《AN ELEMENTARY TREATISE ON COORDINATE GEOMETRY OF THREE DIMENSIONS》

CHAPTER ⅠSYSTEMS OF COORDINATES.THE EQUATION TO A SURFACE1

1.Segments1

2.Relations between collinear segments1

3.Cartesian coordinates1

4.Sign of direction of rotation3

5.Cylindrical coordinates4

6.Polar coordinates4

7.Change of origin6

8.Point dividing line in given ratio7

9.The equation to a surface8

10.The equations to a curve12

11.Surfaces of revolution13

CHAPTER ⅡPROJECTIONS.DIRECTION-COSINES.DIRECTION-RATIOS15

12.The angles between two directed lines15

13.The projection of a segment15

14.Relation between a segment and its projection15

15.The projection of a broken line16

16.The angle between two planes17

17.Relation between areas of a triangle and its projection17

18.Relation between areas of a polygon and its projection18

19.Relation between areas of a closed curve and its projection19

20.Direction-cosines—definition19

21，22.Direction-cosines （axes rectangular）19

23.The angle between two lines with given direction-cosines22

24.Distance of a point from a line24

25，26.Direction-cosines （axes oblique）25

27.The angle between two lines with given direction-cosines27

28，29，30.Direction-ratios28

31.Relation between direction-cosines and direction-ratios30

32.The angle between two lines with given direction-ratios30

CHAPTER ⅢTHE PLANE.THE STRAIGHT LINE.THE VOLUME OF A TETRAHEDRON32

33.Forms of the equation to a plane32

34，35.The general equation to a plane33

36.The plane through three points34

37.The distance of a point from a plane35

38.The planes bisecting the angles between two given planes37

39.The equations to a straight line38

40.Symmetrical form of equations38

41.The line through two given points40

42.The direction-ratios found from the equations40

43.Constants in the equations to a line42

44.The plane and the straight line43

45.The intersection of three planes47

46.Lines intersecting two given lines53

47.Lines intersecting three given lines54

48.The condition that two given lines should be coplanar56

49.The shortest distance between two given lines57

50.Problems relating to two given non-intersecting lines61

51.The volume of a tetrahedron64

CHAPTER ⅣCHANGE OF AXES68

52.Formulae of transformation（rectangular axes）68

53.Relations between the direction-cosines of three mutually perpendicular lines69

