TRANSFORMATIONS OF SURFACES

BY

LUTHER PFAHLER EISENHART

PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY

PUBLISHED WITH THE COOPERATION OF THE NATIONAL RESEARCH COUNCIL

PRINCETON

PRINCETON UNIVERSITY PRESS LONDON: HUMPHREY MILFORD OXFORD UNIVERSITY PRESS

1923

I'iRONiC VERSION Printed hy LIJTCKE & WULFF, Hamburg.

AVAILABLE

Preface

During the past twenty-five years many of the advances in differential geometry of surfaces in euclidean space have had to do with transformations of surfaces of a given type into surfaces of the same type. Before this period Bianchi and Backlund had established their transformations of a pseudo spherical surface into pseudospherical surfaces, the essential feature of which is that a given surface and any transform are the focal surfaces of a W congruence. Furthermore, Bianchi (Lezioni, § 383) established the so-called theorem of permutability of such transformations; that is, if Si and $2 are two transforms of S there can be found a fourth surface 8' which is a transform of both 6\ and Ss. Later (foot- note 41) he showed that there is a similar theorem of permutability for transformations such that a given surface and a transform are the focal surfaces of a W congruence.

In 1899 G-uichard (f. n. 100) announced two theorems con- cerning the deformations of a quadric of revolution which led to the transformations of Darboux of isothermic surfaces. In such a transformation a surface and its transform are the sheets of the envelope of a two-parameter family of spheres with the lines of curvature corresponding on the two sheets. Families of spheres of this type are associated with cyclic systems of circles, which Ribaucour was the first to investigate extensively, and consequently two surfaces which are the sheets of the envelope of a two-para- meter family of spheres with lines of curvature in correspondence are said to be in the relation of a transformation of Ribaucour. Bianchi showed that for transformations of Ribaucour (f. n. 54) and in particular for transformations of Darboux of isothermic surfaces (f. n. 64) there is a theorem of permutability in the sense mentioned above.

When two surfaces are in the relation of a transformation of Ribaucour, the lines joining corresponding points on the surfaces form a congruence whose developables meet the surfaces in their

IV Preface

lines of curvature. The transformations of Darbonx are a particular case of transformations of conjugate systems, or nets, with equal point invariants, such that the lines joining corresponding points of such a net and a transform form a congruence whose deve- lopables meet the surfaces on which the nets He in these nets and corresponding points of the two nets divide harmonically the focal segment of the corresponding lines of the congruence; these trans- formations were first studied by Koenigs (f. n. 17) and are called transformations K.

When two nets and the congruence of the joins of corre- sponding points are so related that the developables of the con- gruence meet the surfaces on which the nets lie in these nets, we say that either net is obtained from the other by a fundamental transformation, or more briefly a transformation F. We have remarked that transformations of Bibaucour and transformations A^ are of this type. The general transformations F for 3-space have been studied by Jonas and the author (f. n. 15) and a theorem of permutability of these transformations has been established.

Most, if not all, of the transformations which have been developed in recent years are reducible to transformations F or to transformations of the type such that a surface and a trans- form are focal surfaces of a W congruence. It is the purpose of this book to develop these two types of transformations and thereby to coordinate the results of many investigations.

October, 1922.

Luther Pfahler Eisenhart.

Contents

• _•

Chapter I.

section Conjugate nets and congruences. Page

1. Geometric entities of euclidean «-space 1

2. Conjugate nets. Normal parameters 3

3. Determination of nets on a surface 5

4. Parallel nets 7

5. Congruences conjugate to a net 12

6. Focal surfaces of a congruence 15

7. Laplace transforms 17

8. Transformations of Levy 18

9. Determination of congruences 20

10. Congruences harmonic to a net 22

11. Derived nets. Derivant nets . 25

12. Determination of nets harmonic to a given congruence 27

13. Congruences harmonic to point nets 28

14. Radial transformations 31

Chapter II. Transformations F.

15. Fundamental equations 34

16. Inverse of a transformation F. Parallel transformations F 37

17. Harmonic congruence of a transformation F 39

18. Transformations F and radial transformations 40

19. Transformations F with .a common conjugate congruence 42

20. Transformations F determined by the same function 6 42

21. The theorem of permutability of transformations F 45

22. Derived nets and transformations F 52

23. Derivant net and derived net of two transformations F 54

24. The extended theorem of permutability 55

25. Transformations K 57

26. Theorem of permutability of transformations K 60

27. Transformations F of applicable nets 61

28. Nets corresponding with orthogonality of linear elements 63

Chapter III. Sequences of Laplace.

29. Homogeneous point coordinates 69

30. Laplace transformations 72

VI Contents

Section Page

31 . Sequences of Laplace 73

32. Periodic sequences of Laplace 76

33. Harmonic congruences 80

34. Levy sequences of the first order HI

35. Levy sequences of higher order. Derived sequences 83

36. Periodic Levy sequences 85

37. Transformations F in homogeneous coordinates 87

38. Transformations F with the same conjugate congruence. Triads of nets 89

39. Theorem of pennutability ' 90

Chapter IV.

Surfaces and congruences in 3-space.

40. Nets in 2-space W\

41. Tangential coordinates of a surface in 3-sp'ace 97

42. Asymptotic lines 98

43. Nets in 3-space. Point .coordinates 100

44. Ray congruence and ray curves 104

45. Nets R 106

46. W congruences 108

47. R surfaces . . . .' Ill

48. R congruences. Transformations 61 113

49. Reciprocally derived nets. Transformations W of nets R 1 15

50. Theorem of pennutability of transformations IT" 119

Chapter V. Transformations J2. W congruences.

51. Tangential coordinates of a net Laplace transforms in tangential coordinates 127

52. Transformations F in tangential coordinates 129

53. Transformations i2 of nets with equal tangential invariants 134

54. Theorems of pennutability of transformations Q and of W congruences 136

55. Nets permanent in deformation 138

56. Transformations & of permanent nets for which C£ 4= 0 and © ^ 0 ... 141

57. Transformations Q of a sequence of permanent nets 143

58. Transformations i2 in point coordinates. Nets in relation O 146

59. Transformations i2 and K of the focal surfaces of a W congruence . . 149

60. Nets with equal point invariants and equal tangential invariants . . . 150

Chapter VI.

Orthogonal nets.

61. Nets 0 and p, O. Congruences I and p, I 156

62. Nets conjugate to congruences I and p, I 159

63. Orthogonal determinants 160

64. Determination of O nets 365

Contents VII

Section Page

65. Congruences conjugate to 0 nets 167

66. Transformations F of an 0 net for which the conjugate congruence

is normal to the net 168

67. Transformations F of 0 nets into 0 nets 171

68. Transformations R 173

69. Transformations R in another form 175

70. Inverse of a transformation R 178

71. Transformations R in tangential coordinates 178

72. Theorem of permutability of transformations R 179

73. Cyclic congruences 182

74. Multiply cyclic congruences 184

75. Transformations F of 0 nets into 0 nets which are not transformations R 185

76. Nets 2, 0 188

Chapter VH.

Transformations of Ribaucour.

t

77. Orthogonal determinants and 0 nets in 3-space 194

78. Transformations R in 3-space 195

79. The cyclic system associated with a transformation R 197

80. When the circle-planes of a transformation R pass through a point 200

81. The circles X and congruence K 201

82. Transformations D,., of isothermic surfaces 206

83. Theorem of permutability of transformations Dm 209

84. Special isothermic nets in 3-space 211

85. Complementary transformations Dm of special isothermic nets 212

86. Transformations Dm of special isothermic nets 214

87. Transformations Dm of minimal surfaces 215

88. Transformations Em of O nets with isothermal spherical representation 215

89. Nets 9. 218

90. Transformations R of nets Q 221

91. Theorem of permutability of the transformations of nets fi 223

92. Surfaces of Guichard of the first kind 224

Chapter VHI. Circles and spheres.

93. Coordinates of a sphere . < 233

94. Pentaspherical coordinates of a point . . . ! 235

95. Pentasplierical coordinates of an 0 net 237

96. Congruences of spheres 237

97. Derived congruences of spheres 240

98. Congruences of circles .' 4. . 241

99. Congruences of spheres and circles in cartesian coordinates 243

100. Congruences R of spheres 245

101. Pliicker line coordinates 247

Contents

Section Page

102. The Lie Hue-sphere transformation 249

103. The Lie transformations of surfaces 252

104. Congruences R of spheres. W congruences of lines 253

105. Harmonic congruences of spheres and circles 255

106. Representation in 5-space 259

107. Conjugate congruences of spheres and circles 261

108. Congruences of circles K determined by transformations /»' 263

109. Orthogonal congruences of spheres and circles 265

110. Transformations F of congruences of circles 267

111. Parallel transformations of congruences of circles -271

112. Congruences of spheres with applicable central nets 271

113. Generation of cyclic systems 272

114. Transformations F of cyclic systems 274

115. Cyclic systems in 3-space and nets O in 5-space 277

116. Cyclic congruences 280

Chapter IX. Rolling surfaces.

117. Congruences harmonic to nets C 2*7

118. Rolling surfaces 288

119. Special isothermic surfaces 201

120. Rolling of a surface applicable to a surface of revolution 292

121. The fundamental theorems of Gui chard 294

122. Deformable transformations R of the first type 296

123. Deformable transformations -R of the second type 298

124. Deformable transformations of the second type of minimal surfaces . 304

125. Deformable transformations Em 305

126. Transformations F of deformable transformations R of the second type 306

127. Converses of the theorems of Guichard 307

128. Theorems of Ribaucour and Bianchi 310

129. The_surface generated by a point in the tangent plane to a surface S

as S rolls on an applicable surface S 313

130. Kinematically conjugate directions on rolling surfaces 314

131. Congruences of rolling 316

Chapter X. Surfaces applicable to a quadric.

132. Transformations F of nets on a quadric 322

133. Permanent nets on a quadric 323

134. The permanent net on a deform of a quadric 326

135. Transformations Ft of permanent nets on a central quadric 329

136. Transformations Fk of surfaces applicable to a central quadric 334

137. Theorem of permutability of transformations Ft of surfaces applicable

to a central quadric 339

IX

Section Page

138. Transformations B* of surfaces applicable to a central quadric 340

139. Pennutability of transformations Fk and Bk 346

140. Theorem of perrautability of transformations B* 351

J41. Transformations Fk of permanent nets on a paraboloid and of surfaces

applicable to a paraboloid . 352

142. Transformations Bk of surfaces applicable to a paraboloid 354

143. Determination of the asymptotic lines on a surface 356

144. Deformations of paraboloids and central quadrics of revolution 358

145. Surfaces conjugate in deformation 360

146. Transformations H of surfaces applicable to a quadric 365

147. Isothermal-conjugate nets on a quadric 366

148. Transformations F and W of isothermal-conjugate nets on a central quadric • 368

Chapter I.

Conjugate nets and congruences.

i. Geometric entities of euclidean n-space. A point in euclidean space of n dimensions is determined by a system of n numbers x1, .... #", called the coordinates of the point, which are a generalization of cartesian coordinates in euclidean 3-space. We refer to the point as P(x). Thus x typifies all of the coordinates. In like manner P(y) is the point whose coordinates are y1, . . . . yn. The distance d between P(x) and P(y) is defined by the equation

As thus used 2 indicates the sum of all terms of the type (yi — a;*)2, but we write it in the above form without subscripts or superscripts, and shall do so in what follows.

If X1, . . . . Xn are n numbers, the points whose coordinates are of the form

where u is a parameter, lie on a line through P(x). The quantities X are called direction-parameters of the line. Evidently they are deter- mined only to within a factor. This equation represents each of the n equations yi = xi-{- uX*. It is important that the reader should become familiar with this notation. It is understood that u is the same for all n equations.

Two non-coincident lines whose corresponding direction-para- meters X and T are proportional are said to be parallel. The angle of inclination of two non-parallel lines of direction-parameters X1 and Yi is defined to be

cos e =

When cos 0 = 0 the lines are said to be perpendicular.

2 I. Conjugate nets and congruences

The locus of the points whose coordinates are of the form

y = x -\-uX-\-vY,

where u and v are parameters, is a plane. The locus of the points whose coordinates satisfy a relation of the form

a V-f aV+. . . .+ anxnjr an+l=Q,

where the a's are constants, is called a hyperplane. For the sake of brevity we write the above equation in the fonn^arr+an+1 = 0. In particular, xi == 0 is the equation of a coordinate hyperplane. Two hyperplanes

are said to be parallel when the corresponding quantities ai and l>1 (i = 1, ____ n) are proportional. The angle of inclination 8 of two non-parallel hyperplanes is defined by

COS0 =

When cos 6 = 0, the two hyperplanes are said to be perpendicular. In particular, any two coordinate hyperplanes are perpendicular. A line is a special type of curve, which by definition is the locus of a point whose coordinates x are functions1) of a parameter u. The tangent to a curve at a point is the line through the point

whose direction-parameters are -=— .

du

A plane is a special case of a surface, the latter being defined to be the locus of a point whose coordinates x are functions of two parameters u and v. The points of the surface for which v has the same value is called a parametric curve v = const. There is a one -parameter family of curves v = const, on a surface.

') In this treatment the parameters may he real or complex, and the only requirements made of the functions is that they and their derivatives (to such order as the latter appear in the development) are uniform and continuous.

2. Conjugate nets. Normal parameters $

When u and v are replaced by functions of two new parameters u' and v', we get new parametric curves, and conversely any two one-parameter families of curves can be made parametric.

As in the case of 3-space [§ 25]2), it can be shown that the tangents to all the curves at an ordinary point P of a surface lie in a plane, called the tangent plane at the point.

2. Conjugate nets. Normal parameters. When the para- meters of a surface are such that the coordinates x are solutions of the same equation of the Laplace form,

8'0 a log a 30 . 9 log 6 90

dudv ' dv du du dv'

the parametric curves are said to form a conjugate net, or simply a net. As a consequence of this definition a net in 3-space consists of a conjugate system of curves [§ 80]. Equation (1) is called the point equation of the net. We speak of the net as N(x). As in the case of ordinary space, if we put

it is readily found that [cf. § 63]

3 log a r dv ' du dlogb * du J dv

dv 2H* du 2H*

The functions E, F, G are called the fundamental coefficients of the net.

The functions -, - and - - are direction-parameters of the tan- du dv

gents to the curves v = const, and u = const, respectively, of a net whose point equation is (1). The same is true of the functions ar and ft' defined by

dx _ , , dx _ ,,, du * a ' dv 2

*) A reference in square brackets, thus [§ 25], is to the author's Differential Geometry, Ginn and Co., 1909; in parentheses, thus (§25), is to the present volume.

4 I. Conjugate nets and congruences

p' and q' being functions of u and v. If these equations be dif- ferentiated with respect to v and u respectively, we have in con- sequence of (1)

(4) ^-mX + n^', ^j£««W»'+%ft

where

a 0' 8log&

y Bi = :y~^'

^

9 log a 8 . &

The coordinates of points on the tangents to curves v = const. and u = const, have the respective forms

By means of (4) we find that the derivatives of these functions with respect to v and u respectively are linear in «' and /?'.

