive and in the negative direction, but separated by an Interval included by the lines a; = a, a; = /3, In which no part of the curve is found. If li^ — ah vanishes, the GENERAL EQUATION OF THE SECOND DEOKEE. 137 quantity under the radical is of the foi'm cither ic — a or a — x. In the one case we have real values of y, provided only that x is greater than a ; in the other, provided only that it is less. The curve therefore consists of a single branch stretching to in- finity either on the right or the left-hand side of the line x = rj.. If the factors a and /3 be imaginary, the quantity under the radical may be thrown into the form [K^ - ah) {(a; — 7)^-1- 8^}. If then K'' — ah is positive, the quantity under the radical is always positive, and lines parallel to the axis of y always meet the curve. Thus in the figure of the hyperbola, p. 13o, lines parallel to the axis of y always meet the curve, although lines parallel to the axis of ic may not. On the other hand, if Ji^ — ab is negative, the quantity under the radical is always negative, and no real figure is represented by the equation. Ex. 1. Construct, as in Art. 16, the figures of the following curves, and determine then- species : 3x- + 4x1/ + 1/- - 3x - 2i/ + '21 = 0. Ans. Hyperbola. 5x2 ^ 4j.^ + ^2 _ 5j. _ 2^ — 19 = 0. Ans. Ellipse. 4x^ + 4x1/ + y^ — 5x — 2y—10 — 0. Arts. Parabola. Ex. 2. The circle is a particular case of the ellipse. For in the most general form of the equation of the circle, a = b, h — a cosoj (Art. 81) ; and therefore h- — ab \s negative, being — — a- sLn-o). Ex. 3. What is the species of the curve when A = ? Ans. An eUipse when a and b have the same sign, and an hyperbola when they have opposite signs. Ex. 4. If either a or 6 — 0, what is the species ? Ans. A parabola if also h - d ; otherwise a hyperbola. "When a = the axis of x meets the curve at infinity ; and when 6 = 0, the axis of y. Ex. 5. What is represented by -2--A+A- -^+1 = 0? a' ab 0' a Ans. A parabola touching the axes at the points x = a, y = b. 139. If in a quadratic Ax' + 2Bx + C=0, the coefficient ii vanishes, the roots are equal with opposite signs. This then will be the case v/ith the equation [a cos"' 0+ 2k cos dsind + b sin" 0) p'^ + 2 [g cos 6 +/ sin^) p + c= 0, if the radius vector be drawn in the direction determined by the equation g cos 6 +f sin ^ = 0. The points answering to the equal and opposite values of p arc equidistant from the origin, and on opposite sides of it ; r 138 GENERAL EQUATION OF THE SECOND DEGREE. therefore, the chord represented by the equation (jx -\-fy = is bisected at the origin. Hence, through any given 'point can in general he drawn one chordj 2vhich icill he hisected at that point. 140. There is one case, however, where more chords than one can be drawn, so as to be bisected, through a given point. If, in the general equation, we liad r/ = 0, /= 0, then the quantity g cos^ + /'sin^ would be = 0, whatever were the value of 6 ; and we see, as in the last article, that in this case every chord drawn through the origin would be bisected. The origin would then be called the centre of the curve. Now, we can in general, by transforming the equation to a new origin, cause the coefficients g and / to vanish. Thus equating to nothing the values given (Art, 134) for the new g and y, we find that the co-ordinates of the new origin must fulfil the conditions ax + Ity' + g = Oj hx -f hy' +f= 0. These two equations are sufficient to determine x and ?/', and being linear ^ can be satisfied by only one value of x and y ; hence, conic sections have in genercd one and only one centre. Its co-ordinates are found, by solving the above equations, to be , hq — hf , of— hq K'-ah' ^ U'-ah' In the ellipse and hyperbola It' — ah is always finite (Art. 137) ; but in the parabola h^ — ah = 0^ and the co-ordinates of the centre become infinite. The ellipse and hyperbola are hence often classed together as central curves, while the parabola is called a non-centred curve. Strictly speaking, however, every curve of the second degree has a centre, although in the case of the parabola this centre is situated at an Infinite distance. 141. To find the locus of the middle points of chords^ parallel to a given line^ of a curve of the second degree. We saw (Art. 139) that a chord through the origin is bisected if ^ cos^+,/sin^ = 0. Now, transforming the origin to any point, it appears, in like manner, that a parallel chord will be GENERAL EQUATION OF THE SECOND DEGREE. 139 bisected at the new origin if the new g multiplied by cos^ + the new/ multiplied by sln^ = 0, or (Art. 134) cos^ [ax + hy' +f/) + sin [lix + hj' +f) = 0. This, therefore, is a relation which must be satisfied by the co- ordinates of the new origin, if it be the middle point of a chord making with the axis of x the angle 6. Hence the middle point of any parallel chord must lie on the right line cos [ax + Ji7/ + (/) + sin 6 [hx + hy +f) = 0, which is, therefore, the required locus. Every right line bisecting a system of parallel chords is called a diameter^ and the lines which it bisects are called its ordinates. The form of the equation shows (Art. 40) that every diameter must pass through the intersection of the two lines ax + hy + fj = 0, and hx + by +f= : but, these being the equations by which we determined the co-ordinates of the centre (Art. 140), we infer that / every diameter ixisses through the centre of the curve. It appears by making 6 alternately = 0, and = 90° in the above equation, that ax + hy -^ y = is the equation of the diameter bisecting chords parallel to the axis of X, and that hx -H hy +f= is the equation of the diameter bisecting chords parallel to the axis of?/.* In the parabola 7r' = ah, '' rrh^ and hence the line * The equation (Art. 138) which is of the form hi/ = - {hx + f) ± ii is most easily constructed by first lajdng down the line hx + hy +f, and then taking on each ordi- nate MP of that line, portions PQ, PQ', above and below P and equal to 7?. Thus also it appears that each ordinate is bisected by h.r + />// + /'. 7 140 GENERAL EQUATION OF TIIH Si'COND DEGREE. ax + hi/+g is parallel to the line hx + %+/; consequently, all dia- meters of a 2)arahola are imrallel to each other. This, indeed, is evident, since we have proved that all diameters of any conic section must pass through the centre, which, in the case of the parabola, is at an infinite distance ; and since parallel right lines may be considered as meeting in a point at infinity.* The familiar example of the circle will sufficiently illustrate to the beginner the nature of the diameters of curves of the second degree. He must observe, however, that diameters do not in general, as in the case of the circle, cut their ordinates at right angles. In the parabola, for instance, the direction of the dia- meter being invariable, while that of the ordinates may be any whatever, the angle between them may take any possible value. 142. The direction of the diameters of a parabola is the same as that of the line through the origin which meets the curve at an infinite distance. For the lines through the origin which meet the curve at in- finity arc (Art. 136) ax^ + ^hxy + hy^ — 0, or, writing for h its value \/[ah)^ y{a)x + s/{h)yY = 0. But the diameters are parallel to ax + hy = (by the last article), which, if we write for h the same value ^/{ab)^ will also reduce to '^{a)x + ^/{h)y = 0. Hence every diameter of the parabola meets the curve once at infinity, and, therefore, can only meet it in one finite point. * Hence, a portion of any conic section being drawn on paper, we can find its centre and determine its species. For if we draw any two parallel chords, and join their middle points, we have one diameter. In like manner we can find another dia- meter. Then, if these two diameters be parallel, the curve is a parabola, but if not, the point of intersection is the centre. It will be on the concave side when the curve is an ellipse ; and on the convex when it is a hyperbola. GENERAL EQUATION OF THE SECOND DEGKEE. 141 143. // hco diameters of a conic section he sucJi, that one of them bisects all chords parallel to the other ^ then^ conversely^ the second will bisect all chords parallel to the first. The equation of the diameter which bisects chords making an angle 6 with the axis of x is (Art. 141) [ax + hy + g)-\- [hx + by +/) tan ^ = 0. But (Art. 21) the angle which this line makes with the axis is 0' where „, a + 7i tan 6 tan V =— --. — ^ n 1 A + & tan ^ ' whence b tan Q tan ^' 4 A (tan Q + tan &) + « = 0. And the symmetry of the equation shows that the chords making an angle & are also bisected by a diameter making an angle 0. Diameters so related, that each bisects every chord parallel to the other, are called conpigate diameters.^ If in the general equation h = 0, the axes will be parallel to a pair of conjugate diameters. For the diameter bisecting chords parallel to the axis of x will, in this case, become ax+g^O^ and will, therefore, be parallel to the axis of y. In like manner, the diameter bisecting chords parallel to the axis of y will, in this case, be by+f=0, and will, therefore, be parallel to the axis of X. 144. If in the general equation c=0, the origin is on the curve (Art. 81) ; and accordingly one of the roots of the quadratic (a cos^ 6 + 2h cos ^ sin ^ + b sin" 6) p" + 2 {g cos 9 +f sin 6) p = is always p = 0. The second root will be also p = 0, or the radius vector will meet the curve at the origin in two coincident points, if _^ cos ^ +y sin ^ = 0. Multiplying this equation hyp, we have the equation of the tangent at the origin, viz.gx+fy = 0.\ The equation of the tangent at any other point on the curve, may be found by first transforming the equation to that point as origin, and when the equation of the tangent has been then found, transforming it back to the original axes. * It is evident tliat none but central curves can have conjugate diameters, since in the parabola the direction of all diameters is the same. t The same argument proves that m an equation of any degi-cc, when the absolute term vanishes the origin is on the curve, and that the terms of the first degi-ee repi-esent the tangent at the origin. 142 (JENEKAL EC^UATIOX OF THE SECOND DEGREE. Ex. The point (1, 1) is ou the curve 3j,-- — ixy + 'ly" + Tj- — 01/ — 3 — ; transform the equation to parallel axes througli that point, and find the tangent at it. Ans. Qx-5y = referred to the new axes, or (x - 1) - 5 (j/ - 1) referred to the old. If this method is applied to the geucral equation, we get for the tangent at any point x'y\ the same equation as that found by a different method (Art. 86), viz. ax'x + h {xy + y'x) + ly'y 4 ,7 (a: + x) +/ [y + 2/') + c = <^'- 145. It Avas proved (Art. 89) that if it be required to draw a tangent to the curve from any point xy not supposed to be on the curve, the points of contact arc the intersections with the curve of a right line whose equation is identical in form with that last written ; and which is called the polar of xy . Consequently, since every right line meets the curve in two points, through any j^oint xy there can he drawn two real^ coin- cident^ or imacjinary tangents to the curved It was also proved (Art. 89) that the polar of the origin is gx+fy-\-c = 0. Now this line is evidently parallel to the chord gx-\fy^ which (Art. 139) is drawn through the origin so as to be bisected. But this last is plainly an ordinate of the diameter passing througli the origin. Hence, the folar of any 2)oint is 2)ciraUel to the ordinates of the diameter jyassing through that j)oint. This includes as a particular case : The tangent at the extremity of any diameter is 'parallel to the ordinates of that diameter. Or again, in the case of central curves, since the ordinates of any diameter are parallel to the conjugate diameter, we infer that, the polar of any j^oint on a diameter of a central curve is ixcrallel to the conjugate diameter. 146. The principal properties of poles and polars have been proved by anticipation In former chapters. Thus It was proved (Art. 98) that If a point A lie on the polar of B^ then B lies on the polar oi A. This may be otherwise stated, If a point move along a fixed line [the polar of B'] its polar p)asses through a fixed point [B'\ ; or conversely, If a line [the polar of A'\ pass * A curve is said to be of the ?i"' class, when through any point n tangents can be drawn to the curve. A conic is therefore a curve of the second degi-ee and of the second class : but in higher curves the degi-ee and class of a ciu-ve are commonly not the same. GENERAL EQUATION OF THE SECOND DEGREE. 143 through a fixed pointy then the locus of its 'pole A is a fixed right line. Or again, The intersection of any two lines is the jjole of the line joining their poles ; and conversely, The line joining any two points is the polar of the intersections of the polars of these points. For if we take any two points on the polar of A^ the polars of these points intersect in A. It was proved (Art. 100) that if two lines he draion through any pointy and the points joined ichere they meet the curve^ the joining lines will intersect on the polar of that point. Let the two lines coincide, and we derive, as a particular case of this. If through a point any line OH be drawn^ the tangents at R' and R" meet on the polar of : o. property which might also be inferred from the last paragraph. For since R'R'\ the polar of P, passes through 0, P must lie on the polar of 0. And it was also proved (Ex. 3, p. 96), that if on any radius vector through the origin, OR be taken a harmonic mean between OR' and 0R'\ the locus of R is the polar of the origin ; and therefore that, any line drawn through a point is cut harmonically hy the pointy the curve^ and the polar of the point ; as was also proved otherwise (Art. 91). Lastly, we infer that, if any line OR be drawn through a point 0, and P the pole of that line be joined to 0, then the lines OP, OR will form a harmonic pencil with the tangents from 0. For since OR is the polar of P, PTPiT is cut harmonically, and therefore OP, (9T, OP, OT form a harmonic pencil. Ex. 1. If a quadrilateral ABCD be inscribed in a conic section, any of tlie points E, F, is the pole of the Une joining the other two. Since EC, ED are two lines drawn thi'ough the point E, and CD, AB, one pair of lines join- ing the points where they n\eet the conic, these lines must intersect on the polar of E ; so must also AD and CB ; therefore, the line OF is the polar of E. In like manner it can be proved that EF is the polar of O, and EO the polar of F. Ex. 2. To draw a tangent to a given conic ^*- J-* section fi-om a point outside, with the help of the iiiler only. Draw any two lines through the given point E, and complete the quadrilateral as 144 GENERAL EQUATION OF THE SECOND DEGREE. in the figure, theu tlie line OF will meet the conic in two points, wliicli, being joined to E, will give the two tangents required. Ex. 3. If a quadrilateral be circumscribed about a conic section, any diagonal is the polar of the intersection of the other two. We shall prove this Example, as we might have proved Ex. 1, by means of the harmonic properties of a quadrilateral. It was proved (Ex. 1, p. 57) that EA, EO, EB, EF are a harmonic pencil. Hence, since EA, EB are, by hjqio thesis, two tangents to a conic section, and EF a line through their point of intereection, by Art. 14G, EO must pass througli the pole of EF; for the same reason, FO must pass through the pole of EF: this pole must therefore be 0. 147. Wc have proved (Art. 92) tliat the equation of the pair of tangents to the curve from any point x'y' is [ax"- 4 2hxy' + hy'^ + 2gx + 2fy' ■+ c) [ax^ + '2hxy + hy^ + 2gx + 2fy -1- c) = [ax'x+ li [x'y + y'x) + hy'y + g [x + x) +f{y' + y) + c}^ The equation of the pair of tangents through the origin may be derived from this by making x=y' = ; or it may be got directly by the same process as that used Ex. 4, p. 78. If a radius vector through the origin touch the curve, the two values of p must be equal, which are given by the equation [a cos"' 6 + 2h cos 6 sin + b sin"'' 0) p' + 2{g cos +/sin 6) p + c=-0. Now this equation will have equal roots if satisfy the equation [a cos^ 6 + 2h cos 6 sin -i b sin^ 6)c = [g cos 6 +f sin 6y. Multiplying by p'"', we get the equation of the two tangents, viz. {ac - f) x'' + 2 [ch - gf) xy + {he -f) / = 0. This equation again will have equal roots ; that is to say, the two tangents will coincide if {ac-g^)ihc-r) = [ch-fg)\ or, c [abc + 2fgh - af - hf - cW) = 0. This will be satisfied if c = 0, that is, if the origin be on the curve. Hence, any point on the curve may he considered as the intersection of two coincident tangents^ just as any tangent may be considered as the line joining two consecutive points. The equation will have also equal roots, if abc + 2fgh - af - bg" - cK' = 0. Now we obtained this equation (p. 72) as the condition, that the equation of the second degree should represent two right lines. To explain why we should here meet with this equation again. GENERAL EQUATION OF THE SECOND DEGREE. 145 It must be remarked that by a tangent wc mean in general a line which meets the curve iu two coincident points ; if then the curve reduce to two right lines, the only line which can meet the locus in two coincident points is the line drawn to the point of intersection of these right lines, and since two tangents can always be drawn to a curve of the second degree, both tangents must in this case coincide with the line to the point of inter- section. 148. If througli any point two cJiords he drawn^ meeting tli^ curve in the points J?', R"^ S\ S'\ then the ratio of the rectangles OR'. OR" .„, . , , , . . ^ . . ^ HQ' n Q" '^^' constant^ whatever be thej^osition ofthejjoint 0, provided that the directions of the lines OR, OS he constant. For, from the equation given to determine p in Art. 136, It appears that OR'. OR" = „^ ^7 ^i-~' — n — 1 — :— y-n • a cos^ + 2h cos6 sm^+ b sm^ In like manner OS'.OS" = hence a cos"' 6' + 2h cos 6' sin 0' + h sin' 6' ' OR'. OR" a cos' 6' + 2h cos 0' sin d' + h sin' 6' OS'. OS" a COS" -f 2h cos sin + b sin' ' But this is a constant ratio : for a, //, h remain unaltered when tlie equation is transformed to parallel axes through any new origin (Art. 134), and 0, 0' are evidently constant while the direction of the radii vectores is constant. The theorem of this Article may be otherwise stated thus : If throiigh two fixed points and 0' any two parallel lines OR 07?' OR" and O'p he drawn, then the ratio of the rectangles _, ,' , „ will ' •' -^ Op.Op be constant^ whatever he the direction of these lines. For, these rectangles are a cos'^ 4- 2h cos sin + h sin"^ ' a cos'^ + 2h cos sin -{-h i\\V0 ' [c being the new absolute term when the equation is transferred to 0' as origin) ; the ratio of these rectangles = - , and is, there- fore, independent of 0. This theorem is the generalization of Euclid iit. 35, 3(5. 14(3 GENERAL EQUATION OF THE SECOND DEGREE. 149. The theorem of the last Article includes under it several particular cases, which it is useful to notice separately. I. Let 0' be the centre of the curve, then 0' p = O p" and the quantity O'p'.O'p" becomes the square of the semidiameter parallel to OR' . Hence, The rectanfjles under the segments of tivo chords ivhich intersect are to each other as the squares of the diameters parallel to those chords. II. Let the line OR be a tangent, then OR' = 0R'\ and the quantity OR'. OR" becomes the square of the tangent ; and, since two tangents can be drawn through the point 0, we may extract the square root of the ratio found in the last paragraph, and infer that Two tangents drawn through any point are to each otlier as the diameters to lohich they are ixirallel. III. Let the line 0' be a diameter, and 07?, 0' p parallel to its ordinates, then OR! — OR" and O'p = O'p". Let the diameter . .1 • .1 ■ . A T, .X. OR' O'p' meet the curve m the pomts A. B. then . ,-. ,, ^ = , ^, ^^, ^ . '■ ' ' AO.OB A O.OB Hence, The squares of the ordinates of any diameter are propor- tional to the rectangles under the segments which they onahe on the diopter. 150. There is one case in which the theorem of Article 148 becomes no longer applicable, namely, when the line OS is parallel to one of the lines which meet the curve at infinity ; the segment OS" is then infinite, and OS only meets the curve in one finite point. AYe propose, In the present Article, to inquire O S" whether, In this case, the ratio ^r» >Qr>» will be constant. Let us, for simplicity, take the line OS for our axis of x^ and OR for the axis of y. Since the axis of x is parallel to one of the lines which meet the curve at Infinity, the coefficient a will = (Art. 138, Ex. 4), and the equation of the curve will be of the form 2hxy + hf + Igx -\-2fy + c = 0. Making y — O, the Intercept on the axis of x Is found to be OS' = — — ; and making x = 0, the rectangle under the Inter- c cepts on the axis of y h = y . GENERAL EQUATION OF THE SECOND DEGREE. 147 OS' _ I Hence OR'.OR" ~ 2 ry ' Now, if we transform the axes to any parallel axes (Art. 134), b Avill remain unaltered, and the new g = ht/' -f g. Hence the new ratio will be Now, if the curve be a parabola, h = 0, and this ratio is con- stant ; hence, Jf a line parallel to a given one meet any diameter (Art. 142) of a parahola^ the rectangle under its segments is in a constant ratio to the intercept on the diameter. If the curve be a hyperbola, the ratio will only be constant while y' is constant ; hence, The inteixej)ts made by tico parallel chords of a hyperbola^ on a given line meeting the curve at infinity^ are proportioyial to the rectangles under the segments of the chords. *151. To find the condition that the line \x + /juy + v may touch the conic represented by the general equation. Solving for y from \x-\- fiy + v^O^ and substituting in the equation/ pf^^ conic ; the abscissse of the intersections of the line and curv^rc determined by the equation (a/i^ - 27«X,/i + ^'^') «" + 2 {SJH^' - hfiv -ffxX + b\v) x + (c/a' - 2//AV 4 bv') = 0. The line Avill touch when the quadratic has equal roots, or when [afi' - 2h\/Jt, + bX') [c/jir - Iffiv + bv') = {giJb' - hfiv -ffi\ -f bXvf. Multiplying out, the equation proves to be divisible by /a", and becomes {be -f) V + [ca -/) yu.'-' + [ab - h') v' + 2 [gh - af] fiv + 2 [hf- bg) v\ + 2{fg- ch) Xfi = 0. We shall afterwards give other methods of obtaining this equation, which may be called the tangential equation of the curve. We shall often use abbreviations for the coefficients, and write the equation in the form AX" + BfM' + Cv' + 2Ftxv + 2 GvX + 2//X/i = 0. The values of the coefficients will be more easily remembered by 148 GENERAL EQUATION OF THE SECOND DEGREE. the lielp of the following rule. Let A denote the dtsa-iminant of the equation ; that is to say, the funetion abc + 2fgh - af - hf - cJi% whose vanishing is the condition that the equation may represent right lines. Then A is the derived function formed from A, regarding a as the variable ; and B, C, 2F, 2 G, 2H are the derived functions taken respectively with regard to J, c, /, (7, A. The coordinates of the centre (given Art. 140) may be written G F 6" C Miscellaneous Examples. Ex. 1, Porm the equation of the conic making intercepts X, X', fx, [x' on the aiea. Since if we make y = 0, or a; =: in the equation, it must reduce to X-- {X + \') a; + XX' = 0, f- - (jx + fx') y + /xf/ = ; the equation is fxfj.'x- + 2Iix!/ + XXy — ix/x (X + X') X — XX' (jx + fx') y + W'/xfx' = 0, and h is undetermined, unless another condition be given. Thus two parabolas can be drawn through the four given points ; for in this case h = ± ^{XX'ixfx'). Ex. 2. Given fo\ir points on a conic, the polar of any fixed point passes through a fixed point. We may choose the axes so that the given points may lie two on each axis, and the equation of the curve is that found in Ex. 1. But the equation of the polar of any point x'l/' (Art. 145) involves the indeterminate h in the first degx-ee^ and therefore passes through a fixed jpoint^ Ex. 3. Find the lociTS of the centre of a conic passing through four fixed points. The centre of the conic in Ex. 1 is given by the equations 2/xix'x + 2hi/ - njx' (\ + X') r; 0, 2\\'y + 2hx - XX' (^ + /i') = ; whence eliminating the indeterminate li, the locus is tfxixlx^ - 2X\'i/" - jxfi (/\ + X') a; + XX' {fx + /x'). y = 0, a conic passing through the intersections of each of the three pairs of lines which can be drawn through the four points, and thi-ough the middle points of these lines. The locus will be a hyperbola when X, X' and fx, fx have either both like^ or both unlike signs ; and an ellipse m the contrary case. Thus it will be an ellipse when the two points on one axis lie on the same side- of the origin, and on the other axis, on opposite sides. In other words, when the quadrilateral formed by the four given points has a re-entrant angle. This is also geometrically evident : for a quadrilateral with a re-entrant angle evidently cannot be inscribed in a figure of the shape of the ellipse or parabola. The circumscribing conic must therefore always be a hj^-perbola, BO that some vertices may he in opposite branches. And since the centre of a hyper- bola is never at infinity, the locus of centres is in this case an ellipse. In the other case, two positions of the centre will be at infinity, corresponding to the two parabolas which can be described through the given points. ( 149 ) CHAPTER XL EQUATIONS OF THE SECOND DEGREE EEFEREED TO THE CENTRE AS ORIGIN. 152. In investigating the properties of the ellipse and hyper- bola, -we shall find our equations much simplified by choosing the centre for the origin of co-ordinates. If we transform the general equation of the second degree to the centre as origin, we saw (Art. 140) that the coefficients of x and y will =0 in the transformed equation, which will be of the form ax^ + 2kxy + htf + c' = 0. It Is sometimes useful to know the value of c in terms of the co- efficients of the first given equation. We saw (Art. 134) that c = ax'^ + llxxy + ly"^ + 'iyx + ify + c, where a;', y are the co-ordinates of the centre. The calculation of this may be facilitated by putting c Into the form c' = [ax + liy ■\ g^x -\ [lix + hy -\rf)y'+ gx +fy' -f c. The first two sets of terms are rendered = by the co-ordi- nates of the centre, and the last (Art. 140) _ hf— hg Jig - af _ ahc + 2^7/i — of- — hg" — cJi^ ^ ~^ ab^f '^^ab^f^ ^"^^ ah-K' *"" 153. If the numerator of this fraction were =0, the trans- formed equation would be reduced to the form ax^ + 2hxy + hy' = 0, and would, therefore (Art. 73), represent two real or imaginary right lines, according as ah — li^ is negative or positive. Hence, * It is evident in like manner tliat the result of substituting x'lj, the co-ordinates of the centre, in the equation of the polar of any point x"y", viz. [ax' + hy' + g) x" + {lix' + hif +/) y" + gx' -\-fy' + c, is the same as the result of substituting x'y' in the equation of the curve. For the first two sets of terms vanish in both cases. 150 CENTKAL EQUATIONS OF THE SECOND DEGREE. as wc have already seen, p. 72, the condition that the general equation of the second degree should represent two right lines, is ahc + 2fgli - af - hg' - cW = 0. For it must plainly be fulfilled, in order that when we transfer the origin to the point of intersection of the right lines, the absolute term may vanish. Ex. 1. Transform Sx'' + ixij + if - 5x - 61/ - 3 = to the centre. (|, - 4). Ans. V2x^ + l&xy + 4/ + 1 = 0. Ex. 2. Transfoi-m x- + 2xy - t/^ + 8a; + 4y - 8 = to the centre (- 3, - 1). Ans. x'^ + 2xy-y'^-22. 154. We have seen (Art. 136) that when 6 satisfies the condition a cos^^ -f 2h cos 6 sin 6 + 1 sm'6 = 0, the radius vector meets the curve at infinity; and also meets the curve in one other point, whose distance from the origin is c g cos 6 +/ sin 6 ' But if the origin be the centre., we have g = 0, f= 0, and this distance will also become infinite. Hence two lines can be drawn through the centre, which will meet the curve in two coincident lyoints at infinity, and which therefore may be considered as tan- gents to the curve whose points of contact are at infinity. These lines are called the asymptotes of the curve ; they are imaginary in the case of the ellipse, but real in that of the hyperbola. We shall show hereafter that though the asymptotes do not meet the curve at any finite distance, yet the further they are produced the more nearly they approach the curve. Since the points of contact of the two real or imaginary tan- gents drawn through the centre are at an infinite distance, the line joining these points of contact is altogether at an infinite distance. Hence, from our definition of poles and polars (Art. 89) the centre may he considered as tlie pole of a line situated altogether at an infinite distance. This inference may be confirmed from the equation of the polar of the origin, gx +fy -f- c = 0, which, if the centre be the origin, reduces to c = 0, an equation which (Art. 67) represents a line at infinity. CENTRAL EQUATIONS OF THE SECOND DEGREE. 151 155. TVe have seen that by taking the centre for origin, the coefficients g and f in the general equation can be made to vanish ; but the equation can be further simplified by taking a pair of conjugate diameters for axes, since then (Art. 143) h will vanish, and the equation be reduced to the form ax^ -\- hif -f c = 0. It is evident, now, that any line parallel to either axis Is bisected by the other ; for if we give to x any value, we obtain equal and opposite values for y. Now the angle between conjugate diame- ters Is not in general right ; but we shall show that there is always one pair of conjugate diameters which cut each other at right angles. These diameters are called the axes of the curve, and the points where they meet It are called its vertices. We have seen (Art. 143) that the angles made with the axis by two conjugate diameters are connected by the relation h tan e tan 6' + li (tan 6 + tan d')+a = 0. But if the diameters are at right angles, tan ^' = — 7 — ^ (Art. 25). Hence h tan'6' + (a - J) tan d-h = 0. We have thus a quadratic equation to determine 6. Multiply- ing by p\ and writing a?, y, for p cos^, p sln^, we get hx" - (« -h)xy- hf = 0. This Is the equation of two real lines at right angles to each other (Art. 74) ; we perceive, therefore, that central curves have two, and only two, conjugate diameters at right angles to each other. On referring to Art. 75 It will be found, that the equation which we have just obtained for the axes of the curve Is the same as that of the lines bisecting the internal and external angles be- tween the real or Imaginary lines represented by the equation ax^ + 2hxy -1- hy^ = 0. The axes of the curve, therefore, are the diameters which bisect the angles between the asymptotes; and (note, p. 71) they will be real whether the asymptotes be real or Imaginary: that Is to say, whether the curve be an ellipse or a hyperbola. 156. We might have obtained the results of the last Article by the method of transformation of co-ordinates, since we can 152 CENTRAL EQUATIONS OF THE SECOND DEGREE. thus prove directly that it is always possible to transform the equation to a pair of rectangular axes, such that the coefficient of xy in the transformed equation may vanish. Let the original axes be rectangular ; then, if we turn them round through any angle ^, we have (Art. 9) to substitute for a;, x co^O-y sin^, and for ?/, x sin ^4?/ cos^; the equation will therefore become a{x C0&9- y sin Of + 2h {x cofid- y sin 6) {x s.md-\-y cos 6) + h [x sin 6 + y cos ^)^ + c = ; or, arranging the terms, we shall have the new a = a cos"''^ + 2h cos 6 smO + b sm^d ; the n«w h = h sin cos 6 -^ h (cos'^^ — sin"'^^) — a sin^ cos 6 ; the new h = a sin^^ — 2h cos 6 sin ^ + ^ cos'^^. Now, if we put the new h = 0, we get the very same equation, as in Art. 155, to determine tan^. This equation gives us a simple expression for the angle made with the given axes by either axis of the curve, namely, a— 157. When it Is required to transform a given equation to the form ax;' + by^ + c = 0, and to calculate numerically the value of the new coefficients, our work will be much facilitated by the following theorem : If ice transform an equation of the second degree from one set of rectangidar axes to another^ the quantities a-\-h^ and ah — W will remain unaltered. The first part is proved immediately by adding the values of the new a and h (Art. 156), when we have a' + 5' = a + h. To prove the second part, write the values in the last article, 2a' = a-^h-\-2h sin 2^ + (a - h) cos 2^, 2h' =a-\-h -2h sin 2^- [a-h) cos2^. Hence ^a'h' = (a + hf - [2h sin 2^ +[a-h) cos 2^}'. But All'' = [2h cos2^ -{a-h) sin 2^}''' ; therefore 4 [ah' - h"') = [a + hf - A¥ - [a -hy = i {ah - h'). When, therefore, we want to form the equation transformed to the axes, we have the new h = 0, CENTilAL EQUATIONS OF THE SECOND DEGREE. 153 a' + h' = a + b^ a'V = ah — h'. Having, therefore, the sura and the product of a and h'^ avc can form the quadratic which determines these quantities. Ex.1. Find the axes of the ellipse lix- — ixy + lly- = GO, and transform the equation to them. The axes are (Art. 155) 4x2 + Qj.y _ 4^2 - q, or (2x - ij) (x + 2y) = 0. We have a' + b' = 26; a'b' = 150 ; «' = 10 ; 6' = 15 ; and the transformed equation is 2a;2 + df = 12. Ex. 2. Transform the hyperbola llx- + 84x^ — 24^^ = 156 to the axes, a' + b' = - 13, a'b' = - 2028 ; a' = 39, b' = - 52. Transformed equation is Sx^ — 4^^ =12. Ex. 3. Transform ax'^ + 2hxy + by'^ = c to the axes. Ans. {a + b - E) x"^ + (a + b + E) y- = 2c : where E'^ = W + (a - b)-. *158. Having proved that the quantities a 4- & and ah — Ic re- main unaltered when we transform from one rectangular system to another, let us now inquire what the.se quantities become if we transform to aa oblique system. We may retain the old axis of x^ and if we take an axis of y inclined to it at an angle oj, then (Art. 9) we are to substitute x-\-y cos w for ar, and y sin co for y. We shall then have d — (7, It — a cos w -\-li sin (w, y = a cos'^^co -i- 27i cosoj sino) + h sin" &). Hence, it easily follows a' 4- J' — 2/i' cos ft) ^ dV — K"^ , ,„ ;— = a-\-o. — r—z, —ab—h. sm 0) sin' CO If^ tJieUj ice transform the equation from one fair of axes to any , - . . a + 5 — 2A cos CO ^oh — W . , , other, the quantities r-s — ■ and — ;— s — remain unaltered. ' -^ sm 6) sin ft) We may, by the help of this theorem, transform to the axes an equation given in oblique co-ordinates, for we can still ex- press the sum and product of the new a and h in terms of the old coefficients. Ex. 1. If coso) = f, transform to the axes, lOx- + &xy + h\f = 10. ^ 285 , 1025 , , 205 a + 6=-, «& = _ , « = D, 6=_. Ans. lGx- + 41/ = 32. Ex. 2. Transform to the axes, x" - 3ry + 7/- -t i = 0, where w = 60°. Ans. X- — 15^- = 3. Ex. 3. Transform ax- + 2lixy '+ by- = c to the axes. Atis. {a + b - 2h cosco - E) x'' + {a + b - 21i cosw + R) y- - 2c sin-o), where ^2 - [2/j - {a + b) cosa)}2 + (a ^ 6)^ sin^o..' X 154 CENTRAL EQUATIONS OF THE SECOND DEGREE. *159. We add the demonstration of the theorems of the last two articles given by Professor Boole [Cambridge Math. Jour.^ III. 1, lOG, and New Series, vi. 87). Let us suppose that we are transforming an equation from axes inclined at an angle w, to any other axes inclined at an angle Q. ; and that, on making the substitutions of Art. 9, the quantity ax^ + 2]ixy -\-hy' becomes a'X' ■\-2h'XY-\-h'Y^. Now we know that the effect of the same substitution will be to make the quantity x^ -{■ 2xy co^s, (o + 1/ become X^ + 2XY coQil+ Y'\ since either is the expression for the square of the distance of any point from the origin. It follows, then, that ax^ -\- 2lixy + hf -f X, [x'^ -f 2xy cos tu ■\-y'' ) = a'X' + 21^XY+ V Y' + \ {X^ + 2XY cosi2 + Y''). And if we determine X so that the first side of the equation may be a perfect square, the second must be a perfect square also. But the condition that the first side may be a perfect square is (a + X) (i + X) = (A + X cos (a)\ or \ must be one of the roots of the equation X" sin"''w -f (a + & - 2h cos oi)\-\-ah- U = 0. We get a quadratic of like form to detennine the value of X, which will make the second side of the equation a perfect square ; but since both sides become perfect squares for the same values of X, these two quadratics must be identical. Equating, then, the coefiicients of the corresponding terms, we have, as before, a + h-2h coso) _ a' + h' -2k' cos 12 _ ah - 7^' _ a'b' - h'^ sin^w ~ sin^I2 ' sin^to ~ sin'^Ii Ex. 1. The sum of the squares of the reciprocals of two semi-diameters at right angles to each other is constant. Let their lengths be a and /3 ; then making alternately a; = 0, y = 0, in the equation of the curve, we have aa'' - c, h(^ - c, and the theorem just stated is only the geo- metrical interpretation of the fact that « + i is constant. Ex. 2. The area of the triangle formed by joining the extremities of two conjugate semi-diameters is constant. The equation referred to two conjugate diameters is ^„ + ^,- 1, and since 'i°~JL a- j3- sin2 co is constant, we have a'jS' sin w constant, Ex. 3. The sum of the squares of two conjugate semi-diameters is constant. Q. „„ « + i-2/«cosa) . , , 1 /I 1\ a'2 + /3'2 . ^^^^ ^fiT^r: ^^ constant, -— — — + _ = , ! - is constant : and since a'/i' sin oi is constant, so must a"^ + (i"^. THE EQUATION REFEERED TO THE AXES. 155 THE EQUATION EEFEERED TO THE AXES. 160. We saw that the equation referred to the axes was of the form B being positive in the case of the elhpse, and negative in that of the hyperbola (Art. 138, Ex. 3). AVe have replaced the small letters by capitals because we are about to use the letters a and h with a different meaning. The equation of the ellipse may be written in the following more convenient form ; — Let the intercepts made by the ellipse on the axes be aj= ff, y = ii then making y = Q and re = a in the equation of the curve, C C we have Aa^=^ (7, and A— —,. In like manner B= yz. Sub- stituting these values, the equation of the ellipse may be written X if h — =1. a b Since we may choose whichever axis we please for the axis of ic, we shall suppose that we have chosen the axes so that a may be greater than h. The equation of the hyperbola, which, we saw, only differs from that of the ellipse In the sign of the coefficient of ?/^, may be written In the corresponding form 2 2 The intercept on the axis of x Is evidently = ± a, but that on the axis of ?/, being found from the equation y^=~h^ is imaginary ; the axis of y, therefore, does not meet the cui've in real points. Since we have chosen for our axis of x the axis which meets the curve in real points, we are not in this case entitled to as- sume that a is greater than h. 161. To find the polar equation of the ellipse, the centre "being the pole. Write p cos6 for a;, and p sin^ for y, in the preceding equa- tion, and we get 1 _ cos"^ sln"'*^ 'p'~ d'"^ ~b'^' 156 THE EQUATION REFERRED TO THE AXES. an equation wliicli we may Avrlte in any of the equivalent forms, ^ ~ a' Bxn'e^ V' zoi'B ~V^ [a' - V) sm'd ~ ci' - {d'- F) cos' 6 ' It is customary to use the following abbreviations : d^ - W and the quantity e Is called the eccentricity of the curve. Dividing by a' tlie numerator and denominator of the fraction last found, we obtain the form most commonly used, viz., P l-e'-'cos'^6'" 162. To investigate the figure of the ellipse. The least value that h'' + {d' — P) sm'6, the denominator in the value of p^, can have, is when ^ = ; therefore the greatest value of p Is the Intercept on the axis of a;, and is = a. Again, the greatest value of h' -\- [a^ — V) '&\\r 6 ^ Is, when sin^=l, or ^ = 90°; hence the least value of p is the intercept on the axis of ?/, and is = h. The greatest line, therefore, that can be drawn through the centre Is the axis of a?, and the least line, the axis of y. From this property these lines are called the axis major and the axis minor of the curve. It is plain that the smaller 6 Is, the greater p will be ; hence, the nearer any diameter is to the axis mojor^ the greater it will be. The form of the curve will, therefore, be that here represented. We obtain the same value of p whether we suppose ^=a, or 6= — a. Hence, Tivo diameters which malce equal angles with the axis will he equal. And it Is easy to show that the converse of this theorem Is also true. This property enables us, being given the centre of a conic, to determine its axes geometrically. For, describe any concen- tric circle Intersecting the conic, then the semi-diameters drawn to the points of intersection will be equal ; and by the theorem just proved, the axes of the conic will be the lines internally and externally bisecting the angle between them. THE EQUATION EEFERRED TO THE AXES. 157 163. The equation of the ellipse can be put into another form, which will make the figure of the curve still more ap- parent. If we solve for y we get Now, if we describe a concentric circle with the radius a, its equation will be y= \l[d'-x^). Plence we derive the following construction : " Describe a circle on the axis major^ and talce on each ordinate LQ a point P, such that LP may he to LQ in the constant ratio h : cr, then the locus of P loill he the required ellipse^ Hence the circle described on the axis major lies wholly withoiit\\\Q. curve. We might, in like manner, construct the ellipse, by describing a circle on the axis minor, and increasing each ordinate in the constant ratio a : h. Hence the circle described on the axis minor lies wholly icithin the curve. The equation of the circle is the particular form which the equation of the ellipse assumes when we suppose h = a. 164. To find the polar equation of the hyperhola. Transforming to polar co-ordinates, as in Art. IGl, we get d^h'' d'b" d'W' P = F cos^ d-a"" sin^ d h' - [d' + ¥) sin' 6 {d'+ b') cos"^ d-d'' 8ince formulae concerning the eUipsc are altered to the corre- sponding formulae for the hyperbola by changing the sign of Z*", we must, in this case, use the abbreviation c^ for d' 4- Z»", and ^ for ^ — , the quantity e being called the eccentricity of the hyperbola. Dividing then by d^ the numerator and denominator of the last found fraction, we obtain the polar equation of the hyperbola, which only differs from that of the ellipse in the sign of h\ viz., ^ e'-'cos^^-L* 158 THE EQUATION REFERRED TO THE AXES. 165. To investigate the figure of the hyperhola. The terms axis major and axis minor not being applicable to the hyperbola (Art. 160), we shall call the axis of x the trans- verse axis, and the axis of?/ the conjugate axis. Now 1/ — [a^ -\-V') sin"'^, the denominator in the value of p\ will plainly be greatest when ^ = 0, therefore, in the same case, p will be least ; or the transverse axis is the shortest line which can he drawn from the centre to the curve. As 6 increases, p continually increases, until sin 9 = or tan 6 = ^/{a'^¥) when the denominator of the value of p becomes = 0, and p be- comes infinite. After this value of ^, p^ becomes negative, and the diameters cease to meet the curve in real points until again sln^ = or tan 6= — si[ci' + h')' when p again becomes infinite. It then decreases regularly as 6 increases, until 6 becomes = 180°, when it again receives its minimum value =a. The form of the hyperbola, therefore, is that represented by the dark curve on the figure, next article. 166. We found that the axis of y does not meet the hyper- bola in real points, since we obtained the equation y'^ = — J^ to determine its point of intersection with the curve. We shall, how- ever, still mark off on the axis of y por- tions CB, CB'=±bj and we shall find that the length CB has an important connexion wltli the curve, and may be conveniently called an axis of the curve. In like manner, if we obtained an equation to determine the length of any other diameter, of the form p'^ = - i?^, although this diameter cannot meet the curve, yet if we measure on it from the centre lengths = ± -S, these lines may be conveniently spoken of as diameters of the hyperbola. THE EQUATION EEFERRED TO THE AXES. 159 The locus of the extremities of these diameters which do not meet the curve is, bj changing the sign of p" in the equation of the curve, at once found to be 1 sin'"^^ cos"6' p' " ~1/ ^ ' or -^ _ = 1. This is the equation of a hyperbola having the axis of y for the axis meeting it in real points, and the axis of x for the axis meeting it in imaginary points. It is represented by the dotted curve on the figure, and is called the hyperbola conjugate to the given hyperbola. 167. We proved (Art. 165) that the diameters answering to tan 6=±- meet the curve at infinity ; they are, therefore, the same as the lines called, in Art. 154, the asymptotes of the curve. They are the lines CK^ CL on the figure, and evidently separate those diameters which meet the curve in real points from those w^hich meet it in Imaginary points. It is evident also, that two conjugate hyperbolee have the same asymptotes. The expression tan 6 = ± - enables us, being given the axes in magnitude and position, to find the asymptotes, for, if we form a rectangle by drawing parallels to the axes through B and A^ then the asymptote CK must be the diagonal of this rectangle. A • /I <^ 1 Agam cos^= —r7-7, — -r^, = - . But, since the asymptotes make equal angles with the axis of .r, the angle which they make with each other must be =20. Hence, being given the eccentricity of a hyperbola^ we are given the angle between the asymptotes^ which is double the angle whose secant is the eccentricity. Ex. To find the eccentricity of a conic given by the general equation. We can (Art. 74) wiite down the tangent of the angle between the Unes denoted by ax- + 2hxy + by- — 0, and thence form the expression for the secant of its half ; or we may proceed by the help of Art. 157, Ex. 3. IGO CONJUGATE DIAMETERS. ^ , 1 « + i - /t! 1 u + h + R a- 2c /32 2c where IT-.- U- + {a- Vf = W - iab + (a -l- h)-. 1 _ i - :? "-""-^ _ 27; Hence CONJUGATE DIAMETERS. 168. We now proceed to investigate some of tbe properties of tbe ellipse and hyperbola. We sball find it convenient to consider botb curves togetber, for, since tbeir equations only diifer in tbe sign of y\ tbey bave many properties in common wbich can be proved at tbe same time, by considering tbe sign oi If as indeterminate. We sball, in tbe following Articles, use tbe signs wbicb apply to tbe ellipse. Tbe reader may tben obtain tbe corresponding formulae for tbe byperbola by changing the sign of V. VYe shall first apply to the particular form — ^ + '4 = 1 , some of the results already obtained for the general equation. Thus (Art. 86) the equation of tbe tangent at any point xy being got by writing xx and y'y for ic^ and y\ is XX ?/'?/ 1- ^^-^ = 1. a b Tbe proof given in general may be repeated for this particular case. Tbe equation of tbe chord joining any two points on tbe curve is (^-^')(^-a^") , {y-y')[y-y") _^l,yl_. d' '^ h^ ~ «^ ^ w ' {x ^x)x [y' + y") y xx y'y" ^ ^? — + — v—^-^^-r^^^ which, when a;', y =x\ y\ becomes tbe equation of the tangent already written. Tbe argument here used applies whether tbe axes be rect- angular or oblique. Now if the axes be a pair of conjugate diameters, tbe coefiicient of xy vanishes (Art. 143) ; the coefficients of X and y vanish, since the origin is tbe centre ; and if a and V be the lengths of the intercepts on tbe axes, it is proved exactly, as in Art. IGO, that tbe equation of tbe curve may be written x' y' a b CONJUGATE DIAMETERS. UJl And It follows from this article, that In the same case the equa- tion of the tangent is - — ■ + ^^^ = 1. a U 169. The equation of the polar, or line joining the points of contact of tangents from any point x'y\ Is similar In form to the equation of the tangent (Arts. 88, 89), and Is therefore xx mi , XX ini ^ the axes of co-ordinates in the Latter case being any pair of conjugate diameters 5 in the former case, the axes of the curve. In particular, the polar of any point on the axis of cc Is -tj = 1 . Hence the polar of any point P is found by drawing a diameter through the point, taking CP. CP' = to the square of the semi- diameter, and then drawing through P' a parallel to the con- jugate diameter. This includes, as a particular case, the theorem proved already (Art. 145), viz., The tangent at the extremity of any diameter is parallel to the conjugate diameter. Ex. 1. To find the condition that Xx + fxy -l may touch — + 75 = !• Comparing ^ + ^ = 1 . \a; + u v = 1 , we find - = \a, ■— - uh, and a'\- + b-p? = I. «■ 6- a o Ex. 2. To find the equation of the pair of tangents from x'y' to the curve (see Art. 92). Ex. 3. To find the angle eft between the pair of tangents from x'y' to the curve. When an equation of the second degree represents two right lines, the three highest terms beuig put = 0, denote two lines through the origin parallel to the two former ; hence, the angle included by the first pair of right Uncs depends solely on the three highest terms of the general equation. AiTanging, then, the equation found in the last Example, we find, by Art. 74, tan^ x"^ + y"- — a- — b- Ex. 4. Find the locus of a pomt, the tangents thi'ough which intereect at right angles. Equating to the denominator in the value of tan <^, we find x- + y' = «' + b-, the equation of a circle concentric with the ellipse. The locus of the intersection of tangents which cut at a given angle is, in general, a curve of the fomth degree. 170. To find the equation^ referred to the axes, of the diameter conjugate to that passing through any point x'y on the curve, Y 162 CONJUGATE DIAMETERS. The line required passes through the origin, and (Art. 169) is parallel to the tangent at xy ; its equation is therefore ^^' A. yy a 2 + 12 = *^' a b Let ^, & be the angles made with the axis of x by the original / diameter and its conjugate; then plainly tan ^="^5 and from ^ Wx the equation of the conjugate we have (Art. 21) tan^' = ^, . Hence tan^ tan^'= — ^ ; as might also be inferred from Art. 143. The corresponding relation for the hyperbola (see Art. 168) is tan d tan & = —.. 171. Since, in the ellipse, tan^ tan^' Is negative, if one of the angles ^, 6\ be acute (and, therefore, its tangent positive), the other must be obtuse (and, therefore, its tangent negative). Hence, conjugate diameters in the ellipse lie on different sides of the axis minor (which answers to ^ = 90°). In the hyperbola, on the contrary, tan^ tan^' is positive, therefore, 6 and 6' must be either both acute or both obtuse. Hence, in the hyperhola^ conjugate diameters lie on the same side of the conjugate axis. In the hyperbola. If tan 9 be less, tan 6' must be greater than - , but (Art. 167) the diameter answering to the angle whose ^ 7 tangent is - , is the asymptote, which (by the same Article) sepa- rates those diameters which meet the curve from those which do not intersect it. Hence, if one of two conjugate diameters meet a hyperbola in real points^ the other will not. Hence also it may be seen that each asymptote is its own conjugate. 172. To find the co-ordinates x"y" of the extremity of the diameter conjugate to that p)o.ssing through, xy . These co-ordinates are obviously found by solving for x and y between the equation of the conjugate diameter, and that of the curve, viz., XX ini ^ x^ V" CONJUGATE dia:\ieters. 163 Substituting in the second the values of x and ?/, found from the first equation, and remembering that x\ y satisfy the equation of the curve, we find without difficulty II I )i t X y y _x a ~ b ^ b a ' 173. 2o express the lengths of a diameter (a'), and its conju- gate {b')j in terms of the abscissa of the extremity of the diameter. (1) We have a:'' = x'^ + y'\ But y--=^^,[d^-x% Hence o!^ = F -] 5— x"' = b^ + e'x'\ a (2) Again, we have u 12 b' = x'+y'=jr,y'+-^,x% or = {d^ - x^) + —2 x'^ ; hence b'~ = a' — e^x'. From these values we have a'2 + J'^ = a'' + 5^; or, The sum of the squares of any pair of conjugate diameters of an ellipse is constant (see Ex. 3', Art. 159). 174. In the hyperbola we must change the signs of Z*'" and V'\ and we get a — b =a — 0^ or, TTie difference of the squares of any pair of conjugate diameters of a hyperbola is constant. If in the hyperbola we have a = Z>, its equation becomes x'-7f = d% and it is called an equilateral hyperbola. The theorem just proved shows that every diameter of an equilateral hyperbola is equal to its conjugate. The asymptotes of the equilateral hyperbola being given by the equation 164 CONJUGATE DIAMETEKS. are at right am/les to each other. Hence this hyperbola is often called a rectanytdur hyperbola. The condition that the general equation of the second degree should represent an equilateral hyperbola is a=—h\ for (Art. 74) this is the condition that the asymptotes [ax^ + 2hxy + hf) should be at right angles to each other ; but if the hyperbola be rectangular it must be equilateral^ since (Art. 167) the tangent of half the angle between the asymptotes = - ; therefore, if this angle = 45°, Ave have h = a. 175. To find the length of the perpendicular from the centre on the tangent. The length of the perpendicular from the origin on the line ^ 4- -^ - 1 d' ^ b' ~ is (Art. 23) ah but we proved (Art. 173) that b" = h'x" + ay\ hence ah 176. To find the angle between any pair ofi conjugate dia- meters. The angle between the diameters is equal to the angle be- tween either, and the tangent parallel to ^>^T the other. Now ~ Hence smcb (or PCF'] = ~ ao The equation a'b' sin(}> = ah proves, that the triangle formed by joining the extremities of conjugate diameters of an ellipse or hyperbola has a constant area (see Art. 159, Ex. 2). CONJUGATE DIAMETERS. 1G5 177. The sum of the squares of any two conjugate diameters of an ellipse being constant, their rectangle is a maximum when they are equal ; and, therefore, in this case, sinc^ is a minimum ; hence the acute angle between the two equal conjugate dia- meters is less (and, consequently, the obtuse angle greater) than the angle between any other pair of conjugate diameters. The length of the equal conjugate diameters is found by making a = h' in the equation «'" + b'^ = a^ -f b'\ whence a'"' is half the sum of a' and h'\ and in this case . , 2ab ^ a- + ¥ The angle which either of the cquiconjugatc diameters makes with the axis of x is found from the equation b'' tixn$ tan^' = „ , by making tan = - tan 6' ; for any two equal diameters make equal angles with the axis of aj on opposite sides of it (Art. 162). Hence j tan 6= -. a It follows, therefore, from Art. 167, that if an ellipse and hyper- bola have the same axes in magnitude and position, then the asymptotes of the hyperbola will coincide with the equiconjugate diameters of the ellipse. The general equation of an ellipse, referred to two conjugate diameters (Art. 168), becomes x^ +y'^ = a'^j when a' = b'. We see, therefore, that, by taking the equiconjugate diameters for axes, the equation of any ellipse may be put into the same form as the equation of the circle, x^ -\ y^ = r\ but that in the case of the ellipse the angle between these axes will be oblique. 178. To exj^ress the peiyendicular from the centre on the tan- fjent in terms of the angles lohich it makes with the axes. If we proceed to throw the equation of the tangent (~2'+"^ = l) ifito the form a; cosa + ?/ sina=p (Art. 23), we find immediately, by comparing these equations, X cos a y sin a of' p ' V p 166 CONJUGATE DIAMETERS. Substituting In the equation of the curve the values of a;', y\ hence obtained, we find jf = a" cos'''a -f W sln^a.* The equation of the tangent may, therefore, be written X cosa + ?/ sin a — sj[(i cos'''a -f V sln'^a) = 0. Hence, by Art. 34, the perpendicular from any point [xy) on the tangent Is \]{d'' cos"a + V sin"'' a) — x cosa — y' sin a, where we have written the formula so that the perpendiculars shall be positive when x'y' is on the same side of the tangent as the centre. Ex. To find the locus of the intersection of tangents which cut at right angles. lMtp,p' be the perpendiculars on those tangents, then p^ = a- cos- a + b- sin^a, p'- = a^ sin- a + IP cos" a, p"^ + p'- — a- + b-. But the square of the distance from the centre, of the intersection of two lines which cut at right angles, is equal to the sum of the squares of its distances from the lines themselves. The distance, therefore, is constant, and the required locus is a circle (see p. 161, Ex. 4). 179. The chords which join the extremities of any diameter to any point on the curve are called supplemental chords. Diameters parallel to any pair of supplemental chords are conjugate. For if we consider the triangle formed by joining the extre- mities of any diameter AB to any point on the curve D ; since, by elementary geometry, the line joining the middle points of two sides must be parallel to the third, the diameter bisecting AD will be parallel to BD^ and the diameter bisecting BD will be parallel to AD. The same thing may be proved analytically, by forming the equations of AD and BD^ and showing that the product of the tangents of the angles made by these lines with the axis is = - — . a This property enables us to draw geometrically a pair of con- jugate diameters making any angle with each other. For if we describe on any diameter a segment of a circle, containing the * In like manner. />- = a'- cos- a + b'" cos-/?, a and p being the angles the perpen- di(?ular makes with any pair of conjugate diameters. CONJUGATE DIAMETERS. 1G7 given angle, and join the points where It meets the curve to the extremities of the assumed diameter, we obtain a pair of supple- mental chords inclined at the given angle, the diameters parallel to which will be conjugate to each other. Ex. 1. Tangents at the extremities of any diameter'are parallel. Theu- equations are — r- + ttt = + !• This also follows from the first theorem of Art. 146, and from considering that the centre is the pole of the line at infinity (Art. 154), Ex. 2. If any variable tangent to a central conic section meet two fixed parallel tangents, it wUl intercept portions on them, whose rectangle is constant, and equal to the square of the semi-diameter parallel to them. Let us take for axes the diameter parallel to the tangents and its conjugate, then the equations of the curve and of the variable tangent wiU be The intercepts on the fixed tangents are fomid by making x alternately = + a' in the latter equation, and we get y and, therefore, their product is — 7; ( 1 ^ ) ; which, substituting for y'- from the equation of the curve, reduces to b'-. Ex. 3. The same construction remaining, the rectangle imder the segments of the variable tangent is equal to the square of the semi-diameter parallel to it, For, the intercept on either of the parallel tangents is to the adjacent segment of the variable tangent as the parallel semi-diameters (Art. 149) ; thei-efore, the rect- angle under the intercepts of the fixed tangents is to the rectangle under the segments of the variable tangent as the squares of these semi-diameters ; and, since the first rectangle is equal to the square of the semi-diameter parallel to it, the second rect- angle must be equal to the square of the semi-diameter parallel to it. Ex. 4. If any tangent meet any two conjugate diameters, the rectangle imder its segments is equal to the square of the parallel semi-diameter. Take for axes the semi-diameter parallel to the tangent and its conjugate ; then the equations of any two conjugate diameters being (Art, 170) „ _ 2/' s"'^' , .'// _ rt " x' ' a'- b- the intercepts made by them on the tangent are foimd, by making x — a', to be v' » n ^'' ^' V = — , « I and V = 7 -. , -^ x' ' •' a' y" whose rectangle is evidently = b'-. We might, in Uke manner, have given a purely algebraical proof of Ex. 3. Hence, also, if the centre be joined to the points where two parallel tangents meet any tangent, the joining lines will be conjugate diameters. Ex. 5. Given, in magnitude and position, two conjugate semi-diametei"s, Oa. Ob, of a central conic, to determine the axes. 168 THE NORMAL. The following construction is founded on the theorem proved in the last Ex- ample : — Tlu'ough a, the extremity of either diameter, draw a parallel to the other ; it must of course be a tan- gent to the curve. Now, on Oa take a point P, such that the rectangle Oa.aP = Ob^ (on the side remote from for the ellipse, on the same side for the hyperbola), and describe a cu-cle through 0, P, having its centre on aC, then the lines OA, OB are the axes of the curve ; for, since the rectangle Aa.aB = Oa.aP — Ob^, the lines OA, OB are conjugate diameters, and since AB is a dia- meter of the cu-cle, the angle A OB is right. Ex. 6. Given any two semi-diameters, if from the extremity of each an ordinate be drawn to the other, the triangles so formed will be equal in area. Ex. 7. Or if tangents be drawn at the extremity of each, the triangles so formed will be equal in area. THE NORMAL. 180. A line drawn through any point of a curve perpen- dicular to the tangent at that point is called the Normal. Forming, by Art. 32, the equation of a line drawn through [x'y') perpendicular to ( — 5- -f —■ = 1 j , we find for the equat of the normal to a conic ion {y-y') = '^^ {x-x) or a X X y (f being used, as in Art. 161, to denote d^ — W. Hence we can find the portion CA^ intercepted by the normal on either axis ; for, making ?/ = in 1 the equation just given, we find — — P c , , or x = ex . We can thus draw a normal to an ellipse from any point on the axis? for given C'A'we can find a;', the abscissa of the point through which the normal is drawn. The circle may be considered as an ellipse whose eccentricity = 0, since c"^ = a^ — H' = 0. The intercept CA, therefore, is con- stantly = in the case of the circle, or every normal to a circle passes througli its centre. THE NORMAL. 169 181. The portion MN intercepted on the axis between the normal and ordinate is called the Suhnormal. Its length is, by the last Article, a" a^ The normal, therefore, cuts the abscissa into parts which are in a constant ratio. If a tangent drawn at the point P cut the axis in T, the in- tercept MT is, in like manner, called the Suhtangent. Since the whole length CT= - (Art. 169), the subtangent — — X — X X The length of the normal can also be easily found. For P^-^ = Pif ^ + ATJ/-'' = /-^ + ^ a)''^ = ^ (^i^ /^ + ^ cc'^') . But if h' be the semi-diameter conjugate to (7P, the quantity within the parentheses =5'^ (Art. 173). Hence the length of the normal PN= — . a If the normal be produced to meet the axis minor, it can be 7 / proved, in like manner, that its length = -j- . Hence, tlie rect- angle under ilie segments of the normal is equal to the sqiiare of the conjugate semi-diameter. Again, we found (Art. 175) that the perpendicular from the centre on the tangent = -p- . Hence, the rectangle wider the normal and the 2}er2}endicular from the centre on the tangent^ is constant and equal to the square of the semi-axis minor. Thus, too, we can express the normal in terms of the angle it makes with the axis, for ~ J> ~ »J[a^ cos''a+ 6' sin' a) ^ ^' ^^' ~ V(i - e" sin^'a) * Ex. 1. To draw a normal to an ellipse or hyperbola passing tlii-ougli a given point. The equation of the normal, a-x'y — h^x'ij = c-x'ij', expresses a i-elation between the co-ordinates x'y' of any point on the curv^e, and xy the co-ordinates of any point on the normal at x'y'. "We express that the .point on the normal is known, and the point on the curve sought, by removing the accents from the co-ordinates of the latter Z 170 THE NORMAL. point, and accentuating those of the former. Thus we find that the points on the cun'e, wliose normals will pass thi-ough {x'y') are the points of intersection of the given cui-ve with the hjiicrbola c^xij — a-x'y — h-ij'x. Ex. 2. If through a given point on a conic any two lines at right angles to each other be dra^\'n to meet the curve, the line joining then- extremities will pass through a fixed point on the nonnal. Let us take for axes the tangent and normal at the given point, then the equation of the cm've must be of the form «x- + 2hxij + hy- + 2fy - (for c = 0, because the origin is on the curve, and ^ = (Art. 144), because the tan- gent is supposed to be the axis of x, whose equation is ?/ = 0). Now, let the equation of any two lines through the origin be x^ + ^pxy + qy"^ — 0. Multiply this equation by «, and subtract it from that of the curve, and we get 2 {h - ap) xy + {b- aq) / + 2fy = 0. This (Art. 40) is the equation of a locus passing through the points of intersection of the lines and conic ; but it may evidently be resolved into y — (the equation of the tangent at the given point), and 2{h- ap) X + {b - aq) y + 2f= 0, which must be the equation of the chord joining the extremities of the given lines. 2f The point where this chord meets the normal (the axis of y) is y = j ; but if the lines are at right angles ^ = — 1 (Art. 74), and the intercept on the normal has the constant length a + b' If the curve be an equilateral hyperbola, a + b = 0, and the line in question is constantly parallel to the normal. Thus then, if through any point on an equilateral hj'perbola be di-awn two chords at right angles, the pei-pendicular let fall on the line joining thek extremities is the tangent to the ciu-ve. Ex. 3. To find the co-ordinates of the intersection of the tangents at the points The co-ordinates of the mtersection of the lines ^,y'y_-, ^ , ^ _ 1 a2 "^ 62 - -^' a2 "^ 62 - ^> a^ («' — y") b- (x' — x") yx' - y X " xy - yx Ex. 4. To find the co-ordinates of the intersection of the normals at the points x'y', x"y" , • _ (g^ - yi) x'x"X _ [V^-<i^)y'y"Y Ji-ns. X - — , y - j4 > where X, Y are the co-ordinates of the intersection of tangents, foimd in the last Example. * This theorem will be equally tiaie if the lines be drawn so as to maie with the normal, angles the product of whose tangents is constant, for, in this case, q is con- 2/ stant ; and, therefore, the intercept — ^—r is constant. '■ aq — b THE FOCI. 171 The values of A' and 1' may be written in other fomis, since by combining the equations -2 + i-z - 1' „. + 12 - 1' we get the results, x'-ij"- - y'-x'"' = ¥ {x"^ — x"-) - — a? {y'- — y'"^) . XT 1- ■' ^y + y'-«" VT _ a-'//" + y'x" Hence A = — y-, — J7~ > -'^ — — r-; — 77— • y + y X + X TVe can also prove „ _ jx' + x") _ (y' + y") x'x" n'ti" ' , x'x" v'u" a- 0- a- 0- THE FOCI. 182. If on the axis major of an ellipse we take two points equidistant from the centre, whose com- rj.' mon distance = ±^/{a^ — ¥), or = + c, these points are called the foci of the curve. The foci of a hyperbola are two points on the transverse axis, at a distance from the centre still =±c, c being in the hyperbola =s/{a' + ¥). To express tlie distance of any point on an ellipse from the focus. Since the co-ordinates of one focus are (a; = + ^, ;y = 0) , the square of the distance of any point from it = [x - cf + y"^ = x'^ + y"^ - lex + c'. But (Art. 173) ^■' ^ y"" ^V" \ d'x% and W^& = a^. Hence FF' = a' - 2cx' + e'x" ; and recollecting that c = ae, we have FF=a — ex'. [We reject the value [ex - a) obtained by giving the other sign to the square root. For, since x is less than a, and e less than 1 , the quantity ex — « is constantly negative, and, there- fore, does not concern us, as we are now considering, not the direction, but the absolute magnitude of the radius vector FF.] AYe have, similarly, the distance from the other focus FF=a + ex\ since we have only to write -c for +c In the preceding formulie. 172 THE FOCI. Hence FP+ F'P= 2aj or, The sum of the distances of any point on an ellipse from the foci is constant^ and equal to the axis major. 183. In applying the preceding proposition to the hyperbola, wc obtain the same value for FP- ; but in extracting the square root we must change the sign in the value of FP^ for in the hyperbola x is greater than a, and e is greater than 1. Hence, a — ex is constantly negative ; the absolute magni- tude, therefore, of the radius vector is FP= ex — a. In like manner, F'P= ex + a. Hence F'P-FP=2a. Therefore, in the hyperhola^ the difference of the focal radii is constant^ and equal to the transverse axis^ The rectangle under the focal radii =±(a^ — eV"*), that is, (Art. 173)= Z-'^ 184. The reader may prove the converse of the above results by seeking the locus of the vertex of a triangle, if the base and either sum or difference of sides be given. Taking the middle point of the base (= 2c) for origin, the ©(juation is V{/ + (c + xY] ± V{/ 4- (c - xY] = 2a, which,, when cleared of radicals, becomes 2 2 x y 2 ~ 2 2 ■• • a a —c Now, if the sum of the sides be given, since the sum must always be greater than the base, a is greater than c, therefore the GoefJicient of y"^ is positive, and the locus an ellipse. If the difference be given, a is less than c, the coefficient of y'^ is negative, and the locus a hyperbola. 185. By the help of the preceding theorems, wc can describe an ellipse or hyperbola mechanically. If the extremities of a thread be fastened at two fixed points F and F'^ it is plain that a pencil moved about so as to keep the thread always stretched will describe an ellipse Avhose foci are F and F'^ and whose axis major is equal to the length of the thread. THE FOCI. 173 In order to describe a hyperbola, lot a ruler be fastened at one extremity (i^), and capable of moving round it, tlieu if a thread, fastened to a fixed point F\ and also to a fixed point on the ruler (7?), be kept stretched by a ring at P, as the ruler is moved round, the point P will describe a hyperbola; for, since the sum of F'P and PR is constant, the difference of FP and F'P will be constant. 