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Casey J. — A treatise on the analytical geometry 


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Abscissae 4
Allersma 218
Andre Desire 28
Angle between asymptotes 166 268
Angle between central vector to, and normal at, a point on ellipse 244
Angle between focal radius vector and tangent 221
Angle between focal vectors bisected by tangent and normal 221
Angle between tangents to a parabola in terms of their lengths and chord of contact 191
Angle between two lines given by a single equation 53
Angle between two lines whose cartesian equations are given 36 37
Angle between two lines whose cartesian equations are given, same for trilinear equations 73
Angle between two tangents to an ellipse 224 234
Angle between two tangents to an ellipse, same expressed in terms of focal vectors to points of intersection 225
Angle of intersection of two circles or curves 107
Angle of intersection of two parabolas 199
Angle, eccentric 206
Angle, first and second, of Steiner 426
Angle, intrinsic 175 177
Angle, intrinsic, same half polar angle of point on parabola 191
Angle, subtended at focus by portion of variable tangent intercepted by two fixed tangents 238
Angle, the Brocard 64 409 411 459
Angles, eccentric, of extremities of conjugate diameters how related 209
Angles, sum of for four concyclic points on conic 241 280
Angles, theorems concerning, how projected 353
Anharmonic ratio of double points and homologous point pairs 376
Anharmonic ratio of four collinear points 55 163
Anharmonic ratio of four conics of a pencil of conics, six of four points 59
Anharmonic ratio of four eollinear points equal to that of pencil formed by their four polars 106
Anharmonic ratio of four lines whose equations are given 59
Anharmonic ratio of four points in which tangent at any variable point meets tangents at four fixed points on conic 344
Anharmonic ratio of four rays of a pencil 57
Anharmonic ratio of four tangents to a conic 385
Anharmonic ratio of pencil formed by two legs of an angle and the isotropic lines of vertex 353
Anharmonic ratio of pencil from any point of conic through four fixed points, to the four points 473
Anharmonic ratio of pencil from variable point to four fixed points on conic 343
Anharmonic ratio of pencil same as that of the four points in which any transversal is cut by rays 58
Anharmonic ratio of pencil unaltered by projection 351
Anharmonic ratio of points in two projective rows 372
Anharmonic ratio of rays in two projective pencils 375
Anharmonic ratio same expressed by trigonometrical ratios 60
Anomaly eccentric 206
Anomaly true 189 236
Antifoci 311 512
Antiparallel 77
Apollonius, circles of 146
Apollonius, first and second theorems of 210
Arcs of conics, theorems concerning 322 323 324
Area of hyperbola and hyperbolic sector 273—275
Area of parabolic sector 199
Area of parallelogram circumscribed to ellipse 210
Area of parallelogram formed by asymptotes and parallels to* them through any point on curve 269
Area of triangle formed by a given line, and a given line pair 80
Area of triangle formed by asymptotes and any tangent 271
Area of triangle formed by asymptotes and normal at any point on an equilateral hyperbola 282
Area of triangle formed by focus and two points on parabola 196
Area of triangle formed by joining extremities of conjugate semidiameters 210 259
Area of triangle formed by three lines whose equations are given 81
Area of triangle formed by three points in tripolar coordinates 306
Area of triangle formed by three points on a conic 11 12
Area of triangle formed by three points on, or three tangents to, an equilateral hyperbola 283
Area of triangle formed by three tangents to a parabola half that formed by points of contact 179
Area of triangle formed by three tangents to hyperbola 271
Area of triangle formed by two tangents and chord of contact 104 106 240
Area of triangle or polygon in terms of coordinates of vertices 10 79
Area of triangle selfconjugate with respect to, inscribed in, or circumscribed to, a conic 243
Area of triangle which is the harmonic transformation of a given triangle 299
Area of triangle which is the pedal triangle of a given point 297
Area of triangle which is the polar reciprocal of a given triangle 299
Area, conic given by general equation 331
Areas, signs of 2
Argument 24
Aronhold's notation 333
Artzt 292 537
Asymptotes are selfconjugate 167
Asymptotes equation of, for hyperbola 268 279
Asymptotes of conic given by general equation 166 465
Asymptotes of equilateral hyperbola are at right angles 106 268
Asymptotes, chord of contact of 167
Asymptotes, constant length intercepted on, by joins of variable to two fixed points on cuivc 271
Asymptotes, defined 166
Asymptotes, divided homographically by parallels to from a series of points on curve 280
Asymptotes, equal intercepts on any chord between curve and 270
Asymptotes, equation of, differs from equation to curve by a constant 268
Asymptotes, hyperbola referred to, as axes 167 269
Asymptotes, intersect in centre 167
Asymptotes, lines joining extremities of any diameter to extremities of conjugate are parallel to 268
Asymptotes, points of intersection of any tangent with, and two foci are con cyclic 281
