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Softcover ISBN:  9780821829134 
Product Code:  CHEL/223.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470469948 
Product Code:  CHEL/223.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9780821829134 
eBook ISBN:  9781470469948 
Product Code:  CHEL/223.S.B 
List Price:  $134.00$101.50 
MAA Member Price:  $120.60$91.35 
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Book DetailsAMS Chelsea PublishingVolume: 223; 1969; 379 ppMSC: Primary 14; Secondary 51;
From the Preface by C.M. Jessop:
“The important character of the extensive investigations into the theory of linegeometry renders it desirable that a treatise should exist for the purpose of presenting these investigations in a form easily accessible to the English student of mathematics. With this end in view, the present work on the Line Complex has been written.”Readership 
Table of Contents

Front Cover

PREFACE.

CONTENTS.

INTRODUCTION.

i. Double Ratio.

ii. Correspondence.

iii. United Points.

iv. Involution.

v. Harmonic Involutions.

vi. Correspondences on different lines.

vii. A ( 1, 1)

viii.

ix. Correspondence between the points of a conic and the lines of a plane pencil.

x. Involution on a conic.

xi .. Corresponding Sheaves.

xii. Systems of Lines.

xiii. Collinear Plane Systems.

xiv. Collineation of systems of space.

xv. General Involution.

xvi. Involution on a twisted cubic.

xvii. [2, 2] Correspondences.

CHAPTER I. SYSTEMS OF COORDINATES.

1. Definition of complex and congruence

29. Systems of coordinates

3. Homogeneous Coordinates.

4.

5. Intersection of two lines.

6. Coordinates of Plücker and Lie.

7. Transformation of Coordinates.

8. Generalized Coordinates.

9. Coordinates of Klein.

1013 Pencil, sheaf and plane system of lines, von Staudt's theorem

10. Plane Pencil of Lines.

11. Double Ratio of four lines of a pencil.

12. Von Staudt's Theorem.

13. Sheaf and plane system of lines.

14. Closed system of 16 points and 16 planes*.

CHAPTER II. THE LINEAR COMPLEX.

1516 The linear complex

15.

16. The Linear Complex.

17. Polar Lines.

18. The Invariant of a linear complex*.

1921 The special complex. coordinates of polar lines

19. The Special Complex.

20. Coordinates of polar lines.

21. Relations between the functions ω and Ω

22. Diameters.

23. Reduction of the complex to its simplest form.

24. Two complexes have one pair of polar lines in common.

2527 Complexes in Involution

25. Complexes in Involution.

26. Three complexes in Involution.

27. Six complexes mutually in Involution*.

28. Transformation of coordinates.

29. The fifteen principal tetrahedra.

CHAPTER III. SYNTHESIS OF THE LINEAR COMPLEX.

3032 Determination of the complex from given conditions

30.

31.

32.

33. Every linear complex contains two lines of any regulus

3436 Collineation and Reciprocity

34. Correlations of Space. Collineation and Reciprocity*.

35.

36. Involutory Reciprocity.

3738 The Null System

37. Null System*.

38.

39. Method of Sylvester*.

40. Automorphic Transformations.

4143 Curves of a linear complex

41. Ruled surfaces and curves of a linear complex.

42.

43.

4445 Polar Surfaces and curves

44. The polar surface.

45.

46. Complex equation of the quadric.

4750 Simultaneous bilinear equations

47. Simultaneous bilinear equations*.

48. Linear transformations which leave a quadric unaltered in form.

49. Collineations which leave a linear complex unaltered in form.

50. Reciprocal transformations.

CHAPTER IV. SYSTEMS OF LINEAR COMPLEXES.

51. The linear congruence

5253 Double ratio of two and of four complexes

52. Double ratio of two complexes

53. Double ratio of four complexes.

54. Special congruence.

5556 Metrical Properties.

55. Metrical Properties.

56. Axes of a system of two terms.

57. The cylindroid.

58. System of three terms.

5960 The generators and tangents of a quadric

59. Expression of the coordinates of a generator of a quadric in terms of one parameter.

60. Complex equation of a quadric.

61. The ten fundamental quadrics.

62. Closed system of sixteen points and planes*.

63. Systems of four and of five terms.

64. Invariants of a system of complexes.

65. Property of the six residuals

CHAPTER V. RULED CUBIC AND QUARTIC SURFACES.