Conversely, if a surface is referred to any system of para- metric curves, a point on the tangent to v = const, has coordinates

dx of the form x + 1 - — . When v varies the direction-parameters of

du

the tangent are

dx . dt dx d*x

~T ~^~~ ~^77 T •'

dv dv du dudv

Hence if this tangent is to be in the tangent plane to the surface,

3 x 9 x the preceding expression must be a linear combination of - — and - — ,

du dv

and we have the theorem:

A necessary and sufficient condition that a system of curves on a surface form a net is that any point on the tangent to a curve v = const, moves in the tangent plane as v varies.

This theorem gives a geometric characterization of a net in the sense that the tangents to v = const, are characteristics of the tangent planes along a curve u = const. It will be convenient at times to refer to the tangent plane to a surface on which a given net lies as the tangent plane of the net.

3. Determination of nets on a surface 5

Evidently any functions proportional to a' and ft' are also direction-parameters. We wish to consider now the particular parameters a and ft, such that

dx dx

in which case equations (4) become

da dfi

(b) ~^-n^ ^"

where m and n are functions of u and v given by db da

Following Guichard we say that the «'s and /tf's are the normal parameters of the net.

Conversely, if we have n pairs of functions a and ft satisfying equations of the form (6), where m and n are given functions of u and v, each pair of functions a and b satisfying (7) leads by quadratures of the form (5) to a net. Thus the complete integration of (7) determines a family of nets, such that at points with the same values of u and v on the nets the tangents to the curves v = const, and n = const, are parallel. A representation of all these nets is given by drawing through the origin lines whose direction-parameters are the «'s and /S's. We call this representation a "point net.

3. Determination of nets on a surface. Consider the differential equation

where A, B, C, D and E are functions of w and v. If we change the independent variables, putting

M' = <?! (u, v), v' — <jp2 (w, v),

6 I. Conjugate nets and congruences

the resulting equation is of the form

where

,/_. A+ 25 1

A h2jB""

D/ . du' dv' . -r^ldu' dvf dv' du'\ , ~du' dv'

J-e' J l_ JJ I _1_ I _l_ fi

du du \ du dv du dv] ' dv dv'

From these expressions it follows that if ^ and y2 are resolutions of

equation (9) is of the form (1). Since ^ and r/2 must be functionally independent, they are obtained by solving the two differential equations of the first order which are factors of

(10) Adv*— 2Bdudv + Cdu* = Q.

Darboux3) has called (10) the differential equation of the characteristics of (8).

There is only one such function y when

(11) AC— B*=0.

If we take it for u' ', equation (9) is reducible to the form

T = 0.

Let 8 be a surface in 3-space whose cartesian coordinates x are functions of any two parameters u and v. We can find by differentiation an equation of the form (8) satisfied by the three coordinates and by any function of them, say F(x\ x*, xs). Then

3) Lemons, vol. 1, p. 193.

4. Parallel nets 7

the curves defined by (10) form a net, which is determined by the character of F; or if (11) is satisfied, a family of asymptotic lines [§ 77].

If 8 is a surface in 4-space4), an equation of the form (8) can be found which is satisfied by the four coordinates, and this equation is unique to within a factor. Consequently there is a unique net on 8, unless (11) is satisfied, in which case equation (10) defines a self-conjugate family of curves.

When S is a surface in a space* of order higher than the fourth, it is not always possible to find an equation of the form (8) satisfied by all of the coordinates of S. Consequently in such spaces there are surfaces upon which there are not any nets.

4. Parallel nets. When the points of two surfaces are in a one-to-one correspondence of any sort, and two corresponding systems of curves are taken as parametric, the parameters can be chosen so that u and v have the same values on the two surfaces at corresponding points. It is understood that this plan will be followed hereafter whenever we are dealing with point-to-point correspondence between two surfaces.

We inquire under what conditions the tangents to the curves of the parametric systems at corresponding points on two surfaces are parallel. When these conditions are satisfied we say that the two systems are parallel. The coordinates x and x' of the two systems, expressed as functions of the parameters u and v, must satisfy the equations

where h and I are functions of u and v such that the conditions

3 ldx'\ d

dv \ du I du\dv

are satisfied. These equations show that the #'s satisfy an equation of the form (1), where now a and & are given by

4) When we speak of a surface in n-space, it is meant that the surface is not contained in any space of order less than n.

I. Conjugate nets and congruences

Hence a necessary condition that a system admit a parallel system is that it be a net. Evidently the parallel system also forms a net. In fact, it follows from (12) that the coordinates x' are solutions of the equation

820' I 8 log a 80' h 8log6 80'

dudv ~~ h dv du ~ I du dv'

which may be written in the form

8*0' 8 log a' BO' , 8log&' 80' dudv ~ dv du du dv'

where

(16) a'=ah, V = bl.

Assume that we have a net whose equation is (1). Each pair of functions satisfying (13) gives by quadratures (12) a parallel net. If 0 is any solution of (1), the function 0' given by the quadratures

(17) ™----h*L M ,90

\ -*- • J - ' v « i — v ~~r

du du' dv dv

is a solution of (14); we call 0 and 0' corresponding solutions of (1) and (14).

The analytical problem of finding parallel nets may be given another form. If we define a function g> by

(18) h — l = y, equations (13) may be replaced by

dl 8log& dl

-

5) A particular solution of these equations is h = I = const., in which case the surfaces are homothetic transforms of one another with respect to the origin, to within a translation. We exclude this case hereafter.

4. Parallel nets 9

The condition of integrability of these equations leads directly to

(20)

Each solution of this equation leads by a quadrature (19) and by (18) to a pair of solutions of equations (13), and consequently to the determination of a parallel net.

Equation (20) is by definition the adjoint of equation (1). Hence we have the theorem:

The determination of nets parallel to a given net N is equivalent to the solution of the adjoint of the point equation of N.

The functions h and I are determined by (18) and (19) only to within the same additive constant k. Hence if h and I are one set of solutions of (18) and (19), and x' the corresponding solution of (12), the other solutions h-\-k, l + k lead to x' + kx.

Suppose now that we have two nets N and N' such that the tangents to the curves of parameter u at corresponding points are parallel. We shall show that N and N' are parallel, unless they are planar nets.

By hypothesis the first of equations (12) holds. Differentiating it whith respect to v and making use of the fact that x satisfies (1) and xf (15), we get

8logfr' dx' ,_8_, ha dx 8log6 dx du " dv ~~ dv g a' du du dv'

dx If the coefficient of - - is zero, the theorem is established. If it

du

is not equal to zero, we express the condition of integrability of this equation and the first of (12). The resulting equation is reducible to the form

d'x _ _ A_dx_. p dx du* du dv '

Expressing the condition of integrability of this equation and (1), we get an equation of the form

9 'a- dx . dx

10 I. Conjugate nets and congruences

In § 40 it will be shown that at most two linearly independent functions can satisfy two such equations and (1). Hence:

If two non-planar nets correspond and the tangents to the parametric curves in one family are parallel, the nets are parallel.

If N(x) and N'(x') are parallel nets, the coordinates of any point on the line L joining corresponding points are of the form

x-\- t(x' — x).

In consequence of (12) the derivatives of this expression are reducible to

dt ,_

du(x x)}

dv ^ dv v

Hence the points for which t has the respective values

1 — h' I — /

describe surfaces 2l and 22 such that the lines L are tangent to the curves v = const, on 2t and u = const, on 2S. The coordinates y and z of J2t and 2% are

x' — hx x — Ix

y- -' * = --

A one-parameter family of straight lines tangent to a curve or meeting in a point, or having constant direction-parameters is called a developable surface*}; Any other one-parameter family of lines constitutes a skew ruled surface. In a two-parameter family of lines each relation between the parameters determines a surface. developable or skew. Each line of the family belongs to an infinity of these surfaces. In space of three dimensions two of these surfaces are developable [§ 163]. In spaces of higher order there

6) cf. [§ 27]. As in 3-space we use the terms cone and cylinder for the second and third types here mentioned.

4. Parallel nets H

are not necessarily two developables of the family through each line. We call a congruence in n-space a two-parameter family of lines such that through each line pass two developable surfaces of the family. Hereafter it is understood that the parameters « and v of the congruence are such that these developables are given by u = const, and v = const.

The preceding results may be stated thus:

The lines joining corresponding points on two parallel nets form a congruence whose developables meet the surfaces on which the nets lie in the nets.

The points, Fl and F^ whose coordinates are given by (21) are called the first and second focal points of the line of the congruence on which they lie; that is, the point at which the line is tangent to the curve of parameter u is called the first focal point. The focal points are also spoken of as being of the first and second rank. The surfaces, 2^ and -22, the loci of Ji and F2 respectively, are called the first and second focal surfaces of the congruence.

We remark that the tangent planes of the second focal surface are the osculating planes of the curves of parameter u of the first focal surface, and the tangent planes of the first focal surface are the osculating planes of the curves of parameter v of the second focal surface [cf. § 163].

By differentiating equations (21), we obtain

( dy x' — x dh	dy I — h	dx . x — x	9loga\
du (1—hY du' (22) dz h — I Idx ,	dv l — h x' — x 8log&\	dv l—h dz x'	dv r — x dl
i and (23)	JM I— I \du { z*y 3 i /	1 — / du ]' a dh\ dy	dv (1 — O2 dv' d /, l — h dy
dudv dv"b\ ^Z 9 \MT(	1 — h du/ du h — l\ dz ,	du \ l — h dv' d / & dl\ dz
\ dudv dv "b \	ai — ll du	du °g\l— I	dv) dv'

From these equations it is seen that the parametric curves on ^! and 22 form nets [cf. § 163]. In the next section we show that

12 I. Conjugate nets and congruences

any congruence consists of the joins of corresponding points on two parallel nets. Hence:

The developables of a congruence meet each of the focal sur- faces in a net.

Following Guichard, we say that a net and a congruence are conjugate when the developables of the congruence meet the surface of the net in the curves of the net, provided that the surface is not a focal surface of the congruence. Accordingly we may state the next to the last theorem in the form:

The lines joining corresponding points of two parallel nets form a congruence conjugate to these nets.

5. Congruences conjugate to a net. We consider a net N(x) and a congruence G of direction-parameters X passing through points of the net. We seek the general conditions to be satisfied by the parameters X in order that N and O shall be conjugate.

A necessary and sufficient condition that N and G be conjugate is that on each line of G there be two points defined by equations of the form (24) y = x—lX, z = x—

such that as u and v vary respectively the corresponding point moves tangentially to the line. This is expressed analytically by

(25) = ,X,

du

where a and T are determinate functions. Substituting the above values, we arrive at equations of the form

/0™ dx 2 dX dx dX

(26) -1-

Expressing the condition of integrability of these equations, we find that the parameters X must satisfy an equation of the form

(97) . 80

dudv " dv du du dv h

5. Congruences conjugate to a net

13

Hence we have the theorem:

The direction-parameters of a congruence referred to its develop- ables satisfy an equation of Laplace.

We call (27) the direction equation of the congruence,

When now we require that the point M of coordinates x describe a net whose equation is (1), we find on differentiating equations (26) with respect to v and u respectively that the coefficients in (27) have the respective values

(28)

_

•

and

(29)

a dlogS _ 1 / 8 log 6

~~

_

g aiogft\

\ „_& r_ g /8 . 6 . v 9loga\

~ ~

„_ r_

~

r

If these two sets of values of the coefficients be equated, we get the following equations of condition:

(30)

9

9 du

j + _5_ + lMi(|_i)=0) ! , joo^/l i\

ft A/i* 9w \[t I)

dv A 9w /* If the last of these equations be replaced by

(31)

du

9 log*

dv '

where t is thus defined, the first two are reducible to

(32)

dv

,

_ = dv t*

d t , 9log6/* t\

- — — j r — — z~l • — U.

'O We note that if we put -y = h, — = /, w = /? = 0, equations (26) and (30) reduce to (12) and (13).

14 I. Conjugate nets and congruences

Moreover, equations (26) can be written

dx 8

Comparing these equations with (12), we see that the locus of the point whose coordinates x' are given by

(34) x' = Xt

is a net parallel to N. Hence by a quadrature (31) we can deter- mine direction-parameters of the congruence which are the cartesian coordinates of a net N' parallel to N, and we have the theorem:

If a net N is conjugate to a congruence G, a net N' parallel to N can be found by quadratures tvhose cartesian coordinates are direction-parameters of the congruence.

Conversely, if N' is any net parallel to N and through points of the latter we draw lines parallel to lines joining the origin to corresponding points of N', that is, lines with direction-parameters x, the coordinates of any point P on such a line are of the form

(35) x-

The first derivatives of this expression are of the form

t*e\ dx dr , 9cc - dr ,

(36) — (1 — rh) -— x. — (1 — rl) — —x.

8ttv du 9v dv

When r takes the values 1/h and I/I respectively, the points are focal points of the line, and hence the lines form a congruence. The coordinates of the focal points are of the form

(37) y = x j~x') z = x —x'.

fl t

From these results follows the theorem:

Any congruence conjugate to a given net N can be obtained by draiving through points of N lines parallel to lines joining the origin to corresponding points of a net N' parallel to the given net; and every parallel net determines in this way a congruence conjugate to N.

6. Focal Surfaces of a congruence 15.

It is evident from the above investigation that lines joining any fixed point to N' determine the directions of a congruence con- jugate to N.

If two congruences with corresponding direction-parameters equal or proportional are said to be parallel, we have, as a corollary of the above results, the theorem:

If two nets are parallel, every congruence conjugate to one is parallel to a congruence conjugate to the other.

From (36) it follows that the point P with coordinates (35) describes a net parallel to N when r is a constant, and only in this case. Hence we have the theorem:

A congru.ence conjugate to a net N is conjugate to an infinity of nets parallel to N.

Combining this result with the third theorem of § 4, we have also :

Any congruence conjugate to a net N consists of the joins of corresponding points of N and of a parallel net.

6. Focal Surfaces of a congruence. From (37) we have by differentiation and reduction by means of (12):

l\fdx x' 8 log a

A2 du dv "" \ */ \»v h d.v

dz L h\ Idx x' 8losr6\ dz 1 82

(38)

(du \^ l! \du I du l} dv Is dv

and

82y _ 8 . la dh\ dy d . ^(Ji-l\ dy

= /i_A\ (dx x> 9I°g&\

\ l/\du I du r

(39)

a dh\ dy -- 1 — — -I

h du) du

- - -- — — - -- - —

dudv dv h du) du du \ h ! dv'

d*z _d_ / l—h\ dz , d , ___

(dudv~~ dvl°s\a I I ^ [og '

du\l dv dv

From these equations we see again that the developables meet the focal surfaces in nets.

In order that the point midway between the focal points shall describe the net N(x) conjugate to the congruence, we must haver as follows from (37), (40) l = — h.

16 I. Conjugate nets and congruences

From (13) and (16) it follows that in all generality we may take

(41) a = b

(42)

We recall that for an equation of the Laplace form

a*0 80 ,,90 ,

(43) — = a — --\-o- - + c0 dudv du dv

the functions

(44) H = -£ +

are called the invariants of the equation, since these functions are invariant when equation (43) is transformed into an equation of the same form in 0l7 where 0! = A0, A being any function of u and v.

From (41) and (42) we have for w-space the theorem announced by Ribaucour for 3-space:

A necessary condition that the developables of a congruence meet the middle surface in a net is that the direction equation of the congruence have equal invariants; in this case the point equation of the net on the middle surface also has equal invariants.