Asymptotes, polar of any point on either is parallel to that one 281
Asymptotes, secant of half angle between, gives eccentricity 268
Axes of conic given by general equation 519
Axes of conics 158
Axes of conics confocal with a given one, and passing through a given point 232 233
Axes of parabola 158 159
Axes of perspective 72 130
Axes of similitude 119
Axes, are parallel to a pair of conjugate diameters of conic in whose equation coefficient of xy vanishes 157
Axes, lengths of, for conic given by general equation 330
Axes, magnitude and direction of, given two conjugate diameters of conic 210 271
Axes, radical 115
Axes, radical, of three circles are concurrent 116
Axes, rectangular and oblique 4
Axes, transverse and conjugate 203
Bisectors of angles between lines given by a single equation 53 54 465
Bisectors of sides or angles of a triangle are concurrent 62
Boscovich, method of generating conics 205 253
Brianchon's theorem 147 319
Briot and Bouquet 21
Brocard 166 174 179 195 198 441 459
Burnside 225
Catncart 510
Cayley 468 499 539
Centre of circle 97
Centre of circle, cutting three circles orthogonally 117
Centre of conic 153 154 203 251
Centre of curvature 185 186 216 264
Centre of incircle, in terms of coordinates of vertices and lengths of sides 16
Centre of inversion 408 411
Centre of involution 379
Centre of parabola 154
Centre of perspective 72 130
Centre of reciprocation 385
Centre of similitude 118 393
Centre of similitude, six of three circles, lie three by three on four right lines 119
Centre recherche of 154
Centre reduction of general equation to 155
Centre, line of centres 155
Centre, pole of line at infinity 167
Centre, radical 117
Centre, theory of mean 14—16
Chasles 219 225 235 324 343
Chord of conic which touches confocal, proportional to square of parallel semi diameter 225
Chord of contact of two tangents to a conic 183 223 256
Chord, joining two points on circle 102 130 133
Chord, joining two points on conic 175 208
Chord, locus of pole of, subtending a right angle at a fixed point 104 195 277
Chord, through a focus 183 222 277
Chords of contact of common tangents to two circles 103
Chords of intersection of two conics 213
Chords, conjugate 158
Chords, locus of middle points of parallel 155 156 179 208 255
Chords, supplemental 213
Circle director 164
Circle of curvature 185 312
Circle of inversion 104
Circle of reciprocation 386
Circle of similitude 119 395
Circle through 3 points 110 111 136
Circle through feet of perpendiculars 144
Circle touching three others 110 111 121
Circle, auxiliary 206 266
 Circle, Boscovich 205
Circle, Brocard 408 411 422 524
Circle, centre of 117
Circle, circum, in barycentric coordinates 131
Circle, circum, is polar conic of symmedian point 127
Circle, circum, of polygon 129
Circle, circum, of triangle formed by three tangents to a parabola, passes through focus 178 190
Circle, circum, of triangle of reference 128
Circle, circumscribed to quadrilateral 143
Circle, cutting three given circles at given angles, or orthogonally 108 149
Circle, diameter of, cutting three given circles orthogonally 148
Circle, Dr. Hart's 147
Circle, equation of 96 108—111
Circle, equation of tangents from any point to 103
Circle, equation of, on join of two points as diameter 77 150
Circle, equation of, referred to two tangents and chord of contact 143 342
Circle, focal 311
Circle, geometrical representation of power of point with respect to 98
Circle, having as diameter chord of contact of tangents to a given circle from a given point 102
Circle, having as diameter the intercept made by a given circle on a given line 100
Circle, having side of triangle of reference as diameter 144
Circle, having triangle of reference as autopolar triangle 339
Circle, inscribed in triangle of reference 131
Circle, Joachimsthal's 185 187 188 218 264
Circle, Lemoine 419
Circle, Lionnets 401
Circle, ninepoints 125 126 147 303 434 435 444 445 451 454
Circle, ninepoints, touches both in and excircles 125 138 149
Circle, orthocentroidal 436
Circle, orthogonal projection of 206
Circle, orthoptic 164 509 510
Circle, osculating 189
Circle, pedal of a given point 137
Circle, Steiner's 315
Circle, tangential equation of 138 139 140 141 143
Circles of Apollonius 146
Circles, a system of tangential 120
Circles, all pass through cyclic points 303
Circles, annex 401
Circles, coaxal system of 114
Circles, concentric, have double contact at infinity 328
Circles, concentric, when 97
Circles, described to triangle of reference 132
Circles, equation of, in pairs, touching three given circles 120
Circles, Fuhrman's 431
Circles, M'Cay's 427 454
Circles, Mutual power of two 107 108
Circles, Neuberg's 423—427 443 444 452 459
Circles, Tucker's 415 419 421
Clehsch 333 337 462
Coates' theorem 68 93 127
Complex variables 24
Concomitant, mixed 463 517
Condition that should be antiparallel to 77
Condition that a circle may be cut orthogonally by four given circles 110
Condition that a line may he cut harmonically by two conics 371
Condition that a line should be cut in involution by three conics 505
Condition that a line should be perpendicular to itself 76
Condition that a line should cut a conic in two points which subtend a right angle at the origin 348
Condition that a line should pass through a given point 86 89