66. Ruled surfaces

67. Ruled Cubics.

68. Ruled Quartics.

69. Ruled quartics of zero deficiency

7072 Analytical classification of Voss

70. Ruled Surfaces whose deficiency is zero.

71. Ruled cubics.

72. Ruled quartics of zero deficiency.

CHAPTER VI. THE QUADRATIC COMPLEX.

73. The quadratic complex

74. The tangent linear complex.

75. Singular points and planes of the complex.

76. Singular Lines.

7778 Identity of the surfaces Φ1 and Φ2

77. Singular points and planes of any line.

78.

79. Polar Lines.

80. The singular lines of the complex of the second and third orders.

81. The complex in Plucker coordinates.

82. The singular surface.

83. Double tangents.

84. A Kummer's Surface and one singular line determine one 02•

85. The singular surface is a general Kummer Surface.

86. The Complex Surfaces of Plücker.

87. Normal form of the equation of a quadratic complex.

8889 Special and harmonic complexes

88. Complex equation of a quadric.

89. Harmonic Complex.

90. Symbolic form of the equation of a quadric in plane coordinates.

91. Plücker surfaces and singular surface of the complex.

CHAPTER VII. SPECIAL VARIETIES OF THE QUADRATIC COMPLEX.

9293 The tetrahedral complex

92. The Tetrahedral Complex.

93. Equation of the tetrahedral complex.

9495 Reguli of the complex.

94. Reguli of the complex.

95.

9697 Other methods of formation of the complex

96. Second method of formation of the complex.

97. Third method of formation of the complex.

98101 Complexes derived from projective pencils

98

99. Complexes determined by two bilinear equations.

100.

101.

102. Reye's Complex of Axes.

103. Ditferential Equation of the Complex.

104106 Curves of the complex

104. The line element.

105. Curves of the Tetrahedral Complex.

106. NonProjective Transformations of the Complex.

107109 The special quadratic complex

107. The Special Quadratic Complex.

108. System of two special complexes*.

109. Covariant tetrahedral complex.

110112 The harmonic complex

110. The Complex of Battaglini or Harmonic Complex*.

111. The Tetrahedroid.

112. There are ∞1 pairs of quadrics such that for each pair the Harmonic complex is the same as the given complex.

113. Painvin's Complex.

CHAPTER VIII. THE COSINGULAR COMPLEXES.

114115 The cosingular complexes

114.

115.

116. Correspondence between lines of cosingular complexes*.

117. The complexes R42, R4'2.

118. The congruence [2, 2].

119. Focal surface of the congruence.

120. Confocal congruences.

121. The quartic surface (02, A, A').

122. Projective formation of C2.

123. Caporali's Theorem.

124. Condition for (1, 1) correspondence in any coordinate system.

125. Equation of the complex referred to a special tetrahedron.

126. Cosingular complexes for this coordinate system.

127128 Involution of tangent linear complexes

127 Involution of tangent linear complexes.

128.

129. Conics determined in a plane by cosingular complexes.

130. Elliptic coordinates of a line.

131. Bitangent linear complexes.

132. Principal Surfaces.

133. Involutory position of two lines.

CHAPTER IX. POLAR LINES, POINTS, AND PLANES.

134135 Polar lines.

134. Polar lines.

135.

136139 Corresponding loci of polar lines.

136. Corresponding loci of polar lines.

137.

138.

139.

140. Polar planes and points of the complex.

141. Polar Point

142. The diameters of the complex.

143. The Centre of the complex.

CHAPTER X. REPRESENTATION OF A COMPLEX BY THE POINTS OF THREEDIMENSIONAL SPACE.

144. Representation of the lines of a qudratic complex by points of threedimensional space

145. The reguli of a congruence[2, 2].

146. Representation of the congruence (2, 2) by the points of a plane.

147. Representation of the lines of a linear complex by points of space.

CHAPTER XI. THE GENERAL EQUATION OF THE SECOND DEGREE.

148153 Reduction of the equation of a quadratic complex to a canonical form

148.