186. The polar of either focus is called the directrix of the conic section. The directrix must, therefore (Art. 169), be a line perpendicular to the axis major at a distance from the centre = + - . Knowing the distance of the directrix from the centre, we can find its distance from any point on the curve. It must be equal to (I , CI . ,, 1 , ,, — —X, or = ~ (a - ex ] = - la — ex . c c e ' But the distance of any point on the curve from the focus = a— ex. Hence we obtain the important property, that the distance of any -point on the curve from the focus is in a constant ratio to its distance from the directrix^ viz., as e to 1. Conversely, a conic section may be defined as the locus of a point whose distance from a fixed point (the focus) is in a con- stant ratio to its distance from a fixed line (the directrix). On this definition several writers have based the theory of conic sections. Taking the fixed line for the axis of x^ the equation of the locus is at once written down {x-xy^{7j-yj = e'y\ which it is easy to see will represent an ellipse, hyperbola, or parabola, according as e is less, greater than, or equal to 1. Ex. If a curve be such that the distance of any point of it from a fixed point can be expressed as a rational function of the first degree of its co-ordinates, then the curve must be a conic section, and the fixed point its focus (see O'Brien's Co-ordinate Geometry, p. 85). For, if the distance can be expressed P = Ax + By + C, since Ax + By + C \s proportional to the perpendicular let fall on the right line whose equation is [Ax + By + C - 0), the equation signifies that the distance of any point of the curve from the fixed point is in a constant ratio to its distance from this hue. 174 THE FOCI. 187. To find the length of tlie i')erj)endicular from the focus on the tangent. The length of the perpendicular from the focus (+<:', 0) on the Ihic f^ + ^ = l) is, by Art. 34, 1- cx a' ^ h* but, Art. 175, Hence Likewise ^%')~ ah' FT=\,{a-ex') = ^,FP. ^ ' F'T'=\-, [a^ex)=\,F'F. ^ ' Hence FT.F' T = W (since «' - d'x'' = ¥% or, The rectangle under the focal ^nr-pendicidars on the tangent is constant^ and equal to the square of the semi-axis minor. This property applies equally to the ellipse and the hyperbola. 188. The focal radii mcdce equal angles loith the tangent. For we had FT=^ \, FP, or CJ = f, ; h ' FF I) ' but FT^ FF = &mFFT. Hence the sine of the angle which the focal radius vector FF makes with the tangent = j, . But we find, in like manner, the same value for sin i^'PT', the sine of the angle which the Other focal radius vector F'F makes with the tangent. The theorem of this article is true both for the ellipse and hyperbola, and, on looking at the figures, it is evident that the tangent to the ellipse is the external bisector of the angle between the focal radii, and the tangent to the hyperbola the internal bisector. Hence, if an ellipse and hyperlola^ THE FOCI. 175 having the same foci^ jjass throucjh the same jyotnt^ they vnll cut each other at right angles^ that Is to say, tli& tangent to the ellipse at that point will be at right angles to the tangent to the hyperbola. Ex. 1. Prove analytically that confocal conies cut at right angles. The co-ordinates of the intersection of the conies ^ + 2^=1 :^ f^^ =1 o- b- ' u"^ b"- ' satisfy the relation obtained by subtracting the equations one from the other, viz. (g^ - cr-) x'^ ^ {l^b'^)j/^ _ ^^ d^a'- b-b"' But if the conies be confocal, «- — a'- = b'^ — b'-, and this relation becomes x'- y'- __ Ifi^^ + 62^2 - "■ But this is the condition (Art, 32) that the two tangents 11^ b- ' a"- b"- ' should be perpendicular to each other. Ex. 2. Pind the length of a line drawn through the centre parallel to either focal radius vector, and terminated by the tangent. This length is found by dividing the perpendicidar from the centre on the tangent , the sine of the angle between the radius vector and tangent, and is therefore = a. (y?) "^ « Ex. 3. Verify that the normal, which is a bisector of the angle between the focal radii, divides the distance between the foci into parts which are proportional to the focal radii (Euc. vi. 3). The distance of the foot of the normal fi-om the centre is (Art. 180) = e-a;'. Hence its distances fi'om the foci are c + e-x' and c — e-x', quantities which are evidently e times a + ex' and ft — ex', Ex. 4. To draw a normal to the ellipse from any point on the axis minor. Ans. The cu-cle through the given pomt, and the two foci, ^vill meet the curve at the point whence the normal is to be di-awn. 189. Another important consequence may be dcilucctl from the theorem of Art. 187, that the rectangle under the focal per- pendiculars on the tangent is constant. For, if we take any two tangents, we have (see figure, next page) FT_ Ft' Tt ~ FT' FT . but -j=r is the ratio of the sines of the parts into which the line FP divides the angle at P, and -prTp, is the ratio of the sines of FT,FT' = Ft.Ft', ov ^^- ^,.,„, 176 THE FOCI. the parts into which F'P divides the same angle ; we have, there- fore, the angle TFF= t'PF'. If we conceive a conic section to pass throngh P, having F and F' for foci, it was proved in Art. 188, that the tangent to it must be equally inclined to the lines FP, F'P', it follows, therefore, from the present Article, that it must be also equally inclined to PT, Pt ; hence we learn that if through any point (P) of a conic section ive draw tamgents [PT, Pt) to a con- focal conic section, these tangents will he equally incliyied to the tangent at P. 190. To find the locus of the foot of the perpendicular let fall from either focus on the tangent. The perpendicular from the focus is expressed In terms of the angles it makes with the axis by putting x = c, y' =■ Q in the formula of Art. 178, viz., p = \/(«^ cos'^a + V^ sin'''a) — x cos a — y sin a. Hence the polar equation of the locus is p = \/{a^ cos'^^a + h' sin^a) — c cos a, or p^ + 2cp cos a + c^ cos^a = a^ cos'^'a + h'^ sin^a, or p^-\- 2cp cosa = &'■'. This (Art. 95) is the polar equation of a circle whose centre is on the axis of x, at a distance from the focus = — c ; the circle is, therefore, concentric with the curve. The radius of the circle is, by the same Article, = a. Hence, If toe describe a circle having for diameter the transverse axis of an ellij^se or hyperbola, the perpendicular from the focus will meet the tangent on the circumference of this circle. Or, conversely, if from any point F (see figure, p. 171) we -draw a radius vector FT to a given circle, and draw TP perpen- dicular to FT, the line TPwill always touch a conic section, having F for its focus, which loill he an ellipse or hyperbola, according as F is tcithin or loithout the circle. It may be inferred from Art. 188, Ex. 2, that the line CT, whose length = a, is parallel to the focal radius vector F'P. THE FOCI. 177 191. To find the angle subtended at the focus hy the tangent drawn to a central conic from any point [xy). Let the point of contact be {x'y')^ the centre being the origin, then, if the focal radii to the points {xy)^ (^V)) be p, /?', and make angles 6', 6\ with the axis, it is evident that cos6= , sin^=-: cos^ = — p- , sin^='-, . P P P P Hence cos {6 - 6') = (^+ ^) (^' + ^) + ^3^' PP but from the equation of the tangent wc must have cc "^ h' ~ Substituting this value oi yy\ we get PP cos (^ — &) = XX + ca; + ex + c* ^ xx + J*, or = ^xx + c^ + ex + a^ = (a + ex) {a + ex) ; or, since p' = a + co;', we have, (see O'Brien's Co-ordinate Geometry,!^. lb&), ^^^^ cos (a— u) = . P Since this value depends solely on the co-ordinates xy^ and does not involve the co-ordinates of the point of contact, either tan- gent drawn from xy subtends the same angle at the focus. Hence, The angle subtended at the focus by any chord is bisected by the line Joining the focus to its jjole. 192. The line Joining the focus to the pole of any chord pass- ing through it is jyerpendicular to that chord. This may be deduced as a particular case of the last Article, the angle subtended at the focus being in this case 180° ; or di- rectly as follows : — The equation of the perpendicular through ft 1 1 n 1 • (xx yy \ . any pomt xy io the polar of that pomt I — ,^ + "^^"^ = 1 j is, as m Art. 180, c^x _b^y _ 1 ^' y ~ ' But if xy be anywhere on the directrix, we have x =■ — ^ and it will then be found that both the equation of the polar and that of the perpendicular are satisfied by the co-ordinates of the focus (a; = c, ?/ = 0). A A 178 THE FOCr. When in any curve we use polar co-ordinates, the portion intercepted by the tangent on a perpendicular to the radius vector drawn through the pole is called the polar suhtangent. Hence the theorem of this Article may be stated thus : The focus heing the pole^ the locus of the extremity of the polar suhtangent is the directrix. It will be proved (Chap, xii.) that the theorems of this and the last Article are true also for the parabola. Ex. 1. The angle is constant which is subtended at the focus, by the portion in- tercepted on a variable tangent between two fixed tangents. By Art. 1 91, it is half the angle subtended by the chord of contact of the fixed tangents. Ex. 2. If any chord PP' cut the direc- trix in Z>, then FD is the external bisector of the angle PFP'. For FT is the internal bisector (Art. 191) ; but D is the pole of FT (since it is the intersection of PP', the polar of T, with the directrix, the polar of F) ; therefore, DF is pei-pendicular to FT, and is therefore the external bisector. [The follo^ving theorems (communica- ted to me by the Kev. W. D. Sadleir) are founded on the analogy between the equations of the polar and the tangent.] Ex. 3. If a point be taken anywhere on a fixed pei-pendicular to the axis, the per- pendicular from it on its polar will pass through a fixed point on the axis. For the intercept made by the perpendicular wiU (as in Art. 180) be e-x', and will therefore be constant when x' is constant. Ex. 4. Find the lengths of the perpendicular from the centre and from the foci on the polar of x'y'. Ex. 5. Prove CM. PX' — b^. This is analogous to the theorem that the rectangle under the normal and the central perpendicular on tangent is constant. Ex. 6. Prove PN'.NN' = -! (a? - e^x'^). When P is on the curve this equation gives us the known expression for the normal = — (Art. 181), Ex. 7. Prove FG.F'G' = CM.NN'. When P is on the curve this theorem becomes FG.F'G' — b-. 193. To find the polar equation of the ellipse or hyperbola, the focus being the pole. The length of the focal radius vector (Art. 182) =a-ex'j but x (being measured from the centre) = p cos + c. Hence p = a — ep cos 6 — ec, = ^(1-^') _ ^ 1 l+€cos^ a'l + ecos^* or /* = THE FOCI. 179 The double ordinate at the focus Is called the imrameter ; its half is found by making 6= 90° in the equation just given, to be I)' = — =a[\ — e^). The parameter is commonly denoted by the letter 2?. Hence the equation is often written ^ 2 * 1 + e cos ^ ' The parameter is also called the Latus Rectum. Ex. 1. The harmonic mean between the segments of a focal chord is constant, and equal to the semi-parameter. For, if the radius vector FP, when produced backwards through the focus, meet the curve again in P', then FP being ^ , , -, FP', wliich answers to (0 + 180°), 1 2 1 + e cos y •n P 1 i 1 — e cos D TT 114 Hence _^ + _=-. Ex. 2. The rectangle under the segments of a focal chord is proportional to tha whole chord. This is merely another way of stating the result of the last Example ; but it may be proved directly by calculating the quantities FP. FP', FP + FP', which are easily seen to be, respectively 5^ 1 ^ 262 1 , and — a- 1 — e^ cos- 6 a 1 — e- cos- Ex. 3. Any focal chord is a third proportional to the transverse axis and the parallel diameter. For it will be remembered that the length of a semi-diameter making an angle with the transverse axis is (Art. IGl) 1 _ e2 cos- Hence the length of the chord FP + FP' found in the last Example = — . Ex. 4. The sum of two focal chords drawn parallel to two conjugate diameters is constant. For the sum of the squares of two conjugate diametei-s is constant (Art. 173). Ex. 5. The sum of the reciprocals of two focal chords at right angles to each other is constant. 194. The equation of the ellipse, referred to the vertex, is Hence, in the ellipse, the square of the ordinate is less than tho rectangle under the parameter and abscissa. The equation of the hyperbola is found in like manner, y =r^^ + a' ^ • IftO THE ASYMPTOTES. Hence, in the hyperbola, the square of the ordinate exceeds the rectangle under the parameter and abscissa. We shall show, in the next chapter, that in the parabola these quantities are equal. It was from this property that the names ^ara&oZa, hyioerhola^ and ellipse, were first given (see Pappus, Math. Coll., Book vii.). THE ASYMPTOTES. 195. We have hitherto discussed properties common to the ellipse and the hyperbola. There is, however, one class of pro- perties of the hyperbola which have none corresponding to them in the ellipse, those, namely, depending on the asymptotes, which in the ellipse are imaginary. We saw that the equation of the asymptotes was always obtained by putting the highest powers of the variables = 0, the centre being the origin. Thus the equation of the curve, re- ferred to any pair of conjugate diameters, being 'i 2 that of the asymptotes is '^z - fr. = 0, or ---L and - + ^ = 0. a o ' a b a Hence the asymptotes are parallel to the diagonals of the paral- lelogram, whose adjacent sides are any pair of conjugate semi- diameters. For, the equation of y V CT'is - = — , and must, therefore, coincide with one asymptote, while the equation of AB | ^ + ^, = 1 j isparallel to the other (see Art. 167). Hence, given any two conjugate diameters, we can find the asymptotes ; or, given the asymptotes, we can find the diameter conjugate to any given one; for if we draw AO parallel to one asymptote, to meet the other, and produce it till 0B= AO, we find B, the extremity of the conjugate diameter. 196. The portion of any tangent intercepted hy the asymptotes is bisected at the curve, and is equal to the conjugate diameter. This appears at once from the last Article, where we have proved A T= b' = AT' ; or, directly, taking for axes the diameter THE ASYMPTOTES. 181 through the point and its conjugate, the equation of the asymp- totes is X- y- '•J 7 '2 — * a Hence, if we take x = a\ we have y = ±l>'] but the tangent at A being parallel to the conjugate diameter, this value of the ordinate is the intercept on the tangent. 197. If any line cut a hyperhola^ the portions DE^ FG^ in- tercepted between the curve and its asymptotes^ are equal. For, if we take for axes a n:::^^ ^^ diameter parallel to DG and its conjugate, it appears from the last Article, that the por- tion DG is bisected by the diameter ; so is also the portion EF; hence BF=FG. The lengths of these lines can immediately be found, for, from the equation of the asymptotes ( -t^j — r;5 = J , we have y{=mi=MG)=± \x. Again, from the equation of the curve •we have y (= EM= FJf) = ±h' /(—^ — Ij . Hence DF{= FG) = h' || - . /d, - 1 and DF{=EG) = h'i'^,-]- 198. From these equations it at once follows, that the rect- angle DE.DF is constant^ and = h''\ Hence, the greater BF'is, the smaller will DE be. Now, the further from the centre we draw DF the greater will it be, and it is evident from the value given in the last article, that by taking x sufficiently large, we can make DF greater than any assigned quantity. Hence, the further from the centre xce draxo any line^ the less will he the intercept between the curve and its asymptote^ and by increasing the distance from the centre^ we can mal'e this intercept less than any assigned quantity. 199. If the asymptotes be taken for axes, the coefficients g and / of the general equation vanish, since the origin is the a a' 182 THE ASYMPTOTES. centre ; and the coefficients a and h vanish, since the axes meet the curve at infinity (Art. 138, Ex. 4) : hence the equation re- duces to the form ^^ _ ]fi^ The geometrical meaning of this equation evidently is, that the area of tlie 'parallclocjram formed hy the co-ordinates is constant. The equation being given in the form xy = Tc\ the equation of any chord is (Art. 86), {x-x'){y-y") = xy-k\ or x'y + y"x = H + xy" . Making x = x" and y' = y'\ we find the equation of the tangent, x'y + y'x = 2lc\ or (writing x'y' for k^) ^ V ^ - + -, = 2. X y From this form it appears that the Intercepts made on the asymptotes by any tangent = 2x' and 2y' ; their rectangle is, therefore, Ak"^. Hence, the trianc/le ichich any tangent forms with the asymptotes has a constant area^ and is equal to double the area of the parallelogram formed hy the co-ordinates. Ex. 1. If two fixed points {x'y', x"y") on a hyperbola be joined to any variable point on the curve {x"'y"'), the portion which the joining lines intercept on either asj'mptote is constant. The equation of one of the joining lines being x"'y + y'x = y'x'" + Ic^, the intercept made by it from the origin on the axis of x is found, by making y = 0, to be x'" + x'. Similarly the intercept from the origin made by the other joining line is x"' + x'', and the difference between these two (a;' — x") is independent of the position of the point x"'y"'. Ex. 2. Find the co-ordinates of the intersection of the tangents at x'y', x"y". Solve for x and y from x'y + y'x — 2Z;-, x"y + y"x — Ih"^, and we find x — y. ; — - , xy — y X k- Z"- 2x'x" which, if we substitute for y', y", — , —^ becomes Similarly x' + x" ' y' + y" ' 200. To express the quantity U^ in terms of the lengths of the axes of the curve. Since the axis bisects the angle between the asymptotes, the co-ordinates of its vertex are found, by putting x = y \n the equation xy = Jc\ to be x = y = k. THE ASYMPTOTES. 183 Hence, If 6 be the angle between the axis and the asymptote, a = 2k cos 0, (since a Is the base of an Isosceles triangle whose sides = k and base angle = 6), but (Art. 165) cos (7 = hence k = And the equation of the curve, referred to its asymptotes, is 201. The ijerpendicular from the focus on the asymptote is equal to the conjugate semi-axis h. For It Is CF sin ^, but CF= V(«' + J'), and sin = » ^2x • This might also have been deduced as a particular case of the property, that the product of the perpendiculars from the foci on any tangent Is constant, and = — h'\ For the asymptote may be considered as a tangent, whose point of contact is at an Infinite distance (Art. 154), and the perpendiculars from the foci on it are evidently equal to each other, and on opposite sides of It. 202. The distance of the focus from anij point on the curve is equal to the length of a line drawn through the point parallel to an asymj)tote to meet the directrix. For the distance from the focus Is e times the distance from the directrix (Art. 186), and the distance from the directrix Is to the length of the parallel line as cos^ (= " ? ^^^' ^^"^ ) ^^ **^ ^' Hence has been derived a method of describing the hyperbola by continued motion. A ruler ABB^ bent at B^ slides along the fixed line DD' ; a thread of a length = BB Is fastened at the two points B and F^ while a ring at F keeps the thread always stretched ; then as the ruler is moved along, the point P will de- scribe an hyperbola, of which i^ is a focus, DD' a directrix, and BB parallel to an asymptote, since PFmust always = BB. £>' ( 184 ) CHAPTER XII. THE PARABOLA, REDUCTION OF THE EQUATION. 203. The equation of the second degree (Art. 137) will re- present a parabola, when the first three terms form a perfect square, or when the equation is of the form {ax + yS?/)' + 2gx + 2/?/ + c = 0. We saw (Art. 140) that we could not transform this equation so as to make the coefficients of x and y both to vanish. The form of the equation however, points at once to another method of simplifying it. We know (Art. 34) that the quantities ax + /3?/, 2gx + ^fy 4 c, are respectively proportional to the lengths of perpendiculars let fall from the point [xy) on the right lines, whose equations are ax + ^y = 0, 2()x + 2/v/ -f c = 0. Hence, the equation of the parabola asserts that the square of the perpendicular from any point of the curve on the first of these lines, is proportional to the perpendicular from the same point on the second line. Now if we transform our equa- tion, making these two lines the new axes of co-ordinates, then since the new x and y are proportional to the perpendiculars from any point on the new axes, the transformed equation must be of the form y^ =px. The new origin is evidently a point on the curve ; and since for every value of x we have two equal and opposite values of y^ our new axis of x will be a diameter whose ordinates are parallel to the new axis of;/. But the ordinate drawn at the extremity of any diameter touches the curve (Art. 145) ; therefore the new axis of ?/ is a tangent at the origin. Hence the line ax + ^y is the diameter passing through the origin, and 2yx-i-2fy + c is the tangent at the point where this diameter meets the curve. And the equation of the curve referred to a diameter and tangent at its extremity, as axes, is of the form y'^^j)^- THE PARABOLA. 185 204. The new axes to which we were led in the last article, are in general not rectangular. We shall now show that it is possible to transform the equation to the form 1/ =px^ the new axes being rectangular. If we introduce the arbitrary constant k^ it is easy to verify that the equation of the parabola may be written in the form [ax + /3y -f ky + 2 (,(/ - ah) x+2 [f- /3k) i/-{- c-k'' = 0. Hence, as in the last article, ax 4- /S?/ -f k is a diameter, 2 {g — ak) x + 2 [f— ^k) y + c — Ic^ is the tangent at its ex- tremity, and if we take these lines as axes, the transformed' equation is of the form 'if =px. Now the condition that these new axes should be perpendicular is (Art. 25) a{g-ak)+^{f-^k) = 0, , ^ aq-\r ^f whence k = -^, — ^. . a' + yS' Since we get a simple equation for k, we see that there is 07\e diameter whose ordinates cut it perpendicularly, and this dia- meter is called the axis of the curve. 205. We might also have reduced the equation to the form y^ =j)x by direct transformation of co-ordinates. In Chap. xi. we reduced the general equation by first transforming to parallel axes through a new origin, and then turning round the axes so as to make the coefficient of xy vanish. We might equally well have performed this transformation in the opposite order ; and In the case of the parabola this is more convenient, since we cannot by transformation to a new origin, make the coeffi- cients of X and y both vanish. We take for our new axes the line ax + /3y, and the line perpendicular to it fix — a?/. Then since the new X and 1^ are to denote the lengths of perpendiculars from any point on the new axes, we have (Art. 34) ,^_ ax -{ fiy Y _ ^^~ ^y -VRTF)' "^"7(^?+^')* If for shortness we write a' + /S^ = 7", the formula of trans- formation become r^Y^ax + /3y , 7 A' = fix - ay ; whence 7a; = a 1"+ fiX^ r^y = fiY— aX. DB 186 THE PARABOLA. Making these substitutions in the equation of the curve, it becomes ry' Y' + 2 (.9/3 -/a) X + 2 {ffa +fi3) Y+ 7c = 0. Thus, by turning round the axes, we have reduced the equation to the form ^,y ^ 2g'x + 2/'7/ + c' = 0. If we change now to parallel axes through any new origin x'y' ; substituting x + x\ ^ + y\ for x and ?/, the equation becomes Vy' + 27'a; + 2 [b'y +/') y + Vy' + 2gx + 2/y + c' = 0. The coefficient of x is thus unaltered by transformation, and ' therefore cannot in this way be made to vanish. But we can evidently determine x and y\ so that the coefficients of y and the absolute term may vanish, and the equation thus be reduced to y =irx. The actual values of the co-ordinates of the new origm are y =— j,^ x — „ ,., — ; and p is evidently — -^ , or in terms of the original coefficients (a'^-f/Sf When the equation of a parabola is reduced to the form y'^ — 'px^ the quantity 'p is called the ixirameter of the diameter which is the axis of x ; and if the axes be rectangular, p is called the principal ^parameter (see Art. 194). Ex. 1. Find the principal parameter of the parabola 9x2 + 2i:xy + 16/ 4. 22a; + 46z/ + 9 = 0. First, if we proceed aa in Art. 204, we determine h = 5. The equation may then be written ^3^ + 4^ + 5)2 = 2 {ix - 3y + 8). Now if the distances of any point from 3a; + 4y + 5, and 4a; — 3y + 8 be Y and A', we ^^"^^ 5r=3a; + 4y + 5, 5A=4a;-3^ + 8, and the equation may be written F" = ■fA'. The process of Art. 205 is first to transfonn to the lines 3a; 4- 4y, 4a; -^ Sy as axes, when the equation becomes 25r2 + 50r-10A'+9 = 0, or 25 (r+l)= = 10A+16, which becomes F^ ^ f A' when transformed to parallel axes thi"ough (- f , — 1). Ex. 2. Find the parameter of the parabola ^'+f- -^' + 1=0. Am. -J . ft- ah ¥ a b (a2 + J2)^ This value may also be deduced directly by the help of the following theorem, wliich will be proved afterwards ; — '• The focus of a parabola is the foot of a perpendi- THE PARABOLA. 187 ciilar let fall from the intersection of two tangents which cut at right angles on their chord of contact ;" and "The parameter of a conic is found by dividing four times the rectangle imder the segments of a focal chord, by the length of that chord" (Art, 193 Ex. 1). Ex, 3, If a and b be the lengths of two tangents to a parabola which intersect at, right angles, and m one quaiter of the parameter, prove n^ b^_ 1_ 206. If in the original equation g^ = /a, the coefScient of x vanishes in the equation transformed as in the last article ; and that equation h'^'^ + 2/'^/ -[- c = 0, being equivalent to one of the represents two real, coincident, or imaginary lines parallel to the new axis of x. We can verify that in this case the general condition that the equation should represent right lines is fulfilled. For this condition may be written c{ah-h'')=af~2hfg^-hg\ But if we substitute for a, A, &, respectively, a*, a/9, ^'\ the left- hand side of the equation vanishes, and the right-hand side becomes [fa—g^y. Writing the condition /a = ^/3 in either of the forms fd^ =-g'x^^ fa^=g^\ Ave see that the general equa- tion of the second degree represents two parallel right lines when ¥ = ah^ and also either af= hg^ or fh = hg. *207. If the original axes were oblique, the equation is still reduced, as in Art. 205, by taking for our new axes the line acc + /3?/, and the line perpendicular to it, whose equation is (Art. 26) (/3 - a cos co) a? - (a - /S cos w)y = 0. And if we write y^ = a'+ /3^'— 2a/3 cosw, the formulie of trans- formation become, by Art. 34, ryY={oix + ^g) sinct), 7^= (/3- a cosw) a; — (a — /9 cosw)y ; whence yx sin w = (a — /3 cos tw) Y+ 13 X sin o) ; 73/ sino) = (/3- a cosw) Y— aX sinw. Making these substitutions, the equation becomes 7'r'^ + 2sin'a)(,^/3~/a)X + 2 sin CO [g (a — /S cos w) +/(/3 — a cos &>)] 1"+ 70 sin'-'w = 0. 188 FIGURE OF THE CUEVE. And the transformation to parallel axes proceeds as in Art. 205. The principal parameter is __ 2^ 2(/a-ff/5) sin'-'ft) (a' + ;8'''-2a;ecosa))' Ex. Find the principal parameter of - ^J-^ ^' + 1=0. Am. («2 + J2 + 2f/Jcostt))'' FiaUEE OF THE CURVE. 208. From the equation y^ =2)x we can at once perceive the figure of the curve. It must be symmetrical on both sides of the axis of a;, since every value for x gives two equal and opposite for y. None of it can lie on the negative side of the origin, since if we make x negative, y will be imagi- nary, and as we give increasing positive values to a;, we obtain increasing values for y. Hence the figure of the curve is that here represented. Although the parabola resembles the hyperbola In having in- finite branches, yet there is an important difference between the nature of the infinite branches of the two curves. Those of the hyperbola, we saw, tend ultimately to coincide with two diverg- ing right lines ; but this Is not true for the parabola, since, if we seek the points where any right line [x = Jcy -\- 1) meets the parabola {y^ =px)^ we obtain the quadratic whose roots can never be Infinite as long as h and I are finite. There Is no finite right line which meets the parabola in two coincident points at infinity; for any diameter (y = wi), which meets the curve once at Infinity (Art. 1J:2), meets It once also In the point x= — ; and although this value Increases as m in- creases, yet it will never become infinite as long as m Is finite. 209. The figure of the parabola may be more clearly con- ceived from the following theorem : If we suppose one vertex THE TANGENT. l^Q and focus of an ellipse given, while its axis major increases with- out limit, the curve will ultimately become a parabola. The equation of the el- lipse, referred to its vertex, t is (Art. 194) We wish to express b in terms of the distance VF{=m)j which we suppose fixe(J. We have m = a — \l[a^ - h^) (Art. 182), whence h^ = 2am — m^^ and the equation becomes ., / 2m\ /2m m\ „ Now, if we suppose a to become infinite, all but the first term of the right-hand side of the equation will vanish, and the equation becomes y'=Amx, the equation of a parabola. A parabola may also be considered as an ellipse whose ecceu- b^ P tricity is equal to 1. For e^ = 1 n . Now we saw that — , , •^ ^ a' a^ ' which is the coefficient of x^ in the preceding equation, vanished as we supposed a increased according to the prescribed condi- tions; hence e'"' becomes finally = 1. THE TANGENT. 210. The equation of the chord joining two points on the curve is (Art. 86) (^_y) (^_y') =f-pa:, or {l/'+f)y=px + 2/y- And if we make 3/" = ?/', and for y'' write its equal px', we have the equation of the tangent If in this equation we put y = 0, we get x = - x : hence TM (see fig. next page) (which is called the Subtangent) is bisected at the vertex. These results hold equally if the axes of co-ordinates are oblique; that is to say, if the axes are any diameter and the tangent at its vertex, in which case we saw (Art. 203) that the equation of the parabola is still of the form ?/' =p'x. 190 DIAMETERS. This Article enables us, there- fore, to draw a tangent at any point on the parabola, since we have only to take TV= VM and join FT- or again, having found this tangent, to draw an ordinate from P to any other diameter, since we have only to take V'3I' = T' V\ and join PJf '. 211. The equation of the polar of any point x't/' is similar in form to that of the tangent (Art. 89), and is, therefore, 27/'i/=2}(x + x'). Putting ?/ = 0, we find that the intercept made by this polar on the axis of cc is —x. Hence the intercejH icMch the 'polar s of any two 'points cut off on the axis is equal to the intercept between jyerjiencliculars from those points on that axis; each of these quantities being equal to {x — x"). DIAMETERS. 212. We have said, that if we take for axes any diameter and the tangent at its extremity, the equation will be of the form y'^ =px. We shall prove this again by actual transformation of the equation referred to rectangular axes (?/'''=j!?a?), because it is de- sirable to express the newp' In terms of the old j). If we transform the equation y'' —px to parallel axes through any point [xy^ on the curve, writing x + a;' and y -^ y iov x and 2/, the equation becomes Now if, preserving our axis of ic, we take a ncAv axis of y, inclined to that of x at an angle 6^ we must substitute (Art. 9), y sind for y, and x + y cos6 for x, and our equation becomes y^ sin'' 6 + 2y'y sin 6 =px -V py cos B. In order that this should reduce to the form if =px^ we must have 2y' &\n6=p) cos^, or tan^= ^, . Now this is the very angle which the tangent makes with the axis of a;, as we see from the equation 2yy=p[x-\-x). THE NORMAL. 191 This equation, therefore, referred to a diameter and tangent, will take the form y sin'6' a;, or y =px. The quantity p is called the parameter corresponding to the diameter F'J/', and we see that the parameter of any diameter is inversely proportional to the square of the sine of the angle xjchich its ordinates inalce with the axis, since p = . ., a . ' ^ sin" V We can express the parameter of any diameter in terms of the p co-ordinates of its vertex, from the equation tan^= ^y-, ; hence. sin^ = P P hence p =p + 4ic'. THE NORMAL. 213. The equation of a line through [xy') perpendicular to the tangent 2yy' =p {x -f x) is piy- y) + 2/ ix - x) = 0. If we seek the intercept on the axis of a-, we have x{=VN)=x'-\-l2,-' and, since VM= x\ we must have MN [\\xQ sulmormal^ Art. 181) = \p. Hence in the parabola the suhnormal is constant^ and equal to the semi-parameter. The normal itself = ^/[PM' + MN'') = Vl/" + i/) = ^{p [x 4- \p)] = i V(i>iV). THE FOCUS. 214. A point situated on the axis of a parabola, at a distance from the vertex equal to one-fourth of the principal parameter, is called ihe Jocks of the curve. This is the point which. Art. 209, has led us to expect to find analogous to the focus of an ellipse ; and we shall show, in the present section, that a parabola may in every respect be considered as an ellipse, having one of Its foci at this distance, and the other at infinity. To avoid trac- 192 THE F0CIJ3. tions wc shall often, in the following Articles, use the abbrevia- tion m = \p. To find the distance of any point on the curve from, the focus. Tiie co-ordinates of the focus being (;«, 0), the square of its distance from any point is [x — m)'^ + 1/"^ = x" — 2mx' + n^ + ^.mx = [x + nif. Hence the distance of any point from the focus =x -\- m. This enables us to express more simply the result of Art. 212, and to say that the parameter of any diameter is four times the distance of its extremity from the focus. 215. The polar of the focus of a parabola Is called the directrix^ as in the ellipse and hyperbola. Since the distance of the focus from the vertex = m, its polar is (Art. 211) a line perpendicular to the axis at the same dis- tance on the other side of the vertex. The distance of any point from the directrix must, therefore, =x' -\-m. Hence, by the last Article, the distance of any point on the curve from the directrix is equal to its distance from the focus. Wc saw (Art. 186) that in the ellipse and hyperbola, the distance from the focus is to the distance from the directrix in the constant ratio e to 1. We see, now, that this is true for the parabola also, since in the parabola e = 1 (Art. 209). The method given for mechanically describing an hyperbola, Art. 202, can be adapted to the mechanical description of the parabola, by simply making the angle ABR a right angle. 216. The point lohere any tangent cuts the axis ^ and its point of contact^ are equally distant from the focus. For, the distance from the vertex of the point where the tangent cuts the axis =x' (Art. 210), its distance from the focus is, therefore, x + m. 217. Any tangent makes equal angles with the axis and with the focal radius vector. This is evident from inspection of the isosceles triangle, which, in the last Article, we proved was formed by the axis, the focal radius vector, and the tangent. This is only an extension of the property of the ellipse (Art. 188), that the angle TPF= TPF'; for, if we suppose the THE FOCUS. 193 focus F' to go off to infinity, the line PF' will become parallel to the axis, and TPF= PTF. (^ee figure, p. 189.) Hence the tangent at the extremity of the focal ordinate cuts the axis at an angle of 45°^ 218. To find the length of the ^perpendicular from the focus on the tangent. The perpendicular from the point (?», 0) on the tangent [yy' = 2w [x + x) [■ is 2m [x + m) 2m fa;' + m) ,, , , ., V(?/ +4m') V(4ma;+4m') ^ ^ ■" Hence (see fig., p. 191) FR is a mean proportional between FV and FP. It appears, also, from this expression, and from Art. 213, that FR is half the normal, as we might have inferred geometrically from the fact that TF= FN. 219. To express the perpendicular from the focus in terms of the angles which it makes with the axis. We have cosa = sini^272=(Art. 212) a/(^^) • Therefore (Art. 218), m FR = \/\m ix + m)| = . •• ^ '■' cosa The equation of the tangent, the focus being the origin^ can therefore be expressed m X cosa + V sma H = 0, cosa and hence we can express the perpendicular from any other point in terms of the angle it makes with the axis. 220. The locus of the extremity of the perpendicular from the focus on the tangent is a right line. For, taking the focus for pole, we have at once the polar equation j,j p = , pcosa = ?H: ^ cos a ' which obviously represents the tangent at the vertex. Conversely, if from any point F we draw FR a radius vector c u 194 THE FOCUS. to a right line VB, and draw PB perpendicular to it, the line BB will always touch a parabola having F for its focus. We shall show hereafter how to solve generally questions of this class, where one condition less than is sufficient to determine a line is given, and it is required to find its envelope^ that is to say, the curve which it always touches. We leave, as a useful exercise to the reader, the investiga- tion of the locus of the foot of the perpendicular by ordinary rectangular co-ordinates. 221. To find the locus of the intersection of tangents which cut at right angles to each other. The equation of any tangent being (Art. 219) X cos'"' a. + y sin a cos a + «i = 5 the equation of a tangent perpendicular to this (that is, whose perpendicular makes an angle = 90° -f a with the axis) is found by substituting cos a for sin a, and —sin a for cos a, or X sin^a — y sin a cosa + m = 0. a is eliminated by simply adding the equations, and we get X + 2m = 0, the equation of the directrix^ since the distance of focus from directrix = 2/>2. 222. The angle between any two tangents is half the angle between the focal radii vectores to their points of contact. For, from the isosceles PFT^ the angle PTF which the tan- gent makes with the axis is half the angle PiuY, which the focal radius makes with it. Now, the angle between any two tangents is equal to the difference of the angles they make with the axis, and the angle between two focal radii is equal to the difference of the angles which they make with the axis. The theorem of the last Article follows as a particular case of the present theorem: for if two tangents make with each other an angle of 90°, the focal radii must make with each other an angle of 180°, therefore, the two tangents must be drawn at the extremities of a chord through the focus, and, therefore, from the definition of the directrix, must meet on the directrix. 223. The line joining the focus to the intersection of two tangents bisects the angle ichich their i^oints of contact subtend at the focus. THE FOCUS. 195 Subtracting one from the other, the equations of two tan- gents, viz., X cos^a + y sina cosa + w = 0, x cos^/3 + ^ sIn/3 cos/S + 5n = 0; we find for the line joining their Intersection to the focus, X sin (a + /31 — ?/ cos (a + /3) = 0. This is the equation of a Hne making the angle a + yS with the axis of X. But since a and y8 are the angles made with the axis by the perpendiculars on the tangent, we have VFP= 2a and VFP' = 2^ ; therefore the line making an angle with the axis = a + /S must bisect the angle PFF. This theorem may also be proved by calculating, as in Art. 191, the angle [6 — 6') subtended at the focus by the tangent to a parabola from the point xy ; when it will be found that cos(^ — ^')= ■ , a value which, being P independent of the co-ordinates of the point of contact, will be the same for each of the two tangents which can be drawn through xi/. (See O'Brien's Co-ordinate Geometry^ p. 156.) COE. 1. If we take the case where the angle Pi^P' = 180°, then PP' passes through the focus ; the tangents TP, TP will intersect on the directrix, and the angle TFP=dO°. (See Art. 192). This may also be proved directly by forming the equa- tions of the polar of any point (— ?«, y') on the directrix, and also the equation of the line joining that point to the focus. These two equations are y'y = 2m [x — ?n), 2m [y - y') + y' {x + m) = 0, which obviously represent two right lines at right angles to each other. p'^ COK. 2. If any chord PP' cut the directrix In Z), then FD is the external bisector of the angle PFP. This is proved as at p. 178. Cor. 3. If any variable tan- gent to the parabola meet two fixed tangents, the angle sub- tended at the focus by the portion of the variable tangent intercepted between the fixed tangents, is the supplement of the angle between the fixed tangents. For (see next figure) 196 " THE FOCUS. the angle QRT is half pFq (Art. 222), anil, by the pre- sent Article, PFQ is obviously also half 2>F(i, therefore, PFQ is = QRT^ or is the supplement of PRQ. Cor. 4. The circle circumscribing the triangle formed hy any three tangents to a parabola will pass r/-<^^^^^^^ through the focus. For the circle de- scribed through T^ PBQ must pass through jP, since the angle contained in the segment PFQ will be the supplement of that contained in PBQ. 224. To find the polar equation of the parabola^ the focus being the pole. p We proved (Art. 214) that the focal radius = cc' -f m = VM+m = FM+ 2wz =p cos 6 + 2m. TT 2 m Hence p = -^ . •^ 1 — cos This is exactly what the equation of Art. 193 becomes, if we suppose e=l (Art. 209). The properties proved in the Ex- amples to Art. 193 are equally true of the parabola. In this equation 6 is supposed to be measured from the side FM] if we suppose it measured from the side FVj the equation becomes 2?n ^^T+cosd' This equation may be written p cos^^6 = mj or p4 cosi^ = (w)*, and is, therefore, one of a class of equations, p COS710 = a , some of whose properties we shall mention hereafter. ( 197 ) CHAPTER XIII. EXAMPLES AND MISCELT.ANEOTTS PROPERTIES OF COXIC SECTIONS. 225. The method of applying algebra to problems relating to conic sections is essentially the same as that employed in the case of the right line and circle, and will present no difficulty to any reader who has carefully worked out the Examples given in Chapters III. and vii. We, therefore, only think it necessary to select a few out of the great multitude of examples which lead to loci of the second order, and wc shall then add some properties of conic sections, which it was not found convenient to insert in the preceding chapters. Ex. 1. Through a fixed pomt P is di-awn a line LK (see fig., p. 40) terminated by two given ILaes. Eind the locus of a point Q taken on the Une, so that PL =: QK. Ex. 2. Two equal rulers, AB, BC, are connected by a pivot at B ; the extremity A is fixed, while the ex- tremity C is made to traverse the right Une A C ; find the locixs described by any fixed point P on BC. Ex. 3, Given base and the product of the tangents of the halves of the base angles of a triangle : find the locus of vertex. Expressing the tangents of the half angles in terms of the sides, it wOl be foimd that the sum of sides is given ; and, therefore, that the locus is an eUipse, of which the extremities of the base are the foci. Ex. 4. Given base and sum of sides of a triangle ; find the locus of the centre of the inscribed circle. It may be immediately inferred, from the last example, and from Ex. 4, p. 47, that the locus is an eUipse, whose vertices are the extremities of the given base. Ex. 5. Given base and simi of sides, find the locus of the intersection of bisectors of sides. Ex. 6. Find the locus of the centre of a circle which makes given intercepts on two given lines. Ex. 7. Find the locus of the centre of a circle which touches two given circles ; or which touches a right line and a given circle. Ex. 8, Find locus of centre of a cu'cle which passes through a given point and makes a given intercept on a given line. Ex. 9. Or which passes thi'ough a given point, and makes on a given line an in- tercept subtending a given angle at that point. Ex. 10. Two vertices of a given triangle move along fixed right lines ; find the locus of the third. 198 EXAMPLES OF CONIC SECTIONS. Ex. 11. A triangle J5C circumecribes a given circle; the angle at C is given, and £ moves along a fixed line ; find the locus of A. Let us use polar co-ordinates, the centre being the pole, and the angles being measured from the perpendicular on the fixed line ; let the co-ordinates of A, B, be p, ; p', 0'. Then we have p' cosf»' =p. But it is easy to see that the angle AOB is given (= a). And since the jjerpendicular of the tnangle AOB is given, we have pp' sin a '' ~ 4(/o2 + p""'- 2pp' cos a) • But 6 + 6' = a; therefore the polar equation of the locus is p-p^ sln^g ~ p^ cos'-* (a - 6) +J)^ - 2pp cos a cos (a - 6) ' which rep-esents a conic. Ex. 12. Find the locus of the pole with respect to one conic A of any tangent to another conic B. Let aji be any point of the locus, and \x + /xy + v its polar with respect to the conic A, then (Art. 89) \, fi, v are functions of the first degi-ee in a, /3. But (Art. 151) the condition that \x + fiy + v should touch 5 is of the second degree in X, /x, v. The locus is therefore a conic. Ex. 13. Find the locus of the intersection of the pei-pendicular from a focus on any tangent to a central conic, with the radius vector from centre to the point of contact. Ans. The con-esponding dii-ectrix. Ex. 14. Find the locus of the intersection of the perpendicular from the centre on any tangent, with the radius vector fi'om a focus to the point of contact. Ans. A circle. Ex. 15. Find the locus of the intersection of tangents at the extremities of conju- gate diameters. . ^ V^ a *" Am. - 4- 1;.= 2. This is obtained at once by squaring and adding the equations of the two tangents, attending to the relations Art. 172. Ex. 16. Trisect a given arc of a circle. The points of trisection are found as the intersection of the circle with a hyperbola. See Ex. 7, p. 47. Ex. 17, One of the two parallel sides of a trapezium is given in magnitude and position ; and the other in magnitude. The sum of the remaining two sides is given ; find the locus of the intersection of diagonals. Ex. 18. One vertex of a parallelogram circumscribing an ellipse moves along one directrix ; prove that the opposite vertex moves along the other, and that the two re- maining vertices are on the circle described on the axis major as diameter. 226. We give in this Article some examples on the focal properties of conies. Ex. 1. The distance of any point on a conic from the focus is equal to the whole length of the ordinate at that point, produced to meet the tangent at the extremity of the focal ordinate. Ex. 2. If from the focus a line be drawn making a given angle with any tangent, find the locus of the point where it meets it. Ex. 3. To find the locus of the pole of a fixed line with regard to a series of con- centric and confocal conic sections. We know that the pole of any line (-+- = 1|, with regard to the conic \m, n J I — 2 + ri = 1 1 , is found from the equations mx = (fi and ny = b- (Art. 169). EXAMPLES OF CONIC SECTIONS. 199 Now, if the foci of the conic are given, a- — b-= c- Ls given ; hence, the locus of the pole of the fixed line is ^j^, _ „„ _ ^2 the equation of a right Une perpendicular to the given line. If the given line touch one of the conies, its pole will be the point of contact- Hence, given two confocal conies, if we draw any tangent to one and tangents to the second where this line meets it, these tangents wiU intersect on the normal to the first conic. Ex. 4. Find the locus of the points of contact of tangents to a series of confocal ellipses from a fixed point on the axis major. Ans. A circle, Ex. 5. The lines joining each focus to the foot of the pei-pendicular from the other focus on any tangent, intersect on the coiTesponding normal and bisect it. Ex. 6. The focus being the pole, prove that the polar equation of the chord thi'ough points whose angular co-ordinates are a + /3, a — /3, is p ~ = e cos B + seep cos (0 — a). This expression is due to Mr. Frost {Cambridge and Duhlin Math, Journal, I., 68, cited by Walton, Examples, p. 375). It follows easily from Ex. 3, p. 37. Ex. 7. The focus being the pole, prove that the polar equation of the tangent, at the poiat whose angrdar co-ordinate is a, is — = e cos0 4- cos(0 — a). 2p This expression is due to Mr. Davies {Philosophical Magcuine for 1842, p. 192, cited by Walton, Examples, p. 3G8). Ex. 8. If a chord PP' of a conic pass through a fixed point 0, then tan^PfCtanJ^P'/^O is constant. The reader wiU find an investigation of this theorem by the help of the equation of Ex. 6 (Walton's Examples, p. 377). I insert here the geometrical proof given by Mr. Mac Cidlagh, to whom, I beUeve, the theorem is due. Imagine a point taken anywhere on PP' (see figure, p. 195), and let the distance FO be e' times the distance of fi-om the directiix ; then since the distances of P and from the dii-ectiis are proportional to PD and OD, we have FP_.FO^__± . smPDF , &mODF _ e FD^ 0D~7" °^" sinPFU " smOFD " e' " cos OFT e Hence (Art. 192) Z^H^FT^ 7' or, since (Art.191) PFT is half the sum, and OFT half the chfference, of PFO andP'FO, tan iPFO . tan iP'FO - ^-^^ . ■' ^ e + e' It is obvious that the product of these tangents remains constant if be not fixed, but be anywhere on a conic having the same focus and directxix as the given conic. Ex. 9. To express the condition that the chord joining two points x'y', x"g" on the curve passes through a focus. This condition may be expressed ia several equivalent forms, two of the most useful of which are got by expressing that 0" = 6' + 180° where 6', 6" are the angles made with the axis by the Unes joining the focus to the points. The condition sin 6" — - sin t)' gives — ^, + '•' „ = ; a (y' + y") - e {x'g" + x"y'). 200 EXAMPLES OF CONIC SECTIONS. The condition cos 6" = — cost)' gives '^ ~ , + — —,, = : 2ex'x" - (rt + ce) (x' + x") + 2ac = 0. a — ex a — ex ' \ / \ Ex. 10. If normals be drawn at the extremities of any focal chord, a line drawn through their intereection parallel to the axis major will bisect the chord. [This solution is by Larrose, Nouvelles Annahs xix. 85]. Since each normal bisects the angle between the focal radii, the intersection of normals at the extremities of a focal chord is the centre of the circle inscribed in the triangle whose base is that chord, and sides the lines joining its extremities to the other focus. Now M a,b, c be the sides of a triangle whose vertices are x'y', x"y", x"'y"'i then, Ex. 6, p. 6, the co-ordinates of the centre of the inscribed cii'cle are _ ax' + hx" -f ex'" _ ay' + hy" 4- cy'" "^ ~ a + b + c ' ^~ a+TTc ' In the present case the co-ordinates of the vertices are x', y' ; x", y" ; — c, ; and the lengths of opposite sides are a + ex", a + ex', 2a — ex' — ex". We have therefore _{a + ex') y" + {a + ex") y' y_ _ ^ or, reducing by the first relation of the last Example, y = i{y' + y"), which proves the theorem. In like manner we have _{a + ex") x' + {a + ex') x" — (2a — ex' — ex") c ^ - 4a ' which, reduced by the second relation, becomes (a + ec) (x' + x") — 2ac X = ^^— r . 2a We conld find, similarly, expressions for the co-ordinates of the intersection of tangents at the extremities of a focal chord, since this point is the centre of the circle exscribed to the base of the triangle just considered. The hne joining the intersection of tangents to the corresponding intersection of normals evidently passes through a focus, being the bisector of the vertical angle of the same triangle. Ex. 11. To find the locus of the intersection of normals at the extremities of a focal chord. Let a, /3 be the co-ordinates of the middle point of the chord, and we have, by the last ExamiJle, a = i{x' + x") = ^-t±^; p = i{y' + y")=y. a- + c' If, then, we knew the equation of the locus described by a/3, we should by making the above substitutions have the equation of the locus described by xy. Now the polar equation of the locus of middle point, the focus being origin, is (Art. 193) — 62 e cos 6 which transformed to rectangular axes, the centre being origin, becomes b-u- + a"ji- — b-ca. The equation of the locus sought is, therefore, a-b^ {x + c)- + {a- + cYy" = b-c (a^ + c-) {x + c). EXAMPLES ON CONIC SECTIONS. 201 Ex. 12. If 6 be the angle between the tangents to an ellipse from any point P ; and if p, p' be the distances of that point from the foci, prove that cos 6 — — -. . 2pp Eor (Art. 189) FT.F'T' h- sm TPF. sin tPF = ^77^^77,, = — . PF.Pl' pp But cos FPF' - cos TPt = 2 sin TPF. sin tPF; and 2pp' cos FPF' - p"- -v p"^ - dc^. Ex. 13. If fi-om any point two lines be drawn to the foci (or touching any confocal conic) meeting the conic \a. R, R' ; S, S' ; then UR-m'^US-WS'- [Mr. M. Roberts.] It appears from the quadratic, by which the radius vector is determined (Art. 136), that the difference of the reciprocals of the roots will be the same for two values of 6, which give the same value to {ac — g-) cos^ 6 + 2 (c/« - gf) cos 6 sin + {be —f-) sin- 0. Now it is easy to see that A cos- -f- 2H cos 6 sin Q + B sin- has equal values for any two values of 0, which correspond to the directions of lines equally inclined to the two represented by Ax^ + 2Hxy + By- = 0. But the function we are considering becomes = for the dii-ection of the two tangents thi-ough (Art. 147) : and tangents to any confocal are equallj' inclined to these tangents (Art. 189). It follows fi-om this example that chords which touch a confocal conic are proportional to the squares of the parallel diameters (see Ex. 15, p. 210). 227. We give in this Article some examples on the parabola. The reader will have no difficulty in distinguishing those of the examples of the last Article, the proofs of which apply equally to the parabola. Ex. 1. Find the co-ordinates of the intersection of the two tangents at the point3 x'y', x"y", to the parabola y"^ — px, _ ^' + y" _ .'/.'/" A.ns. y — ^ , X — . i p Ex. 2. Find the locus of the intersection of the pei-pendicular from focus on tan- gent with the radius vector fi'om vertex to the point of contact. Ex. 3. The three peiTJendiculars of the ti-iangle formed by three tangents intersect on the directrix (Steiner, Gergonne, Annales, XIX. 59 ; Walton, p. 119), The equation of one of those pei"pendiculars is (Art. 32) yV" - y'y" /^ _ y"!/"' \ ^ y'" - y" (y _ y" + y"' \ _ q . which, after dividing by y'" — y", may be written „' (x +p\ - y'y"y"' +py p(^'+ y" + y'"^ _ o. \ 4/ ^ 2 4 The symmetry of the equation shows that the three perpendiculars intersect on the directrix at a height _ -y'u"y"' , y' + y" + y'" ■' p^ "*" 2 Ex. 4. The area of the triangle formed by three tangents is half that of the tri- angle formed by joining theii- points of contact (Gregory, Cambridge Journal, II. 16 : Walton, p. 137. See also Lessons on Higher Algebra, Ex. 12, p. 14). DD 202 EXAMPLES ON CONIC SECTIONS. Substituting the co-ordinates of the vertices of the triangles in the expression of 27/ Art. 36, we find for the latter area, it- {y' - y") {y" - y'") W" - 2/0 j ^^^ ^°^' ^^^ former area half this quantity. Ex. 5. Find an expression for the radius of the circle circumsciibing a triangle inscribed in a parabola. The radius of the cu-cle circunisci-ibing a tiiangle, the lengths of whose sides are def il, e, f, and whose area = S is easily proved to be -j^ . But if d be the length of the chord joining the points x"y", x"'y"', and 6' the angle which this chord makes with the axis, it is obvious that d sin 6' = y" — y'". Using, then, the expression for the P area found in the last Example, we have E = 9 g^p g' r^q q" cjj, ^"' • ^^ might ex- press the radius, also, in terms of the. focal chords parallel to the sides of the triangle. For (Art. 193, Ex. 2) the length of a chord making an angle 6 with the axis P -TT r^ C'C"C"' is c = -—-;. , Hence E^ = — . sm- ip It follows from Art. 212 that c', c", c'" are the parameters of the diameters which bisect the sides of the triangle, Ex. 6. Express the radius of the circle circumscribing the triangle formed by three tangents to a parabola in terms of the angles which they make with the axis. P p'p"p'^' Ans. R - o ■ a- ■ an ■ A»/ ; ox R"- - —— — where »', ;/', ;/" are the para- 8 sin e sm % sm t) 64;j ' i > x > 1 1 meters of the diameters through the points of contact of the tangents (see Art. 212). Ex. 7. Find the angle contained by the two tangents through the point x'y' to the parabola y'^ = 4mx. The equation of the pair of tangents is (as in Art. 92) found to be (^'2 _ 4„ja;') (j^2 _ 4_„ix) = {y?/' — 2m {x + x')}^. A parallel pair of Hnes through the origin is x'y'^ — y'xy + mi? — 0. i(«'2 _ ^inx'\ The angle contained by which is (Art. 74) tan <^ = , ■ . Ex. 8. Find the locus of intersection of tangents to a parabola which cut at a given angle. Ans. The hyperbola, y" — ^mx — {x 4- nCf- tan- ^, or ?/- + (a; — inf = (a; + ot)^ sec'^ <^. From the latter foi-m of the equation it is evident (see Art. 186) that the hyperbola has the same focus and directrix as the parabola, and that its eccentricity = sec <p. Ex. 9. Find the locus of the foot of the peii^endicular from the focus of a parabola on the normal. The length of the perpendicular from (?«, 0) on 1m {y — y') + y' (x — a;') = is But if 6 be the angle made with the axis by the perpendicular (Art, 212) sin0= If-^V cose= If^). ■ ^J\a; 4- mj 4\x -J- mj Hence the pokr equation of the locus is TO cos „ EXAMPLES ON CONIC SECTIONS. 203 Ex. 10. Find the co-ordinates of the intersection of the normals at the points Ans. x-lm+ ^^ , y- g^^^. Or if a, /3 be the co-ordinates of the corresponding intersection of tangents, then (Ex.1) p, „^j Ex. 11. Find the co-ordinates of the points on the curve, the normals at which pass through a given point x'l/'. SolvLag between the equation of the normal and that of the ciu"ve, we find 2y^ + {p'- 2px') y =p'^y\ and the three roots are connected by the relation yi + «/2 + ^3 = 0- The geometric meaning of this is, that the chord joining any two, and the line joining the third to the vertex, make equal angles with the axis. Ex. 12, Find the locus of the intersection of normals at the extremities of chords which pass through a given point x'y'. "We have then the relation (iy' = 2in [x' + a) ; and on substituting in the results of Ex. 10 the value of a derived from this relation, we have 2m,x + Py' = 4m2 + 2^^ ^ ^inx' ; 2vihj — 2j37nx' — fPy' ; whence, eliminating j8, we find 2 {2m (j/ - y^ + y' {x - x')}^ = {imx' - y'^) {t/'y + 2x'x - imx' - 2x"^), the equation of a parabola whose axis is perpendicular to the polar of the given point. If the chords be parallel to a fixed line, the locus reduces to a right line, as is also evident from Ex. 11. Ex. 13. Find the locus of the intersection of normals at right angles to each other. In this case a — — m, x — 3??i -i — , y = /3, y- — m {x — 3»j). Ex. 14, If the lengths of two tangents be a, b, and the angle between them w ; find the parameter. Draw the diameter bisecting the chord of contact ; then the parameter of that y- y- sin- d rn-y" . diameter is p = — , and the principal parameter is j) — = -j-j- (where w is the length of the perpendicular on the chord fi-om the intersection of the tangents). But 2my = ab sinoi, and 16x- = a'' + b- + 2ab cos to ; hence 4rt-62 sin^ (0 , , OQ. p = (see p. 188), (a^-t- b^ + 2ab cos to) '^ Ex. 15. Show, from the equation of the circle circumscribing three tangents to a parabola, that it passes through the focus. The equation of the circle circumscribing a triangle being (Art, 124) /3y sin A -f ya sin £ -I- a/3 sin C = 0, the absolute term in this equation is foimd (by wi-iting at full length for a, X cos a->r y ^m.a— p, &c.) to be p'p" sin (/3 — y) -I- p"p sin (y — a) -I- pp' sin (a — /3), But if the line a be a tangent to a parabola, and the origin the focus, we have (Art, 219) m P — > and the absolute term cos a cos /J cos y which vanishes identicallv. sin {fi - y) cos a + sin (y - ft) cos/? + sin (« - p) cos y ■, 204 EXAMPLES ON CONIC SECTIONS. Ex. 16. Find the locus of the intersection of tangents to a parabola, being given either (1) the product of sines, (2) the product of tangents, (3) the sum or (4) difference of cotangents of the angles they make with the axis. Alts. (1) a cu-cle, (2) a right line, (3) a right Ime, (4) a parabola. 228. "VVe add a few miscellaneous examples. Ex. 1. If an equilateral hjijerbola circumsciibe a triangle, it will also pass through the intersection of its pei-pendiculars (Brianchon and Poncelet; Gergonne, Annales, XI., 205 : Walton, p. 283). The equation of a conic meeting the axes in given points is (Ex. 1, p. 143) fxfx'j? + 2hxy + XXy — n/x' (X + X') x — W (jx + fx) y + XX.'/ui/x' = 0. And if the axes be rectangular, this will represent an equilateral hyperbola (Art. 174) if W = — fxfi'. If, therefore, the axes be any side of the given triangle, and the perpendicular on it from the opposite vertex, the portions X, A.', /x are given, there- fore, fi' is also given ; or the curve meets the perpendicular in the fixed point y = — — > which is (Ex. 7, p. 27) the intersection of the perpendiculars of the triangle. Ex. 2. What is the locus of the centres of equilateral hyperbolas through three given points ? A71S. The circle through the middle points of sides (see Ex. 3, p. 148). Ex. 3. A conic being given by the general equation, find the condition that the pole of the axis of x should lie on the axis of i/, and vice versa, Ans. he —Jff. Ex. 4. In the same case, what is the condition that an asymptote should pass through the origin ? Ans. aj"^ — 2fgh + bg'^ — 0. Ex. 5. The circle circumscribing a tiiangle, self -con jugate with regard to an equi- lateral hyperbola (see Art. 99), passes through the centre of the curve. (Brianchon and Poncelet ; Gergonne, XI. 210 ; Walton, p. 304). [This is a particular case of the theorem that the six vertices of two self -con jngate triangles lie on a conic (see Ex. 1, Art. 375).] The condition of Ex. 3 being fulfilled, the equation of a circle passing through the origin and thi-ough the pole of each axis is h {x- -t- 2xy cos w 4- f/^) -'rfx + gy = 0, or X {hx + hy +f) + y (ax + hy + g) — (a + b — 2h cos oi) a-y, an equation which will evidently be satisfied by the co-ordinates of the centre, pro- vided we have a + b = 2k cos tu, that is to say, provided the cm-ve be an equilateral hyperbola (Arts. 74, 174). Ex. 6. A chcle described through the centre of an equilateral hyperbola, and through any two points, mil also pass through the intersection of lines drawn through each of these jDoints parallel to the polar of the other. Ex. 7. Find the locus of the intersection of tangents which intercept a given length on a given fixed tangent. The equation of the pah of tangents from a point x'y' to a conic given by the general equation, is given Ai-t. 92. Make y =.0, and we have a quadratic whose roots are the intercepts on the axis of x. Forming the difference of the roots of this equation, and putting it equal to a constant, we obtain the equation of the locus reqmi-ed, which will be in general of the fourth degree ; but if ^- = ac, the axis of x will touch the given conic, and the equation of the locus will become di^'isible by y'^, and will reduce to the second EXAMPLES ON CONIC SECTIONS. 205 degi'ee. We could, by the help of the same equation, find the locus of the intersection of tangents ; if the sum, product, d'C, of the intercepts on the axis be given. Ex. 8. Given four tangents to a conic to find the locus of the centre. [The solution here given is by P. Serret, Nouvelhs Annales, 2nd series, iv. 145.] Take any axes, and let the equation of one of the tangents be x cos a + ^ sin a ~p — 0, then a is the angle the perpendicular on the tangent makes vrith the axis of x ; and if © be the unknown angle made with the same axis by the axis major of the conic, then a — 6 is the angle made by the same perpendicular with the axis major. If then X and y be the co-ordinates of the centre, the formula of Art. 178 gives us {x cosa + ?/ sina — pf — '*" COS^(a — 6) + i- sin-(a — 6). We have four equations of this form from which we have to eliminate the three unknown quantities o^, 6-, 6. Using for shortness the abbreviation a for X cos a-\-y sma— p (Art. 53), this equation expanded may be written a^ = (a- cos-e + 6- sin^O) cos-a + 2 (a- — h") cos 6 sin % cos a sin a + (a- sin-0 + 6- cos-e) sin-a. It appears then that the three quantities a- cos- % -\-h- sin= 0, (a^ — J-) cos 6 sin 0, «- sin^ + 62 cos^ 0, may be eliminated linearly from the four equations ; and the result comes out in the form of a determinant a-, cos^ a, cos a sin a, sin- a /3^, cos^/S, cosj3 sin/3, sin^/S y^, cos- y, cos y sin y, sLa- y 0^, cos- 5, coso sin (5, sin- 5 = 0, which expanded is of the form yla^ + jB/3- -f- Cy- -1- Bc^ = 0, where A, B, C, D are known constants. But this eqiiation though apparently of the second degi-ee is in reahty only of the first ; for if, before expanding the determinant, we write a-, <tc., at full length, the coefficients of x- are cos-a, cos-j3, cos-y, cos-o; but these being the same as one column of the determinant, the part multipUed by x- vanishes on expansion. Similarly, the coefficients of the terms xy and y- vanish. The locus is therefore a right line. The geometrical determination of the line depends on prin- ciples to be proved afterwards : namely, that the polar of any point with regard to the conic is Aa'a + BjS'lS + Cy'y + M'o = ; and therefore that the polar of the point ap passes through yS. But when a conic reduces to a line by the vanishing of the three highest tei-ms in its equation, the polar of any point is a parallel line at double the distance from the point. Thus it is seen that the hne represented by the equation bisects the lines joining the points a/3, yS ; ay, (i8 ; ad, |8y. Conversely, if we are given in any form the equations of four lines a — 0, &c., the equation of the hne joining the three middle points of diagonals of the quadi-ilateral may, in practice, be most easily formed by determining the constants so that Aa- + Bl3r + Cy" + Do- = shall represent a right line. Ex. 9. Given three tangents to a conic and the sum of the squares of the axes, find the locus of the centre. We have thi-ee equations as in the last example, and a fourth a- + b'^ = k-, which may be ^vlitten k- = (a- cos 6 + b- sin-0) -f- {a- siu-0 + b- cos- 6), and, as before, the result appears m the form of a detennurant a", cos- a, cos a sin a, sin- a /S^, C0S-/3, cos/3 sin/3, sin-/3 y-, cos- y , cos y sin y , sin- y X--. 1 . , 1 ' = 206 THE ECCENTRIC ANGLE. which expanded is of the form Aa- + BjP + Cy^ + D-O. It is seen, as in the last example, that the coefficient of a-// vanishes in the expansion, and that the coefficients of a;* and y" are the same. The locus is therefore a circle. Now it Aa'^ + Bji" + Cy^ = represents a circle, it will afterwards appear that the centre is the intersection of perpendiculars of the triangle formed by the lines a, /8, y. The present equation there- fore, which differs from this by a constant, (Art. 81) represents a circle whose centre is the intersection of perpendiculars of the triangle formed by the three tangents. Ex. 10. Given four points on a conic to find the locus of either focus. The distance of one of the given points fi-om the focus (see Ex., p. 173) satisfies the equation p- Ax' + By' + C. We have four such equations from which we can linearly eliminate A, B, C, and we get the determinant p x' » y' , 1 p x" , y" , 1 p" x'" > y'" , 1 p"' x'" , y"" , 1 = 0, which expanded is of the form Ip + mp' + np" +pp"' — 0. If we look to the actual values of the coefficients /, m, n, ]), and their geometric meaning (Art. 36), this equation geometrically intei-preted gives us a theorem of Mobius, viz. OA.BCD+ OC.ABD= OB. ACD + OD.ABC, where is the focus, and BCD the area of the triangle formed by three of the points (compare Art. 94). It is seen thus that l + m + n + p — 0. If we substitute for p its value J{(a; — x'Y + (y — y'Y], <fcc., and clear of radicals, the equation of the locus is found to be of the sixth degree. If the four given points be on a circle, Mr. Sylvester has remarked that the locus breaks iip into two of the third degree, as Mr. Bm-nside has thus shewn. We have by a theorem of Feuerbach's, given Art. 94, Ip- + mp'- + np"- -i-pp'"- = 0. We have then (? + m) {Ip'^ + mp'-) = (« +p) {np"- -^ pp'"-), {/p + mp'y-= (np" +pp'"y-, whence, subtracting hu (j> — p')- = np {p" — p'")", which obviously breaks up into factors. THE ECCENTEIC ANGLE.* 229. It is always advantageous to express the position of a point on a curve, if possible, by a single independent variable, rather than by the two co-ordinates x'y'. We shall, therefore, find it useful, in discussing properties of the ellipse, to make a substitution similar to that employed (Art. 102) in the case of the circle ; and shall write X =a cos ^, y' = l> sin 0, a substitution, evidently, consistent with the equation (f - ^ = 1. * The use of this angle was recommended by Mr, O'Brien, Cambridge Mathematical Journal, vol. iv., p. 99c THE ECCENTRIC ANGLE. 207 The geometric meaning of the angle </> is easily explained. If we describe a circle on the axis major as diameter, and produce the ordinate at P to meet the circle at Q^ then the angle QCL= (f}^{or CL=CQ cosQCL, or x' = a cos(})', andPZ=- QL (Art. 163) ; or, since QL = a sin^, we have y' =^h sin^. 230. If we draw through P a parallel PNto the radius CQ, then pj/. CQ :: PL : QL :: I ; a, but CQ=^ a, therefore PM= h. PN parallel to CQ is, of course, = a. Hence, if from any point of an ellipse a line = a be inflected to the minor axis, its intercept to the axis major =h. If the ordinate PQ were produced to meet the circle again in the point Q\ it could be proved in like manner, that a parallel through P to the radius CQ' is cut into parts of a constant length. Hence, con- versely, if a line MN^ of a constant length, move about in the legs of a right angle, and a point P be taken so that MP may be constant, the locus of P is an ellipse, whose axes are equal to MP and NP. (See Ex. 12, p. 47). On this principle has been constructed an instrument for de- scribing an ellipse by continued motion, called the Elliptic Com- jyasses. CA^ CD' are two fixed rulers, MN a third ruler of a constant length, capable of sliding up and down between them, then a pencil fixed at any point of MN will describe an ellipse. If the pencil be fixed at the middle point of MN it will de- scribe a circle. (O'Brien's Co-ordinate Geometry^ p. 112). 231. The consideration of the angle ^ affords a simple me- thod of constructing geometrically the diameter conjugate to a given one, for ^ ' j tan^= ■^ = - tanrf). X a Hence the relation If tan^ tan^'=--^ (Art. 170) becomes tan ^ tan </>' = - 1 , or (j)- (ji' = 90°. 208 THE ECCENTRIC ANGLE. Hence we obtfiin the following construction. Let the ordi- nate at the given point P, when produced, meet the semicircle on the axis major at Q, join CQ^ and erect CQ' perpendicular to it; then the perpendicular let fall on the axis from Q' will pass through P', a point on the conjugate diameter. Hence, too, can easily be found the co-ordinates of P' given in Art. 172, for, since It I cosd)' = sin 6, we have — = i- , ^ ^^ a b and since sin 6' = — cos 6, we have 4- = • "^ ' o a From these values it appears that the areas of the. triangles PCMj P'CM' are equal. Ex. 1. To express the lengths of two conjugate semi-diameters in terms of the angle <^. Aiis. a'- = a- cos- (p + b- sin- (f> ; b'^ — a- shi-(p + b'^ cos-(t). Ex. 2. To express the equation of any chord of the ellipse in terms of 4> ^''■'^^ 'P' (see p. 94). ^^^^ | ^^^^ ^^ + <^') + 1 sin i (</> + </>') = cos^ (c^ - <p'). Ans. - COS<p + 't sin (^ = 1, Ex. 3. To express similarly the eqiiation of the tangent, Ana — r»nc A\ M Ex. 4. To express the length of the chord joining two points a, (3, Ifi — a^ (cos a — cos/3)- -I- 6^ (sin a — sin/3)-, D = 2 stH^ (a - /3) {a2 sui^i {a + fi)-^ i^ cos2i (« + /?)