We are in position now to establish the theorem:

When two congruences are conjugate to a net, the joins of corresponding focal points of the same rank form a congruence conjugate to the nets described by these focal points.

Let N(x) be the net, and let the direction-parameters of the congruences be the coordinates x' and x" of nets N' and N" parallel to N, determined by solutions h, I and h:, li of (13). The coor- dinates of the focal points of the congruences are of the respective forms (37) and

" /v."

(45) yi = x — , z\, = x - — .

The coordinates of any point on the line joining the focal points of coordinates y and y± are of the form y-\-t(y — yj. If we differentiate this expression with respect to u and v and make use

.7. Laplace transforms

of (38) and similar equations for ylt we find that its first derivatives are proportional to the corresponding first derivatives of y when t = />j. Consequently the points of coordinates y and

generate parallel nets, and hence by the third theorem of § 4 the congruence is conjugate to these nets. In like manner we can show that it is conjugate to the net of coordinates y:.

7. Laplace transforms. In [§ 165] we have derived for 3-space the expressions for the cartesian coordinates of the second focal surface of the congruence of tangents to the curves v = const. or tc =» const, of a net. The method followed is equally valid for spaces of higher order. If (1) is the point equation of the net, the coordinates of these respective second focal nets are of the form

_ _1 __ 9x_ 1 dx

~ 31og& du' Xl~

du dv

In fact we have (48) dx-i b d \~dudx dxi' a d . \ dv \dx

dv 8_b_ dv & \ ab I duj du ' 9«. du b \ ab / dv'

du dv

The nets NI and N-i with the respective coordinates^ anda;_i are called the first Laplace transform of N and the minus first Laplace transform respectively. If the point equation of N— i is written in the form

„ _i

(49J

we find that

86

ab d du bK

/RAs .(00)

The Laplace transfonns of a net parallel to N are defined by equations of the form

18 1- Conjugate nets and congruences

1 fix' ,_ , 1

x x~

du dv

From these, (14) and (48) we have

dx-i	dX-i	dx-i	l*K' dX-!
du	du '	dv	hK dv
dx[	h*H' 3.	v, dx[	- 7; dXl
( du	IH d	w' dv	~h dv'

(52)

where H' and K' are the invariants of (15).

Hence we have the theorem:

If N and N' are parallel nets, their respective Laplace trans- forms are parallel.

8. Transformations of Levy. By means of (38) equations (37) can be written in the form

= y—

It is readily shown that l/h and l/l are solutions of the respective equations (39). By a change of notation the second of these equations may be written

8 dx

1 h dy	1 I dz
d 1 du' X du h	d 1 dv' dv I

~

dv

where now the congruence consists of the tangents to the curves u = const, of N(x) and 0 is a solution of its point equation, and the y's are the coordinates of a net conjugate to this congruence. In like manner the first of the above equations may be written

6 dx dO d'

Since N(x) in (37) may be any net conjugate to the tangents to the curves u = const, of N(y) or v = const, of N(z\ we have the first part of the following theorem of Levy:

8. Transformations of Levy 19

TJie coordinates of any net conjugate to the congruence of tangents to the curves u = const, or v = const of a net N(x) whose point equation is (1) may be expressed in the respective forms *

6 dx _e_ dx_

y~ > z~~

M. - -

dv du

where 6 is a solution of (1). Conversely, every solution of (I) gives two nets conjugate to tlie congruences of tangents.

In order to prove the latter part of this theorem, we note that if 6 is a solution of (1), we have from (53):

_., dy d logB , dz d log A

~d~u ~ du (y~z'i ~37 " dv ** y)t

where

(5V A--a° 7?- - bd

A--~d([> ^'"W'

du dv

From (54) we have by differentiation

9 log B dy

(56)

I

dudv dv & \ du I du du dv'

d*z dlogA dz , d I 9log^i\ dz

dudv dv du ' 9w &\ dv I dv

Hence the points of coordinates (53) describe nets. We call these nets the Leiy transforms of N by means of 6.

As a corollary of the above theorem we have:

There are nets conjugate to any congruence.

It is evident that, if 6 is a solution of (1), the function

du

is a solution of (49), the point equation of the minus first Laplace transform of N. We call B-\ the minus first Laplace transform of e. From these equations. (47), (48) and (53), we have

2*

20 I. Conjugate nets and congruences

0-! dx-

__

dv du

Consequently the net of coordinates z in (53) is the Levy transform of the minus first Laplace transform of ATby means of 0_i. Similar results follow when we consider the net of coordinates y. Hence :

The Levy transforms of a net determined ~by a solution 6 of the point equation of N are Levy transforms of the minus first and first Laplace transforms of N by means of the corresponding Laplace transforms of 6.

g. Determination of congruences. We saw in § 5 that the direction-parameters X of a congruence are solutions of an equation of the form (27). If in (27) we put

we obtain

8V

(07)

where

1 IdlogA 36 8logJ? 80 d*0 \

~~e \~to~ ~^~ ~^T~~87H ~dudvl'

Hence a necessary and sufficient condition that the quantities x' are the cartesian coordinates of a net is that 6 be a solution of (27).

By the above corollary of the theorem of Levy there are nets conjugate to any congruence. From the second theorem of § 5 it follows that any one of these nets is parallel to a net whose coordinates are direction-parameters of the congruence. Hence:

In order to obtain a congruence with a given set of direction- parameters X, we find a solution 6 of the equation of Laplace satisfied by the X's; then the quantities

(59) *'=4-

u

are the coordinates of a net N' ; through points of a net N parallel to N' draw lines with parameters X; these lines form a congruence conjugate to N; all congruences with direction-parameters X can be found in this way.

9. Determination of congruences 21

From (38) we find by means of (12) and (16), that the direction- parameters of the tangents to the curves u = const, and v = const, respectively, on the first and second focal surfaces defined by (37) are expressible in the form

dxr , 9 log a' dxf , 8log&' dv dv du du

When x' is replaced by the value (59), the resulting expressions are reducible to the same form in terms of X and the coefficients of (27) to within the factor 1/6. Hence:

If the direction-parameters 'X of a congruence G are solutions oj an equation (27), the functions

tan dX iMAv dX — *- y

~d^ dv du~ ~~d^~A

are direction-parameters of the tangents to tJie curves u = const, and v = const., respectively, on the first and second focal surfaces of G.

We say that these congruences of tangents are the first derived and minus first derived congruences of G, and we denote them by GI and G—I.

As a corollary of this theorem we have:

WJten two congruences are parallel, their focal nets of the same rank are parallel.

Let M0 (XQ) be a generic point on the middle surface of a con- gruence with direction- parameters X, the parameters u and v being those of the developables of the congruence. The coordinates of the focal points are of the form

(61) y^XQ-^-gX, z — XQ — gX.

Expressing that these values must satisfy (25), we have equations of the form (26). In order that these equations be consistent, they must reduce, on the assumption that the X's satisfy (27), to

9% dX

(62)

dxo

22 I. Conjugate nets and congruences

and Q must satisfy the equation

which is the adjoint of equation (27).

Conversely, each solution of (63) and n linearly independent solutions of (27) determine a congruence for which the surface of coordinates x0 is the middle surface.

As a consequence of these results and the preceding corollary we have:

The determination of nets parallel to the focal nets oj a con- gruence is equivalent to the integration of the adjoint of the direction equation of the congruence.

10. Congruences harmonic to a net. From (54) it follows that the points of coordinates y and z defined by (53) are the focal points of first and second rank respectively of the congruence of lines joining these points. Hence:

The two Levy transforms of a net N by means of the same solution B of the point equation of N are the focal points of the con- gruence of the joins of corresponding points of the transforms; that is, the points defined by (53) are Laplace transforms of one another.

The mutual arrangement of N R( } and the congruence is shown in fig. 1 where u and v indicate the parameter varying along the curve; this notation is used in all sub- sequent figures.

A net and a congruence are said to be harmonic when the foci of the EiffJL &(*•> congruence lie on the tangents of the

net, and the developables of the con-

gruence correspond to the curves of the net. As a consequence of the above theorem and the first one of § 8 we have:

When a net N is conjugate to a congruence G, the congruence of tangents to one family of curves of N is harmonic to one of the focal nets of G and the congruence of tangents of the other family of curves is harmonic to the other focal net of G.

10. Congruences harmonic to a net

23

A'

This situation is illustrated by fig. 2. We have also the theorem :

If N is conjugate to a congruence G, the osculating planes of tJie curves of parameter u(v) of the first (second) focal net ofG are determined by the lines ofG and the tangents of the curves u(v) of N. Suppose that we have a net N and a congruence whose lines lie in tan- gent planes of N and the developable* of G correspond to the curves of N, taken as parametric. As u varies, the corresponding focus of a line of the con- gruence must lie on the characteristic of the developable of the tangent planes along a curve v = const.8). Since these characteristics are tangent to the curves u - = const., we have the theorem:

If lines of a congruence lie in tangent planes of a net and developdbles of the congruence correspond to the curves of the net, the congruence is harmonic to the net.

We shall prove that any congruence harmonic to a net N(x) may be obtained as in the first theorem of this section. In fact, the coordinates z and y of the foci R and S of a congruence harmonic to N(x) are of the form

(64) Now

dx

dx -r—,

dv

dv

— — r

dx alogra -- —

du dv

dx L 9log&\

I -- 1 1 — r —-— — I

dv \ du /

dx dx Expressing the condition that this is proportional to r—— — t~jr~t

we get (65)

JLJL- dv r r dv

t du

rt

s> This can be shown analytically by making use of the fact that the x'x cannot satisfy (1) and two equations of the form referred to in § 4.

24 I- Conjugate nets and congruences

In similar manner the condition that the expression for ~- shall

du

fix dx . be proportional to r— — — t — - is

du dv

81 1 9 log a . 1 8log& 1

_l ---------

du t r dv t du tr

Hence there must exist a function 0 such that i -_L_9JL J_ 1 90

7" " ~e du' t " e QV'

Substituting these values in (65), we find that 6 is a solution of equation (1), and consequently equations (64) become equivalent to (53). Hence we have the theorem:

A necessary and sufficient condition that a congruence be harmonic to a net N(x) is that the focal nets of the congruence he Levy transforms of N by means of the same solution 6 of the point equation of N.

Since the direction-parameters of the harmonic congruence are

of the form

90 dx dd dx

du dv dv du'

it follows that if a second harmonic congruence, determined by a function 6it is to be parallel to the given one, 0X must be a function of 0. Since both must satisfy (1), 0j is a linear function of 0 with constant coefficients. Hence we have:

A necessary and sufficient condition that tivo congruences, harmonic to a given net N and determined by solutions 0 and 61 of the point equation^ of N, be parallel is that 0j be a linear function of 9 with constant coefficients.

Consider now a congruence G and two nets N^ and Ns harmonic to G. Corresponding tangents to NI and N2 meet in a point of a focal net of G, and the congruences of these tangents are con- jugate to this focal net, by the theorem of Levy. These two con- gruences and the nets NI and Na are in the relation discussed in the last theorem of § 6. Hence we have the theorem:

11. Derived nets. Derivant nets

25

If two nets are harmonic to a congruence, the joins of corre- sponding points of the nets form a congruence conjugate to the nets.

ii. Derived nets. Derivant nets. Let O^ and #2 be con- gruences harmonic to a net N, determined by solutions Ql and 02 of the point equation (1) of Nt it being understood that 02 is not a linear function of 0t. The function

9 0g

9u

is a solution of the second of equations (56) with 0 replaced by 6l} that is, the point equation of the jsecond focal net of Gt. The coordinates of the Levy transform N(x) of this focal net by means of y conjugate to GI are of the form

x = z-

dv

dz dv'

which in consequence of (53) and (54) is reducible to

90* n 90, \ dx [n 90, A 90, \ dx

1 66) a; = x —

dv

dv

90, 90S 90, 90S

dv

du

The coordinates ?/2 and z^ of the focal points of 6?2 are given by (53) when 0 is replaced by 02. The point equation of the second

o t\ I fl A

focal surface of 6r2 admits the solution 0! — 02 — — /— — , which deter-

9 u I ou

mines a Levj7 transform of this surface conjugate to G*. The expressions for the coordinates of this transform are reducible to (66). Hence:

If two congruences are harmonic to a net N, the point of intersection of corre- sponding lines of the two congruences de- scribes a net conjugate to the two congruences.

This result is illustrated by fig. 3, where L\u and L\v are Levy transforms

2(J I. Conjugate nets and congruences

of N\)j means of Oi and L2u and L2v by means of 02. Nis called the corresponding derived net by Guichard.

We shall prove the following converse of the above theorem: If two congruences are conjugate to a net N, the planes determined by pairs of corresponding lines of the congruences envelop a net harmonic to the congruences.

Let the coordinates of the focal points of the congruences be taken in the forms (37) and (45). We have seen that the point of coordinates (46) describes a net parallel to the net of coordinates y, and is conjugate to the congruence G of the lines joining the points of coordinates y and y^. If the expression (46) is differentiated with respect to v, the resulting expression is reducible by means of (12)

— dv

and (38) to l—^-. where ov

— . nli h\l

^~~'

If we apply the formulas (21) to his congruence G, we find that the coordinates of the second focal net are in the form

y-\-hi(y— yi)— ly

i c which is reducible to

{67) ^l_fll)_^(Z_ft)

In like manner we find that the coordinates of the first focal point of the congruence conjugate to the nets of coordinates z and z\ are of the form (67). Hence the above theorem is proved. We say that the net of coordinates (67) is the derivant net of N. From (66) we have by differentiation

(68 aK

A 9& a 90,
dx °L dv	U2 9v
du 902 90i	902 90!
du dv /920, 902 920Z 90i \ dx	dv du /920! 90S 9202 90i\ dx~
d2x , I 9«2 dv du2 dv 1 du	\ du2 du du2 du I dv
9u* 902 90,	90S 90,
9w dv	dv du

' 12. Determination of nets harmonic to a given congruence 27

(68 b)

a 00, 0 ou\
dx	1 du	"2 du
dv r 1 820, 80, 8J0,	80, 80,	80, 80,
du dv 80, \ dx	dv du i 8*0, 80, 820, 80, \ dx I
d*x , 1 8v2 dv dv*	dv ! du	' dv* du dv* du I dv
dv*	80, 80,	80, 80,
du dv	dv du •

We remark that if we replace 02 by 02 -f- const, the expressions in the parentheses are unaltered. Consequently:

The oo l derived nets of N determined by 8^ and 02 + c, where c is a parameter, are parallel to one another and conjugate to the congruence harmonic to N determined by 8l.

12. Determination of nets harmonic to a given con- gruence. We establish the following theorem which may be looked upon as a limiting case of the second theorem of the preceding section:

If two congruences are parallel, the point of intersection of lines joining corresponding focal points generates a net harmonic to the congruences.

Let N(x) and N'(x') be the first focal nets of the congruences. The second focal nets are the minus first Laplace transforms of N and N', and their coordinates are given by (47) and (51). From these expressions we find that the coordinates of the points of intersection of the lines joining the focal points of the first and

x ~~ Ix

second ranks respectively are of the form — r-. By (21) this

1 — L

is the second focal point of the lines joining corresponding points on N and N'. In like manner it can be shown that it is the first focal point of the congruence conjugate to N—i and N'—i.