149.

150.

151.

152.

153.

154. Arbitrary constants of a canonical form.

155. Complexes formed by linear congruences.

156. Double Lines.

157158 The cosingular complexes and the correspondence between lines of two cosingular complexes

157. The Cosingular Complexes.

158. Correspondence between lines of cosingular complexes.

159. The singular surface of the complex.

160. Degree of a complex.

161214 Varieties of the quadratic complex

161. The varieties of the quadratic Complex.

162.

163.

164.

165.

166.

167.

168.

169.

170.

171.

172.

173.

174.

175.

176.

177.

178.

179.

180.

181.

182.

183.

184.

185.

186.

187.

188.

189.

190.

191.

192.

193.

194.

195.

196.

197.

198.

199.

200.

201.

202.

203.

204.

205.

206.

207.

208.

209.

210.

211.

212.

213. Number of constants in a canonical form.

214.

CHAPTER XII. CONNEXION OF LINE GEOMETRY WITH SPHERE GEOMETRY.

215. Coordinates of a sphere

216. Intersection of lines corresponds to contact of spheres.

217. Points of Λ correspond to minimal lines of Σ.

218219 Definition of a surface element. A surface element of Λ defines a surface element of Σ

218. Surface Element.

219. Corresponding surfaces in Λ and Σ,

220. Principal tangents and principal spheres.

221. Pentaspherical Coordinates.

CHAPTER XIII. CONNEXION OF LINE GEOMETRY WITH HYPERGEOMETRY.

222. Definition of point, line, hyperplane of space of four dimensions

223. Equations connecting lines of A and points of S4

224. Correlation of Schumacher*.

225. Correlatives of the lines of any plane system and sheaf of Λ.

226227 Metrical Geometry.

226. Metrical Geometry.

227. Automorphic transformations in Λ correspond to anallagmatic transformations of S4.

228. Principal Surfaces of A and Lines of Curvature of S4.

229. Line Geometry is point geometry of an S42 in an S5.

230. Line Geometry in Klein coordinates is point geometry of S4 with hexaspherical coordinates.

231232 The congruence (m, n)

231. Congruences of the mth order and nth class.

232. Rank of a congruence.

CHAPTER XIV. CONGRUENCES OF LINES.

233. Order and class of a congruence.

234. Halphen's Theorem.

235. Characteristic numbers of a congruence.

236. Focal points, planes and surface.

237. Degree and Class ofthe Focal Surface.

238. Singular Points.

239240 Determination of a ray by two coordinates

239. Expression of the coordinates of a ray in termsof two variables*.

240.

241248 Appplication of Schumacher's method of projection to determine the degree, class, and rank of the focal surface

241. Schumacher's method*.

242. Tangents to F m+n.

243. Triple secants of F.

244. The Focal Surface.

245. Degree and Class of the Focal Surface.

246. Double and Cuspidal curves of the focal surface.

247. Rank of the Focal Surface.

248. Determination of r and t for the intersection of two complexes.

CHAPTER XV. THE CONGRUENCES OF THE SECOND ORDER WITHOUT SINGULAR CURVES.

249. The rank of the congruence (2, n) is n2

250. The Surfaces (P).

251. Each singular point of the congruence is a double point of Φ.

252. Double rays of the congruence.

253. The class of a congruence (2, n) cannot be greater than seven.

254256 Number and distribution of the singular points

254. Number of singular points.

255. Distribution of the singular points*.

256. Conjugate singular points.

257. Equation of a surface (P).

258. Tetrahedral complexes of the congruences (2, n).

259. Nonconjugate singular points.

260262 Reguli of the congruences (2, n).

260. Reguli of the congruences (2, n).

261.

262.

263. Confocal congruences.

CHAPTER XVI. THE CONGRUENCE OF THE SECOND ORDER AND SECOND CLASS.

264. The congruence (2, 2) is contained in 40 tetrahedral complexes

265. Confocal congruences (2, 2).

266. Distribution of the Singular Points.

267. Every (2, 2) is included in 40 tetrahedral complexes*.

268. The Kummer Configuration.

269. The Weber groups.

270271 Reguli of the congruence.

270. Reguli of the congruence.

271. A congruence (2, 2) includes ten sets of ∞1 reguli.

272. Focal surface of the intersection of any two complexes.

273. Double rays of special congruences (2, 2)*.

CHAPTER XVII. THE GENERAL COMPLEX.