}^ But (Ex. 1) the quantity between the parentheses is the semi-diameter conjugate to that to the point ^ (a -I- /3) ; and (Ex. 2, 3) the tangent at the point l{a + (i) is parallel to the chord joining the points a, /3 ; hence, if b' denote the length of the semi- diameter parallel to the given chord, D = 2b' sin^ (a - /3). Ex. 5. To find the area of the triangle formed by three given points a, /3, y. By Art. 36 we have 22 = ab (sin (a - (3) + sin {(3 - y) + sin (y - a)} = ab {2 sini (a - ft) cos^ (a -ft) -2 sm^ (a - /3) cos i (a -)- /3 - 2y)} = 4a6 sini (a - /3) sin ^ (/3 - y) sini (y - a) S = 2a6 sin^ (a - /3) smi (/3 - y) sui^ (y - a). Ex. 6. If the bisectors of sides of an inscribed triangle meet in the centre, its area is constant. Ex. 7. To find the radius of the circle circumscribing the triangle formed by three given points n, (3, y. THE ECCENTRIC ANGLE. 209 If d, e,f be the sidcf? of the tiiangle formed by the three points, where h', b", h'" are the semi-diameters parallel to the sides of the triangle. If c', c", c'" be the parallel focal chords, then (see Ex. 5, p. 202) B? ip (These expressions are due to Mr. MacCullagh, Dublin Exam, Papers, 1836, p. 22.) Ex. 8. To find the equation of the circle cii'cum scribing this triangle. 2 (a2 - J2) X Ans. X- + y^ cosj (a + /8) cos J (/3 + y) cos J- (y + a) 2 (62 - a2) V - ^ sm j (a + /?) sini (/3 + y) sin^ (y + a) = i («2 + b") - i (a2 _ J2) {cos (a + /3) + cos (/3 + y) + COS (y + a)]. From this equation the co-ordinates of the centre of this cu-cle are at once obtained. Ex. 9. The area of the triangle formed by three tangents is, by Art. 39, ab tan i{a — (3) tan .} (y3 — y) tan^ (y — a). Ex. 10. The area of the triangle formed by thi-ee normals is c* |^tan^(a-^) tangos -y) tan^(y- o) {sin(/3 + y) -|-sin(y-l- o) + sin{a + fi)]^, consequently three normals meet in a point if sin (J3 + y) + sin (y + a) + sin {a + (i) = 0. [Mr. Bumside.] Ex. 11. To find the locus of the intersection of the focal radius rector IT with the radius of the circle CQ. Let the central co-ordinates of P be x'//', of 0, xy, then we have, from the similar triangles, FOX, FPM, y _ y . X + c x' + c 6 sin</) a {e + cos ip) Now, since (p is the angle made with the axis by the radius vector to the point 0, we at once obtain the polar equation of the locus by writing p cos (f> for x, p sin (fi for y, and we find p _ b c + p cos (p a {e + cos <^) ' be c + {a — b) cos <p Hence (Art. 193) the locus is an ellipse, of which C is one focus, and it can easily be proved that F is the other. Ex. 12. The normal at P is produced to meet CQ ; the locus of their intersection is a circle concentric ■with the ellipse. The equation of the normal is ax by _ , cos (f> sin (p ' but we may, as in the last example, write p cos <p and p sin (p for x and y, and the equation becomes / .\ „•> „„ „ , i ' [a — 0) p = C-, or p = a + 0. Ex. 13. Prove that tun iPFC = \(z — ^") tan i</>. EE 210 THE ECCENTRIC ANGLE. Ex. 14. If from the vertex of an ellipse a radius vector be drawn to any point on the cuiTe. find the locus of the point where a pai^llel radins through the centre meets the tangent at the point. The tangent of the angle made with the axis by the radius vector to the vertex = -j^ — ; therefore, the equation of the parallel radius through the centre is y y' _ ^ ^^ */' _ ^ 1 — cos <p _ x~ x' + a~ a{\ + cos<p) ~ a sin<p ' or V sin + - cos rf> = - , baa and the locus of the intersection of this line with the tangent r- sin A + - cos </> = !, a is, obviously, - = 1, the tangent at the other extremity of the axis. The same investigation will apply, if the first radius vector be drawn through any point of the curve, by substituting a' and b' for a and b ; the locus \vill then be the tangent at the diametrically opposite point. Ex. 15. The length of the chord of an ellipse which touches a confocal ellipse, 2hb'^ the squares of whose semiaxes are «- — h-, b- — h", is —r— [Mr. Bumside] . The condition that the chord joining two points a, (3 should touch the confocal conic, is " ~ ' - cos-^ {a + p)+ ~ '' ■ sin2^ (a + /3) = cos^i (a - /3), or siii2^(a-/3) = -^2 {i2cos-^(a + j9) + «-sm2J^(a +/3)}--^, 6'2. (Ex.4.) But the length of the chord is 2i'sini(a-,3)=.?M^. ab By the help of this Example several theorems concerning chords through a focus may be extended to chords touching confocal conies. 232. The methods of the preceding Articles do not apply to the hyperbola. For the hyperbola, however, we may substitute a;' = asec0, i/' = h tancj), since (- ) - l^j = 1. This angle may be represented geometrically by drawing a tangent MQ from the foot of ^-----..^ ^ ^ the ordinate M to the circle de- scribed on the transverse axis, then the angle QCM=^, since CM=CQsecQCM. SIMILAR CONIC SECTIONS. 211 We have also QM=atiin(f), but FM=b tanc^. Hence, if from the foot of any ordinate of a hyperbola we draw a tangent to the circle described on the transverse axis, this tangent is in a constant ratio to the ordinate. Ex. If any point on the conjugate hjT^jerbola be expressed similarly y" = b sec</>', a;" = a tan <!>', prove that the relation connecting the extremities of conjugate dia- meters is ^ = (j)'. [Mr. Tiu-ner.] SIMILAR CONIC SECTIONS. 233. Any two figures are said to be similar and similarly placed^ if radii vectores drawn to the first from a certain point are in a constant ratio to parallel radii drawn to the second from another point o. If it be possible to find any two such points and 0, we can find an infinity of others ; for, take any point C, draw oc parallel to OC, and in the constant Q^ ratio -Jypi tb^n from the similar triangles OCP, oc/>, cp is parallel to CP and in the given ratio. In like manner, any other radius vector through c can be proved to be proportional to the parallel radius through G. If two central conic sections be similar and similarly placed, all diameters of the one are proportional to the parallel diameters of the other, since the rectangles OP.OQ^ op.oq^ are propor- tional to the squares of the parallel diameters (Art. 149). 234. To find the condition that two conies, given by the general equations, should be similar and similarly placed. Transforming to the centre of the first as origin, we find (Art. 152) that the square of any semi-diameter of the first is equal to a constant divided by a cos'"* 6 -\- 2 A cos 6 sin ^ + J sin'* ^, and in like manner, that the square of a parallel semi-diameter of the second is equal to another constant divided by a! cos^d + 2h' cos 6 s'md + b' sm'0. The ratio of the two cannot be Independent of 6 unless a h b a h' b' ' Hence, two conic sections ivill be similar ^ and similarly placed^ if the cof'fficients of the highest poicers of the variables are the same in both^ or only differ by a constant midtiplier. 212 SI.MILAU CONIC SECTIONS. 235. It is evident that the directions of the axes of these conies must be the same, since the greatest and least diameters of one must be parallel to the greatest and least diameters of the other. If the diameter of one become infinite, so must also the parallel diameter of the other, that is to say, the asymptotes of similar and similarly placed hyperbolas are parallel. The same thing follows from the result of the last Article, since (Art. 154) the directions of the asymptotes are wholly determined by the highest terms of the equation. 2 7 2 Similar conies have the same eccentricity; for — ^ — must 2 2 27 2 be = 2~^ . Similar and similarly placed conic sections have hence sometimes been defined as those whose axes ai'e parallel, and which have the same eccentricity. If two hyperbolas have parallel asymptotes they are similar, for their axes must be parallel, since they bisect the angles be- tween the asymptotes (Art. 155), and the eccentricity wholly depends on the angle between the asymptotes (Art. 167). 236. Since the eccentricity of every parabola is =1, we should be led to infer that all parabolas are similar and similarly placed, the direction of whose axes is the same. In fact, the equation of one parabola, referi'ed to its vertex, being 3/^ =i>a;, or _p cos6 it is plain that a parallel radius vector through the vertex of the other will be to this radius in the constant ratio p : p?. Ex. 1. If on any radius vector to a conic section through a fixed pomt 0, OQ be taken in a constant ratio to OP, find the locus of Q. We have only to substitute mp for p in the polar equation, and the locus is found to be a conic similar to the given conic, and similarly placed. The point may be called the centre of similitude of the two conies; and it is obviously (see also Art. 115) the point where common tangents to the two conies intersect, since when the radii vectores OP, OP' to the first conic become equal, so must also OQ, OQ' the radii vectores to the other. Ex. 2. If a pair of radii be drawn through a centre of similitude of two similar conies, the chords joining their extremities wUl be either parallel, or wiU meet on the chord of intersection of the conies. This is proved precisely as in Art. 116. Ex. 3. Given three conies, similar and similarly placed, their six centres of simili- tude will lie three by three on right lines (see figure, page 108), SIMILAR CONIC SECTIONS. 213 Ex. 4. If any line cut two similar and concentric conies, its parts intercepted between the conies will be equal. Any chord of the outer conic which touches the ulterior wUl be bisected at the point of contact. These are proved in the same manner as the theorems at page 181, which are but particular cases of them ; for the asymptotes of any hyperbola may be considered as a conic section similar to it, since the highest terms in the equation of the asymp- totes are the same as in the equation of the curve. Ex. 5. If a tangent drawn at V, the vertex of the inner of two concentric and similar ellipses, meet the outer in the points T and T', then any chord of the inner dra\vn through V is half the algebraic sum of the parallel chords of the outer through T and T'. 237. Two figures will be similar, although not similarly placed, if the proportional radii make a constant angle with each other, instead of being parallel ; so that, if we could imagine one of the figures turned round through the given angle, they would be then both similar and similarly placed. To find the condition that two conic sections^ given hij the general equations^ should he similar, even though not similarly ■placed. We have only to transform the first equation to axes making any angle 6 with the given axes, and examine whether any value can be assigned to d which will make the new a, h, &, pro- portional to a', A', h'. Suppose that they become ma, mh', vih'. Now, the axes being supposed rectangular, we have seen (Art. 157) that the quantities a + b, ah-K\ are unaltered by transformation of co-ordinates ; hence we have a-\-h = m{a -\- h'), ah-Ji' = m''{ab'-h"), and the required condition is evidently ab — li^ db' — h"' If the axes be oblique it is seen In like manner (Art. 158) that the condition for similarity Is ab-K ' a'b' - K^ {a + b— 2h cos oif (a -\- b' — 2h' cos coY ' It will be seen (Arts. 74, 154) that the condition found ex- presses that the angle between the (real or imaginary) asymptotes of the one curve is equal to that between those of the other. 214 THE CONTACT OF CONIC SECTIONS. THE CONTACT OF CONIC SECTIONS. 238. Two curves of the m^ and n'" degrees respectively^ inter- sect in mn points. For, if we eliminate either x or y between the equations, the resulting equation in the remaining variable, will in general be of the inn^ degree [Higher Algebra^ p. 58 ; Todhunter's Theory of Equations^ p. 169). If it should happen that the resulting equation should appear to fall below the W2n"' degree, in consequence of the coefficients of one or more of the highest powers vanishing, the curves would still be considered to inter- sect in mn points, one or more of these points being at infinity (see Art. 135). If account be thus taken of infinitely distant as well as of imaginaiy points, it may be asserted that the two curves always intersect in mn points. In particular tioo conies always intersect in four 'points. In the next Chapter some of the cases will be noticed where points of intersection of two conies are infinitely distant ; at present we are about to consider the cases where two or more of them coincide. Since four points may be connected by six lines, viz. 12, 34 ; 13, 24 ; 14, 23 ; tico conies have three pairs of chords of intersection. 239. AVhen two of the points of intersection coincide, the conies touch each other, and the line joining the coincident points Is the common tangent. The conies will in this case meet in two real or imaginary points X, M distinct from the point of contact. This is called a contact oftJie first order. The contact is said to be of the second order when three of the points of intersection coincide, as for instance, if the point ilf move up until it coincide with T. Curves which have contact of an order higher than the first are also said to osculate ; and it appears that conies which osculate, must intersect in one other point. Contact of the third order is when two curves have four consecutive points common ; and since two conies cannot have more than four points common, this is the highest order of contact they can have. THE CONTACT OF CONIC SECTIONS. 215 Thus, for example, the equations of two conies, both passing through the origin, and having the line x for a common tangent are, (Art. 144) ax^ + 'ihxy + ly' + ^gx = ; ax^ + 2h'xy + h'y^ -f 2g'x = 0. And, as in Ex. 2, p. 170, x {{ah' - a'b) x + 2 {Jib' - h'h) y + 2 {gh'-c/h')] = 0, represents a figure passing through their four points of inter- section. The first factor represents tlie tangent which passes through the two coincident points of intersection, and the second factor denotes the line L3I passing through the other two points. If now ffb'=g'bj LM passes through the origin, and the conies have contact of the second order. If in addition hb' = h'b^ the equation of LM reduces to a; = ; iyJ/ coincides with the tangent, and the conies have contact of the third order. In this last case, if we make by multiplication, the coefficients oi y^ the same in both the equations, the coefficients of xy and x will also be the same, and the equations of the two conies may be reduced to the form ax^ + 'Ihxy + hy^ + 2gx = 0, a'x^ + 2hxy -f by"^ + 2gx = 0. 240. Two conies may have double contact^ if the points of intersection 1, 2 coincide and also the points 3, 4. The condition that the pair of conies, considered in the last article, should touch at a second point, is found by expressing the condition that the line LM^ whose equation is there given, should touch either conic. Or, more simply, as follows : Multiply the equa- tions by g and g respectively, and subtract, and we get {ag - a'g) x' -f 2 {hg - lig) xy + {bg - b'g) f = 0, which denotes the pair of lines joining the origin to the two points in which L3f meets the conies. And these lines will coincide if («^' _ ,,'^) (j^' _ j'^) ^ (7,^' _ j.'^y^ 241. Since a conic can be found to satisfy any five conditions (Art. 133), a conic can be found to touch a given conic at a given point, and satisfy any three other conditions. If it have contact of the second order at the given point, it can be made to satisfy two other conditions ; and if it have contact of the third order, it can be made to satisfy one other condition. Thus 216 THE CONTACT OF CONIC SECTIONS. wc can determine a parabola having contact of the third order at the origin with ax^ + ^hxij + hif + "igx = 0. Referring to the last two equations (Art. 239), we see that it is only necessary to write a instead of a, where a is deter- mined hy the equation ah = li\ We cannot, in general, describe a circle to have contact of the third order with a given conic, because two conditions must be fulfilled in order that an equation should represent a circle ; or, in other words, we cannot describe a circle through four consecutive points on a conic, since three points are sufficient to determine a circle. We can however easily find the equation of the circle passing through three consecutive points on the curve. This circle is called the osculating circle^ or the circle of curvature. The equation of the conic to oblique or rectangular axes, being, as before, ^^^ + ^hxy + hf + 2gx = 0, that of any circle touching it at the origin is (Art. 84, Ex. 3) x^ + 2xy cos (o + y^ — Irx sin o) = 0. Applying the condition gl>'=g'h (Art. 239), we see that the condition that the circle should osculate is g = — rh sin oj, or r= ./ % h sino) ' The quantity r is called the radius of curvature of the conic at the point T. 242. To find the radius of curvature at any point on a central conic. In order to apply the formula of the last Article, the tangent at the point must be made the axis of y. Now the ♦ In the Examples which follow we find the absolute magnitude of the radius of curvature, without regard to sign. The sign, as usual, indicates the direction in which the radius is measured. For it indicates whether the given curve is osculated by a circle whose equation is of the form X- + 2xy cos to + 2/- + 2rx sin oj = 0, the upper sign signif3-ing one whose centre is in the positive direction of the axis of X ; and the lower, one whose centre is in the negative direction. The formula in the text then gives a positive radius of curvature when the concavity of the curve is turned in the positive direction of the axis of x, and a negative radius when it is turned in the opposite direction. THE CONTACT OF CONIC SECTIONS. 217 equation referred to a diameter through the point and its con- jugate ( — a + 7T2 = 1 ) 5 is transferred to parallel axes through the given point, by substituting cc + a' for x^ and becomes -75 + 4 + -r = 0. a b a Therefore, by the last Article, the radius of curvature is ¥'■ Now a sin CO is the perpendicular from the centre on a sin CO the tangent ; therefore the radius of curvature = - , or (Art. XiS) = —: . ji «'' 243. Let ^ denote the length of the normal P.Y, and let -v/r denote the angle FPN between the normal ^.,^ — "^ ^ and focal radius vector; then the radius of curvature is — t^-t • For A^= — (Art. 181), cos^'tfr a ^ '^ and cos-v/r=— (Art. 188), whence the truth of the formula is manifest. Thus we have the following construction : Erect a perpen- dicular to the normal at the point where it meets the axis, and again at the point Q^ where this perpendicular meets the focal radius, draw CQ perpendicular to it, then G will be the centre of curvature, and CP the radius of curvature. 244. Another useful construction is founded on the principle that if a circle intersect a conic^ its chords of intersection will make equal angles icith the axis. For, the rectangles under the segments of the chords are equal (Euc. III. 35), and therefore the parallel diameters of the conic are equal (Art. 140), and, therefore, make equal angles with the axis (Art. 1G2). Now in the case of the circle of curvature, the tangent at T (see ligure, p. 214) is one chord of intersection, and the line TL the other; we have, therefore, only to draw TL^ making the same angle with the axis as the tangent, and wc have the point Z-; then the circle described through the points T^ X, ami, touching the conic at T, is the circle of curvature. FF 218 THE CONTACT OF CONIC SECTIONS. This construction shows that the osculating circle at either vertex has a contact of tlic third degree. Ex. 1. Using the notation of the eccentric angle, find the condition that four points a, fi, y, S should lie on the same circle (Joachimsthal, Crelle, xxxvi. 95). The chord joining two of them must make the same angle with one side of the mis as the chord joining the other twodoea with the other ; and the chords being ' C03^ (a + /3) + I sin4 (a + /3) = cos^ (a - /3) ; a - cos^(y + ^) +T sin|(y + o) = 003^(7- o) ; we have tan J- (a + /3) + tan J {y + 6)=zO; a + /3 + y + o = 0; or = 2mTr. Ex. 2. Find the co-ordinates of the point where the osculating circle meets the conic again. We have a = /3 = y ; hence = - 3a ; or X = -„ — 3j;' ; F = -|^ — 3y'. Ex. 3. There are three points on a conic whose osculating circles pass through a given point on the curve ; these lie on a circle passing through the point, and form a triangle of which the centre of the curve is the intersection of bisectors of sides (Steiner, Crelle, XXXII. 300 ; Joachimsthal, Crelle, xxxvi. 95). Here we are given c, the point where the circle meets the curve again, and from the last Example the pomt of contact is a = — ^6. But since the sine and cosine of 6 would not alter if were increased by 360°, we might also have a = — ^6 + 120°, or = — ^5 + 240°, and from Ex. 1, these three points lie on a circle passing through o. If in the last Example we suppose X, Y given, since the cubics which determine x' and y' want the second terms, the sums of the three values of x' and of y' are respectively equal to nothing ; and therefore (Ex. 4, p. 5) the origin is the intersection of the bisectors of sides of the triangle formed by the three points. It is easy to see that when the bisectors of sides of an inscribed triangle intersect in the centre, the normals at the vertices are the three perpendiculars of this triangle, and therefore meet in a point. 245. To find the radius of curvature of a parabola. The equation referred to any diameter and tangent being y^ = p'x. the radius of curvature (Art. 241) is ^ . - ^ ^ where d ^ ' 2 sin t' N is the angle between the axes. The expression — 5— , and the construction depending on it, hold for the parabola, since N=\p' sin^ (Arts. 212, 213) and i|r = 90°-^ (Art. 217). Ex. 1. In all the conic sections the radius of curvature is equal to the cube of the normal divided by the square of the semi-parameter. Ex. 2._ Express the radius of curvature of an ellipse in terms of the angle which the normal make« with the axi?. THE CONTACT OF CONIC SECTIONS. 219 Ex. 3. Find the lengths of the chords of the circle of curvature which pass through the centre or the focus of a central conic section. . Vi"^ . W- ° Ans. —r 1 and . a a Ex. 4. The focal chord of curvature of any conic is equal to the focal chord of the conic drawn parallel to the tangent at the point. Ex. 5. In the parabola the focal chord of curvature is equal to the parameter of the diameter passing through the point. 246. To find the co-ordinates of the centre of curvature of a central conic. These are evidently found by subtracting from the co-ordi- nates of the point on the conic the projections of the radius of curvature upon each axis, ^^ow It Is plain that this radius is to its projection on y as the normal to the ordinate y. We find the projection, therefore, of the radius of curvature on the axis of y f by multiplying the radius — by *-^^ j = —j~ . The y of the 72 _ 7 '2 2 centre of curvature then Is — jz — y. But h'^ = 1/ + — y"^^ there- 72 2 fore the y of the centre of curvature Is — —^ — y'^. In like 6* ^2 7 2 . . a —b ^ manner its x is — r — x a We should have got the same values by making a = /3 = 7 in Ex. 8, p. 209. Or again, the centre of the circle circumscribing a triangle Is the intersection of perpendiculars to the sides at their middle points ; and when the triangle Is formed by three consecutive points on a curve, two sides are consecutive tangents to the curve, and the perpendiculars to them are the corresponding normals, and the centre of curvature of any curve is the intersec- tion of two consecutive normals. Now if we make x =x" = X^ y =y" = Y^ In Ex. 4, p. 170, we obtain again the same values as those just determined. 247. To find the co-ordinates of the centre of curvature of a parabola. The projection of the radius on the axis of y Is found In like manner fby multiplying the radius of curvature ^^rh by ^J 220 THE CONTACT OF CONIC SECTIONS, and subtracting tliis quantity from ?/', wc have r=-- ■^l. = --'^r(Art. 212). tan'^ y ^ ' In like manner Its X is »'+ iz~-^n\ = ^ +^ — ^^ — = 3a;' + J ». 2 sin (7 2 "-^ The same values may be found from Ex. 10, p. 203. 248. The evolute of a curve is the locus of the centres of curvature of Its different points. If it were required to find the evolute of a central conic, we should solve for x'y' In terms of the X and y of the centre of curvature, and, substituting in the equa- / . . e^ c^ \ tlon of the cun-e, should have (writing — = ^, j = ^J , 3 3 X y •i 2 In like manner the equation of the evolute of a parabola is found ^0 be 21pf = \(!>[x-y)% which represents a curve called the semi-cubical imrahola^ ( 221 CHAPTER XIY. METHODS OF ABRIDGED NOTATION. 249. If ;S'=0, >S"=0, be the equations of two conies, then tlie equation of any conic passing through their four, real or imaginary, points of intersection, can be expressed in the form S = kS'. For the form of this equation shows (Art. 40), that it denotes a conic passing through the four points common to S and S' j and we can evidently determine Jc so that S=kS' shall be satisfied by the co-ordinates of any fifth point. It must then denote the conic determined by the five points.* This will of course still be true, if either or both the quan- tities S, S' be resolvable into factors. Thus ;S=/ia/3, being evidently satisfied by the co-ordinates of the points where the right lines a, /8, meet S, represents a conic passing through the four points where S is met by this pair of lines ; or, in other words, represents a conic having a and /3 for a pair of chords of intersection with S. If either a or /3 do not meet S in real points, it must still be considered as a chord of imaginary inter- section, and will preserve many important properties in relation to the two curves, as w^e have already seen in the case of the circle (Art. 106). So again, c/.y = Jt^S denotes a conic circum- scribing the quadrilateral a^yS, as we have already seen (Art. 1'22.'\ It is obvious that in what is here stated, a need not * Since five conditions determine a conic, it is evident that the most general equation of a conic satisfj-ing four conditions must contain one independent constant, whose vahie remains undetermined until a fifth condition is given. In like manner, the most general equation of a conic satisfying three conditions contains two in- dependent constants, and so on. Compare the equations of a conic passing through three points or toucliuig three hues (Arts. 124:, 129). If we are given any four conditions, in the expression of each of which the co- efficients enter only in the first degi'ee, the conic passes through four fixed points : for by ehminating all the coeificients but one, the equation of the conic is reduced to the form S = kS'. t If a/3 be one pair of chords joining four points on a conic i'!', and 70 another pair of chords, it is immaterial whether the general equation of a conic passing through the four points be expressed in any of the forms S — Za/3, S — kyc, afi — kyS, where A- is indeterminate ; because, in virtue of the general principle, S is itself of the form afi — kyo. 222 METHODS OF ABRIDGED NOTATION. be restricted, as at p. 53, to denote a line whose cqnation has been reduced to the form £c cosa + ?/ sina=^j» ; but that the argument holds if a denote a line expressed by the general equation. 250. There are three values of 7c, for which ;S^— kS' re- presents a pair of right lines. For the condition that this shall be the case, is found by substituting a — ha\ h — ]cb\ &c. for a, i, &c. In ale + 2fijh - af - hf - c//' = 0, and the result evidently Is of the third degree In Z;, and Is therefore satisfied by three values of h. If the roots of this cubic be h\ k", k"\ then S—k'S', /S— A;">S", S—k"'S\ denote the three pairs of chords joining the four points of Intei-sectlon of /Sand S' (Art. 238). Ex. 1. What is the equation of a conic passing through the points where a givea conic S meets the axes ? Here the axes x = 0, y = 0, are the chords of intersection, and the equation mnst be of the form S = kxy, where k is indeterminate. See Ex, 1, p. 148. Ex. 2. Form the equation of the conic passing through fire given points ; for example (1, 2), (3, 5), (— 1, 4), (— 3,-1), (— 4, 3). Forming the equations of the sides of the quadrilateral formed by the firet four points, we see that the equation of the required conic must be of the form {3x -2i/+ 1) (5x - 2y + 13) = k{x-4y h 17) (3x - i>/ + 5). Substituting in this, the co-ordinates of the fifth point (— 4, 3), we obtain k = — ^^. Substituting this value and reducing the equation, it becomes 79x2 _ 320a;2/ + 301/ + llOl.r - 1665y + 1586 = a^ 251. The conies S, S — 7caP will touch; or, In other words, two of their points of intersection will coincide ; if either a or yS touch 8, or again, If a and /3 intersect in a point on S. Thus If T= be the equation of the tangent to /S at a given point on It a;y, then S= T[lx + my + n)^ is the most general equation of a conic touching S at the point x'y' ; and If three additional con- ditions are given, we can complete the determination of the conic, by finding /, ?«, n. Three of the points of Intersection will coincide If lx-\-my + n pass through the point x'y' ; and the most general equation of a conic osculating >S at the point x'y\ is S=T[lx + my — Ix — my). If it be required to find the equation of the osculating circle^ we have only to express that the coefficient xy vanishes In this METHODS OF ABRIDGED NOTATION. 223 equation, and that the coefficient of x^ = that of y'3 when we have two equations which determine I and m. The conies will have four consecutive points common if lx^my-\- n coincide with 7", so that the equation of the second conic is of the form S=kT\ Compare Art. 239. Ex. 1. If the axes of S "be parallel to those of S', so ^-fll also the axes of S — kS'. For if the axes of co-ordinates be parallel to the axes of S, neither iS' nor S' will contain the term xy. If »S' be a circle, the axes oi S — kS' are parallel to the axes of S. Ji S — kS' represent a pair of right lines, its axes become the internal and external bisectors of the angles between them ; and we have the theorem of Art. 244. Ex. 2. If the axes of co-ordinates be parallel to the axes of <S^, and also to those oi S — kaji, then a and /3 are of the forms Ix + my + n, Ix — my + n'. Ex. 3. To find the equation of the circle osculating a central conic. The equation must be of the form Expressing that the coeflS.cient of xy vanishes, we reduce the equation to the form , fx- ■ifl , \ Ixx' mi' , \ fxx' yrf x"^ y'- , And expressing that the coeflBcient of x- — that of y", we find X = 7:^ ^ , and the equation becomes , „ 2 (n2 - 6=) x'^x 2 (6= - a"-) y'^y ,„ „,„ . x- + y- !^ ^ ^ ,,■ ^ ^ -I- a'- - W- - 0. Ex. 4. To find the equation of the circle osculating a parabola. Am. {p- + 4px') {1/- -px) = {2yy' - p [x + x'j] [2yy' + px - 3/>x'}. 252. We have seen that S = ha^ represents a conic passing through the four points P^Q] ^?, 5j where a, /3 meet S\ and it is evident that the closer to each other the lines a, yS are, the nearer the point P is to ^j>, and Q to q. Suppose then that the lines a and /S coincide, then the points P, p'^ Q., q coincide, and the second conic will touch the first at the points P, Q. Thus, then, the equation S=ka'' represents a conic having double contact loith S, a being the chord of contact. Even if a do not meet S^ it Is to be regarded as an imaginary chord of contact of the conies S and S—k'x'. In like manner a7 = Jc^'^ represents a conic to which a and 7 are tangents and /3 the chord of contact, as we have already seen (Art. 123). The equation of a conic having double contact with S at two given points x'g', x'y" may be also written in the 224 METHODS OF ABRIDGED NOTATION. form S=kTT\ where 2" and T' represent the tangents at these pouits. 253. If the line a be parallel to an asymptote of the conic S, it will also 1)0 parallel to an asymptote of any conic repre- sented by S= koL/3j which then denotes a system passing throngh three finite, and one infinitely distant point. In like manner, if in addition /3 were parallel to the other asymptote, the system wonld pass through two finite and two infinitely distant points. Other forms which denote conies having points of intersection at infinity, will be recognized by bearing in mind the prin- ciple (Art. 67) that the equation of an infinitely distant line is 0.x + 0.i/-{ C=0; and hence (Art. 69) that an equation, appa- rently not homogeneous, may be made homogeneous in form, if in any of the terms which seem to be below the proper degree of the equation we replace one or more of the constant multipliers by 0.x + 0.1/ + C. Thus, the equation of a conic referred to its asymptotes xy = U (Art. 199), is a particular case of the form a7 = /3'^ referred to two tangents and the chord of contact (Arts. 123, 252). Writing the equation xy = {() .x + .y + 1^)\ it is evident that the lines x and y arc tangents, Avhose points of contact arc at infinity (Art. 154). 254. Again, the equation of a parabola y'' =2^x is also a par- ticular case of ay=/3~. Writing the equation x[0.x+ 0.y+p)—y^] the form of the equation shows not only that the line x touches the curve, its point of contact being the point where x meets ?/, but also that the line at infinity touches the curve, its point of contact also being on the line y. The same inference may be drawn from the general equation of the parabola [ax + l3yY + {:2gx + 2fy + c){0.x + 0.y +1) =0, which shews that both 2gx + 2fy + c-, and the line at Infinity are tangents, and that the diameter ax + f3y joins the points of con- tact. Thus, then, every iKwahola has one tanyent aJtoyether at an infinite distance. In fact, the equation which determines the direction of the points at infinity on a parabola is a perfect square (Art. 137) ; the two points of the curve at infinity therefore coincide ; and therefore the line at infinity is to be regarded as a tangent (Art. 83). METHODS OF ABRIDGED NOTATION. 225 Ex. The general equation ax^ + 2hx)j + by- + 2gx + 2/// + c = 0, may be regarded as a particular case of the form (Art. 122) ay - kftS. For the first three terms denote two lines a, y passing through the origin, and the last three terms denote the line at infinity /3, together with the line o, 2gx + 2ft/ + c. The foinn of the equation then shows that the lines a, y meet the curve at infinity, and also that 6 re- presents the line joming the finite points in which ay meet the curve, 255. In accordance with Art. 253, tbe equation S=k^ is to be regarded as a particular case of S = a/3, and denotes a system of conies passing through the two finite points where /9 meets 8^ and also through the two infinitely distant points where S is met by 0.x-\- O.y + Jc. Now it is plain that the coefiicients of x\ of xi/^ and of y'\ are the same in S and in S—k^^ and there- fore (Art. 234) that these equations denote conies similar and similarly placed. We learn therefore that tivo conies similar and similarly jplaced raeet each other in hvo infinitely distant points^ and consequently only in tioo finite -points. This is also geometrically evident when the curves are hyperbolas : for the asymptotes of similar conies are parallel (Art. 235), that is, they intersect at in- finity ; but each asymptote intersects its own curve at infinity ; consequently the infinitely distant point of intersec- tion of the two parallel asymptotes is also a point common to the two curves. Thus, on the figure, the infinitely distant points of meeting of the lines OX^ Ox, and of the lines OY, Oy, are common to the curves. One of their finite points of intersection is shown on the figure, the other is on the opposite branches of the hyperbolas. If the curves be ellipses, the only difference is that the asymptotes are imaginary instead of being real. The directions of the points at infinity, on two similar ellipses, are determined from the same equation {ax^ -\- 2hxy -^ oy' = 0) (Arts. 136,234). Now although the roots of this equation are imaginary, yet they are, in both cases, the same imaginary roots, and therefore the curves are to be considered as having two imaginary points at infinity common. In fact, it was observed before, that even when the line a does not meet S in real points, it is to be re- el Ct 226 ]\[ETHODS OF ABRIDGED NOTATION. garded as a chord of imaginary intersection of ;S^ and S - ^ayS, and this remains true when the line a is infinitely distant. If the curves be parabolas, they are both touched by the line at infinity (Art. 254) : but the direction of the point of contact, depending only on the first three terms of the equation, is the same for both. Hence, tioo similar and similarly jjlaced para- bolas touch each other at infinity. In short the two infinitely distant points common to two similar conies, are real, imaginary, or coincident, according as the curves are hyperbolas, ellipses, or parabolas. 256. The equation >9=A:, or >S'= Z;(0. a; + 0.?/+ I)"'' is mani- festly a particular case of S=hd\ and therefore (Art. 252) de- notes a conic having double contact with S^ the chord of contact being at infinity. Now 8—h differs from 8 only in the constant term. Not only then are the conies similar and similarly placed, the first three terms being the same, but they are also con- centric. For the co-ordinates of the centre (Art. 140) do not involve c, and therefore two conies whose equations differ only in the last term are concentric (see also Art. 81). Hence, tivo similar and concentric conies are to he regarded as touching each other at tioo infinitely distant points. In fact, the asymptotes of two such conies are not only parallel but coincident ; they have therefore not only two points at infinity common, but also the tangents at those points ; that is to say, the curves touch. If the curves be parabolas, then, since the line at infinity touches both curves, 8 and S—Jc^ have with each other, by Art. 251, a contact at infinity of the third order. Two para- bolas whose equations differ only in the constant term will be equal to each other; for the curves y^=px^ y^=p[x-\-n) are obviously equal, and the equations transformed to any new axes will continue to differ only in the constant term. We have seen, too, (Art. 205) that the expression for the parameter of a para- bola does not involve the absolute term. The parabolas then, 8 and 8—Ic^, are equal, and we learn that tioo equal and similarly placed p>tt'>'o^olas whose axes are coincident may he considered as having with each other a contact of the third order at infinity. 257. All circles are similar curves, the terms of the second degree being the same in all. It follows then, from the last METHODS OF ABRIDGED NOTATION. 227 Articles, that all circles j^ass through the same tioo imaginary points at infinity^ and on that account can never intersect in more than two finite points, and that concentric circles touch each other in tico imaginary jioints at infinity; and on that account can never intersect in any finite point. It will appear hereafter that a multitude of theorems concerning circles are but parti- cular cases of theorems concerning conies which pass through two fixed points. 258. It is important to notice the form Vd^ -\-m^^^ = i^'f^ which denotes a conic with respect to which a, yS, 7 are the sides of a self-conjugate triangle (x\rt. 99). For the equation may be written in any of the forms The first form shews that W7 + «2/5, ^7 — m^ (which intersect in /37) are tangents, and a their chord of contact. Consequently the point ^'^ is the pole of a. Similarly from the second form 7a is the pole of /3. It follows then, that a.B is the pole of 7 ; and this also appears from the third form which shows that the two imaginary lines Za + m^ v (— 1) are tangents whose chord of contact is 7. Now these imaginary lines intersect in the real point a/3 which is therefore the pole of 7 ; although being within the conic, the tangents through it are imaginary. It appears, in like manner, that ad' + 2^ayS + W = c^" denotes a conic, such that a/3 is the pole of 7 ; for the left-hand side can be resolved into the product of factors representing lines which intersect in ajS. 279*. If a? = 0, ?/ = be any lines at right angles to each other through a focus, and 7 the con-esponding directrix, the equation of the curve is a particular form of the equation of Art. 258. Its form shows that the focus [xy^ is the pole of the directrix 7, and that the polar of any point on the directrix is perpendicular to the line . I * In changing the place of this Article I retain the numbering of the last edition for the reason given p. 113. 228 METHODS OF ABRIDGED NOTATION. joining It to the focus (Art. 192) ; for ?/, the poUir of [xy) Is perpendicular to a*, but x may be any line drawn through the focus. The form of tlie equation shows that the tAVO imaginary lines x^ + ?/'^ are tangents drawn through the focus. Now, since these lines are the same whatever 7 be, it appears that all conies lokich have the same focus have tivo imaginary common tanqents jyassing through this focus. All conies, therefore, which have both foci common, have four imaginary common tangents, and may be considered as conies inscribed in the same quadrilateral. The imaginary tangents through the focus (,«'■' + y-* = 0) are the same as the lines drawn to the two Imaginary points at Infinity on any circle (see Art. 257). Hence we obtain the following general conception of foci : " Through each of the two imaginary points at infinity on any circle draw two tangents to the conic ; these tangents will form a quadrilateral, two of whose vertices will be real and the foci of the curve, the other two may be considered as imaginary foci of the curve." Ex. To find the foci of the conic given by the general equation. We have only to express the condition that x — x' + {y — y') J(— 1) should touch the curve. Substituting then in the formula of Art. 151, for \, fx, h respectively, 1, J(— 1), — [x' + y' J(— 1)} ; and equating separately the real and imaginary parts to cypher, we find that the foci are determined as the intersection of the two loci C [3? - /) + 2/> - 2(?a; + yl - I? = 0, Cxy- Fx-Gy + n= 0, which denote two equilateral hyperbolas concentric with the given conic. "Writiag the equations {Cx - GY - {Cy - Fy- =G^--AC- {F^ - BC) = A {a - b), (Cx -G){Cy-F) = FG-Cn= Ah ; the co-ordinates of the foci are immediately given by the equations (Cx - G)2 = iA (i? + « - J) ; {Cy-FY = ^A{R + b- a), where A has the same meaning as at p. 148, and i2 as at p. 153, If the cui-ve is a parabola, C = 0, and we have to solve two linear equations which give (F2+ G'')x=Fn+^{A-B) G; (F2+ G^)y= Gn+i{B-A) F. 259. We proceed to notice some inferences which follow on interpreting, by the help of Art. 34, the equations we have already used. Thus (see Arts. 122, 123) the equation ay = 7cfi'^ implies that the jproduct of the perpendiculars from any point of a conic on tioo fixed tangents is in a constant ratio to the square of the perpendicular on their chord of contact. The equation ay='k^h^ similarly interpreted, leads to the METHODS OF ABRIDGED NOTATION. 229 important theorem : The product of the perpendiculars let fall from any point of a conic on two opposite sides of an inscribed quadrilateral is in a constant ratio to the 'product of the perpen- dicidars let fall on the other tivo sides. From this property we at once infer, that the anharmonic ratio of a pencil^ whose sides pass throucjh four fixed points of a conic^ and whose vertex is any variable point of it^ is constant. For the perpendicular OA . OB. BmA OB OG.OD. sin COD - J 7= — a = AB CD &c. Now if we substitute these vakies in the equation a'y + A-^S, the con- tinued product OA.OB.OC.OD will appear on both sides of the equation, and may therefore be suppressed, and there will remain QmAOB. s'm COD s'mBOC.smAOD = k. AB.CD BC.AD' but the right-hand member of this equation Is constant, while the left-hand member is the anharmonic ratio of the pencil OAj OB, OC, OD. The consequences of this theorem are so numerous and im- portant, that we shall devote a section of another chapter to develope them more fully. 260. If >S= be the equation to a circle, then (Art. 90) S is the square of the tangent from any point xy to the circle ; hence S— ha^ = (the equation of a conic whose chords of intersection with the circle are a and /3) expresses that the locus of a pointy such that the square of the tangent from it to a fixed circle is in a constant ratio to the product of its distances from tico fixed hnes, ^s a conic passing through the four points in which the fixed lines intersect the circle. This theorem is equally true whatever be the magnitude of the circle, and whether the right lines meet the circle in real or imaginary points ; thus, for example, if the circle be infinitely small, the locus of a point, the square of whose distance from a fixed, point is in a constant ratio to the product of its distances from 230 METHODS OP ABRIDGED NOTATION. two fixed lines ^ is a conic section; and the fixed lines may be considered as chords of imaginary intersection of the conic with an infinitely small circle whose centre is the fixed point. 261. Similar inferences can be drawn from the equation >S— Z.a^ = 0, where 8 \s o, circle. We learn that the locus of a pointy such that the tangent from it to a fixed circle is in a constant ratio to its distance from a fixed line, is a conic touching the circle at the two points where the fixed line meets it ; or, conversely, that if a circle have douhle contact with a conic^ the tangent drawn to the circle from any point on the conic is in a constant ratio to the Ijerpendicular from the])oint on the chord of cmitact. In the particular case where the circle is infinitely small, we obtain the fundamental property of the focus and directrix, and we infer that the focus of any conic may he considered as an in- finitely small circle^ touching the conic in two imaginary poi/ats situated on the directrix. 262. In general, if in the equation of any conic the co-ordi- nates of any 2)oint he suhstituted^ the result will be proportional to the rectangle under the segments of a chord drawn through the point piarallel to a given line* For (Art. 148) this rectangle a cos"' 6 + 2 A cos 6 ^md-\-h sin'' 6 ' where, by Art. 134, c is the result of substituting in the equa- tion the co-ordinates of the point ; if, therefore, the angle d be constant, this rectangle will be proportional to c. Ex. 1. If two conies have double contact, the square of the peipendicular from any point of one upon the chord of contact, is in a constant ratio to the rectangle under the segments of that perpendicular made by the other. Ex. 2. If a line parallel to a given one meets two conies in the points P, Q, p, q, and we take on it a point 0, such that the rectangle OP.OQ may be to Op.Oq in a constant ratio, the locus of is a conic through the points of intersection of the given conies. Ex. 3. The diameter of the circle circumscribing the triangle formed by two b'b" tangents to a central conic and their chord of contact is ; where b', b" are the P semi-diameters parallel to the tangents, and p is the perpend iciUar fi-om the centre on the chord of contact. [Mr. Bumside] . * This is equally true for curves of any degi-ee. METHODS OF ABKIDGED NOTATION. 231 It will be convenient to suppose the equation divided by such a constant, that the result of substituting the co-ordiuates of the centre shall be unity. Let t', t" be the lengths of the tangents, and let S' be the result of substituting the co-ordinates of their intersection ; then r- ib'^-i-.S': 1, <"2 : 6"2 : : ,S' : 1. But also if t3 be the perpendicular on the chord of contact from the vertex of the triangle, it is easy to see, attending to the remark. Note, p. 149, IS -.p i-.S' -.1. _ t't" h'h" Hence — = . vs p But the left-hand side of this equation, by Elementary Geometiy, represents the diameter of the cu-cle circumscribing the triangle. Ex. 4. The expression (Art. 242) for the radius of the osculating circle may be deduced from the last example by supposing the two tangents to coincide ; or also from the following theorem due to Mr. Roberts : If «, «' be the lengths of two in- tersecting normals ; ^j, 2^' the corresponding central perpendiculars on tangents ; b' the semi-diameter parallel to the chord joining the two points on the curve, then 7i/> + n'2}' — 2b'-. For if ;S' be the result of substituting in the equation the co-ordi- nates of the middle point of the chord, ra, zs' the perpendiculars from that point on the tangents, and 2/3 the length of the chord, then it can be proved, as in the last example, that (i- = b'-S', m—ijS', vs'=p'S', and it is very easy to see that nv! + n''C5' — 2/3^, 263. If two conies have each doiible contact with a third^ their chords of contact with the third conic^ and a pair of their chords of intersection with each other.^ loill cdl pass through the same pointy and will form a harmonic pencil. Let the equation of the third conic be >S'=0, and those of the other two conies, Now, on subtracting these equations, we find U - M"^ = 0, which represents a pair of chords of intersection [L±M=Q) passing through the intersection of the chords of contact (L and J/), and forming a harmonic pencil with them (Art. 57). Ex. 1. The chords of contact of two conies vri th their common tangents pass through the intersection of a pair of their common chords. This is a paiticulai' case of the preceding, S being supposed to reduce to two right lines. Ex. 2. The diagonals of any inscribed, and of the corresponding circumscribed quadrilateral, pass through the same point, and form a harmonic pencil. This is also a particular case of the preceding, S being any conic, and <S -f Z^, S + 31- being supposed to reduce to right lines. The proof may also be stated thus : Let «„ U, Cj ; tj, f^, e, be two pairs of tangents and the corresponding chords of contact. In other words, Ci, Cj are diagonals of the coiTCsponding inscribed quadi'Uateral. Then the equation of S may be wiitten in either of the forms t/, - Ci- = 0, t/^ - c,- - 0. 232 METHODS OF ABRIDGED NOTATION. The second equation must therefore be identical with the first, or can only differ from it by a constant multiplier. Hence tit^ — ^t^U must be identical with c,^ — \co^. Now c,^ — Xco- = represents a pair of right lines passing thi-ough the intersection of c,, Cj, and harmonically conjugate with them ; and the equivalent form shows that these right lines join the points ^,^3, tj^ and tlt^, t^^. For tit, — Xt^ti = must denote a locus passing through these points. Ex. 3. If 2a, 2/3, 2y, 2o be the eccentric angles of foiu- points on a central conic, form the equation of the diagonals of the quadrilateral formed by their tangents. Here we have «, =-cos2a + y-sin2a - 1, «, = " cos2i3 + | sin2/3 - 1, a a b c, = -COs(a + /3) +|sin(a + ^) -C0s(a-/3), and we easily veiify hh -Cy' = - Sin2(a - ^) {J + I* - 1} . Hence reasoning, as in the last example, we find for the equations of the diagonals r. n. -= + sin (a — /8) ~ sin (y — b) 264. If three conies have each double contact with a fourth^ six of their chords of intersection will pass three hy three through the same points^ thus forming the sides and diagonals of a quadrilateral. Let the conies be By the last Article the chords will be L-3f=0, 3f-N=0, N-L = 0', L + M=0, 3I-N = 0, N+L = 0; X-i)/=0, il/+^=0, N+L = 0. As in the last Article, we may deduce hence many particular theorems, by supposing one or more of the conies to break up into right lines. Thus, for example, if S break up into right lines, it represents two common tangents to S+M'\ S + N'''] and if Z denote any right line through the intersection of those common tangents, then S+L^ also breaks up into right lines, and represents any two right lines passing through the intersec- tion of the common tangents. Hence, if through the intersection of the common tangents of two conies v^e draw any pair of right lineSj the chords of each conic joining the extremities of those lines will meet on one of the common chords of the conies. This is the METHODS OF ABRIDGED NOTATION. 238 extension of Art. IIG. Or, again, tangents at the extremities of either of these right lines will meet on one of the common chords. 265. If 8+L\ 8+M\ S-vN% all break up into pairs of right lines, tliey will form a hexagon circumscribiug >S', the chords of intersection will be diagonals of that hexagon, and we get Brianchon's theorem : " The three opposite diagonals of every hexagon circumscrihing a conic intersect in a pointy By the opposite diagonals we mean (if the sides of the hexagon be numbered 1, 2, 3, 4, 5, 6) the lines joining (1,2) to (4, 5), (2, 3) to (5, 6), and (3, 4) to (6, 1) ; and by changing the order in which we take the sides, we may consider the same lines as forming a number (sixty) of different hexagons, for each of which the present theorem is true. The proof may also be stated as in Ex. 2, Art. 263. If ^X^4-C>0, <A-^.>0, y«-c/=:0, be equivalent forms of the equation of S^ then c^ = c^ = c^ re- presents three intersecting diagonals.* 266. If three conic sections have one chord common to all^ their three other common chords will p)ass through the same point. Let the equation of one be >S'=0, and of the common chord X = 0, then the equations of the other two are of the form which must have, for their intersection with each other, L[M-N) = 0', but 31— N is a line passing through the point [MN). According to the remark in Art. 257, this is only an extension of the theorem (Art. 108), that the radical axes of three circles meet in a point. For three circles have one chord (the line at infinity) common to all, and the radical axes are their other common chords. * Mr. Todliimter has -ndth justice objected to tliis proof, that since no rule is priven which of the diagonals of ^1*4^5 is i\ = + c^, all that is in strictness proved is that the lines joining (1, 2) to (4, 5) and (2, 3) to (5, f>) intersect either on the line joining (3, 4) to (6, 1), or on that joining (1, 3) to (4, 6). But if the latter were the case the triangles 123, 456 would be homologous (see Ex. 3, p. 59), and therefore the inter- sections 14, 25, 36 on a right hue; and if we suppose five of these tangents fixed, the pixth instead of touching a conic would pass tlirongh a fixed point. 1111 234 METHODS OF ABRIDGED NOTATION. The theorem of Art. 2G4 Diay be considered as a still further extension of the same theorem, and three conies which have each double contact with a fourth may be considered as having four radical centres, through each of which pass three of their common chords. The theorem of this Article may, as in Art. 108, be other- wise enunciated : Given four jj)oints on a conic section^ its cliord oj intersection 7vith a fixed conic passing through two of these j^oints ioillj)ass through a fixed point. Ex. 1, If through one of the points of intersection of two conies we draw any lin meeting the conies in the points P, p., and through any other point of intersection B a Ime meeting the conies in the points Q. y, then the lines PQ., pq, will meet on CZ*, the other chord of intersection. This is got by supposing one of the conies to reduce to the pair of lines OA, OB. Ex. 2. If two right lines, drawn through the point of contact of two conies, meet the cvirves in points P, p, Q, q, then the chords PQ, 2>^h '^^'i^l meet oia the chord of inter- section of the conies. This is also a particular case of a theorem given in Art. 264, since one intersection of common tangents to two conies which touch, reduces to the point of contact (Cor., Art. 117). 267. The equation of a conic circumscribing a quadrilateral (ciy = 7cl3B) furnishes us with a proof of " Pascal's theorem," tha the three intersections of the opposite sides of any hexagon inscrihei in a conic section are in one right line. Let the vertices be abcdef and let ah = denote the equation of the line joining the points a, h ; then, since the conic circum- scribes the quadrilateral ahcd^ its equation must be capable of being put into the form • ah . cd — he . ad = 0. But since it also circumscribes the quadrilateral defa^ the same equation must be capable of being expressed in the form de .fa — ef. ad = 0. From the identity of these expressions, we have ah.cd— de .fa = {he — ef) ad. Hence we learn that the left-hand side of this equation (which from its form represents a figure circumscribing the quadrilateral formed by the lines a&, f?e, cc?, af) is resolvable into two factort-, which must therefore represent the diagonals of that quadri- lateral. But ad is evidently the diagonal which joins the vertices METHODS OF ABETDGED Nt^TATION. 285 a and fZ, tlierefore Ic — ef must be the other, and must join the points [nh^ de), {cd^ af) ; and since from its form it denotes a line through the point (ic, e/), it follows that these three points are in one right line. 268. We may, as in the case of Brianchon's theorem, obtain a number of different theorems concerning the same six points, according to the different orders in which we take them. Thus since the conic circumscribes the quadrilateral hcef^ its equation can be expressed in the form he . cf— he . cf= 0. Now, from identifying this with the first form given in the last Article, we have ^j , ^c? - he . cf= {ad - ef) he ; whence, as before, we learn that the three points (a&, c/"), (cJ, Je), («fZ, ef) lie in one right line, viz. ad— ef=0. In like manner, from identifying the second and third forms of the equation of the conic, we learn that the three points (f/e, c/"), (/a, he), {ad, he) lie in one right line, viz. hc — ad = 0. But the three right lines he — ef= 0, ef— ad = 0, ad — Z-c = 0, meet in a point (Art. 41). Hence we have Steiner's theorem, that " the three Pascal's lines which are obtained by taking the vertices in the orders respectively, ahedef adcfeh, afehed, meet in a point." For some further developments on this subject we refer the reader to the note at the end of the volume. Ex. 1 . If a, h, c be tlu'ee points on a right line ; a', b', c' three points on another line, then the intersections {be', b'c), {ca', c'a), [ab', a'b) lie in a right line. This is a particular case of Pascal's theorem. It remains true if the second line be at infinity and the lines ba', ca' be parallel to a given line, and similaily for cb', ab' ; ac', be', Ex. 2. From foiu- lines can be made four triangles, by leaving out in turn one line. The four intersections of peiiiendiculars of these triangles lie in a right line. Let a, b, c, d be the right lines ; a', b', c', d' lines peii^endicular to them ; then the theorem follows by applj-ing the last example to the three points of intersection of a, b, c with d, and the thi-ee points at infinity on a', b', c'.* * Tliis proof was given me independently by Prof. De Morgan and by Mr. Buniside. The theorem itself follows at once from Steiner's theorem, Ex. 3, p. 201. For the four intersections of perpendiculars must lie on the du-ectrbc of the parabola, which has the four lines for tangents. It follows in the same way from Cor. 4, p. lOG, that the circles circumscribing the four triangles pass through the same point, viz. the focus of the same parabola. If we are given five lines M. Anguste Miqucl has proved (sec Catalan's Thcoremes et Problemes de Geometric Elcmentaire, p. 93) that the foci of the five parabolas which have four of the given lines for tangents lie on a circle. 23: MCTIIOL.S OF AliRIDGED NOTATIOxV. Ex. 3. Steiner's theorem, that the pei-peucHculars of the triangle formed by three tangents to a parabola intei-sect on the directrix is a particular case of Brianchon's theorem. For let the three tangents be n, b, c; let three tangents perpendicular to thera be «', b', c', and let the line at infinity, which is also a tangent, (Art. 254) be oo . Then consider the six tangents a, b, c, c', <x>, a' ; and the lines joinuig ab, c' « ; be, a' 00 ; cc', aaf meet in a point. The first two are perpendiculars of the triangle ; and the last is the directi-ix on which intersect every pair of rectangular tangents (Art. 221). Tills proof is by Mr. John C. Moore. Ex. 4. Given five tangents to a conic, to find the point of contact of any. Let ABODE be the pentagon formed by the tangents; then if ^C and BE intersect in 0, DO passes through the ix)int of contact of AB, This is derived from Brianchon's theorem by supposing two sides of the hexagon to be indefinitely near, smce any tangent is intersected by a consecutive tangent at its point of contact (p. 144). 269. Pascal's theorem enables us, given five points A^ B^ C, i>, Ej to construct a conic j for if we draw any line AP through D jp r one of the given points we can find the point F in which that line meets the conic again, and can so determine as many points on the conic as we please. For, by Pascal's theorem, the points of intersection [AB^ DE), [BC, EF), [CD, AF) are in one right line. But the points [AB, DE), {CD, AF) are by hypothesis known. If then we join these points 0, P, and join to E the point Q in which OP meets BC, the intersection of QE with AP determines F. In other words, F is the vertex of a triangle FPQ ivhose sides pass through the fixed points A, E, 0, and whose base angles P, Q move along the fixed lines CD, CB (see Ex. 3, p. 42). The theorem was stated in this form by MacLaurin. Ex. 1. Given five points on a conic, to find its centre. Draw AP parallel to BC and determine the point F. Then AF and BC are two parallel chords and the line joining their middle jwints is a diameter. In like manner, by drawing QE parallel to CD we can find another diameter, and thus the centre. Ex. 2. Given five points on a conic, to draw the tangent at any one of them. The point F must then coincide with A, and the line QF drawn through E must therefore take the position qA. The tangent therefore must be jf^.l. METflODS OF ABRIDGED NOTATION. ^37 Ex. 3, Investigate by trilinear co-ordinates (Art. 62) Mac Laurin's method of generating conies. In other words, find the locus of the vertex of a triangle whose sides pass through fixed points and base angles move on fixed lines. Let <t, (3, y be the sides of the triangle formed by the fixed points, and let the fixed lines be la + mj3 + ny — 0, I'a + ?rt'/3 + n'y = 0. Let the base be a = /i^, Theii the line joining to /3y, the intersection of the base with the first fixed line, is (//i + ?») /3 + «y = 0. And the hne joining to ay, the intersection of the base with the second line, is {I'fi + m') a + n'fiy — 0. Eliminating /ix from the last two equations, the equation of the locus is found to be Im'a^ - {mp + ny) {I'a + n'y), a conic passing through the points /3y, ya, (a, lu + m(i + ny), (Ji, I'a + m'(i + n'y). EQUATION REFEllRED TO TWO TANGENTS AND THEIR CHORD. 270. It much facilitates computation (Art. 229) when the position of a point on a curve can be expressed by a single variable : and this we are able to do in the case of two of the principal forms of equations of conies already given. First, let Z-, M be any two tangents and R their chord of contact. Then the equation of the conic (Art. 252) is L3I= E' ; and if /xL = R be the equation of the line joining LB to any point on the curve, (which we shall call the point /i), then substituting in the equation of the curve, we get M= [xR and /x"''X = M for the equations of the lines joining the same point to MR and to L2L Any two of these three equations therefore will determine a point on the conic. The equation of the chord joining two points on tlie curve /^j /*'j is ^fj,'L _ (^ 4- ^') i? + M= 0. For it is satisfied by either of the suppositions {ju,L = R, ixR = M\ {fM'L = R, fi'R^3I). If fi and /J,' coincide we get the equation of the tangent, viz. fi'^L - 2fiR 4- M= 0. Conversely, if the equation of a right line [/jb'L — 2fji,R + 2I=0) involve an indeterminate /j, in the second degree^ the line will always touch the conic LM= R\ 271. To find the equation of the polar of anij point. The co-ordluatcs L\ il/', E of the point substituted in the equation of either tangent through it, give the result Ix^L - 2fiR' + M' = 0. 238 METHODS OF ABRIDGED NOTATION. i\.r T* Now at the point of contact /^^ = ^ , and y^= y (-^^t. 270). 1j -Li Therefore the co-ordinates of the point of contact satisfy the equation ML' -2RR' + LM' ^0, which is tliat of the polar required. If the point had been given as the intersection of the lines aL = B, hli = 31, it is found by the same method that the equa- tion of the polar is ahL - 2aR + M= 0. 272. In applying these equations to examples it is useful to take notice that, if we eliminate R between the equations of tAvo tangents fi'L - 2iiR + M= 0, ix'^L - 2ijlR + M= 0, we get ixim'L = M for the equation of the line joining LM to the intersection of these tangents. Hence, if we are given the product of two /u,'s, /Lt/i' = a, the Intersection of the corresponding tangents lies on the fixed line aL = M. In the same case, sub- stituting a for jxix in the equation of the chord joining the points, we see that that chord passes through the fixed point [aL + J/, R). Again, since the equation of the line joining any point fx to LM is iJu'L = M, the points + /n, — fj, lie on a right line passing through ZJ/. Lastly, if LM= R\ LM= E'^ be the equations of two conies having L, M for common tangents ; then since the equation fi'L = M does not involve R or R\ the line joining the point + fj, on one conic to either of the points + /x on the other, passes through LM the intersection of common tangents. We shall say that the point + y". on the one conic corresjtonds directly to the point + yu. and inversely to the point — /* on the othei*. And we shall say that the chord joining any two points on one conic corresponds to the chord joining the corresponding points on the other. Ex. 1. CoiTesponding chords of two conies intersect on one of the chords of intei-section of the conies. The conies LM-Ii-, L.V - R'- have Jt- - li'- for a pair of common chords. But the chords /i/xL - in + fji') E + M= 0, nfx'L - (jjL + fi') E' + M= 0, evidently intersect on R - R'. And if we change the signs of n, /x in the second equation, they intersect on R + R', METHODS OF ABIIIDGED NOTATION. 239 Ex. 2. A triangle is circumscribed to a given conic ; two of its vertices move on fixed right lines : to find the locus of the third. Let us take for lines of reference- the two tangents through the intersection of the fixed Unes, and their choixl of contact. Let the equations of the fixed lines be aL~M-0, bL-M=0, while that of the conic is i J/ — E- = 0. Now we proved (Art. 272) that two tangents which meet on aL — M must have the product of their /x's = a ; hence, if one side of the triangle touch at the point jx, a b the othere will touch at the points — , — , and their equations will be MM %L-2-R + 2I=0, -L-2-R + M^Q. fx can easily be eliminated from the last two equations, and the locus of the vertex is found to be (rt + bf the equation of a conic havmg double contact ■\\'ith the given one along tlie line R.* Ex. 3. To find the envelope of the base of a triangle, inscribed in a conic, and whose two sides pass through fixed points. Take the line joining the fixed pomts for R, let the equation of the conic be LM = Rr, and those of the lines joining the fixed pomts to LM be al, + il/=0, bL + M=Q. Now, it was proved (Art. 272) that the extremities of any chord passing thi'ough {aL + M, R) must have the product of their fis = a. Hence, if the veilex be /u, the base angles must be — and — and the equation of the base must be MM abL - (rt + b)jjiR + fjT-Jf = 0. The base must, therefore (Art. 270), always touch the conic iab a conic having double contact with the given one along the line joining the given points, Ex. 4, To inscribe in a conic section a triangle whose sides pass thi-ough three given points. Two of the points being assumed as in the last Example, we saw that the equa- tion of the base must be abL - (rt + b) fxR + fj?M= 0, Now, if this line pass through the point cL — R = 0, cJR — M = 0, we must have ab — (rt + b) fxc + fJL-cd = 0, an equation sufficient to determine fx. Now, at the point /x we have fxL = R, fx-L = M ; hence the co-ordinates of this point must satisfy the equation abL - {a + b)cR + cdM = 0. * This reasoning holds even when the point LM is within the conic, and therefore the tangents L, M imaginary. But it may also be proved by the methods of the next section, that when the equation of the conic is L- + J/- = R-, that of the locus is of the fonn L- + M- = k-R-, 240 METHODS OF ABRIDGED NOTATION. The question, therefore, admits of two sohitions, for either of the points in which this line meets the curve may be taken for the vertex of the requu-ed triangle. The geo- metric construction of this Hne is given Ai-t, 297, Ex. 7. Ex. 5. The base of a triangle touches a given conic, its extremities move on two fixed tangents to the conic, and the other two sides of the triangle pass through fixed points : find the locus of the vertex. Let the fixed tangents be L, M, and the equation of the conic L3I — B}. Then the point of intersection of the line L with any tangent (yu-L — 1fj.R + J/) will have its co-ordinates L, R, M respectively proj^ortional to 0, 1, 2/i. And (by Art. 65) the equation ©f the line joining this point to any fixed point L'R'M' will be LM' - L'M= 2ix {LR' - L'R). Similarly, the equation of the line joining the fixed point L"R"M" to the point (2, fi, 0), which is the intersection of the line M with the same tangent, is 2 {RM" - R"M) - ,x {LM" - L"M). Ehminating fx, the locus of the vertex is found to be {LM' - L'M) {LM" - L"M) = 4 {LR' ~ L'R) {RM" - R"M), the equation of a conic through the two given points. 