From these two theorems it follows that the problem of finding nets harmonic to a given congruence G is equivalent to the deter- mination of congruences conjugate to a net conjugate to G, or of congruences parallel to G, or of nets parallel to either focal net of G. In this section we give another means of finding nets harmonic to G, arising from the solution of the last problem.

If 6? and G' are parallel congruences, and we use the notation of the above paragraph, we have from (47), (51), (52) and (14).

28 I- Conjugate nets and congruences

(69) x^x-^-^x-!

where x is thus defined. Consequently for the congruences conjugate to N and N—\ with direction-parameters x' and x'-\ corresponding lines meet in the points of coordinates x, which is the second and first focal point respectively of these two congruences, in con- sequence of (52). Hence:

If G and O' are parallel congruences, and through the focal points of Q lines are drawn parallel to the lines joining a fixed point to the corresponding focal points ofG', these lines are tangent to the curves of a net harmonic to O.

From these results and the last theorem of § 9 we have:

The determination of nets harmonic to a congruence is equivalent to the integration of the adjoint of the direction equation of the congruence.

13. Congruences harmonic to point nets. If y and z are the coordinates of the first and second focal points of a congruence, we have

where p and q are determinate functions. These equations are reducible to the normal form (6), if we put

(71) a = zp, fi = yq

with

Hence we have the theorem:

The lines joining the origin to the foci of a congruence constitute a point net.

We say that the congruence is harmonic to the point net.

The direction-parameters of the congruence are given by

(73) Z=aq — fip.

13. Congruences harmonic to point nets 29

Conversely, suppose we have any point net of parameters « and ft, and a pair of solutions p and q of the equations (6). The functions y and z given by (71) satisfy the conditions (70). Hence the joins of the points whose coordinates are y and z form a con- gruence for which these are the focal points, and consequently the congruence is harmonic to the point net. Accordingly we have the theorem:

If the parameters of a point net are in the normal form, each pair of solutions of the corresponding equations (6) gives directly a congruence harmonic to the point net, and all such harmonic congruences are so determined.

It is readily seen that if the parameters are not in the normal form the determination of harmonic congruences reduces to the solution of the equations (4) of the parameters.

Consider a net N with normal parameters of its tangents giv^en by (6). If p and q are a pair of solutions of (6), it follows from (5) that 0, given by the quadrature

30 30

is a solution of the point equation (1) of N. Making use of this function 0, we get a family of parallel congruences harmonic to N whose direction-parameters are of the form

dx 30 dx 30

From (73) it is seen that these congruences are parallel to those harmonic to the parallel point net determined by p and q.

Conversely, when a congruence harmonic to a net N is known, we have by a quadrature at most a solution of the point equation of Nin consequence of the theorem of Levy (§ 8). If the parameters of the tangents of N are in the normal form, then p and q given by (74) satisfy the corresponding equations (6). Hence:

When the congruences harmonic to a point net are known, all the congruences harmonic to a parallel net can he found by quadratures; when a congruence harmonic to any net is known, by a quadrature n 1 moat a congruence harmonic to the parallel point net can be found.

30 I- Conjugate nets and congruences

Because of this theorem and the second one of § 12 we have:

If N is a net harmonic to a congruence G, and lines be drawn through the focal points of a parallel congruence O' parallel to the corresponding tangents to N, these lines are tangent to a net harmonic to Gf.

We have also:

Of all the parallel nets harmonic to the family of congruences parallel to a given congruence, one is the point net of the family.

Since the direction-parameters of any congruence harmonic to a net can be given the form (73), we have:

Each pair of solutions p, q of equations (4) satisfied by the direction-parameters of the tangents to a net N determine a con- gruence harmonic to N; its direction-parameters are of the form a'q — ft'p; all congruences harmonic to N can be obtained in this way.

From (73) we have by differentiation, and with the aid of (6? and (72),

du du du ' dv ' ' dv dv '

^Z_ _l£_^__lP.M . , nZ. dudv dv du du dv

Hence the direction equation of the congruence is _^L dlogg VZ dlogp dZ i dlogp dlogq]

-- --~---- --

This is of the form (27), where now (76) q

U and V being functions of u and v alone respectively. Hence: The direction-parameters of any congruence whose direction equation is (27) are expressible in the form

(77) Z=aAU— ftBV,

14. Radial transformations. Exercises 31

where a and fi are normal parameters of a net harmonic to the congruence.

14. Radial transformations. Suppose we have a net whose point equation is (1) and let 6 be any solution of (1). From (57), and (58) it follows that the locus of the points of coordinates ~x. given by

(78) x - |>

is a net N, whose point equation is

dudv ' 9v '9 du du 0 dv'

Conversely it follows from (58) that only when 6 is a solution of (1) does the point x describe a net. We call N the radial transform of N by means of 6.

The tangents to the curves v = const, at corresponding points on N and N meet in the point whose coordinates are

	6 —	1	dx -.	1-	1 0	9;r
x>	80 du		du	^(.	i)	9 1*'

and the curves u = const, in the point

0 — 1 dx 1 ~6 dx

x —

90. dv ~ A/M dv

dv dv

(i

These points generate the Levy transforms of N by means of the function 0 — 1, and of A7 by 1 — 1/0. Hence we have the theorem: The lines of intersection of the tangent planes of two nets N and N in the relation of a radial transformation generate a congruence harmonic to both nets.

Exercises.

1 . The coordinates of any point on a line joining two points of coordinates x\ and x''2 are expressible in the form (l\a?i+ l»afdf(li+ It).

2. The coordinates of any point of a plane determined by three points of coordinates x\, a?,, x*, are expressible in the form (lixfi + lt

32 I. Conjugate nets and congruences

3. The tangents to the curves v = const, of all nets conjugate to a given congruence at points of the same line of the congruence are coplanar; likewise for the tangents to the curves u = const.

4. A necessary and sufficient condition that a point P on the join of corre- sponding points of two parallel nets describe a net parallel to them is that P divide the segment between points of the nets in constant ratio.

5. Show that if h in (12) is a function V of v alone, then (1) must be of

the form

920 T,, 90 , aiogyo 90

dudv ~ du ' du 8r'

one of whose invariants is zero. In this case I — F+l//>.

6. When in equation (1) we have a = U and b = V, where U and F are

920 functions of u and v alone respectively, the point equation is -^ — x— =0. In

this case we say that N(x) is a net of translation [cf. § 81]. Show that all nets parallel to a net of translation are nets of translation.

7. The curves on the surface S of a net N which are defined by Eduz -j- 2Fdu dv-\-G dvz = 0 are called the minimal curves of S [cf. § 35]. When the curves of N are the minimal curves of S, Nis called a minimal net. Show that every net parallel to a minimal net is a minimal net.

8. If a net^'(;r') has equal point invariants, that is b'= a', the equations

1 dx' dx 1 dx'

a'2 du1 dv a'2 dv

are consistent, and the x's are coordinates of a net N. The congruence conjugate to N and of direction-parameters x' has for focal points x — or'/a'2, x-}-x'/a'2. Consequently N lies on the middle surface of the congruence.

9. If N and N' are parallel nets, and 0 and 0' are corresponding solutions of their point equations (§4), the point of coordinates (x6' — x'0)/(0' — 0) describes a net conjugate to the congruence of the lines joining corresponding points on N and j\".

10. If N and N' are parallel nets, and 0 and 0' are corresponding solutions of their point equations (§4), the corresponding Levy transforms of N and N' by means of these respective functions are parallel nets: also the lines joining corresponding Levy transforms meet in the points of the net of Ex. 9.

Martin, Comptes Rendus, vol. 139 (1904), p. 32.

rif) I rl ft r)f) I ftf)

11. If 0, is a solution of (1), then e^—B^l — and 0i-0-| - are

ov I ov ou I ou

solutions of the respective equations (56); and the former admits the latter as its minus first Laplace transform.

12. To each solution y> of the first of equations (56) there correspondends

a solution QI of (1) such that v = 0i— 0 -^-1-^- .

ov I ov

13. If N is a derived net of a net N, the osculating planes of the curves of parameter u and v of N pass through the corresponding points of the minus first and first Laplace transforms of N.

Exercises 33

14. If N is a derived net of a net N, the first and minus first Laplace trans- forms of N are derived nets of the first and minus first Laplace transforms respectively of N. Tzitzeica, Comptes Rendus, vol. 156 (1913), p. 374.

15. If N is the derived net of #by means of solutions 0, and 6t of (1), the

.. _ /9x80, dx 80, \/ / 80, 80, 90, 80, \

quantities x = -5 — 5 ~ — 5 — / I -~ ~ 5 o — are the coordinates

\du ov ov dull \ ou ov ov ou I

of a net parallel to N.

16. A necessary and sufficient condition that a net .AT and a parallel net N' defined by (12) he radial transforms of one another, to within a translation of Cither net, is that h = I = const., say c; then if N' is a radial transform, x1 = ex.

17. A necessary and sufficient condition that two nets, N' and JV", parallel to JY and determined by pairs of solutions hi, h, and /tj, lt of (13) be radial trans- forms of one another, to within a translation of either, is that -j— = -^- = const.

AI li

18. If more than two ruled surfaces of a congruence are developable, all the ruled surfaces are developable and the lines of the congruence are concurrent.

19. If N is a radial transform of a net N by means of a solution B_ of the point equation (1) of N, the minus first and first Laplace transforms of N are radial transforms of the corresponding Laplace transforms of N. the respective functions being

„ b 80 a dO

db_ 8«' da, dv'

du dv

20. If N is a radial transform of a net N by means of a solution 0 of the point equation of N, and 0t is any other solution of this equation, then 0i/0 is a solution of the point equation of N. Show that the Levy transforms of N and N by means of B\ and 0,/0 are radial transforms of one another.

21. If G and Gi are parallel congruences and lines be drawn through the focal points of each parallel to the lines joining the corresponding foci of the other to the origin, the two nets determined by the intersections of these pairs of lines are in the relation of a radial transformation.

Chapter n.

Transformations F.

•

15. Fundamental equations. In this chapter we are concerned with the determination of all nets NI such that for a net Ni and a given net N the lines joining corresponding points form a congruence G conjugate to N and Ari9). These transformations of N into nets NI are fundamental in a general theory of nets, and we call them the fundamental transformations, or for the sake of brevity transformations F. We say also that N and Nt are in relation F. We call G the conjugate congruence of the trans- formation10). An example of this relation is afforded by two parallel nets and the lines joining corresponding points (§ 4). Also the second theorem of § 6 and the last theorem of § 11 may be stated as follows:

When two congruences are conjugate to a net, corresponding focal nets are in relation F, or are radial transforms of one another.

When two nets are harmonic to a congruence, they are in relation F. or are radial transforms of on another.

We turn now to the general study of this relation. From the second theorem of § 5 it follows that if N and Ari are in relation F, the direction-parameters of the conjugate congruence of the transformation are proportional to the coordinates x' of a net A7' parallel to N, and also to the coordinates x( of a net A"[ parallel to N^. Hence these coordinates must satisfy a relation of the form

(1) <- -

9) A statement of the history of these transformations is given in the Preface.

10) Two nets in the relation of a radial transformation (§ 14) satisfy this requirement, since all the lines of the congruence meet in a point, and then every ruled surface of the congruence is developable. However, we exclude this exceptional case from the definition of transformations F.

15. Fundamental equations 35

where, as follows from § 14, 6' is necessarily a solution of the point equation of Nf.

The coordinates x^ of ATi are necessarily of the form

0 , (2) Xi = x —f x ,

where 6 is to be determined. The coordinates x' are given by equations of the form (cf. I, 12)

dx' _ dx dx' dx

~8V: h~d^' ~d^'' '7?

Hence the first derivatives of Xi are reducible to the forms

(4)

*1

h i du \h du du!

dxi__le' n\ dx'i _ '/J_l^. _JL?_

I 1 17 Q „. Q „,

I / 8t;

From these expressions it follows that NI is parallel to N[, if, and only if, 0 and 0' satisfy

80' . 80 80' , 30

(o) - = h— = / .

8w 8w' 8v dv

Expressing the condition of integrability of these equations, we find from (I, 13) that 0 is a solution of the point equation of N, namely

320 8loga 80 8log6 80

(D) == .

8w9r dv du du dv

Moreover, from (5) it follows that 0' is the corresponding solution of the point equation of N' (cf. § 4). Conversely, if 0 is any solution of (6) and 0' the corresponding solution of the point equation of N'. then (2) defines an F transform of N. Hence:

Ant/ transformation F of a net N is determined by a net Nr parallel to N. such that the joins of corresponding points of N and N' are not concurrent, and by a solution of the point equation of N; awl fni// .<-nr/t )/>'f \' "??<7 a solution determine an F transform.

36 II- Transformations I*1

In consequence of (5) equations (4) may be written

dxi _ ff dx[

dxi^= T dx[

du h du' dv ~

where we have put

(8) T = hQ — 6' a =

I dv '

By means of (1), (3), and (5) equations (7) are reducible to

(9)

r I , 90 , dx\ dxl

I *Xs ~ V

du 8f* r du du)' dv ' ' 0'2

which, in consequence of (2), are equivalent to

_^= . dd

(10)

dxl

du 90

From (9) it follows that the point equation of JVi is reducible to

(ID

where

(12)

dudv

dv du

«i == a -fir, »i

90! . 9lOg&! 90!

du 8tT'

a

V

This equation may also be put in the form

h 90\ 90! T /9log& I 90

dudv

\dv

du

In consequence of (I, 37) equations (2) can also be given the forms

(14)

T, - • -- T

1 0'

Shy ff

- ^— — — --

0' 0'

Biz

-

0' '

16. Inverse of a transformation F. Parallel transformations F 37

From (8) and (I, 18) follows (15) r — a = <f B.

Incidentally we observe that T and a satisfy

(16)

( dr 9,7. 9* 8.0

= ^_logj9P) _=9,,__Iog_

da 8 . 0 dff 8 .

— = — By— -log-r, — = — 0<jp — — logay.

I du ' 8t< 6 &' 8v r 8v

Suppose we have any congruence G. There is a net N con- jugate to G (§ 8), and a net N' parallel to N whose coordinates x' are direction-parameters of G (§ 5). Each radial transform of N', say N'l} determines an F transform NI of N, and ^\ is parallel to N(. Since there is an infinity of parallel nets JVt satisfying this condition (§ 5), we have the theorem:

If the coordinates of a net are the direction-parameters of a congruence, there are an infinity of nets parallel to the former net and conjugate to the congruence.

As a corollary we have:

If two congruences are parallel, every net conjugate to the, one is parallel to oo1 nets conjugate to the other.

For, if N is a net conjugate to the first congruence, there is a net N't parallel to N, whose coordinates are direction-parameters of both congruences, and by the theorem there are cc 1 nets conjugate to the second congruence and parallel to N'.

1 6. Inverse of a transformation F. Parallel trans- formations F. Evidently N can be looked upon as a transform of NI and now we seek the functions 0~x and (0')-1 giving this transformation. Since the roles of N' and N( are interchanged it follows from (1) that

(17) (fl')-1--^-

Hence if we make (2) conform to

0-1 4 x == Xi— ./_ x(,

38 H. Transformations F

we find that

(19) «-' = -\.