274. The general complex

275277 The Singular Surface.

275. The Singular Surface.

276.

277.

278. The Principal Surfaces.

279. Independent constants of the complex.

280. The Special Complex.

281283 Congruences and their Focal Surfaces*.

281. Congruences and their Focal Surfaces*.

282.

283. Degree and class of the Focal Surface.

284285 The ruled surface which is the intersection of three complexes

284. The ruled surface common to three complexes.

285. Rank of the surface.

286. Clifford's Theorem.

287288 Symbolic form of the equations of the complex and its singular surface

287. Symbolic form of the equation of the complex*

288. Symbolic forms for the Complex surface and Singular surface.

CHAPTER XVIII. DIFFERENTIAL EQUATIONS CONNECTED WITH THE LINE COMPLEX.

289. Application of the surface element to partial differential equations

290. The characteristic curves of a partial differential equation.

291. The Monge equation of a line complex.

292. The characteristics on an Integral surface are principal tangent curves.

293294 Partial Differential equation corresponding to a linecomplex

293. Form of the partial differential equation corresponding to a line complex.

294.

295. Contact transformations of space*.

296299 The trajectory circle. The equations D11, D12, D13.

296. The trajectory circle.

297. Partial differential equations whose characteristics are geodesics.

298.

299.

300. The complex of normals.

301302 Partial differential equations of the second order associated with line and spherecomplexes.

301. Partial differential equations of the second order connected with line and spherecomplexes.

302.

303304 Partial differential equations of the second orderon whose Integral surfaces both sets of characteristics areprincipal tangent curves or lines of curvature.

303. Partial differential equations of the second orderon whose Integral surfaces both sets of characteristics areprincipal tangent curves or lines of curvature.

304. The curves s and σ of a D22".

305306 Equations D21", D22" with one and with two general first integrals

305. To determine when a D2111 or a D22" has a general first integral.

306. D21" and D22" with two general first integrals.

307. Application to the quadratic complex*.

MISCELLANEOUS RESULTS AND EXERCISES.

INDEX.

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From the Preface by C.M. Jessop:
“The important character of the extensive investigations into the theory of linegeometry renders it desirable that a treatise should exist for the purpose of presenting these investigations in a form easily accessible to the English student of mathematics. With this end in view, the present work on the Line Complex has been written.”

Front Cover

PREFACE.

CONTENTS.

INTRODUCTION.

i. Double Ratio.

ii. Correspondence.

iii. United Points.

iv. Involution.

v. Harmonic Involutions.

vi. Correspondences on different lines.

vii. A ( 1, 1)

viii.

ix. Correspondence between the points of a conic and the lines of a plane pencil.

x. Involution on a conic.

xi .. Corresponding Sheaves.

xii. Systems of Lines.

xiii. Collinear Plane Systems.

xiv. Collineation of systems of space.

xv. General Involution.

xvi. Involution on a twisted cubic.

xvii. [2, 2] Correspondences.

CHAPTER I. SYSTEMS OF COORDINATES.

1. Definition of complex and congruence

29. Systems of coordinates

3. Homogeneous Coordinates.

4.

5. Intersection of two lines.

6. Coordinates of Plücker and Lie.

7. Transformation of Coordinates.

8. Generalized Coordinates.

9. Coordinates of Klein.

1013 Pencil, sheaf and plane system of lines, von Staudt's theorem

10. Plane Pencil of Lines.

11. Double Ratio of four lines of a pencil.

12. Von Staudt's Theorem.

13. Sheaf and plane system of lines.

14. Closed system of 16 points and 16 planes*.

CHAPTER II. THE LINEAR COMPLEX.

1516 The linear complex

15.

