An introduction to the theory of multiply periodic functions

Cambridge University Press......Page 1
Title page......Page 2
Preface......Page 4
TABLE OF CONTENTS......Page 8
Corrigenda......Page 15
The parameter at any place of the surface......Page 16
Algebraic integrals of the first, second and third kinds......Page 17
2. The sum of the logarithmic coefficients for an algebraic integral is zero......Page 18
3. Algebraic forms of the integrals of the first and third kind......Page 19
4. Certain fundamental theorems obtained by contour integration......Page 20
The normal integrals of the third kind, their periods......Page 22
5. The elementary integral of the second kind ; the normal elementary integral of the second kind, obtained by differentiation from the normal elementary integral of the third kind......Page 23
6. A fundamental identity furnishing an elementary integral of the third kind which allows interchange of argument and parameter......Page 25
7. Introductory description of the matrix notation......Page 26
8. The relations connecting the periods of the integrals of the first and second kind......Page 28
9. Integral functions of two variables, elementary properties......Page 32
Quasi-periodic integral functions of two variables, their expression by a finite number......Page 34
Particular quasi-periodic integral functions of two variables ; the theta functions......Page 37
10. The zeros of Riemann\'s theta function......Page 41
Jacobi\'s inversion problem has definite solutions......Page 44
Half-periods defined by integration between branch places......Page 46
Necessary and sufficient form of the arguments of a vanishing theta function ; identical vanishing of a theta function......Page 48
11. The cross-ratio identity between theta functions and integrals of the third kind......Page 49
The algebraic expression of the zeta functions......Page 51
The algebraic expression of the $\\\\mathcal{P}$ functions, and the identity connecting them......Page 53
The identities connecting the squares of the differential coefficients of the $\\\\mathcal{P}$ functions with the $\\\\mathcal{P}$ functions themselves......Page 54
The geometric interpretation of the parametric expressions for the Rummer and Weddle surfaces......Page 55
12. The Rummer Surface, an associated hyperelliptic surface and the finite integrals of total differentials......Page 56
Deduction of the zeta and sigma functions......Page 61
And of the differential equations satisfied by the sigma functions......Page 62
Converse integration of these equations......Page 63
13. The covariantive form and transformation of the differential equations......Page 64
14. Illustration to explain a method......Page 70
The four fundamental quadrics ; the Rummer matrix in covariantive form ; the linear transformations in space......Page 71
15. Employment of the transformation to obtain the nodes and singular planes of the Rummer surface......Page 75
16. Reversion of the Rummer matrix to obtain the Weddle matrix ; element ary properties of the Weddle surface......Page 80
A construction for the tangent plane of the surface......Page 83
17. Projection of the Weddle surface from a particular node......Page 84
The generalisation to any node by means of the transformation ; the six skew symmetrical matrices......Page 86
The bitangents of a Rummer surface......Page 90
Satellite points ; a parametric expression of the Weddle surface ; the forms of the surface integrals......Page 92
18. The 32 birational transformations of the Rummer surface expressed by the six skew symmetrical matrices......Page 94
19. The first terms, for an even function and for an odd function......Page 98
20. Proof that the differential equations determine the terms of fourth and higher dimensions in an even function, when the quadratic terms are given......Page 99
21. Similarly for an odd function, when the linear terms are given ; determination of terms to the ninth dimension for the fundamental odd function......Page 102
22. The same put in connexion with the invariants of a particular sextic......Page 105
24. An even function with the special sextic......Page 108
25.26. Comparison of the arguments employed......Page 109
27. The fundamental sigma function, and $\\\\mathcal{P}$ functions......Page 112
28. The number of linearly independent theta functions of the second order......Page 113