273. The chord joining the points fi tan0, fi cot(f> (where (f) is any constant angle) will always touch a conic having double contact with the given one. For (Art. 270) the equation of the chord is ^-^ jT _ ^^ ^^^^ ^ _^ ^^^ ^^ _^ j^^ Q^ which, since tan ^ + cot (^ = 2 cosec2^, is the equation of a tan- gent to LM sm^ 2^ = B^ at the point //, on that conic. It can be proved, In like manner, that the locus of the intersection of tan- gents at the points fi tanc/), /a cot^ is the conic LM= E^ sin'''20. Ex. If in Ex. 5, Art. 272, the extremities of the base lie on any conic having double contact with the given conic, and passing through the given points, find the locus of the vertex. Let the conies be LM- R^-0, LMsm^2(ji - R^ - 0, then, if any line touch the latter at the point /n, it will meet the former in the points ,u tan f/) and n cot </) ; and if the fixed points are /u', /x", the equations of the sides are /x/ji' tan <(>L — {fx' + fx tan <f>) R + 3f = 0, fx/x" cot <I>L - {/x" + fx cot(l)) R + M-0. Eliminating fx, the locus is found to be {M - fx'R) {n"L -R) = tan"-.t> {M - ,x"R) {fx'L - R). 274. GlveM four points of a conic^ the anharmonic ratio of the pencil joining them to any fiftli point is constant (Art. 259). The lines joining four points /*', yu,", i.(!"^ //,"" to any fifth point /i, are / {fxL-R)-]-{M-fiJR) = (}, fi" {fiL-R) + {M-firi) = 0, fx'" {fiL -R) + {3{- ixB) = 0, fi!'" [ixL - i?) -[- [M - fiB) = 0, METHODS OF ABRIDGED NOTATION:. 241 and their anharmonic ratio is (Art. 58) / f in\ I II iiii\ 5 (/*-/* ) (/^ - /^ ) and is, therefore, independent of the position of the point /i. We shall, for brevity, use the expression, " the anharmonic ratio of four points of a conic," when we mean the anharmonic ratio of a pencil joining those points to any fifth point on the curve. 275. Four fixed tangents cut any fifth in points loliose anhar- monic ratio is constant. Let the fixed tangents be those at the points yu.', /i", /i'", ^"", and the variable tangent that at the point ^ ; then the anhar- monic ratio in question is the same as that of the pencil joining the four points of intersection to the point LM. But (Art. 272) the equations of the joining lines are fj,'liiL-M=0, /u,>Z-ii/=0, iu,"'p,L-II=0, fi""fML-M=0, a system (Art. 59) liomograijliic with that found in the last Article, and whose anharmonic ratio is therefore the same. Thus, then, the anharmonic ratio of four tangents is the same as that of their points of contact. 276. The expression given (Art. 274) for the anharmonic ratio of four points on a conic, /a', /a", /i,'", /a"", remains unchanged if we alter the sign of each of these quantities ; hence (Art. 272) if ice draw four lines through any point LM^ the anharmonic ratio of four of the points (/*', /i", /i'", //."") ivhere these lines meet the conicj is equal to the anharmonic ratio of the otlwr four ptoints (— yu,', —/a", — a*-": ~ H'"") '^^here these lines meet th-e conic. For the same reason, the anharmonic ratio of four points on one conic is equal to thai of the four corresponding points on another ; since corresponding points have the same /i. (Art. 272). Again, the expression (Art. 274) remains unaltered, if w^e multiply each fjb either by tanc^ or cot^: hence we obtain a theorem of Mr. Townsend's, "TjT two conies have double contact^ the anharmonic ratio of four of the points in which any four tangents to the one meet the other is the same as that of the other four points in which the four tavgents meet the curve^ and also the same as that of the j-Qur points of contact. II 242 METHODS OF ABRIDGED NOTATION. 277. Conversely, given three fixed chords of a conic aa\ hh', cc ; a fourth chord dd', such that the anharmonic ratio of ahcd is equal to that of ah'c'd\ will always touch a certain conic having double contact with the given one. For let a, Z>, c, a', h\ c denote the values of /* for the six given fixed points, and /x, ft those for the extremity of the variable chord, then the equation (g -l)[c- fi) ^ {a - h') [c - fj.') (a — c){b — fi) {a — c) {b' - jm) ' when cleared of fractions, may, for brevity, be written Aixix + Bfji, + C/x' + D = 0, where A, Bj C^D are known constants. Solving for /i' from this equation and substituting in the equation of the chord it becomes /jL{B^ + I))L + Ii{fM{Afi+C)-{Bfi + D)]-3I{A,M+C) = 0j or fi'{BL-^AE)+fi{DL + {G-B)B-AM]-[DB + CM) = {), which (Art. 270) always touches {I)L+{C-B)B-AMY + A{BL + AB){CjM+I)R) = 0, an equation which may be written in the form 4 [BC-AD) [LM-B') + [DL + {B+ O) J? + ^ilf f = 0, showing that it has double contact with the given conic. In the particular case when B=Cj the relation connecting /i, >' becomes j^^' + B{fM + fj,') +B = 0, which (Art. 51) expresses that the chord fifju'L- {fjt, + p,') B + M passes through a fixed point. EQUATION REFERRED TO THE SIDES OF A SELF-CONJUGATE TRIANGLE. 278. The equation referred to the sides of a self-conjugate triangle T'a* -f m^fi'' = nV (Art. 258) also allows the position of any point to be expressed by a single indeterminate. For if we write la = n<y cos cj)^ OT;Q = ?i7 sln^, then, as at pp.94, 206, the chord joining any two points is la cos^ {(f) + <^') + m/3 sin | (^ + ^') = ny cos^ {(f) - ^'), and the tangent at any point Is la cos(f) -f m^ slxKf) = ny. METHODS OF ABRIDGED NOTATION. 243 If for symmetry we write the equation of the conic then It may be derived from the last equation, that the equation of the tangent at any point a'/3'7' is aarx + h^^' + C77' = 0, and the equation of the polar of any point a'/3'7' Is necessarily of the same form (Art. 89). Comparing the equation last written with Xa -I- fx^ + V7 = 0, we see that the co-ordinates of the pole of the last line are - , y- , - : and, since the pole of a c any tangent Is on the curve, the condition that Xa + fJ^^ + vy -\ 2 'i a may touch the conic Is — 1- S- -f — =0. When this condition a c is fulfilled the conic Is evidently touched by all the four lines Xa + ixl3 ± 1/7, and the lines of reference are the diagonals of the quadrilateral formed by these lines (see Ex. 3, p. 144). In like manner, If the condition be fulfilled aa!^ + h^''' + 07''* = 0, the conic passes through the four points a', + /S', + 7'. Ex. 1. Find the locus of the pole of a given line Xa + ju/3 + vy vcith regard to a conic which passes through four fixed points a', + /3', + y'. Xa'- u^'2 i/y'2 „ Ans. + ^ + -^ = 0. a p y Ex. 2. Eind the locus of the pole of a given Hue \a + fj.13 + vy, with regard to a conic which touches four fixed Unes la + viQ + ny. . Pa vi-li n^y Ans. -r- H -\ ^ = 0. K fi. V These examples also give the locus of centre ; since the centre is the jxjle of the line at infinity a smA + /3 sinB + y sin (7. Ex. 3. What is the equation of the circle having the triangle of reference for a self -con jugate triangle? Ans. (See Ex. 2, p. 122) a2sin2.4 + pT- &\n'2B + ■y2sin2C=0. It is easy to see that the centre of the circle is the intersection of pei-pendiculars of the triangle, the square of the radius being the rectangle under the segments of any of the perpendiculars, (taken with a positive sign when the triangle is obtuse angled, and with a negative sign when it is acute angled). In the latter case, there- fore, the circle is imaginary. 280.* The equation (Art. 279) a;'' + / = eV, (where the origin is a focus and 7 the corresponding directrix), is a parti- cular case of that just considered. The tangents through (7, x) to the curve are evidently e<y + x and «7 — x. If, therefore, the curve be a parabola, e = 1 ; and the tangents are the Internal * For Art. 279, see p. 227. 244 METHODS OF ABRIDGED NOTATION. and external bisectors of the angle (7^:;). Hence, " tangents to a parabola from any point on the directrix are at right angles to each other." In general^ since x = e'y cos^, y = 67 sin^, we have ^=tan<^; or expresses the angle which any radius vector makes with x. Hence we can find the envelope of a chord which subtends a constant angle at the focus, for the chord X cosi (0 + 0') 4- y sin| {(fi + (})'] = ey cosi (<^— 0'), if — 0' be constant, must, by the present section, always touch a conic having the same focus and directrix as the given one. 281. Tlie line joining the focus to the intersection of two tangents is found by subtracting X cos (f) + 7/ s,in(f> — ey = 0, X cos </)' + y sin (}}' — ey = 0, to be X sin | {(f) + (f>')—y cos |(<^ + (f>') — ^j the equation of a line making an angle ^ (_(f> + ^') with the axis of a:, and therefore bisecting the angle between the focal radii. The line joining to the focus the point where the chord of contact meets the directrix is X cos \ [^ -\- <\>') -^ y sin |(<^ + 4>') — 0, a line evidently at right angles to the last. To find the locus of the intersection of tangents at points which subtend a given angle 'ih at the focus. By an elimination precisely the same as that in Ex. 2, p. 94, the equation of the locus is found to be (x^ + y^) cos""* S = e'Y^ which represents a conic having the same focus and directrix as the given one, and whose eccentricity = ^ . •^ cosd If the curve be a parabola, the angle between the tangents is in this case given. For the tangent (ic cos ^ + ?/ sin (^ — 7) bisects the angle between x cos (p -\-y sin (p and 7. The angle between the tangents is, therefore, half the angle between x cos^-f ?/ sin (p and METHODS OF ABRIDGED NOTATION. 245 X cos(f)' + 7/ sin(jE)', or = h{(f> — (f>')- Hence, the angle letween two tan<jcnts to a parahola is half the. angle ivhich the j^oints of contact suhtend at the focus ; aud again, the locus of the intersection of tangents to a parahola^ ichich contain a given angle^ is a hyperhola icith the same focus and directrix^ and lohose eccentricity is the secant of the given angle., or whose asymptotes contain double the given angle (Art. 167). 282. Any two conies have a common self-conjugate triangle. For (see Ex. 1, p. 143) if the conies intersect in the points -4, B^ C, D, the triangle formed by the points E, Fj 0, in which each pair of common chords intersect, is self-conjugate with re- gard to either conic. And if the sides of this triangle be a, /3, 7, the equations of the conies can be expressed in the form aa^ + h/3' + cf = 0, a'a" + L'/3" + c'y' = 0. We shall afterwards discuss the analytical problem of reducing the equations of the conies to this form. If the conies intersect in four imaginary points, the lines a, /8, 7 are still real. For it is obvious that any equation with real coefficients which is satisfied by the co-ordinates x +x" \/[—l), 2/' + 2/" V(— l)? ""'iH also be satisfied by x' — x" \/(— 1)? V —y" V(— 1)? •'^ud that the line joining these points is real. Hence the four imaginary points common to two conies consist of two pairs x ±x" \J[— 1), 2/'±y V(-l); x" ±x"' '^{-l)^ J/"'±3/"" V(— I)- Two of the common chords are real and four imaginary. But the equa- tions of these imaginary chords are of the form L±M \J[—\)^ L' ±M' \/(- 1)5 intersecting in two real points XJ/, Z'J/'. Con- sequently the three points E^ F^ are all real. If the conies intersect in two real and two imaginary points, two of the common chords are real, viz. those joining the two real and two imaginary points; and the other four common chords are imaginary. And since each of the imaginary chords passes through one of the two real points, it can have no other real point on it. Therefore, in this case, one of the three points E^ F^ is real and the other two imaginary ; and one of the sides of the self-conjugate triangle is real and the other two imaginary. Ex. 1. Find the locus of vertex of a triangle wliose base angles move along one conic, and whose sides touch another. [The following solution is Mr. Burnside's.] 246 J[ETHODS OF ABRIDGED NOTATION. Let the conic touched by the sides be a-- + y- — z", and the other ax- + hy* — cz-. Then, as at Ex. 1, p. 94, the co-ordinates of the intereection of tangents at points a, y, are cos J (o + y), sin^ (a + y), cos^ (a — y) ; and the conditions of the problem give a cos-4 {a + y) + b sin-} {a + y) — c cos-} (a — y) ; or {a {-b — c) + {a — b — c) cos a cos y -I- {b — c — a) sin a sin y = 0. In like manner {a + b — c) + {a — b — c) cos /? cos y + {b — c — a) sin (i sin y = 0, whence {a + b — c) cos} {a + p) = {b + c — a) cos} (a — /3) cos y ; {a + b — c) sin} (a -f /3) = (a -)- c — S) cos} (a — /3) siny, and since the co-ordinates of the point whose locns we seek are cos}(a -)- /3), sin } (a -}- /3), cos}(a — /3), the equation of the locus is x^ y^ _ s* (jb + c- aY "*" {c + a- bf " {a + b- cf ' Ex. 2. A triangle is inscribed in the conic a;- -t- ?/^ = s- ; and two sides touch the conic ax'^ + by- = cz^ • find the envelope of the third side. Alls, (ca + ab — be)- x- + {ab + be — ca)- y"^ — {be + ca — aby z'^. ENVELOPES. 283. If the equation of a right line involve an indeterminate quantity in any clcgTce, and If we give to that indeterminate a series of different values, the equation represents a series of different lines, all of which touch a certain curve which is called the envelope of the system of lines. We shall illustrate the general method of finding the equation of an envelope, by proving, independently of Art. 270, that the line fi^L—2fiR-\-M, where /t is Indeterminate, always touches the curve LM—Ii'\ The point of Intersection oFthe lines answering to the values yu. and fM + A;, Is determined by the two equations /x'^L - 2fiR -h M=- 0, 2 {ixL -B)+kL = 0; the second equation being derived from the first by substituting fi -f /.; for /i, erasing the terms which vanish in virtue of the first equation, and then dividing by k. The smaller k is, the more nearly does the second line approach to coincidence with the first ; and if we make Jc = 0, we find that the point of meeting of the first line with a consecutive line of the system is de- termined by the equations /x'iy-2/i7? + J/=0, flL-E = 0', or, what comes to the same thing, by the equations fxL-E = 0^ fiR- M= 0. METHODS OF ABRIDGED NOTATION. 247 Now since any point on a curve may be considered as the inter- section of two of its consecutive tangents (p. 144), the point where any line meets its envelope is the same as that where it meets a consecutive tangent to the envelope; and therefore the two equations last written, determine the point on the envelope which has the line jji^L — 2/xi2 + M for its tangent. And by eliminating jj. between the equations we get the equa- tion of the locus of all the points on the envelope, namely LM=E\ A similar argument will prove, even if Z, J/, R do not re- present right lines, that the curve represented by /i^Z— 2/a^+J/, always touches the curve LM—E\ The envelope of L cos (^ + il/ sin (^ — i?, where ^ is indeter- minate, may be either investigated directly in like manner ; or may be reduced to the preceding by assuming tan 1 = /a, when on substituting 1 — At'' . 2 II cos(p = - — ^, s\n6= -—^. and clearing of fractions, we get an equation in which /x only enters in the second degree. 284. We might also proceed as follows : The line fi'^L - 2fxB + M is obviously a tangent to a curve of the second class (see note, p. 142) ; for only two lines of the system can be drawn through a given point : namely, those answering to the values of /u. de- termined by the equation fu:'L'-2/xR' + M' = 0, where L\ R\ M' are the results of substituting the co-ordinates of the given point in X, i?, M. Now these values of ya will evidently coincide, or the point will be the intersection of two consecutive tangents, if its co-ordinates satisfy the equation LM—R^. And, generally, if the indeterminate fj. enter alge- braically and in the n^ degree, into the equation of a line, the line will touch a curve of the n^ class, whose equation is found by expressing the condition that the equation in jm shall have equal roots. 248 METHODS OF ABRIDGED NOTATION. Ex. 1, The vertices of a triangle move along the three fixed lines a, /3, y, and two of the sides pass through two fixed points a'/i'y'j o."ii"y"i fi"^^ ^^^^ envelope of the third side. Let a + fift be the line joining to aji the vertex which moves along y, then the equations of the sides through the fixed points are y' (a + /u/3) - (a' + fi(i') y = 0, y" (a + fift) - {a" + ^ft") y = 0. And the equation of the base is (a' -f ^/3') y"a + {a" + ix(i") fiy'li - (a' + /x/3') (a" + ^/3") y = 0, for it can be easily verified, that this passes through the intersection of the first line with a, and of the second line with /3. Arranging according to the powers of fi, we find for the envelope (a/3'y" + jSy'a" — ya'/3" — ya"/3')2 = 4a'/3" (ay" — a"y) Q3y' — /3'y). Tliis example may also be solved by arranging according to the powers of a, the equation in Ex. 3, p. 49. Ex. 2. Find the envelope of a line such that the product of the perpendiculars on it from two fixed points may be constant. Take for axes the line joining the fixed points and a peipendicular through its middle point, so that the co-ordinates of the fixed points may be y = 0, x = + c ; then if the variable line be t/ — mx + 7i = 0, we have by the conditions of the question {n + mc) (n — vie) = J- (1 + 7ft-), or n- = b'^ + b-m- + c^m^, but nP' —y" — 2mxy -f- m^x^, therefore m- {x^ — b- — c-) — 2mxt/ + y- — b- = 0; and the envelope is x'^y- = {x- — b- — c") (j/" — b-), b- + 0- b Ex. 3. Find the envelope of a line such that the sum of the squares of the per- pendiculars on it from two fixed points may be constant. 2^;^ 2y'^ _ ^"^- F^W^ + T^" - ^• Ex. 4. Find the envelope if the difference of squares of pei-pendiculars be given. Ans, A parabola. Ex. 5. Tkrough a fixed point any line OP is drawn to meet a fixed line ; to find the envelope of PQ drawn so as to make the angle OPQ constant. g Let OP make the angle 6 with the perpendicular on the fixed line, and its length is p sec 6; but the perpendicular from ots PQ makes a fixed angle /3 with OP, therefore its length is =p sec 6 cos/i; and since this pei-pendicular makes an angle = t) -f- /3 with the perpendicular on the fixed Une, if we assume the latter for the axis of X the equation of PQ is X cos {Q + li) + y sin (6 -F /3) = ;j sec 6 cos/3, or X cos (26 -f- /3) + y sin(20 + fi)=2p cos/3 - x cos/3 - y sin/3, an equation of the form L cos <^ -f- J/ sin ^ = i?, whose envelope, therefore, is a:^ -f ^2 - (_j, C03/3 + y sin/3 - 2p cosjS)^, the equation of a parabola having the point for its focus. A B Ex. 6. Find the envelope of the line — i — , = 1, where the indetenninates are connected by the relation n + /x — C ^ ^ METHODS OF ABRIDGED NOTATION. 249 We may substitute for fx', C - /x, and clear of fractions ; the envelope is thus found to be ^2 + ^ + (72 _ 2AB -2AC- 2BC = 0, an equation to wliich the following form will be foimd to be equivalent, Thus, for example, — Given vertical angle and sum of sides of a triangle, to find the envelope of base. The equation of the base ia * y _ 1 a where a + b = c. The envelope is, therefore, x^ ■\-]p — 2xy — 2cx — 2cy + c- = 0, a parabola touching the sides x and y. In Hke manner, — Given in position two conjugate diameters of an ellipse, and the Slim of their squares, to find its envelope. If in the equation "^ + ^ = ^' we have a"^ + b"^ = c-, the envelope is X ±y ±c = 0, The ellipse, therefore, must always touch four fixed right Unes. 285. If the coefficients in the equation of any right line Xa + /iyS + vy he connected hy any relation of the second order in X, )i4, V, AX' 4 ^/i' + Cv' + 2F1JLV + 2 (?;/X + 2H\ix = 0, the envelope of the line is a conic section. Eliminating v between the equation of the right line and the given relations, we have {Aj^ -2Gya+ Co?) X' + 2 {Hrf - Fya - Gj/3 -f C(xJ3] Xfi and the envelope is {Ay^-2Gya+ Ca:'){By''-2Fy^+Cfi') = {Hy'-Fya- Gjl3+ Ca^)'. Expanding this equation, and dividing by 7''', we get {BC-F") a'+iCA- G') /3'+ {AB-H') y' + 2 {GH- AF) /37 + 2 {HF- BG) 7a + 2 [FG - CH) a^ = 0. The result of this article may be stated thus : Any tangential equation of the second order in X, yu., v represents a conic^ whose trilinear equation is found from the tangential hy exactly the same process that tlie tangential is found from the trilinear. For it is proved (as in Art. 151) that the condition that Xa + ^l^ + vy shall touch aoi' + h^"" 4 cy' 4 2fl3y 4 2.77a 4 2/^ a/3 = 0, 1< K 250 ]\IETIIODS OF ABRIDGED NOTATION. or, ill other words, the tangential equation of that conic, is {he -f) V + {ca - /) 11^ + [ah - 7r) v" + 2 {(jh - af) /iv + 2 [hf- hg) v\ + 2 [fy - ch) \/j, = 0. Conversely, the envelope of a line whose coefficients X, //., v fulfil the condition last written, is the conic aa' + &c. = ; and this may be verified by the equation of this article. For, if we write for A, B, &c., hc-f^, ca-ff', &c.j the equation [BC- F') a' + &c. = becomes {ahc+2fgh-af-hf-cK') {aa'+h^''+cy'+2f^y+2gya-\-2ha/3)=0. Ex, 1. "We may deduce as particular cases of the above, the results of Arts. 127, 130, namely, that the envelope of a line which fulfils the condition -r H 1 = is 4{Fa) + 4^G/3) + J(-^y) = ; and of one which fulfils the condition 4{F\) + ^{Gf.) + 4iHu) =0i3-^ + -^ + "=0. Ex. 2. What is the condition that Xa + n(3 + vy should meet the conic given by the general equation, in real points ? Ans. The line meets in real points when the quantity (5c -f^) X- + &c. is negative ; in imaginary points when this quantity is positive ; and touches when it vanishes. Ex. 3, What is the condition that the tangents drawn thi'ough a point a'/S'y' should be real? Ans. The tangents are real when the quantity {BC—F') a'^ + &c. is negative; or, in other words, when the quantities abc + 2fgh + &c. and aa'^ + 6/3'^ + &c. have opposite signs. The point \vill be inside the conic and the tangents imaginary when these quantities have like signs. 286. It is proved, as at Art. 76, that if the condition be fulfilled ABG+ 2FGH- AF' -BO- CH' = 0, then the equation AX-" + Bfj,' -1- Cv'' + 2Ffiv + 2 6^v\ + 2HXfi = 0, may be resolved into two factors, and is equivalent to one of the form (^^'x 4 13' fj, + ry'y) [a"\ + 13" fi + i'v) = 0. And since the equation is satisfied if either factor vanish, it denotes (Art. 51) that the line \ol + fi^ -\- vy passes through one or other of two fixed points. If, as in the last article, we write for A, hc-f^^ &c., it will be found that the quantity ABC+2FGH+&C. is the square of ahc -\- 2fgh + &c. METHODS OP ABRIDGED NOTATION. 251 Ex. If a conic pass through two given points and have double contact with a fixed conic, the chord of contact passes through one or other of two fixed points. For let S be the fixed conic, and let the equation of the other he S — {\a + fj-ft + vyY. Then substituting the co-ordinates of the two given points, we have S' = (Xa' + /x/3' + vy'f ; S" = (Xo" + ;u)3" + uy")^ ; whence (Xa' + ju/3' + vy') .1{S") = ± (Xa" + ju/3" + vy") 4(8'), showing that Xa + fxfi + vy passes through one or other of two fixed points, since S', S" are known constants. 287. To find the equation of a conic having double contact with two given conies, S and S'. Let E and F he a pair of their chords of intersection, so that S— 8' = EF'^ then represents a conic having double contact with S and S' ; for it may be written {fj,E+Fy = 4.f^S, or {fiE-Fy = 4.fiS'. Since fi is of the second degree, we see that through any point can be drawn tivo conies of this system ; and there are three such systems, since there are three pairs of chords E, F. If 8' break up into right lines, there are only two pairs of chords distinct from >S", and but two systems of touching conies. And when both 8 and 8' break up into right lines there is but one such system. Ex. Find the equation of a conic touching four given lines. Ans. fx^E^ - 2fi. {AC+ BD) + F« = 0, where A, B, C, D are the sides; E, F the diagonals, and AC — BD — EF. Or more symmetrically if L, M, N be the diagonals, L ± M ± N the sides, /x2Z2 _ ^ (1,2 + j»/2 - JV2) + 3/2 = 0. For this always touches (i^ + M^ - N^)'^ - 'WM^ = {1 + M+ N) (M+ iV- L) {L + N-M) (J/+ L - X). L- 31- Or again, the equation may be ^mtten N^ = , • + ^. , (see Art. 278). 288. The equation of a conic having double contact with two circles assumes a simpler form, viz. fM'-2fi[C-\-C') + {C-C'Y = 0. The chords of contact of the conic with the circles arc found to be C-G' + /M = 0, and C-C'-fM = 0, which are, therefore, parallel to each other, and equidistant from the radical axis of the circles. This equation may also be written in the form ^ C± x/ C = s/,u. 252 METHODS OF ABRIDGED NOTATION. Hence, the locus of a pointy the sum or difference of whose tangents to tioo given circles is constant^ is a conic having double contact loith the two circles. If we suppose both circles infinitely small, Ave obtain the fundamental property of the foci of the conic. If /i be taken equal to the square of the intercept between the circles on one of their common tangents, the equation de- notes a pair of common tangents to the circles. Ex. 1. Solve by this method the Examples (pp. 105, 106) of finding common tangents to circles. Ans. Ex. 1. JC + JC" = 4 or = 2. Ans. Ex. 2, JC + JC" = 1 or = J - 79. Ex. 2. Given three circles; let L, L' be the common tangents to C'C", M, M' to C", C; N, N' to C, C" ; then if L, M, N meet in a point, so will U, M', N'* Let the equations of the pairs of common tangents be Then the condition that L, M, N should meet in a point is t' ±t = t" ; and it is obvious that when this condition is fulfilled, L', 31', N' also meet in a point. Ex. 3. Three conies having double contact with a given one are met by three common chords, which do not pass all through the same point, in sis points which lie on a conic. Consequently, if three of these points lie in a right hne, so do the other three. Let the three conies be S — L^, S — 31^, S — N^ ; and the common diords L + M, 31 + N, N + L, then the ti-uth of the theorem appears fi-om inspee- tion of the equation S + MX + NL + L3I ={S- U) + (Z + 31) [L + X). GENERAL EQUATION OF THE SECOND DEGREE. 289. There is no conic whose equation may not be written in the form rta' + 5/3" + C7' + 2/^7 + 2gyoL + 2ha^ = 0. For this equation is obviously of the second degree ; and since it contains five independent constants, we can determine these constants so that the curve which It represents may pass through five given points, and therefore coincide with any given conic. * This principle is employed by Steiner in his solution of Malfatti's problem, viz. '• To inscribe in a triangle three circles which touch each other and each of which touches two sides of the triangle." Steiner's construction is "Inscribe circles in the triangles formed by each side of the given triangle and the two adjacent bisectors of angles : these circles having three common tangents meeting in a point wUl have three other common tangents meeting in a point, and these are common tangents to the circles required." For a geometrical proof of this by Dr. Hart, see Quarter!?/ Journal of 3Iathematksy Vol. I., p. 219. "We may extend the problem by substituting for the word "circles," "conies having double contact with a given one." In this extension, the theorem of Ex. 3, or its reciprocal, takes the place of Ex. 2. METHODS OF ABRIDGED NOTATION. 253 The trlllnear equation just written includes the ordinary Car- tesian equation, if we write x and y for a and /3, and if we suppose the line 7 at infinity, and therefore write 7 = 1, (see Art. 69 and note, p. 72). In like manner the equation of every curve of any degree may be expressed as a homogeneous function of a, y3, 7. For it can readily be proved that the number of tenns in the complete equation of the 71*" order between two variables is the same as the number of terms in the homogeneous equation of the n^ order between three variables. The two equations then, con- taining the same number of constants are equally capable of representing any particular curve. 290. Since the co-ordinates of any point on the line joining two points a'/3'7', a"/3"7" are (Art. 66) of the form la! + »2a", 7/3' + ?«/3", I'y' + iwy'^ we can find the points where this joining line meets any curve by substituting these values for a, /3, 7, and then determining the ratio I : m by means of the resulting equation.* Thus (see Art. 92) the points where the line meets a conic are determined by the quadratic r (aa" + h^"' + c^"' + 2//3'7' + 2^7'a' + 2Aa'/3') + 2lm [aaa + 5/3'^" -f 077" +/()S'7" + ^'V) +.^ (7'a" + 7"a') + 1i (a'/3" + a"/3')} + m' {aa'"' + h/S'"' + cy'"' -f 2f/3'Y + %ji'd' 4- 27^a"/3") = ; or, as we may write it, for brevity, V&' -\-'ilmF-k- vi^S" = Qt. ^Yhen the point a'yS'7' is on the curve, B' vanishes, and the quadratic reduces to a simple equation. Solving it for I : /?2, we see that the co-ordinates of the point where the conic is met again by the line joining a'^"<y" to a point on the conic a'/SV, are S"a - '2Pd\ S"^' - 2P/3", S"j' - 2Py". These co-ordinates reduce to a'/Sy if the condition P= be fulfilled. Writing this at full length, we see that if a'^"y" satisfy the equation aaa'+ 5yS/3' + C77' +f{fiy' ^ jS'y) +g{y'a -f 7a') + h (a'/S + a/3') = 0, then the line joining a!'^"y" to a^'y meets the curve in two points coincident with a'^'y : in other words, a"/3"y" lies on * This method was introduced by Joachimsthal. 254 METHODS OF ABRIDGED NOTATIO>T. the tangent at a'fi'y'. The equation just written is therefore the equation of the tangent. 291. Arguing, as at Art. 89, from the symmetry between a/37, o'/^V of the equation just found, we infer that when a.'13'j is not supposed to be on the curve, the equation represents the polar of that point. The same conchision may be drawn from observing, as at Art. 91, that P=0 expresses the condition that the line joining a'fi'j'j a"^"y" shall be cut harmonically by the curve. The equation of the polar may be written a (rta + A/3 + .77) + /3' {ha + &/3 4/7) + 7' {ga + f/3 + 07) = 0. But the quantities which multiply a', /S', 7' respectively, are half the differential coefficients of the equation of the conic with re- spect to a, /3, 7. We shall for shortness write /S,, /S'j,, 8^^ instead „ do do do , , , • /. 1 1 • ot -y- , — , y- ; and we see that the equation or the polar is In particular, if /S', 7' both vanish ; the polar of the point /3y is ^'j, or the equation of the polar of the intersection of two of the lines of reference is the differential coefficient of the equation of the conic considered as a function of the thii-d. The equation of the polar being unaltered by interchanging a^7, o!fi'y\ may also be written a>SV + /3/S; 4 7/^; = 0. 292. When a conic breaks up into two right lines, the polar of any point whatever passes through the intersection of the right lines. Geometrically it is evident that the locus of har- monic means of radii drawn through the point is the fourth harmonic to the pair of lines, and the line joining their inter- section to the given point. And we might also infer, from the formula of the last article, that the polar of any point with respect to the pair of lines a/8 is /3'a + a'/3, the harmonic con- jugate with respect to a, /8 of yS'a- a'/3, the line joining a^ to the given point. If then the general equation represent a pair of lines, the polars of the three points /37, 7a, a/3, aa + h^+jy = 0, hoL + h^+fy = 0, r/a+f/3 + cy = 0, are three lines meeting in a point. Expressing, as in Art. 38, the condition that this should be the case, by eliminating a, /3, 7 METHODS OF ABRIDGED NOTATION. 255 between these equations, we get the condition, already found by other methods, that the equation should represent right lines; which we now see may be written in the form of a determinant, a, h, g 9i fi (^ = ; or, expanded abc + '^fgh - af^ — hg"' — clt' = 0. The left-hand side of this equation is called the discriminant'^ of the equation of the conic. We shall denote it in what follows by the letter A. 293. To find the co-ordinates of the pole of any line \a + yLt/S + v<y. Let a'/3'7' be the sought co-ordinates, then we must have acL + h^' + g^' = X, lia + 5/3' +//' = /*, g'x 4//3' + cy = v. Solving these equations for a', /5', 7',*We get Aa' = X [he -f) ■\-ix.[fg- ch) + v{hf- hg), Afi' = \{fg-ch) + fi{ca- g')-]-v{gh-af), Ay' ^\{hf- hg) + fi{gh-af)-]- V {ab - F)', or, if we use A^ 5, 0,t &c. in the same sense as in Art. 151, we find the co-ordinates of the pole respectively proportional to AX + H/jL+Gv, Hx^Bfi + Fv, G\ -\- F/m -\- Cv. Since the pole of any tangent to a conic is a point on that tangent, we can get the condition that Xa + /u./3 + vy may touch the conic, by expressing the condition that the co-ordinates just found satisfy Xa + /a/3 + V7 = 0. We find thus, as in Art. 285, ^X' + Bfji' + Cv' + 2Ffiv + 2 Gv\ + 2E\fj, = 0. If we write this equation 2 = 0, it will be observed that the co-ordinates of the pole are 2j, 2^, 2,, that is to say, the diffe- rential coefficients of 2 with respect to X, /i, v. Just, then, as the equation of the polar of any point is aS^' + ^S,^ + yS^ = 0, so the condition that Xa + /iyS -f vy may pass through the pole of X'a + /Lt'/3 + v'7, (or, in other words, the tangential equation of * See Lessons on Modern Higher Algebra, Lesson XI. t A, B, C, &c, are the viinoj's of the detenninaut of the last article. 256 METHODS OF ABRIDGED NOTATION. this pole) Is \2/ + /aS,' + vSg' = 0. And again, the condition that two lines Xa + fi^ + vy, Va -f yu-'/S -f v'7 may be conjugate with respect to the conic ; that Is to say, may be such that the pole of either lies on the other, may obviously be written In either of the equivalent forms X2^ + /.'2, -f v'23 = 0, xs; -f /*s; + vs; = o. From the manner In which S was here formed. It appears that 2 Is the result of eliminating a', yS', 7', p between the equations aa + h^' + gi + p\ = 0, hd + h^' +//' + p/i = 0, goL+f^' + C7' -f pv = 0, Xa' + /^yS' + V7' = ; in other words, that S may be written as a determinant «, A, g, \ h h /) H- Ex. 1. To find the co-ordinates of the pole of \a + fjift + vy with respect to ^{la) + 4('»/3) + .liny). The tangential equation, in this case, (Art. 130), being Ifxv + invK + tikfi — 0, the co-ordinates of the pole are a' =:mv + fhfi, 13' = nX + Iv, y' = I/j. + mX, Ex. 2. To find the locus of the pole of \a + fifS + vy with respect to a conic being given three tangents, and one other condition.* Solving the preceding equations for I, m, n, we find /, m, n proportional to X (/i/3' + vy' — Xa'), fx {vy' + Xa — fx^'), v (Xa' + fifi' — vy'). Now ,](?«) + ■i{ni(i) -)- ^(wy) denotes a conic touching the three lines a, /3, y ; and any fourth condition estabhshes a relation between I, in, n, in which, if we substitute the values just found, we shall have the locus of the pole of Xa + /xfi + vy. If we write for X, fj., v the sides of the triangle of reference a, b, c, we shall have the locus of the pole of the line at infinity aa + bl3 + cy; that is, the locus of centre. Thus the condition that the conic should touch Aa + B^+Cy, being it+d + tv— ^j (Art. 130), we infer that the locus of the pole of Xa + /xji + vy with respect to a conic touching the four lines a, (3, y, Aa + B^ + Cy, is the right line X (ji^ -f vy — Xa) fx (vy + Xa — /u/S) v (Xa + /x^ — vy) _ A + B ^ C -"• Or again, since the condition that the conic should pass through a'(3'y' is 4{la') + J(ffi/3') + J(«y') = 0, the locus of the pole of Xa + /i/3 + vy with respect to a conic which touches the thi-ee lines a, fi, y, and passes through a i^oint a'^'y', is 4{Xa' {fxfi + vy- Xa)] + ^[ixji' {vy + Xa - /u/3)} + J{i/y' (Xa -|- (u/3 - i/y)} = 0, which denotes a conic touching /^/J + vy — Xa, vy + Xa — /xjS, Xa + /xfi — vy. In the * The method here u?ed i« taken from Heam's Researches on Conic Sections. JIETHODS OF ABRIDGED NOTATION. 257 case where the locus of centi-e is sought, these three Unes are the lines joining the middle pomts of the sides of the triangle fonued by a, (3, y. Ex. 3. To find the co-ordinates of the pole of Xa + fifi + vy with respect to Ijiy + viya + iiafi. The tangential equation in this case being, Art. 127, l-X- + m-fi- + n-v- — 2iiiii/jLv — 2nlv\ — 2foiX/x = 0, the co-ordinates of the pole are a' — I {T\ — vifi — nv), /J' = m {m/x - nv — //\), y' - n {nv — l\ - 7»/x\ whence my' + n(i' — — 2lmn\, na' -\- ly' — — 2hnnii, Ij3' + ma' = — 2hnnv ; and, as in the last example, we find I, m, n respectively proportional to a' (jj.fi' -t- vy' — \a'), /3' {vy* + \a! — jj.fi'), y' {Xa' + fj.fi! - vy). Thus, then since the condition that a conic circumscribing afiy should pass through a fourth. point a'fi'y' is — + o> "! — 7=0, the locus of the pole of \a + m/3 + vy, with regard to a conic passing through the four points, is -, {fJ-fi + vy- Xa) + §-, {vy + Xa- ixfi) + '^, (Xa + /nfi - vy) = 0, afiy which, when the locus of centre is sought, denotes a conic passing through the middle points of the sides of the triangle. The condition that the conic should' touch Aa + Bfi+ Cy, being 4{AI) + 4{Bm) + J(C») = 0, the locus of the pole of Xa + nfi + vy, with regard to a conic passing tlu-ough three points and toucliing a fixed line, is J{Aa (jjfi + vy- Xa)} + 4{Bfi {»y + Xa~ /xfi)} + JCy {Xa + fxfi - vy) = 0, which, in general, represents a curve of the fourth degree. 294. If a"/3"7" be any point on any of the tangents drawn to a curve from a fixed point a^''y\ the Tine johiuig a'/SV, <x'^"<^" meets the curve in two coincident points, and the equation in I : m (Art. 290), which determines the points where the joining line meets the curve, will have equal roots. To find, then, the equation of all the tangents which can be drawn through a'/3V? "^^'^ must substitute Ia + ma\ I/3+ml3\ ly + my' in the equation of the curve, and form the condition that the resulting equation in I : m shall have equal roots. Thus, (see Art. 92) the equation of the pair of tangents to a conic is SS' = F^', where S=aa^ + &c.^ /S" = aa'^ + &c., P=aaa4-&c. This equation may also be written ia another form ; for since any point on either tangent through a'^'y' evidently possesses the property that the line joining it to a'/3'y' touches the curve, we have only to express the condition that the line joining two points (Art. 65) a {/3'y" - I3"y') -f (3 {y'a" - V'a') + 7 («'/3" - a"/3') = LL 258 METHODS OF ABRIDGED NOTATION. should touch the curve, and then consider a'/S'V' variable, when we shall have the equation of the pair of tangents. In other words, we are to substitute /37' — /S'y, 7a' — 7'a, a/3' — a'yS for X, /x, V In the condition of Art. 285, AX' + B/j:' + Cv' + 2Ffiv + 2 GvX + 2H\ii = 0. Attending to the values given (Art. 285) for A^ B^ &c., it may easily be verified that [aoc + &c.) [aa' + &c.) - (aaa + &c.)' = A {/3y' - fi'yY + &c. Ex. To find the locus of mtersection of tangents which cut at right angles to a conic given by the general equation (see Ex. 4, p. 161). We see now that the equation of the pair of tangents through any point (Art. 147) may also be ■written ■^{y-yy + B {x- x'Y + C {xy' - yx')- -2F{x- x') {xy' - yx') + 2G{y-y') {xy' - x'y) -2E{x- x') {y - y") = 0. This -will represent two right lines at right angles when the sum of the coefficients of x" and y- vanishes, which gives for the equation of the locus C {x^ + /) - 2Gx - 2Fy + A + B = Q. This circle has been called the director circle of the conic. When the curve is a parabola, C = 0, and we see that the equation of the directrix is Gx + Fy = ^{A + B), 295. It follows, as a particular case of the last, that the pairs of tangents from /S7, 7a, ayS are By'-^C^-'-2F^r^, Cce-vArf-2Gria, A^'' + Ba' - 2Eal3, as indeed might be seen directly by throwing the equation of the curve into the form (aa + 7i/3_+ gy)' + ( C^' + By' - 2F0y) = 0. Now If the pair of tangents through /S7 be /3 — 7cy, yS — Icy, it appears from these expressions that JcJc = -^ , and that the corre- . C A sponding quantities for the other pairs of tangents are -7 > "b r and these three multiplied together are =1. Hence, recollecting the meaning of Jc (Art. 54), we learn that If A, F, B, i>, (7, F be the angles of a circumscribing hexagon, smFAB.smFAB.smFBC.smBBC.smDCA.f^'mFCA _ smEAC.s\nFAC.smFBjLsmBBA.sinI)CB.smFCB~ Hence also three pairs of lines will touch the same conic if their equations can be thrown Into the form 2P+ N'+ 2f'MX= 0, N'+U+2fj'NL = 0, U+ M''-^ 2h'LM= 0, METHODS OF ABRIDGED NOTATION. 259 for the equations of the three pairs of tangents, ah-eady found, can be thrown into this form by writing L \J[A) for a, «&;c. 296. If we wish to form the equations of the lines joining to a'/3'7' all the points of intersection of two curves, we have only to substitute la. + «?«', l^ + m/8', ?7 + iwy' in both equations, and eliminate I : m from the resulting equations. For any point on any of the lines in question evidently possesses the property that the line joining it to a'/3'7' meets both curves in the same point ; therefore the equations in I : ?«, which determine the points where one of these Hues meets both curves, must have a common root; and therefore the result of elimination between them is satisfied. Thus, the equation of the pair of lines joining to a'/3'7' the points where any right line L meets >S', is L'^S- 2LL'P+ US' = 0. If the point a'/3'7' be on the curve the equation reduces to L' S— 2LP—0. Ex. A chord which subtends a right angle at a given point on the curve, passes through a fixed point (Ex. 2, p. 170). We use ordinaiy rectangular co-ordinates, and, as above, form the equation of the lines joining the given point to the intersection of the conic mth Xx + fiy + v. These lines wUl be at right angles if the sum of the coefficients of x- and ij- vanish, which gives the condition {\x' + fxij' + v) (a + Z*) = 2 {aXx' + hjxy'). And since X, /x, v enter in the first degree, the chord passes through a fixed point, b — a , a — b ■VIZ. , X , —— y , If the point on the curve vary, this other point will describe a conic. If the angle subtended at the given point be not a right angle ; or, if the angle be a right angle, but the given point not on the ciu've, the condition found in like manner ■will contain \, fx, v in the second degi'ce ; and the chord A^ill envelope a conic. 297. Since the equation of the polar of a point involves the coefficients of the equation In the first degree, If an indeterminate enter in the first degree into the equation of a conic It will enter In the first degree into the equation of the polar. Thus, if P and P' be the polars of a point with regard to two conies /S, 8' ) then the polar of the same point with regard to >S-f IcS' will be P-1- hP. For {a + ha) aa + &c. = aaa! + &c. + h [a'aa! + &:g.]. Hence, given four points on a conic^ the polar of any given point lyasses through a fixed point (Ex. 2, p. 148). If Q and Q be the polars of another point with regard to S and /S", then the polar of this second point with regard to S-\-kS' 260 METHODS OF ABPIDOED NOTATION. is Q-\-hQ', Thus, then, (see Art. 59) the polars of two points ■with regard to a system of conies through four points, form two homographie pencils of lines. Given two homograj)1iic iiencils of lineSj the locus of the inter- section of the corref^ponding lines of the pencils is a conic through the vertices of the jpe^xcils. For, if we eliminate h between F-^hP^ Q-{-hQ\ we get PQ' = P'Q. In the particular case tmder consideration, the intersection of P-\-kP\ Q + kQ' is the pole with respect to S + Jc8' of the line joining the two given points. And we see that, given four jwints on a conic^ the locus of the pole of a given line is a conic (Ex. 1, p. 243). If an indeterminate enter in the second degree into the equation of a conic, it must also enter in the second degree into the equation of the polar of a given point, which will then envelope a conic. Thus, if a conic have double contact with two fixed conies, the polar of a fixed point will envelope one of three fixed conies ; for the equation of each system of conies in Art. 287 contains yu, in the second degree. We shall in another chapter cuter into fuller details re- specting the general equation, and here add a few examples illustrative of the principles already explained. Ex. 1. A point moves along a fixed line ; find the locus of the intersection of its polars with regard to two fixed conies. If the polars of any two points a'(i'y% a"^^"y" on the given Une with respect to the two conies be P', P" ; Q', Q," ; then any other point on the Una is Xa'+/ia", XjQ'^ + /n/S", X"/ + fxy" ; and its iiokrs \P''+t*.P", X.Q' + ft-Q", which intersect on the conic P'Q" = P"Q.'. Ex. 2. The anhaannonlc ratio of four points on a right line is the same as that of their four polars. For the anhai-monic ratio of the four points la + ma", I'a' + m'a", I" a' + »i"a", l"'a! + m"'a", is evidently the same as that of the four lines IP' + mP", I'P' + m'P", l"P' + m"P", l"'P' + m"'P". Ex. 3. To find the equation of the pair of tangents at the points where a conic S is met by the line y. The equation of the polar of any point on y is (Art. 291) a'Si + /S'/Sfj = 0. But the points where y meets the curve, are found by making y =• in the general equation, whence ^,^„ ^ ^,^ ^^,^, ^ j^^ ^ 0^ Eliminating a', (3', between these equations, we get for the equation of the pair of tangents ^gj, _ ^j^g^^^ ^ j^^, ^ 0^ Thus the equation of the asymptotes of a conic (given by the Cartesian equation) is METHODS OF ABRIDGED NOTATION. 2G1 for the asymptotes are the tangents at the points where the curve is met by the Uno at infinity z. Ex. 4. Given three points on a conic ; if one asymptote pass tlu-ough a fixed point, the other will envelope a conic touching the sides of the given triangle. If <,, to be the asymptotes, and aa-\-h^ -^ cy the line at infinity, the equation of the conic is <,<2 = («« + 6^ + cyf. But since it passes througli ^y, ya, a/3, the equa- tion must not contain the terms a^, /3-, y-. If therefore t^ be \a. + fifi + try, t, must be yaH — /3-I — y; and if <2 pass through a'/^'y' ^^^^ (^^- 1? P- ^50) «, touches a J(aa') + b ^{(ifi') + c J(yy') = 0. Tlie same argument proves, that if a conic pass through three fixed points, and if one of its chords of intersection with a conic given by the general equation be \a + fx(i + vy, the other ■will beY" + ~ /3 + -y. Ex. 5. Given a self conjugate triangle -vath. regard to a conic; if one chord of intersection with a fixed conic (given by the general equation) pass thi-ough a fixed l)oint, the other will envelope a conic [Mr. Bumside]. The terms a/3, /3y, ya are now to disappear fi'om the equation, whence if one chord be Xa + /a/3 4- vy, the other is found to be Xa {jxg + vli - \f) + fMl3 {vh + \f~ jxg) + vy (X/+ jxr; - vh). Ex. 6, A and A' {a^^^y^, anfiiy^z) ^^^ the points of contact of a common tangent to two conies U, V; P and P' are variable points, one on each conic ; find the locus of C, the intersection of AP, A''P', if PP' pass through a fixed point on the common tangent [Mr. Williamson]. . Let P and Q denote the polars of ai/?jyj, a.,/32y2) 'with respect to U and V respec- tively ; then (Art. 290) if a/3y be the co-ordinates of C, those of the point P where AC meets the conic again, are Ua^ — 2Pa, U(3i — 2P/3, Uy^ — 2Py ; and those of the point P' are, in hke manner, Va^ — 2Qa, &c. If the line joining these points pass, through 0, which we choose as the intersection of a, /3, we must have Uai - 2Pa _ Va^ - 2Qa Up, - 2P/3 ~ 1% - 2(2/3 ' and when A, A', are unrestricted in position, the locus is a curve of the fourth' order. If, however, these points be in a right line, we may choose this for the line a,- and making a, and oj = 0, the preceding equation becomes divisible by a, and re- duces to the cm-ve of the third order PVfio— QU(i^. Further, if the given points are points of contact of a common tangent, P and Q represent the same line ; and another factor divides out of the equation which reduces to one of the fonn U — kV, representing a conic through the intersection of the given conies. Ex. 7. To inscribe in a conic, given by the general equation, a triangle whose sides pass thi-ough the thi-ee points /Jy, ya, a/3. We shall, as before, write S^, S„, S^ for the thi-ee quantities, aa + hjS + gy, ha + J/3 +fy, ga +fji + cy. Xow we have seen, in general, that the hue joining any point on the curve a/3y to another point a'/3'y' meets the curve again in a point, whose co-ordinates are S'a — 2P'a', S'/i — 2P'l3', S'y — 2P'y. Now if the point a'/3'y' be the intersection of lines /3, y, we may take a' = 1, /3' = 0, y' = 0, which gives S' = a, P' — Si, and the co-ordinates^ of the point where the line joining a/3y to fSy meets the curve, ai-e aa — 28^, ap, ay^ In hke manner, the line joinmg a/3y to ya, meets the cm-ve again in ba, 6/3 — 2^2, by. The line joining these two pouats ^vill pass through afi, if aa - 2.Sr, _ ba fl/3 "" J/3 - 2^2 ' or, expanding 2,S',.S^2 = co-^z + b^^i, 2G2 METHODS OF ABRIDGED NOTATION". which is the condition to be fulfilled by the co-ordinates of the vertex. Writing in this equation aa- Si — h(i — ijy, bji = S., — ha —fy, it becomes h[aS,+(^,)+y{fS,+fjS,) = 0. But since «/3y is on the curve, a<S, + jiS.^ + yS^ = 0, and the equation last written '•'^^'"'^^^0 y{fS,+ffS,-kS,} = 0. Now the factor y may be set aside as irrelevant to the geometric solution of the problem ; for although either of the points where y meets the curve fulfils the con- dition which we have expressed analytically, namely, that if it be joined to /Sy and to ya, the joining lines meet the curve again in points which he on a Kne with afi ; yet, since these joining lines coincide, they cannot be sides of a triangle. The vertex of the sought triangle is therefore either of the points where the curve is met by ^"8^+ ffS^ — hSj. It can be verified immediately that fSi= gS^ — hSj denote the lines joining the con-csponding vertices of the triangles afiy, SiS^S^. Conseqiientlj'- (see Ex. 2, p. 58), the hne J'Si + gS-^ — hS^ is constructed as follows : " Form the tri- angle DEF whose sides are the polars of the given points^, B, C; let the lines joining the coiTesponding vertices of the two triangles meet the opposite sides of the polar triangle in L, M, N; then the Hues LM, MN, NL pass through the vertices of the requked triangles." The truth of this eonstruction is easily shown geometrically : for if we suppose that we have drawn the two triangles 123, 456 which can, be drawn through the points A, B, C; then appljnng Pascal's theorem to the hexagon 123456, we see that the line BC passes through the intersection of 16, 34. But this latter point is the pole of AL (Ex. 1, p. 143). Conversely, then AL passes thi-ough the pole of BC, and L is on the polar of A (Ex. 1, p. 143). This construction becomes indeterminate if the triangle is self conjugate in which case the problem admits of an infinity of solutions. Ex. 8. If two conies have double contact, any tangent to the one is cut har- monically at its point of contact, the points where it meets the other, and where it meets the chord of contact. If in the equation S + Ji^ =z 0, we substitute lu + ma", 7/3' + ot/3", ly' + my", for «> Pj 7) (where the points a'fi'y', a"^"y" satisfy the equation 8 = 6), we get iJR' 4- mE'f + IhnP = 0. Now, if the Hne joining a'(3'y', a"/3"y", touch S + Ji-, this equation must be a perfect square : and it is evident that the only way this can happen is if P = - 2E'Ii", when the equation becomes {IE' — mR'f = ; when the truth of the theorem is manifest. Ex. 9. Find the equation of the conic touching five lines, viz. a, /3, y, Aa + Bp+ Cy, A'a + B'p + C'y. Ans. (/a)* + (»?/3) + («y)'', where /, 7h, n are determined by the conditions I m n ^ I m m „ n + 5+(7=°' A'^B'-'-C'-'- METHODS OF ADraDGED NOTATION. 263 Ex. 10. Find the equation of the conic touching the five lines, a, /?, y, a + ft + y, 2a + ft-y. We have I + in + n = 0, ^l + m — n = : hence the required equation is 2(-a)^+(3/?)' + (7)i = 0. Ex. 11. Find the equation of the conic toHching u, ft, y, at their middle points. Ans. {aa)i + [hft)^ + (cy)^ = 0. 1 1 1 Ex. 12. Find the condition that {lay + {mfty + {ny)' = should represent a para- bola. , r^, , , ,• . ^ . I ril 11 ^ Ans, The curve touches the line at mfimty when — + -r -I — =0. •' a b c Ex. 13. To find the locus of the focus of a parabola touching a, ft, y. Generally, if the co-ordinates of one focus of a conic inscribed in the triangle afty be a'ft'y, the lines joining it to the vertices of the triangle will be a _ ft ft _'/ y _ a a'~ft" ft'~y" Y^a" and since the lines to the other focus make equal angles with the sides of the triangle (Art, 189), these lines will be (Art. 55) aa = ft' ft, ft' ft = y'y, y'y = a'a ; and the co-ordinates of the other focus may be taken — , , —., — , . a" ft" y' Hence, if we are given the equation of any locus described by one focus, we can at once write down the equation of the locus described by the other; and if the second focus be at infinity, that is, if a"BmA+ft"smB + y"smC=0, the first , . , sLn^4 sinB sinC must he on the circle — — + -—r- H — - 0. The co-ordinates of the focus of afty ,,,.„., I in n a parabola at mfanity are ■ . ^ , ""^t^j ■^-rr;^,, smcc (remembeiing the i-elation in sm-A sux-B sm-C/ ^ ° Ex. 12) these values satisfy both the equations, a sin.4 + /3 sin5 + y sinC = 0, ^Ja + ,Jmft + >y = 0. _, ,. , . , - . sin-.4 sin- 5 sin-C The co-ordinates, then, of the fimte focus ai-e — -, — , , . I m n Ex. 14. To find the equation of the directrix of this parabola. Forming, by Art. 291, the equation of the polar of the point whose co-ordinates have just been given, we find la (sxD-B + sin-C— sin^^) + mft {sin'C'+ sin-A — sin-5) + wy {sm-A + sva-B — sLn-C) = 0, or la sinSsinCcosJ + mft sinCsin^4 cosS + ny sin A sinB cosC= 0. Substituting for n fi-om Ex. 12, the equation becomes I sinB sinC (a cos^ — y cosC) + in sinC sin J (ft cosB — y cosC) = ; hence the directrix always passes thi-ough the intei-section of the perpendiculai-s of the triangle (see Ex. 3, p. 54). Ex. 15. Given four tangents to a conic find the locus of the foci. Let the four tangents be a, ft, y, o, then, since any line can be expressed in terms of three othei's, these must be connected by an identical relation aa + bft + cy + do = 0. This relation must be satisfied, not only by the co-ordinates of one focus a'ft'y'S', but also by those of the other — , — , — , - . The locus is therefore the curve of the third degixje a ft y a b c d ^ " P 7 « ( 264 ) CHAPTER XV. THE PRINCIPLE OF DUALITY; AND THE METHOD OF RECIPROCAL POLARS. 298. The methods of abridged notation, explained in the last chapter, apply equally to tangential equations. Thus, if the constants X, /n, v in the equation of a line be connected by the relation (aX-f&/i4 ci') [a'X+h'fi + c'v) = [a'\-\-h"[i + c"v) (a"'X + &"'/i + c"V), the line (Art. 285) touches a conic. Now it is evident that one line which satisfies the given relation is that whose X, /x, v are determined by the equations aX + J//. + cv = 0, a"X + V'fx, + c"v = 0. That is to say, the line joining the points which these last equations represent (Art. 70), touches the conic in question. If then a, /S, 7, S represent equations of points, (that is to say, functions of the first degree in A, /*, v) ay = J>:^S is the tangential equation of a conic touched by the four lines ttyS, /3y, 7S, Sa. More generally, if 8 and S' in tangential co- ordinates represent any two curves, S—kS' represents a curve touched by every tangent common to S and >S". For, whatever values of \, /x, v make both S=0 and /S" = 0, must also make S—kS' — O. Thus, then, if 8 represent a conic, 8—ka^ re- presents a conic having common with ;S' the pairs of tangents drawn from the points a, ^. Again, the equation ay = l'j3'' represents a conic such that the two tangents which can be drawn from the point a coincide with the line a/3; and those which can be drawn from 7 coincide with the line 7^. The points a, 7 are therefore on this conic, and /S is the pole of the line joining them. In like manner, ^— a'' represents a conic having double contact with 8, and the tangents at the points of contact meet in a ; or, in other words, a is the pole of the chord of contact. So again, the equation ay = Jc^^'^ may be treated in the same manner as at Art. 270, and any point on the curve may be THE METHOD OF RECIPROCAL POLARS. 265 represented by ix'a. + 2/i^•y8 + 7, while the tangent at that point joins the points fxa + k^^ fik^ 4- 7.* Ex. 1. To find tlie locus of the centre of conies touching four given Unes. Let 2 = 0, 2' = be the tangential equations of any two conies touching the four lines ; then, by Art. 298, the tangential equation of any other is 2 + ^'2' = 0. And (see p. 148) the co-ordmates of the centre are tttt?-' ' c + J-C ' *^^ ^°™^ °^ which shows (Art. 7) that the centre of the variable conic is on the line joining the centres of the two assumed conies, whose co-ordinates are 7; , 7, ; w,, ■;Q,i and that it divides the distance between them in the ratio C -.kC Ex. 2. To find the locus of the foci of conies touching four given lines. We have only in the equations (p. 228) which determine the foci to substitute A + kA' for A, &c., and then eliminate k between them, when we get the result in the form {C (a:2 - if) + 2Fy -2Gx + A-B] {C'xy - F'x - G'y + H'} = {C (x2 - f) + 2F'y - 2G'x + A' - B'] [Cxy - Fx-Gy + U). This represents a curve of the third degi-ee (see Ex. 15, p. 263), the terms of higher order mutually destroying. If, however, 2 and 2' be parabolas, 2 + ^■2' denotes a system of parabolas having three tangents common. We have then C and C" both = 0, and the locus of foci reduces to a ckcle. Again, if the conies be concentric, taking the centre as origin, we have F, F', G, G' all = 0. In this case 2 4- k'2' re- presents a system of conies touching the four sides of a parallelogram and the locus of foci is an equilateral hyperbola.f Ex. 3. The director circles of conies touching four fixed lines have a common radical axis. This is apparent fi'om what was proved, p. 258, that the equation of the director circle is a linear function of the coefficients A, B, &c., and that therefore when we substitute A + kA' for A, &.c. it will be of the form S 4- kS' — 0. This theorem includes as a particular case, "The circles having for diametei"s the thi'ee diagonals of a complete quadrilateral have a common radical axis." 299. Thus we see (as in Art. 70) that each of the equations used in the last chapter is capable of a double interpretation, according as it is considered as an equation in trilinear or in tangential co-ordinates. And the equations used in the last chapter, to establish any theorem, will, if interpreted as equations * In other words, if in any system x'y'z', x"y"z", be the co-ordinates of any two points on a conic, and x"'y"'z"' those of the pole of the line joining them, the co- ordinates of any point on the curve may be wi-itten firx' -(- Ifxkx"' + x", firy' + 2fiky"' + y", fj?z' -)- 2fxkz"' -I- s", wliile the tangent at that point divides the two fixed tangents in the ratios /u : k, fxk : 1. When k = 1, the curve is a parabola. Want of space prevents us from giving illustrations of the gi-eat use of this principle in solving examples. The reader may try the question : — To find the locus of the point where a tangent meeting two fixed tangents is cut in a given ratio. t It is pi'oved in like manner that the locus of foci of conies passing through four fixed points, which is in general of the sixth degree, reduces to tlic fourth when tlie points form a parallelogram. M M 266 THE METHOD OF llECIPROCAL POLARS. in tangential co-ordinates, yield another tlieorcm, the reciprocal of the former. Thus (Art. 266) we proved that if three conies (/S, S+LM^ S-\-Ly) have two points [S, L) common to all, the chords in each case joining the remaining common points (il/, Nj M—N), will meet in a point. Consider these as tangential equations, and the pair of tangents drawn from L is common to the three conies, while J/, lY, M— N denote in each case the point of intersection of the other two common tangents. We thus get the theorem, " If three conies have two tangents common to all, the intersections in each case of the remaining pair of common tangents, lie in a right line." Every theorem of position (that is to say, one not involving the magni- tudes of lines or angles) is thus twofold. From each theorem another can be derived by suitably interchanging the words " point" and " line" ; and the same equations differently inter- preted will establish either theorem. We shall in this chapter give an account of the geometrical method by which the attention of mathematicians was first called to this " principle of duality."* 300. Being given a fixed conic section (U) and any curve {S)y we can generate another curve [s] as follows: draw any tangent to >S', and take its pole with regard to ?/; the locus of this pole will be a curve s, which is called the polar curve of S with regard to U. The conic Z7, with regard to which the pole is taken, is called the auxiliary conic. We have already met with a particular example of polar curves (Ex. 12, p. 198), where we proved that the polar curve of one conic section with regard to another is always a curve of the second degree. We shall for brevity say that a point corresponds to a line when we mean that the point is the pole of that line with regard to U. Thus, since it appears from our definition that every point of s is the pole with regard to U of some tangent to S^ we shall * The method of reciprocal polars was introduced by M. Poncelet, whose account of it will be found in Crelle's Journal, Vol. iv. M. Pliicker, in Ms "System der Analytischen Geometrie," J 835, presented the principle of duality in the purely ana- lytical point of view, from which the subject is treated at the beginning of this chapter. But it was Mobius who, in his " Bai-ycentrische Calcul," 1827, had made the important step of introducing a system of co-ordinates in which the position of a right line was indicated by co-ordinates and that of a point hy an equation. THE METHOD OF RECIPROCAL POLARS. 267 briefly express this relation by saying that every point of s cor- responds to some tangent of S. 301. The 'point of intersection of two tangents to S loill corre- spond to the line joining the corresponding -points of s. This follows from the property of the conic ?7, that the point of intersection of any two lines is the pole of the line joining the poles of these two lines (Art. 146). Let us suppose that in this theorem the two tangents to S are indefinitely near, then the two corresponding points of s will also be indefinitely near, and the line joining them will be a tangent to s ; and since any tangent to S intersects the con- secutive tangent at its point of contact, the last theorem be- comes for this case : If any tangent to S correspond to a point on s, the point of contact of that tangent to S will correspond to the tangent through the point on s. Hence we see that the relation between the curves is reci- procal^ that is to say, that the curve S might be generated from s in precisely the same manner that s was generated from 8. Hence the name " reciprocal polars.'^ 302. We ai*e now able, being given any theorem of position concerning anycurve 8^ to deduce another concerning the curve s. Thus, for example, if we know that a number of points con- nected with the figure S lie on one right line, we learn that the corresponding lines connected with the figure s meet in a point (Art. 146), and vice versa ; if a number of points connected with the figure ^9' lie on a conic section, the corresponding lines connected with s will touch the polar of that conic with regard to Z7; or, in general, if the locus of any point connected with ;S^ be any curve S\ the envelope of the corresponding line connected with s is s', the reciprocal polar of S'. 303. The degree of the polar reciprocal of any curve is equal to the class of the curve (see note, p. 142), that is, to the number of tangents which can he drawn from any point to that curve. For the degree of s is the same as the number of points in which any line cuts s ; and to a number of points on s, lying on a right line, correspond the same number of tangents to S passing through the point corresponding to that line. Thus, if 6' be a 268 THE METHOD OF RECIPROCAL POLARS. conic section, two, and only two, tangents, real or imaginary, can be drawn to it from any point (Art. 145) ; therefore, any line meets s in two, and only two points, real or imaginary ; we may thus infer, independently of Ex. 12, p. 198, that the reci- procal of any conic section is a curve of the second degree. 304. We shall exemplify, in the case where S and s are conic sections, the mode of obtaining one theorem from another by this method. We know (Art. 267) that " if a hexagon be in- scribed in S, whose sides are A^ B, C, Dj JE, F^ then the points of intersection, AD^ BE, CF, are in one right line.'''' Hence we infer, that " if a hexagon be ciVc?«nscribed about 5, whose vertices are a, J, c, fZ, e,yj then the lines ad, he, cf, will meet in a pioint'''' (Art. 265). Thus we see that Pascal's theorem and Brianchon's are reciprocal to each other, and it was thus, in fact, that the latter was first obtained. In order to give the student an opportunity of rendering him- self expert in the application of this method, we shall write in parallel columns some theorems, together with their reciprocals. The beginner ought carefully to examine the force of the argu- ment by which the one is inferx-ed from the other, and he ought to attempt to form for himself the reciprocal of each theorem before looking at the reciprocal we have given. He will soon find that the operation of forming the reciprocal theorem will reduce itself to a mere mechanical process of interchanging the words "point" and "line," "inscribed" and "circumscribed," "locus" and "envelope," &c. If two vertices of a triangle move If two sides of a triangle pass through along fixed right Unes, while the sides fixed points, while the vertices move on pass each through a fixed point, the locus fixed right lines, the envelope of the third of the third vertex is a conic section, side is a conic section. (Art. 2G9). If, however, the points through wluch If the lines on which the veitices move the sides pass lie in one right line, the meet in a point, the third side will pass locus will be a right line. (Ex. 2, p. 41). through a fixed point. In what other case wiU the locus be In what other case will the third side a right line ? (Ex. 3, p. 42). pass through a fixed point ? (p. 49). If two conies touch, their reciprocals will also touch ; for the first pair have a point common, and also the tangent at that point common, therefore the second pair will have a tangent common and its point of contact also common. So likewise if two conies have double contact their reciprocals will have double contact. THE METHOD OF RECIPROCAL POLARS. 269 If a triangle be circumscribed to a If a triangle be inscribed in a conic conic section, two of whose vertices move section, two of whose sides pass througli on fixed lines, the locus of the third ver- fixed points, the envelope of the third side tex is a conic section, having double con- is a conic section, having double contact tact with the given one. (Ex. 2, p. 239). with the given one. (Ex. 3, p. 239). 305. We proved (Art. 301, see figure, p. 270) If to two points P, P', on /S, correspond the tangents pt^ p't\ on s, that the tan- gents at P and P' will correspond to the points of contact^, ^', and therefore Q^ the intersection of these tangents, will corre- spond to the chord of contact 2-)p\ Hence we learn that to any point Q^ and its polar PP\ loiih respect to Sj correspond a line pp and its pole q loith respect to s. Given two pomts on a conic, and two of its tangents, the line joinuig the points of contact of those tangents passes thi'ough one or other of two fixed points. (Ex., Art. 286, p. 251). Given four points on a conic, the polar of a fixed point passes through a fixed point. (Ex. 2, p. 148). Given fom- points on a conic, the locus Given two tangents and two points on a conic, the point of intersection of the tangents at those points wiU move along one or other of two fixed right hues. Given four tangents to a conic, the locus of the pole of a fixed right line is a right line. (Ex. 2, p. 243). Given four tangents to a conic, the of the pole of a fixed right line is a conic envelope of the polar of a fixed point is section. (Ex. 1, p. 243). a conic section. The hnes joining the vertices of a tri- The points of intersection of each side angle to the opposite vertices of its polar of any triangle, with the opposite side of triangle with regard to a conic, meet in the polar triangle, lie in one right line. a i^oint. (Art. 99). Inscribe in a conic a triangle whose Circumscribe about a conic a triangle sides pass through three given points, whose vertices rest on three given lines. (Ex, 7, Art. 297). 306. Given two conies, S and S\ and their two reciprocals, s and s \ to the four points u4, P, (7, P common to S and 8' correspond the four tangents a, &, c, d common to s and s', and to the six chords of intersection of S and /S", -4P, CD ; A C, BD ; AD^ BC correspond the six intersections of common tangents to s and s ; aZ», cd ; ac^ hd ; ad^ he.* If three conies have two common tan- If three conies have two points com- gents, or if they have each donble contact mon, or if they have each double contact with a fourth, their six chords of inter- with a fourth, the six points of inter- section will pass three by three through section of common tangents lie three by the same points. (Art. 264). three on the same right lines. Or, in other words, three conies, having Or, thi'ce conies, having each double each double contact with a fourth, may be contact with a fourth, may be considerctl * A sj-stem of four points connected by six lines is accurately called a quadrangle.. as a system of four lines intersecting in six points is called a quadrilateral. 