16. The Linear Complex.

17. Polar Lines.

18. The Invariant of a linear complex*.

1921 The special complex. coordinates of polar lines

19. The Special Complex.

20. Coordinates of polar lines.

21. Relations between the functions ω and Ω

22. Diameters.

23. Reduction of the complex to its simplest form.

24. Two complexes have one pair of polar lines in common.

2527 Complexes in Involution

25. Complexes in Involution.

26. Three complexes in Involution.

27. Six complexes mutually in Involution*.

28. Transformation of coordinates.

29. The fifteen principal tetrahedra.

CHAPTER III. SYNTHESIS OF THE LINEAR COMPLEX.

3032 Determination of the complex from given conditions

30.

31.

32.

33. Every linear complex contains two lines of any regulus

3436 Collineation and Reciprocity

34. Correlations of Space. Collineation and Reciprocity*.

35.

36. Involutory Reciprocity.

3738 The Null System

37. Null System*.

38.

39. Method of Sylvester*.

40. Automorphic Transformations.

4143 Curves of a linear complex

41. Ruled surfaces and curves of a linear complex.

42.

43.

4445 Polar Surfaces and curves

44. The polar surface.

45.

46. Complex equation of the quadric.

4750 Simultaneous bilinear equations

47. Simultaneous bilinear equations*.

48. Linear transformations which leave a quadric unaltered in form.

49. Collineations which leave a linear complex unaltered in form.

50. Reciprocal transformations.

CHAPTER IV. SYSTEMS OF LINEAR COMPLEXES.

51. The linear congruence

5253 Double ratio of two and of four complexes

52. Double ratio of two complexes

53. Double ratio of four complexes.

54. Special congruence.

5556 Metrical Properties.

55. Metrical Properties.

56. Axes of a system of two terms.

57. The cylindroid.

58. System of three terms.

5960 The generators and tangents of a quadric

59. Expression of the coordinates of a generator of a quadric in terms of one parameter.

60. Complex equation of a quadric.

61. The ten fundamental quadrics.

62. Closed system of sixteen points and planes*.

63. Systems of four and of five terms.

64. Invariants of a system of complexes.

65. Property of the six residuals

CHAPTER V. RULED CUBIC AND QUARTIC SURFACES.

66. Ruled surfaces

67. Ruled Cubics.

68. Ruled Quartics.

69. Ruled quartics of zero deficiency

7072 Analytical classification of Voss

70. Ruled Surfaces whose deficiency is zero.

71. Ruled cubics.

72. Ruled quartics of zero deficiency.

CHAPTER VI. THE QUADRATIC COMPLEX.

73. The quadratic complex

74. The tangent linear complex.

75. Singular points and planes of the complex.

76. Singular Lines.

7778 Identity of the surfaces Φ1 and Φ2

77. Singular points and planes of any line.

78.

79. Polar Lines.

80. The singular lines of the complex of the second and third orders.

81. The complex in Plucker coordinates.

82. The singular surface.

83. Double tangents.

84. A Kummer's Surface and one singular line determine one 02•

85. The singular surface is a general Kummer Surface.

86. The Complex Surfaces of Plücker.

87. Normal form of the equation of a quadratic complex.

8889 Special and harmonic complexes

88. Complex equation of a quadric.

89. Harmonic Complex.

90. Symbolic form of the equation of a quadric in plane coordinates.

91. Plücker surfaces and singular surface of the complex.

CHAPTER VII. SPECIAL VARIETIES OF THE QUADRATIC COMPLEX.

9293 The tetrahedral complex

92. The Tetrahedral Complex.

93. Equation of the tetrahedral complex.

9495 Reguli of the complex.

94. Reguli of the complex.

95.

9697 Other methods of formation of the complex

96. Second method of formation of the complex.

97. Third method of formation of the complex.

98101 Complexes derived from projective pencils

98

99. Complexes determined by two bilinear equations.

100.

101.

102. Reye's Complex of Axes.

103. Ditferential Equation of the Complex.

104106 Curves of the complex

104. The line element.

105. Curves of the Tetrahedral Complex.

106. NonProjective Transformations of the Complex.

107109 The special quadratic complex

107. The Special Quadratic Complex.

108. System of two special complexes*.

109. Covariant tetrahedral complex.

110112 The harmonic complex

110. The Complex of Battaglini or Harmonic Complex*.

111. The Tetrahedroid.

112. There are ∞1 pairs of quadrics such that for each pair the Harmonic complex is the same as the given complex.

113. Painvin's Complex.

CHAPTER VIII. THE COSINGULAR COMPLEXES.

