A Treatise on the Analytic Geometry of Three Dimensions

George Salmon

Hodges, Smith, and Company, 1865 - Geometry, Analytic - 520 pages

Contents

• THE POINT: 1
• Directioncosines of a lines: 7
• Every plane section of a surface of the nth degree is a curve of the nth degree: 14
• Every equation of the first degree represents a plane: 16
• Volume of tetrahedron the equations of whose faces are given: 22
• Conditions that a line may lie in a given plane: 28
• CHAPTER IV: 34
• Locus of harmonic means of radii through a point: 37
• A quadric has three principal diametral planes: 44
• Functions of coefficients which are unaltered by rectangular transformation: 50
• Paraboloids: 56
• Also sum of squares of projections of conjugate diameters on any line and on: 63
• RECTILINEAR GENERATORS: 69
• No two lines of the same system intersect: 71
• Four generators of one system cut any generator of the other in a constant: 77
• Method of finding equation of cone given its vertex and a curve through which: 83
• Properties of surfaces of revolution obtained by reciprocation: 89
• Four cones pass through the intersection of two quadrics: 96
• FOCI AND CONFOCAL SURFACES: 101
• Focal lines of a cone: 107
• Focal conics the limits of confocal surfaces: 114
• PD constant along the intersection of two confocals: 120
• The focal lines of these cones are the generators of the hyperboloid through: 126
• Locus of vertices of right cones enveloping a quadric: 132
• Locus of points of contact of parallel planes touching a series of confocals: 139
• Condition that a tetrahedron can be inscribed in one quadric having two opposite: 152
• Contravariants of a system of two quadrics: 158
• A system of confocals is touched by a common developable: 164
• Equation of sphere inscribed in a tetrahedron: 170
• Invariants of a system of three quadrics: 177
• CONES AND SPHEROCONICS: 183
• Sum of focal distances constant: 190
• Equation of a right cone: 196
• Section of surface by tangent plane has point of contact for a double point: 202
• Application of Joachimsthals method: 209
• Discriminant of a surface: 215
• Eulers formula: 221
• Conditions for an umbilic: 227
• Their differential equation: 234
• Geodesic lines defined: 240
• CHAPTER XII: 246
• Tangent planes to developables touch along a line: 253
• Nodal curve on developable: 260
• Their different species: 269
• Four quartics of second family through eight points: 278
• Singularities of a double curve connected with those of its complementary: 284
• Condition that four points may lie in a plane: 291
• Osculating right cone: 297
• Characteristics of polar developable: 303
• PD constant for a geodesic on a quadric: 309
• Chasless proofs of this theorem: 315
• Dr Harts proof of Mr Robertss expressions: 323
• Gausss theory of curvature of surfaces: 329
• CHAPTER XIII: 339
• Conoidal surfaces: 345
• Surfaces generated by lines parallel to a fixed plane: 352
• Differential equations of ruled surfaces: 356
• Partial differential equation of developables: 363
• Differential equation of geodesic on a cone: 369
• Nature of contact along any generator: 375
• Surfaces generated by a line which meets a curve twice and another curve once: 381
• CHAPTER XIV: 387
• Equation in elliptic coordinates: 394
• Its sections by principal planes: 402
• Lines of curvature of surface of elasticity: 408
• Cubics having double points: 415
• Corresponding points on the Hessian: 418
• Schäflis scheme for the twentyseven lines: 425
• Method of obtaining contravariants in five letters: 431
• CHAPTER XVI: 437
• And of triple tangents: 443
• Locus of points of contact of double tangent planes: 449
• Effect of multiple lines on degree of reciprocal: 458
• On triple orthogonal systems: 476
• On the order of systems of equations: 489
• On the properties of systems of surfaces: 507
• Its sections by principal planes 113: 515

Bibliographic information

• Title: A Treatise on the Analytic Geometry of Three Dimensions
• Author: George Salmon
• Edition: 2
• Publisher: Hodges, Smith, and Company, 1865
• Original from: Harvard University
• Digitized: Aug 16, 2007
• Length: 520 pages