It is readily verified that equations (7) are satisfied by these values of 0"1 and (0')"1- Hence 0-1 is a solution of (11). Moreover, we have

9 / 1 \ 9/1

T

du \0~1/ du \ 0

<.L(M_M1

. dv \e-ll dv \ 0

Hence equations (9) can be written

(21)

d ( & \ d lx

= f~ — i-r-

9w\0~1/ 9w\0

9 / x, \ _ d lx

a dv\e]'

From these equations follows:

If N and N± are in the relation F determined by a solution 6 of the point equation of N, and S~l is the solution of the point equation of Nt likewise determining the transformation, the radial transforms of N and Nj, by means of 6 and 6~l respectively are parallel.

We have observed in § 15 that two parallel nets are in relation F, since they are conjugate to the congruence of lines joining corresponding points. We wish to find the form of equation (2) in this case.

From (9) we see that:

A necessary and sufficient condition that NI be parallel to N is that 6 be a constant.

Now 0' also is a constant, which must be different from zero. Hence the equations of the parallel transforms are of the form

/y*- ^' '^ np /"/*

^,j --^-- ^ t/*A/ *

where c is an arbitrary constant.

In a general transformation F the function 6 corresponding to a given 0' is determined by (5) only to within an additive

17. Harmonic congruence of a transformation F 39

constant. Suppose we consider the transforms Nt and iV2, corre- sponding to the values 0 and 0 -f c, where c is a constant and to the same 0'. Now the coordinates of JVi are given by (2) and those of Ns by

= x e^Cx' From these follows

Ju% Ji/1 ~~ (sJb+m

In consequence of the above results we have:

When the conjugate congruence of two transformations is the same, and the two functions 0 differ by an additive constant while S' is the same, the two transforms are parallel to one another.

In the definition of transformations F we have required that N' be not a radial transform of N. It is readily shown that in order that Nf, parallel to N, be a radial transform of N it is necessary and sufficient that x' = ex, where c is a constant. In this case corresponding solutions 0 and 0' of the point equations of N and Nf are in the relation 0' - = cO-\-d. Substituting in (2), we have

xd

cO -f- d'

that is NI is a radial transform of N. Conversely, in order that (2) define a radial transform of N it is necessary that N* be a radial transform of N. Consequently, if the restriction is removed from the definition of transformations F, radial transformations form a sub-group of transformations F. But we shall retain the restriction and thus distinguish between the two types of trans- formations.

17. Harmonic congruence of a transformation F. From (20) and (21) we have

(22)

0"1 dxi 0 dx

Xl ~'

0- dX 0 dx

dv dv

Hence the corresponding tangents to the curves u = const, of the nets N and N\ meet in points of a net which is a Levy transform

40 II. Transformations .F

of Nky 6 and JVi by 0"1, and likewise the tangents to the curves v= const. Furthermore as follows from the theorem of Levy (§ 8) the line joining these points of intersection generates a con- gruence harmonic to the nets N and NL. Hence we have the following converse of the second theorem of § 15:

When two nets N, Ni are in. the relation of a transformation F, tJieir corresponding tangent planes meet in a line generating the congruence harmonic to N determined by the function 6 and harmonic to NI determined by 6~1.

We call this the harmonic congruence of the transformation.

From the above theorem and the second of § 15 follows:

If N is a net and O a congruence harmonic to it, the nets harmonic to G are obtainable from N by transformations F involving the same function Q, or by radial transformations of N.

Since 0' is determined by (5) only to within an additive constant, we have as a corollary to this theorem:

All the nets Nt obtained from N by transformations F deter- mined bg*the same function 6, and differing only in the additive constant of 6' , are harmonic to the same congruence, and consequently their tangent planes form linear pencils.

The coordinates of the point of intersection of the conjugate congruence of a transformation F and the hyperplane xl = 0 (cf. § 1) are of the form

_^_ , " of :

Since xl and a;*" are corresponding solutions of the point equations of N and N', we have the result:

The developables of any congruence meet a hyperplane in a net.

In the above case the ith coordinate of N( is 1, as follows from (1). Consequently N( also is a net in a hyperplane.

1 8. Tansformations F and radial transformations. Let N&) be a radial transform of N(x) by means of a solution « of the point equation (6) of N, so_that x = x/a>. From § 14 we have that the point equation of N is

(23) 82^ _9_, JLllj_J_i A

' ' °g ~ °g

dudv dv a du du w dv

18. Transformations F and radial transformations 41

If 6 is any other solution of (6), then 6 = 0/« is a solution of (23). Also it can be shown that if y is a solution of the adjoint of (6), then <p = <f><*> is a solution of the adjoint of (23).

From (16) it follows that if t and a are functions of the trans- formation F of N by means of 6 and y, these functions, T and tf, serve alsojtor the transformation F of N into a net jVi(#i), deter- mined by 6 and gT . Hence similarly to (20) and (21) we have

.94) 1 \ - r-t /-1\ - r J- W 9 / 1 \ -ffd(w

(.**) r~ — ^o.. ' — ro../>> o.. — ff~^r

/nc\ ui-*'i\ u/-^i ui-^i ^i -^i \ v i X

From these equations and (21) we have »26) ^r = -7PT.

to within an additive constant of integration. If we define a function ^ by

fl-1^! = tf-1, equations (24) become

(27)

Comparing these equations with (21), we note that «i is a solution of the point equation of ATi. Hence from (26) it follows that NI is the radial transf ormj)f NI by means of «i. Thus by the quadrature (27) we obtain a net NI which is an F transform of N. Moreover, there are an infinity of such nets NI, since wc = MI-J- c0—1, where c is an arbitrary^ constant, satisfies (27). Hence:

If N and N are nets in the relation of a radial transformation, and NI is an F transform of N, there can be found by a quadrature oo l nets NI, which are F transforms of N. and radial trans- forms of NI.

When in particular 6 = w, then 0 = 1 and consequently N and the nets NI are parallel, the functions &>i being c6~1. Hence:

42 II- Transformations F

A transformation F is equivalent to ilie combination of a radial, a parallel and a radial transformation11}.

19. Transformations F with a common conjugate congruence. Suppose we have two solutions 0i and 02 of equation (6). We seek the two nets obtained from N by trans- formations F determined by these respective functions and by a net N' parallel to N. We denote these nets by .2Vi,i and ^.i12).

The point coordinates of Ni,i and N-t,i are expressible in the forms

(28) #1,1 = # — tf-x', x*ti = x — rr x',

01 02

where 0i, 01 and 02, 0a are pairs of corresponding solutions of the point equations of N and N', that is 01 and 02 are obtained from 0i and 02 respectively by quadratures of the form (5). We consider the functions

From (2) and (1) it follows that these functions are solutions of the point equations of -ZVi,i and N[,i respectively, and their derivatives are in relations analogous to (7). Hence a transformation of Ni, i is given by

0201—0102 ,

(30) xi, i

-TJ

02

By substitution we find that this expression is reducible to that of X2,i, given by (28). Hence JV^.i is the transform of JVi,i by means of N{,i. and the functions (29). It follows then from § 16 that ^i,i is obtained from .A/2,1 by the functions 0i — 0201/02 and 0i/02.

20. Transformations F determined by the same func- tion 0. Let N1 and N" be two nets parallel to a given net N

n) In fact Jonas developed the transformations from this point of view (see Preface); this theorem follows also from the first of § 16.

12) In this notation the first subscripts refer to the subscripts of 0i and 0* and the second to the subscript of common function, 9*1, determining the con- jugate congruence.

20. Transformations F determined by the same function 0

43

which are not radial transforms of one another. The coordinates x' and x" of N* and N" respectively are given by quadratures of the form

( dx' dx dx' dx

(31)

du dx"

du

du'

dv

dx"

dv

dv'

dx

~dv'

where hi, l\ and hz, k are pairs of solutions of equations (1, 13). The coordinates x' and x" are the direction-parameters of two congruences, G-' and O" respectively, conjugate to N. By means of them and a solution 0i of the point equation of N we obtain two transforms JVi.i and Ari,2 of N, whose point coordinates are of the respective forms

#1,1 == x —

#1,2 — X

•#1.

It is our purpose to show that Ni,i and Nitz are in relation F. There is a net which is the F transform of N" by means of e[f and N'. Its coordinates are of the form

Differentiating and making use of (19), (31) and similar equations for B'I and 0", we get

(34)

0 V

du 3 #1,1

Hence ^i''i as defined by (33) is parallel to JVi.i13).

We have seen that the solution of the point equation of Ni, i giving N by the inverse transformation is — 0i/0i. When x\ ,1 in (34) is replaced by this value, we have by a quadrature a solution of the point equation of ATi'i. By means of (20) we find that the

13) We have used the notation x^ to mean that the net is parallel to JV, j and determines a congruence G-'" conjugate to N^v

44 H. Transformations F

•

corresponding solution of (34) is — 0i70i. Hence a transformation F of NI,I is given by equations of the form

(35) *M— Jr*£i.

By substituting the above values we find that this expression is reducible to the second of (32). Thus Ni,2 is a transform of Ni,i by means of the same function, — 0i/0{, which gives the trans- formation of Niti into N. Hence we have the theorem:

If a net N is transformed into two nets NI,I and NIJ by means of the same function Bv the latter two nets are in the relation of a transformation F; moreover, in the triad of nets N, N^i andNitz, any two are the transforms of the third by means of the same solution of its point equation.

Hereafter we say that three nets so related form a triad under transformations F. Now equation (33) may be interpreted as follows :

If the nets N, Nlf and jV2 form a triad, and if N' and N" are the nets parallel to N determining the transformations from N to Nj_ and NZ respectively, the net N["i determining the congruence of the transformation from ^/i to NS can be so placed in space that it is an F transform of N", the conjugate congruence of the latter transformation being determined by N' .

As a particular case of this result, suppose we use for 0! the coordinate x(n). Then the two transforms are the nets in which the hyperplane x(n) = 0 is met by the lines of the two congruences, and in accordance with the above theorem these two nets in the hyperplane x(n) = 0 are in the relation of a transformation F. In general, we have:

If a net N is conjugate to two congruences Or and G" , the developables of these congruences meet any hyperplane in two nets in the relation F.

We shall prove the converse of the above theorem:

If two nets Ni and Nz, transforms of a net N by means of congruences G' and G', are F transforms of one another, the three nets form a triad, unless Ni and N% are parallel transforms of N.

Let the coordinates of xi and x% of Ni and N% be given by

0i / 02 n

x\. — x t x , Xz — x - n x ,

PI #2

21. The theorem of permutahility of transformations F 45

If these nets are to be in relation F, it is necessary and sufficient that on the lines joining corresponding points there be focal points, that is that there exist functions A1? A2, /*i, /*2 such that

du

When the above values are substituted in these equations, we get equations of the form

, .

ou ov

Evidently we must have

At = B! = Cj. = Az = jB2 = (72 = 0. These conditions are equivalent to

(1 + ^i) *, »2f- ^^ 0; = 0, - (1 + Ax) tfl 02'— ^tf, 0; — 0,

If Tl = ffl=TS.=a^== 0, then Ax = l± = const, and 7^ = Z2 = const. (§ 4), and consequently #' and G-" are not distinct. If 0X and 02 are constants, then N± and ^ are parallel (§ 16). Excluding these cases, we find that the above equations necessitate 02/0i = const., that is N, Nlf N% form a triad.

21. The theorem of permutability of transformations F. The equations (1) and (2) apply to any pairs of solutions of the point equations of N and JV~i,i. Making use of (1), (2) and (18) we can show that any solution of the point equation of ATi,i is expressible in the form

(36) ea = 09—-0't,

46 n. Transformations F

where 02 is a solution of (6). Incidentally we remark that from (21) it follows that

I-

I du

(37)

(012 — ) == — *, ^— (— ).

I dv \ 6j l dv \01!

This function 0i2 determines a transform of N\.,i such that its points lie on the lines joining corresponding points on JVi,! and NI,%, that is, G'" is the conjugate congruence. From (33) it is seen that the corresponding function 0i'2' is given by

ft"

(38) 012 == 02 ~7J~ ^2-

Hence the coordinates of the transform N-u are of the form

/om ^2^! — ^!^2 '"

(39) xu — xi,i— — 17-7 - ^rr^r #1,1.

02 vi — 0i t/2

The function 02 and the congruences G' and G" define two transforms of N, namely ^2,1 and ^2,2, whose coordinates are respectively of the forms

f) 0

"

(40)

"2

Corresponding points of the nets N, Ni,2 and JVT2,2 lie on a line, and from (29), (32) and (40) we have that JV2,2 is a transform of ^Vi,2 by means of the function

(41) »,— &-C

In like manner it follows from (29) and (35) that jVi2 is obtain- able from Ni,2 by means of the function 0i2 — 0i0i2/0i'. But by means of (36) and (38) we show that this expression is reducible to (41). Hence ^2,2 and Nu, being transforms of N^z by means of the same solution of the latter's point equation are themselves in relation F. We wish to show further that ATi2 bears to ^2,2 and JVg.i a relation analogous to that born to JVi,i and Ni.* used to determine ^Vi2.

21. The theorem of permutability of transformations F

47

Since Nz,z and Nit\ are obtained from N by 02, they are in relation F. The corresponding net parallel to N2,z is defined by equations of the form (cf.(33))

(42)

x

A solution of the point equation of ^2,2 is

(43)

021 01 ~7jT

02

and the corresponding function 02i" is given by

/ A A \ tttt t ff

(44) 02i =0j rr #1 •

02

The net N^. obtained by this transformation is defined by equations of the form

(*K:\ 01^2 — ' 0201 //// (40) X21 = X2,Z — -^77- TJT^r %2, 2 .

Making use of the above values, we find

(Aft) .— • _ (01 02 0~/fl ^

coincide and the congruences Gr

Vz.,

M

Hence the nets Nia and ^2i and G"" are conjugate to -ZV^.i15)-

In view of the above results we have that when two nets N' and N" parallel to N are known, and two solutions 6l and 02 of equation (6) are given, the four functions 0i, 01', 02 and 0% (each involving an additive constant of integration) can be found by as many quadratures. When these are known, we have a group of

u) The mark " " indicates that the net is the parallel to N2t 2 determining: the conjugate congruence G"",

13) Of. Jonas, Sitzungsberichte Berl. Math. Gesell., vol.14 (1915), p. 103; also Transactions, vol. 18 (1917), p. 111.

48 n. Transformations!''

six nets pictured schematically in the fig. 4 by corresponding points of these nets. We say that any four nets as N, Niti, AT2,2, NM, such that the first and fourth are in relations F with each of the second and third form a quatern.

We note that N^ and N2t2 determine the additive constants in 0i and 6%, but that the additive constants of 02 and 0" are arbitrary and consequently there are oo2 transforms Ni2 of a given Niti and N2j2. However, it follows from (33) and (42) that when one of these constants is fixed, the oo1 transforms N& are conjugate to the same congruence.

We may gather together the foregoing results into the following fundamental theorem of per mutability:

If NI,I and -$2,2 are two transforms of N by means of functions 0X and 02 and congruences 0' and G" , there exist oo2 nets N-&, each of which is an F transform of N^ and N2,2, their determination involving two quadratures; there are obtained incidentally two other nets JV2,i and Ni.2 such that N2)i, N^i, N2>2, Ni2is a quatern, and also N, N%ti, Nitz, NU, Moreover, the six nets can be associated into the triads N, Nlti, N^2; N, N2^, N2>2; Nlt2, N2,2, N12; N2ti, Niti, N&.