114115 The cosingular complexes

114.

115.

116. Correspondence between lines of cosingular complexes*.

117. The complexes R42, R4'2.

118. The congruence [2, 2].

119. Focal surface of the congruence.

120. Confocal congruences.

121. The quartic surface (02, A, A').

122. Projective formation of C2.

123. Caporali's Theorem.

124. Condition for (1, 1) correspondence in any coordinate system.

125. Equation of the complex referred to a special tetrahedron.

126. Cosingular complexes for this coordinate system.

127128 Involution of tangent linear complexes

127 Involution of tangent linear complexes.

128.

129. Conics determined in a plane by cosingular complexes.

130. Elliptic coordinates of a line.

131. Bitangent linear complexes.

132. Principal Surfaces.

133. Involutory position of two lines.

CHAPTER IX. POLAR LINES, POINTS, AND PLANES.

134135 Polar lines.

134. Polar lines.

135.

136139 Corresponding loci of polar lines.

136. Corresponding loci of polar lines.

137.

138.

139.

140. Polar planes and points of the complex.

141. Polar Point

142. The diameters of the complex.

143. The Centre of the complex.

CHAPTER X. REPRESENTATION OF A COMPLEX BY THE POINTS OF THREEDIMENSIONAL SPACE.

144. Representation of the lines of a qudratic complex by points of threedimensional space

145. The reguli of a congruence[2, 2].

146. Representation of the congruence (2, 2) by the points of a plane.

147. Representation of the lines of a linear complex by points of space.

CHAPTER XI. THE GENERAL EQUATION OF THE SECOND DEGREE.

148153 Reduction of the equation of a quadratic complex to a canonical form

148.

149.

150.

151.

152.

153.

154. Arbitrary constants of a canonical form.

155. Complexes formed by linear congruences.

156. Double Lines.

157158 The cosingular complexes and the correspondence between lines of two cosingular complexes

157. The Cosingular Complexes.

158. Correspondence between lines of cosingular complexes.

159. The singular surface of the complex.

160. Degree of a complex.

161214 Varieties of the quadratic complex

161. The varieties of the quadratic Complex.

162.

163.

164.

165.

166.

167.

168.

169.

170.

171.

172.

173.

174.

175.

176.

177.

178.

179.

180.

181.

182.

183.

184.

185.

186.

187.

188.

189.

190.

191.

192.

193.

194.

195.

196.

197.

198.

199.

200.

201.

202.

203.

204.

205.

206.

207.

208.

209.

210.

211.

212.

213. Number of constants in a canonical form.

214.

CHAPTER XII. CONNEXION OF LINE GEOMETRY WITH SPHERE GEOMETRY.

215. Coordinates of a sphere

216. Intersection of lines corresponds to contact of spheres.

217. Points of Λ correspond to minimal lines of Σ.

218219 Definition of a surface element. A surface element of Λ defines a surface element of Σ

218. Surface Element.

219. Corresponding surfaces in Λ and Σ,

220. Principal tangents and principal spheres.

221. Pentaspherical Coordinates.

CHAPTER XIII. CONNEXION OF LINE GEOMETRY WITH HYPERGEOMETRY.

222. Definition of point, line, hyperplane of space of four dimensions

223. Equations connecting lines of A and points of S4

224. Correlation of Schumacher*.

225. Correlatives of the lines of any plane system and sheaf of Λ.

226227 Metrical Geometry.

226. Metrical Geometry.

227. Automorphic transformations in Λ correspond to anallagmatic transformations of S4.

228. Principal Surfaces of A and Lines of Curvature of S4.

229. Line Geometry is point geometry of an S42 in an S5.

230. Line Geometry in Klein coordinates is point geometry of S4 with hexaspherical coordinates.

231232 The congruence (m, n)

231. Congruences of the mth order and nth class.

232. Rank of a congruence.

CHAPTER XIV. CONGRUENCES OF LINES.