270 THE JIEinOD OF RECIPROCAL POLARS. considered as having four radical centres. K through the point of contact of two conies which touch, any chord be drawn, tangents at its extremities will meet on the common chord of the two conies. If, through an intersection of common tangents of two conies any two chords be drawn, lines joining tlieir extremities will intei'sect on one or other of the common chords of the two conies. (Ex. 1, p. 238). If .4 and B be two conies having each double contact with S, the chords of con- tact of A and B wdth S, and their chords of intersection with each other, meet in a point, and form a harmonic pencil. (Art. 263). If A, B, C, be three conies, ha^ong each double contact with S, and if A and B both touch C, the tangents at the points of contact will intersect on a common chord of A and B. as having four axes of similitude, (See Art. 117, of which this theorem is an ex- tension). If from any point on tlie tangent at the point of contact of two conies wliich touch, a tangent be drawn to each, the line joining their points of contact will pass through the intersection of common tangents to the conies. If, on a common chord of two conies, any two points be taken, and from these tangents be drawn to the conies, the dia- gonals of the quadrilateral so fonned will pass through one or other of the intersec- tions of common tangents to the conies. If A and B be two conies having each double contact with S, the intersections of the tangents at their points of contact ■with S, and the intersections of tangents common to A and B, lie in one right line, which they divide harmonically. If A, B, C, be thi-ee conies, having each double contact with .S", and if A and B both touch C, the line joining the points of contact will pass through an intersec- tion of common tangents of A and B, 307. We have hitherto supposed the auxiliary conic U to be any conic whatever. It is most common, however, to suppose this conic a circle ; and hereafter, when we speak of polar curves, wc intend the reader to understand polars with regard to a circle^ imless we expressly state otherwise. We know (Art. 88) that the polar of any point with regard to a circle is perpendicular to the line joining this point to the centre, and that the distances of the point and its polar are, when multiplied together, equal to the square of the radius ; hence the relation between polar curves with regard to a circle is often stated as follows : Being given __ any point 0, if from it we let full a perpendicular OT on any tan- gent to a curve S, and produce it until the rectangle OT.Op is equal to a constant J>f^ then the locus of the point p is a curve 5, ichich is called the polar recipro- cal of S. For this is evidently THE i\[ETIIOD OF RECIPROCAL POLARP. 271 equivalent to saying that p is the pole of P2\ witli regard to a circle whose centre is and radius h. We see, therefore (Art. 301), that the tangent pt will correspond to the point of contact P, that is to say, that OP will be perpendicular to pt^ and that 0P.Ot = ¥. It is easy to show that a change in the magnitude of Ic will affect only the size and not the shape of 5, which is all that in most cases concerns us. In this manner of considering polars, all mention of the circle may be suppressed, and s may be called the reciprocal of 8 with regard to the ^int 0. We shall call this point the origin. The advantage of using the circle for our auxiliary conic chiefly arises from the two following theorems, which are at once deduced from what has been said, and which enable us to trans- form, by this method, not only theorems of position, but also theorems involving the magnitude of lines and angles : The distance of any p)oint P from the origin is the reciprocal of the distance from the origin of the corresponding line pt. TJie angle TQT' between any two lines TQ^ T' Q^ is equal to the angle jyOp subtended at the origin by the corresponding points J), p ; for Oj) is perpendicular to TQj and (9/^' to T' Q. W^e shall give some examples of the application of these principles when we have first Investigated the following problem : 308. To find the polar reciprocal of one circle ivifh regard to another. That is to say, to find the locus of the pole jy with re- gard to the circle {0) of any tangent PT Xo the circle (C). Let MN be the polar of the point G with regard to 0, then having the points C, />, and their polars JlfiV, PT, we have by Art. 101, ,, . 00 Op , , ^ the ratio -77^ = -<,, but the first UP ^jA ' ratio is constant, since both 00 and CP are constant ; hence the distance of ^; from is to its distance from MN in the constant ratio OC:OP'y its locus is therefore a conic, of which is a focus, J/A^ the corresponding directrix, and whose eccentricity Is OC 272 THE METHOD OF RECIPROCAL POLARS. divided by CP. Hence the eccentricity is greater, less than, or = 1, according as is without, within, or ou the circle C. Hence the polar reciprocal of a circle is a conic section^ of which the origin is the fpcus^ the line corresponding to the centre is the directrix^ and xohich is an ellipse^ hyperhola^ or parabola^ according as the origin is within^ without^ or on the circle. 309. We shall now deduce some properties concerning angles, by the help of the last theorem given in Art. 307. Any two tangents to a circle make Tlie line clra%vn fi'om the focus to the equal angles with their chord of contact. intersection of two tangents bisects the angle subtended at the focus by their chord of contact. (Art. 191). For the angle between one tangent PQ (see fig., p. 270) and the chord of contact PP' is equal to the angle subtended at the focus by the corresponding points p^ q ; and similarly, the angle QPP is equal to the angle subtended by p\ q ; therefore, since QPP ' = QP 'P, pOq = 'p' Oq. Any tangent to a circle is pei-pen- Any point on a conic, and the point dicular to the line joining its point of where its tangent meets the directrix, contact to the centre. subtend a right angle at the focus. This follows as before, recollecting that the directrix of the conic answers to the centre of the circle. Any line is perpendicular to the line Any point and the intersection of its joining its pole to the centre of the circle, polar with the directrix subtend a right angle at the focus. The line joining any point to the If the point where any line meets the centre of a circle makes equal angles with directrix be joined to the focus, the join- the tangents tlu'ough that point. ing line will bisect the angle between the focal radii to the points where the given hue meets the curve. The locus of the intersection of tan- The envelope of a chord of a conic, gents to a circle, which cut at a given which subtends a given angle at the focus, angle, is a concentric circle. is a conic having the same focus and the same directiix. The envelope of the chord of contact The locus of the intersection of tan- of tangents which cut at a given angle gents, whose chord subtends a given angle is a concentric circle. at the focus, is a conic having the same focus and directiix. If from a fixed point tangents be If a fixed line intersect a series of drawn to a series of concentric circles, conies having the same focus and same the locus of the points of contact will be directrix, the envelope of the tangents to a circle passing through the fixed point, the conies, at the points where this Ime and through the common centre. meets them, will be a conic having the same focus, and toucliing both the fixed line and the common directrix. THE METHOD OF KECTPEOCAL POLARS. 273 In the latter theorem, if the fixed line be at uifinlty, we find the envelope of the asymptotes of a series of hyperbolas having the same focus and same directrix, to be a parabola having the same focus and touching the common directrix. If two chords at right angles to each The locus of the intersection of tan- other be drawTi through any point on a gents to a parabola which cut at right circle, the line joining their extremities angles is the dii'ectrix. passes through the centre. We say a parabola, for, the point through which the chords of the circle are drawn being taken for origin, the polar of the circle is a parabola (Art. 308). The envelope of a chord of a circle The locus of the intersection of tan- which subtends a given angle at a given gents to a parabola, which cut at a given point on the curve is a concentric circle. angle, is a conic having the same focus and the same directrix. Given base and vertical angle of a Given in position two sides of a tri- triangle, the locus of vertex is a circle angle, and tlie angle subtended by the passing through the extremities of the base at a given point, the envelope of the base. base is a conic, of which that point is a focus, and to which the two given sides will be tangents. The locus of the intersection of tan- The envelope of any chord of a conic gents to an ellipse or hj'perbola which which subtends a right angle at any fixed cut at right angles is a circle. pomt is a conic, of wliich that point is a focus. " If from any point on the circumference of a circle perpen- diculars be let fall on the sides of any inscribed triangle, their three feet will lie in one right line" (Art. 125). If we take the fixed point for origin, to the triangle inscribed in a circle will correspond a triangle circurtiscribed about Si^^ara- hola ; again, to the foot of the perpendiculai' on any line corre- sponds a line through the corresponding point perpendicular to the radius vector from the origin. Hence, " If we join the focus to each vertex of a triangle circumscribed about a parabola, and erect perpendiculars at the vertices to the joining lines, those perpendiculars will pass through the same point." If, therefore, a circle be described, having for diameter the radius vector from the focus to this point, it will pass through the vertices of the circumscribed triangle. Hence, Given three tangents to a para- bola, the locus of the focus is the circumscribing circle (p. 196). The locus of the foot of the perpen- If fi-om any point a radius vector be dicular (or of a line making a constant drawn to a circle, the envelope of a jier- angle with the tangent), from the focus pendicular to it at its extremity (or of a 274 THE METHOD OF RECIPROCAL POLARS. of an ellipse or hjiierbola on the tangent line making a constant angle with it) is a is a circle. conic having the fixed point for its focus. 310. Having sufficiently exemplified in the last Article the method of transforming theorems involving angles, we proceed to show that theorems involving the magnitude of lines j^nssing through the origin are easily transformed by the help of the first theorem in Art. 307. For example, the sum (or, in some cases, the difference, if the origin be without the circle) of the perpen- diculars let fall from the origin on any pair of parallel tangents to a circle is constant, and equal to the diameter of the circle. Now, to two parallel lines correspond two points on a line passing through the origin. Hence, " the sum of the reciprocals of the segments of any focal chord of an ellipse is constant." We know (p. 179) that this sum is four times the reciprocal of the parameter of the ellipse, and since we learn from the present example that it only depends on the diameter, and not on the position of the reciprocal circle, we infer that the reci- ^Jrocals of equal circles, ivith regard to any origin, have the same imrameter. The rectangle under the segments of The rectangle under the perpendiculars any chord of a circle through the oi"igin let fall from the focus on two parallel is constant. tangents is constant. Hence, given the tangent from the origin to a circle, we are given the conjugate axis of the reciprocal hyperbola. Again, the theorem that the sum of the focal distances of any point on an ellipse is constant, may be expressed thus : Tlie sura of the distances fi-om the The sum of the reciprocals of perpen- focus of the points of contact of parallel diculars let fall fi-om any point within a tangents is constant. circle on two tangents whose chord of con- tact passes through the point, is constant. 311. If we are given any homogeneous equation connecting the perpendiculars FA, PB, &c. let fall from a variable point P on fixed lines, we can transform it so as to obtain a relation connecting the perpendiculars ap, hp , &c. let fall from the fixed points «, h, &c. which correspond to the fixed lines, on the variable line which corresponds to P. For we have only to divide the equation by a power of OP, the distance of P from the origin, and then, by Art. JO], substitute for each term THE METHOD OF RECIPROCAL TOLARS. 275 ^, ^ . For example, if PA, PB, PC, PD be the perpen- diculars let fall from any point of a conic on the sides of an inscribed quadrilateral, PA.PG=kPB.PD (Art. 259). Divid- ing each factor by OP, and substituting, as above, we have -^ . ^ =^ -^, . -^; and Oa, Oh, Oc, Od being constant, we (Ja (Jc Uo Ud infer that if a fixed quadrilateral he circumscribed to a conic, tJie product of the perpendiculars let fall from two opposite vertices 071 any variable tangent is in a constant ratio to the product of the perpendiculars let fall from the other two vertices. The product of the periDcndiculars from The product of the pei-pendiculars from any point of a conic on two fixed tangents, two fixed points of a conic on any tan- is in a constant ratio to the square of the gent, is in a constant ratio to the square per^jendicular on their chord of contact, of the perpendicular on it, from the inter- (Art. 259). section of tangents at those points. If, however, the origin be taken on the chord of contact, the reciprocal theorem Is, " the intercepts, made by any variable tangent on two parallel tangents, have a constant rectangle." The product of the peipendiculars on The square of the radius vector from any tangent of a conic from two fixed a fixed point to any point on a conic, is in points (the foci) is constant. a constant ratio to the product of the per- pendiculars let fall from that point of the conic on two fixed right lines. Generally since every equation in trillnear co-ordinates Is a homogeneous relation between the perpendiculars from a point on three fixed lines, we can transform It by the method of this article, so as to obtain a relation connecting X, //., v, the per- pendiculars let fall from three fixed points on any tangent to the reciprocal curve, which may be regarded as a kind of tan- gential equation* of that curve. Thus the general trillnear equation of a conic becomes, when transformed, a — + h ^, + c -r,7, + 2/ ^-, -f 27 -7- + 2/i —, = 0, P P' P ' PP P P PP where p, p, p" are the distances of the origin from tlie vertices of the new triangle of reference. Or, conversely. If we are given any relation of the second degree AX~ + &c. = 0, con- * Sec Appendix on Tangential Equations. 27G THE :\[ETIIOD OF RECIPROCAL POLAES. nccting the three perpendiculars \, yu., v, the trillnear equation of the reciprocal curve is a /3 7 y37 7a a^S where a', IB\ 7' are the trilinear co-ordinates of the origin. Ex. 1. Given the focus and a triangle circumscribing a conic, the perpendiculars let fall from its vertices on any tangent to the conic are connected by the relation Sim -^ + sin 6' '^ + sin 6" ^ = 0, where C, 6', 6" are the angles the sides of the triangle subtend at the focus. This is obtained by forming the reciprocal of the trilinear equation of the circle circum- scribing a triangle. If the centre of the inscribed circle be taken as focus, we have = 90° + ^.-1, p sin J^ = r, whence the tangential equation, on this system, of the inscribed circle is uv cot\A + vX cot IB + \^ cot^C= 0. In the case of any of the exscribed circles two of the cotangents are replaced by tangents. Ex. 2. Given the focus and a triangle inscribed in a conic, the pei-pendiculars let fall fi-om its vei-tices on any tangent are connected by the relation slni.J(i) + si„ie-J(5) + »„J."J(il) = 0. The. tangential equation of the circumscribing circle takes the fonn sinJ J(\) + sin£ J(/i) + sinC J(w) = 0. Ex. 3. Given focus and three tangents the trilmear equation of the conic is ^'»»J(j)-'°»'J©-'°«"J&) = «- This is obtained by reciprocating the equation of the circumscribing circle last found. Ex, 4. In like manner from Ex. 1, we find that given focus and three points the trilinear equation is tan 10 - + tan.}6' ^ + tan^e" ^ = 0. a ^ y 312. Very many theorems concerning magnitude may be reduced to theorems concerning lines cut harmonically or an- harmonically, and are transformed by the following principle : To any four points on a right line correspond four lines passing through a pointy and the anharmonic ratio of this pencil is the same as that of the four points. This is evident, since each leg of the pencil drawn from the origin to the given points is perpendicular to one of the corre- sponding lines. We may thus derive the anharmonic properties of conies in general from those of the circle. The anharmonic ratio of the pencil The anharmonic ratio of the points in joining four points on a conic to a variable which four fixed tangents to a conic cut fifth is constant. any variable fifth is constant. THE METHOD OF RECiritOCAL POLARS. 277 The first of these theorems is true for the circle, since all the angles of the pencil are constant, therefore the second is true for all conies. The second theorem is true for the circle, since the angles which the four points subtend at the centre are constant, therefore the first theorem is true for all conies. Bj observing the angles whicli correspond in the reciprocal figure to the angles which are constant in the case of the circle, the student will perceive that the angles which the four points of the variable tangent subtend at either focus are constant, and that the angles are constant which are subtended at the focus by the four points in which any inscribed pencil meets the directrix. 313. The anharmonic ratio of a line is not the only relation concerning the magnitude of lines which can be expressed In terms of the angles subtended by the lines at a fixed point. For, if there be any relation which, by substituting (as in Art. 56) . . ,. ,„ . 11.., OA.OB.s'mAOB , tor each Ime AB mvolved m it, -yp , can be re- duced to a relation between the sines of angles subtended at a given point 0, this relation will be equally true for any trans- versal cutting the lines joining to the points A^ B^ &c. ; and by taking the given point for origin a reciprocal theorem can be easily obtained. For example, the following theorem, due to Carnot, is an immediate consequence of Art. 148 : " If any conic meet the side AB of any triangle in the points c, c ; BC in «, a' ; AC in Z», h' ; then the ratio Ac.Ac'.Ba.Ba'.Ch.Ch' _ „ Ab.Ah'.Bc.Bc.Ca.Ca~ ' Now, it will be seen tliat this ratio Is such that we may substitute for each line Ac the sine of the angle AOc^ which it subtends at any fixed point ; and if we take the reciprocal of this theorem, we obtain the theorem given already at p. 258. 314. Having shown how to form the reciprocals of particular theorems, we shall add some general considerations respecting reciprocal conies. We proved (Art. 308) that the reciprocal of a circle is an ellipse, hyperbola, or parabola, according as the origin is within, 278 THE METHOD OF KECIPROCAL P0LAR9. Avitliout, or ou the curve ; we shall now extend this conclusion to all the conic sections. It is evident that, the nearer any line or point is to the origin, the farther the corresponding point or line will be ; that if any line passes through the origin, the corre- sponding point must be at an infinite distance ; and that the line corresponding to the origin itself must be altogether at an infinite distance. To two tangents, therefore, through the origin on one figure, will correspond two points at an infinite distance on the other ; hence, if two real tangents can be drawn from the origin, the reciprocal curve will have two real points at infinity, that is, it will be a hyperbola ; if the tangents drawn from the origin be imaginary, the reciprocal curve will be an ellipse ; if the origin be on the curve, the tangents from it coincide, therefore the points at infinity on the reciprocal curve coincide, that is, the reciprocal curve will be a parabola. Since the line at infinity corresponds to the origin, we see that, if the origin be a point on one curve, the line at infinity will be a tangent to the reciprocal curve ; and we are again led to the theorem (Art. 254) that every parabola has one tangent situated at an infinite distance. 315. To the points of contact of two tangents through the origin must correspond the tangents at the two points at infinity on the reciprocal curve, that is to say, the asymptotes of the reciprocal curve. The eccentricity of the reciprocal hyperbola depending solely on the angle between its asymptotes, depends, therefore, on the angle between the tangents drawn from the origin to the original curve. Again, the intersection of the asymptotes of the reciprocal curve [i.e. its centre) corresponds to the chord of contact of tangents from the origin to the original curve. We met with a particular case of this theorem when we proved that to the centre of a circle corresponds the directrix of the reciprocal conic, for the directrix is the polar of the origin which is the focus of that conic. Ex. 1. The reciprocal of a parabola with regard to a point on the directrix is an equilateral hyperbola. (See Art. 221). Ex. 2. Prove that the following theorems are reciprocal : The intersection of peiiicndiculars of The intersection of pei-pendiculars of a triangle circnmscribing a parabola is a a triangle inscribed in an equilateral hy- point on the directrix. perbola lies on the ctu've. THE METHOD OF liECIPKOCAL POLARS. 279 Ex. 3. Derive the last fi-om Pascal's theorem; (see Ex. 3, p. 23G). Ex. 4. The axes of the reciprocal curve are parallel to the tangent and normal of a conic drawn through the origin confocal with the given one. For the axes of the reciprocal curve must be parallel to the internal and external bisectors of the angle between the tangents drawn from the origin to the given curve. The tlieorem stated follows by Art. 189. 316. Given two circles, we can find an origin such that the reciprocals of botli shall be confocal conies. For, since the reci- procals of all circles must have one focus (the origin) common ; in order that the other focus should be common, it is only necessary that the two reciprocal curves should have the same centre, that is, that the polar of the origin with regard to both circles should be the same, or that the origin should be one of the two points determined in Art. 111. Hence, given a system of circles, as in Art. 109, their reciprocals with regard to one of these limiting points will be a system of confocal conies. The reciprocals of any two conies will, in like manner, be concentric if taken with regard to any of the three points (Art. 282) whose polars with regard to the curves are the same. Confocal conies cut at right angles. The common tangent to two circles (Art. 188). subtends a right angle at either limit- ing point. The tangents from any point to two If any hne intersect two circles, its confocal conies are equally inclined to two intercepts between the circles subtend each other. (Art. 189). equal angles at either Hmiting point. The locus of the pole of a fixed line The polar of a fixed point, with regard with regard to a series of confocal conies to a series of circles ha'N'ing the same is a line perpendicular to the fixed line, radical axis, passes through a fixed pomt ; (p, 199), and the two points subtend a right angle at either limiting point. 317. We may mention here that the method of reciprocal polars affords a simple solution of the problem, " to describe a circle touching three given circles." The locus of the centre of a circle touching two of the given circles (1), (2), is evidently a hyperbola, of which the centres of the given circles are the foci, since the problem is at once reduced to — " Given base and difference of sides of a triangle." Hence (Art. 308) the polar of the centre with regard to either of the given circles (1) will always touch a circle which can be easily constructed. In like manner, the polar of the centre of any circle touching (1 ) and (3) must also touch a given circle. Therefore, if we draw a common tangent to the two circles thus determined, and take the pole 280 THE METHOD OF RECIPROCAL POLARS. of this line with respect to (1), we have the centre of the circle touching the three given circles. 318. To find the equation of tlie recip'ocal of a conic with regard to its centre. We found, In Art. 178, that the perpendicular on the tangent could be expressed in terms of the angles it makes with the axes, / = «"^cos''^ + i-^sin'^. Hence the polar equation of the reciprocal curve is -^ = d^ cos'^ d-^-l)^ sin' 0^ a concentric conic, whose axes are the reciprocals of the axes of the given conic. 319. To find the equation of the reciprocal of a conic with regard to any point {x'y). The length of the perpendicular from any point is (Art. 178) 2) = —= f^{a^ cos^ -J- h'^ sin^ 0) — x cos — y sin 6 ; r therefore, the equation of the reciprocal curve is {xx + yy + hy = aV + Vy'- 320. Given the recijirocal of a curve with regard to the origin of co-ordinates^ to find the equation of its reciprocal loith regard to any point {x'y'). If the perpendicular from the origin on the tangent be P, the perpendicular from any other point is (Art. 34) P— x cos — y' sin ^, and, therefore, the polar equation of the locus is P = y^- X COS a — y sm c/ ; , Jv xx + y7/-\- k' , P cos p COS Jience ^ = -^ and — ^., — = — f^ , — r-z \ M p k XX -i- yy + fc we must, therefore, substitute, in the equation of the given reciprocal, — ; — - — ; — tt, for x. and — ; —, — ^ for y. ^ ' XX -^ yy -\- k ' xx ■\-yy -^ k TUE METHOD OF RECIPROCAL TOLARS. 281 The effect of this substitution may be very simply written as follows: Let the equation of the reciprocal with regard to the origin be ,^ + ^, , ,, ^ &c. = 0, II ' n-l ' ii— 2 ' where ?f,, denotes the terms of the n" degree, &c., then the reciprocal with regard to any point is fxx + yy -I- U\ (XX + yy + 'U^\'' „ "« + u,., (^ f j + ^^„., (^ -|| j + &c. = 0, a curve of the same degree as the given reciprocal. 321. To find (lie reciprocal loith respect to x^ -\-y^ — Tc' of the conic given hy the general eqiiation. We find the locus of a point whose polar xx +yg' — Tc' shall touch the given conic by writing x\ y\ — ^■■' for X, /a, v in the tangential equation (Art. 151). The reciprocal is therefore Ax' + 2Hxy + By' - 2 GFx - 2F1c'y + Ck' = 0. Thus, if the curve be a parabola, or ah - h^ = 0, and the reciprocal passes through the origin. We can, in like manner, verify by this equation other })roperties proved already geo- metrically. If we had, for symmetry, written k^ = — z'j and looked for the reciprocal with regard to the curve x' + y^ + z' — 0, the polar would have been xx + yy + zz ^ and the equation of the reciprocal would have been got by writing x, ?/, z for X, ^, v in the tangential equation. In like manner, the condition that "Xx-^- fiy-\- vz may touch any curve, may be considered as the equation of its reciprocal with regard to x' ■\-y'' -^ z''. A tangential equation of the n^ degree always represents a curve of the n^ class; since if we suppose Xx-'t iiy-{ vz to pass through a fixed point, and therefore have \x' + ^ly + vz — ; eliminating v between this equation and the given tangential equation, we have an equation of the n^ degree to determine \ : fjb] and therefore n tangents can be drawn through the given point. 322. Before quitting the subject of reciprocal polars, we w^ish to mention a class of theorems, for the transformation of which M. Chasles has proposed to take as the auxiliary conic ajjarabola instead of a circle. We proved (Art. 211) that the intercept made on the axis of the parabola between any two ou 282 THE METHOD OF RECIPROCAL POLARS. lines is equal to the intercept between perpendiculars let fall on the axis from the poles of these lines. This principle, then, enables us readily to transform theorems which relate to the magnitude of lines measured parallel to a fixed line. We shall give one or two specimens of the use of this method, premising that to two tangents parallel to the axis of the auxiliary parabola correspond the two points at infinity on the reciprocal curve, and that, consequently, the curve will be a hyperbola or ellipse, according as these tangents are real or imaginary. The reci- procal will be a parabola if the axis pass through a point at infinity on the original curve. " Any variable tangent to a conic Intercepts on two parallel tangents, portions whose rectangle is constant." To the two points of contact of parallel tangents answer the asymptotes of the reciprocal hyperbola, and to the intersections of those parallel tangents with any other tangent answer parallels to the asymptotes through any point ; and we obtain, in the first instance, that the asymptotes and parallels to them through any point on the curve intercept on any fixed line portions whose rectangle is constant. But this is plainly equivalent to the theorem : " The rectangle under parallels drawn to the asymp- totes from any point on the curve is constant." Chords drawn from two fixed points If any tangent to a parabola meet two of a hyperbola to a variable third point, fixed tangents, perpendiculars from its ex- intercept a constant length on the asymp- tremities on the tangent at the vertex wiU tote. (p. 179). intercept a constant length on that line. This method of parabolic polars is plainly very limited in its application. ( 283 ) CHAPTER XVI. HARMONIC AND ANHARMONIC PROPERTIES OF CONICS.* 323. The harmonic and anharmonic properties of conic sec- tions admit of so many applications in the theory of these curves, that we think it not unprofitable to spend a little time in point- ing out to the student the number of particular theorems either directly included in the general enunciations of these properties, or which may be inferred from them without much difficulty. The cases w^hich we shall most frequently consider are, when one of the four points of the right line, whose anharmonic ratio we are examining, is at an infinite distance. The an- harmonic ratio of four points, A^ _5, (7, D, being in general /i ^ ..^ ^B AD ' , M . 1 ,. -4i? , (Art. 56) = -^-^ -f- jr^ reduces to the simple ratio — -^ when D is at an infinite distance, since then AD ultimately = — DC. If the line be cut harmonically, its anharmonic ratio = — 1 ; and if D be at an infinite distance AB = BC, and ^C is bisected. The reader is supposed to be acquainted with the geometric investigation of these and the other fundamental theorems con- nected with anharmonic section. 324. We commence with the theorem (Art. 146) : " If any line through a point meet a conic in the points E\ R'\ and the polar of in M, the line OR'RR" is cut harmonically." First. Let R" be at an infinite distance ; then the line OR must be bisected at R' ; that is, if through a fixed j^oint a line be draion parallel to an asymptote of an hyperbola^ or to a diameter of a parabola^ the portion of this line between ilie fixed point and its polar will be bisected by the curve (Art. 211). * The fvmdamental property of anharmonic pencils was given by Pappus, Math. Coll, VII., 129. The name "anharmonic" was given by Chasles in his History of Geometrij, from the notes to which the followimg pages have been developed. Further details will be fovmd in his Traite de Gcomctrie Supcrieitre ; and in his recently pubhshed Treatise on Conies, The anharmonic relation, however, had been studied by Mobius in his Barycentvic Calculus, 1827, under the name of " Doppelschnitts- vcrhiiltniss." 284 AXHARMONIC rilOPEETIES OF CONICS. Secondly. Let R be at an Infinite distance, and R'R" must be bisected at ; that is, if through, any iwint a cliord he drmon parallel to the jyolar of that pointy it will he hisected at the point. If the polar of be at infinity, every chord through that point meets the polar at infinity, and is therefore bisected at 0. Hence this point is the centre, or the centre may he considered as a point whose polar is at infinity (p. 150). Thirdly. Let the fixed point itself be at an infinite distance, then all the lines through it will be parallel, and will be bisected on the polar of the fixed point. Hence every diameter of a conic may he considered as the polar of the point at infinity in which its ordinates are supposed to intersect. This also follows from the equation of the polar of a point (Art. 145) {ax + hy^g) + [hx + hj +/) |! + ^J^±^ = 0. Now, if xy be a point at infinity on the line my = nx^ we must y n make — , = — , and x infinite, and the equation of the polar becomes .^^ (,,^ ^. j^y _^ ^^-^ ^ ,, ^^^ _^ ^ +y) ^ o^ a diameter conjugate to my — nx (Art. 141). 325. Again, it was proved (Art. 146) that the two tangents through any point, any other line through the point, and the line to the pole of this last line, form a harmonic pencil. If now one of the lines through the point be a diameter, the other will be parallel to its conjugate, and since the polar of any point on a diameter is parallel to its conjugate, we learn that the portion between the tangents of any line drawn parallel to the polar of the point is bisected by the diameter through it. Again, let the point be the centre, the two tangents will be the asymptotes. Hence the asymptotes^ together with any pair of conjugate diameters^ form a harmonic pencil^ and the portion of any tangent intercepted between the asymptotes is bisected by the curve (Art. 196). 326. The anharmonic property of the points of a conic (Art. 259) gives rise to a rnucli greater variety of particular theorems. For, the four points on the curve may be any whatever, and ANIIARMONIC I'liOPEKTIES OF CONK'S. 285 either one or two of them may be at an Infinite distance ; the fifth point 0, to which the pencil is drawn, may be also cither at an infinite distance, or may coincide with one of the four points, in which latter case one of the legs of the pencil will be the tangent at that point ; then, again, we may measure the anharmonic ratio of the pencil by the segments on any line drawn across it, which we may, if we please, draw parallel to one of the legs of the pencil, so as to reduce the anharmonic ratio to a simple ratio. The following examples being Intended as a practical exercise to the student in developing the consequences of this theorem, we shall merely state the points whence the pencil is drawn, the line on which the ratio is measured, and the resulting theorem, recommending to the reader a closer examination of the manner in which each theorem is inferred from the general principle. We use the abbreviation [O.ABCD] to denote the anhar- monic ratio of the pencil OA , OB^ (7, OD. Ex. 1, [A.ABCD] = {B.ABCD}. Let these ratios be estimated by the segments on the line CD; let the tangents at .4, B meet CD in the points T, T', and let the chord AB meet CD in K, then the ratios are TK.DC _ KT'.DC TD.KC~ KD.T'C' that is, if any chord CD meet two tangents in T, T', and their chord of contact in K. KC. KT . TD = KD .TK.T'C. (The reader mnst be careful, in this and the following examples, to take the points of the pencil in the same order on both sides of the equation. Thus, on the left- hand side of this equation we took K second, because it answers to the leg OB of the iDencU ; on the right hand we take ^ first, because it answers to the leg OA). Ex. 2. Let T and 7" coincide, then KC.TD^-KD.TC, or, any chord through the intersection of two tangents is cut hannonically by the chord of contact. Ex. 3. Let T" be at an infinite distance. Or the secant CD drawn parallel to FT', and it will be foimd that the ratio -nill reduce to 7X2 = TC. TD. Ex.4. Let one of the points be at an infinite distance then {O.ABC cc] is con- stant. Let this ratio be estimated on the line C co . Let the lines AO. BO cut C oo in «, b ; then the ratio of the pencil -will reduce to j^ ! •'^^'^ '^^s learn, that if two fixed points, A. S. on a hyperbola or parabolaj, be joined to any variable point 0, 286 ANHARMONIC PROPERTIES OF CONICS. and the joining line meet a fixed parallel to an asymptote (if the curve be a hyper- bola), or a diameter (if the curve be a parabola), in a, b, then the ratio Ca : Cb will be constant. Ex. 5. If the same ratio be estimated on any other parallel line, lines inflected from any three fixed points to a variable point, on a hyperbola or parabola, cut a fixed parallel to an asymptote or diameter, so that ab : ac is constant. Ex. G. It follows from Ex. 4, that if the lines joining A, B to any fourth point 0' meet C oo in a', b', we must have ab _ aC a'b' a'C' Now let us suppose the point C to be also at an infinite distance, the line C <x> becomes an asymptote, the ratio ab : a'b' becomes one of equality, and lines joining two fixed points to any variable point on the hyperbola intercept on either asymptote a constant portion (p. 182). Ex.7. {A.ABC(a}-{B.ABC'x>]. Let these ratios be estimated on Coo; then if the tangents at A, B, cut Coo in a, b, and the chord of contact AB in K, we have Ca _CK CK~ Cb (observing the caution in Ex. 1). Or, if any parallel to an asymptote of a hyperbola, or a diameter of a parabola, cut two tangents and their chord of con- tact, the intercept from the curve to the chord is a geometric mean between the intercepts from the curve to the tangents. Or, conversely, if a Une ab, parallel to a given one, meet the sides of a triangle in the points a, b, K, and there be taken on it a point C such that CK"^ — Ca.Cb, the locus of C will be a parabola, if Cb be parallel to the bisector of the base of the triangle (Art. 211), but otherwise a hyperbola, to an asymptote of wliich ab is parallel. Ex. 8. Let two of the fixed points be at infinity, { 00.^5 00 co'] = {<x>' .AB 00 oo'} ; the lines oo oo, oo' oo', are the two asymptotes, while oo oo' is altogether at infinity. Let these ratios be estimated on the diameter OA ; let this line meet the parallels to the asymptotes B <x, B oo', , , , , . OA Oa' ^ in a and a ; then the ratios become Tf - TTl' ' parallels to the asymptotes through any point on a hyper- bola cut any semi-diameter, so that it is a mean propor- tional between the segments on it from the centre. Hence, conversely, if through a fixed point a line be drawn cutting two fixed lines, Ba, Ba', and a point A taken on it so that OA is a mean between Oa, Oa', the locus of ^ is a hyperbola, of which is the centre, and Ba, Ba', parallel to the asymptotes. Ex.9. {cx).AB 00 Oi'] = [cn'.AB oc a'}. Let the segments be measured on the asymptotes, and we have ^ = y^-, {0 being the centre), or the rectangle under parallels to the asjTnptotes through any pomt on the curve is constant (we invert the second ratio for the reason given in Ex. 1). ANHARMONIC TROPERTIES OF CONICS. 287 327. We next examine some particular cases of the anhar- monic property of the tangents to a conic (Art. 275). Ex. 1. This property assumes a very simple form, if the curve be a parabola, for one tangent to a para- bola is always at an in- finite distance (Art. 254), Hence three fixed tan- gents to a parabola cut any fourth in the points A, B, C, so that AB : AC is always constant. If the variable tangents co- iucijie in turn with each of the given tangents, we obtain the theorem, pQ^RP _Qr QR" Fq~ rP' Ex. 2. Let two of the four tangents to an ellipse or hyi^erbola be parallel to each other, and let the variable tangent coincide alter- nately with each of the parallel tangents. In the first case the ratio is —r- , and m the second -^c, , . Ac 1)0 Hence the rectangle Ab . DV is constant. It may be deduced from the anharmonic pro- perty of the points of a conic, that if the lines joining any point on the curve to A, D, meet the parallel tangents in the points b, b', then the rectangle Ab.Dl' will be constant. 328. We now proceed to give some examples of problems easily solved by the help of the anharmonic properties of conies. Ex. 1. To prove MacLaiu-in's method of generating conic sections (p. 236), viz. — To find the locus of the vertex F of a triangle whose sides pass through the points A, B, C, and whose base angles move on the fixed Unes Oa, Ob. Let us suppose four such triangles di-a-^^-n, then since the pencil {C.aa'a"a"'} is the same pencil as { C. bb'b"b"'}, we have {aa'a"a"'} — \bb'b"b"'], and, therefore, {A.aa'a"a"'} - {B .bb'b"b"'] ; or, from the nature of the question, [A.VV'V'V'"] = {B.VV'V'V'"] ; and therefore A, B, V, V, V", V" lie on the same conic section. Now if the first three triangles be fixed, it is evident that the locus of V" is the conic section passing through ABVV'V". Or the reasoning may be stated thus: The systems of lines through -1, and through B, being both homographic vnth the system through C, are homographic with each other : and therefore (Art. 297) the locus of the intersection of con-espond- 288 ANHARMONIC PKOPERTIES OF CONICS. ing lines \s a conic through A and B. The following examples are, in like manner, Uliistrations of the application of this piinciple of Art. 297. Ex. 2. M. Chasles has showed that the same demonstration will hold if the side ab, instead of passing through the fixed point C, touch any conic wliich touches Oct, Ob; for then any four positions of the base cut Oa, Ob, so that {aa'a"a"'] - {bb'b"b"'] (Art, 275), and the rest of the proof proceeds the same as before. Ex. 3. Newton's method of generating conic sections :— Two angles of constant A A' K A" magnitude move about fixed points P, Q ; the mtersection of two of theu- sides tra- verses the right line AA' ; then the locus of V, the intersection of their other two sides, will be a conic passing through r, Q. For, as before, take four positions of the angles, then {P. A A' A" A'"] = {Q.AA'A"A"'] ; but {P. A A' A" A'"] = {P.VV'VV'"}, {Q.AA'A"A"'} = {Q.VV'V'V'"}, since the angles of the pencils are the same ; therefore {P.VV'V"V"'}=z {Q.VV'V'V'"}; and, therefore, as before, the locus of V" is a conic through P, Q, V, V, V". Ex. 4. M. Chasles has extended this method of generating conic sections, by supposing the point A, instead of moving on a right line, to move on any conic passing through the points P, Q ; for we shall still have {P.AA'A"A"'} - {Q.AA'A"A"'}. Ex. 5. The demonstration would be the same if, in place of the angles APV, AQV being constant, APV and AQV cut o£E constant intercepts each on one of two fixed lines, for we should then prove the pencil {P.AA'A"A"'} - {P.VV'V'V'"}, because both ijencils cut off intercepts of the same length on a fixed line. Thus, also, given base of a triangle and the intercept made by the sides on any fixed line, we can prove that the locus of vertex is a conic section. Ex. 6. "We may also extend Ex. 1, by supposing the extremities of the line ab to move on any conic section passing through the points AB, for, taking four positions of the triangle, we have, by Art. 276, {aa'a"a"'} — {bb'b"b"'} ; therefore, {,1 . aa'a"a"'} = {B . bb'b"b"'\, and the rest of the proof proceeds as before. Ex. 7. The base of a triangle passes through C, the intersection of common tangents to two conic sections ; the extremities of the base ab He one on each of the conic sections, while the sides pass through fixed points A, B, one on each of the conies : the locus of the vertex is a conic through A, B. The proof proceeds exactly as before, depending now on the second theorem proved. Art. 276. We may mention that this theorem of Art. 276 admits of a simple geometrical proof. Let the pencil {O.ABCD\ be drawn from points coiTesponding to {o,abc(l\. Now, the lines OA, on, intersect at r on one of the common chords of the conies ; in like manner, BO, bo, intersect in r' on the same chord, &c. ; hence {rr'r"r"'] measures the anharmonic ratio of both these pencils. AN HARMONIC PROPERTIES OF CONICS. 289 Ex. 8. In Ex. 6 the base, instead of passing through a fixed point C, may Ix: sup- jwsed to touch a conic having double contact with the given conic (see Art. 27C). Ex. 9, If a polygon be inscribed in a conic, all whose sides but one pass through fixed points, the envelope of that side will be a conic having double contact with the given one. For, take any four positions of the polygon, then, if «, b, c, &c. be the vertices of the polygon, we have {aa'a"a"'] = {bb'b"b"'] = {cc'c"c"'], &c, The problem is, therefore, reduced to that of Art, 277, — " Given three pairs of points, rtft'rt", dd'd", to find the envelope of a"'d"', such that {««'«"«'") = {dd'd"d"']." Ex. 10. To inscribe a polygon in a conic section, all whose sides pass through fixed points. If we assume any point (a) at random on the conic for the vertex of the polygon, and form a polygon whose sides pass through the given points, the point z, where the last side meets the conic, will not, in general, coincide with a. If we make four such attempts to inscribe the polygon, we mi;st have, as in the last example, {aa'a"a"'] — {zz'z"z"']. Now, if the last attempt were successfiil, the point «'" would coincide with s'", and the problem is reduced to, — " Given three pairs of points, aa'a", zz'z", to find a point A' such that {Kaa'a"] = {Kzz'z"}." Now if wo make az"a'za"z' the vertices of an inscribed hexagon (in the order here given, taking an a and z alternately, and so that (iz, a'z', a"z", may be opposite vertices), then either of the points in wliich the line joining the inter- sections of opposite sides meets the conic may be taken for the point K. For, in the figui-e, the points A CE are aa'a", DFB are zz'z" ; and if we take the sides in the order ABCDEF, L, M. N are the intersections of opposite sides. Now, since [KPXL] measures both [D.KACE] and {A.KDFB}, we have {KA CE} = {KDFB}. Q. E. d.* It is easy to sec, from the last example, that A' is a point of contact of a conic having double con- tact ynth the given conic, to which az, a'z', a"z" are tangents, and that we have therefore just given the solution of the question, "To describe a conic touching three given linos, and having double contact with a given conic." Ex. 11. The anharmonic property affords also a simple proof of Pascal's thooi-cni. alluded to in the last example. We have {E.'CBFB} = {A.CDFB}. Now, if we examine the segments made by the first pencil on BC, and by the second on DC, we have {CRMB] = {CDNS]. * This construction for inscribing a polygon in a conic is due to M. Poncelct {TraUv des Proprletes Projectives, p. 351). The demonstration here used is Mr. Townsend"s. It shows that Poncelet's construction will equally solve the problem, '• To inscribe a polygon in a conic, each of whose sides shall touch a conic having double contact M'iih the given conic." The conies touched by the sides may be all different. PP 290 ANHARMONIC PROPERTIES OF CONICS: Now, if we draw lines from the point L to each of these points, we form two pencils which have the three legs, CL, DE, AB, common, therefore the fom-th legs, NL, LM, must form one right line. In like manner, Brianchon's theorem is derived from the anharmonic property of the tangents. Ex. 12. Given four points on a conic, ADFB, and two fixed lines through any one of them, DC, DE, to find the envelope of the line CE joining the points where those fixed Hues again meet the cui-ve. The vertices of the triangle GEM move on the fixed lines DC, DE, NL, and two of its sides pass through the fixed points, B, F; therefore, the third side envelopes a conic section touching DC, DE (by the reciprocal of MacLaurin's mode of generation). Ex. 13. Given four points on a conic ABDE, and two fixed lines, AF, CD, pass- ing each through a different one of the fixed points, the line CF joining the points where the fixed lines again meet the curve will pass through a fixed point. For the triangle CFM has two sides passing through the fixed points B, E, and the vertices move on the fixed lines AF, CD, NL, which fixed lines meet in a point, therefore (p. 268) CF passes through a fixed point. The reader will find in the Chapter on Projection how the last two theorems are suggested by other well-known theorems. (See Ex. 3 and 4, Art. 355). Ex. 14. The anharmonic ratio of any four diameters of a conic is equal to that of theu- four conjugates. This is a particular case of Ex. 2, p. 260 that the anharmonic ratio of four points on a line is the same as that of theii" fom- polars. We might also prove it directly, from the consideration that the anharmonic ratio of four chords proceeding from any point of the curve is equal to that of the supplemental chords (Art. 179). Ex. 15. A conic circumscribes a given quadrangle, to find the locus of its centre. (Ex. 3, p. 148). Draw diameters of the conic bisecting the sides of the quadrangle, their anhar- monic ratio is equal to that of their four conjugates, but this last ratio is given, since the conjugates are pai'allel to the four given lines; hence the locus is a conic passing tlu'ough the middle points of the given sides. If we take the cases where the conic breaks up into two right lines, we see that the intersections of the diagonals, and also those of the opposite sides, are points in the locus, and, therefore, that these points lie on a conic passing through the middle points of the sides and of the diagonals. 329. Wc think it unnecessary ta go through the theorems, which arc only the polar reciprocals of those investigated in the last examples ; but we recommend the student to form the polar reciprocal of each of these theorems, and then to prove it directly by the help of the anharmonic property of the tangents of a conic. Almost all are embraced in the following theorem : // there he any numher of points a, &, c, cZ, dsc. on a right line^ and a homographic system a', h\ c', d\ &c. on another line, the lines Joining corresponding points will envelope a conic. For if we construct the conic touched by the two given lines and by three lines ad, W , cc, then, by the anharmonic property of the tangents of a conic, any other of the lines dd' must touch the ANHARMONIC PROPERTIES OF CONICS. 291 same conic* The theorem here proved is the reciprocal of that proved Art. 297, and may also be established by interpreting tangentially the equations there used. Thus, if P, P' ; Q^ Q' re- present tangentially two pairs of corresponding points, P+ XP', Q + \Q' represent any other pair of corresponding points ; and the line joining them touches the curve represented by the tangential equation of the second order, PQ' = P' Q. Ex. Any transversal tlu'ough a fixed point P meets two fixed lines OA, OA', in the points AA' ; and portions of given length Aa, A! a' are taken on each of the given lines; to fixid the envelope of aa'. Here, if we give the transvei"sal four positions, it is evident that {ABCD\ = {A'B'O'D'], and that {ABCD\ — [ahcd], and {A'B'C'D'\ - {a'b'c'd']. 330. Generally when the envelope of a moveable line is found by this method to be a conic section, it is useful to take notice whether in any particular position the moveable line can be altogether at an infinite distance, for if it can, the envelope is a parabola (Art. 254). Thus, in the last example the line aa cannot be at an infinite distance, unless in some position AA' can be at an infinite distance, that is, unless P is at an infinite distance. Hence wx see that in the last example if the trans- versal, instead of passing through a fixed point, were parallel to a given line, the envelope would be a parabola. In like manner, the nature of the locus of a moveable point is often at once perceived by observing particular positions of the moveable point, as we have illustrated in the last example of Art. 328. 331. If we are given any system of points on a right line we can form a homographic system on another line, and such that three points taken arbitrarily a', h\ c shall correspond to three given points a, &, c of the first line. For let the distances of the given points on the first line measured from any fixed * In the same case if P, P' be two fixed points, it follows fi'om the last ai-ticlc that the locus of the intersection of Pd, P'd' is a conic thi-ough P, P'. We saw (Art. 277) that if a, b, c, d, &c., a', b', c', d' be two homographic systems of points on a conic, that is to say, such that {abed] always = {a'b'c'd'}, the envelope of dd' is a conic having double contact with the given one. In the same case, if P, P' be fixed points on the conic, the locus of the mtersection of Pd, P'd' is a conic through P, P'. Again, two conies are cut by the tangents of any conic having double con- tact with both, in homogi-aphic systems of points, or such that [abed] — {n'b'c'd'\ (Art. 276) ; but it is not true conversely, that if we have two homognqihic systems of points on different conies, the lines joining correspontliug points necessarily en- velope a conic. 292 ANHARMONIC PROPERTIES OF CONICS. origin on the line be a, J, c, and let the distance of any vari- able point on the line measured from the same origin be x. Similarly let the distances of the points on the second line from any origin on that line be a', h\ c', x\ then, as in Art. 277, we have the equation [a — b){c — x) _ (a — b') {c — x) \a - c) {h -x)^ [a! - c) {h' - x) ' which expanded is of the form Axx +Bx-{-Cx' ^D = 0* This equation enables us to find a point x in the second line corresponding to any assumed point x on the first line, and such that [ahcx] = [a'h'c'x]. If this relation be fulfilled, the line joining the points a?, x envelopes a conic touching the two given lines ; and this conic will be a parabola if ^ = 0, since then x is Infinite when x is infinite. The result at which we have arrived may be stated, con- versely, thus: Two systems of lyoints^ connected hy any relation^ loill he homocp'apMc^ if to one point of either system always corre- sponds one^ and hut one, ^^02n^ of the other. For, evidently, an equation of the form Axx + Bx + Cx' + D = Q Is the most general relation between x and x that we can write down, which gives a simple equation whether we seek to deter- mine X in terms of x\ or vice versa. And when this relation is fulfilled, the anharmonic ratio of four points of the first system Is equal to that of the four corresponding points of the second. For the anharmonic ratio ) ~4 { is unaltered (x — z) [y^w) * M, Chasies states the matter thus : The points x, x belong to homographic systems, if o, i, a', h' being fixed points, the ratios of the distances ax : hx, a'x' : h'x', be connected by a linear relation, such as , ax a'x' ox ox Denoting, as above, the distances of the points from fixed origins, by a, b. x; a', h'l x', tliis relation is ^ a — x a' — x' ^b^x + "V^+'' = ^' xvhich, expanded, gives a relation between x and x' of the form Axx' + Bx + Cx' + D - 0, ANIIARMONIC PROPERTIES OF CONICS. 293 if Instead of x we write ^ 7-, , and make similar siibstltu- tlons for ?/, ^j w. 332. The distances from the ori(]in of a iiair of points Aj B on the axis of x heincj given hy the equation^ ax^-'r^hx + 6 = 0, and those of another pair of points A\B' hy a'x^ -\- '2h'x-\-h' = 0^ to find the condition that the two pairs should he harmonically con- jugate. Let the distances from the origin of the first pair of points be a, yS ; and of the second a', /3' ; then the condition Is AA' _ _ AR a -a _ _ a- /3' ^ A'B ~ B'B ' ^^ a' - /3 ~ /3' - ^3 ' which expanded may be written (a + ;S)(a'+/8') = 2a/S-f 2a'/3'. Ti , n 27i r, ^'^ f I r,n 27i' , _, b' But a + yS = , a/3 = -; a + yS 1 = r, a/3=-. a a a a The required condition Is therefore ah' + ah - 2h]i = * It Is proved, similarly, that the same Is the condition that the pairs of lines aoi' + 2ha^ + h^% ad' + 2/i'ayS + V ^\ should be harmonically conjugate. 333. If a pair of points ax'' + 'ilix + J, be harmonically con- jugate with a pair doi? + '2lix + V ^ and also with another pair a^ + iWx -h J", it will be harmonically conjugate with every pair given by the equation (aV + 'llix + J') -f X {ax' + 27i"a; + U') = 0. For evidently the condition a [h' + \h") 4- h [a + \a") - 2h [h' + \h") = 0, will be fulfilled if we have separately ah' + ha - "Ihli = 0, ah" + ha" - 2hh" = 0. * It can be proved that the anharmonic ratio of the system of four points will be given, ii {ab' + a'b - 2hh')- be in a given ratio to {nb - /(-) {n'b' - h'-). 294 ANHAR]\[ONIC PROPERTIES OF CONICS. 334. To find the locus of a point such that the tangents from it to two given conies may fur ni a harmonic pencil. If four lines form a harmonic pencil they will cut any of the lines of reference harmonically. Now take the second form (given Art. 294) of the equation of a pair of tangents from a point to a curv^e given by the general trilinear equation, and make 7 = when we get ( C^"' + Bi^ - 2F^'j') a^ - 2 ( Ca'^' - Fa'ry' - G^'y' 4 Hy") a/3 ■i{CoL" + Ay"'-2Gay')^' = 0. We have a corresponding equation to determine the pair of points where the line 7 is met by the pair of tangents from a'/3'y' to a second conic. Applying then the condition of Art. 332 we find that the two pairs of points on 7 will form a harmonic system, provided that a'/3'y' satisfies the equation ( C^' + By' - 2Fl3y) ( C'd' + ^'7^ - 2 G'ay) -f ( Cd' + ^7' - 2 Gay) ( C'/S' + B'y^ - 2F'/3y) = 2 ( Cal3 - Fay - G^y + Hf) ( C'a^ - Fay - G'^y + E'y') . On expansion the equation is found to be divisible by y\ and the equation of the locus is found to be [BC'^B' C-2FFy+{CA'+ C'A-2 G G')^'+{AB'+A'B-2HH')y' + 2[GH'-\-G'H-AF'-A'F)^y+2{HF-\-H'F-BG'-B'G)ya + 2 [FG' -{-F'G- GH' - C'H) a^S = ; a conic having important relations to the two conies, which will be treated of further on. If the anharmonic ratio of the four tangents be given, the locus is the curve of the fourth degree, F'' = kSS'j where >S', >S', F^ denote the two given conies, and that now found. 335. To find the condition that the line \a + fi^ + vy should he cut harmonically hy the two conies. Eliminating 7 between this equation and that of the first conic, the points of inter- section are found to satisfy the equation (cV + av" - 2g\v) a' + 2 (cX/i - f^y - gi^v + hv") a/3 + (c/i' 4 Iv' - 2ffjiv) /S' = 0. We have a similar equation satisfied for the points where the line meets the second conic; applying then the condition of ANHARMONIC PKOPERTIES OF CONICS. 295 Art. 332, we find, precisely as in the last article, that the re- quired condition is {ho + h'c - 2ff) V + {ca + o'a - 2gg') yu-' + [ah' + ah - 2hh') v' + 2 igh' 4- g'k - af - a'f) f,v + 2 (//' + h'f- hg' - Vg) v\ + 2 ifg' +f'g - cli - c'h) \/j, = 0. The line consequently envelopes a conic* INVOLUTION'. 336. Two systems of points «, Z*, c, &c., «', h'j c', &c., situ- ated 071 the same right line^ will be homographic (Art. 331) if the distances measured from any origin, of two corresponding points, be connected by a relation of the form Axx + Bx + Cx +D = 0. Now this equation not being symmetrical between x and re', the point which corresponds to any point of the line considered as belonging to the first system, will in general not be the same as that which corresponds to it considered as belonging to the second system. Thus, to a point at a distance x considered as belonging to the first system, corresponds a point at the dis- Bx + D tance — -, ^; but considered as belonging to the second ^^^ ^ Cx + B system, coi'responds — -^ ^ . Two homographic systems situated on the same line are said to form a system m involution^ when to any point of the line the same point corresponds whether it be considered as belonging to the first or second system. That this should be the case it is evidently necessary and suflScicnt that we should have B= C In the preceding equation, in order tliat the relation connecting x and x may be symmetrical. We shall find it * If svibstituting in the equations of two conies U, V, for a, Xa + ixa'. i'c. we obtain results \- U + 2\fiP + fj} U', \2 r + 2X/za + M- J^''. then it is easy to see, as above, that UV + U'V — 2PQ, represents the pair of lines which can be di-awn thi-ough a'/3'y', so as to be cut hai-monically by the conies. In the same case (Art. 296), the equation of the system of four hues joining a'fi'y' to tlie intei-sections of the conies, is {UV + U' r- 2PQ)- = 4 {UW - P2) ( W - Q"). ri" - P2 and VV - Q- denote the paii-s of tangents from a'li'y' to the conies. 296 ANHARMONIC PROPERTIES OF CONICS. convenient to write the relation connecting any two correspond- ing points ^^^' ^ ^(^ + x') + B=0] and if the distances from the origin of a pair of corresponding points be given by the equation ax^ + 2hx + Z* = 0, we must have - Ah + Ba — llili — 0. 337. It appears, from what has been said, that a system in invoUition consists of a number of pairs of points on a line a, a ; h, V ; &c., and such that the anharmonic ratio of any four is equal to that of their four conjugates. The expression of this equality gives a number of relations connecting the mutual distances of the points. Thus, from \ahcd\ = [aY/o'a-, we have cib . ca ah' . c'a aa i be a' a . b'o ' or ab.ca Jj'c = — ah' .c'a.hc. The development of such relations presents no difficulty. 338. The relation Axx + II{x + x') + B=0^ connects the distances of two corresponding points from any orvjin chosen arhitrarily • but by a proper choice of origin this relation can be simplified. Thus, If the distances be measured from a point at the distance a; = a, the given relation becomes A[x-^a) [x + a) + II{x -f a;' + 2a) + i?= ; or Axx + (// -f ^a) (cc + x) + Ad' + 2//a + i? = 0. And if we determine a, so that //4-^a = 0, the relation reduces to a;.c' = constant. The point thus determined is called the centre of the system ; and we learn that tlie product of the dis- tances from the centre of two corresjponding points is constant. X \B — - • when J\.x + 11=0^ the corresponding point Is 339. Since, in general, the point corresponding to any point nx^ B ix^ infinitely distant : or the centre is the 2'>oint ivhose conjugate is infinitely distant. The same thing appears from the relation [ahcc] = [a'h'c'c]^ or ac . he a'c . h'c ac .he a'c. h'c ' ANHARMONIC PROPEKTIES OF CONICS. 297 Let c' be infinitely distant, he ultimately = ac\ and ac — h'c\ and this relation becomes ac.ac = hc.h'c\ or, in other words, the product of the distances from c of two conjugate points is con- stant. The relation connecting the distances from the centre may be either ca.ca = + Tc^ or ca.cd = — 1^. In the one case two conjugate points lie on the same side of the centre 5 in the other case they lie on opposite sides. 340. A point which coincides with its conjugate is called a focus of the system. There are plainly two foci j^/' equidistant from the centre on either side of it, whose common distance from the centre e, is given by the equation cf'' = + /.;'''. Thus, when W is taken with a positive sign, that is, when two con- jugate points always lie on the same side of the centre, the foci are real. In the opposite case they are imaginary. By writing x = x in the general relation connecting corresponding points, we see that in general the distances of the foci, from any origin, are given by the equation Ax'-^'2Hx + B=0. 341. We have seen (Art. 336) that if a pair of corresponding points be given by the equation aa? •\- 2hx -{■ h = 0^ we must have Ab + Ba- 2Hh = 0. Now this equation signifies (sec Art. 332) that ani/ two corresjjonding points are harmonicaUy conjufjate with the tivo foci. The same inference may be drawn from the relation [of a] = {a'J^'a]j which gives af.af af. af fa fa' aa'.ff' a a .ff' ' fa fa ' or the distance between the foci f is divided internally and ex- ternally at a and o! into parts which are in the same ratio. Cor. When one focus is at infinity, the other bisects the distance between two conjugate points; and it follows hence that in this case the distance ah between any two points of the system is equal to a'h\ the distance between their conjugates. 342. Tivo imirs of points determine a si/steni in involution. We may take arbitrarily two pairs of points ax^ + 2hx ■+ hj a'x'^ + 2h'x + //, QQ 298 ANHARMONIC PROPERTIES OF CONICS. anil we can then determine A^ 11^ B from tlie equations Ah + Ba - inii = 0, Ah' + Ba - ^Hh' = 0. We see, as in Art. 333, that any other pair of points in in- volution witli the two given pairs may be represented by an equation of the form {ax"^ -f 2hx + Z>) -f X [a'x^ -f 2h'x + h') = 0, since, when A^ 11^ B are determined so as to satisfy the two equations wzntten above, they must also satisfy A[h^ W) + ^ (« 4 \d) - 2H[li + \li') = * The actual values of A, B, Hj found by solving these equations, are 2 {ak' — ah), 2{hb' — h'h)^ ah' — ah. Consequently the foci of the system determined by the given pairs of points, arc given by the equation [ah' - ah) x" + {ah' - ah) x + [hV - h'h) = 0. This may be otherwise written if we make the equations homogeneous by introducing a new variable ?/, and write U= ax' + 2hxy + hif^ V= ax' + 2h'xy + h'y\ The equation which determines the foci is then dUdV _dUdV _^ dx dy dy dx The foci of a system given by two pairs of points «, a ; h^ h' may be also found as follows, from the consideration that {afha'] = [a'fh'a]^ or af. ha a'f. h'a ^ a'f. ha af. h'a ' whence af : a'f : : ah.aV : a'h.a'h' -^ or f is the point where aa is cut either internally or externally in a certain given ratio. 343. The relation connecting six points in involution is of the class noticed in Art. 313, and is such that the same relations * It easily follows from this, that the condition that three pairs of points fr.r- + 2hx + b, a'x^ + 2h'x + b', a"x^ + 2h"x + b" sliould belong to a system in in- volution, is the vanishing of the determinant I «, /'•, ^> I I "', '«', /'' I rt", h", h" . ANIIARMONIC PKOPERTIES OF CONICS. 209 will subsist between the sines of the angles subtended by them at any point as subsist between the segments of the lines them- selves. Consequently, if a 'pencil he drawn from any point to six points in involution^ any transversal cuts this pencil in six points in involution. Again, the reciprocal of six points in in- volution is a pencil in involution. The greater part of the equations already found apply equally to lines drawn through a point. Thus, any pair of lines a — /i^, a — /i.'/3 belong to a system in involution, if and if we are given two pairs of lines U= ar^ + 2/^a/3 + h^\ V= ad' H- 2A'a/3 + h'l3'\ they determine a pencil in involution whose focal linos are {ah' - ah) d' + [aV - ah) a/3 + (JiV - h'h) /3' = 0, (UJdV _dUdV da d^ d/3 da '' or 0. o44. A system of conies passing through four fixed p)oints meets any transversal in a system of points in involution. For, If >S, S' be any two conies through the points, ^'4 X'S' will denote any other; and if, taking the transversal for axis of X and making y — m the equations, we get ax^ + 'igx + c, and ax' + 2^'« + c to determine the points in which the trans- versal meets S and S'y it will meet S-\- \S' in ax' + 2gx + c -f A. [ax' + 2g'x + c'), a pair (Art. 342) In Involution with the two former pair. This may also be proved geometrically as follows : By the anharmonic proper- ties of conies, {a.AdbA'} = {c.AdhA']i but if we observe the points In which these pencils meet AA', we get [aCBA'} = {AB'C'A'\ = [A'C'B'A]. Consequently the points A A' belong to the system in in- volution detcjinined by BB\ CC, the pairs of points in which 300 ANHARMONIC PROPERTIES OF CONICS. the transversal meets the sides of the quadrilateral joining the given points. Reciprocating the theorem of this article we learn that, the j)airs of tangents drawn from avy point to a system of conies touching four fixed lines^form a system in involution. 345. Since the diagonals ae^ hd may be considered as a conic through the four points, it follows, as a particular case of the last Article, that any transversal cuts the four sides, and the diagonals of a quadrilateral in points BB\ CC\ DD\ which are in invo- lution. This property enables us, being given two pairs of points BB\ DD' of a system in involution, to construct the point con- jugate to any other G. For take any point at random, a ; join «i?, «i), aC; construct any triangle hcd^ whose vertices rest on these three lines, and two of whose sides pass through B'D\ then the remaining side will pass through C", the point conjugate to C. The point a may be taken at infinity, and the lines aB^ aD^ aC will then be parallel to each other. If the point C be at infinity the same method will give us the centre of the system. The simplest construction for this ease is, — " Through B^ D^ draw any pair of parallel lines Bb, Dc ; and through B'^ D\ a different pair of parallels D'h^ B'c ; then be will pass through the centre of the system.'^ Ex. 1. If three conies circumsGri'be the same quaxirilateral, the common tangent to any two is cut harmonically by the third. For the points of contact of this tangent are the foci of the system La involution. Ex. 2. If through the intersection of the common chords of two conies we draw a tangent to one of them, this line will be cut harmonically by the other. For in this case the points D and D' in the last figure coincide, and will therefore be a focus. Ex. 3.. If two: conies have double contact with each other, or if they have a con- tact of the third order, any tangent to the one is cut harmonically at the points where it meets the other, and where it meets the chord of contact. For in this case the common chords coincide, and the point where any transversal meets the chord of contact is a focus. Ex. 4. To describe a conic through four points a, b, c, dy to touch a given right line. The point of contact must be one of the foci of the system BB', CC, &c., and these points caa be determined by Art. 342. This problem, therefore, admits of two solutions. Ex. 5. If a parallJel toi an asymptote meet the curve in C, and any inscribed quadiilateral in points abcd^ Ca .Cc — Cb .Cd. For C is the centre of the system. Ex. 6. Solve the examples, p. 285, &c., as cases of involution. In Ex. 1, K is a focus : in Ex. 2, T is also a focus : in Ex. 3, 7" is a centre, kc. Ex. 7. The intercepts on any line between a hyperbola and its asymptotes are equal. For in this case one focus of the system is at infinity (Cor., Art. 341). ANHARMONIC TROPERTIES OF COXICS. .'jOl 346. If there he a system of conies liamnrj a common self-con- jugate triangle^ any line passing through one of the vertices of this triangle is cut hy the system in involution. For, if in ad^ + h^'^ + cy'^ we write a = k/3^ we get {ak'-^h)fi' + cy% a pair of points evidently always harmonically conjugate with the two points where the line meets /3 and 7. Thus, then, in particular, a system of conies touching the fyur sides of a fixed quadrilateral cuts in involution any transversal which passes through one of the intersections of diagonals of the quadrila- teral (Ex. 3, p. 143). The points in which the transversal meets diagonals are the foci of the system, and the points where it meets opposite sides of the quadrilateral are conjugate points of the system. Ex. 1. If two conies U, V toiich their common tangents A, B, C, D va. the points a, b, c, d; a', h\ c', cV ; a conic S through the points a, b, c, and touching D at d', •ttT.ll have for its second chord of intersection with V, the line joining the intersections of A with be, B with c«, C with ab. Let V meet ab in a, (3, then, by this ai-ticle, since ab passes through an intersection of diagonals of ABCD (Ex. 2, p. 231), «, b; a, (3 belong to a system in involution, of which the points where ab meets C and D are conjugate points. But (Art. 345) the common chords of S and V meet ab in points belonging to this same system in involution, detennined by the points a,b; a, 13, in which S and V meet the Une ab. If then one of the common chords be £>, the other must pass through the intersection of C with ab. Ex. 2. If in a triangle there be inscribed an ellipse touching the sides at their middle points a, b, c, and also a circle touching at the points a', b', c', and if the fourth common tangent D to the eUipse and cu'cle touch the circle at d', then the circle de- scribed through the middle points touches the inscribed cu-cle at d'. By Ex. 1, a conic described through a, b, c. wOl touch the circle at d', if it also pass through the points where the circle is met by the line joining the intersections of A, be ; B, ea ; C, ab. But this line is in this case the line at infinity. The touching conic is therefore a circle. Sir W. R. Hamilton has thus deduced Feuerbach's theorem (p. 126) as a par- ticular case of Ex. 1. The point d' and the line D can be constructed without drawing the elUpse. For since the diagonals of an inscribed, and of the corresponding cii-cumscribing quad- rilateral meet in a point, the lines ab, cd, a'b', c'd', and the lines joining .