Any set of corresponding points in the above configuration are the vertices of the complete quadrilateral formed by corre- sponding lines of the four conjugate congruences of the trans- formations. Each point generates a net which may be taken in place of N as the given net from which the configuration is obtained.

With each net there are associated two parallel nets whose coordinates are direction-parameters of the two congruences con- jugate to the former net. These twelve auxiliary nets may be so chosen that they may be constituted into four groups such that corresponding points of the three nets of a group lie on a line through the origin (cf. (1)). These four groups are N', Nl,i, -ZV^i;

AT"" A7" A7" . AT7" A7"' A7'". A7"" AT7'" AT""16N

iv , -tVi,2, -^2,2; -Wi,i, JMi,2, xVu; «tVa,i» -"2,2, -iVia )•

In consequence of (32), (36), (38), (40) and (43) equation (46) can be written

-- 021— 012 x.

16) In this notation N'tj means that the radius vector is parallel to a line of G'" and the net is parallel to Nij.

21. The theorem of permutability of transformations F From the expressions (36), (38), (43) and (44) we find

01 02

(48) 61 Og = 0? 0%'

010s

(02 021 ~r 01 012 012 02l).

49

Hence the above equation may be written 0"

(49) 0j 012 Xja = -. ~ (02 021 Xi, 1 + Ol 012 X2, 2 — 012 021 X).

02

From (34) it follows that the functions hi? and l\i of the parallel transformation determining G'" the congruence conjugate to N^ and NIZ have the values

(50)

The functions r12, oi2, yi2 of the transformations from N^ into NV2 are given by equations similar to (8) and (15), namely

(51)

12

In consequence of the above values these functions have the expressions

01 02' /02

(52)

2 2 Q 0i „ » \

12 - -- 1T~Z~ I "Z>" 021^1 + -T77 012 ^2 +012 021 , "l "z \01 02

*

tfl "I 02

1 01

In an analogous manner the functions /^i and ^1 of the trans- formation from JV^ to Ni2 are of the form

(53)

and the values of the corresponding functions T2i, er2i, 9321, defined by

(54) ^21=^21021 02l", 021= ^21021 02l". 9>21 021 = r21 °21j

4

50 II. Transformations F

are reducible to

02021 =

< a'a ^

21 = - — (*2 01 - Tl °2 + f/2 01 012) •

02 r2 "l

Consider in particular the case when ^2,2 is a parallel trans- form of N. If we take

02=02'=1, 02=0, then

012 == &12 == 1?

and consequently j^2 is a parallel transform of N^I. From (43) we have

021=0!— 01',

and consequently from (49)

(56) 0i (#12—2:2,2) — (0i— 0") (#1,1— #).

Since 0" involves an additive arbitrary constant, AVC have the theorem: If NI is any transform of a net N and Nz is parallel to N, there

are oo1 transforms N& of NI and N2 which are parallel to NI; corre-

sponding points oftliese nets lie on the line through Mz parallel toMM^ Consider also the case where 02 = 0i + c, c being a constant.

We have accordingly

oi=oi+<f, e'^e'l+c",

012= c-^-c', 9*= -c + ±c", 0%= c»-£c',

Ol t?2 Vl

where c and c" are constants. If c = 0, 0i2 is constant; it follows from § 16 that -ZVi2 is parallel to N^i. Also ^2,2 is parallel to -ZVi-> if c"= 0.

In order to determine the effect of the additive arbitrary constants of 6-2 and 01', we replace them by 02+(c — 1) and 01'+ (e — 1) in (33) and (36). If we denote the new functions by x'e,( and 0C2, we find in consequence of (1) and (19)

(58) xe',( = aft— (e — 1)4, 0C2 = 0i2+ (c —

21. The theorem of permutability of transformations F 51

Hence the oo1 nets N& obtained by varying c and holding e fixed are conjugate to the same congruence conjugate to ATi,i also; similarly as e varies and c remains fixed the oo1 nets Nu and N2,z are conjugate to the same congruence.

The foregoing formulas are interesting also in another connection. Thus if we look upon N and a net Ni2 as F transforms of ^2,2, there are oo2 nets, including Ni,i, each of which forms a quatern with Ni,z, N and N^. The above results lead to a means of finding these nets NCte. In fact we replace Nltl by the transform of N by means of the functions

(59) ec= 0x + (c — 1)02, x® = x'+(e — l)x",

the quantities x(e) being taken as direction-parameters of the con- gruence of the transformation. Now we have

— e ~ ) — e ~ >

du du dv dv

where

(61) he=h1 + (e-l)hs, l,= li+(e—

also

e'c' == ey + (c- 1) o?,

When these values are substituted in the following expression which is the analogue of the right-hand member of (46), it is found that the result is reducible to the latter:

(e'c' e, - oy 0C) afl> + (e? ec - #* e,}x"

Hence:

Let -/VTi5i and N-^-, be F transforms ofNby means of functions 0X and 6%, x and x" being the direction-parameters of the congruences of the transformations; if Nc^is the transform of N by means of 0i -f- (c — 1) 02 and a congruence of direction-parameters x'-{- (e — 1)#", c and e being constants, the oo2 nets Ni2 forming quaterns ivith Nf JVi,i and -A/2, 2 form quaterns also irith N, Nc^ and -Ar2,2 whatever be c and e.

52 II- Transformations F

22. Derived nets and transformations F. If N(x) is a net with the point equation (6), and 0t and 02 are solutions of (6), they determine a derived net JV(x) of N (cf. § 11), whose equations may be written

dx , dx .

x = x where

(65)-

t A m £\

*- = ~J\9lW~ ~e*~d

II ci^l ~ /J \ C3 /) C3 /i Q £1 Q /i

/ 0 u± 0 "2 \ . 0 u% O MI 0 t/2 0 t/i

J \ du 1 du i du dv dv du'

Let 08 be another solution of (6), and JV' a net parallel to A". An F transform N9 of N is given by

(Rfi\ r» — r — r'

\\j\Jj 1X^-3 *X/ i\9

The functions 0ai and 032 defined by

(67) 08i = Oi — ~Y Bi (i=l, 2), where

(68) M = AM, M = ^(,-=1>2?3),

are solutions of the point equation of Ns. They determine a derived net JVS of JV^, whose coordinates are of the form

/en\ 9^

(69) ;ra =

du dv'

where p$ and q3 are given by (65) when 0X and 02 are replaced by 03i and 032 respectively.

From (66) and (67) we have

'<— 03 / , 903 a, 9 ^ [iff— 03-

22. Derived nets and transformations F ' — 03/^,903 >, 90;

53

du dOsi

— 03 tn, 90

81;

On substituting these expressions in (69), the resulting expression in reducible to

(70)

where

01 02 03		01 02 03		01 02 03
90! 902 903	dx	01 02 03	dx	01 02 03
9 u du du	1 du	dv
90i 902 90s		90i 90g 903		90j 902 903
dv dv dv	9 v dv dv	du du du
90s 903 902 903 \ . , /903 90, 903 90X \
dv du du dv/ * \ dv du du dv/

, , /90i 902 90! 908 h 3 \ 9y 9w ~~ du dv!'

Since 0i and 6'2 are solutions of the point equation of N', a derived net N' of N' is given bjr equations of the form

, dx' . , dx'

du "*" q dv'

where p' and #' are obtained from (65) by replacing 0i and 02 by 0i and 02. From equations analogous to (I, 68), we find

where

dx'	r dx dx'	7 dx
du Q' 8<?2	— ll * 1 du dv ,,90i	, 904 , 90,
1 dv	2 dv j_	1 du * du
a 902	ft 8dl'	90s 90i

Consequently 2^' is parallel to N.

54 n. Transformations I*7

The functions 03 and 6% defined by

are corresponding solutions of the point equations of N and N'. Hence quantities of the form

03-,

x — ^^x

08

are the coordinates of an F transform of N. When the above expressions are substituted in this quantity, it is reducible to (70). Since 01 and 02, as given by (68) involve additive arbitrary constants, there are_oo2 nets Ns. Hence we have the theorem:

If N is a derived net of N and N3 is any F transform of N, there can be found ~by two quadratures oo2 nets Nz, each of ivhicli is a derived net of N% and an F transform of N.

23. Derivant net and derived net of two transfor- mations F. We note that the corresponding points of six nets in the relation of the theorem of permutability are coplanar. From the second theorem of § 11 it follows that these planes envelope a derivant net N of N. Since the four congruences of the con- figuration are harmonic to N, it is a derivant net of each of the six nets. By means of (32), (40) and equations similar to (8) and (15) the expressions (I, 67) for the coordinates of N are reducible to

(71)

I *

01 0g 02 01

0<j' 01

ft f) ft Jl""*}^ f\ JS" 'i, J."

From § 17 it follows that corresponding tangent planes of N, Niti and -A^,2 meet in the point which generates the derived net N of N determined by the solutions 0X and 02 of the point equation of N. We shall show that the corresponding tangent planes of the oo 2 nets N^ pass through this point. In fact from (I, 66) it follows that the coordinates x of this point may be given the form

x = x-\-

__ _ dx 9_ _

dv e du du\e dv

rJL| _L)_L(A)_.J_f_L\JL|«ifl ll8« \9ii dv \ej dv \ej du \ein

24. The extended theorem of permutability 55

Consider now the net Niti and look upon N and a net -ZVi2 as F transforms of it by means of 0f1 = — BjQ[ and 0i2. The corre- sponding formula for this case is

/} / /•}' \ /5 'V*

.^r12"0t)~9V~

dxt

0! a / i | a /„ 0'i\ 9 / i \ 9 / 0; "9^ V0^~) "97 r12 07) ~"97 \~0r~l "97 ri:r0;

By means of (9), (20) and (37) this is reducible to the preceding- form. Hence:

The corresponding tangent planes of N, N^i, AT2i2 and tJie oo2 nets N& meet in the point which generates the derived net of N by means of the functions B\ and 02 of the transformations of N into NI,I and N2>2-

24. The extended theorem of permutability. In this section we extend the theorem of permutability so as to involve three transforms of N. Let NI, N2 and N8 be these transforms of N by means of the functions 0» and wi (i == 1, 2, 3), where

0i 02 0r_' 02i

01 02 012

>-2\

Applying the theorem of permutability to the three pairs of these nets, we get three families of nets N^, N2s and JVgi, since NH = Njt. From (48) and (49) we have

,-n\ I 8i 8iJ WiJ = WJ fy ®J* ~^ ®* 8iJ — ^*V ^/f)'

= M;; (0; 0;^ + 0i QijXj— Oy QjiX) (i $ j).

Since a net -ZV12 and a net Nw are transforms of -ZVi, there exist oo 2 nets N for each of which NI, N&, N^ and ^ form a quatern. It is our purpose to show that one of these nets N is such that •#2, NIZ, N2S, ^forn^ a quatern; and likewise N9, N13, N2S and N. We denote by 6ij and Wij the functions by means of which Nij is _transformed into JV; from their definition it_follows that Bij = Bji and W^ = wji. According as we look upon N as belonging to the_one or the other of the quaterns, Ni} Ni2, NIS, -^V; NtJ Niz, NW, N, the coordinates x of N are given by the respective equations

56 II. Transformations F

\ 612OuWi2X — Wl8 ( 013 013^12 + 012 012 #13— 012 018^1 ),

(73) \ __ _

{ 021021^21^ = W23 (023 028 #12 +021 021 #28 - 021028#2),

which are analogous to (72). When we equate these two values for x, the resulting equation is reducible by means of (72) to an equation of the form Ax\-\-Ex^-\-Cx = 0. Since A, B and C are necessarily equal to zero, we have the following equations of condition : _ / ft ft \ ft f)

01 ^'12 01 2 I 01 M>18 013 - 08 WS — — I - 01 M>13 013 02 W* — L -f 02 W2S 023 01 M>2 023 = 0,

\ 018 / 012

01 M>12 012 I 02 M>23 023 - 03 U>3 —JT ~ I - 02 ^28 023 01 M>2 028 + 01 ^'18 013 BaW2 ~ = 0,

\ 023 / 012

ft ft

0lM>12 012 (01 ^3 032 - 02 ^3 03l) - 01^13 013 0g Wt - — + 02 M>23 023 01^2 023 = 0.

012

By subtracting the last of these equations from the first and adding it to the second we find that the above system is equivalent to

( 01 012 Wi2 012 ^ ^2 (02 021 013 ~]~ 01 012 023 - 03 012 02l)>

(74) < 01 013Wl30i8 = W3 (08 081 012 + 01 018 082" 02 013 «8l), V 02 023 W23 023 === U '3 (0$ 032 021 ~1~ 02 023 031 - 01 023 032)-

From (38) and (44) it follows that

(75) li\ Wi-2 01 012 = WZ W%\ 02 021 ,

so that the_above equations are consistent with the requirement that Oij — Oji. When equations (74) are compared with the second of (72), it is found that Oij is a solution of the point equation of N^. which is a necessary condition. The analogue of the first of (72) is

012 012 Wl2 = M'18 (013 013 ~f~ 012 012 - 012 01s)-

Substituting in the right-hand member the expressions for 0i2 and 0is from (74), we have

25. Transformations K 57

O=— - = 0i (012023+ #13032 032 #23 )

102^3

H~ ^S (023 081 4~ 021 013 013 03l)

-f~ 03 (031 012 ~h 032 021 021 012)

— 012 023 031 021 013 032?

which in consequence of (75) is consistent with the requirement that ttty— Wji. By means of these results equations (73) reduce to

(76)

— Xi (02 023 031+ 03 032 021 ~ 01 023 032)

~h ^ (03 031 012 H- 01 013 032 02 031 0is)

~h #3 (01 012 023 H- 02 021 013 03 012 02l)

X (012 023 031 ~h 021 013 082)-

Since this expression is symmetrical in the functions involved, it follows that N defined by (76) forms a quatern with Nt, N12 and jV28 and also with N$, N& and ^3.

25. Transformations K. We inquire under what conditions a net N and a transform NI meet the lines of the congruence in points harmonic to the focal points. From (I, 37) and (2) we have that the necessary and sufficient condition is that

(77) 0'==j(A + 00.

When we require that this function satisfy (5), the resulting' equations are reducible, in consequence of (I, 13, 18), to

8 e

I

From this it follows that the point equation of N is necessarily of the form

(79) 9*0 _ 9 log Vg" 90 8 log V~Q 9 0 dudv ~ dv du du dv'

where Q is defined by

(80) gy = 20.

58 II- Transformations F

Then from (I, 18) and (77) we have

A /•)' A A

(8i) 7z==7 + T' l= "7 + T-

Now the equations (5) may by written

0'\ 1 90 9 I0'\ 1 90

9M \ 0 / Q du' dv \ 0

From (8) it follows that (83) *==-,= £

Hence equations (10) become

/ dxi _ !:•/•> s 90

(84)

. ( V^x •"/*__

1 0 L 90

-rfe — a?)- +0

dv Q 0 and the point equation of Ni is

(85) 920i = 9 log Vgi 9 0i , 9 log i7^ 9 0i

9?<9v ~ 9y du du dv'

where

(86)

We note that the invariants H and ^T of equation (79) are equal, or, in other words, N has equal point invariants. Since the same in true of Ni} as shown by (85), we have the theorem:

In order that a net N and a F transform N± meet the lines of the conjugate congruence in points harmonic to the focal points, it is necessary that both N and Nt have equal point invariants.