Condition that a line should pass through origin 33
Condition that a line should touch a conic 161 163 261 335
Condition that a triangle may be circumscribed to one conic and hare its vertices on three other conics 485
Condition that a triangle may be inscribed in one conic, and have its sides touching three other conics 484
Condition that a triangle selfconjugate with respect to one conic may be inscribed in, or circumscribed to, another 475
Condition that any number of circles may have one common tangential circle 122
Condition that four circles should cut a fifth orthogonally or be tangential to it 112
Condition that four conics should cut a fifth orthogonally or be tangential to it 497
Condition that four convergent rays should form a harmonic pencil 59
Condition that four points on conic should be concyclic 241 280
Condition that four points should be concyclic 110
Condition that general equation should represent a circle (in Cartesian coordinates) 96
Condition that general equation should represent a circle in barycentric coordinates 137
Condition that general equation should represent a circle in normal coordinates 134
Condition that general equation should represent an ellipse, a parabola, or hyperbola 340
Condition that general equation should represent an equilateral hyperbola or a parabola 509
Condition that general equation should represent two right lines 51 52 334
Condition that intercept made by circle on line should subtend a right angle at a fixed point 100
Condition that joins of vertices of triangle of reference to points in which general conic meets sides should form two concurrent triads 527
Condition that normals at three points on ellipse, should be concurrent 215
Condition that normals at three points on parabola should be concurrent 188
Condition that three conics may have a common point 517
Condition that three lines should he concurrent 48
Condition that three points may be collinear 8
Condition that threepoint pairs should form an involution 379
Condition that triangle of reference may be in perspective with one the coordinates of whose vertices are given 82
Condition that two circles should be concentric 97
Condition that two circles should cut orthogonally 107
Condition that two circles should touch 107
Condition that two conics are so related that a triangle may be inscribed in one and circumscribed to the other 468
Condition that two conics inscribed in the same conic should cut orthogonally 495
Condition that two conics should be homothetic 326 327
Condition that two conics should touch, osculate 309 469 471
Condition that two lines should be parallel 36 74
Condition that two lines should he at right angles 36 37 53 74
Condition that two lines should intersect on a conic 336 520
Condition that two points may be conjugate with respect to a conic 334
Condition that two points may be conjugate with respect to two lines 336
Condition that twopoint pairs should be harmonic conjugates 57 368
Condition that when general equation represents two lines they should be parallel or perpendicular 77
Cone, right and oblique, sections of 363
Confocal conics 224—227 232—236 239 247 248 279 311 324 331 332
Confocal conics are inscribed in the same imaginary quadrilateral 311
Confocal conics, cut at right angles 333
Confocal conics, general equation of 312 512
Confocal conics, length, of are intercepted between tangents from 322—324
Conic through, five points, description of 172
Conic, eight points of contact of common tangents to two conics lie on a 488
Conic, equation of, given focus and three tangents 391
Conic, fourteenline 490
Conic, fourteenpoint 487
Conic, isoptic curve of 184 520
Conic, isotomic transformation of 296
Conic, ninepoint of quadrangle 165
Conic, number of conditions sufficient to determine a 170
Conic, polar conic of point, and pole of 27
Conic, which reciprocates the Brocard ellipse into Kiepert's hyperbola 509
Conics, classification of 165
Conics, conjugate with respect to quadrilateral 514
Conics, diametral 445
Conics, for which and vanish 482
Conics, harmonic properties of 479
Conics, harmonic system of 482
Conics, homothetic 326
Conics, invariant angles of two 471
Conics, invariant theory of 462
Conics, mutual power of two 493
Conics, orthogonal 499
Conics, osculation of two 471
Conics, pencil and net of, tangential and trilinear 463
Conics, point and line harmonic conics of two 371
Conjugate area of triangle included by 210 259
Conjugate diameters 157 258
Conjugate diameters are parallel to a pair of supplemental chords 213
Conjugate diameters, equation of conic referred to as axes 211 259
Conjugate eccentric angles of extremities of 209
Conjugate hyperbola 257
Conjugate points and lines 106 334 336
Conjugate sum of squares of 209
Conjugates, harmonic 13 56 117 368 369
Conjugates, isogonal 63
Conjugates, isotomic 13 65
Constants 31
Contact, double 318 513
Contact, fourpointic 318 321
Contact, of different orders 309
Contravariants 508
Coordinates of double points 334
Coordinates of incentre of triangle in terms of coordinates of vertices and lengths of sides 16
Coordinates of orthocentre of triangle formed by tangents to ellipse and chord of contact 245



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