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This text contains an early exposition of tetrahedra and algebraic curves on surfaces.

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Cornell University LibraryQA 608.H88Kummer's quartic surface.

3 1924 001 544 406

Cornell UniversityLibrary

The original of this book is inthe Cornell University Library.

There are no known copyright restrictions inthe United States on the use of the text.

http://www.archive.org/details/cu31924001544406

KUMMER'S QUARTIC SURFACE

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,C. F. CLAY, Manager.

ILonJon: AVE MABIA LANE, E.C.

ffilaBBofo: 50, WELLINGTON STREET.

Iclpjifl: F. A. BROCKHAUS.

#ei lord: THE MACMILLAN COMPANY.

BombsE ani (Mcutta: MACMILLAN AND CO., Ltd.

[All Rights reserved.]

KUMMER'S QUARTIC SURFACE

BY

R? W: H: IT HUDSON, M.A., D.Sc.Late Fellow of St John's College, Cambridge, and Lecturer in Mathematics

at the University of Liverpool.

Cambridge :

at the University Press

J 9 5

1ty

ffiam&rtoge:

PRINTED BY JOHN CLAY, M.A.

AT THE UNIVERSITY PRESS.

PEEFATOEY NOTE.

Ronald William Henry Tuenbull Hudson would havebeen twenty-nine years old in July of this year; educated atSt Paul's School, London, and at St John's College, Cambridge,he obtained the highest honours in the public examinations of theUniversity, in 1898, 1899, 1900 ; was elected a Fellow of St John's

College in 1900 ; became a Lecturer in Mathematics at UniversityCollege, Liverpool, in 1902; was D.Sc. in the University of Londonin 1903 ; and died, as the result of a fall while climbing in Wales,in the early autumn of 1904.

This book was then in course of printing, and the writer had

himself corrected proofs of the earlier sheets, assisted in this

work by Mr T. J. I'A. Bromwich, Professor of Mathematics in

Queen's College, Galway, and by Mr H. Bateman, of Trinity

College, Cambridge; for the remaining portion Mr Bateman andmyself are responsible ; we have followed the author's manuscript

unaltered throughout ; and gratefully acknowledge the care given

to the matter by the University Press.

Attentive readers can judge what devotion, what acumen,went to the making of a book of such strength and breadth;

a book whose brevity grows upon one with study. To those who

knew the writer it will be a reminder of the enthusiasm and3

VI PREFATORY NOTE

brilliance which compelled their admiration, as the loyalty of his

nature compelled their regard. A many-sided theory such as thatof this volume is generally to be won only by the work of many

lives ; one who held so firmly the faith that the time is well spent

could ill be spared.

H. F. BAKER.

27 March 1905.

CONTENTS.

CHAPTER I.

KUMMERS CONFIGURATION.

SECT. PAGE

1. Desmic tetrahedra 12. The group of reflexions 43. The 166 configuration . 54. The group of sixteen operations 65. The incidence diagram 76. Linear construction from six arbitrary planes ... 87. Situation of coplanar points ....... 12

CHAPTER II.

THE QUARTIC SURFACE.

8. The Quartic surface with sixteen nodes 149. Nomenclature for the nodes and tropes 16

10. The equation of the surface 1711. The shape of the surface 19

CHAPTER III.

THE ORTHOGONAL MATRIX OF LINEAR FORMS.

12. Preliminary account of matrices . . ... 24

13. Orthogonal matrices 2614. Connection between matrices and quaternions ... 27

viii CONTENTS

SECT. FAGE

15. The sixteen linear forms 2816. Quadratic relations 30

17. The ten fundamental quadrics 3218. The six fundamental complexes 3319. Irrational equations of Kummer's surface .... 34

CHAPTER IV.LINE GEOMETRY.

20. Polar lines 37

21. Apolar complexes 38

22. Groups of three and four apolar complexes .... 3923. Six apolar complexes 40

24. Ten fundamental quadrics 41

25. Klein's 6016 configuration 42

26. Kummer's 16 configuration 44

27. Line coordinates 4528. Fundamental quadrics 4729. Fundamental tetrahedra 48

CHAPTER V.

THE QUADRATIC COMPLEX AND CONGRUENCE.

30. Outline of the geometrical theory 5031. Outline of the algebraical theory 5332. Elliptic coordinates 55

33. Conjugate sets 5634. Klein's tetrahedra 5735. Eelations of lines to * 5836. Asymptotic curves 6037. Principal asymptotic curves 6238. The congruence of second order and class .... 6339. Singularities of the congruence 6340. Relation between 4> and A 6541.

" Confocal congruences 66

CHAPTER VI.

flicker's complex surface.

42. Tetrahedral complexes 68

43. Equations of the complex and the complex surface , . 6944. Singularities of the surface 71

45. The polar line 7246. Shape of the surface 73

CONTENTS. IX

CHAPTER VII.SETS OF NODES.