-ID, BC; AC, BD all intei-sect in the same point. If then a, /3, y be the vertices of the triangle formed by the intersections of be, b'c' ; ca, c'a' ; ab, a'b' ; the lines joining a'a, b'l3, c'y meet in d'. In other words, the triangle a/3y is homologous with abc, a'b'c', the centres of homology behig the points d, d'. In hke manner, the triangle afiy is also homologous with ABC, the axis of homology being the line I>. ( 302 ) CHAPTER XVII. THE METHOD OF PROJECTION* 847. We have already several times had occasion to point out to the reader the advantage gained by taking notice of the number of particular theorems often included under one general enunciation, but we now propose to lay before him a short sketch of a method which renders us a still more impor- tant service, and which enables us to tell when from a particular given theorem we can safely infer the general one under which it is contained. If all the points of any figure be joined to any fixed point in space (0), the joining lines will form a cotie, of which the point is called the vertex, and the section of this cone, by any plane, will form a figure which is called the jjrojection of the given figure. The plane by which the cone is cut is called the plane of projection. To any point of one figure loill correspond a point in the other. For, if any point A be joined to the vertex 0, the point a, in which the joining line OA is cut by any plane, will be the projection on that plane of the given point A. A right line will always he projected into a right line. For, If all the points of the right line be joined to the vertex, the joining lines will form a plane, and this plane will be inter- ' sected by any plane of projection in a right line. Hence, if any number of points in one figure lie in a right line, so will also the corresponding points on the projection ; and if any number of lines in one figure pass through a point, so will also the corresponding lines on the projection. * This method is the invention of M. Poncelet. See his Traite des Projjrictes Pvojectives, published in the year 1822, a work which, I believe, may be regarded as the foundation of the Modern Geometiy. In it vi^ere taught the principles, that theorems concerning infinitely distant points may be extended to finite points on a right line; that theorems concerning systems of circles may be extended to conies having two points common ; and that theorems concerning imaginary points and lines may be extended to real points and lines. THE METHOD OF TROJECTION. 303 348. Any plane curve will always he jjrojected info another fCurve of the same degree. For it is plain that, if tbc given curve be cut by any right line in any number of points, ^, 7?, C, jD, &c. the projection will be cut by the projection of that right line in the same numher of corresponding points, «, />, c, d^ &c. ; but the degree of a curve is estimated geometrically by the number of points in which It can be cut by any right line. If AB meet the curve in some real and some imaginary points, ah will meet the projection in the same number of rea-l and the same number of imaginary points. In like manner, if any two curves intersect, their projections will intersect in the same number of points, and any point common to one pair, whether real or imaginary, must be con- sidered as the projection of a corresponding real or imaginary point common to the other pair. Any tancjent to one curve loi'II he projected into a tangent to the other. For, any line AB on one curve must be projected into the line ah joining the corresponding points of the projection. Now, if the points ^, _B, coincide, the points a, /;, will also coincide, and the line ah will be a tangent. More generally, if any two curves touch each other in any number of points, their projections will touch each other in the same number of points. 349. If a plane through the vertex parallel to the plane of projection meet the original plane in a line AB^ then any pencil of lines diverging from a point on AB will be projected into a_ system of parallel lines on the plane of projection. For, since the line from the vertex to any point of AB meets the plane of projection at an infinite distance, the intersection of any two lines which meet on AB is projected to an Infinite distance on the plane of projection. Conversely, any system of parallel lines on the original p)lane is j^^'ojccted into a system of lines meeting in a point on the line DF^ where a plane through the vertex parallel to the original plane is cut hy the p^lane of projection. The method of projection then leads us naturally to the conclusion, that any system of parallel lines may be considered as passing througii a point at an infinite distance, for their projections on any plane 304 THE METIIOD OF PROJECTION. pass tliroiigh a point In general at a finite distance ; and again, that all the j^oinis at infinity on any ])lane may he considered as^ lying on a right line^ since we have showed, that the projection of any point In which parallel lines intersect must lie somewhere on the right line DF in the plane of projection. 350. We see now, that if any property of a given curve does not involve the magnitude of lines or angles, but merely relates to the position of lines as drawn to certain points, or touching- certain curves, or to the position of points, &c., then this property will be true for any curve Into which the given curve can be pro- jected. Thus, for Instance, " if through any point in the plane of a circle a chord be drawn, the tangents at its extremities will meet on a fixed line." Now since we shall presently prove that every curve of the second degree can be projected Into a circle, the method of projection shows at once that the properties of poles and polars are true not only for the circle, but also for all curves of the second degree. Again, Pascal's and Brianchon's theorems are properties of the same class, which It Is sufficient to prove In the case of the circle, in order to know that they are true for all conic sections. 351. Properties which, If true for any figure, are true for its projection, are caWed 2)^ojective jJ^operties. Besides the classes of theorems mentioned In the last Article, there are many projective theorems which do involve the magnitude of lines. For instance, the anharmonic ratio of four points In a right line \ABCD]^ being measured by the ratio of the pencil [O.ABCD] drawn to the vertex, must be the same as that of the four points {abcd}j where this pencil Is cut by any transversal. Again, if there be an equation between the mutual distances of any number of points in a right line, such as AB.CD. EF+ k.AC.BE.DF^l.AD. CE. BF-\- &c. = 0, where In each term of the equation the same points are men- tioned, although In different orders, this property will be pro- jective. For (see Art. 311) If for AB yva substitute OA.OB.smAOB each term of the equation will contain OA.OB.OC.OD.OE.OF THE METHOD OF PROJECTION. 305 in the numerator, and OP'^ in the denominator. Dividing, then, by these, there will remain merely a relation between the sines of angles subtended at 0. It is evident that the points A^ B, C, i), E^ F^ need not be on the same right line ; or, in other words, that the perpendicular OP need not be the same for all, provided the points be so taken that after the substitution, each term of the equation may contain in the denominator the same product, OP.OP'.OP'\ &c. Thus, for example, "If lines meeting in a point and drawn through the vertices of a triangle ABC meet the opposite sides in the points a, J, c, then Ah. Be. Ca = Ac.Ba. Ch.''^ This is a relation of the class just mentioned, and which it is sufficient to prove for any projection of the triangle ABC. Let us suppose the point C projected to an infinite distance, then ACj BCj Cc are parallel, and the relation becomes Ah.Bc = Ac.Ba, the truth of which is at once perceived on making the figure. 352. It appears from what has been said, that if we wish to demonstrate any projective property of any figure, it is sufficient to demonstrate it for the simplest figure into which the given figure can be projected ; e.g. for one in which any line of the given figure is at an infinite distance. Thus, if it were required to investigate the harmonic pro- perties of a complete quadrilateral ABCD^ whose opposite sides intersect in E^ F^ and the intersection of whose diagonals is G^ we may join all the points of this figure to any point in space 0, and cut the joining lines by any plane parallel to OEF., then EF is projected to infinity, and we have a new quadrilateral, whose sides a5, cd intersect in e at infinity, that is, are parallel ; while ad^ he intersect in a point f at infinity, or are also parallel. We thus see that any quadrilateral may he 2)rojectcd into a parallelogram. Now since the diagonals of a parallelogram bisect each other, the diagonal ac is cut harmonically in the points a, g, c, and the point where it meets the line at in- finity ef. Hence AB is cut harmonically in the points A, G^ C\ and where it meets EF. Ex. If two triangles ABC, A'B'C, be such that the points of intersection of AB, A'B'; BC, B'C ; CA, C'A' ; lie in a right line, then the hues AA', BB', CC meet in a point. RR 306 THE METHOD OF PROJECTION. Project to infinity tlie line in which AB, A'B', A-c, intersect; then the theorem becomes : " If two triangles abc, a'b'c' have the sides of the one respectively parallel to the sides of the other, then the lines an\ bb', cc' meet in a point." But the truth of this latter theorem is evident, since aa', bb' both cut cc' in the same ratio. 353. In order not to interrupt the account of the applications of the method of projection, we place in a separate section the formal proof that every curve of the second degree may be projected so as to become a circle. It will also be proved that by choosing properly the vertex and plane of pro- jection, we can, as in Art. 352, cause any given line EF on the figure to be projected to infinity, at the same time that the projected curve becomes a circle. This being for the present taken for granted, these consequences follow : Given any conic section and aj^oint in its plane ^ ice ca7i j)roJect it into a circle^ of which the projection of that point is the centre^ for we have only to project it so that the projection of the polar of the given point may pass to infinity (Art. 154). Any fivo conic sections may he projected so as both to become circles^ for we have only to project one of them into a circle, and so that any of its chords of intersection with the other shall pass to infinity, and then, by Art. 257, the projection of the second conic passing through the same points at infinity as the circle must be a circle also. Any two conies which have double contact with each other may be projected into concentric circles. For we have only to project one of them into a circle, so that its chord of contact with the other may pass to infinity (Art. 257). 354. We shall now give some examples of the method of deriving properties of conies from those of the circle, or from other more particular properties of conies. Ex. 1. "A line through any point is cut harmonically by the cui've and the polar of that pomt." Tliis property and its reciprocal are projective propei-ties (Art. 351), and both being true for the circle, are true for every conic. Hence all the properties of the circle depending on the theory of poles and polars are true for all the conic sections. Ex. 2. The anharmonic properties of the points and tangents of a conic are pro- jective properties, which, when proved for the circle, as in Art. 312, are proved for all conies. Hence, every property of the circle which results from either of its anharmonic properties is true also for all tlie conic sections. Ex. 3. Camot's theorem (Art. 313), that if a conic meet the sides of a triangle, A b . A b'.Bc. Be'. Co . Cu' -Ac. Ac'. Ba . Ba'. Cb . Cb', THE METHOD OF PEOJECTION. 307 is a projective property which need only be proved in the case of the circle, in which case it is evidently true, since Ab.Ab' — Ac. Ac', &c. The theorem can evidently be proved in like manner for any polygon. Ex. 4. From Camot's theorem, thus proved, could be deduced the properties of Art. 148, by su23posing the point C at an infinite distance; we then have Ab.Ab' _Ba.Ba' Ac. Ac' ~ Be'. Be' ' where the line Ab is parallel to Ba. Ex. 5. Given two concentric circles, Given two conies having double con- any chord of one wliich touches the tact with each other, any chord of one other is bisected at the i)oint of con- which touches the other is cut harmo- tact. nically at the point of contact, and where it meets the chord of contact of the conies. (Ex. 3, p. 300). For the line at infinity in the first case is projected into the chord of contact of two conies having double contact mth each other. Ex. 4, p. 213, is only a particular case of this theorem. Ex. G. Given three concentric circles, Given three conies all touching each any tangent to one is cut by the other other in the same two points, any tan- two in four points whose anharmonic gent to one is cut by the other two in ratio is constant. four points whose anharmonic ratio is constant- The first theorem is obviously true, since the four lengths are constant. The second may be considered as an extension of the anharmonic property of the tangents of a conic. In like manner, the theorem (in Art. 27G) with regard to anharmonic I'titios in conies having double contact is immediately proved by projecting the conies into concentric circles. Ex. 7. We mentioned already, that it was sufficient to prove Pascal's theorem for the case of a circle, but, by the help of Art. 353, we may still further simplify our figure, for we may suppose the line joining the intersection of AB, DE, to that of BC, EF, to pass ofi: to infinity ; and it is only necessaiy to prove that, if a hexagon be inscribed in a cii'cle having the side AB parallel to DE, and BC to EF, then CD win be ^mraUel to AF; but the truth of tliis can be shown from elementary considerations. Ex. 8. A triangle is inscribed m any conic, two of whose sides pass throxigh fixed points, to find the envelope of the third (p. 239). Let the line joining the fixed points be projected to infinity, and at the same time the conic into a circle, and this pro- perty becomes, — " A triangle is inscribed in a circle, two of whose sides are parallel to fixed lines, to find the envelope of tlie third." But this envelope is a concentric circle, since the vertical angle of the triangle is given ; hence, in the general case, the envelope is a conic touching the given conic in two points on the luie joinmg the two given points. Ex. 9, To investigate the projective properties of a quadrilateral inscribed in a conic. Let the conic be projected into a circle, and the quadrilateral into a ixirallclo- gi-am (Art. 352). Now the intersection of the diagonals of a parallelogram inscribed in a circle is the centre of the circle; hence the intersection of the diagonals of a quadrilateral inscribed in a conic is the pole of the line joining the intersections of the opposite sides. Again, if tangents to the circle be drawn at the vertices of this parallelogi-am, the diagonals of the quadrilateral so formed wiU also pass through the centre, bisecting the angles between the first diagonals; hence, "the diagonals 308 THE METHOD OF PROJECTION. of the inscribed and corresponding circumscribing quadrilateral pass through a point, and form a harmonic pencil." Ex. 10. Given four points on a conic, the locus of its centre is a conic through the middle points of the sides of the given quadrilateral. (Ex. 15, p. 200). Given four points on a conic, the locus of the pole of any fixed line is a conic passing through the fourth harmonic to the point in which this line meets each side of the given quadrilateral. Ex. 11. The locus of the point where If through a fixed point a line be parallel chords of a circle are cut in a given ratio is an ellipse having double contact with the circle. (Art. 1G3). drawn meeting the conic in ^, JB, and on it a point P be taken, such that [OABP] may be constant, the locus of P is a conic having double contact with the given conic. 355. We may project several properties relating to foci by the help of the definition of a focus given p. 228, viz. that if -F be a focus, and A^ B the two imaginary points in which any circle is met by the line at infinity y then FA^ FB are tangents to the conic. Ex. 1. The locus of the centre of a circle touching two given circles is a hy- jierbola, having the centres of the given circles for foci. If a conic be described through two fixed points A, B, and touching two given conies which also pass through those points, the locus of the pole of ^B is a conic touching the four lines CA, CB, C'A, C'B, where C, C, are the poles of AB vnth regard to the two given conies. In this example we substitute for the word 'circle,' "conic through two fixed points A, B" (Art. 257), and for the word ' centre,' " pole of the line AB." (Art. 154). Ex. 2. Given the focus and two points Given two tangents, and two points of a conic section, the intersection of tan- on a conic, the locus of the intersection gents at those points will be on a fixed of tangents at those points is a right line, line. (Art. 191). Ex.3. Given a focus and two tan- Given two fixed points ^, i?; two tan- gents to a conic, the locus of the other gents FA, FB passing one through each focus is a right line. (This follows from point, and two other tangents to a conic ; Art. 189). the locus of the intersection of the other tangents from A, B, is a right line. Ex. 4. If a triangle circumscribe a parabola, the circle circumscribing the triangle passes through the focus, p. 196. For if the focus be F, and the two cir FAB is a second triangle whose three sides Ex. 5. The locus of the centre of a circle passing through a fixed point, and touching a fixed line, is a parabola of which the fixed point is the focus. If two triangles circumscribe a conic, their six vertices lie on the same conic. ■cular points at infinity A, B, the triangle touch the parabola. Given one tangent, and three points on a conic, the locus of the intersection of tangents at any two of these points is a conic inscribed in the triangle formed Vjy those points. THE METHOD OF PKOJECTION". 309 Ex. 6. Given four tangents to a conic, Given foiu* tangents to a conic, the the locus of the centre is the Una joining locus of the pole of any line is the line the middle points of tlie diagonals of the joining tlie fourth harmonics of the points quacMlateral. where the given line meets the diagonals of the quadrilateral. It follows from our definition of a focus, that if two conies have the same focus, this point will be an intersection of common tangents to them, and will possess the properties mentioned at the end of Art. 26-4. Also, that if two conies have the same focus and directrix, they may be considered as two conies having double contact ■with each other, and may be projected into concentric circles. 356. Since angles wliicli are constant in any figure will in general not be constant in the projection of that figure, we pro- ceed to show what property of a projected figure may be inferred when any property relating to the magnitude of angles is given ; and we commence with the case of the right angle. Let the equations of two lines at right angles to each other be a; = 0, ?/ = 0, then the equation which determines the direction of the points at infinity on any circle is d^ -1- y^ = 0, or x + y ^/ — 1 =0j x — y^/ — l = 0. Hence (Art. 57) these four lines form a harmonic pencil. Hence, given four points, A^ B^ C, Z), of a line cut harmonically, where A^ B may be real or imaginary, if these points be trans- ferred by a real or imaginary projection, so that yl, B may become the two imaginary points at infinity on any circle, then any lines through C, D will be projected into lines at right angles to each other. Conversely, any two lines at right angles to each other loill he inojected into lines ichich cut harmonically the line joining the two fixed i^oints which are the jpi'ojections of the imaginary jyoints at infinity on a circle. Ex. 1. The tangent to a cu'cle is at Any chord of a conic is cut harmoni- right angles to the radius. cally by any tangent, and by the line joining the point of contact of that tan- gent to the pole of the given chord. (Art. 14G). For the chord of the conic is supposed to be the projection of the line at infinity in the plane of the circle ; the points where the chord meets the conic will be the projections of the imaginary points at infinity on the circle ; and the pole Of the chord will be the projection of the centre of the circle. Ex. 2. Any right line di-awn through Any right line through a point, the the focus of a conic is at right angles Ime joinmg its pole to that point, and to the line joining its pole to the focus, the two tangents from the point, form (Art. 192). a harmonic pencil. (Art. 14G). It is evident that the first of these properties is only a particular case of the 310 THE METHOD OF rROJECTION. second, if wc recollect thai the tangents from the focus are the hnes joining the focus to the two imaginary points on any circle. Ex. .". Let us apply Ex. 6 of the last Article to determine the locus of the pole of a given line with regard to a system of confocal conies. Being given the two foci, we are given a quadrilateral circumscribing the conic (Art. 279) ; one of the diagonals of this quadrilateral is the line joining the foci, therefore (Ex. 6) one point on the locus is the fourth harmonic to the point where the given line cuts the dis- tance between the foci. Again, another diagonal is the line at infinity, and since the extremities of this diagonal are the jDoints at infinity on a circle, therefore by the present Article, the locus is perpendicular to the given Hne. The locus is, therefore, completely determined. Ex. 4. Two confocal conies cut each If two ccnics be inscribed in the same other at right angles. quadrilateral, the two tangents at any of their points of intersection cut any dia- gonal of the circumscribing quadi'ilateral harmonically. The last theorem is a case of the reciprocal of Ex. 1, p. 300. Ex. 5. The locus of the mtersection The locus of the intersection of tan- of two tangents to a central conic, wliich gents to a conic, which di\'ide harmo- cut at right angles, is a cu'cle. nically a given finite riglit line AB, is a conic through A, B. The last theorem may, by Art. 1-iG, be stated otherwise thus : " The locus of a point 0, such that the line joining to the pole oi AO may pass through B, is a conic through A, B" and the tnith of it is evident directly, by taking four positions of the line, when we see, by Ex. 2, p. 260, that the anhannonic ratio of four lines, AO, is equal to that of four correspondmg lines, BO. Ex. G. The locus of the intersection If in the last example AB touch the of tangents to a parabola, which cut at given conic, the locus of vdW. be the right angles, is the directrix. line joining the points of contact of tan- gents from A, B. Ex. 7. The circle circumscribing a tri- If two triangles are both self-con- angle self -con jugate with regard to an jugate with regard to a conic; their six equilateral hyperbola, jjasses through the vertices lie on a conic, centre of the curve, (p. 204). The fact that the asymptotes of an equilateral hyperbola are at right angles, may be stated, by this article, that the line at infinity cuts the ciu-ve in two points which are harmonically conjugate with respect to A, B, the imaginary circular jjoints at infinity. And since the centre C is the pole of AB, the triangle CAB is self -conjugate with regard to the equilateral hyperbola. It follows by reciprocation, that the six sides of two self -conjugate triangles touch the same conic. Ex. 8. If from any point on a conic If a harmonic pencil be drawn through two lines at right angles to each other be any point on a conic, two legs of which drawn, the chord joining their extremities are fixed, the chord joining the extremities passes thi-ough a fixed point, (p. 170). of the other legs will pass through a fixed point. In other words, given two points, o, c, on a conic, and {ahcd] a harmonic ratio, Id will pass thi'ough a fixed point, namely, the intersection of tangents at a, c. But the truth of this may be seen directly : for let the line ac meet hd in K, then since {a . abed] is a harmonic pencil, the tangent at a cuts bd in the fourth harmonic to K : but so Ukewise must the tangent at c, therefore these tangents meet bd in the same point. As a particular case of this theorem we have the follo\nng : " Through a fixed THE METHOD OF PROJECTION. 311 point on a conic two lines are drawn, making equal angles with a fixed line, the chord joining their extremities will pass through a fixed point." 357. A system of 'pairs of rigid lines drawn through a pointy so that the lines of each jpair mahe equal angles with a fixed line^ cuts the line at infinity in a system of points in involution^ of which the two jpoints at infinity on any circle form one pair of con- jugate points. For tliey evidently cut any right line in a system of points in involution, the foci of which are the points where the line is met by the given internal and external bisector of every pair of right lines. The two points at infinity just mentioned belong to the system, since they also are cut harmonically by these bisectors. The tangents from any point to a The tangents from any point to a system of confocal conies make equal system of conies inscribed in the same angles with two fixed lines. (Art. 18'J). qviadrilateral cut any diagonal of that quadrilateral in a system of points in involution of which the two extremities of that diagonal are a pair of conjugate points. (Art. 34i). 358. Two lines ivhich contain a constant angle^ cut the line joining the two points at infinity on a circle^ so that the anhar- monic ratio of the four points is constant. For the equation of two lines containing an angle 6 being a; = 0, 2/ = 0, the direction of the points at infinity on any circle is determined by the equation a;''^ + y" + 2a:?/ cos ^ = ; and, separating this equation into factors, we see, by Art. 57, that the anharraonic ratio of the four lines is constant if 6 be constant. Ex. 1. "The angle contained in the same segment of a circle is constant."' "We see, by the present Article, that this is the form assumed by the anharmonic property of four points on a circle when two of them are at an infinite distance. Ex. 2. The envelope of a chord of a K tangents through any point meet conic which subtends a constant angle the conic in T, 7", and there be taken at the focus is another conic havmg the on the conic two points A, B, such that same focus and the same directrix. {O.ATBT'] is constant, the envelope of AB is a conic touching the given conic m the points T, T', Ex. 3. The locus of the intersection If a finite line AB, touching a conic, of tangents to a parabola which cut at be cut by two tangents in a given an- a given angle is a hyperbola having the harmonic ratio, the locus of their inter- same focus and the same directrix. section is' a conic touching the given conic at the pomts of contact of tangents from .1. B. 312 THE METHOD OF PROJECTION. Ex. 4. If from the focus of a conic a If a variable tangent to a conic meet line be drawn making a given angle with two fixed tangents in T, T', and a fixed any tangent, the locus of the point where line in M, and thei'e be taken on it a it meets it is a circle. point P, such that [PTMT'] may be con- stant, the locus of P is a conic passing through the points where the fixed tan- gents meet the fixed line. A particular case of this theorem is : " The locus of the jioint where the intercept of a variable tangent between two fixed tangents is cut in a given ratio, is a hyper- bola whose asymptotes are parallel to the fixed tangents." Ex. 5. If from a fixed point 0, OP be Given the anharmonic ratio of a pencil, di-a'wn to a given circle, and the angle three of whose legs pass tlu'ough fixed TPO be constant, the envelope of TP is points, and whose vertex moves along a a conic having for its focus. given conic, passing through two of the points ; the envelope of the fourth leg is a conic touching the lines joining these two to the third fixed point. A particular case of this is : "If two fixed points A, B, on a conic be joined to a variable point P, and the intercept made by the joining chords on a fixed line be cut in a given ratio at J/, the envelope of PM is a conic touching parallels through A and B to the fixed line." Ex. 6. If from a fixed point 0, OP be Given the anharmonic ratio of a pencil, drawn to a given right line, and the angle three of whose legs pass through fixed TPO be constant, the envelope of TP is points, and whose vertex moves along a a parabola having for its focus. fixed hne, the envelope of the fom-th leg is a conic touching the three sides of the triangle formed by the given points. 359. We have now explained the geometric method by which from the properties of one figure may be derived those of another figure which corresponds to it, (not as in Chap. XV. so that the points of one figure answer to the tangents of the other, but) so that the points of one answer to the points of the other, and the tangents of one to the tangents of the other. All this might be placed on a purely analytical basis. If any curve be represented by an equation in trilinear co-ordinates, referred to a triangle whose sides are «, 5, c, and if we interpret this equation with regard to a different triangle of reference whose sides are a', l)\ c', we get a new curve of the same degree as the first ;* and the same equations which establish any pro- perty of the first curve will, when diff"erently interpreted, establish * It is easy to see, that the equation of the new cmwe refen-ed to the old triangle, is got by substituting in the given equation for a, /3, y ; lu + m(i + ny, I'a + m'ji + n'y, I" a + »j"/3 + n"y ; where la + mji + 7iy represents the hne which is to correspond to a, (fee. For fuller information on this method of transformation, see llirjher Plane Curves, Chap. VI. THE METHOD OF PROJECTION. 313 a corresponding property of the second. In this manner a right line in one system always corresponds to a right line in the other, except in the case of the equation aa + h/3 + cj = 0, which in the one system represents an infinitely distant line, in the other a finite line. And, in like manner, a'a + h'^ -\- c'7, which represents an infinitely distant line in the second system represents a finite line in the first system. In working with trilinear co-ordinates the reader can hardly have failed to take notice, how the method itself teaches him to generalize all theorems in which the line at infinity is concerned. Thus (see p. 243) if it be required to find the locus of the centre of a conic, when four points or four tangents are given, this is done by finding the locus of the pole of the line at infinity aa + h^ + cjj and the very same process gives the locus under the same conditions of the pole of any line Xa 4 fi^ + V7. We saw (Art. 59) that the anharmonic ratio of a pencil P—kP', P—IP\ &c. depends only on the constants A;, ?, and is not changed if P and P' are supposed to represent different right lines. We can infer then that In the method of transformation which we are describing, to a pencil of four lines in the one system answers in the other system a pencil having the same anharmonic ratio ; and that to four points on a line correspond four points whose anharmonic ratio is the same. An equation, /S=0, which represents a circle in the one system will, in general, not represent a circle In the other. But since any other circle in the first system Is represented by an equation of the form S f {aa + Z'/S + 07) (Xa + /a/3 + V7) = 0, all curves of the second system answering to circles In the first will have common the two points common to S and aa. + h^ + C7. oGO. In this way we are led, on purely analytical grounds, to the most important prmclplca, on the discovery and application of which the merit of Poncclct's great Avork consists. The principle of continuity (in virtue of which properties of a figure in which certain points and lines arc real, arc asserted to Xn." true even wlien some of these points and lines are imaginary), ss 314 THE METHOD OF PROJECTION. is more easily established on analytical than on purely geo- metrical grounds. In fact, the processes of analysis take no account of the distinction between real and imaginary, so im- portant in pur,e geometry. The processes for example by which, in Chap. xiv. we obtained the properties of systems of conies represented by equations of forms S = Jca^ or S=ka^ are un- aflfected, whether we suppose a and /3 to meet S in real or imaginary points. And though from any given property of a system of circles, we can obtain, by a real projection, only a property of a system of conies having two imaginary points common, yet it is plainly impossible to prove such a property by general equations without proving it at the same time for conies having two real points common. The analytical method of transformation, described in the last article, is equally applicable if we wish real points in one figure to correspond to imaginary points on the other. Thus, for example, a^ 4 yS^ = 7^ denotes a curve met by 7 in imaginary points ; but if we substitute for a, /3; F±Q\/ {- 1), and for 7, i?, where P, Q^ R denote right lines, we get a curve met in real points by R the line corre- sponding to 7. The chief difference in the application of the method of projections, considered geometrically and considered algebrai- cally, is that the geometric method would lead us to prove a theorem first for the circle or some other simple state of the figure, and then infer a general theorem by projection. The algebraic method finds it as easy to prove the general theorem as the simpler one, and would lead us to prove the general theorem first, and afterwards infer the other as a particular case. THEORY OF THE SECTIONS OF A CONE. 361. The sections of a cone hy parallel planes are similar. Let the line joining the vertex to any fixed point A in one plane, meet the other in the point a ; and let radii vectorcs be drawn from ^, a, to any other two corresponding points i?, h. Then, from the similar triangles OAB^ Oab, ^5 is to ab in the constant ratio OA : Oa ; and since every radius vector of the one curve is parallel and in a constant ratio to the corresponding radius vector of the other, the two curves are similar (Art. 233). THE METHOD OF PROJECTION. 315 Cor. If a cone standing on a circular base be cut by any plane parallel to the base, the section will be a circle. This is evident as before : we may, if we please, suppose the points A, «, the centres of the curves. 362. The sections of a concj standing on a circular base^ may he either an ellipse^ hyperbola^ or p)cirahola. A cone of the second degree Is said to be riyht if the line joining the vertex to the centre of the circle which is taken for base be perpendicular to the plane of that circle ; ki which case this line is called the axis of the cone. If this line be not per- pendicular to the plane of the base, the cone is said to be oblique. The investigation of the sections of an oblique cone Is exactly the same as that of the sections of a right cone, but we shall treat them separately, because the figure in the latter case being more simple will be more easily understood by the learner, who may at first find some difficulty in the conception of figures in space. Let a plane ( OAB) be drawn through the axis of the cone C perpendicular to the plane of the section, so that both the section MSsN and the base A SB are supposed to be perpendicular to the plane of the paper: the line BS, In which the section meets the base. Is, therefore, also supposed perpendicular to the plane of the paper. Let us first suppose the line il/iV, in which the section cuts the plane OAB to meet both the sides OA^ OB, as In the figure, on the same side of the vertex. Now let a plane parallel to the base be drawn at any other point s of the section. Then we have (Euc. iii. 35) the square of BS, the ordinate of the circle, = AB.BB, and In like manner rs^ = ar.rb. But from a comparison of the similar triangles ABM, arM; BBN, hrN, it can at once be proved that AB.BB : MB.BN'. : ar.rb : Mr.rN. Therefore BS'' : rs^ : : MB . BN : Mr . vN. Ilcnce the section MSsN is such that the square of any ordinate / -X^ «^ ■:--^. ?^. -^ 316 THE METHOD OF PROJECTION. \ \ 1 / j/ M. /i\ ,N // \! \ H" ' ^ 1 r' ~~\\ A/ i •; -^'^N /^ f-T— 1 t's is to the rectangle under the parts in which it cuts the line MNin the constant ratio BS' : ME.BK Hence it can immediately be inferred (Art. 149) that the section is an ellipsey of which MN is the axis major, while the square of the axis minor is to MN^ in the given ratio RS' : MR.RK Secondly. Let MN meet one of the sides OA produced. The proof proceeds exactly as before, only that now we prove the square of the ordinate rs in a constant ratio to the rectangle Mr.rN under the parts into which it cuts the line MN pro- duced. The learner will have no difficulty in proving that the locus will In this case be a hyperhola^ consisting evidently of the two opposite branches NsS^ Ms'S'. \ Thirdly. Let the line MN be parcdlel to one of the sides. In this case, since AR = ar^ and RB : rh :: RN: riV, we have the square of the ordinate rs {=ar.rh) to the abscissa rN in the constant ratio RS'{=AR.RB) :RN. The section is therefore a, ^yarahola.* 363. It Is evident that the projections of the tangents at the points Aj B of the circle are the tangents at the points M, N of 7 * Those who first treated of conic sections only considered the case when a right cone is cut by a plane perpendicular to a side of the cone : that is to say, when JAV is peq^endicular to OB. Conic sections were then divided into sections of a right- angled, acute, or obtuse-angled cone ; and according to Eutochius, the commentator on Apollonius, were called parabola, ellipse, or hyperbola, accorduig as the angle of the cone was equal to, less than, or exceeded a right angle. (See the passage cited in full, 'lVako7t's Examphx. p. 428). It was Apollonius who first showed that all three sections could be made from one cone ; and who, according to Pappus, gave them the names parabola, elhpse, and hyperbola, for the reason stated, p. 180. The authority of Eutochius, who was more than a century later than Pappus, may not be very great, but the name parabola was used by Archimedes, who was prior to Apollonius, THE METHOD OF PROJECTION. 817 tlie conic section (xVrt. 348) ; nowJn the case of the parabola the point M and the tangent at it go oif to iiifinity ; we are therefore again led to the conclusion that every iiarabola has one tangent altogether at an injinite distance. 364. Let the cone now be supposed oblique. The plane of the paper is a plane drawn through the line OC, perpendicular to the plane of the circle AQSB. Kow let the section meet the base in any line QS^ draw a diameter LK bisecting ^^S", and let the section meet the plane OLK in the line MN^ then the proof proceeds exactly as before ; we have the square of the ordi- nate RS equal to the rectangle LB.BK- if we conceive a plane, as before, draAvn parallel to the base (which, however, is left out of the figure in order to avoid render- ing it too complicated), we have the square of any other ordinate, rs^ equal to the corresponding rectangle Ir.rk] and we then prove by the similar triangles KEM^ Jci-M; LRN^ hNj in the plane OLK^ exactly as in the case of the right cone, that RS"^ : rs-^, as the rectangle under the parts into which each ordinate divides MN^ and that therefore the section is a conic of which MNh the diameter bisecting QS^ and which is an ellipse when MN meets both the lines OL^ OK on the same side of the vertex, a hyperbola when it meets them on different sides of the vertex, and a parabola when it is parallel to eitlier. In the proof just given Q8 is supposed to intersect the circle in real points ; if it did not, we have only to take, instead of the circle AB^ any other parallel circle aZ*, which docs meet the sec- tion in real points, and the proof will proceed as before. 365. We give formal proofs of the two following theorems, though they are evident by the principle of continuity : \. If a circular section he cut hy any idane in a line QS, the diameters conjugate to QS in that j^lane^ and in the plane of the circle^ meet QS in the same point. When qs meets the circle in real points, the diameter conjugate to it in every plane nnist evidently pass through its middle point r. "\Vo have therefore 318 THE METHOD OF PROJECTION. only to examine the case wlierc QS does not meet in real points. It was proved (Art. 8G1) that the diameter df which bisects chords, parallel to qs, of any circular section, will be pro- jected into a diameter i)i^ bisecting the parallel chords of any parallel section. The locus therefore of the middle points of all chords of the cone parallel to qs is the plane Odf. The diameter therefore, conjugate to QS in any section is the inter- section of the plane Odf with the plane of that section, and must pass through the point JR in which QS meets the plane Odf. II. In the 9ame case^ if the diameters conjugate to QS in the circle^ and in the other section^ he cut into segments RD^ RF ; Rg^ Rk ; the rectangle DR.RF is to gR.Rlc as the square of the dia- tneter of the section parallel to QS is to the square of the conjugate diameter. This is evident when qs meets the circle in real points; since rs^ = dr.rf In general, we have just proved that the lines gh^ df DF^ lie in one plane passing through the vertex. The points i), d are therefore projections of g ; that is to say, they lie In one right line passing through the vertex. AVe have therefore, by similar triangles, as in Art. 364, dr.rf: DR.RF: : gr.rk : gR.Rlc, and since dr.rf is to gr.rk as the squares of the parallel semi- diameters, DR.RF is to gR.Rk in the same ratio. If the section gskq and the line QS be given, this theorem enables us to find DR.RF^ that is to say, the square of the tangent from R to the circular section whose plane passes through ^^S*. 366. Given any conic gskq and a line TL in its plane not cutting it, we can j^roject it so that the conic mag become a circle, and the line may he jyrqjected to infinity. To do this, it is evidently necessary to find the vertex of a cone standing on the given conic, and such that its sections parallel to the plane OIL shall be circles. For then any of THE METHOD OF PROJECTION. ;:j19 these parallel sections would be a projection fulfilling the con- ditions of the problem. Now, if TL meet the conjugate dia- meter in the point L^ it follows from the theorem last proved that the distance OL is given: for, since the plane OTL is to meet the cone in an infinitely small circle, OU is to gL.Lh in the ratio of the squares of two known diameters of the section. OL must also lie in the plane perpendicular to TL^ since it Is parallel to the diameter of a circle perpendicular to TL. And there Is nothing else to limit the position of the point 0, which may lie anywhere in a known circle in the plane perpendicular to TL. 367. If a sphere . he inscribed in a right cone touching tlie plane of any section^ the point of contact ivill he a focus of that section^ and the corresponding directrix will he the intersection of the plane of the section with the plane of contact of the cone with the sphere. Let spheres be both Inscribed and exscribed between the cone and the plane of the section. Now, If any point P of the section be joined to the vertex, and the joining line meet the planes of contact In Dd^ then we have PD = PF^ since they are tangents to the same sphere, and, similarly, Pd=PF\ therefore PF+PF' = Dd, which Is constant. The point [R) where FF' meets AB produced, is a point on the direc- trix, for by the property of the circle, NFMR is cut harmonically, therefore i? Is a point on the polar of F. It Is not difficult to prove that the parameter of the section ]\fPN is constant, if the distance of the plane from the vertex be constant. CoK. The locus of the vertices of all right cones, out of which a given ellipse can be cut. Is a hyperbola passing through the foci of the ellipse. For the difference of MO and XO is constant, being equal to the difference between MF' and NF'.* * By the help of this principle, Mr. Mulcahy showed how to derive properties of angles subtended at the focus of a conic from properties of small circles of a sphere. For example, it is known that if through any pohit P, on the sm-face of a sphere, a gi-eat circle be drawn, cutting a small circle in the points A, B, then tan^AP tan^BP is constant. Now, let us take a cone whose base is the small cu-cle, and whose vertex 320 THE METirOD OF rROJECTTON'. ORTHOGONAL PKOJECTION. 368. If from all the points of any figure perpendiculars be let fall on any plane, their feet will trace out a figure which is called the orthogonal projectmi of the given figure. The ortho- gonal projection of any figure is, therefore, a right section of a cylinder passing through the given figure. All ijaralhl lines are in a constant ratio to their orthogonal 2>roject{ons on any plane. For (see fig., p. 3) MM' represents the orthogonal projection of the line PQ^ and it is evidently =PQ multiplied by the cosine of the angle which PQ makes with MM'. All lines parallel to the intersection of the plane of the figure with the plane on which it is projected^ are eqibal to their orthogonal projections. For, since the intersection of the planes is itself not altered by projection, neither can any line parallel to it. The area of any figure in a given plane is in a constant ratio to its orthogonal projection on another given plane. For, if we suppose ordinates of the figure and of its pro- jection to be drawn perpendicular to the intersection of the planes, every ordinate of the projection is to the correspond- ing ordinate of the original figure in the constant ratio of the cosine of the angle between the planes to unity ; and it will be proved, in Chap, xrx., that if two figures be such that the ordinate of one is in a constant ratio to the corresponding ordinate of the other, the areas of the figures are in the same ratio. Any ellipse can he orthogonally projected into a circle. For, if we take the intersection of the plane of projection with the plane of the given ellipse parallel to the axis minor of that ellipse, and if we take the cosine of the angle between the planes is the centre of the sphere, and let us cut this cone by any plane, and we leani that " if through a point 7>, in the plane of any conic, a line be drawn cutting the conic in the points a, b, then the product of the tangents of the halves of the angles which «;j, hj) subtend at the vertex of the cone will be constant." This property will be true of the vertex of any right cone, out of which the section can be cut, and, therefore, since the focus is a point in the locus of such vertices, it must be true that tan.]rt/yj tanJ/;./}) is constant (sec p. 199). THE METHOD OF PROJECTION. 321 = - , then every line parallel to the axis minor will be unaltered by projection, but every Ihie parallel to the axis major will be shortened in the ratio h : w, the projection will, therefore (Art. 163), be a circle, whose radius is b. 369. We shall apply the principles laid down in the last Article to investigate the expression for the radius of a circle circumscribing a triangle inscribed in a conic, given Ex. 7, p. 209.* Let the sides of the triangle be a, yS, 7, and its area A^ then, by elementary geometry, P a/37 Now let the ellipse be projected into a circle whose radius is J, then, since this is the circle circumscribing the projected triangle, we have 'oy AA But, since parallel lines are in a constant ratio to their projec- tions, we have a' ; a : : ^ : h\ 13' : ^::h: h", <y' : <y :: h : b'" ; and, since (Aii. 368) A' is to A as the area of the circle {=7rb") to the area of the ellipse [=7rah), (sec chap, xix.) we have A'-. A::b:a. Hence « '^j' : ^ : : «Z."' : Z. W, 4A 4:A ' and, therefore, li = — = — . ao * This proof of Mr. Mac CuUagh's theorem is due to Dr. Graves. TT ( '"^22 ) CHAPTER XVIIL INVARIANTS AND COVARIANTS OF SYSTEMS OF CONICS. 370. It was proved (Art. 250) tliat if S and S' represent two conies, there are tliree values of k for which kS+S' re- presents a pair of right lines. Let S = ax^ + hf + cz^ -V 'ifyz + 2gzx 4 '2hxy^ S' = ax' + Ijif + cz^ + 2/"3/s + Ig'zx \ =llixy. We also write A = aZ'C + 2^7i - af - hcf - ch% A' = ah'c' + 2/'^'A' - a'P - Vf - d¥\ Then the values of h in question are got by substituting ha + a', Tcb 4 &', &c. for a, 5, &c. in A = 0. We shall write the resulting cubic ^^3 ^ q;^.. _^ Q,j^ + A' = 0. The value of 9, found by actual calculation, is {he -f) a' + [ca - /) Z»' + [ab - P) c' ^ 2 {yh-af)f + 2{hf-hy)g' + 2 ifg-ch) h' ', or, using the notation of Art. 151, Aa' -{ Bb' + Cc' + 2Ff + 2Gff' + 2Eh' ; or, again, , dA ,, dA , dA ., dA , dA ,, dA ^^■^^ ^+^^+-^^ + ^^ + ^^^' as is also evident from Taylor's theorem. The value of 9' is got from 9 by interchanging accented and unaccented letters, and may be written 9' = A'a + B'b + C'c + 2F'f+ 2 G'g + 2E'h. If we eliminate h between kS-\- S' = 0j and the cubic which determines Z;, the result AS" - oS"S + e'S'S'' - A'S' = 0, (an equation evidently of the sixth degree,) denotes the three pairs of lines which join the four points of intersection of the two conies (Art. 238). INVARIANTS AND COVARIANTS OF CONICS. S23 Ex. To find the locus of the intersection of normals to a conic, at the extremities of a chord which passes through a given point a/3. Let the curve be S- -^^ + ^^-i.; then the points whose normals pass through a given point x'y' are determined (p. 168), as the intersections of S with the hj^perbola S' = 2 {c^xy + bhj'x - a^x'y). We can tlien, by this article, form the equation of the six chords which join the feet of normals thi-ough ay/, and expressing that this equation is satisfied for the point a/?, we have the locus required. We have A = - ^tt. , = 0, 0' = - (aV^ + V-y'- - c% A' = - 2aWc^x'y'. a-b- The equation of the locus is then ^ («=/?« - b'^-ay - c"-atiY + 2 (a^x^^ + b^' - C«) {a"-^ - IP-ay - c'-ap) (^ + ^ " l) + 2a*iVxyg + f;-iy = 0, which represents a curve of the third degree. If the given point be on either axis, the locus reduces to a conic, as may be seen by making a := in the preceding equa- tion. It is also geometrically evident, that in this case the axis is part of the locus. The locus also reduces to a conic if the point be infinitely distant : that is to say, when the preblem is to find the locus of the intersection of normals at the extremities of a chord parallel to a given line. 371. If on transforming to any new set ^f co-ordinates, Cartesian or trilinear, S and S' become S and S\ it is manifest that kS+ S' becomes kS-{- S\ and that the coefficient k is not affected. It follows that the values of k, for which kS -1- S' represents right lines, must be the same, no matter in what system of co-ordinates ^S' and S' are expressed. Hence, then, the ratio between any two coefficients in the cubic for /i, found in the last Article, remains unaltered when we transform from any one set of co-ordinates to another.* The quantities A, G, e', A' are on this account called invariants of the system of conies. If then, in the case of any two given conies, having by transformation brought S and S' to their simplest form, and having calculated A, 9, e', A', we find any homogeneous rela- tion existing between them, w^e can predict that the same relation will exist between these quantities no matter to what axes the equations are referred. It will be found possible to express in * It may be proved by actual transformation that if in S and S' we substitute for .r, ?/, z; Ix + my + nz, I'x + m'y + n'z, l"x + m"y + n"z, the quantities A, 9, 0', A' for the transformed system, are equal to those for the old, respectively midtiplied by the square of the determinant I h '") " /', m', h I /", m", h 324 INVARIANTS AND COVARIANTS terms of the same four quantities the condition that the conies should be connected by any relation, independent of the position of the axes, as is illustrated in the next Article. The following exercises in calculating the invariants A, O, G', A', include some of the cases of most frequent occurrence. Ex. 1. Calculate the invariants when the conies are refeiTed to their common self -con jugate triangle. We may take S=ax^ + hy^' + cz^, S ' = a'x- + b'y- + c's^; and we may further simplify the equations by writing a-, i/, z, instead of x J(n'), y J(6'), z 4(c'), so as to bring S' to the form x'' + y^ + z^. We have then A = abc, Q = bc + ca + ab, O' — a + b + c, A' = 1. And S + kS' will represent right lines, if Jcr' + F (a + 6 + c) + k {be + ca + ab) + abc = 0. And it is otherwise evident that the three values for which S + kS' represents right lines, are —a, —b, — c. Ex. 2. Let S' as before be x" + y- + 2^, and let S represent the general equation. Ans. e = {be -P) + {ea - g^) + {ab - h'^) = A + B + C; B' = a + b + c. Ex. 3. Let S and S' represent two circles x'^ + y^ — r'', {x — of + {y — (3y — »•'-. Ans. A-- 7-2, G = a2 + /32 - 2r2 - r'% O' = a' + ^ - r'' - 2r'^, A' = - r'-. So that if Z> be the distance between the centres of the circles, S + kS' will represent right lines, if ,.2 + (2r2 + r'2 - Z»2) k + (r^ + 2r'^ - D^) k'^ + r'^k"^ = 0. Now since we know, that S — S' represents two right lines (one finite, the other infinitely distant), it is evident that — 1 must be a root of this equation. And it is in fact divisible by k + 1, the quotient being ,.2 + (^2 ^ ,.'2 _ x»2) k + r'^-k"- = 0. Ex. 4. Let S represent — + f:; — 1, while ^S*^ is the circle {x - of + {y — /3)- - r^. Ans. A = - 4r:; , 9 = ^- (a2 + /32 - a2 - &2 _ ^2) a-b- a^b- a- b- \a- b-j Ex. 5, Let S represent the parabola y''- — imx, and S' the cii-cle as before. Ans. A = — 'im-, = — 4m {a + m), Q'' — p- — ima — r'', A' = — r-. 372. To find the condition that two conies S mid S' should touch each other. When two points, A^ JB, of the four inter- sections of two conies coincide, it is plain that the pair of lines AC\ BD is identical with the pair AD., BC. In this case, then, the cubicy ^¥ + Ql^ + Q'h + A' = 0, must have two equal roots. But it can readily be proved that the condition that this should be the case is (ee' - 9 AA')' = 4 (e' - 3 Ae') (e'^ - SA'e), OF SYSTEMS OP CONICS. 325 or eV'+ 18AA'ee'-27A'A"- 4Ae"-4A'e'' = 0, ■which is the required condition that the conies should touch. It is proved, in works on the theory of equations, that the left-hand member of the equation last written is proportional to the product of the squares of the differences of the roots of the equation in h ; and that when it is positive the roots of the equation in h are all real, but that when it is negative two of these roots are imaginary. In the latter case (see Art. 282), S and S' intersect in two real and two imaginary points : in the former case, they intersect either in four real or four imaginary points. These last two cases have not been distin- guished by any simple criterion. Ex. 1. To find by this method the condition that two circles should tonch. Forming the condition that the reduced equation (Ex. 3, Art. 371), »■« + {r^ + r'^ — IF) k + r'^k- = 0, should have equal roots, we get ?"^ + r'^ — D^ = ± 2r?'' ; i) = »• + r' as is geometrically evident. Ex. 2. Find the locus of the centre of a circle of constant radius toucliing a given conic. We have only to ■write for A, A', 9, 0' in the equation of this article, the values Ex. 4 and 5, Art. 371 ; and to consider a, /3 as the running co-ordinates. The locus is in general a curve of the eighth degree, but reduces to the sixth in the case of the parabola. This curve is the same which we should find by measuring from the curve on each normal, a constant length, equal to r. It is sometimes called the curve parallel to the given conic. Its evolute is the same as that of the conic. The following are the equations of the parallel curves given at fuU length, which may also be regarded as equations giving the length of the normal distances from any point to the curve. The parallel to the parabola is ,.6 _ (3^2 + 2.2 .^ g/nx - 8m2) r* + {Zy^ + y" {2x^ - 2mx + 20m-) + Sinx^ + im^x"- - Z2m'^x + 16»i^} r- - (j/"^ - imx)- [y- -\- {x - m)"] - 0. Tlie parallel to the ellipse is c4,.8 - 2c2r« {c2 («« H- &2) + (a2 - 2b-) x^ + (2ff2 - V) if\ + r< (c* (a* + ^o"-W- + &•") - 2e («■» - a^fi^ + Z¥) x^ + 2e- (3rt' - a-^lT- + h^) >/- + {a* - Qa"-b"- + Gb*) X* + {Ga* - Ga"-^- + b*) y* + {Ga* - lOa^-lt'^ + 66') x"-y-} + 7-2 {- 2«26V (a2 + J-) + 2c-x- {oa* - a-b- + b*) - 2c-y- {a* - a-b- + 3b*) - b^-x* {Ga* - 10a"-b^ + G¥) - a-y* {Ga* - lOa^J^ + Gb*) + a;-/ (4o« - Ga^b"^ - Ga-b* + 4b") + 2V- («2 _ 262) ^6 _ 2 (a* - (i^W- + 3¥) xY - 2 (3a* - a"-b"- + b*) a^y* + 2rt2 (^2 _ 2a-) y"} + {b-x- + a"-y^ - a-b-)'^ {(.c - c)^ + /} {(x + c)- + y-} - 0. Thus the locus of a point is a conic, if the sum of squares of its normal distances to the curve be given. If we form the condition that the equation in /•- should have equal roots, we get the squares of the axes multiplied by the cube of the evolute. If we make /• = 0, we find the foci appearing as points whose normal distance to tlie curve vanishes. This is to be accounted for by remembering that the distance from the origin vanishes of any point on either of the lines x- -i- y- — 0. Ex. 3. To find the equation of the evolute of an ellipse. Since two of the normals coincide which can be drawn through every point on the evolute, we have only to 32G INVARIANTS AND COVARIANTS express the condition that in Ex., Art. 370 the curves S and S' touch. Now when the term k- is absent from an equation, the condition that Ak^ + Q'k + A' should have equal roots, reduces to 27AA"- + 46'^ = 0. The equation of the evolute is therefore (a^x^ + hhf- - c^Y + 21ci^b"-c*xY- - 0. (See Art. 248). Ex. 4. To find the equation of the evolute of a parabola. We have here S — y^ — 4.mx, S' — 2xt/ + 2 {2m — x') y — imy', A-- 4to-, e = 0, e' = - 4m {2m - x), A' = Amy, and the equation of the evolute is 21 my" = 4 (a; — 2;?i)'. It is to be observed, that the intersections of S and S' include not only the feet of the three normals which can be drawn through any point, but also the point at infinity on y. And the six chords of intersection of S and S', consist of three chords joining the feet of the normals, and three parallels to the axis through these feet. Consecjuently the method used (Ex., Art. 370) is not the simplest for solving the corresponding problem in the case of the parabola. We get thus the equation found (Ex. 12, p. 203), but multiplied by the factor 4ot {2my + y'x — 2my') — y'^. 373. If S' break up into two right lines we have A' = Oy and we proceed to examine the meaning in this case of 9 and 9'. Let us suppose the two right lines to be x and y ; and, by the principles already laid down, any property of the invariants^ true when the lines of reference are so chosen, will be true in general. The discriminant of 8+2lcxi/ is got by writing h-\-k for h in A, and is A + 2k [frj — ch) — cJc^ Now the coefficient of Jc^ vanishes when c = 0, that is, when the point xi/ lies on the curve S. The coefficient of Jc vanishes when f(j = ch; that is (see Ex. 3, p. 204), when the lines x and y are conjugate with respect to S. Thus, then, when S' represents two rigid lineSy A' vanishes; 9' = represents the condition that the intersection of the two lines should lie on S; and Q = is the condition that the two lines should he conjugate with respect to 8. The condition that A -f 9^' + Q'k^ should be a perfect square Is 9'"' = 4A9', which, according to the last Article, is the condition that either of the two lines represented by S' should touch S. This Is easily verified in the example chosen, where 9^ — 4A9' is found to be equal to (^c— /") [ca — g'^). Ex. 1. Given five conies S^, S.,, &c. it is of course possible in an infinity of ways to determine the constants /j, Zj, &c. so that Z,-S. + hS, + 1,,% + ks, + ks, may be either a perfect square L-, or the product of two lines MN : prove that the lines L aU touch a fixed conic V, and that the lines M, N are conjugate with regard to V. We can determine V so that the invariant shall vanish for V and each of the five conies, since we have five equations of the form Aoi + Bb^ + Cci + 2FJ\ + 2Ggy + 2//Aj = 0, which are sufficient to deteiinine the mutual ratios of A. B. &c., the coefficients in OF SYSTEMS OF CONICS. 327 the tangential equation of V. Now if we have separately ^Iw, + &c. = 0, .!«._, + &c. = 0, Aa^ + &c, — 0, &c., we have plainly also A (7i«, + La^ + l^a^ + I^a^ + l^a^) + &c. = ; that is to say, 9 vanishes for V and every conic of the system 7,5. + I,S, + l,S, + ?,5, + J,.%, whence by tliis article the theorem stated immediately follows. If the line M be given, N passes through a fixed point; namely, the pole of J/ with respect to V. Ex. 2. If six lines x, y, z, u, v, w all touch the same conic, the squares are con- nected by a linear relation Iix- + l^"^ + h^' + h^- + h^- + h'"' = 0. This is a particiJar case of the last example ; but may be also proved as follows : "Write down the conditions. Art. 151, that the six hues should touch a conic, and eliminate the unknown quantities A, B, &c., and the condition that the lines should touch the same conic is found to be the vanishing of the determinant ^1^ /"i^ "i^ /^I'^i) "'Ad K/^i V) /f2^ "2^ IH^i, "2^2) \f^ K"^, Ms^ "s'l /'3'^3. "3^3. ^3^3 ^4^ ^4*. "i^J /«4''4> "4^4) ^4i"4 ^5^ M3^ "5^ M5''3) V^K, ^5^5 V) ^6") "6^) ^ei'e) "e^^e, Kf^e But this is also the condition that the squares should be connected by a linear relation. Ex. 3. If we are only given fom- conies Si, S^, S3, (S'4, and seek to determine V, as in Ex. 1, so that shall vanish, then, since we have only four conditions, one of the tangential coefiBcients A, &c. remains indeterminate, but we can detennine all the rest in terms of that ; so that the tangential equation of V is of the form S + i-2' = 0, or V touches four fixed lines. We shall afterwards show du'ectly that in four ways we can determine the constants so that liS^ + I2S2 + I3S3 + l^S^ may be a pei-fect square. It is easy to see (by taking for M the line at infinity) that if J/ be a given line it is a definite problem admitting of but one solution to determine the constants, so that IjSi + &c. shall be of the form J/iV. And Ex. 1 shows that N is the locus of the pole of J/ with regard to I^ Compare Ex. 8, p. 205. 374. To Jind the equation of the imir of tangents at the jyoi'nts ichere S is cut hy any line \x-\- fiy -'r vz. The equation of any conic having double contact with 8^ at the points where it meets this line, being hS-\- {kx -\- jxy + vzj = \ it is required to deter- mine k so that this shall represent two right lines. Now it will be easily verified that in this case not only A' vanishes but O' also. And If we denote by S the quantity AX' + Bfj:' + Cv' + iFixv + 2 GvX + 2H\fi ; the equation to determine k has two roots = 0, the third root being given by the equation h^ + S = 0. The equation of the pair of tangents is therefore ^S= A {Xx + fxy ■{ vzY- It is plain that when \x + fiy + vz touches S, the pair of tangents coincides 328 TNVAUIANTS AND COVARIAXTS with \x + fiJ/ + vz Itself; and the condition that this should be the case Is plainly 2 = 0; as Is otherwise proved (Art. 151). Under the problem of this Article Is included that of finding the equation of the asymptotes of a conic given by the general trillnear equation. 375. We now examine the geometrical meaning, in general, of the equation = 0. Let us choose for triangle of reference any self-conjugate triangle with respect to Sj which must then reduce to the form ax^ + 'by'' + cz' (Art. 258). We have there- fore /= Q^g = 0^h = 0. The value then of 6 (Art. 370) reduces to ica + cah' + abc\ and will evidently vanish if we have also a =0, h' = 0, c = 0, that is to say, if 8\ referred to the same triangle, be of the iorm. f'yz+g'zx-Yh'xy, Hence, 6 vanishes whenever any triangle inscribed in S' is self-conjugate with regard to S. If we choose for triangle of reference any triangle self- conjugate with regard to /S", we have /' = 0, g' = 0, h' = 0, and e becomes [be ~f) a' + {ca -/) b' + {ab - W) c ; and will vanish if we have be =/^, ca =g% oh = h^. Now be =f' is the condition that the line x should touch S] hence, 6 also vanishes if any triangle circumscribing S is self-conjugate with regard to S'. In the same manner it is proved that, e' = is the condition either that it should be possible to inscribe in S a tri- angle self-conjugate with regard to S\ or to circumscribe about S' a triangle self-conjugate with regard to S. When one of these things is possible, the other is so too. A pair of conies connected by the relation 6 = 0, possesses another property. Let the point In which meet the lines joining the corresponding vertices of any triangle and of its polar tri- angle with respect to a conic, be called the pole of either triangle with respect to that conic; and let the line joining the intersections of corresponding sides be called their axis. Then if 6 = 0, the pole with respect to S of any triangle inscribed in >S" will lie on >S" ; and the axis with respect to S' of any tri- angle circumscribing >S' will touch S. For eliminating «•, ^, z in turn between each pair of the equations ax + hy+gz = 0^ hx + by+fz = Oj gx+fy + cz = Oj we get {gh -af)x= [hf- bg) y = [fg- ch) z, OF SYSTEMS OF COXICS. 329 for tlie equations of the lines joining the vertices of the triangle xyz to the corresponding vertices of its polar triangle with respect to 8. These equations may be written Fx = Gy = IIz^ and the co-ordinates of the pole of the triangle are -r, , -r, , -77 • Substituting these values in S\ in which it Is supposed that the coefficients a', &', c vanish, we get 2i^' + 2 6*^^' + 2////,' = 0, or 9 = 0. The second part of the theorem is proved in like manner. Ex. 1. If two triangles be self -conjugate -vrith regard to any conic S', & conic can be described passing through their six vertices ; and another can be described touch- ing their six sides (see Ex. 7, p. 310). Let a conic be described through the three vertices of one triangle and through two of the other, which we take for a-, ;/, z. Then because it circumscribes the first triangle, 9' = 0, or a 4- 6 + c = (Ex. 2, Art. 371), and because it goes through two vertices of xyz, we have a = 0, b — Q, therefore c = 0, or the conic goes through the remaining vertex. The second part of the theorem is proved in like manner. Ex. 2. The square of the tangent drawn fi-om the centre of a conic to the circle circumscribing any self -conjugate triangle is constant, and = a- + b- [M. Faure]. This is merely the geometrical intei-pretation of the condition 9 — found (Ex. A, Art. 371), or a^ + ^ — r- = n- + b-. The theorem may be otherwise stated thus : "Every circle which circumscribes a self -con jugate triangle, cuts orthogonally the circle which is the locus of the intersection of tangents mutually at right angles." For the square of the radius of the latter circle is a- + b-, Ex. 3. The centre of the circle inscribed in eveiy self -con jugate triangle ^vith respect to an equilateral hyperbola, lies on the curve. This appears by making b- = — (fl in the condition 9' = (Art. 371, Ex. 4). Ex. 4. If the rectangle under the segments of one of tlie i>erpendiculars of the triangle formed by three tangents to a conic be constant and equal to M, the locus of the intersection of perpendiculai-s is the circle x- + y- — a- + b- + M. For 9 = (Ex. 4, Art. 371) is the condition that a triangle self-conjugate with i-egard to the circle can be circumscribed about S. But when a triangle is self -conjugate with regard to a circle, the intersection of peiiiendiculai's is the centre of the circle and M is the square of the radius (Ex. 3, p. 243). The locus of the intersection of rect- angular tangents is got from this example, by making M — 0. Ex. 5. If the rectangle under the segments of one of the pcrpcndiculai-s of a triangle inscribed in S be constant, and = M, the locus of intersection of perpen- diculai-s is the conic concentric and similar with S, »?