30. The expression of $ \\\\sigma(u+v)\\\\sigma(u—v) / \\\\sigma^2(u)\\\\sigma^2(v) $......Page 114
31. The expression of $ \\\\sigma^2(u,q) / \\\\sigma^2(u) $, when $\\\\sigma(u,q)$ is an even function......Page 115
33. The linear transformation for the functions $\\\\mathcal{P}(u)$ obtained by addition of even half-periods......Page 116
34. The same for odd half-periods, expressed by the six fundamental skew symmetric matrices......Page 117
35. The formulae for $ \\\\sigma(u+v,q)\\\\sigma(u—v,q) / \\\\sigma^2(u)\\\\sigma^2(v) $, and for $ \\\\sigma(u+v,q)\\\\sigma(u—v,q) / \\\\sigma^2(u,q)\\\\sigma^2(v,q) $ similarly expressed......Page 120
Deduction of an orthogonal matrix of sigma functions......Page 121
36. Irrational forms of the equation of the Kummer surface......Page 123
37. The transcendental definition of a tangent section ...........Page 125
The asymptotic lines of the Kummer surface, in the form $ t^2-t\\\\mathcal{P}_{22}(2u)-\\\\mathcal{P}_{21}(2u) = 0 $......Page 128
38. Twin or satellite points ; conjugate points ; the irrational finite integrals of the Kummer surface......Page 129
Asymptotic line contains satellite points......Page 132
39. Relations among the arguments of four collinear points......Page 133
40. Expressions for $\\\\mathcal{P}_{22}(2u)$, etc., rational in $\\\\mathcal{P}_{22}(u)$, etc. deduced from equation of asymptotic lines......Page 134
41. These expressions deduced from converse of Abel\'s Theorem; geometrical construction for argument 2u in connexion with Weddle\'s surface......Page 136
Expressions for $\\\\mathcal{P}_{22}(2u)$, etc., symmetrical in regard to a point and its satellite......Page 139
42. The asymptotic lines of the Weddle surface......Page 140
The Kummer surface and the Weddle surface are so related that asymptotic directions on either correspond to conjugate directions on the other......Page 142
43. The expressions for $\\\\mathcal{P}_{22}(2u)$ etc., determined from the formula for $ \\\\sigma(u+v)\\\\sigma(u—v) / \\\\sigma^2(u)\\\\sigma^2(v) $......Page 144
44. The 32 transformations of the Weddle surface and the invariants $\\\\mathcal{P}_{22}(2u)$, $\\\\mathcal{P}_{21}(2u)$......Page 145
45. The formulae for $\\\\mathcal{P}_{22}(u+v)$, etc......Page 147
The same deduced from Abel\'s Theorem ; geometrical interpretation......Page 148
A Kummer surface with nodes on the original, with singular planes tangent of the original, and having a singular conic common with this; geometrical interpretation of the associated Weddle surface. Comparison with known case......Page 151
46. 47. Cubic surface with four nodes reciprocal to Steiner\'s Roman surface, in connexion with the determinantal form of Kummer\'s equation ; the asymptotic lines......Page 154
48. Examples, references......Page 165
Degenerations of the cubic surface with four nodes......Page 167
Kummer surface referred to a Rosenhain tetrahedron and to a Goepel tetrad of nodes......Page 168
A hyperelliptic surface whose plane section possesses defective integrals......Page 170
Another case of the cubic surface with four nodes......Page 171
The tetrahedroid......Page 172
Plueker\'s complex surface as a case of Kummer\'s surface when two roots of the fundamental sextic are equal......Page 173
The principal asymptotic curves of the Kummer surface......Page 177
Note I. Some algebraical results in connexion with the theory of linear complexes Representation of a straight line by a single matrix; the condition for intersection......Page 178
Fundamental algebraic theorem for invariant factors......Page 180
Three lines in one plane or through one point......Page 181
Representation of a linear complex by a single matrix......Page 182
Six linear complexes in involution; the identities connecting the matrices......Page 183
Reduction of the matrices to a standard form......Page 185
Deduction of a general orthogonal matrix......Page 189
Note II. Introductory proof of Abel\'s Theorem and its converse......Page 191
49. Power series in two variables......Page 198
50. An inequality of importance ; zero points of the series......Page 199
51. Weierstrass\'s implicit function theorem......Page 201
52. Monogenic portion of an algebraic construct near the origin......Page 204
53. A simultaneous system of power-series equations near the origin ; they define a set of irreducible, independent, constructs......Page 207