SECT> PAGE47. Group-sets 7548. Comparison of notations 7649. Pairs and ootads 7750. Eighty Eosenhain odd tetrads 7851. Sixty Gbpel even tetrads 795~2. Odd and even hexads 80

CHAPTER VIII.EQUATIONS OF RUMMER'S SURFACE.

53. The equation referred to a fundamental tetrahedron . . 8154. The equation referred to a Eosenhain tetrahedron . . 8355. Nodal quartic surfaces 86

CHAPTER IX.SPECIAL FORMS OF RUMMER'S SURFACE.

56. The tetrahedroid 8957. Multiple tetrahedroids 9158. Battaglini's harmonic complex 9459. Limiting forms 98

CHAPTER X.THE WAVE SURFACE.

60. Definition of the surface 100

61. Apsidal surfaces 10162. Singularities of the Wave Surface 10263. Parametric representation 10464. Tangent planes ... 10665. The four parameters 10866. Curvature , 109

67. Asymptotic lines 11068. Painvin's complex 112

CHAPTER XI.REALITY AND TOPOLOGY.

69. Eeality of the complexes .

70. Six real fundamental complexes

71. Equations of surfaces Ia , Ih , Ic72. Four real and two imaginary complexes

73. Two real and four imaginary complexes74. Six imaginary complexes .

115

118121

122

125

126

X CONTENTS

CHAPTER XII.

GEOMETRY OF FOUR DIMENSIONS.SECT. PAGE

75. Linear manifolds 12776. Construction of the 15

econfiguration from six points in four

dimensions 12977. Analytical methods 13078. The 16

econfiguration 131

79. General theory of varieties 13280. Space sections of a certain quartic variety . . . . 134

CHAPTER XIII.

ALGEBRAIC CURVES ON THE SURFACE.

81. Geometry on a surface 13782. Algebraic curves on Kummer's surface 13883. The e-equation of a curve 14184. General theorems on curves 14285. Classification of families of curves 14586. Linear systems of curves 146

CHAPTER XIV.

CURVES OF DIFFERENT ORDERS.

87. Quartic curves88. Quartics through the same even tetrad89. Quartics through the same odd tetrad90. Sextics through six nodes

.

91. Sextics through ten nodes92. Octavic curves through eight nodes.93. Octavic curves through sixteen nodes

149151

153154157

158

159

CHAPTER XV.

WEDDLE'S SURFACE.

94. Birational transformation of surfaces 16095. Transformation of Kummer's surface 16296. Quartic surfaces into which Kummer's surface can be trans-

formed 16597. Weddle's surface 16698. Equation of Weddle's surface 169

CONTENTS XI

CHAPTER XVI.

THETA FUNCTIONS.SECT.

99. Uniformisation of the surface.

100. Definition of theta functions ...101. Characteristics and periods102. Identical relations among the double theta functions103. Parametric expression of Rummer's surface .104. Theta functions of higher order105. Sketch of the transcendental theory

PAGE

173175

176179180182

184

CHAPTER XVII.

APPLICATIONS OF ABEL'S THEOREM.

106. Tangent sections107. Collinear points

.

108. Asymptotic curves109. Inscribed configurations

188

190

194196

CHAPTER XVIII.

SINGULAR KDMMER SURFACES.

110. Elliptic surfaces

111. Transformation of theta functions112. The invariant113. Parametric curves

114. Unicursal curves

115. Geometrical interpretation of the singular relation kr12 -

116. Intermediary functions117. Singular curves .

118. Singular surfaces with invariant 5

119. Singular surfaces with invariant 8

120. Birational transformations of Kummer surfaces into themselves

Index

Plate (Rummer's Surface, see pp. 21, 22)

221

. Frontispiece

CHAPTER I.

RUMMER'S CONFIGURATION.

1. DESM1C TETRAHEDRA.

The eight corners of a cube form a very simple configuration

;

yet by joining alternate corners by the diagonals of the faces weget two tetrahedra such that each edge of one meets two oppositeedges of the other, and the figure possesses all the projectivefeatures of the most general pair of tetrahedra having thisproperty.

Take an arbitrary tetrahedron of reference XYZT, and anypoint S whose homogeneous coordinates are as, y, z, t. Draw threelines through this point to meet the pairs of opposite edges, andon each line take the harmonic conjugate of S with respect tothe intercept between the edges; in this way three new pointsP, Q, R are obtained, making in all the set of four

P, ( *, -y, -z, t ),Q, (-, y, -z, t ),p. (-, -y, z,t ),S, ( x, y, z, t ).

Then PQRS and XYZT are a pair of tetrahedra possessing theabove property, for PS and QR meet both XT and YZ, and so on

;

they are the most general pair, for the preceding harmonic con-struction is deduced from the fact that, by hypothesis, any face ofone tetrahedron cuts the other in a complete quadrilateral whose,diagonals