Koenigs solved this problem for 3-space17). Accordingly we call a transformation of this sort for space of any order a trans- formation K.

") Comptes Rendus, vol. 113 (1891), p. 1022.

25. Transformations K 59

When two nets in the relation of the above transformation are parallel we say they are associate18). In all generality equations (84) are in this case

dxi I dx dx± 1 dx

(87)

du Q du' dv Q dv

Suppose now that N has equal point invariants. The know- ledge of a parallel net Nr gives h and I, and consequently a solution y of the adjoint (I, 20) of the point equation which in this case is reducible to

Since equation (79) can be written

a solution of this equation is given by (80). It is readily shown that 6' given by (77) satisfies (5), and thus we have:

When N has equal point invariants, each parallel net N' deter- mines without quadrature a transformation K into a net Ni, the function 6 of the transformation having the value y> • Q/2, 6' being

given by (77).

2 In particular, q> = - - is a solution of (88). In this case

Q 10 = 1 from (80) and consequently Ari is associate to N.

Suppose conversely that we have a solution 0 of the point equation. From (88) and (89) it follows that y> = 26/Q is a solution of the adjoint equation. From § 4 we know that by means of y> we find oo x nets N' parallel to N, of the form x' -\- ~kx, where x' are the coordinates of one of these nets and A; is a constant. Accordingly we have the theorem:

When N has equal point invariants, each solution of the point equation determines by a quadrature, an infinity of transformations K,

18) This is a generalization of the idea of associate surfaces in 3-space [§ 155 and Ex. 22, p. 425].

60 H. Transformations!'1

such that corresponding lines of the conjugate congruences of the transformations which pass through a point of N are coplanar.

26. Theorem of permutability of transformations K. If NI and NZ are two nets, each in the relation of a transformation AT with a given net N, we apply the theorem of permutability of general transformations F, and seek the nets N&, which are K transforms of NI and N%. For this case we must have

Tl2 === 012, ^21 == 021«

Since

(90) TI = — GI = — , TZ === — 02 == — j

these conditions are equivalent, in consequence of (52), to

(91) 0^4.^ = 0.

In consequence of (90) equations (37) for the case of trans- formations E are reducible to

(92)

0i \ I 901 90

I i ft _ n _

I I I/O \J\

0i / Q \ du di

Oi2-^} = —— (024^-0i

dv V12 6,1 Q

From these equations and similar ones in 02i we find that the left- hand member of (91) is necessarily constant. Since each term of this expression is determined to within an additive constant, there are oo1 sets of solutions satisfying (91). In fact, in consequence of (36) and (43) we can put (91) in the form

/l" /j fi? f\

/r»Q\ "2 t/i „ . aitfg „;

(93) — — 0i -) * — 0o = 0.

From (52), (86) and (93) it follows that

O 012 o 021 <JP12 & , 9>21 == & •

Hence each JVia for which (93) is satisfied is a K transform of and JVg.

27. Transformations F of applicable nets 61

Now equations (48) and (49) become

(94)

1 / 0" 0' fi>

01012*12= 012 01^*2,2— 02^*1,1+ 012^*. \ »2 01 01 /

By means of these results the coordinates (71) of the point of contact Mot the plane of the quatern with its envelope are reducible to

* __ 02X *2,2— 0102*1,1 = _ 01201 *12— 01 012*

02 01 - 0102 012 01 - 01012

i

Hence M is the intersection of the lines MMIS and l/i-M"2; con- sequently the points If12 of the oo1 nets lie on a line, MM12- There- fore in consequence of the theorem of permutability of general transformations F (§§ 21, 23), we have the following theorem of permutability of transformations K:

If NI and N2 are K transforms of a net N with equal point invariants, there can be found by a quadrature oo1 nets NH which are K transforms of Nt and N2; corresponding points M^ of these nets NIS lie on a line I through the corresponding point M of N and in the plane re determined, by M and the coi'responding points Mt and M2 of N-L and N2; the plane n touches its envelope at the inter- section of I and the line M^M*; the parametric lines on the envelope form a net to which are harmonic the congruences generated by the lines MMi, MMZ, M^M^, MSM12, and the tangents to this net are harmonic to I and MiMz.

27. Transformations F of applicable nets. We say that a net N in n-space and a net N in ^?-space are applicable, when their coordinates x and x satisfy the condition

which is equivalent to

E = y* (— H = y (— —)» F ='

^ \ou! i \ou] i ou ov ^r1 ou

n /ar\2

o =y i— —i =

1

62 n. Transfonnations F

From (I, 3) it follows that the point equations of the two nets are the same. Hence_a pair of solutions h and I of equations (1, 13) determine a net N' parallel to N, as well as a net N' parallel to N. Moreover, it is_evident that N' and N' are applicable. Hence:

If N and N are applicable nets, the knowledge of a net parallel to either enables one to find by quadratures a net parallel to the other, to which it is applicable.

Suppose now that we subject j^to a transformation ^determined by a parallel net N' and a solution 6 of the point equation of N. From (9) it follows that the fundamental coefficients of the new net JVi are given by

We transform N by means of the net N', applicable to Nr, and the same function 6 used above. In order that Nt and N^ shall_be applicable, the expressions for the fundamental coefficients for NI must be equal to the above. Equating the corresponding expressions, we get three equations, which in fact are equivalent to the two

— 2 x'— — x'— 6' - J * du ^x g 6

'

By integration we find, to within a negligible constant factor,

(95) 0'==2y*— 2£".

It is readily shown that this function is a solution of the common point equation of the applicable nets Nr and Nf. Consequently the F transforms of N and N by means of 6' and 6 given by the quadrature

28. Nets corresponding with orthogonality of linear elements 63

dv dv

are applicable. _Hence19):

If N and N are applicable nets, each net parallel to N deter- mines a pair of applicable^ nets NI and NI which are respectively F transforms of N and N.

We shall establish a theorem of permutability of these trans- formations. Let two applicable nets N and N be transformed into pairs of applicable nets_2V"i, NI and N2) Nz be means of parallel nets N't N' and N", N" and functions 01 and 0" given by

(96) 8{ =2at* -2*'*, 0* =2*"* —2*"*.

We apply the results of § 21 to this case and seek whether jV12 and N12 are applicable. To this end we take

nin _ V'/v,"'2 \\T'"2 />"" _ ^\,""2 ^^""2

012 - £,X\ — £jX\ , 021 ~ ^X% — Z-.X1 .

Substituting the values of these functions from §§ 20, 21, we get

(97) 0i' + 02-2 (^x1 x" —2*' *") = 0.

In consequence of (96) we find by differentiation that the left-hand member of (97) is constant. We have observed that in the general case 0i' and 02 involve additive arbitrary constants. Hence they can be chosen in an infinity of ways to satisfy (97), and we have:

Of the oo 2 transforms Ni2 and N12, cx>1 pairs are applicable, these cases arising when the constants in 0" and 02 are chosen so that (97) is satisfied.

28. Nets corresponding with orthogonality of linear elements. Two nets in n-space are said to correspond with orthogonality of linear elements, if corresponding directions on the surfaces of these nets are orthogonal to one another. We say that two such nets are in relation 0. A necessary and sufficient condition that N(x) and N(x) are in relation 0 is [cf. § 153]

dx -sdx dx idx dx dx dx

19) Transactions, vol. 19 (1918) p. 170.

64 II- Transformations F

Let (6) be the point equation of N(x) and let

920 _ 9 logo" 30 8 log 6 38

dudv ~ dv du du dv

be the point equation of N(x). If the first and third of (98) be differentiated with respect to v and u respectively, we have in consequence of the second of (98) that N and N have the same point equation. Hence:

When N and N are two nets in relation 0, they have the same point equation.

From this result and § 4 it follows that if h and I are a pair of solutions of

(99) »

the nets N'(x') and N'(x'), whose coordinates are given by

f

dx dx -,dx dx dx

du ~ du' dv dv' du ~ l du' dv ' dv

are parallel to N and N respectively. Moreover, from (98) it follows that TV7' and N' are in relation 0. Ifjwe say that Nf and N' are corresponding parallel nets" of N and N, we have :

If N and N are two nets in relation 0, corresponding parallel nets are in relation 0.

We seek transformations F of nets N and^ in relation O into nets NI and NI in relation 0. Let N' and N' be two corre- sponding parallel nets of N and N, and let 6 and 6' be corresponding- solutions of the point equations of N and N' respectively, that is

(loi)

du du' dv dv

The equations of the transformations are

ft ft

(102) Xj = X 77"^ > X\ = X 77"^ •

V u

28. Nets corresponding with orthogonality of linear elements 65

From the first we have by differentiation equations (9) and similar

expressions for - - and - ^-.

du dv

When we express the condition that these quantities satisfy equations of the form (98), the resulting equations give, to within a negligible constant factor,

(10^} ft' — TV^'

\l\JtJj ^^^iAs tji, •

Since AT/ and N' are in the relation 0, this function 0' satisfies their common point equation.

'Hence:

If N(x) and N(x) are nets in relation 0, and N' (x') and JV'Or') are two corresponding parallel nets, the nets NI and Nt ivhich are F transforms of N and N respectively by means of the equations

i\ i\

(104) jc — x x< % %

2*x x

•where 6 is given by

, dS _ 1 8 v ,_, 30 _ 1 9 v ,_,

^r = STl^^^C:' ~to = '"T~to^X

are in relation 0.

We shall establish a theorem _o_f pel-mutability of these trans- formations. Let two nets N and JVjn relation 0 be transformed into pairs of nets N1} NI and N2, N2 by means of parallel nets N1, N' and 'N", N" and the functions e( and 0" given by

(106) 0'1 = 2x'xt, e'J = 2x"x">

We apply the results of § 21 to this case and seek under what condition N12 and Nis are in relation 0. For this to be the case we must have

nin X1 in — in aiiri "^ nn — nn

B\-2 ~— £jXi X\ , 021 -— £jX-2 X-2, .

where x1" and x%" are given by (33) and (42) and x"' and x'l" have analogous expressions. Substituting the values of 012' and 02i" from § 21. we have

(107) 01' + d't — 2(x" x' + x' x") = 0.

66 II. Transformations F

In consequence of (106) \ve find by differentiation that the left-hand member of (107) is constant. We have observed (§ 21) that in the general case of the theorem of permutability the functions 6" and 02 are determined to within additive arbitrary constants. Since these can be chosen in an infinity of ways so that (107) holds, we have:

Of the oo 2 F transforms Niz and JV12 arising from the thc.orci/i of pet-mutability of any nets N and N in relation 0, there are oo1 pairs in relation 0\ they arise when the constants in 6-> and 6" are chosen so that (107) is satisfied; all these oo1 pairs of transforms can be found ~by two quadratures, when Nif NI and JV2, N£ are known.

Exercises.

1. If N is a net of translation (cf. I, Ex. 6), the only F transforms of N which are nets of translation are parallel to N.

2. If N is a minimal net (cf. I, Ex. 7), the only F transforms of N which are minimal nets are parallel to N.

3. If N is any net and NI is the F transform defined by (2), and (N*)i, (3>Ti)i and (N')i are the first Laplace transforms of N, NI and N' respectively, then

where

i — nl "7; — i \ i — --- oT -- 1 -- o — •

8 log a ov ologah ov

Qv dv

4. If NI is an F transform of a net N, any Laplace transform of N admits as F transform the corresponding Laplace transform of NI, the equations of the transformation being

r r

where (0)r, (0'V and (x')r are the rth Laplace transforms of 6, Q' and x'.

5. If N and JVi are in relation F, and y and ?i are corresponding solutions of their point equations, the Levy transforms of N and NI on the tangents to the curves v = const, of N and Nj. are in relation F, and also the Levy trans- form on the tangents to the curves u = const.; moreover, the lines of the conjugate congruences of these transformations are tangent to the net which is the F transform of N by <f> conjugate to the congruence of the transformation from JV into J\ri.

6. By means of Ex. 5 and § 11 show that if N and NI are nets in relation F, and <?, a>i and </>, </>i are corresponding pairs of solutions of the point equations

Exercises g7

of N and Ni, the derived net of N by y> and <p is in relation .F with the derived net of Ni by <PI and </>i.

7. If N and JVi are nets in relation F, and JV2 and JV12 are parallel to N and J\Ti respectively in accordance with the second theorem of § 21, corresponding lines of the congruences of parameters x2 and x^ conjugate to N and Ni meet in a point describing a net forming a triad with N and NI.

8. Show that if in § 24 we take 0, = «?3 = 1, 0'3 = 0'3' = 0, the nets Na, Nit, Ar2g, ^ are parallel to N, NI, N3, Nlt respectively.

9. If N, Ni, Ni, Nu are nets of a quatern, the respective functions 02, 012, — 0s/0", —014/012' determine radial transforms N, NI, Na, N12, forming a quatern under transformations F. Transactions, vol. 18 (1917), p. 123.

10. If N is a net and 0 any solution of its point equation, the equations xi = x — c0, where the c's are constants define a net N\ which is a transform of N, the congruence of the transformation consisting of parallel lines. The tangent planes to .iV and NI meet in the harmonic congruence of N determined by 0.

11. If N and N are applicable nets, the equations x\ —x — c0, Hci — oT — c~0 determine applicable nets N\ and Ni, if 0 = 2 (~~cx — 2«aj)/(^eJ — ^c2).

12. When the functions a and b in equation (6) satisfy the condition

92a ; 826 _/a b\da 96'

dudv dudv \b a! bv 8u' the equations

91ogyr/o 1 36 Slog"!//* 1 da

011 a 011 ov b ot'

are consistent, and the functions (ii) /»., = £—-, Ii =

, .1 — , , i- — , , »z —

a Ypa

satisfy equations (I, 13). The two nets N and JV0 pai-allel to a net N' with equation (6) determined by the solutions (ii) have equal point invariants, and are associate (§ 25) to one another.

Conversely, if a net N' with point equation (6) admits a parallel net with equal point invariants equation (i) must be satisfied.

13. When the condition (i) of Ex. 12 is satisfied, equation (6) admits the

solution 0' = -T- (a2 — 62). If we put 0 = -^- j/~/o (a — 6), we have -^ — = -~—

\/ o

-« — = -ft—. These values satisfy (81), and consequently 0 and 0' and the

congruence conjugate to N whose direction-parameters are the coordinates x' of N' determine a K transform of N.

14. Let M, MI, Mi, MII be corresponding points of four nets N, NI, Ni, Nu of a quatern under transformations F. Show that a necessary and sufficient condition that another net N^ obtained by varying the additive constants of 0i' and 02 be such that its points lie on the corresponding lines MM^ is that the transformations be K.

68 II- Transformations F

15. If X is a net with equal point invariants, NI, JV2, Ns are A' transforms of N, and NU, Ni3, N^ are the nets with equal point invariants which together with the respective groups^7, NI, Nt; N, Ni, Na; N, N», Ns form quaterns under trans- formations A', then N defined by (76) is a A' transform of NV2, NiS, -A'-,.

Transactions, vol. 16 (1915), p. 29(5.

16. Determine whether transformations K are the only transformations F of a net with equal point invariants into nets with equal point invariants.