233. Order and class of a congruence.

234. Halphen's Theorem.

235. Characteristic numbers of a congruence.

236. Focal points, planes and surface.

237. Degree and Class ofthe Focal Surface.

238. Singular Points.

239240 Determination of a ray by two coordinates

239. Expression of the coordinates of a ray in termsof two variables*.

240.

241248 Appplication of Schumacher's method of projection to determine the degree, class, and rank of the focal surface

241. Schumacher's method*.

242. Tangents to F m+n.

243. Triple secants of F.

244. The Focal Surface.

245. Degree and Class of the Focal Surface.

246. Double and Cuspidal curves of the focal surface.

247. Rank of the Focal Surface.

248. Determination of r and t for the intersection of two complexes.

CHAPTER XV. THE CONGRUENCES OF THE SECOND ORDER WITHOUT SINGULAR CURVES.

249. The rank of the congruence (2, n) is n2

250. The Surfaces (P).

251. Each singular point of the congruence is a double point of Φ.

252. Double rays of the congruence.

253. The class of a congruence (2, n) cannot be greater than seven.

254256 Number and distribution of the singular points

254. Number of singular points.

255. Distribution of the singular points*.

256. Conjugate singular points.

257. Equation of a surface (P).

258. Tetrahedral complexes of the congruences (2, n).

259. Nonconjugate singular points.

260262 Reguli of the congruences (2, n).

260. Reguli of the congruences (2, n).

261.

262.

263. Confocal congruences.

CHAPTER XVI. THE CONGRUENCE OF THE SECOND ORDER AND SECOND CLASS.

264. The congruence (2, 2) is contained in 40 tetrahedral complexes

265. Confocal congruences (2, 2).

266. Distribution of the Singular Points.

267. Every (2, 2) is included in 40 tetrahedral complexes*.

268. The Kummer Configuration.

269. The Weber groups.

270271 Reguli of the congruence.

270. Reguli of the congruence.

271. A congruence (2, 2) includes ten sets of ∞1 reguli.

272. Focal surface of the intersection of any two complexes.

273. Double rays of special congruences (2, 2)*.

CHAPTER XVII. THE GENERAL COMPLEX.

274. The general complex

275277 The Singular Surface.

275. The Singular Surface.

276.

277.

278. The Principal Surfaces.

279. Independent constants of the complex.

280. The Special Complex.

281283 Congruences and their Focal Surfaces*.

281. Congruences and their Focal Surfaces*.

282.

283. Degree and class of the Focal Surface.

284285 The ruled surface which is the intersection of three complexes

284. The ruled surface common to three complexes.

285. Rank of the surface.

286. Clifford's Theorem.

287288 Symbolic form of the equations of the complex and its singular surface

287. Symbolic form of the equation of the complex*

288. Symbolic forms for the Complex surface and Singular surface.

CHAPTER XVIII. DIFFERENTIAL EQUATIONS CONNECTED WITH THE LINE COMPLEX.

289. Application of the surface element to partial differential equations

290. The characteristic curves of a partial differential equation.

291. The Monge equation of a line complex.

292. The characteristics on an Integral surface are principal tangent curves.

293294 Partial Differential equation corresponding to a linecomplex

293. Form of the partial differential equation corresponding to a line complex.

294.

295. Contact transformations of space*.

296299 The trajectory circle. The equations D11, D12, D13.

296. The trajectory circle.

297. Partial differential equations whose characteristics are geodesics.

298.

299.

300. The complex of normals.

301302 Partial differential equations of the second order associated with line and spherecomplexes.

301. Partial differential equations of the second order connected with line and spherecomplexes.

302.

303304 Partial differential equations of the second orderon whose Integral surfaces both sets of characteristics areprincipal tangent curves or lines of curvature.

303. Partial differential equations of the second orderon whose Integral surfaces both sets of characteristics areprincipal tangent curves or lines of curvature.

304. The curves s and σ of a D22".

305306 Equations D21", D22" with one and with two general first integrals

305. To determine when a D2111 or a D22" has a general first integral.

306. D21" and D22" with two general first integrals.

307. Application to the quadratic complex*.

MISCELLANEOUS RESULTS AND EXERCISES.

INDEX.

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The best introduction to the subject.
Virgil Snyder