54.Transformation to examine the section of a given surface by a given plane72

55.Formulae of transformation（oblique axes）75

EXAMPLES Ⅰ.76

CHAPTER ⅤTHE SPHERE81

56.The equation to a sphere81

57.Tangents and tangent plane to a sphere82

58.The radical plane of two spheres83

EXAMPLES Ⅱ.85

CHAPTER ⅥTHE CONE88

59.The equation to a cone88

60.The angle between the lines in which a plane cuts a cone90

61.The condition of tangency of a plane and cone92

62.The condition that a cone has three mutually perpendi-cular generators92

63.The equation to a cone with a given base93

EXAMPLES Ⅲ.95

CHAPTER ⅦTHE CENTRAL CONICOIDS.THE CONE.THE PARABOLOIDS99

64.The equation to a central conicoid99

65.Diametral planes and conjugate diameters101

66.Points of intersection of a line and a conicoid102

67.Tangents and tangent planes102

68.Condition that a plane should touch a conicoid103

69.The polar plane104

70.Polar lines105

71.Section with a given centre107

72.Locus of the mid-points of a system of parallel chords108

73.The enveloping cone108

74.The enveloping cylinder110

75.The normals111

76.The normals from a given point112

77.Conjugate diameters and diametral planes114

78.Properties of the cone119

79.The equation to a paraboloid122

80.Conjugate diametral planes123

81.Diameters124

82.Tangent planes124

83.Diametral planes125

84.The normals126

ExAMPLES Ⅳ.127

CHAPTER ⅧTHE AXES OF PLANE SECTIONS.CIRCULAR SECTIONS131

85.The determination of axes131

86.Axes of a central section of a central conicoid131

87.Axes of any section of a central conicoid134

88.Axes of a section of a paraboloid137

89.The determination of circular sections138

90.Circular sections of the ellipsoid138

91.Any two circular sections from opposite systems lie on a sphere139

92.Circular sections of the hyperboloids139

93.Circular sections of the general central conicoid140

94.Circular sections of the paraboloids142

95.Umbilics143

EXAMPLES Ⅴ.144

CHAPTER ⅨGENERATING LINES148

96.Ruled surfaces148

97.The section of a surface by a tangent plane150

98.Line meeting conicoid in three points is a generator152

99.Conditions that a line should be a generator152

100.System of generators of a hyperboloid154

101.Generators of same system do not intersect155

102.Generators of opposite systems intersect155

103.Locus of points of intersection of perpendicular genera-tors156

104.The projections of generators156

105.Along a generator θ±φ is constant158

106.The systems of generators of the hyperbolic paraboloid161

107.Conicoids through three given lines163

108.General equation to conicoid through two given lines163

109.The equation to the conicoid through three given lines163

110，111.The straight lines which meet four given lines165

112.The equation to a hyperboloid when generators are co-ordinate axes166

113.Properties of a given generator167

114.The central point and parameter of distribution169

EXAMPLES Ⅵ.172

CHAPTER ⅩCONFOCAL CONICOIDS176

115.The equations of confocal conicoids176

116.The three confocals through a point176

117.Elliptic coordinates178

118.Confocals cut at right angles179

119.The confocal touching a given plane179

120.The confocals touching a given line180

121.The parameters of the confocals through a point on a central conicoid181

122.Locus of the poles of a given plane with respect to confocals181

123.The normals to the three confocals through a point182

124.The self-polar tetrahedron183

125.The axes of an enveloping cone183

126.The equation to an enveloping cone184

127.The equation to the conicoid184

128.Corresponding points186

129.The foci of a conicoid187

130.The foci of an ellipsoid and the paraboloids189

EXAMPLES Ⅶ.193

CHAPTER ⅪTHE GENERAL EQUATION OF THE SECOND DEGREE196

131.Introductory196

132.Constants in the equation196

133.Points of intersection of line and general conicoid197

134.The tangent plane198

135.The polar plane201

136.The enveloping cone202

137.The enveloping cylinder203

138.The locus of the chords with a given mid-point203

139.The diametral planes204

140.The principal planes204

141.Condition that discriminating cubic has two zero-roots206

142.Principal planes when one root is zero206

143.Principal planes when two roots are zero207

144.The roots are all real208

145.The factors of （abc fgh）（xyz）2-λ（x2＋y2＋z2）209

146.Conditions for repeated roots210

147.The principal directions212

148.The principal directions at right angles212

149.The principal directions when there are repeated roots212

150.The transformation of（abc fgh）（xyz）2214

151.The centres215

152.The determination of the centres216

153.The central planes216

154.The equation when the origin is at a centre217

155-161.Different cases of reduction of general equation219

162.Conicoids of revolution228

163.Invariants231

EXAMPLES Ⅷ.233

CHAPTER ⅫTHE INTERSECTION OF TWO CONICOIDS.SYSTEMS OF CONICOIDS238

164.The quartic curve of intersection of two conicoids238

165.Conicoids with a common generator239

166.Conicoids with common generators241

167.The cones through the intersection of two conicoids245

168.Conicoids with double contact246

169.Conicoids with two common plane sections248

170.Equation to conicoid having double contact with a given conicoid248

171.Circumscribing conicoids249

172.Conicoids through eight given points251

173.The polar planes of a given point with respect to the system251

174.Conicoids through seven given points252

EXAMPLES Ⅸ.253

CHAPTER ⅩⅢCONOIDS.SURFACES IN GENERAL257

175.Definition of a conoid257

176.Equation to a conoid257

177.Constants in the general equation259

178.The degree of a surface260

179.Tangents and tangent planes261

180.The inflexional tangents261

181.The equation ζ＝f（ξ，η）262

182.Singular points263

183.Singular tangent planes265

The anchor-ring266

The wave surface267

184.The indicatrix270

185.Parametric equations271

EXAMPLES Ⅹ.273

CHAPTER ⅩⅣCURVES IN SPACE275

186.The equations to a curve275

187.The tangent275

188.The direction-cosines of the tangent277

189.The normal plane277

190.Contact of a curve and surface278

191，192.The osculating plane279

193.The principal normal and binormal282

194.Curvature284

195.Torsion284

196.The spherical indicatrices285

197.Frenet’s formulae285

198.The signs of the curvature and torsion288

199.The radius of curvature288

200.The direction-cosines of the principal normal and binormal289

201.The radius of torsion289

202.Curves in which the tangent makes a constant angle with a given line291

203.The circle of curvature292

204.The osculating sphere292

205.Geometrical investigation of curvature and torsion298

206.Coordinates in terms of the arc301

EXAMPLES Ⅺ.303

CHAPTER ⅩⅤENVELOPES.RULED SURFACES307

207.Envelopes—one parameter307

208.Envelope touches each surface of system along a curve308

209.The edge of regression309

210.Characteristics touch the edge of regression309

211.Envelopes—two parameters311

212.Envelope touches each surface of system312

213.Developable and skew surfaces313

214.The tangent plane to a ruled surface315

215.The generators of a developable are tangents to a curve316

216.Envelope of a plane whose equation involves one para.meter316

217.Condition for a developable surface318

218.Properties of a skew surface320

EXAMPLES Ⅻ.322

CHAPTER ⅩⅥCURVATURE OF SURFACES326

219.Introductory326

220.Curvature of normal sections through an elliptic point326

221.Curvature of normal sections through a hyperbolic point327

222.Curvature of normal sections through a parabolic point329

223.Umbilics330

224.Curvature of an oblique section—Meunier’s theorem330

225.The radius of curvature of a given section331

226.The principal radii at a point of an ellipsoid332

227.Lines of curvature333

228.Lines of curvature on an ellipsoid333

229.Lines of curvature on a developable surface333

230.The normals to a surface at points of a line of curvature334

231.Lines of curvature on a surface of revolution335

232.Determination of the principal radii and lines of curvature337

233.Determination of umbilics342

234.Triply-orthogonal systems，—Dupin’s theorem344

235.Curvature at points of a generator of a skew surface346

236.The measure of curvature346

237.The measure of curvature is 1/ρ1ρ2347

238.Curvilinear coordinates348

239.Direction-cosines of the normal to the surface349

240.The linear element350

241.The principal radii and lines of curvature350

EXAMPLES ⅩⅢ.354

CHAPTER ⅩⅦASYMPTOTIC LINES—GEODESICS358

242.Asymptotic lines358

243.The differential equation of asymptotic lines358

244.Osculating plane of an asymptotic line359

245.Torsion of an asymptotic line359

246.Geodesics362

247.Geodesics on a developable surface363

248.The differential equations of geodesics363

249.Geodesics on a surface of revolution365

250.Geodesics on conicoids367

251.The curvature and torsion of a geodesic369

252.Geodesic curvature370

253.Geodesic torsion373

EXAMPLES ⅩⅣ.375

INDEX378