The analytical geometry of the conic sections

Definitions ......Page 19 Formulae connecting Cartesians and Polars ......Page 24 Distance between two points ......Page 25 Area of triangle—Condition of collinearity of three points ......Page 27 Definitions ......Page 31 Relativity of equations of loci ......Page 32 Line through two given points ......Page 35 General linear equation ......Page 37 Standard forms ......Page 39 Length of perpendicular on a given line ......Page 47 Parallel lines. Perpendicular lines ......Page 50 Intersection of lines ......Page 53 Polar equation ......Page 56 Area of triangle ......Page 58 Angle between two lines ......Page 63 General equation representing two lines ......Page 66 Bisectors of angles between two lines ......Page 70 General equation of the circle ......Page 77 Constant rectangle of segments of chords through a point ......Page 79 Imaginary points ......Page 81 Tangent ......Page 84 Circle referred to its centre ......Page 85 Poles and polars ......Page 89 Conjugate points and lines ......Page 92 Equation of chord in terms of its middle point ......Page 93 Trigonometrical notation ......Page 95 Systems of circles—Coaxial circles ......Page 98 Formulae of transition ......Page 109 Invariants ......Page 113 Oblique axes ......Page 116 Definitions ......Page 120 General equation ......Page 121 Standard forms for parabola, ellipse and hyperbola ......Page 122 Criteria of discrimination for General Equation ......Page 137 Tangent. Chord of contact. Poles and polars. Conjugate points and lines ......Page 140 Equation of chord in terms of its middle point ......Page 148 Pair of tangents ......Page 150 Special equations and properties ......Page 154 Normals ......Page 157 Equation of parabola referred to diameter and tangent ......Page 159 The parabola and the general equation ......Page 161 Special equations and properties ......Page 172 Director circle ......Page 175 Eccentric angle ......Page 177 Properties of normals ......Page 180 Conjugate diameters ......Page 187 Special equations ......Page 203 Asymptotes ......Page 205 Conjugate hyperbola ......Page 210 Conjugate diameters ......Page 212 Hyperbola referred to its asymptotes ......Page 216 General polar equation ......Page 221 Focus pole ......Page 222 Chord, tangent and normal ......Page 224 Intersections of two conies ......Page 231 Contact of conies ......Page 232 Conic through five points ......Page 233 Conies through points of intersection of two conies ......Page 235 Pair of tangents ......Page 238 Foci of conies ......Page 239 Equation of axes ......Page 242 Length of axes ......Page 244 Eccentricity ......Page 248 Director circle ......Page 251 A theorem of Newton's ......Page 254 Contact of conies and circles of curvature ......Page 256 Conic referred to two tangents ......Page 263 System of conies through four points ......Page 265 Definitions and propositions respecting similarity ......Page 274 Confocals and their properties ......Page 283 Definitions ......Page 291 Formulae of transition from Cartesians to areals ......Page 292 Area of triangle—Distance between points—Equation of line ......Page 294 The line at infinity ......Page 297 Parallel lines—Length of perpendicular ......Page 298 Special form of equation of line ......Page 301 General equation of second degree ......Page 302 Criteria of discrimination ......Page 305 Tangents &c ......Page 308 Centre. Foci, Axes. Asymptotes ......Page 311 Conies circumscribing the triangle of reference ......Page 315 Conies inscribed in the triangle of reference ......Page 316 Conies for which the triangle of reference is self-polar ......Page 318 Equations of special circles ......Page 319 The circular points at infinity ......Page 322 Radical axis ......Page 324 Orthogonal circles ......Page 326 Definitions ......Page 335 Geometrical interpretation ......Page 337 Transformation to Cartesians ......Page 338 General equation of second degree ......Page 339 Criteria of discrimination ......Page 341 Foci. Axes ......Page 343 Trilinears ......Page 344 Cartesians as a homogeneous system ......Page 345 Polar reciprocals ......Page 347 Conies expressed by a single parameter ......Page 349 The equations of two conies ......Page 350 Double contact ......Page 351 Single contact ......Page 353 Three point contact. Four point contact ......Page 354 Analytical representation of cross ratios ......Page 360 Harmonic conjugates ......Page 362 The representation of four coplanar points ......Page 364 Conies through four points ......Page 366 The representation of four coplanar lines ......Page 367 Conies touching four lines ......Page 369 Constant cross-ratio property of conies ......Page 370 Involution ......Page 372 Homographic ranges ......Page 377 Invariants of a single conic ......Page 387 Invariants of two conies ......Page 392 Invariants and projection ......Page 393 Illustrations ......Page 397 Conditions for single, double, three point contact ......Page 402 Invariants and reciprocation ......Page 406 Reciprocal relation between point and tangential equations ......Page 412 Envelopes ......Page 416 Tangential equation of conies touching the common tangents of two conies ......Page 420 Points of intersection of two conies ......Page 421 Equation of four common tangents ......Page 423 Circular points at infinity ......Page 426 Confocal conies ......Page 428 Foci of conies ......Page 430 Condition for rectangular hyperbola ......Page 432 Perpendicularity of lines ......Page 433 Definition ......Page 440 Geometrical interpretation ......Page 441 The F conic ......Page 442 Conditions for double contact ......Page 445 Polar reciprocal ......Page 446 The Ф conic ......Page 450 Contravariants ......Page 452 Jaoobian of three conies ......Page 454 The cubic covariant of two conies ......Page 457 The cubic contravariant ......Page 458