= J/( — -I- tt.) [Dr. Hart]. This follows in the same way from 9' = 0. Ex. 6. Find the locus of the intersection of perpendiculai-s of a triangle inscribed in one conic and circumscribed about another [Mr. Burnside]. Take for origin the centre of the latter conic, and equate the values of M found from Ex. 4 and 5 ; then if a', h' be the axes of the conic S hi which the triangle is inscribed, the equation of the locus is x- + y" - a" -b- — -^'---t>7. S. The locus is therefore a conic, whose sixes a - -f- b- are parallel to those of .9, and which is a circle wlieu ,S^ is a ciivlo. UlT 330 INVARIANTS AND COYARIANTS Ex. 7, The centre of the circle circumscribing evei-y triangle, self-conjugate with regard to a parabola, lies on the directrix. This and the next example follow from e = 0, (Ex. 5, Art. 371). Ex. 8. The intersection of pcqiendicnlars of any triangle circumscribing a para- bola, lies on the directrix. Ex. 9. Given the radius of tlie cu-cle inscribed in a self -con jugate triangle, the locus of centre is a parabola of equal parameter with the given one. 376. If two conies be taken arbitrarily it Is In general not possible to inscribe a triangle in one which shall be circum- scribed about the other; but an infinity of such triangles can be drawn If the coefficients of the conies be connected by a certain relation which we proceed to determine. Let vis suppose that such a triangle can be described ; and let us take it for triangle of reference; then the equations of the two conies must be reducible to the form S = x' -f- f -j- z' - 2f/z - 2zx - 2x7j = 0, S' = 'Ifyz -h Igzx -f ^hxy = 0. Forming then the invariants, we have A = -4, e = 4(/+,7 + A), e' = -{f+g-\-h)'\ A' = 2fgh', values which are evidently connected by the relation 9''= 4Ae'.* This is an equation of the kind (Art. 371) which is unaffected by any change of axes ; therefore, no matter what the form in which the equations of the conies have been originally given, this relation between their coefficients must exist, if they are capable of being transformed to the forms here given. Con- versely, it Is easy to show, as in Ex. 1, Art. 375, that when the relation holds e''' = 4A0', then if we take any triangle circum- * This condition was first given by Mr. Cay ley [Philosoijliical Magazine, Vol. TI., p. 99) who derived it from the theoiy of elliptic functions. He also proved, in the same way, that if the square root of ^•'A -f ^-9 -f IcQ' + A', when expanded in powers of k, be A -f Bh + Ck- + &c., then the conditions that it should be possible to have a polygon of n sides inscribed in V and circumscribing V, are for n — .3, 5, 7, iSrc. respectively c=o, c, D A E and for w = 4, fi, S, tc. are /; = o. D, E 1 E, F = 0, c, A E A E, F E, F, G D. E, F E, F. G F. G. II = 0, &c., = 0. Ac. OF SYSTEMS OF CONICS. 331 scribing /S, and two of whose vertices rest on S\ the third must do so likewise. Ex. 1. Find the comlition that two circles may be such that a triangle can be inscribed in one and cu'cumscribed about the other. Let D- — j'^ — ?•'- — G ; then tlie condition is (see Ex. 3, Art. 371) ((? _ ,.2)2 + 4^.2 (G _ r'2) - 0, or {G + r^f = irh-'' ; whence D- = r'^ ± 2rr', Euler's well known expression for the distance between the centre of the circumscribing circle and that of one of the circles which touch the three sides. Ex. 2. Find the locus of the centre of a circle of given radius, cii'cumscribing a triangle cii-cumscribing a conic, or inscribed in an inscribed triangle. The loci are cui-ves of the foiu-th degree except that of the centre of the circumscribing circle in the case of the parabola, which is a circle whose centre is the focus, as is other- wise evident. Ex. 3. Find the condition that a triangle may be inscribed in ,S" whose sides touch respectively S+IS', S + mS', S + nS'. Let S = x^ + y'^ + Z'-2{l + If) yz-2{l + my) zx - 2 (1 + nh) xy, S' = 2fyz + 2gzx + 2hxy ; then it is evident that <S + ?>S" is touched by x, Ac. We have then A = - (2 + lf+ mcj + nlif - 2lmnfgh, = 2 (/+ g + h) (2 + lf+ mg + nh) + 2fgli {mn + nl + hn), G' = - (/+ fir + hf - 2 (? + wi + n)fgh, A' = 2fgh. Whence obviously {9 - A' {mil + nl + lm)Y = 4 (A + ImnA') [Q' + A' (/ + m + n)], which is the reqiiired condition. 377. To find the condition that the line \x + /^^ 4 vz should pass through one of the four points common to S and S'. This is, in other words, to find the tangential equation of these four points. Now we get the tangential equation of any conic of the system S+kS' by writing a + ka\ &c. for «, &c. in the tangential equation of S, or 2 = {be -f) V + [ca - g') /^'^ + [ah - h') v' + 2 [gh - af) fxv + 2 (hf- hg) v\ ■i-2{fg- ch) \/m = 0. We get thus S + A;4> + Jc'l' = 0, where * = [he + h'c - 2f') V + [ca + ca - 2gg) yi' + [aV + ah - 2hh') v' + 2 [gh' -^-gh - af - af) fxv + 2 [hf + hf- hg' - h'g) v\ + 2 [fg \fg - ch' - ch) \fi. The tangential equation of the envelope of this system is there- fore (Art. 298) 4>''' = 422'. But since S+hS', and the corre- sponding tangential equation, belong to a system of conies 332 INVARIANTS AND COVARIANTS passing through four fixed points, the envelope of the system is nothing but these four points, and the equation 4>''' = 42S' is the required condition that the line \x + fi?/ + vz should pass through one of the four points. The matter may be also stated thus : Through four points there can in general be described two conies to touch a given line (Art. 345, Ex. 4) ; but if the given line pass thi'ough one of the four points, both conies coincide in one whose point of contact is that point. Now 4>''' = 4SS' is the condition that the two conies of the system S+kS', which can be drawn to touch \x + /jl^ -\- vs, shall coincide. It will be observed that <I> = is the condition obtained (Art. 335), that the line \x + fiy + vZj shall be cut harmonically by the two conies. 378. To find the equation of the four common tangents to two conies. This is the reciprocal of the problem of the last Article, and is treated in the same way. Let 2 and 2' be the tangential equations of two conies, then (Art. 298) 2 + 1^'%' represents tan- gentially a conic touched by the four tangents common to the two given conies. Forming then, by Art. 285, the trilinear equation corresponding to 2 + ^'S' = 0, we get where 'F = {BC' + B'C-2FF')x'-^{CA'+C'A-2GG')if + {AB' + A'B-2lIH')z' + 2{GB' + G'E-AF'-A'F)yz + 2{HF' + H'F-BG'-B'G)zx + 2 [FG' -\-F'G- GIF - G'H) xy, the letters A^ B^ &c. having the same meaning as in Art. 151. But A/S+ZcF + ^'^A'/S' denotes a system of conies whose en- velope is F^ = 4AA'>S'/S"5 and the envelope of the system evi- dently is the four common tangents. The equation F'' = 4AA'/S'>S", by its form denotes a locus touching S and >S", the curve F passing through the points of contact. Hence, the eight points of contact of two conies with their common tangents^ lie on another conic F. Reciprocally, the eight tangents at the points of intersection of two conies envelope another conic ^. It will be observed that F = is the equation found, Art. 334, OF SYSTEMS OF CONICS. 333 of the locns of point^s, whence tangents to the two conies form a harmonic penciL* If S' reduces to a pair of right hnes, i^ represents the pair of tangents to S from their intersection. Ex. Find tlie eqnation of the four common tangents to the pair of conies ax- + ly- + cz- = 0, a'x- + l/y'- 4- c'z" = 0, Ilere A = be, B ~ ca, C = ab, whence F = "«' {be + b'c) X- + bU {ca' + c'rt) y- + cc' {ab' + u'b) 2^, and the required equation is \iia' {b'c + b'c) X- + bb' {ca' + c'a) y- + cc' {ab' + a'b) s^p = iabca'b'c' {ax- + by"^ + c.i-} {a'x- + b'y- + c's^, 379. The former part of this Chapter has sufficiently shown what is meant by invariants, and the last Article will serve to illustrate the meaning of the word covariant. Invariants and covariauts agree in this, that the geometric meaning of both is independent of the axes to which the questions are referred ; but invariants are functions ,of the coefficients only, while covariants contain the variables as well. If we are given a curve, or system of curves, and have learned to derive from their general equations the equation of some locus, Z7=0, whose relation to the given curves is independent of the axes to which the equations are referred, U is said to be a covariant of the given system. Now if we desire to have the equation of this locus referred to any new axes, we shall evidently arrive at the same result, whether we transform to the new axes the equation U= 0, or whether Ave transform to the new axes the equations of the given curves themselves, and from the trans- formed equations derive the equation of the locus by the same rule that U was originally formed. Thus, if we transform the equations of two conies to a new triangle of reference, by writing instead of a?, y^ z, Ix + my 4- nz^ I'z + m'y + n'z^ l"x -\- in"y + n"z ; and if we make the same substitution in the equation F''=iA A'aS'/S", we can foresee that the result of this last substitution can only differ by a constant multiplier from the equation Y^ = -^^^' SS\ formed with the new coefficients of ;S' and S'. For either form * I beUeve I was the first to dii-ect attention to the importance of this conic in the theoiy of two conies. 334 INVARIANTS AND COVARIANTS represents the four common tangents. On this property is founded the analytical definition of covarlants. "A derived function formed by any rule from one or more given functions is said to be a covarlant, if when the variables in all are trans- formed by the same linear substitutions, the result obtained by transforming the derived differs only by a constant multiplier from that obtained by transforming the original equations and then forming the corresponding derived." 380. There is another case in which it is possible to predict the result of a transformation by linear substitution. If we have learned how to form the condition that the line \x + [xy -\- vz should touch a curve, or more generally that it should hold to a curve, or system of curves, any relation independent of the axes to which the equations are referred, then it is evident that when the equations are transformed to any new co-ordinates, the corresponding condition can be formed by the same rule from the transformed equations. But it might also have been obtained by direct transformation from the condition first ob- tained. Suppose that by transformation \x ■\- [ly -\- vz becomes X, [Ix + my + nz) + [m [Tx + my + n'z) + v [l"x + m"y + n"z)j and that we write this \'x •+ /j,'y + v'z, we have X' = ?A 4 I' /J' + l"v, fjb' = m\ + m'fjb + ?n'V, v' = 7i\ -f n'/x, + n"v. Solving these equations, we get equations of the form \=LX^L'fi^L"v\ /x=MX'+3r/u,'+3rv\ v=NX'+N'/ji,'+A^"v'. If then we put these values Into the condition as first obtained in terms of X, yu., v, we get the condition in terms of V, yu,', v\ which can only differ by a constant multiplier from the condition as obtained by the other method. Functions of the class here considered are called contravariants. Contravariauts are like covarlants in this : that any contravariant equation, as for example, the tangential equation of a conic, {be —f^) X^ + &c. = can be transformed by linear substitution into the equation of like form (/'V— /'''^) V'' + &c. = 0, formed with the coefficients of the transformed trilinear equation of the conic. But they differ in that X, yu., v are not transformed by the same rule as x^ y^ z] that is, by writing for A,, /X4 mjx + nv^ &c., but by the different rule explained above. OF SYSTEMS OF CONICS. 335 The condition <I> = founcl, Art. 377, is evidently a contra- variant of the system of conies ^S*, S'. 381. It will be found that the equation of any conic co- variant with S and S' can be expressed in terms of S, S', and F ; while its tangential equation can be expressed in terms of S, 2', <P. Ex. 1. To express in terms of S, S', F the equation of the polar conic of >Si with respect to S'. From the nature of covariants and invariants, any relation found con- necting these quantities, when the equations are referred to any axes, must remain tnie when the equations are transformed. We may therefore refer S and <S' to their common self -conjugate triangle and write S = ax- + hif -h cz-, S' — x^ -'r y- + z^. It wOl be foimd then that "E — a {b + c) x- + b {c + a) y^ + c (« + V) z"^, Now since the condition that a line should touch S is bc)\? + ca/C^ + abv^ = 0, the locus of the poles ■«ith respect to S' of the tangents to *S is hcx"^ 4- cay- + abz- = 0. But this may be written {be + ca + ab) {x- + ^^ + 2^) = F. The locus is therefore (Ex. 1, Art. 371) QS' — 'p. In like manner the polar conic of S' with regard to /S is Q'S—'F, Ex. 2. To express in terms of S, S', F the conic enveloped by a line cut har- monically by S and S', The tangential equation of this conic 4» = is {b + c) X- + (c + a) ,j? +(« + &) 1/2 = 0, Hence its tiilinear equation is (c + a) {a + b) x--\- (« + b) (6 + c) 2/- + (c + a) {b + c) z- = 0, or {be + ca + ab) {x^ + y^ + z^) -{- {a -{- b + c) {ax- + by^ + cz-) - P = 0, or 05' + e'/S - F = 0. Ex. 3. To find the condition that F should break up into two right lines. It is abc {b + c) {c + a) {a + b) = 0, or abc {{a + b + c) {be + ca + nh) — abc] = 0, or AA' (GG' - AA') = 0, which is the requu-ed formula. GG' = AA' is also the condition that ^ should break up into factors. This condition wUl be found to be satisfied in the case of two circles which cut at right angles, in which case any line through either centre is cut har- monically by the circles, and the locus of points whence tangents form a hannonic pencil also reduces to two right Unes. The locus and envelope will reduce similarly if Z)2 = 2 (j-2 + r'2). Ex. 4. To reduce the equations of two conies to the forms .T- + y- + .-- = 0. rt.r^ + by- + cz- = 0. The constants a, b, c are determined at once (Ex. 1, Art. 371) as the roots of A/>:3 -Qk"- + Q'k - A' = 0. And if we then solve the equations x2 + 2^2 + g2 _ g^ (,x" + by" + cz- = S', a {b + c) x- -\- b {c + a) y- + c {a + b) z- = F, we find as-, y^, z- in tenns of the known functions S, S', F. Strictly speaking, wc ought to commence by dividing the two given equations by the cube root of A, since we want to reduce them to a form in which the discriminant of (S shall be 1. But it v.iU be seen that it will come to the same thing if leaving /Sand S' imclianged, we calculate F from the given coefficients and divide the result by A. Ex. ."). Reduce to the above form .B.<-2 - Cxy + 0//2 - 2x + \y - 0, o.r- - U.ry + 8/- - G.r - 2 = 0. 33G INVARIANTS AND COTAKTANTS It is convenient to begin by forming the coefficients of the tangential equation, A, B, &c. These are - 4, - 1, 18 ; - 3, 3, - 2 ; - IG, - 19, - 9 ; 21, 24, - 14. We have then A = - 9, e = - 54, e' = - 99, A' = - 54, whence (u, 6, c arc 1, 2, 3. We next calcnlate F which is - 9 {2ox" - 50a-.!/ + 44/ - 18a; + 12y - 4). Writing then X- + Y- + Z-= ox- - QxTj + 9/y- - 2x + 4i/, X^ + 2Y^ + 3Z-= 5a;2-14a;y+ 8/- 6x - 2, 5X2 + 8F2 + 9^2 = 23*2 _ ^Oxy + 44/ - 18a; + 12?/ - 4. Wc get from GS + S' - F, X^ = {3>/ + 1)-, from F - 3S - 2S', I'^ = (2x - yY; from 2^ + BS'-F, Z^ = - {z + y + ly-, Ex. C. To find the equation of the four tangents to /S at its intersections with S'. Ans. (eS- ASy = 4:AS{e'S-F). Ex. 7. A triangle is circumscribed to a given conic ; two of its vertices move on fixed right lines \x + fiy + vz, Vx 4- /u'y + v'z : to find the locus of the third. It was proved (Ex. 2, p. 239) that when the conic is s^ — xy, and the lines ax — y, hx — y. the locus is (a + 6)^ (2- — a;^) — {a — V)- z^. Now the right-hand side is the square of the polar with regard to S of the intersection of the lines, which in general would be F = (ax + by +gz) {fiv — fiv) + {hx + by +fz) {v\' — v'X) + {gx +fy + cz) {Xfj.' — X'fx) = ; and a + 5 = is the condition that the lines should be conjugate \\4th respect to aS', which in general (Art. 373) is = 0, where e = ^XX' + B/ifi.' + Cuv' + F (jxv' + n'p) + G {v\' + v'X) + H (X/x' + X» = 0. The particular equation, found p. 239, must therefore be replaced in general by Q--U+ AF'-^O. Ex. 8. To find the envelope of the base of a triangle inscribed in S and two of wliose sides touch S'. Take the sides of the triangle in any position for hues of reference, and let <S=2 {fyz+gzx+ lixy), S' = x"^ + y"^ + z^ — 2yz — 2zx — 2xy — 2hkxy, where x and ;/ ai-e the lines touched by S'. Then it is obvious that kS + S' will be touched by the third side s, and we shall show by the invariants that tliis is a fired conic. We have A = 2fgh, e = -{f+g+ hf - 2fghh, G' = 2 (/+ ^ + h) (2 + hk), A' = - (2 + hh)\ whence 6'- — 40A = 4AA'A-, and the equation kS + ;S' = may be written in the form (9'2 - 4eA) -S + 4AA'/S" = 0, which therefore denotes a fixed conic touched by the third side of the triangle. It is obvious that when 0'- = 40 A the third side will always touch S', Ex. 9. To find the locus of the vertex of a triangle whose three sides touch a conic U and two of whose vertices move on another conic V. We have slightly altered the notation, for the convenience of being able to denote by U' and V the results of substituting in U and V the co-ordinates of the vertex x'y'z'. The method we pursue is to form the equation of the pair of tangents to U through x'y'z' ; then to form the equation of the Imes joining the points where this pair of lines meets F; and, lastly, to form the condition that one of these lines (which must be the base of the triangle in question) touches V. Now if P be the polar of x'y'z', the pair of tangents is TJV - P-. In order to find the chords of intersection with V of the pair OF SYSTEMS OF CONICS. 337 of tangents, we form the condition that VU' - P"- + W may represent a pair of lines. This discriminant wU be found to give us the following quadratic for determining X, X'^A' + XF' + A U' V — 0. In order to find the condition that one of these chords should touch U, we must, by Art. 372, form the discriminant of yuf/ + {UV — P^ + XT), and then form the condition that this considered as a function of /x should have equal roots. The discriminant is ^i^A + fx {2U'A + Xe) + {U'^-A + X{QU' + AV') + X'-O'}, and the condition for equal roots gives X (4Ae' - 9=) + 4An" = 0. Substituting this value for X in X-A' + \F' + AU'V - 0, we get the equation of the required locus 16A3A'F- 4A (4 AG' - Q"-) F + U (4Ae' - 6-)^ = 0, whiclr, as it ought to do, reduces to V when 4A9' = 9-.* Ex. 10. Find the locus of the vertex of a triangle, two of whose sides touch U, and the third side aU + bV, while the two base angles move on V. It is found by the same method as the last, that the locus is one or other of the c&uios, touching the four common tangents of U and T'^, AA'X- V + \fiF + n-U = 0, where \ : fiis given by the quadratic a {ab - 13a) X^ + « (4Aa + 29i) X/i - &V = 0, where a - 4AA', ft = 0- - iA-Q'. Ex. 11. To find the locus of the free vertex of a polygon, all whose sides touch U, and all whose vertices but one more on V. This is reduced to the last ; for the line joining two vertices of the polygon adjacent to that whose locus is sought, touches a conic of the form aU+bV. It will be found if X', fi' ; X", fx." ; X'", fi'" be the values for polygons of « — 1, n, and w + 1 sides respectively, that X'" = MV"^ fx" — A'X'X" {a/j." — A'/3X"). In the case of the triangle we have X' = a, ^' = A'/J ; in the case of the quadiilateral X" = fi-, /x" = a (4Aa + 2/J9), and from these we can find, step by step, the values for every other polygon. (See rhUosophical .^layazine, Vol. XIII., p. 337). Ex. 12. The triangle formed by the polai-s of middle points of sides of a given triangle with regard to any inscribed conic has a constant area [M. Faurc]. Ex. 13. Fmd the condition that if the points in which a conic meets the sides of the triangle of reference be joined to the opposite vertices, the joining lines shall form two sets of thi-ee each meeting in a point. Ans. abc — y§h — af — by- — cli' — 0. 382. The theory of covarlants and invariants enables us readily to recognize the equivalents in trilinear co-ordinates of certain well-known formula in Cartesian. Since the general expression for a line passing through one of the imaginary circular points at infinity is x±y \J[—\) + c^ the condition that * The reader will find {Quarterly Journal of Mathemntkg, Vol. I., p. 344) a dis- cussion by Mr. Cayley of the problem to find the locus of vertex of a triangle circum- scribing a conic S, and whose base angles move on given curves. "When the curves are both conies, the locus is of the eighth degi-ee, and touches S at the iwints where it is met by the polars with regard to S of the intersections of the two conies. XX 338 INVARIANTS AND COVARIANTS \x -\- fiy ■\- V should pass through ono of these points is V + fx^ = 0. In other words, this is the tangential equation of these points. If then 2 = be the tangential equation of a conic, we may form the discriminant of S -f 7c (\^ + /a^) . Now it follows from Arts. 285, 286, that the discriminant in general of 2 + h'2! is A' + ^-Ae' + ^'^A'e + ^^A". But the discriminant of 2 + ^ (X^ + /*"') is easily found to be A' + 7<;A (a + J) + F {ah - ¥). If, then, in any system of co-ordinates we form the Invariants of any conic and the pair of circular points, 6' = is the con- dition that the curve should be an equilateral hyperbola, and 9 = that it should be a parabola. The condition {a + hf = 4.{ah-¥), or (a - Jf + 4A'^ = 0, must be satisfied if the conic pass through either circular point ; and it cannot be satisfied by real values except the conic pass through hoth^ when a = 5, A = 0. Now the condition X,^ + /i' = 0* implies (Art. 34) that the length of the perpendicular let fall from any point on any line passing through one of the circular points is always infinite. The equivalent condition in trilinear coordinates is therefore got by equating to nothing the denominator in the expression for the length of a perpendicular (Art. 61). The general tan- gential equation of the circular points is therefore X^ + yu," + v^ — 2/;tv cosu4 -2vX C03B-2\/m cosC=0. Forming then the Q and Q' of the system found by combining this with any conic, we find that the condition for an equilateral hyperbola, 6' = 0, is a + b-\- c — 2f cos A — 2g cosB— 2h cos (7=0; while the condition for a parabola, 9 = 0, Is A sm'A -\r B sm'B + C sin' C^2F sin5 sin G + 2(?sinCsin^ + 2Zrsin4 sln^=0. * This condition also implies (Art. 25) that every line dra^\'n through one of these two points is perjDendicular to itself. This accounts for some apparently irrelevant factors which appear in the equations of certain loci. Thus if we look for the equa- tion of the foot of the peii^endicular on any tangent from a focus a/3, {x— a)' + (j/ — fty' will appear as a factor in the locus. For the perpendicular from the focus on either tangent through it coincides with the tangent itself. This tangent therefore is part of the locus. OF SYSTEMS OF CONICS. 339 The condition that the curve should pass through either circular point is G'^ = 40, which can in various ways be resolved into a sura of squares. 383. If we are given a conic and a pair of points, the covariant F of the system denotes the locus of a point such that the pair of tangents through it to the conic are harmoni- cally conjugate with the lines to the given pair of points. When the pair of points is the pair of circular points at in- finity, F denotes the locus of the intersection of tangents at right angles. Now, referring to the value of F, given Art. 378, it is easy to see that when the second conic reduces to \^ + fj/'] that is, when A' = B'=lj and all the other coefficients of the tangential of the second conic vanish, F is C[x^ + f) -2Gx-2F7/ + A + B=0, which is, therefore, the general Cartesian equation of the locus of intersection of rectangular tangents. (See p. 258). When the curve is a parabola 0=0, and the equation of the directrix is therefore 2[Gx + Fxj) — A-\-B. The corresponding trilinear equation found in the same way is (i?+(7+2i^cos^)a;' + (a+^ + 26^cos^)/+(^+7?+2Zrcos(7)2' + 2 (^ cos^-F- G co%C-Hco&B)yz + 2 (J5 cosB - G - II cos A - F cos C) zx. + 2{C cosC -H-FcosB-G cosA)xij = 0. It may be shown, as in Art. 128, thgit this represents a circle, by throwing it into the form / . . . „ . ^, (B+C+2FC0&A C-\-A^2GcosB {xsmA-\-y s\nB-\zsmC] -. — -. x-\ -. — ^ y ^ ^ ' \ m\A s\nB ^ A+B-\-2lIcOsG \ e , . ^ . -n ■ n^ H -. — :^ z = -; — J—. — :rr-- — 7^ [vzsmA+zxsmB4-xy sm G ), smC J sm^smi^smC^*^ ^ ^' where is the condition (Art. 382) that the curve should be a parabola. When 9 = 0, this equation gives the equation of the directrix. 384. In general, 2 + /i;S' denotes a conic touching the four tangents common to 2 and 2' ; and when k is determined so that 2+/v2' represents a pair of points, those points arc two 340 INVARIANTS AND COVAKIANTS opposite vertices of tlic quadrilateral formed by the common tangents. In the case where S' denotes the circular points at infinity, when 2 -|- /cS' represents a pair of points, these points are the foci (Art. 279). If then it be required to find the foci of a conic, given by a numerical equation in Cartesian co-ordi- nates, we first determine Jc from the quadratic {ab - A'O U' + A (« + J) ^• + A' = 0. Then, substituting either value of h in 2 + ^ {X^ + jjl^)^ it breaks up into factors [\x' + iiy + vz) [\x" + /xy" + vz") ; and the foci t t n tf CC 'U tJC It are — , ^ ; -77 , ^, . One value of h gives the two real foci, z z z z and the other two imaginai*y foci. The same process is appli- cable to trilinear co-ordinates. In general, 2 + h (A,'*' + jji^) represents tangentially a conic confocal with the given one. Forming, by Art. 285, the corre- sponding Cartesian equation, we find that the general equation of a conic confocal with the given one is From this we can deduce that the equation of common tangents is [C {x' + f) -2Gx- 2Fy + A-V B\' ^^^S, By resolving this into a pair of factors [{^x-ay+[y-m[{^-^r+[y-m^ ■we can also get a, /S j a', /8' the co-ordinates of the foci. Ex.1, Find the foci of 2x" - 2.ry + 2>f - 2x — ^■ij + \l. The quadratic here is Si- + 4A-A + A2 = 0, whose roots are k = — A, k = - ^A, But A - - 9. Using the value k = 3, C\- + 2I/tt2 + 3j/2 + ISfJiv + 12:;\ + 30X/X + 3 (\2 + ^2) - 3 {X + 2fi + i>) {3\ + fji + p), sliDwing that the foci are 1, 2 ; S, 1. The value 9 gives the imaginary foci 2 + J(- 1), 3 + 4{- 1). Ex. 2. Find the co-ordinates of the focus of a jiarabola given by a Cartesian equation, Tlic quadratic here reduces to a simple equation, and we find that (a + h) {AX' + Bp? + 2Ffxv + 2Gv\ + 2H\p.} - A (X2 + /x^) is resolvable into factors. But these evidently must be (« + I.) i2GX + 2Fp) and ^-^^ i:,r«"^+ 2 (oi jR" ^ + "' The fh'st factor gives the infinitely distant focus, aad shows that the axis of the eui-ve is parallel to Fx — Gtj. The second factor shows that the co-ordinates of the focus ate the coeflicieiits of \ and /x in that factor. OF SYSTEMS OF CONICS. 341 Ex. 3. Find the co-ortlinates of the focus of a parabola given by the triUnear equation. The equation wliich represents the pair of foci is 9'2 = A {\- + /Jr + V- - 2/ii/ cosvl - 'IvX cos 5 - 2\u cosC). But the co-ordinates of the infinitely distant focus are known, from Art. 293, since it is the pole of the line at infinity. Hence those of the finite focus are 6'^- A e'B- A A ehxA + Hs.inB + G sinC" II smA + B sinB + F sinC" e'C- A G siaA + FsinB + C sinC" 385. The condition (Art. Gl) that two lines should be mutually perpendicular XA.' -f /j,fi' + vv — (/iv + /iV) cos^ — (vV + v'X,) cosi? — (X/x' + X'/a) cos C = 0, is easilj- seen to be the same as the condition (Art. 293) that the lines should be conjugate with respect to X^ + yu.'-' + v' - 2yu,v cos^-2vX cosB-2\fj, cosC=0. The relation, then, between two mutually perpendicular lines Is a particular case of the relation between two lines conjugate with regard to a fixed conic. Thus, the theorem that the three perpendiculars of a triangle meet in a point, is a particular case of the theorem that the lines meet in a point which join the corresponding vertices of two triangles conjugate with re- spect to a fixed conic, &c. It is proved [Geometry of Three Dimensions^ Chap. IX.) that, in spherical geometry, the two imaginary circular points at iufiuity are replaced by a fixed imaginary conic : that all circles on a sphere are to be considered as conies having double contact with a fixed conic, the centre of the circle being the pole of the chord of contact ; that two lines are perpendicular if each pass through the pole of the other with respect to that conic, &c. The theorems then, which in the Chapter on Projection, were extended by substituting, for the two imaginary points at infinity, two points situated anywhere, may be still further extended by substituting for these two points a conic section. Only these extensions are theorems suggested, not proved. Thus the theorem that the intersection of perpendiculars of a triangle inscribed in an equilateral hyperbola Is on the curve, suggested the property of conies connected by the relation = 0, proved at the end of Art. .375. 342 INVARIANTS AND COVARIANTS. It has been proved (Art. 303), that to several theorems concern- ing systems of circles, correspond theorems concerning systems of conies liaving double contact with a fixed conic. We give now some analytical investigations concerning the latter class of systems. 386. To form the condition that the line \x + fii/ + vz may touch S+ [\'x + /h't/ + v'zy\ We are to substitute in 2, a + X'", b + fji'\ &c. for o, hj &c. The result may be written 2 + [a {fxv - fjivf + &c.} = 0, where the quantity within the brackets is intended to denote the result of substituting in S fiv' — /iV, vV — v'X, \p! — Vyu, for a;, ?/, z. This result may be otherwise written. Tor it was proved (Art. 294), that {ax' + &c.) {ax' 4- &c.) - {axx -j- &c.)^ = A {yz -yz)' + &c. And it follows, by parity of reasoning, and can be proved in like manner, that {A\^ + &c.) [A\"' + &c.) - {AX\'+ &c.y = A {a {ixv - fi'vY+ &c.|, where AW' + &c. is the condition that the lines Xx + fMy + vz^ \'x + fi'y + v'z may be conjugate 5 or AXX!+Bfifi -\-Cvv' + F{[xv' + fjJv)+G{v'\' + v'\) + H {Xf^' + X'fx). If then we denote AX''' + &c. by 2', and AXX' -+ &c. by n ; and if we substitute for a [ixv — fi'v)' + &c. the value just found, the condition previously obtained may be written (A + 2') 2-n' = o. If we recollect (Art. 321) that X, /i, v may be considered as the co-ordinates of a point on the reciprocal conic, the latter form may be regarded as an analytical proof of the theorem that the reciprocal of two conies which have double contact, is a pair of conies also having double contact. This condition may also be put into a form more convenient for some applications, if instead of defining the line Xx + fxy + vz by the coefliclents \, /i, V, we do so by the co-ordinates of its pole with respect to /S', and if we form the condition that the line P' may touch S+ P"'\ where P' is the polar of xy'z'^ or axx -\-&c. Now the polar of x'y'z will evidently touch S when x!y'z' Is on the curve ; and OF SYSTEMS OF CONICS. 343 in fact If In S we substitute for \, /a, v ; 8^^ S,^, S^ the coefficients of cc, _y, z in the equation of the polar, we get AaS". And again two lines will be conjugate with respect to S, when their poles are conjugate ; and In fact if we substitute as before for X, /i, v in n we get Ai?, where R denotes the result of substituting the co-ordinates of either of the points x't/'z\ x'y"z\ in the equation of the polar of the other. The condition that P' should touch >S'+ F"' then becomes (1 + 8") S' = E\ 387. To find the condition that the two conies S^ (^'x + fi'y + v'z)'\ S-\- {\"x + iil'y + v"z)\ shoidd touch each other. They will evidently touch if one of the common chords, (X'a; + [jcy + v'z) ± {\"x + y"-"?/ -}- v"z), touch either conic. Substituting, then, in the condition of the last Article X' ± X" for X, &c., we get (A + r) (s' ± 2n + 2") = (2' + n)^ which reduced may be written in the more symmetrical form (A + S')(A + 2") = (A±n)^ The condition that S+P'' and S+P"'^ may touch Is found from this as in the last Article, and is [l + S'){l + S") = {l±E)\ Ex. 1. To draw a conic having double contact witli S and toucbing three given conies S + P'2, S + P"2, S + P"'-, also having double contact with S. Let xijz be the co-ordinates of the pole of the chord of contact with S of the sought conic S + P'-', then we have (1 + ^)(1 + ;S') = (1+PT; il-\-S){l + S") = {l + P'r-; {l + S){l+S''') = {l + P"r-; where the reader will obsei-ve that S', S", S'" are known constants, but S, P', &c, involve the co-ordinates of the sought point xf/;:. If then we write 1 -f aST = k", ic, we get L-k' =1 + P', M" = 1 -I- P", hk'" = 1-1- P"\ It is to be observed that P', P", P'" might each have been written with a double sign, and in taking the square roots a double sign may, of com-se, be given to k', k", k'". It will be found that these varieties of sigu indicate that the problem admits of thirty-two solutions. The equations last written give k {k' - k") = P' - P" ; k ik" - k'") = P" - P'" ; whence eliminating k, we get P' {k" - k'") + P" [k'" - k') + P'" {k' - k") = 0, the equation of a line on which must lie the pole with regard to S of the chord of contact of the sought conic. This equation is evidently satisfied by the point P' = P" — P"', Bat this point is evidently one of the ruilical centres (see p. 270) of the conies S + P'-, S + P"-, S + P"'\ 344 INVARIANTS AND CO VARIANTS p, pn pnt The equation is also satisfied by the point tj = 17; — 'jJTi • ^^ order to see the geometric interpretation of this we remark that it may be deduced from Art. 386 that the tangential equations oi S + P"^, S + P'"^ are respectively (1 + >S') 2 = A {\x' + fxij' + vz')-, (1 + S") 2 = A (\a;" + yuy" + vz")": Hence \x' + ,.^ + vz' _^ \x;^+ ^.v'^+j^ represent points of intei-section of common tangents to S + P'-, S + P"-, that is to .t' x" say, the co-ordinates of these points are t> ± p; , <tc., and the polars of these points, p' p" p' pii pill with respect to 5, are 77 + 7:77 • It follows that --7 = -jj, — =_,„ denote the pole, with respect to S, of an axis of similitude (p. 270) of the three given conies. And the theorem we have obtained is, — the pole of the sought chord of contact lies on one of the lines joining one of the four radical centres to the pole, with regard to S, of one of the four axes of similitude. This is the extension of the theorem at the end of Art. 118. To complete the solution, we seek for the co-ordinates of the point of contact of S+ P^ with S + P'2. Now the co-ordinates of the point of contaict, which is a centre of similitude of the two conies, being j — j^, &c., we must substitute x + j-,^' ^or k k k X, &c. in the equations kh' = 1 + J", itc, and we get kk' = 1 -f- P' -f. |, S' ; i/L" = 1 -h P" -t- 1^ /2 ; K-'" = 1 -I- P'" -(- 1 PJ, where 7?, R' are the results of substituting x"g"z", x"'ii"'z"' respectively in the polar of x'g'z'. We have then k {k' - k") = P'- P" + I (5' -R); k {k' - k'") = P' - P'" + ^{S'- R'), whence eliminating k, we have - {- - f - (- - f )} - - {-• - f - ('■ - I)} - - {- - F - (- - ")} . the equation of a line on which the sought point of contact must lie ; and which evidently joins a radical centre to the point where P', P", P"' are respectively pro- portional to k' — —■ , k" — J, , k'" ~ jr, or to 1, k'k" — R, k'k'" — R'. But if we form the equations of the polars, with respect to ^S" -I- P"^, of the three centres of simUitude as above, we get {k'k" -R) P' = P", (k'k'" -R')P' - P'", &c., showing that the line we want to construct is got by joining one of the four radical centres to the pole, with respect to S + P"^, of one of the four axes of similitude. This may also be derived geometrically as in Art. 121, from the theorems proved, p. 271. The sixteen lines wliich can be so di-awn, meet S + P'^ in the thirty-two points of contact of the different conies which can be drawn to fulfil the conditions of the problem.* * The solution here given is the same in substance (though somewhat simplified in the details) as that given by Mr. Cayley, Crelle, Vol. xxxix. Mr. Casey {Proceedings of the Royal Irish Academy, 18G6) has arrived at another solution from considerations of spherical geometiy. He shows by the method used, p. 113, that the same relation wliich connects the common tangents of foiu- circles touched by the same fifth connects also the sines of the halves of the common tan- OF SYSTEMS OF CONICS. 345 Ex. 2. The four conies having double contact \\-ith a given one S, which can bo drawn through three fixed pomts, are all touched bj' four other conies also having double contact with ^.f Let S = x- + 1/- + z- - 2tjz COS A - Izx cosB - 2xij coaC, then the four conies are S = ix±i/ ± zf, which are all touched by >Sr = {a; cos (B - C) + 2/ cos (C - ^) + 2 cos (.1 - 5)}^ and by the thi-ee others got by changing the sign of A, B, or C, in tliis equation. Ex. 3. The four conies which touch x, y, z, and have double contact with S are all touched by four other conies having double contact with S, Let 3I=l{A + B + C), then the four conies are S={x sin (J/- .-1) + y sin(.1/- B) + z sin (.1/ - C)}«, together with those obtained by changing the sign of yl, i?, or C in the above ; and one of the touching conies is Jx sin ^iB sin ^ 6' »/ sin ^ C sin ^A z sin \A sin \B^ ~ \ sinp ^ sin^B "*" siu^C J ' tlae others being got by changing the sign of x, and at the same time increasing B P.nd C by 180°, &e. Ex. 4. Find the condition that three conies V, T, W shall all have double contact with the same conic. The condition, as may be easily seen, is got by eliminating \, \x, V between AX3 _ Q-y-^ +e'X/i2 _ AV =: 0, and the two corresponding equations which expi-esa that (xV ~ vW, vW - \U break up into right Unes, 388. The theory of invariants and covariants of a system of three conies cannot be fully explained without assuming some knowledge of the theory of curves of the third degree. Given three conies U, F, TF, tJie locus of a point ichosc j^olars loith respect to the three meet in a point is a curve of the third cleqree ; which we call the Jacobian of the three conies. For we have to eliminate x^ y, z between the equations of the three polars U^x + V.^y + l\z = 0, T> + p;y + V^z = 0, Tl> + TT> 4- Tr> = 0, and we obtain the determinant u, ( V, w, - n it;) + u, ( t; if, - 1\ w,) + u, ( i; tf, - 1; tfj = o. It is evident that when the polars of any point with respect to gents of four such circles on a sphere ; and hence, as in Art. 121 (b), that if the equations of three circles on a sphere (see Geoinetry of Three Dimensions, Chap. IX.) be S - L'' = 0, S - M- = 0, S - N- = 0, that of a gi-oup of circles touching all thi-ee will be of the form 4[X (si - X)} + 4{|u (si - M)] + 4{v {S^ - X)} = 0. This evidently gives a solution of the problem in the text, but I have not succeeded in an-iving at it directly. The constants X, /u, v are, I believe, found by fonning for each pair of conies A - n - J{(A - 2') (A - S")}. ■|- This is an extension of Feuerbacli's theorem (p. 12G) and itself admits of further extension, Sec Quarterly Journal of Mathaiiatics, Tol. VI., p. 67. YY 846 INVARIANTS AND COVARIANTS U^ F, TT^mcet in a point, the polar with respect to all conies of the system JU-rmV+nW will pass through the same point. If the polars with respect to all these conies of a point A on the Jacobian pass through a point i?, then the line AB is cut harmonically by all the conies; and therefore the polar of B will also pass through A. The point B is, therefore, also on the Jacobian, and is said to correspond to A. The line AB is evidently cut by all the conies in an involution whose foci are the points -4, B. Since the foci are the points in which two corresponding points of the involution coincide, it follows that if any conic of the system touch the line AB^ it can only be in one of the points -4, B] or that if any break up into two right lines intersecting on AB^ the points of intersection must be either A or i?, unless indeed the line AB be itself one of the two lines. It can be proved directly, that if lU-\- onV+nW represent two lines, their intersection lies on the Jacobian. For (Art. 292) it satisfies the three equations whence, eliminating ?, m, n^ we get the same locus as before. The line AB joining two corresponding points on the Jacobian meets that curve in a third point ; and it follows from what has been said that AB is itself one of the pair of lines passing through that point, and included in the system lU -\- mV ■\- nW. The general equation of the Jacobian is - {(5c'A") + [hfg")]fz - [{ca'f)+[cg'h")] r'x- {[ho'g") + {ch'f'Wy -[[aVc") + 2[fg'h")]xyz = Q, where [ag'h") &c. are abbreviations for determinants. Ex. 1. Through four points to draw a conic to touch a given conic W, Let the four points be the intersection of two conies U, V; and it is evident that the problem admits of six solutions. For if we substitute a + ka', &c. for a in the condition (Art. 372) that U and W should touch each other, h, as is easily seen, enters into the result in the sixth degree. The Jacobian of U, V, W intersects W in the six points of contact sought. For the polar of the point of contact with regard to W being also its polar with regard to a conic of the form \U + fiV passes through the intersection of the polars with regard to U and V. Ex. 2. If three conies have a common self-conjugate triangle, their Jacobian i3 three right lines. For it ia verified at once that the Jacobian of ax^ + by^ + cz", OP SYSTEMS OP CONICS. 347 a' 3? + h'y^ + c'z^, «"«* + 6"j/2 + o"2- ia xyz = 0. We can hence find at once the equa- tion of the sides of the common self -conjugate triangle of two conies, by forming the Jacobian of S, S' and the covariant F; since this triangle is also self -conjugate with respect to p (Art. 381, Ex. 1). Comparing this with the result obtained by Art. S81, Ex. 4, we get the identical equation j2 _ J.3 _ p2 (9^/ + e'^S") + F (A'eS"- + AQ'S'') + (09' - 3AA') fSS' - A'-AS^ - AA'-'S'^ + A' (2Ae' - 8^) -S^/S' + A (2A'e - 0'^) SS'^. Ex. 3. If three conica have two points common, their Jacobian consists of a line and a conic through the two points. It is evident geometrically that any point on the line joining the two pouits fulfils the conditions of the problem, and the theorem can easily be verified analji;ically. In particular the Jacobian of a system of three ch'cles is the ckcle cutting the three at right angles. Ex. 4. The Jacobian also breaks up into a line and conic if one of the quantities /S be a pei-fect square i^. For then Z is a factor in the locus. Hence we can describe four conies touching a given conic S at two given points {S, L) and also touching S" ; the intersection of the locus with S" determining the points of contact. When the three conies are a conic, a circle, and the square of the line at infinity, the Jacobian passes through the feet of the normals v/hich can be drawn to the conic through the centre of the circle. 388a. To find the condition that a line \x + fiy -f vz should he cut in involution hy three conies. It appears from Art. 335 and from the Note, p. 298, that the required condition is the vanishing of the determinant cV —2gv\ +av% c/u.^ -'^fvf^ +^''" c\/a -f^'^ —9^1^ +^^^' c'X^ —2g'v\-\-a'v\ cyi^ —2/^/4 ■\-Vv\ c'\[i -f'v\ -ffvfi +h'v- c"X^—2g"yX+a"v', c'jm^- ■2f"vfi,-ih"v% c"\fi—f"v\-c/'vfx + h"v'' When this is expanded, it becomes divisible by v\ and may be written V {hc'f") + fi' [ca'g") + v' {ah'h") + X'> [2 [ch'f") - {hc'g")] + XV {2 [hfg") - {Ic'h")] + fi'X [2 {cg'h") - {ca'f"]} + /.V {2 {afg") - {cah")] + v'X [2 [hg'h") - {ah'f")] + vV {2 [ah'f") - [ah'g")] + V^ ((«J'c'') " ^ (i/'^*")} = 0, From the form of this condition, it is immediately inferred that any line cut in invokition by three conies Z7, T", Tl' is cut in involution by any three conies of the system lU^mV-\-nW. The locus of a point whence tangents to three conies form a system in involution, is got by writing cr, ?/, z for X, /x, v in the preceding, and the reciprocal coefficients J, i>, &c. instead of a, &, &c. 348 INVARIANTS AND COVAUIANTS 389. If WO form the discriminant of lU+viV+nW^ the co- efficients of the several powers of ?, m^ n will be invariants of the system of conies. All these belong to the class of invariants already considered, except the coefficient of hmi^ in which each term ahc of the discriminant of Uis replaced by ah'c" + ah"c' + a'b"c + dhc' + ahc + al'h'c^ &c. Another remarkable invariant of the system of conies, first obtained by a different method by Mr. Sylvester, is found by the help of the principle [Higlier Algebra^ p. 110), that when we have a covariant and a contravariant of the same degree, we can get an invariant by substituting differential symbols in cither, and operating on the other. By the help of the Jacoblan and the contravariant of the last article we get the invariant T, T= [aVd'y-Vi {ciVf") {ac'f") + 4 {hcf) {hag") + 4 {ca'k") {chli') + 8 Wfj") Wfj') + 8 {^<m {rfh") + 8 {cg'h") hg'h") - 8 [ag'K') [hc'f") - 8 [hh'f") {ca'g") - 8 (c/V/') {al'h") + Hah'c"){fg'h"]-8{Jr/hJ. 389a. Some of the properties of a system of three conies can be studied with advantage by expressing each in terms of four lines x, y, z^ 2v : thus U= ax^ 4- Ig' + cz^ -+ dw\ F= a'x^ + h'g" + cz" + d'io'\ W= a'x^ + h"y' + c'z' + d!'i6\ It is always possible, in an infinity of ways, to choose x^ ?/, z^ lo so that the equations can be brought to the above form: for each of the equations just written contains explicitly three in- dependent constants : and each of the lines a?, ?/, «, ^v contains implicitly two Independent constants. The form, therefore, just written puts seventeen constants at our disposal, while Z7, F, W contain only three times five, or fifteen, independent constants. The equations of four lines are always connected by a relation of the form w = \x -^^ jxy ■\- vz^ and we may suppose that the constants X, &e. have been included In a?, &c., so that this rela- tion may be written In the symmetrical form x-\-y ■\-z-\-w = Q. Let It be required now to find the condition that ?7, F, W may have a common point. Solving for x\ y\ z\ w^ between the equations i7=0, F=0, TF=0, and denoting by A^B^G^D OF SYSTEMa OF CONICS. 349 the four (letermlnants [he'd"), {dc'a"), [dab"), [ba'c"), wc get a;", ?/', z'^j 10^ proportional to A^ B, 6', D] and substituting in a; + ?/ + s + w = 0, we obtain the required condition or {A'+B'-\-C'+D'-2AB-2BC-2CA-2AD-2BD-2CDY = GiABCI). The left-hand side of this equation is the square of the invariant T already found ; the right-hand side ABCD is an invariant which we shall call J/, whose vanishing expresses the condition that it may be possible to determine ?, «?, n, so that Z£/'+?nF+ wTF shall be a perfect square. Since M is of the fourth degree in the coefficients of each conic, it follows that four conies of the system 8+111+ inV+7iW can be determined so as to be perfect squares (see Ex. 3, p. 327), for if we equate to nothing the invariant 31 found for S+IU, T", IF, we have an equation of the fourth degree for determining I. 3895. Any three conies may in general be considered as the polar conies of three points with regard to the same cubic ; or, in other words, their equations may all be reduced to the form a [x' - 2yz) + (3 (f - 2zx) + 7 [z' - 2xy) = 0. If we use for the equations of the conies the forms given in the last article, the equation of the cubic whence they ai-e derived will be x^ f z' w' ^ A^ B^ G^ D ' and it appears that if the invariant J/ vanish (in which case either A, B, C or D vanishes), an exception occurs, and the conies cannot all be derived from the same cubic. In the general case the equation of the cubic may be obtained by forming the Hessian of the Jacobian of the thi'ce conies, and subtracting the Jacobian itself multiplied by T. If we operate with the conies on the cubic conti'avariant, or with their reciprocals on the Jacobian, we obtain linear contravariants and covariauts which geometrically represent the points of which the given conies are polar conies, and the polar lines of these points with respect to the cubic. ( 350 ) CHAPTER XIX. THE METHOD OF INFINITESIMALS. 390. Eefereing the reader to other works where it is sliown how the differential calculus enables us readily to draw tangents to curves, and to determine the magnitude of their areas and arcs, we wish here to give him some idea of the manner in which these problems were investigated by geometers before the invention of that method. The geometric methods are not merely interesting in a historical point of view ; they afford solutions of some questions more concise and simple than those furnished by analysis, and they have even recently led to a beautiful theorem (Art. 400) which had not been anticipated by those who have applied the integral calculus to the recti- fication of conic sections. If a polygon be inscribed in any curve, it is evident that the more the number of the sides of the polygon is increased, the more nearly will the area and perimeter of the polygon approach to equality with the area and perimeter of the curve, and the more nearly will any side of the polygon approach to coincidence with the tangent at the point where it meets the curve. Now, if the sides of the polygon be multiplied ad infinitum^ the polygon will coincide with the curve, and the tangent at any point will coincide with the line joining two indefinitely near points on the curve. In like manner, we see that the more the number of the sides of a circumscribing polygon is increased, the more nearly will its area and perimeter approach to equality with the area and peri- meter of the curve, and the more nearly will the intersection of two of its adjacent sides approach to the point of contact of either. Hence, in investigating the area or perimeter of any curve, we may substitute for the curve an inscribed or circumscribing polygon of an indefinite number of sides; we may consider any tangent of the curve as the line joining two indefinitely near points on the curve, and any point on the curve as the inter- section of two indefinitely near tangents. THE METHOD OF INFINITESIMALS. 351 391. Ex. 1. To find the direction of the tangent at any point of a circle. In any Isosceles triangle A OB^ either base angle OB A Is less than a right angle by half the vertical angle ; but as the points A and B approach to coincidence, the vertical angle may be supposed less than any assignable angle, therefore the angle OBA which the tangent makes with the radius is ultimately equal to a right angle. We shall frequently have occasion to use the principle here proved, viz., that two indefinitely near lines of equal length are at right angles to the line joining their extremities. Ex. 2. The circumferences of two circles are to each other as their radii. If polygons of the same number of sides be Inscribed in the circles, it is evident, by similar triangles, that the bases «&, AB^ are to each other as the radii of the circles, and, therefore, that the whole perimeters of the polygons are to each other In the same ratio ; and since this will be true, no matter how the number of sides of the polygon be Increased, the circumferences are to each other In the same ratio. Ex. 3. The area of a circle is equal to the radius multiplied hy the semi-circumference. For the area of any triangle OAB Is equal to half Its base multiplied by the perpendicular on It from the centre ; hence the area of any Inscribed regular polygon is equal to half the sum of its sides multiplied by the perpendicular on any side from the centre ; but the more the number of sides Is increased, the more nearly will the perimeter of the polygon approach to equality with that of the circle, and the more nearly will the perpen- dicular on any side approach to equality with the radius, and the difference between them can be made less than any assignable quantity ; hence ultimately the area of the circle is equal to tlie radius multiplied by the semi-circumference; or = 7r/-". 392. Ex. 1. To determine the direction of the tangent at any 2wint on an ellipse. 352 THE METHOD OF INFINITESIMALS. Let P and F be two indefinitely near points on the curve, then FF+FF' = FF' + F'F'', or, taking FB = FF, FE = F'F\ Ave have FF = FE'', but in the tri- angles FRF\ FR'F\ we have also the base FF' common, and (by- Ex. 1, Art. 391) the angles FRF' FR'F' right ; hence the angle FFR = F'FR'. Now TFF is ultimately equal to FFF^ since their difference FFF' may be supposed less than any given angle ; hence TFF= T'FF\ or the focal radii make equal angles with the tangent. Ex. 2. To determine the direction of the tangent at any j^oint on a hyperhola. We have F'F-F'F^FF~FF, or, as before, FR = FR'. Hence the angle FF'R = FFE, or, the tangent is the internal bisector of the angle FFF. Ex. 3. To determine the direction of the tangent at any 2^oint of a parahola. We have FF=FN, and FF = F'N'', hence FR = F8, or the angle N'F'F= FFF. The tangent, there- fore, bisects the angle FFN. 393. Ex. 1. To find the area of the para- holic sector FVF. Since FS = FR, and FN= FF^ we have the triangle FFR half the parallelogram FSNI^'. Now if we take a number of points F'F'\ &c. between V and P, it is evident that the closer we take them, the more nearly will the sum of all the parallelogrrans FSXN\ &c., approach to equality with the area DVFN^ and the sum of all the tri- angles FFR^ &c., to the sector VFF] hence ultimately the sector FFV is half the area DVFN^ and therefore one-third of the quadrilateral DFFK THE METHOD OF INFINITESIMALS. 353 Ex. 2. To find the area of the segment of a paralola cut off hy any right line. Draw the diameter bisecting it, then the parallelogram PR' is equal to PM\ since they are the com- plements of parallelograms about the dia- gonal; but since TM is bisected at F', the parallelogram PN' is half PS'; if, therefore, we take a number of points P, P', P", &c., it follows that the sum of all the parallelograms PM' is double the sum of all the parallelograms PN\ and therefore ultimately that the space V'PM is double VPN] hence the area of the parabolic segment F'Pili" is to that of the parallelogram V'NPM in the ratio 2 : 3. 394. Ex. 1. The area of an eUijyse is equal to the area of a circle ivhose radius is a geometric mean between the semi-axes of the ellipse. For if the ellipse and the circle on the transverse axis be divided by any number of lines parallel to the axis minor, then since mh : md : : nih' : m'd' : : 5 : a, the quadrilateral mhh'm Is to mdd'm in the same ratio, and the sum of all the one set of quad- rilaterals, that is, the polygon Bhh'U'A inscribed in the ellipse is to the corresponding polygon Ddd'd"A inscribed In the circle, in the same ratio. Now this will be true whatever be the number of the sides of the polygon : If we suppose them, therefore, increased indefinitely, we learn that the area of the ellipse is to the area of the circle as Z» to « ; but the area of the circle being = 7ra", the area of the ellipse = irab. Cor. It can be proved, In like manner, that if any two figures be such that the ordinate of one is in a constant ratio to the corresponding ordinate of the other, the areas of the figures are in the same ratio. zz 354 THE METHOD OF INFINITESIMALS. Ex. 2. Every cHameter of a conic bisects the area enclosed hy the curve. For if we suppose a number of ordinates drawn to this dia- meter, since the diameter bisects them all, it also bisects the trapezium formed by joining the extremities of any two adjacent ordinates, and by supposing the number of these trapezia in- creased without limit, we see that the diameter bisects the area. 395. Ex. 1. The area of the sector of a hyperhola made hy joining any two "points of it to the centre^ is equal to the area of the segment made hy drawing i^arallels from them to the asymjitotes. For since the triangle PKQ= QLC^ the area PQC=PQKL. Ex. 2. Any two segments PQLK^ RSNM^ are equal, if PK : QL :: RM : SN. yy For // PK : QL '.'. CL : CK, A^ but (Art. 197) ^X/o \ CL = MT\ CK=NT- /^^y^'Vv \ we have, therefore, /yy/C- ---^?^^-~ -N.^ BM : SN:: 31 T : NT, c kl """"""" mT^T ~ and therefore QR is parallel to PT. We can now easily prove that the sectors PCQ, RCS are equal, since the diameter bisect- ing PS, QR will bisect both the hyperbolic area PQRS, and also the triangles PCS, QCR. If we suppose the points Q, R to coincide, we see that we can bisect any area PKNS by drawing an ordinate QL, a geo- metric mean between the ordinates at its extremities. Again, if a number of ordinates be taken, forming a continued geometric progression, the area between any two is constant. 396. The tangent to the interior of two similar, similarly placed, and concentric conies cuts off a constant area from the exterior conic. For we proved (p. 213) that this tangent is always bisected at the point of contact ; now if we draw any two tangents, the angle AQA' will be equal to BQD' and the nearer we suppose the point Q to P, the more nearly will the sides AQ,A'Q approach to equality with the sides BQ, B'Q; if, therefore, the two THE METHOD OF INFINITESIMALS. 355 tangents be taken indefinitely near, the triangle A QA' will be equal to £QB', and the space A VB will be equal to A' VB' ; since, therefore, this space remains constant as we pass from any tangent to the consecutive tangent, it will be constant whatever tangent we draw. COE. It can be proved, in like manner, that if a tangent to one curve always cuts off a constant area from another, it will be bisected at the point of contact ; and, conversely, that if it be always bisected it cuts off a constant area. Hence we can draw through a given point a line to cut off from a given conic the minimum area. If it were required to cut off a give7i area it would be only necessary to draw a tangent through the point to some similar and concentric conic, and the greater the given area, the greater will be the distance between the two conies. The area will therefore evidently be least when this last conic passes through the given point ; and since the tan- gent at the point must be bisected, the line through a given point which cuts off the minimum area is bisected at that point. In like manner, the chord drawn through a given point which cuts off the minimum or maximum area from any curve is bisected at that point. In like manner can be proved the following two theorems, due to the late Professor MacCullagh. Ex. 1. If a tangent AB to one curve cut off a constant arc from another^ it is divided at the iwint of contact^ so that AP : PB in- versely as the tangents to the outer curve at A and B. Ex. 2. If the tangejit AB he of a constant lengthy and if the 'perjyendicular let fall on AB from the intersection of the tangents at A and B meet AB in J/, then AP ivill = 3IB. 397. To find the radius of curvature at any point on an ellipse. The centre of the circle circumscribing any triangle is the intersection of perpendiculars erected at the middle points of the sides of that triangle ; it follows, tlie:*efore, that the centre of the circle passing through three consecutive points on the curve is the intersection of two consecutive normals to the curve. Now, given any two triangles FIF\ FP'F\ and PX^ P'X, the two bisectors of their vertical angles, it is easily proved, by elementary geometry, that twice the angle PXP'=PFP' + PFP', (See the first figure, p. 352). 356 THE METHOD OF INFINITESIMALS. Now, since the arc of any circle is proportional to the angle it subtends at the centre (Euc. VI. 33), and also to the radius, (Art. 391), if we consider PP' as the arc of a circle, whose centre PP is N^ the angle PNP' is measured by -pTv- In like manner, PR taking FR = FP^ PFP is measured by -^^ , and we have 2PP PN PR P'R\ FP ^ FP' ' but PR = P'E = PP smPP'F', therefore, denoting this angle by 0, PN by R, FP^ F'P, by />, /»', we have 2 11 ^ sin^ p p ' Hence it may be inferred that the focal chord of curvature is clouhJe the harmonic mean hetiveen the focal 7'acUi. Substituting p for sin 6 J 2a for p 4 p', and h'^ for pp\ we obtain the known value, ah ' The radius of curvature of the hyperbola or parabola can be investigated by an exactly similar process. In the case of the parabola we have p' infinite, and the formula becomes 2 _ 1 R s'md p ' I owe to Mr. Townsend the following investigation, by a different method, of the length of the focal chord of curvature : Draw auT/ parallel QR to the tangent at P, and describe a circle through PQR meeting the focal r ^ chord PL of the conic at C. Then, by -—==--= the circle PS.SC=QS.SR, and by q^ the conic (Ex. 2, p. 179) PS.SL : QS.SR :: PL : illzY; therefore, whatever be the circle, SC: SL'.'.MN:PL', but for the circle of curvature the points S and P coincide, therefore PC \ PL w MN : PL', or, the THE METHOD OF INFINITESIMALS. 857 focal chord of curvature is equal to tlie focal chord of the, conic drawn parallel to the tangent at the point (p. 219, Ex. 4). 398. The radius of curvature of a central conic may other- wise be found thus : Let Q be an indefinitely near point on the curve, QU a parallel to the tangent, meeting the normal in 8\ now, if a circle be de- scribed passing through P, Q^ and touching PT at P, since QS is a per- pendicular let fall from Q on the diameter of this circle, we have PQ^=PS multiplied by the diameter ; PQ' or the radius of curvature =^-po. Now, since QR is always drawn parallel to the tangent, and since PQ must ultimately coincide with the tangent, we have PQ ultimately equal to QR ; but, by the property of the ellipse (if we denote CP and its conjugate by «', Z*'), h" : therefore :: QR':PR.RP'{=2a'.PR), 2h'\PR QR' = J'2 PPl Hence the radius of curvature = — r . ^,7 . a Pb small PR^ PS are taken, we have, by similar triangles, their Now, no matter how ratio PR Hence radius of curvature = h" CP _c P8~7JT~'p p It is not difficult to prove that at the intersection of two con- focal conies the centre of curvature of either is the p>ole with respect to the other of the tangent to the former at the intersection. 399. If two tangents he draivn to an ellipse from any point of a confocal ellipse^ the excess of the sum of these two tangents over the arc intercepted hetiveen them is constant.^ For, take an indefinitely near point J", and let fall the per- pendiculars TP, P'/S', then (see figure next page) PT= PR = PP' + P'R * This beautiful theorem was discovered by Dr. Graves. See his Translation of ChasUss Mcnwirs on Cones and Spherical Conies, p. 77, 358 THE METHOD OF INFINITESIMALS. (for P'R may be considered as the continuation of the line PF) ; in like manner -^-T' Q'T' = QQ + QS. Again, since, by Art. 194, the angle TT'R=TT8, we have TS=TTi', and therefore PT+TQ' = PT' + T'Q'. Hence {PT+ TQ) - {F T + T Q) = PF-QQ' = PQ- F Q. Cor. The same theorem will be true of any two curves which possess the property that two tangents, 7LP, TQ^ to the inner one, always make equal angles with the tangent TT to the outer. 400. If two tangents he drawn to an ellipse from any point of a confocal hyperhola^ the difference of the arcs PK^ QK is eqiial to the difference of the tangents TP^ TQ." For it appears, precisely as before, TF' - FK over TP-PK= T'P, and that the excess of T'Q'-Q'K over TQ - QK is TS, which is equal to T'R, since (Art. 189) TT bisects the angle P TS. The dif- ference, therefore, between the that the excess of excess of TP over PA'', and that of TQ over QK^ is constant ; but in the particular case where T coincides with A", both these ex- cesses, and consequently their dif- ference, vanish; in every case, therefore, TP- PK=TQ—QK. Cor. FagnanVs theorem^ " That an elliptic quadrant can be so divided, that the difference of its parts may be equal to the difference of the semi-axes," follows immediately from this Article, since we have only to draw tangents at the extremities of the axes, and through their intersection to draw a hyperbola * This extension of the preceding theorem was discovered by Mr, MacCullagh. Dublin Exam, Pajiers, 1841, p. 41 ; 1842, pp. 68, 83. M. Chasles afterwards inde- pendently noticed the same extension of Dr. Graves's theorem. Comjites Jiendus, October, 1843, torn, xvii., p. 838. THE METHOD OF INFINITESIMALS. 359 confocal with the giveu elhpse. The co-ordinates of the points where it meets the elhpse are found to be X = 7 , y = 7 . 401. If a polygon circumscribe a conic^ and if all the vertices hid one move on confocal conies^ tlie locus of the remaining vertex will he a confocal conic. In the first place, we assert that if the vertex T of an angle PTQ circumscribing a conic, move on a confocal conic (see fig.. Art. 399) ; and if we denote by «, 5, the diameters parallel to TP, TQ ; and by a, /S, the angles TPT\ TQ'T\ made by each of the sides of the angle with its consecutive position, then a% = h^. For (Art. 399) TR = T'S] but TR = TP.a', T'S=T'Q'./3, and (Art, 149) TP and TQ are proportional to the diameters to which they are parallel. Conversely, if aa = 5jS, T moves on a confocal conic. For by reversing the steps of the proof we prove that TR = T'S\ hence that TT makes equal angles with TP, TQ^ and therefore coincides with the tangent to the confocal conic through T] and therefore that J" lies on that conic. ^^ If then the diameters parallel to the sides of the polygon be a, 5, c, &c., that parallel to the last side being J, we have aa=i/3, because the first vertex moves on a confocal conic; in like manner 5/3 = cy, and so on until we find aa = fZS, which shows that the last vertex moves on a confocal conic* * This proof is taken fi-om a paper by Dr. Hart ; Cambridge and Dublin Mathe- matical Journal, Vol. iv,, 193. NOTES. Pascal's Theorem, Page 235. M. Steiner was the first ■who (in Gergonne's AnnaUs) directed the attention of geometers to the complete figure obtained by joining in every possible way six points on a conic. M. Steiner's theorems were coiTected and extended by M. PlUcker (CreZ/e's Journal, Vol. v., p. 274), and the subject has been more recently investigated by Messrs. Cayley and Ku-kman, the latter of whom, in particular, has added several new theorems to those already known (see Cumh'idge and Dublin Mathematical Journal, Vol. v., p. 185). We shall in this note give a sUght sketch of the more important of these, and of the methods of obtaining them. The greater part are derived by joining the simplest principles of the theory of combinations with the following elementary theorems and their reciprocals : " If two triangles be such that the lines joining con-esponding vertices meet in a point {the centre of homology of the two triangles), the intersections of corresponding sides will lie in one right line (their axis)." "If the intersections of opposite sides of three triangles be for each pair the same three points in a right line, the centres of homology of the first and second, second and third, third and first, \vill he in a right line." Xow let the six points on a conic be a, b, c, d, e, f, which we shall call the points P. These may be connected by fifteen right lines, ab, ac, &c., which we shall call the lines C. Each of the lines C (for example ab) is intersected by the fomteen others ; by four of them in the point a, by four in the point b, and consequently by six in points distinct from the points P (for example the points {ab, cd), &c.) These we shall call the points j}. There are forty-five such points ; for there are six on each of the hnes C. To find then the number of points j}, we must multiply the number of lines C by 6, and divide by 2, since two Imes C pass through eveiy point p. If we take the sides of the hexagon in the order abcdef, Pascal's theorem is, that the three p points, {ab, de), {cd, fa), {be, ef), lie in one right line, which we may call fab.cd.ef] either the Pascal abcdef, or else we may denote as the Pascal -j , ^ h I' ^ form which we sometimes prefer, as showing more readily the three points tlu-ough which the Pascal passes. Through each point p four Pascals can be drawn. Thus through {ab, de) can be dra^^m abcdef, abfdec, abcedf abfedc. "We then find the total number of Pascals by midtiplying the number of points p by 4, and dividing by 3, since there are thi-ee points j^ on each Pascal. We thus obtain the number of Pascal's lines = 60. We might have derived the same dii'ectly by considering the number of different ways of aiTanging the letters abcdef. Consider now the three triangles whose sides are ab, cd, ef, (1) de, fa, be, (2) cf, be, ad. (3) NOTES; 361 iThe intersections of corresponding sides of 1 and 2 lie on the same Pascal, therefore the lines joining corresponding vertices meet in a f)oint, but these are the three Pascals, (ab .de.cf] (cd .fa . he l \ef . be . ad\ \cd .fa Mi' \ef. be . adj ' [ab . de . cfj ' This is Steiner's theorem (p. 235) ; we shall call this the g point, Iab.de .cf^ cd .fa . be ef.bc. ad) The notation shows plainly that on each Pascal's Une there is only one g point ; for (ab . de . c/*^ given the Pascal i ,' ~ ' , \ the g point on it is found by writing under each term the two letters not already found in that vertical line. Since then three Pascals intersect in every point g, the number of points g — 20. If we take the triangles 2, 3 ; and 1, 3 ; the lines joining corresponding vertices are the same in all cases : therefore, by the reciprocal of the second preliminary theorem, the three axes of the fab .cd .ef\ three triangles meet in a point. This is also a g point < de .fa . be V , and Steiner \cf . be . ad) has stated that the two g points just \\Titten are harmonic conjugates with regard to the conic, so that the 20 g points may be distributed into ten pairs. The Pascals which pass through these two g points correspond to hexagons taken in the order respectively, abcfed, afcdeb, adcbef; abcdef, afcbed, adcfeb; three alternate vertices holding in all the same position. Let us now consider the triangles, ab cd ef (1) ab.ce.df] cd.bf.ae] ef.bd.ac]^ de.bf.ac)' af.ce.bdi' bc.ae.dfj' ^' ab.ce.df^ cd.bf.ae'] ef .bd.ac"\ cf.bd.ae]' be.ac.df]' ad.ce.bfr ^'^'' The intersections of corresponding sides of 1 and 4 are three points which lie on the same Pascal ; thei"efoie the lines joining con^esponding vertices meet in a point. But these are the three Pascals, ab.ce.df 't cd.bf.ae^ ef.ac.bd] cd.bf.ae)' ef.ac.bd J ' ab.df.ce) ab . ce . df\ "We may denote the point of meeting as the h point, cd.bf.ae > , ef.ac. bd) The notation differs from that of the g points in that only one of the vertical columns contains the six letters without omission or repetition. On every Pascal there are three h points, viz., there ai-e on ab . cd .ef^ ab . cd . ef \ ab .cd . ef ^ '^ ff}' '^^•"f-'"'\> de.af.bc>, de.af.'bc]-, •^ ' cf.bd.aeJ ac.be.dfJ bf, ce.ad) where the bar denotes the complete vertical column. We obtain then Mr. Kirkman'a extension of Steiner's theorem: — The Pascals intersect three by three, not only in Steiner^s twenty points g, but also in sivty other points h. The demonstration of Art. 268 appUes alike to Mr. Kirkman's and to Steiner's theorem. In like manner if we consider the triangles 1 and 5, the lines joining corresjwnding vertices are the same as for 1 and 4 ; therefore the corresponding sides intersect on a right line, as they manifestly do on a Pascal, In the same manner the corie- AAA 362 NOTES. spending sides of 4 and 5 must intei'sect on a right line, but tliese intersections iire the thi-ee h points, (lb . ce . (If \ ae . cd . bf \ (ic . bd . ej' de . bf .