54. Definition of a meromorphic function of several variables......Page 214
55. Analysis of the definition ; zero and infinity construct......Page 215
56. Comparison with meromorphic functions of one variable......Page 216
57. Limitation to periodic functions ; exclusion of infinitesimal periods......Page 217
58. Limitation to arguments that are functions of one complex variable......Page 220
59. The resulting construct of two dimensions, defined as an open aggregate ; its limiting points......Page 223
Analytic expression of the construct, near an ordinary point and near a limiting point ; the limiting points are isolated......Page 224
Analytical continuation ; definition of a monogenic construct......Page 228
60. Limitation to a monogenic portion of the construct......Page 231
Proof that a periodic function takes every complex value upon this portion......Page 232
61. Proof that the function takes every value the same finite number of times......Page 234
62. Introduction of an algebraic construct in correspondence with the analytic construct......Page 236
63. Period relations ; defective integrals on the algebraic construct......Page 238
Construction of theta functions......Page 242
64. General considerations in the light of the preceding chapter......Page 244
66. The theta function of $n$ variables has $nr$ zeros on the Riemann surface......Page 247
The sum of the vanishing arguments......Page 250
67. The theta function of $n$ variables is a factor of a transformed theta function of $p$ variables belonging to the Riemann surface......Page 251
Existence of a complementary system of defective integrals, also of index $r$......Page 255
68. The transformed theta function of the Riemann surface has $rp$ zeros, and is expressible as a polynomial of the $r$th degree......Page 256
70. The multiplicity ; determination of its value......Page 260
71. The case of one integral reducing to an elliptic integral; proof of the Weierstrass-Picard Theorem......Page 265
72. Kowalevski\'s example of a quartic curve with four concurrent bitangents......Page 270
73. The Legendre-Jacobi example......Page 271
74. A particular case of Kowalevski\'s example; verification of the index and multiplicity......Page 274
Canonical and normal systems of periods......Page 278
75. The quartic curve of 168 collineations; proof that its integrals are defective......Page 280
References ; further problems......Page 285
76. A plane section of a hyperelliptic surface......Page 287
77. Kronecker\'s reduction of a system of rational equations......Page 288
78. Association of systems of $n$ places on a Riemann surface with points of a surface in space of $n$ dimensions......Page 291
79. Expression of the functions of a corpus in terms of a limited number of functions......Page 294
80. Proof of the theorem : the most general single-valued multiply-periodic meromorphic function is expressible by theta functions......Page 296
81. Two alternative methods of argument......Page 298
82. Jacobian functions ; definition ; necessary period relations......Page 301
83. Number of simultaneous zeros of a set of Jacobian functions ; sum of these zeros......Page 304
84. Expression of the Jacobian function by means of theta functions......Page 313
85. The derivatives of the Jacobian function on the Riemann surface......Page 317
Note I. The reduction of a matrix to one having only principal diagonal elements......Page 318
Note II. The cogredient reduction of a skew symmetric matrix......Page 322
Note III. Two methods for the expansion of a determinant......Page 329
Note IV. Some curves lying upon the Kummer surface, in connexion with the theory of defective integrals The factorial integrals of a Riemann surface......Page 332
Reduction of the integrals of a Kummer surface to the factorial integrals of a plane section of the surface......Page 334
The Kummer surface through an arbitrary plane quartic curve......Page 335
Algebraic curves on a Kummer surface for which the integrals of the surface are integrals of the first kind......Page 336
The curve of contact of a Weddle surface with the tangent cone drawn from a node (Principal Asymptotic curve of a Kummer surface); lies on five cones of the third order ; is of deficiency 5 and has 5 elliptic integrals of the first kind......Page 337
Additional Bibliographical Notes......Page 342
Index of Authors......Page 346
General Index......Page 347