Treatise on the Line Complex

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Treatise on the Line Complex

C. M. Jessop

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

Softcover ISBN:	978-0-8218-2913-4
Product Code:	CHEL/223.S

List Price:	$69.00
MAA Member Price:	$62.10
AMS Member Price:	$62.10

eBook ISBN:	978-1-4704-6994-8
Product Code:	CHEL/223.E

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MAA Member Price:	$58.50
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Softcover ISBN:	978-0-8218-2913-4
eBook: ISBN:	978-1-4704-6994-8
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Treatise on the Line Complex

C. M. Jessop

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

Softcover ISBN:	978-0-8218-2913-4
Product Code:	CHEL/223.S

List Price:	$69.00
MAA Member Price:	$62.10
AMS Member Price:	$62.10

eBook ISBN:	978-1-4704-6994-8
Product Code:	CHEL/223.E

List Price:	$65.00
MAA Member Price:	$58.50
AMS Member Price:	$58.50

Softcover ISBN:	978-0-8218-2913-4
eBook ISBN:	978-1-4704-6994-8
Product Code:	CHEL/223.S.B

List Price:	$134.00 $101.50
MAA Member Price:	$120.60 $91.35
AMS Member Price:	$120.60 $91.35

Add to cart

Book Details
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AMS Chelsea Publishing

Volume: 223; 1969; 379 pp

MSC: Primary 14; Secondary 51

From the Preface by C.M. Jessop:

“The important character of the extensive investigations into the theory of line-geometry 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

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
2-9. 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.
10-13 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.
15-16 The linear complex
15.
16. The Linear Complex.
17. Polar Lines.
18. The Invariant of a linear complex*.
19-21 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.
25-27 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.
30-32 Determination of the complex from given conditions
30.
31.
32.
33. Every linear complex contains two lines of any regulus
34-36 Collineation and Reciprocity
34. Correlations of Space. Collineation and Reciprocity*.
35.
36. Involutory Reciprocity.
37-38 The Null System
37. Null System*.
38.
39. Method of Sylvester*.
40. Automorphic Transformations.
41-43 Curves of a linear complex
41. Ruled surfaces and curves of a linear complex.
42.
43.
44-45 Polar Surfaces and curves
44. The polar surface.
45.
46. Complex equation of the quadric.
47-50 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
52-53 Double ratio of two and of four complexes
52. Double ratio of two complexes
53. Double ratio of four complexes.
54. Special congruence.
55-56 Metrical Properties.
55. Metrical Properties.
56. Axes of a system of two terms.
57. The cylindroid.
58. System of three terms.
59-60 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
70-72 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.
77-78 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.
88-89 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.
92-93 The tetrahedral complex
92. The Tetrahedral Complex.
93. Equation of the tetrahedral complex.
94-95 Reguli of the complex.
94. Reguli of the complex.
95.
96-97 Other methods of formation of the complex
96. Second method of formation of the complex.
97. Third method of formation of the complex.
98-101 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.
104-106 Curves of the complex
104. The line element.
105. Curves of the Tetrahedral Complex.
106. Non-Projective Transformations of the Complex.
107-109 The special quadratic complex
107. The Special Quadratic Complex.
108. System of two special complexes*.
109. Covariant tetrahedral complex.
110-112 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.
114-115 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.
127-128 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.
134-135 Polar lines.
134. Polar lines.
135.
136-139 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 THREE-DIMENSIONAL SPACE.
144. Representation of the lines of a qudratic complex by points of three-dimensional 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.
148-153 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.
157-158 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.
161-214 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.
218-219 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.
226-227 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.
231-232 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.
239-240 Determination of a ray by two coordinates
239. Expression of the coordinates of a ray in termsof two variables*.
240.
241-248 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 complex-es.
CHAPTER XV. THE CONGRUENCES OF THE SECOND ORDER WITHOUT SINGULAR CURVES.
249. The rank of the congruence (2, n) is n-2
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.
254-256 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. Non-conjugate singular points.
260-262 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.
270-271 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
275-277 The Singular Surface.
275. The Singular Surface.
276.
277.
278. The Principal Surfaces.
279. Independent constants of the complex.
280. The Special Complex.
281-283 Congruences and their Focal Surfaces*.
281. Congruences and their Focal Surfaces*.
282.
283. Degree and class of the Focal Surface.
284-285 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.
287-288 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.
293-294 Partial Differential equation corresponding to a line-complex
293. Form of the partial differential equation corresponding to a line complex.
294.
295. Contact transformations of space*.
296-299 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.
301-302 Partial differential equations of the second order associated with line- and sphere-complexes.
301. Partial differential equations of the second order connected with line- and sphere-complexes.
302.
303-304 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".
305-306 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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