(ic ]■ , bd . (if. ce > , df. (te . be cf.(ie.bd) ac.be.df) ce.bf.ad Moreover, the axis of 4 and 5 must pass through the intersection of the axes of ab .cd. ef\ 1, 4, and 1, 5, namely, through the (/ point, de . af. be cf. be , ad) In this notation the g point is found by combining the complete vertical columns of the three h points. Hence we have the theorem, " There are twenty lines G, each of rchich passes through one g and three h points." The existence of these lines was observed independently by Mr. Cayley and myself. The proof here given is Mr. Cayley'p. It can be proved similarly that "The twentt/ lines G pass four by four through fifteen points i." The four lines G whose g points in the preceding notation have a common vertical column will pass through the same point. Again, let us take three Pascals meeting in a point h. For instance, ab.ce.df] de.bf.ac] cf.ae.bd) de.bf.ae)' cf.ae.bdy ab.df.cei' We may, by taking on each of these a point 2h form a triangle whose vertices are {(If ac), {bj, ae), {bd, ce), and whose sides are, therefore, ac . bf. de'] bJ .ce . ad] bd . nc . ef] df. (le . cbj ' ae . bd . cf) ' ce . df. ab) ' Again, we may take on each a point h, by writing under each of the above Pascals (f .cd.be, and so form a triangle whose sides are ac . bf. de i cf.ae. bd'] df. ab . ce "1 be . cd . af) ' be . cd . af) ' be . cd . af) ' But the intersections of corresponding sides of these triangles, which must therefore be on a light line, are the three g points, be . cd . af \ be . cd . af \ be . cd . af '\ be . cd . af ^ ac . bf. de I , cf. ae . bd y , df. ab . ce r , cf .ab . de > , (If. ae . be J ad . bf. ce J ac . ef . bd ) ad . ef. be ) I have added a fourth g point, which the symmetry of the notation shows must lie on the same right line ; these being all the g points into the notation of which be .cd . ff/'can enter. Now there can be formed, as may readily be seen, fifteen different products of the form be . ed . af; we have then Steiner's theorem, The g points lie four by four on ffteen right lines I. Hesse has noticed that there is a certain reci- procity between the theorems we have obtained. There are 60 Kirkman points /(, and GO Pascal lines // corresponding each to each in a definite order to be explained presently. There are 20 Steiner points g, through each of wliich passes three Pascals // and one line G ; and there are 20 hues G, on each of which lie three Kirkman points h and one Steiner g. And as the twenty lines G pass four by four through fifteen points i, so the twenty points g lie four by four on fifteen lines /, The following investigation gives a new proof of some of the preceding theorems and also shews what h point corresponds to the Pascal got by taking the vertices in the order abcdef. Consider the two inscribed triangles ace, bdf; their sides touch a conic (see Ex. 4, p. 308) ; therefore we may apply Brianchon's theorem to the bexiigon whose sides are ce, (If, ae, bf, ac, bl. Taking them in this order, the dia- NOTES. 363 ce . hf. «f/) gonals of the hexagsn are the three Pascals intersecthig in the h point, df. ac . be ae .hd .cf . And since, if retaining the alternate sides ce, ae, ac, we permntate cyclically the other three, then by the reciprocal of Steiner's theorem, the three resulting Brianchon points lie on a right line, it is thus proved that three h points lie in a right line (r. From the same circumscribing hexagon it can be inferred that the lines joining the point a to {be, dj"] and d to {ac, ef} intersect on the Pascal abcdef, and that there are six such intersections on every Pascal. More recently Mr. Cayley has deduced the properties of this figure by consider- ing it as the projection of the lines of intersection of six planes. See Quarterly Journal, Vol, IX., p. 348. Systems op Tangential Co-ordinates, Page 275. Through this volume we have ordinarily understood by the tangential co-ordinates of a line la + n;)3 + «y, the constants I, m, n in the equation of the line (Art. 70) ; and by the tangential equation of a curve the relation necessary between these constants in order that the line should touch the curve. "We have preferred this method because it is the most closely connected with the main subject of this volume, and because all other systems of tangential co-ordinates may be reduced to it. We wish now to notice one or two points in this theory which we have omitted to mention, and then briefly to explain some other systems of tangential co-ordinates. "We have given (Ex. C, p. 128) the tangential equation of a circle whose centre is a'jS'y' and radius r, viz. {la + mft' + ny'Y = r"^ {P + m- + n- — Imn cos A - 2nl cos 5 - 2/ot cosC) : let us examine what the right-hand side of this equation, if equated to nothing, would represent. It may easily be seen that it satisfies the condition of resolvability into factors, and therefore represents two points. And what these points are may be seen by recollecting that this quantity was obtained (Art. 41) by writing at full length la + TO/3 + ny, and taking the sum of the squares of the coefficients of x and ;/, I cos a + m cos (i + n cos y, I sin a + m sin /3 + « sin y. Now if «- + 4- = 0, the line ax + by + c is parallel to one or other of the lines x±y 4(— 1) = 0, the two points therefore are the two imaginary points at infinity on any circle. And this appears also from the tangential equation of a circle which we have just given : for if we call the two factors w, w', and the centre o, that equation is of the form a^ = r-woo', showing that w, w' are the points of contact of tangents from a. In like manner if we form the tangential equation of a conic whose foci are given, by expressing the condition that the product of the perpendicidars fi-om these points on any tangent is constant, we obtain the equation in the form {la' + w/3' + ny') {la" + mji" + ny") — b'-totu', showing that the conic is touched by the lines joining the two foci to the points w, lo' (Art. 279). It appears fi-om Art. 61 that the result of substituting the tangential co-ordinates of any line in the equation of a point is proportional to the perpendicular from that point on the line ; hence the tangential equations a/3 = kyo, ay = k^ when inter- preted give the theorems proved by reciprocation Art. 311. If we substitute the co-ordinates of any line in the equation of a circle given above, the result is easily seen to be proportional to the square of the chord intercepted on the line by the circle. Hence if 2, S' represent two circles, we learn by iuteq^reting the equation 2 = k"'2.' that the envelope of a line on which two given circles intercept chords having to each other a constant ratio is a conic touching the tiuigents common to the two circles. 364 NOTES. Lastly, it ia to lie remarked that a system of two points cannot be adequately represented by a trilinear, nor a system of two lines by a tangential equation. If^ we are given a tangential equation denoting two points, and form, as in Art. 285, the corresponding trilinear equation, it will be found that we get the square of the equation of the line joining the pouits, but all trace of the points themselves has dis- appeared. Similarly if we have the equation of a pair of lines intersecting in a point "'/^'y'j the corresponding tangential equation will be found to be {la + mj3' + ny')^ — 0. In fact, a line analytically fulfils the conditions of a tangent if it meets a curve in two coincident points ; and when a conic reduces to a pair of lines, any line through their intersection must be regarded as a tangent to the system. The method of tangential co-ordinates may be presented in a form which does': not presuppose any acquaintance with the trilinear or Cartesian systems. Just as , in trilinear co-ordinates the position of a point is determined by the mutual ratios of the perpendiculars let fall from it on three fixed lines, so (Art. 311) the position of a line may be determined by the mutual ratios of the perpendiculars let fall on it from three fixed points. If the perpendiculars let fall on a line from two points A, B be X, /u, then it is proved, as in Art. 7, that the perpendicular on it from the point which cuts the line AB in the ratio of to : Ms -, , and consequently that if the line pass through that point we have l\ + m^ — 0, which therefore may be regarded as the equation of that point. Thus X + ^ = is the equation of the middle point of AB, X — fi. = that of a point at infinity on AB. In like manner (see Art. 7, Ex. 6) it is proved that A. + m/x + nv = ia the equation of a point 0, which may be constructed (see fig., p. 61) either by cutting BC in the ratio n : m and AD in the ratio m + n : I ; or hj cutting AC :: I : n and BE : : I + n : m, or by cutting AB ■.-.m : I and CF : : I + m : n. Since the ratio of the triangles AOB : AOC is the same as that of BD : BC,we may write the equation of the point in the form BOC.X + COA . fi + AOB .V = 0. Or, again, substituting for each triangle BO Cits value p'p" sin0 (see Art. 311) X sin 6 fi sind' v sin 6" _ P P' P" Thus, for example, the co-ordinates of the line at infinity are X = /i = i/ since all finite points may be regarded as equidistant from it : the point TK + m/t. + nv will be at infinity when l + m + n — Q; and generally a curve will be touched by the line at infinity if the sum of the coefficients in its equation = 0. So again the equations of the intersection of bisectors of sides, of bisectors of angles, and of the perpendiculars, of the triangle of reference are respectively \ -f /u -1- v = 0, X sin .4 + /u sin ZJ 4- 1/ sinC = 0, X tan 4 4- /x tan B + v tanC =0. It is unnecessary to give further illustrations of the application of these co-ordinates because they differ - only by constant multipliers from those we have used already. The length of the perpendicular from any point on la + mjS + ny is (Art. 61) la' + m/3' + ny'^ 4{P + m^ + «2 _ 2.mn coaA — 2nl coaB — 2lm coaC) ' the denominator being the same for every point. If then p, p', p" be the perpen- diculars let fall from each vertex of the triangle on the opposite side, the perpen- diculars X, JUL, V from these vertices on any line are respectively proportional to Ip, mp', np" ; and we see at once how to transform such tangential equations as were used in the preceding pages, viz. homogeneous equations in /, m, n, into equations expressed in terms of the perpendiculars X, fx, v. It is evident from the actual values that \, p., V are connected by the relation X2 , p' i;« 2pv , 2v\ „ 2X,ii ^ , -i + hi + -Hi — , ,/ cos A TT- cos B , cosC = 1. P^ P^ p 2 pp p p pp NOTES. 365 It was shown (Art. 311) how to deduce from the trilinear equation of any curve the tangential equation of its reciprocal. The system of three point tangential co-ordinates just explained includes under it two other methods at first sight very different. Let one of the points of re- ference C be at infinity, then both v and p" become infinite, but their ratio remains finite and = sin COE. where DOE is any line drawn through the point 0. The equation then of a point already given becomes in this case ^ T+sine" = 0. p BinCOE p' BvaVOE When is given every thing in this equation is constant except the two variables . „„„ , . ^^c^ , but since smCOE smCOE sinCOE = 9inOZ>^, these two variables are respectively AD, BE. In other words, if we take as co-ordinates AD, BE the intercepts made by a variable Une on two fixed parallel lines, then any equation a\ + bfi + c = 0, denotes a point ; and this equation may be considered as the form assumed by the homogeneous equation a\ + bp. + cv = when the point v = is at infinity. The following example illustrates the use of co-ordinates of this kind, "We know from the theory of conic sections that the general equation of the second degree can be reduced to the form aft = k-, where a, ft are certain hnear functions of the co-ordinates. This is an analytical fact wholly independent of the inter- pretation we give the equations. It follows then that the general equation of curves of the second class in this system can be reduced to the same form aft = k", but this denotes a curve on which the points o, ft lie and which has for tangents at these points the parallel lines joining a, ft to the infinitely distant point k. We have then the well known theorem that any variable tangent to a conic intercepts on two fi^ed parallel tangents portions whose rectangle is constant. Again, let two of the points of reference be at infinity, then, as in the last case, the equation of a line becomes ^^^^ + sin e' . sin BOD + sin 6" . sinCOE, or, as may be easily seen, sin 6 • n, 1 , • n'/ 1 n h sme' -77-, -f sme ' --^ = 0. p AD AE When the point is given, the only things variable in this equation are AD, AE, and we see that if we take as co-ordinates the reciprocals of the intercepts made by a variable line on the axes, then any linear equation between these co-ordinates denotes a point, and an equation of the n^^ degi-ee denotes a curve of the n*'' class. It is evident that tangential equations of this kind are identical v,-ith that form of the tangential equations used in the text where the co-ordmates are the coefficients /, m, in the Cartesian equarion Ix + my = 1, or the mutual ratios of the coefficients in the Cartesian equation Ix + my 4- « - 0. 366 NOTES. On Mr. Casey's form of the Equation of a Conic having Double Contact with a given one and touching three others, page 345. Since p. 345 was printed I have succeeded in proving the truth of Mr. Casey's equation by a method analogous to that used in the case of three circles, Ex. 4, p. 130. Let the conic S he x- + y- + z", and let L — lx-^ my + m, M — I'x + m'y + n'z ; then the condition that S - U, S - 3f^ should touch is (Art. 387) {l-S'){l- S") = (1 - li)'' where S' = P + m^ + ri^, S" = P + m'^ + n'% R = ll' + mm' + nn'. I write now (12) to denote J(l - S') (1 - S") - (1 - R). Let us now, according to the rule of multiplication of determinant"?, form a deter- minant from the two matrices containing five columns and six rows each. 1, 0, 0, 0, 1, /, m, n, J(l - S') 1, r, m', n', 4(1 — S") 1, I", m", n", J(l - ;S"') 1, V", to'", n"', J(I - S^ 1, ?4, TO^, n^, J(l - ^,) 0, 0, 0, 0, 1, - 1, I, m, n, ,i(l - *S'), - 1, ?', m', n', 4(1 — S"), - 1, I", to", n", 4(1 — S'"), - 1, V", to"', n'", 4(1 - Si), The resulting determinant which must vanish since there are more rows than columns, is 0, 1, 1, I, 1, 1 4(1 - S'), 0, (12), (13), (14), (15) 4(1 - S"), (12), 0, (23), (24), (25) 4(1 - S'"), (13), (23), 0, (34), (35) 4(1 - S,), (14), (24), (34), 0, (45) 4(1 - .%), (15), (25), (35), (45), =0, an identical relation connecting the invariants of five conies all having double contact with the same conic S. Suppose now that the conic (5) touches the other four, then (15), &c. vanish ; and we learn that the invariants of four conies all having double contact with S and touched by the same fifth are connected by the relation 0, (12), (13), (14) (12), 0, (23), (24) (13), (23), 0, (34) (14), (24), (34), = 0, or, 4{(12) (34)) ± 4((13) (24)} ± 4[(14) (23)} = 0. We may deduce from this equation as follows the equation of the conic touching three othei-a. If the discriminant of a conic vanish, S =1, and then the condition of contact with any other reduces to ^ = 1. If then a, /3, y be the co-ordinates of any point satisfying the relation S — L^ = Q, or x"^ + y^ + z- — {Ix + my + nzf ~ 0, then x^ + y- + z- f ax + /3.y + yz ^ l4(«' + /3-+'y'')i ' evidently denotes a conic whose discriminant vanishes and which touches S — 1?. If then we are given three conies S — L?, S — M^, S — N^, take any point a, /3, y on the conic which touches all three and take for a fourth conic that whose equa- tion has just been written, then the functions (14), (24), (34) are respectively 1 — 77-57 ' 1 — /Ten ) 1 ~ TTm" > ^^^ '^^ ^^^ ^^^^ ^"7 Poiut on the conic touching •JK^J 4{^) 4('3; all three satisfies the relation J([(23)} U{S) - L]] ± 4;[(31)} U(S) - J/]] + 4;[(12)} (4(5) - A'}] = 0. NOTES. 367 On the Problem to describe a Conic under Five Conditions. We saw (p. 131) that five conditions determine a conic ; we can, therefore, in general describe a conic being given m points and n tangents where m + 71 — b. We shall not think it worth while to treat separately the cases where any of these are at an infinite distance, for which the constructions for the general case only require to be suitably modified. Thus to be given a parallel to an asymptote is equivalent to one condition, for we are then given a point of the curve, namely, the point at infinity on the given parallel. If, for example, we were required to describe a conic, given four points and a parallel to an asymptote, the only change to be made in the construction (p. 236) is to suppose the point E at infinity, and the lines DE, QE therefore drawn parallel to a given line. To be given an asymptote is equivalent to two conditions, for we are then given a tangent and its point of contact, namely, the point at infinity on the given asymptote. To be given that the curve is a parabola is equivalent to one condition, for we are then given a tangent, namely, the line at infinity. To be given that the curve is a circle is equivalent to two conditions, for we are then given two points of the cuiwe at infinity. To be given a Joans is equivalent to two conditions, for we are then given two tangents to the curve (p. 228), or we may see otherwise that the focus and any three conditions will determine the cm-ve ; for by taking the focus as origin, and reciprocating, the problem becomes, to describe a circle, three conditions being given ; and the solution of this, obtained by elementary geometry, may be again reciprocated for the conic. The reader is recommended to construct by this method the directrix of one of the four conies which can be described when the focus and three points are given. Again, to be given the pole, with regard to the conic, of any (jiven right line, is equivalent to two conditions ; for thi'ee more will determine the curve. For (see figure, p. 143) if we know that P is the polar of R'R", and that T is a pomt on the curve, T', the fourth harmonic, must also be a point on the curve ; or if OThe a tangent, OT' must also be a tangent ; if then, in addition to a line and its pole, we are given three points or tangents, we can find three more, and thus determine the curve. Hence, to be given the centre (the pole of the line at infinity) is equivalent to two conditions. It may be seen likewise that to be given a point on the polar of a given point is equivalent to one condition. For example, when we are given that the curve is an equilateral hyperbola, this is the same as sajang that the two points at infinity on any circle lie each on the polar of the other with respect to the curve. To be given a self -con jugate triangle is equivalent to three conditions ; and when a self-conjugate triangle with regard to a parabola is given three tangents are given. Given five points. — We have shown, p. 236, how by the ruler alone we may deter- mine as many other points of the curve as we please. We may also find the polar of any given point with regard to the curve ; for by the help of the same Example we can perform the construction of Ex. 2, p. 143. Hence too we can find the pole of any line, and therefore also the centre. Five tangents. — We may either reciprocate the construction of p. 236, or reduce this question to the last by Ex. 4, p. 236. Four points and a. tangent. ^We have already given one method of solving this question, p. 300. As the problem admits of two solutions, of course we cannot expect a consti-uction by the rviler only. We may therefore apply Carnofs theorem (Art. 313), Ac . Ac'. Ba . Ba'. Cb .Cb' = Ab. Ab'. Be . Be'. Ca . Ca'. Let the four points a, a', h, b' be given, and let AB be a tangent, the points c. c' will coincide, and the equation just given determines the mtio Ac- : Be'-, everything else in the equation being known. This question may also be reduced, if we please, to those which follow ; for given four points, there are (Art. 282) thi'ee points whose polai-s are 368 NOTES. given ; having also then a tangent, we can find three other tangents immediately, and thus have four points and four tangents. Four tangents and a point. — This is either reduced to the last by reciprocation, or by the method just described ; for given four tangents, there are three points whose polars are given (p. 143). Three points and two tangents. — It is a particular case of Art. 344, that the pair of points where any Une meets a conic, and where it meets two of its tangents, belong to a system in involution of which the point where the line meets the chord of contact is one of the foci. If, therefore, the line joining two of the fixed points a, b, be cut by the two tangents in the points A, B, the chord of contact of those tangents passes through one or other of the fixed points F, F', the foci of the system (a, b, A, B), (see Ex., Art. 286). In like manner the chord of contact must pass through one or other of two fixed points G, G' on the hne joining the given points a, c. The chord must therefore be one or other of the four lines, FG, FG', F'G, F'G' ; the problem, there- fore, has foiu" solutions. Two points and three tangents. — The triangle formed by the three chords of contact has its vertices resting one on each of the three given tangents ; and by the last case the sides pass each through a fixed point on the line joining the two given points ; therefore this triangle can be constructed. To be given two points or two tangents to a conic is a particular case of being given that the conic has double contact with a given conic. For the problem to describe a conic having double contact with a given one, and touching three lines, or else passing through three points, see pp. 289, 345. Having double contact with two, ^nd passing through a given point, or touching a given line, see p. 251. Having double contact with a given one, and touching three other such conies, see p. 344. On systems of Conics satisfying Four Conditions. If we are only given four conditions, a system of diiEerent conics can be described satisfj'ing them all. The properties of systems of curves, satisfying one condition less than is sufficient to determine the curve, have been studied by De Jonquieres, Chasles, Zeuthen, and Cayley. References to the original memoirs will be found in Mr. Cayley's memoir [Phil. Trans., 1867, p. 75). Here it will be enough briefly to state a few results following from the application of M. Chasles' method of characteristics. Let ft. be the number, of conics satisfying four conditions, which pass through a given point, and v the number which touch a given line, then fi, v are said to be the two characteristics of the system. Thus the characteristics of a system of conics passing through four points are 1, 2, since, if we are given an additional point, only one conic will satisfy the five conditions we shall then have ; but if we are given an additional tangent two conics can be determined. In like manner for three points and a tangent, two points and two tangents, a point and three tangents, four tangents, the characteristics are respectively (2, 4), (4, 4), (4, 2), (2, 1). We can determine a priori the order and class of many loci connected with the system by the help of the principle that a curve will be of the «'" order, if it meet an arbitraiy line in n real or imaginary points, and will be of the «'•> class if through an arbitrary point there can be drawn to it n real or imaginary tangents. Thus the locus of the pole of a given Ime with respect to a system, whose characteristics are /x, V wUl be a curve of the order v. For. examine in how many points the locus can meet the given line itself. When it does, the pole of the line is on the hne, or the Ime is a tangent to a conic of the system. By hypothesis this can only happen in V cases, therefore v is the degree of the locus. This result agi-ees with what has been already found in particular cases, as to the order of locus of centre of a conic through four points, touching four lines, &c. In like manner let us investigate ^'•OTES. 309 the order of the locus of the foci of conies of the system. To do this let us generalize the question, by the help of the conception of foci explained Art. 279, and we shall see that the problem is a particular case of the follow'ing : Given two points A, B to find the order of the locus of the intersection of either tangent drawn from A to a conic of the system with one of the tangents drawn from R. Let us examine in how many points the locus can meet the line AB; and we see at once that if a point of the locus be on AB, this line must be a tangent to the conic. Consider then any conic touching AB in a point T, then the tangent AT meets the tangent BT in the point T, which is therefore on the locus : and likewise the tangent AT meets the second tangent from B in the point B, and the tangent BT meets the second tangent from A in the point A, Hence every conic which touches AB gives three points of the locus on AB. The order of the locus is therefore 3i/, and A and B are each multiple points of the order v. Thus the locus of foci of conies touching four lines is a cubic passing through the two circular points at infinity. If one of the con- ditions be that all the conies should touch the Une AB, then it will be seen that any transversal through A is met by the locus in v points distinct from A, and that A itself also counts for v : hence the locus is in this case only of the order 2v ; which is therefore the order of the locus of foci of parabolas satisfpng three conditions. An iraportant principle iu these investigations is that if two points A, A' on a right Hue so correspond that to any position of the point A coiTCspond m positions of A', and to any position of A' correspond n positions of A, then in m + n cases A and A' -vnW coincide. This is proved as ;ui Aits. 336, 340. Let the line on which A, A' he be taken for axis of x ; then the abscissae x, x' of these two points are con- nected by a certain relation, which by hypothesis is of the 7«"' degi^ee in x and the n^^ in x, and wO.1 become therefore an equation of the {m + «)"' degree if we make x = x'. To illustrate the application of this principle, let us examine the order of the locus of points whose polar with respect to a fixed conic is the same as that with respect to some conic of the sj'stem ; and let us enquire how many pomts of the locus can lie on a given Une. Consider two points A, A' on the Une, such that the polar of A v^-ith respect to the fixed conic comcides with the polar of A' with respect to a conic of the system, and the problem is to know in how many .cases A and A' can coincide. Now first if A be fixed, its polar with respect to the fixed conic is fixed ; the locus of poles of this Jast line with respect to conies of the system, is, by therfirst theorem, of the order /', and therefore determines by its intersections with the given Unej/ positions of A'. Secondly, exaniine how many positions of .1 con'espond to any fixed position of A', By the reciprocal of the first theorem, the polars of A' wdth respect to conies of the system, envelope a cui-ve whose class is /u, to which therefore fi tangents can be drawn through the pole of the given Une AA' with respect to the fixed conic. It follows then, that yu. positions of A correspond to any position of -1'. Hence, in /u + i/ cases the two coincide, and this wiU be the order of the required locus. Hence we can at once determine how many conies of the system can touch a fixed conic : for the point of contact is one which has the same polar with resiiect to the fixed conic and to a conic of the system ; it is therefore one of the intei-sections of the fixed conic with the locus last found ; and there may evidently be 2 Qu + v) such intei-sections. We have thus the number of conies which touch a fixed conic, and satisfy any of the systems of conditions, four points, three points and a tangent, two points and two tangents, d'c, the numbers being respectively 6, 12, 16, 12, 6. From these numbei-s again we find the characteristics of the system of conies which touch a fixed conic and also satisfy three other conditions, three points, two points and a tangent, A'C. ; these characteristics being respectively (6, 12), (12, 16), (16, 12), (12, 6). We find hence in the same manner the number of conies of the respective systems which will touch a second fixed conic, to be 36, 56, 5G, 36. And thus again we have B B B 370 NOTES. the characteristics of systems of conies touching two fixed conies, and also satisfying the conditions two jwints, a point and a tangent, two tangents ; -vtz. (3G, 56), (56, 56), (56,36). In like manner we have the number of conies of these respective sj'stems which will touch a third fixed conic, viz. 184, 224, 184. The characteristics then of the systems three conies and a point, three conies and a line are (184, 224), (224, 184). And the numbers of these to touch a fourth fixed conic, are in each case 816, so that finally we ascertain that the number of conies which can be described to touch five fixed conies is 3264. For further details, I refer to the memoirs already cited, and only mention in conclusion that 2v — fx conies of any system reduce to a pair of lines, and 2/* — j/ to a pair of points. INDEX. Angle, between two lines whose Cartesian equations are given, 21, 22. ditto, for tiilinear equations, 60. between two lines given by a single equation, 69. between two tangents to a conic, 161, 201, 258. between two conjugate diameters, 164. between asymptotes, in terms of ec- centricity, 159. between focal radius vector and tan- gent, 174. subtended at focus by tangent from any point, 177, 195. subtended at limit points of system of circles, 279. theorems respecting angles subtended at focus proved by reciprocation, 272. by spherical geometrj^ 319. theorems concerning angles liow pro- jected, 309, 811. Anharmonic ratio, fundamental theorem proved, 55. what, when one point at infinity, 283. of four lines whose equations are given, 56, 293. property of four points on a conic, 229, 240, 276, 306. of four tangents, 241, 276. of three tangents to a parabola, 287. these properties developed, 284, itc. properties derived from projection of angles, 311, 312. of four points on a conic when equal to that of fom* others on same conic, 241, 242. on a different conic, 241, 291. of four points equal that of their polars, 260. of four diameters equal that of their conjugates, 290. of segments of tangent to one of tliree conies having double contact, by other two, 307. ApoUonius, 316. Arc, line cutting off constant arc from curve where met by its envelope, 355. theorems concerning arcs of conies, 358. Area, of a polygon in tenns of co-ordinates of its vertices, 31, 128. of a triangle, the equations of whose sides are given. .32. Area, of triangle inscribed in, or cu'cum- scribing a conic, 201, 208. of triangle formed by three normals, 209. constant, of triangle fonned by join- ing ends of conjugate diameters, 164, 164, constant, between any tangent and asymptotes, 182. of polar triangles of middle points of sides of fixed triangle with regard to inscribed conic, 337. of triangles equal, formed by drawing from end of each of two diameters a parallel to the other, 168. found by infinitesimals, 352. constant, cut from a conic by tangent to similar conic, 354. line cutting off fi-om a curve constant ai-ea bisected by its envelope, 355. Asymptotes, defined as tangents through centre whose points of contact are at in- finity, 150. are self -con jugate, 162. are diagonals of a pai-allelogi'am whose sides are conjugate diameters, 180. general equation of, 200, 328. and pair of conjugate diameters form harmonic pencil. 284. portion of tangent between, bisected by cmwe, 180. equal intercepts on any chord between cui-ve and, 181, 300. constant length intercepted on by chords joining two fixed points to vai-iablc, 182, 282, 286. parallel to, how cut by same chords, 286. by two tangents and their chord, 286. bisected between any point and its polar, 283. parallels to. through any point on curve include constant area, 182, 282, 286. how divide any semi-diameter, 286. Axes, of conic, equation of, 151. lengths, how found, 153. constructed geometrically, 156. how foixnd when two conjugate dia- meters are given, 168. of reciprocal curve, 279. axis of parabola, IHj. 372 TXDESr. of similitude, 108, •21-2, 270. radical, 99, 126. Bisectors of angles between lines given by a single equation, 71. of sides or angles of a triangle meet in a point, 5. 34, 54. BobUlier on equations of conic inscribed in, or circumscribing a triangle, 120. Boole on invariant functions of coefBcients of a conic, 154, Brianclion's theorem, 233, 268, 362. Bumside, theorems or proofs bv, 80, 209, 210, 231, 235, 245, 261, 329. Camot, theorem of transversals, 277, 300, 367. Cartesian, equations, a case of trilinear, 64. Casey, theorems by, 113, 126, 130,344,366. Cayley, theorems and proofs by, 129, 330, 337, 344, 360, 362, 368, Centre, of mean position of given points, 50. of homology, 59. radical, 99, 270, of similitude, 105, 213, 270. _ chords joining ends of radii through c. s. meet onradicalaxis,107,212, 238. of conic, coordinates of, 138, 148. pole of line at infinity, 150, 284. how found, given five points, 236. of system in involution, 296. of cur\'ature, 219. 357. Chasles, theorems by, 283, 288, 292, 358, 368. Chord of conic, perpendicular to line join- ing focus to its pole, 177, 309. which touches confocal conic, propor- tional to square of parallel semi- diameter, 201, 210. Cliords of intersection of two conies, equa- tion of, 322. Circle, equation of, 14, 75, 87, tangential equation of,120,124,128,363. passes through two fixed imaginary points at infinity, 227, 313. circumscribing a trianglfe, its centre and equation, 4, 86, 118, 128, 276. inscribed in a triangle, 122, 276. having triangle of reference for self- conjugate triangle, 243. through middle points of sides (see Feuerbach), 86, 122. which cuts two at constant angles, touches two- fixed circles, 103. touching three others, 110, 114, 130, 279. cutting thi-ee at right angles, 102, 128, 347. circumscribing triangle formed by three tangents to a parabola, passes through focus, 196, 203, 263, 273, 308. circumscribing triangle formed by two tangents and chord, 231 . circumscribing trianglie inscribed in a conic, 209, 321'. circumscribing, or inscribed, irt a self- conjugate triangle, .32!>; Circles circumscribing triangles formed 6y' four lines, meet in a point, 235. when five lines are given, the five such points lie on a circle, 235. tangents, area, and arc found by in- finitesimals, 351. Circtimscribing triangles, six vertices of two lie on a conic, 308, 362. Class of a curve, 142. Common tangents to two circles, 104, 106, 252. to two conies, 332. their eight points of contact lie on a conic, 332. Condition that,- three points should be on a right line, 24. thfea lines meet in a point, 32, 34. four convergent lines should form harmonic pencil, 56. two lines should be perpendicular, 21, 59, .341. a right line should pass through a fixed point, 50. equation of second degree should re- present right lines, 72, 144, 148, 150, 255. a circle. 75, 121, 339. a parabola, 136, 263, 338. an equilateral hjperbola, 164, 338. equation of any degi-ee represent right lines, 74. two circles should be concentric, 77. four points should he on a circle, 86. intercept by circle on a line should subtend a right angle at a given point, 90. two circles should cut at right angles, 102, 335, a line shoidd touch a conic, 81, 147, 255, 328. two conies should be similar, 213. two conies should touch, 324, 343. a point should be inside a conic, 250. two lines should be conjugate with respect to a conic, 256. two paire of pouits should be harmonic conjugates, 293. four points on a conic should lie on a circle, 218. a Une be cut harmonically by two eonics, 294. in involution by three conies, 347. three paii-s of lines touch same conic, 258. three pairs of points form system in involution, 298. a triangle may be inscribed in one conic and circumscribed to another, 330. a triangle self-conj"ugate to one may be inscribed or circumscribed to another, 328. three conies have double contact with same conic, 345. have a common point, 348. may include a perfect sqtiare ia their syzygy, 349. Index. 373 Condition that, lines joining to vertices of triangle points wTiere conic meets sides should form two sets of three, 337, Cone, sections of, 314. Confocal conies, cut at right angles, 175, 310. may be considered as inscribed in same quadrilateral, 228. most general equation of, 310. tangents from point on (1) to (2) equally inclined to tangent of (1), 176. pole with regard to (2) of tangent to (1) Ues on a normal of (1), 198. used in finding axes of reciprocal cui-ve, 279. in finding centre of curvature, 357. properties proved by reciprocation, 279. length of arc intercepted between tangent from, 357. Conjugate diameters, 141. their lengths, how related, 154, 163. triangle included by, has constant area, 154, 164. form harmonic pencil with asj-mp- totes, 284. at given angle, how constructed, 166. construction for, 207. Conjugate hyjierbolas, 159. Conjugate lines, conditions for, 256. Conjugate triangles, homologous, 91, 92. Continuity, principle of, 313. Covariants, 333. Criterion, whether three equations repre- sent lines meeting in a point, 34. whether a point be within or without a conic, 250. whether two conies meet in two real and two imaginary points, 325. Cui-vature, radius of, expressions for its length, and construction for, 217, 357. circle of, equation of, 223. centre of, co-ordinates of, 219; De Jonquieres, 368. Determinant notation, 128. Diagonals of quadrilateral, middle points lie in a hne, 26. 62, 205. circles described on, as diameters, have common radical axis, 231. Diameter, polar of point at infinity on its conjugate, 284. Director circle, 258, 339. when four tangents are given, have common radical axis, 265. Directrix, 173. of parabola, equation of, 258, 339. is locus of rectangular tangents, 194, 258, 339. passes through intersection of per- pendiculars of circumscribing tri- angle, 201, 236, 263, 278, 329. Discriminant defined, 255. method of forming, 72, 144, 148, 1.50. Distance between two points, 3, 10, 128. Distance of two points from centre of circle proportional to distance of each from polar of other, 03 when a rational function of co-ordi- nates, 173. of four points in a plane,- how con- nected, 129. Double contact, 215, 223. equation of eoais having d. c- with two others, 251. tangent to one cut harmonically by other, and chord of contact, 300, 307. properties of two conies having d. c withathu-d, 231, 269. of three ha^'ing d. c. with a fourth, 232, 252, 270. tangential equation of, 342. condition two should touch, 343-, problem to describe one such conic^ touching three others, 343, 345; 366,- Duality, principle of, 266. Eccentric angle, 206, &c., 232. in terms of- corresponding focal angle, 209. of four points on a circle, how con- nected, 218. Eccentricity, of conic given by general equation, 159. depends on angle between asymp- totes, 159. Ellipse, origin of name, 180, 316. mechanical descrijjtion of, 172, 207.. area of, 353. Envelope of line whose equation involves indeter-- minates in second degree, 246, &c. line on which sum of pei-pendiculars from several fixed points is con- stant, 95. given product or sum or difference of squares of perpendiculars from two- fixed points, 248. base of triangle given vertical angle and sum of sides, 249. whose sides pass through fixed points and vertices move on fixed lines, 248.- and inscribed in given conic, 239, 269,- 307. ■R-hich subtends constant angle at fixed point, two sides being given in position, 273. polar of fixed point with regard to a conic of which four conditions are given, 260, 269. polar of centre of circle touclung two given, 279. chord of conic subtending constant angle at fixed point. 244, 272, 273. perpendicular at extremity of radius vector to circle, 194. asymptote of hj-perbolas having same focus and du-ectri.x, 273. given three points and other a.symp-- tote, 261. line joining corresponding points of two homogi-aphic systems on different lines, 290. 374 TNDEX. Envelope of (in a conic, 24-2, 201. free side of inscribed polygon, all the rest passing through fixed points, 239, 289. base of triangle inscribed in one conic, two of whose sides touch another, 336. leg of given anhannonic pencil under different conditions, 312. ellipse given two conjugate diameters and sum of their squares, 249. Equation, its meaning when co-ordinates of a given point are substituted in it ; for a right line, circle, or conic, 29, 84, 127, 230. ditto for tangential equation, 363. pair of bisectors of angles between two lines, 71. * of radical axis of two circles, 98, 127. common tangents to two circles, 104, 106, 2.52. circle thi-ough three points, 86, 128. cutting tkree cii-cles oi-thogonally, 102, 128. touching tkree circles, 114, 130, 366. inscribed in or circumscribing a tri- angle, 118, 125, 276. having triangle of reference self- conjugate, 243. tangential of circle, 128. 363. tangent to circle or conic, 80, 141, 253. polar to circle or conic, 82, 142, 254. pair of tangents to conic from any point, 85, 144, 257. where conic meets given line, 260. asymptotes to a conic, 260, 328. chords of intersection of two conics,322. circle osculating conic, 223. conic through five points, 222. touching five lines, 262. having double contact with two given ones, 251. having double contact with a given one and touching three others, 345, 366. through three points, or touching three lines, and having given centre, 256. and having given focus, 276. reciprocal of a given one, 281, 335, 342. directrix or director circle, 258, 339. lines joining pomt to intersection of two curves, 259, 295. four tangents to one conic where it meets another, 336. curve parallel to a conic, 325, evolute to a conic, 220, 326. Jacobian of three conies, 316. Equilateral hyperbola, 163. general condition for, 338. given three points, a fourth is given, 204, 278, 329. circle circumscribing self-conjugate triangle passes through centre, 204, 329. Elder, expression for distance between centres of inscribed and ch'cum- scribing circles, 33 1 . E volutes of conies. 220. 326. Fagnani"s theorem on arcs of conies, 358, Faure, theorems by, 329, 337. Feuerbach, relation connecting four points on a circle,. 87, 206. theorem on circles touching four lines, 126, 128, 301, .345. Fixed point, the following lines pas:* through a coefficients in who.se equation are con- nected by relation of first degi-ee, 50.- base of triangle, given vertical angle' and sum of reciprocals of sides, 4iS. ■whose sides pa-ss through fixed points, and vei-tices move on three converging lines, 48, line sum of whose distances from fixed points is constant, 49. polar of fixed point with respect to circle, two points given, 100. with respect to conic, four points- given, 148, 259, 269. chord of intersection with fixed centre- of circle through two points, 100. of two fixed lines with conic through four points, one lying on each line, 290. chord of contact given two points and tW'O lines, 251, chord subtending right angle at fixed point on conic, 170, 259. when product is constant of tangents of parts into wliich normal divides subtended angle, 170. given bisector of angle it subtends at fixed point on cm-ve, 310. perpendicular on its polar, from point on fixed perpendicular to axis, 178. Focus, see Contents, pp. 171-179, 198-200. infinitely small circle having double contact with conic, 230. intersection of tangents fi'om two fixedi imaginary points at infinity, 228. equivalent to two conditions, 367. co-ordinates of, given three tangents,. 263. when conic is given by general equa- tion, 228, 340. focus and directrix, 173, 229, theorems concerning angles subtended; at, 272, 319. focal properties investigated by pro- jection, 308. focal radii vectores from any point have equal difference of reciprocals, 201. line joining intersections of focal nor- mals and tangents passes through other focus, 200. locus of, given three tangents to a parabola, 196, 203, 263, 273, 308. given four tangents, 263, 265. given four points, 206, 276. given three tangents and a point, see Ex. 3, p. 276. of section of right cone, how found, 319. of systems in involution, 297. Gaultier of Toiu-s. 09. INDEX. O t D Gergorme, on circle toucKng three others, 110. Graves, theorems by, 321, 357, Harmonic, section, 56. what when one point at infinity, 283. properties of quadrilateral, 57, 305. property of poles and poiai-s, 85, 143, 283, 285, 306. pencil fonned by two tangents and two co-polar lines, 143, 284. by asj-mptotes and two conjugate diameters, 284. by diagonals of inscribed and circum- scribing quadrilateral, 231. by chords of contact and common chords of two conies having double contact with a third, 231. properties derived from projection of right angles, 309, 310. condition for haiTQonic pencil, 203. condition that line should be cut har- monically by two conies, 294. locus of points whence tangents to two conies form a harmonic i:)encil, 294. Hart, theorems and proofs by, 123, 125, 126, 252, 359, Heame, mode of finding locus of centre, given four conditions, 256. Hermes, on equation of conic circumscrib- ing a triangle, 120, Hesse, 362. Hexagon (see Brianchon and Pascal), property of angles of circumscribing, 258, 277, Homogeneous, equations in two variables, meaning of, 67. triUnear equations, how made, 64, Homogi'aphic systems, 57, 63. criterion for, and method of forming, 292. locus of intersection of corresponding Unes, 260. envelope of line joining corresponding points, 290, 291. Homologous triangles, 59. Hyperbola, origin of name, 180, 316. area of, 354, Imaginary, lines and points, G9, 77. circular points at infinity, equation of, 338. every line through either perpen- dicular to itself. 338. Infinity, fine at, equation of, 64. touches parabola, 224, 278, 3 18, centre, pole of. 150, 2«4. Inscription in conic of triangle or polygon whose sides pass through fixed points, 239, 261, 269. 289. Intercept on chord between curve and asjTuptotes equal, 181, 300. on asjTnptotes constant by fines join- ing two variable points to one fixed, 182. 236. on axis of parabola by two lines, equal to projection of distance between their poles, 190, 281. Intercept on parallel tangents bv variable tangent, 167, 275, 287, 305. Invariants, 154, 323. Inversion of cur^'es, 114. Involution, 295. Jacobian of three conies, 345, iSrc, Joachimsthal, relation between eccentric angles of four points on a circle, 218. method of finding points where line meets curve, 283. Ku-kman's theorems on hexagons, 360. Latus rectum, 179. Limit jxiints of system of circles, 101, 279. Locus of vertex of triangle given base and a relation between lengths of sides, 39, 47, 172. and a relation between angles, 39, 47, 88, 107. and mtercept by sides on fixed line, 288. and ratio of parts into which sides divide a fixed parallel to base, 41. vertex of given triangle, whose base angle moves along fixed Imes, 197. vertex of triangle of which one base angle is fixed and the other moves along a given locus, 51, 96. whose sides pass thi-ough fixed points and base angles move along fixed lines, 41, 42, 237, 268, 287. generalizations of the last problem, 288. of vertex of triangle which cii-cum- scribes a given conic and whose base angles move on fixed lines, 239, 307, 336. generahzations of this problem, 337. common vertex of several triangles given bases and sum of areas, 40. vertex of right cone, out of which given conic can be cut, 319. point cutting in given ratio parallel chords of a circle, 157. intercept between two fixed fines, on various conditions, 39, 40, 47. variable tangent to conic between two fixed tangents, 265, 311. point whence tangents to two cii'cles have given ratio or sum, 99, 252. taken according to different laws on radii vectores through fixed pomt, 52. such that 2n?>'- = constant, 88. whence square of tangent to circle is as product of distances from two fixed lines, 229. cutting in given anhannonic ratio, chords of conic through fixed point, 308. on perpendicular at height from base equal a side, given base and sum of sides, 59. such that triangle formed by joining feet of peipendiculars on sides of triangle lias constant area, 119. 376 IKDEX. Locus ol l)oint on line of given direction meeting sides of triangle, so that oc- - oa . ob, 286. on lines cut in given anhai-monic ratio, of which other three describe riglit lines, and line itself touches a conic, 311. chords through wliich subtend right angle at point on conic, 259. ■whence tangents to two conies fonn harmonic pencil, 294, whose polars with respect to three conies meet in a point, 345. middle point of rectangles inscribed in triangle, 43. of parallel chords of conic, 1 38. of convergent chords of circle, 96. intersection of bisector of vertical angle with perpendicular to a side, given base and sum of sides, 51. of perpendicular on tangent from centre, or focus, with focal or central radius vector, 198. focal radius vector with con-esponding eccenti-ic vector, 209. of pei-pendiculars to sides at extremity of base, given vertical angle and another relation, 47. of peipendiculars of triangle given base and vertical angle, 88. of pei-pendiculars of triangle inscribed in one conic and circumscribing another, 329. eccentric vector ^^-ith con-esponduig normal, 209. corresponding lines of two homogi-a- phic pencils, 260. polai-s \rith respect to fixed conies of points which move on right lines, 260. intersection of tangents to a conic which cut at right angles, 161, 166, 258, 339. to a parabola which cut at given angle, 202, 245, 273. at extremities of conjugate dia- meters, 198. whose chord subtends constant angle at focus, 272. from two points, which cut a given .line harmonically, 340. -each or both on one of four given tangents. 290, 308. at two fixed points on a conic satisfy- •ing two other conditions, 209, 308. •various other conditions, 204. intersection of normals at extremity of focal chord, 200. ■or chord through fixed point, 203, 323. foot of perpendicular fi'om focus on tangent, 176, 193. on normal of parabola, 203. on chord of circle subtending right angle at given point, 91. extremity of focal subtangcnt, 178. centre of circle making given inter- cepts on given lines, 197. Locus of. centre of insciibed circle given base and sum of sides, 197. of circle cutting three at equal angles, 108. of circumscribing circle given vertical angle, 89. of circle toucliing two given circles, 279, 308. centre of conic (or pole of fixed line) given four points, 148, 243, 256, 260, 290, 308. given four tangents, 243, 256, 265, ^69, 809, 327. given three tangents and sum of squares of axes, 205. four conditions, 256, 368. pole of fixed line with regard to sys- tem of confocals, 198, 310. pole with respect to one conic of tan- gent to another, 198, 266. focus of parabola given three tan- gents, 196, 203, 263, 273, 308. focus given four tangents, 203, 265. given four points, 206, 276. given three tangents and a point, 276. given four conditions, 369. vertices of self-conj ugate triangle, com- mon to fixed conic, and variable of which four conditions are given, 369. MacCullagh, theorems by, 209, 319, 355, 357. MacLaurin's mode of generating conies, 236, 240, 287, 288. Malfatti's problem, 252. Mechanical construction of conies, 172, 183, 192, 207. Middle points of diagonals of tjuadrilate- ral in one line, 26, 02. Miguel, on circles circumscribing triangles formed by five lines, 238. Mobius on tangential co-ordinates, 266. on harmonic properties, 283. Moore, deduction of Steiner's theorem from Brianchon's, 236. Mulcahy, on angles subtended at focus, 319. Newton's method of generating conies. 288. Normal, 168, &c. 323. Number of terms in general equation, 74. of conditions to determine a conic, 131 . of inter.sections of two curves, 214. of solutions of problem to describe a conic touching five others, 370. Orthogonal systems of circles, 102, 128, 335, 347. Osculating circle, 216, 223. three pass through given point on cui-ve, 218. Pappus, 180, 283, 316. • Parabola (see Contents, pp. 184—196, 201-203). origin of name, 180, 316. has tangent at infinity, 224, 278, 318. co-ordinates of focus, 228, 263, 341. equation of directrix, 258, 339. INDEX. o< / Parallel to conic, equation of, o-2i>. Parameter, 179, 18G, 191. same for reciprocals of equal circles, 274. Pascal's hexagon, 234, 268, 289, 307. Perpendicular, equation and length, 26, 60. condition for, 59. extension of relation, 342. fi'om centre and foci on tangent, 16-1, 174. 193. Plucker, 266, 360. Polar co-ordinates and equations, 9. 86, 87, 95, 156, 179, 196. poles and polare, properties of, 92, 143. polar, equation of, 82, 142, 254. pole of given line, co-ordinates of, 255. polar reciprocals, 266, ifcc. point and polar equivalent to two conditions, 367. Poncelet. 101, 266, 289, 302. Projection, 303, 321. Quadrilateral, middle points of diagonals lie on a right line, 26, 62. circles having diagonals for diameters have common radical axis, 265. harmonic properties of, 57, 305. inscribed in conies, 143, 307. sides and diagonals of inscribed quad- rilateral cut transversal in involu- tion, 300. diagonals of inscribed and circum- scxibed form harmonic pencil, 231. Radical axis aud centre, 99, 122, 212, 270. Eadius of circle circumscribing triangle insciited in conic, 202, 321. Radius of curvature, 216. Reciprocals, method of, 66, 264—282, 342. Sadleir, theorems by, 178. Self-conjugate triangles, 91. circle having triangle of reference for, 243. of equilateral hjqierbola, 204. vertices of two lie on a conic, 310, 329. equation of conic refen-ed to, 227, 242. common to two conies, 246. determination of, 335, 347. Scrret on locus of centre given four tangents. 205. SimiUtude, centre of. 105. 212. 270. Similar conies, 211. condition for, 213. have points common at infrnit}', 225. tangent to one cuts constant area from other, 354. Stciner, theorem on triangle circumscriVjing parabola, 201, 236, 263, 278, 329. on points whose osculating circle passes through given point, 218. theorems on Pascal's hexagon, 235, 360. solution of Malfatti's problem, 252. Subnormal of parabola constant, 191. Supplemental chords, 1G6. Systems of circles ha\ing common radical axis, 100. of conies through four points cut a transversal in uivolution. 300, Tangent, general definition of, 78. to circle, length of, 84. to conic constracted geometricallv, 143. detemiination of points of contact, five tangents given, 236. variable, makes what intercepts on two parallel tangents, 167, 175. or on two conjugate diameters, 167. of parabola, how divides three fixed tangents, 287. Tangential equations, 65, 264. <fcc., 363, etc. of inscribed and cii'cumscribing circles. 120, 124, 275. of cii'cle in general, 128, 363. of conic in general, 147, 249. of imaginary circular points. 337. of confocal conies, 340, 363. of points common to foiur conies, 33 1 . interpretation of, 363. Townsend, theorems and proofs by, 241, 289, 355. Transformation of co-ordinates, 6, 9, 151. 323. Transversal, how cuts sides of triangle, 35. Carnot's theorem of met by system of conies in involu- tion, 300. 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Ayre, M.A. IVith Maps, 15 Plates, and numerous Wood- cuts. Fcp. %vo. 6s. The Theory and Prac- tice of Banking. By H. D. Macleod, M.A. Third Edition, revised throughout, ^zo prici I2j'. The Elements of Bank- ing. By Henry Dunning ULtc- leod, Esq. M.A. Crcwn Sz'f. 7 J'. 6d. 40 NEW WORKS PUBLISHED BY LONGMANS & CO. Modern Cookery for Pri- vate Families, reduced to a System of Easy Practice in a Series of carefUly-tested Receipts. By Eliza Acton. With 8 Plates 0-150 Woodcuts. Ftp. Zvo. 6s. ^ Practical Treatise on Breiuiiig ; with For mules for Public Brewers, and Instructio7is for Private Families. By W. Black. Fifth Edition. Zvo. \os. 6d. English Chess Problems. Edited by J. Pierce, ALA. and IV. T. Pieire. With 60S Diagrams. Crown Zvo. I2s. Q>d. The Theory of the AIo- dern Sciejitific Game of Whist. By W. Pole, F.R.S. Hcvciith Edition. Fcp. %vo. is. 6d, The Correct Card ; or, PIozu to Play at Whist : a Whist Catechism. By Captain A. Campbell- Walker. Fcp. Zvo. [.Vcar/j' ready. The Cabinet Lawyer ; a Popular Digest of the Laws of England, Civil, Crimi- nal, and Constitutional. Twenty-fourth Edition, corrected and ex- tended. Fcp. Zvo. gs. Pewtners Comprehensive Specifier ; a Guide to the Pi'actical Specification of every kind of Bidlding- Artificers Work. Edited by W. Young. Croivn Zvo. 6s. Chess Openings. By F. W. Longman, Bal- liol College, Oxford. Second Edition, revised. Fcp. Zvo. 2s. 6d. Hints to Mothers on the Management of their Health ditring the Period of Pregnancy and in the Lying-in Room. By Thomas Bull, ALD. Fcp. Zvo. 5j. The Maternal Manage- ment of Children in Health and Disease. By Thomas Bull, AI.D. Fcp. Zvo. 5J-. INDEX. Acton's Modem Cookery 40 Aird's Blackstone Economised 39 W/n'' J Hebrew Scriptures 29 Alpine Club Map of Switzerland 34 Alpine Guide (The) 34 Amos' s Jurisprudence 10 Primer of the Constitution 10 Anderson's Strength of Materials 20 ^/vwy/rf?;/^^ Organic Chemistry 20 Arnold's (Dr.) Christian Life 29 Lectures on Modem History 2 Miscellaneous Works 13 School Sermons 29 Sermons 29 (T. ) Manual of English Literature 1 2 Atherstone Priory 36 Autumn Holidays of a Country Parson ... 14 Ayre's Treasury of Bible Knowledge 39 Bacon's Essays, by Whately 11 Life and Letters, by Spedding ... 11 Works 10 Bain's Mental and Moral Science 12 on the Senses and Intellect 12 Emotions and Will 12 Baker's Two Works on Ceylon 34 ^a//'j Guide to the Central Alps 34 Guide to the Western Alps 35 Guide to the Eastern Alps 34 Bancroft's Native Races of the Pacific 23 Barry on Railway Appliances 20 Becker's Charicles and Gallus 35 Black's Treatise on Brewing 40 Blackley's German- English Dictionary 16 Blaine's Rural Sports 37 Bloxam s Metals 20 Botiltbce on 39 Articles 29 Bourne s Catechism of the Steam Engine . 27 Handbook of Steam Engine 27 Treatise on the Steam Engine ... 27 Improvements in the same 27 Bawdier s Family Shakspeare 37 Brantley-Moore s Six Sisters of the Valley . 36 Brandc's Dictionary of Science, Literature, and Art 23 Brinkley s AsVconomy 12 Browne's Exposition of the 39 Articles 29 Buckle's History of Civilisation 3 Posthumous Remains 12 Buckton's Health in the House 24 Bull's Hints to Mothers 4° Maternal Management of Children . 4° Burgomaster's Family (The) 34 Burke's Rise of Great Families 8 Burke s Vicissitudes of Families 8 Busk's Folk-lore of Rome 35 Valleys of Tirol 33 Cabinet Lawyer 4° Campbell's Norsvay 35 Gates' s Biographical Dictionary 8 and Woodward' s Encyclopsedia ... 5 Changed Aspects of Unchanged Tmths ... 14 Chesney's Indian Polity 3 Modern Military Biography 4 Waterloo Campaign 3 Codrington s Life and Letters 7 Colenso on Moabite Stone &c 32 's Pentateuch and Book of Joshua. 32 Collier's Demosthenes on the Crown 13 Commonplace Philosopher in Town and Country, by A. K. H. B 14 Comte s Positive Polity » Congreve's Essays 9 Politics of Aristotle n Conington's Translation of Virgil's ^neid 37 Miscellaneous Writings 13 Co«i'(Z«j(r(7K'5 Two French Dictionaries ... 15 Conybeare and Howson's Life and Epistles of St. Paul 30 Corneille's Le Cid 3^ Counsel and Comfort from a City Pulpit... 14 Cox's (G. W.) Aryan Mythology 4 Crusades 6 History of Greece 4 • Genenal History of Greece 4 School ditto 4 Tale of the Great Persian War 4 Tales of .Ancient Greece ... 36 Crawley's Thucydides 4 Creighfoti's Age of Elizabeth 6 Cresy's Encyclopcedia of Civil Engineering 27 Critical Essays of a Country Parson 14 C"rw;f«'.f Chemical Analysis 25 ■ Dyeing and Calico-printing 28 Culley's Handbook of Telegraphy 27 Davidson's Introduction to the New Tes- tament 3^ Z3'x-1 //^/V"'" -f Reform.ition 31 De Cai'sne and Lc Afaout's Botany 24 De Aforgan's VAnxAo^cs 13 De J'(;f./««77/<;' J Democracy in America... 9 Disraelis Lord George Bentinck 3 a6 42 NEW WORKS PUBLISHED BY LONGMANS & CO. /)/j-r(7r//'j Novels and Tales 35 Ddbson on tlie Ox 38 Dove's Law of Storms 18 Doyle's {\l.) Fairyland 25 Eastlakes Hints on Household Taste 26 Edwards's Rambles among the Dolomites 34 Nile 32 Elements of Botany 23 FJlicott's Commentary on Ephesians 30 Galatians 30 — ■ Pastoral Epist. 30 Philippians, &c. 30 Thessalonians . 30 Lectures on Life of Christ 29 Elsa : a Tale of the Tyrolean Alps 36 Evans' (J.) Ancient Stone Implements ... 23 (A. J.) Bosnia 33 Ewald s History of Israel 30 Antiquities of Israel 31 Fairbairii s Application of Cast and Wrought Iron to Building... 27 Information for Engineers 27 Life 7 Treatise on Mills and Millwork 27 Farrar's Chapters on Language 13 Families of Speech 13 Fiisiuygra77i on Horses and Stables 38 Forbes's Two Years in Fiji 33 Francis's Fishing Book 37 /vvrOT^y^'j Historical Geography of Europe 6 Frcshficld's Italian Alps 33 /■><?«a'^'j- English in Ireland 2 History of England 2 Short Studies 12 Gairdncr's Houses of Lancaster and York 6 Ganot' s Elementary Physics 20 ' Natural Philosophy 19 Gardiner's Buckingham and Charles 3 ■ Thirty Years' War 6 Gcffcken' s Church and State 10 German Home Life 13 Gibson's Religion and Science 29 Gilbert cr' Churchill' s Dolomites 34 Girdlesionc s Bible Synonyms 29 Goodeve s Mechanics 20 Mechanism 20 Grant' s Ethics of Aristotle 11 Graver Thoughts of a Country Parson 14 (7rcc77/f'j Journal 2 Griffin's Algebra and Trigonometry 20 Grohman's Tyrol and the Tyrolese 32 Grove (Sir W. R.) on Correlation of Phy- sical Forces 19 (F. C. ) The Frosty Caucasus 32 Gwilt's Encyclopaedia of Architecture 26 Harrison's Order and Progress 9 y/iz;//^^ on the Air 19 hart-wig's Aerial World 22 Polar World 22 Hartwigs Sea and its Living Wonders ... 22 Subterranean World 22 Tropical World 22 Hai/ghton s AmmAl Mechanics 20 //(y7«'(7ra''j Biographical and Critical Essays 7 Heathcole s Fen and Mere 28 Heine s Life and Works, by Stigand 7 Hclmholtz on Tone 23 Helmholtz's Scientific Lectures 19 Hebnsley's Trees, Shrubs, and Herbaceous Plants 24 //frj^cA^/'^ Outlines of Astronomy 18 Hiiuhliff's Over the Sea and Far Away ... 33 /^t'//i77/ a'' J Fragmentary Papers 21 Holms on the Army 4 Hullak's History of Modern Music 23 Transition Period 23 Hume's Kssnys 12 Treatise on Human Nature 12 /,%««r'j History of Rome 5 Indian Alps 32 Ingelaw's Poems 37 Jameson's Legends of Saints and Martyrs . 26 Legends of the ^Tadonna 26 Legends of the Monastic Orders 26 Legends of the Saviour 26 y^^ on Confession 30 Jenkins Electricity and Magnetism 20 y^Vcr/w'^ Lycidas of Milton 35 Jerrold's Life of Napoleon 2 Johnston's Geographical Dictionary 17 Jukes' s Types of Genesis 31 on Second Death 31 Kaliscli s Commentary on the Bible 30 AVZ/^'j Evidence of Prophecy 30 Kerl' s Metallurgy, by Crookes ^vlA Rohrig. 27 A'/V/f'j/ty'j' American Lectures 13 Kirby and Spence's Entomology 21 Kirkvians Philosophy 11 Knatc/ibull-Hugesscn's Whispers from Fairy-Land ... 35 Higgledy-piggledy 35 Lamartine' sToussaSni Louverture 36 Landscapes, Churches, &c. by A. K. H. B. 14 Lang's Ballads and Lyrics 36 Latham' s English Dictionary 15 Handbook of the Enghsh Lan- guage 15 Laughton s Nautical Surveying 19 Lawrence on Rocks 22 Lecky's History of European Morals 5 • Rationalism 5 Leaders of Public Opinion 8 Lee s Kesslerloch 22 L-^cfroy s Bermudas 33 Leisure Hours in Town, by A. K. H. B 14 Lessons of Middle Age, by A. K. H. B 14 Lewes' s Biographical History of Philosophy 6 NEW WORKS PUBLISHED BY LONGMANS & CO. 43 ZLd'TcvV on Authority 12 Liddell and Scott's Greek-English Lexicons 16 Liiidlcy and Moore's Treasury of Botany... 23 Lloyd's Magnetism 21 Wave-Theory of Light 21 Lonzman's (F. W. ) Chess Openings 40 German Dictionary ... 15 (W. Edward the Third. Lectures on History of England Old and New St. Paul's Loudon's Encyclopasdia of Agriculture ... 28 Gardening 28 Plants 24 Lubbock's Origin of Civilisation 22 Lyra Germanica 32 Macaulay s (Lord) Essays History of England -~ Lays of Ancient Rome 25, Life and Letters Miscellaneous Writings Speeches Works McCullock' s Dictionary of Commerce Macleod's Principles of Economical Philo- sophy Theory and Practice of Banking Elements of Banking Afademoiselle Mori Malct's Annals of the Road Malleson' s Genoese Studies Native States of India Marshall s Physiology Marshman' s History of India Life of Havelock Martineau's Christian Life H)'mns. 16 Maunder s Biographical Treasury Geographical Treasury Historical Treasury .Scientific and Literary Treasury Treasury of Knowledge Treasui-y of Natural History ... Maxwells Theory of Heat May's History of Democracy History of England Melville's Digby Grand General Bounce • — Gladiators Good for Nothing Holmby House Interpreter Kate Coventry Queens Maries Forest Trees and Woodland Alenzies' Scenery Merivale's Fall of the Roman Republic General History of Rome ... Romans under the Empire . jl/«-r//f<-/if'j Arithmetic and Mensuration... Allies on Horse's Foot and Horse Shoeing on Horse's Teeth and Stables Mill(].) on the Mind (J. S.) on Liberty on Representative Government Utilitarianism Autobiography 31 39 39 39 39 39 39 36 36 36 36 36 36 36 36 24 S 4 4 38 10 9 9 9 7 Mill's Dissertations and Discussions 9 Essays on Religion &c 29 Hamilton's Philosophy 9 S)'stem of Logic 9 Political Economy 9 Unsettled Questions 9 Miller's Elements of Chemistry 24 Inorganic Chemistry 20 Miuto's (Lord) Life and Letters 7 Mitchell's Manual of Assaying 28 Modern Novelist's Library 36 ./1/6'//j-^//'j ' Spiritual Songs ' 32 ^l/(?tfrj'j Irish Melodies, illustrated 26 Morant' s ij^vciQ Preservers 22 Morcll's Elements of Psychology 11 Mental Philosophy n Mailers Chips from a German Workshop. 13 Science of Language 13 Science of Religion 5 Neisoii on the Moon 18 New Reformation, by Theodoms 4 New Testament, Illustrated Edition 25 Northcott' s Lathes and Turning 26 0'Cw/c7;-'^ Commentary on Hebrews 31 — — Romans 31 -St. John 31 Owen's Comparative Anatomy and Ph3sio- logy of Vertebrate Animals 21 Pi/c^'c'j Guide to the PjTenees 35 Paget' s Naval Powers .' 28 Pattisoii's Casaubon 7 Payen's Industrial Chemistry 26 Pewt/ier's Comprehensive Specifier 40 Pierce's Chess Problems 40 Plaiiket's Travels in the Alps 33 Pole's Game of Whist 40 Preece & 5/i'rtti;7>/!/'j Telegraphv 20 P/-6V/(3'frf,«'j^'j Mastery of Languages 16 Present-Day Thoughts, by A. K. H. B. ... 14 Proctor s Astronomical Essays 17 ■ Moon 17 Orbs around Us 18 Other Worlds than Ours 18 ■ Saturn 17 Scientific Essays (New Series) ... 21 Sun 17 Transits of Venus 17 Two Star Atlases 18 Universe 17 Public Schools Atlas of Ancient Geography 17 Atlas of Modern Geography 17 Manual of Modern Geo- graphy 17 Rawlinson' s Parthia 5 Sassanians 5 Recreations of a Country Parson 14 AV</iT.nr'.r Dictionarj' of Artists 25 Reilly's Map of Mont Blanc 34 Monte Ros-T. 34 Reresbys Memoirs 8 41 NEW WORKS FLELisHED BY LONGMANS & CO. Reynardsoii s Down the Road 37 JiicKs Dictionary of Antiquities 15 JP/t'fr'j Rose Amateur's Guide 23 Hogers s Eclipse of Faith 30 Defence of Eclipse of Faith 30 Essays 9 Hoget's Thesaurus of English Words and Phrases iS Monald' s Fly-Fisher's Entomolojjy 38 JPtjj-i-f'ir' J Outlines of Civil Procedure 10 Jiothschild' s Israelites 30 JiiisscU's Recollections and Suggestions ... 2 ^S(z«a'<?r/j Justinian's Institutes 10 Sdvik o\\ Apparitions 13 on Primitive Faith 30 Schcllen s Spectrum Analysis 19 Scott's Lectures on the Fine Arts 25 Poems 25 Papers on Civil Engineering 28 ■Seaside Musing, by A. K. H. B 14 Sccbohnis Oxford Reformers of 1498 4 Protestant Revolution 6 ^^ci^/Z'i- Questions of the Day 31 Preparation for Communion 31 Stories and Tales 36 Thoughts for the Age 31 History of France 3 Shelley s Workshop Appliances 20 Short's Church History 6 Sivipson s Meeting the Sun 34 Smith's [Sydney] Essays 12 Wit and Wisdom 13 (Dr. R. A.) Air and Rain 19 Sottthey's Doctor 13 Poetical Works 37 Stanley's History of British Birds 22 Stephen's Ecclesiastical Biography 8 Stockmar's Memoirs 7 Stonehenge on the Dog 38 on the Greyhound 38 ^/i?//;*/ on Strains 28 Sunday Afternoons at the Parish Church of a University City, by A. K. H. B 14 Supernatm-al Religion 32 Svjhibournc's Picture Logic 11 Taylor's History of India 3 Manual of Ancient History 6 Manual of Modern History 6 (y«;rOT_v) Works, edited by £'(f^«. 31 Text-Books of Science 20 7y/0OTj(7«'5 Laws of Thought 11 jyz6';;/^'j Quantitative Analysis 20 Thorpe and Mit'ir s Qualitative Analysis ... Todd (A.) on Parliamentary Government... Trench's Realities of Irish Life Trollope's Barchester Towers Warden Twiss's Law of Nations Tyndall's American Lectures on Light ... Diamagnetism Fragments of Science Lectures on Electricity Lectures on Light ■■ Lectures on Sound Heat a Mode of Motion Molecular Physics 26' 2 13 36 Ucberweg's System of Logic Ure's Dictionary of Arts, Manufactures, and Mines Voltaire's Zaire. Walker on Whist Warburton's Edward the Third Watson's Geometry Watts's Dictionary of Chemistry , Webb's Objects for Common Telescopes , We in hold's Experimental Physics Wellington's Life, by Gleig Whately's English Synonymes Logic Rhetoric White and Riddle's Latin Dictionaries Wilcocks's Sea-Fisherman , Williams's Aristotle's Ethics , Wood's {T. G.) Bible Animals Homes without Hands Insects at Home ■ Insects Abroad Out of Doors Strange Dwellings Ephesus (J. T Wyatt's History of Prussia 'Vonge' s English-Greek Lexicons Horace Youatt on the Dog .. on the Horse Zellers Plato Socrates Stoics, Epicureans, and Sceptics... Zinunern's Life of Schopenhauer 7 LONDON : PRINTED BV SPOTTISWOODE AND CO., KEW-STKEET SQUARE AND PARLIAMENT STREET SCIENCE AND ENGINEERING LIBRARY University of California, San Diego Please Note: This item is subject to RECALL after one week. DATE DUE SE 7 (Rev. 7/82) UCSD Libr. UC SOUTHERN REGIONAL LIBRARY FACIL TY A A 001 414 018 SCIENCE AND ENGINEERING LIBRARY University of California, San Diego .-h-^ :; S i9?^ (ipR 1 '^ ^y^^ JUN 3 1974 APR 10 REC'O FEBTri975 — — fmiujm DEC31J978 I98G- ijiWTrmi JUfN 1 ^'fi' 7^ 4i PEC 3 IMliJ985__ \ - [ ^ 1985 MAR 1 7 1986 ?^