17. In order that the Levy transforms conjugate to the tangents to the curves of parameter u of a net IV with the point equation (6) and determined by solutions 6 and 0t of (6) be K transforms of one another, it is necessary that 661 = 62F, where V is a function of v alone.

18. If N and NI are two nets in relation K. their respective associates can be so placed in space that they shall be in relation K.

1 9. If in (46) 0i' and 0'2 are replaced by O'j -f- c and 0'2 — c, where c is a parameter, the corresponding points of the oo1 nets JNri2 lie on a conic which passes through the corresponding points of N, NI and N» ; this conic is degenerate when the transformations are K, and only in this case (cf. Ex. 14).

Chapter III.

Sequences of Laplace.

29. Homogeneous point coordinates. If the cartesian coordinates of a point P are x1, .... xn, the n -{- 1 quantities y, of which yn J ^ 0, satisfying the conditions

are called homogeneous coordinates of P. If the coordinates y are given, P is determined, but if the cartesian coordinates x are given, the homogeneous coordinates y are defined only to within a factor. In homogeneous coordinates the equation of a hyperplane is of the form (§1)

Now */? = 0 (i == 1, ... ., ii) is the equation of a coordinate hyper- plane. Also //n+1= 0 is taken as the equation of a hyperplane, namely the hyperplane at infinity. This hyperplane likewise is a coordinate hyperplane in homogeneous coordinates. Thus we have n-}-l hyperplanes forming a coordinate (?j-f-l)-hedron. More- over, a point all of whose homogeneous coordinates save one, say y', are zero lies in all of these hyperplanes except the hyper- plane ^=0; it is a vertex of the coordinate (w-f-lj-hedron. Suppose now that we have any n-\-l hyperplanes, say

"y+^r-f - - ..4;a«+1^1= 0 (t ==!,;..., «-f 1),

subject to the single condition that they do not have a point in common, that is, the determinant of the «'s is not equal to zero; thus c^ ^0. If we put

70 HI. Sequences of Laplace

where (> is a factor independent of i, the quantities zl serve also as point coordinates. For when the y's are given for a point the z's are uniquely determined except for a factor. Conversely, when the z's are given, the determination of the corresponding y's requires the solution of a linear system of equations, whose determinant is different from zero. We call the z's general homogeneous paint coordinates, and as such they are distinguished from the particular system of ?/'s, corresponding to the case where n of the coordinate planes are mutually perpendicular, and the other is at infinity. As in the case of tetrahedral coordinates in 3-space, the transformation is not completely determined by the (w-f-l)-hedron, but is defined when it is required that a particular point, not on the faces of the new (n -f- l)-hedron is to have the coordinates (1, 1, . . ., 1) in the new system.

Since the steps used in the preceding are reversible, we have that there exist line'ar transformations P0 by means of which from a general system of homogeneous coordinates z we pass to a system y so that yl/yn^~l. . . yn/yn+1 are cartesian coordinates. Hereafter w.e understand that when homogeneous coordinates are used they are of the general type.

We are prepared to prove the theorem:

The homogeneous coordinates of any point on the line joining two points PI fo) and P2 (z») are of the form

and conversely.

Consider the point with these coordinates and apply a trans- formation P0 (referred to above). The resulting expressions will be of the form hyi-\- py*. Hence the cartesian coordinates of the point are

114 9

which shows that the point lies on Pj P2 (cf. I, Ex. 1), Evidently the converse also is true.

In like manner it can be shown (cf. I. Ex.2) that: Tlie homogeneous coordinates of any point on the plain- through three points P± (zv). P» (z%) Ps (z3) are of the form

29. Homogeneous point coordinates 71

and conversely.

From § 1 the locus of a point whose homogeneous coordinates z are functions of a parameter u is a curve. We wish to prove:

TJie Jwmogeneous coordinates of any point on the tangent to a curve z (u) are of the form

r du and conversely.

If the coordinates of a point are in this form, and we apply a transformation P0 we get for the new coordinates of the point

ty H~ /* "» • Hence the cartesian coordinates of the point are du

, ,; , dy* dx'

'• If ~\~ frb ~^i

du • i ( du

~~d~tT+r ~~ X ~"~ ~~T~i <

A^"1 + p

dx As these are of the form x -}- A — — , the point lies on the tangent

Cv IL

(§1). Conversely, the cartesian coordinates of any point on the tangent being of this form are readily transformable into the form of the theorem.

Consider a point on the tangent to a curve with its coordinates

in the form lz -}- /*~T~- Two functions Q and t are defined by

dt

In terms of Q and t the coordinates of the point are of the form

Q — - (tz). Hence we have: du

The homogeneous point coordinates z of a, curve can be chosen so that the coordinates of a given point on the tangent other than

dz

the point of contact are of the form-j—.

The osculating plane of a curve z(it) at a representative point is by definition the locus of points whose coordinates are of the form

72 m. Sequences of Laplace

fly (7 ?*

/. ,?-}-# — \- v -, "„. If we apply a transformation P0 to the du du*

coordinates and proceed as above, we can show that the cartesian coordinates of any point of the osculating plane are of the form

du 'In'

We observe that this is the result previously found for 3-spacc \§1\. 30. Laplace transformations. We have seen in § •> that a necessary and sufficient condition that a system of parametric curves be a net is that any point on the tangent to a curve v = const, of a system moves in the tangent plane to the surface as v varies. Given a net JVwith n -f 1 homogeneous coordinate x. The coordinates of any point P on the tangent to a curve v = const.

are of the form kx -\- w — . As v varies, the point moves in the

9tt

tangent plane, provided— - Ux -\-.f*-f — j is expressible linearly in

0 I/ \ O c* /

dx dx

terms of x. -— , and-—. Hence a net is characterised analyti- ' du' dv

cally by the condition that its homogeneous coordinates are solutions of the same equation of the form

80 . . e

dudv dv du du dv

We call this the point equation of the net.

If, in particular, the point P moves tangentially to the tangent to v = const., that is if P describes the minus first Laplace trans- form of N, we must have

, Blogfr * -r !• — r— - U.

8w

Since similar results follow for the first Laplace transform, we have: The coordinates of the minus first and first Laplace transform* of a net with the point equation (1) can be taken in the form

dx d log b _ dx

~d^~ ~d^TJ'' Xl~~~Jr~~ dv

31. Sequences of Laplace 73

111 consequence of the fourth theorem of § 29 the equations of a Laplace transformation can be put in the simple form indicated in the following1 theorem:

The homogeneous coordinates z' and ij of the focal points of

tt cftD/nicitct' ran be chosen so that

(3) = "•* =

From (2) we have by differentiation

(4) -./•_, 4- A>?

9r or ou vu

where H and K are the invariants of (1) namely _ 9 2 log a , 9 log a 8 log b .

H- — ~~p ''.

OUOV 0V OU

(5)

„ _ 8*logO , 9 log a 9 log b .

"T^T" r'r"~W

If K -- 0, we get on integration x~\ == Ua, where U is an arbitrary function of u alone. As the arbitrary function U varies with the integral but a remains the same, the point J/_i describes a curve, and not a net; we say that A7-! is degenerate. Substituting in the first of (2) and integrating the resulting equation, we find

'Ua. \

TH

V being an arbitrary function of v alone.

In like manner, if H = 0. the integral of (1) can be given the form

(7)

f-

Hence we have the theorem:

When either invariant of an equation of Laplace is equal to zero, tJte equation can be integrated by quadratures.

31. Sequences of Laplace. When the invariants H and K of (1) are different from zero, by the iteration of the first Laplace

74 HI- Sequences of Laplace

transformation upon a net N with (1) for its point equation, and also of the minus first transformation, we get a sequence of nets

N-j, N-(J-I), N-i, N, NI, , Nit

such that any net of the sequence is a first Laplace transform of its predecessor (sense being from left to right), and a minus first transform of its successor. We say that these nets form a sequence of Laplace.

We are interested in finding the Laplace equation of each net of the sequence. In order to write these equations and others associated with (1) in abbreviated form, we denote by

(8) [Or, at, li,

the equation

. , dudv dv du du dv

(y) <

8s 8 log Oi 8 log 6, . 8 log a 8 log b

Then from (5) it follows that (10)

Tf 9* e/ 3lQg« 8 log?;

Hi ~ " ~ ~

If in equation (1) we put (11) ,=^,

where / is a function of // iind v, the Laplace equation satisfied by x' is denoted by

(12) [0'; «;.,. &;., ;.].

Then from (10) we have H'=H, K' = K, showing that H and K are invariants of (1) for transformations of the form (11).

31. Sequences of Laplace 75

In order to find the Laplace equation of Ni7 we differentiate the second of (4) with respect to v. Making use of (2), we find that the Laplace equation of Nt is denoted by

(13) «i; a-H, &,

and that

Proceeding in a like manner with the first of (4). we find that the Laplace equation of N-i is denoted by

(15) [0_i; a, bK,~] and that

(16) *-! = **_!= Jr

The coordinates ./;» of N» are given by the equations

(17) a* = -j-~

analogous. to (2), and the point of equation of N* is denoted by (18) »s; aHHlt I,

In general the coordinates of Nr are given by

(19) XT = - — --logiaHH, .... jy,-2) xr-i.

or ov

and the point equation is

(20) k; aHH, .... JSTr-i, &, ^r^^^jf

L Of JO. Jl

76 III. Sequences of Laplace

The equations analogous to (4) and (17) arc

s 8log& ^ ; Hf_iXf_

9 n 3 u

(21)

The coordinates of N-r are given by

(22) X-r

and the point equation is (28)

Also we have

dx-r 8

du, 8n

. . . .K-r+l)

32. Periodic sequences of Laplace. Ordinarily a sequence of Laplace is unlimited in both directions. If Hr or K-r is zero, the sequence terminates in the positive or negative sense (cf. Ill, Ex. 5). In the present section we are concerned with the case when the sequence is periodic, that is when a certain net Sp coincides with iV. In this case we must have

(25) yp = mx,

where m is at most a function of u and v which is the same for all n coordinates xl. Since n is at least equal to 3, the coefficients of (12) and (20) for r=p must be equal. Hence we must have

(27) ~l°e-*-* ^

32. Periodic sequences of Laplace 77

Differentiating the first of (26) with respect to u and making use of the second, we find that

(28) HH1....Hp-i^= UV,

where U and V are functions of u and v respectively. If we change the independent variables in accordance with the equations

we find that the invariants H',H(, — H'pof. the transformed equations of the nets N, NI, ---- Np are given by

where </ and <// denote the derivates of <JP and fy. Hence y and ty can be chosen so that (28) becomes

(29) HHl....Hp-i==l.

Then from (26) it follows that m in (25) is constant20).

Suppose now that (27) and (29) are satisfied for equation (1). Applying (10) to (20) for r =p — 1. we get, in consequence of (27),

-.!-=!*— - log

(30) <

— 77

dudv

TT- d~ . b

= H — - -log—

a a a j. ° //

In like manner, making use of (27), (29) and (30), we obtain

T_ T_ u , it- JLJ-- . . . , j-im-

"»-!= H-^^log- ~1^-

^ » 70 T7- r- 9 7 TS"

9- , b"Hn—i ^ a- . uA = H- -log— - = K- ~log-

ditdv 6 a~ dudv a

2U) Cf. Tzitzeica, Comptes Rendus, vol. 157 (1913), p. 908; also Hammond. Annals of Mathematics, ser. 2, vol. 22 (1921), p. 245.

78 HI. Sequences of Laplace

Hence in general we have

(31) Hp-i -=

If we differentiate (25) with respect to u, and make use of (2) and (21), we can reduce the resulting equation to

(32) Hp-iXp—i = mx—i

which gives the analytical form of the condition that the nets Np-i and N—i coincide. Again differentiating (32) with respect to w, we get, in consequence of (21), (24), (30) and (32),

Hp—iHjj—2 Xp—2 = nix—-). And in general because of (31) we have

(33) Hp-iHp—2. . . . Hp—i Xp-i = mx-i, which in consequence of (29) is equivalent to

(34) xp—i

Finally we have

(35) x = mx—p,

showing that ^Y and N—p coincide.

Conversely if we differentiate (35) with respect to v we get (34) with i = p — 1. In general, if we differentiate (34) with respect to v, we get (34) with i replaced by i — 1. Hence according as we differentiate (33) or (34) with respect to u or v we increase or diminish / by 1. In order to point out the significance of this observation, we note that the expressions (19) and (22) for xr and x— r are expressible in the forms

drx

32. Periodic sequences of Laplace 79

where A's and B's are determinate functions of the derivates of a, b and c. obtained by repeated use of (19) and (22). Hence equation (33) involves the derivatives of x with respect to v of orders 1, . . . . , p — i, and of x with respect to u of orders 1, .... i. If this equation is differentiated with respect to u and all derivatives with respect to u and v are eliminated by means of (1) and the equations obtained by differentiating (1), we obtain an equation involving derivatives of x with respect to v of orders 1, — p — i — 1 and with respect to u of orders 1, ....-/+ 1? which necessarily is (33) with i replaced by i — 1.

Suppose now that we consider a periodic net of odd order, and write p = 2n-}-l. If in (34) we put i = n and in (33) i == n-\-l, we get

Xn+l = mHHi ---- HnX-n,

1

X—n — l — •"§»•"*»— 1 • • • •

Qll-rly. gn-l/p

These two equations express — - and - - linearly in terms

dvn~f- dun r

of the 2 n + 1 quantities

We have seen that the consistency of (36) and (1) leads to other equations of the series (33) and (34), by means of which and (1) we can express all the derivatives of order higher than n in terms of (37). Since m does not appear in (27) and (29), for each value of m there exist p independent solutions of (1), (33) and (34) including the condition (25); and not more than p independent solutions.

"When jp is even, and we put p = 2n, we have in place of (36),

#»— mHHi .... Hn—i x—n, x — n—i — — ~ffan—i • • • Hn—\xn—\.

m

Then all the derivatives of order n and higher are expressible linearly in terms of the p quantities

dnx dx dn~lx dx

x.

du»*' ' du' at;"-1' ' dv'

8() III. Sequences of Laplace

Hence we have the theorem:

When an equation of Laplace (1) satisfies the conditions (27) and (29), an infinity of sequences of Laplace of order p exist in space of order p — 1.

33. Harmonic congruences. We wish to establish the theorem :

The homogeneous coordinates x of a net N can be so chosen that homogeneous coordinates of the focal points of any harmonic con-

dx , dx gnience are — — and -r— respectively.

Let N(x) be a net with homogeneous coordinates x satisfying (1). Let F! and F2 be the foci of any harmonic congruence, these points being on the tangents to the curves v = const., u = const., respectively. Evidently x can be chosen so that the coordinates

dx of FI are - - (§ 29). Then the coordinates of F» according as

0 Cv

it is looked upon as on the tangent to the curves u = const, at M or on the tangent to the curves u = const, at Fj. are of the respective

dx dx d^x

forms Ix -f /<- -— . tf — p r - — , These forms must be pro-

dv' du dudr

portional to one another in consequence of x being a solution of the corresponding equation (1). Expressing this requirement, we

9 lofif b d x find that both of these must be proportional to - : + ex.

du dv

Evidently - — — ^ 0, otherwise F3 and M coincide. Looking upon

ou

Ft as the minus first Laplace transform of F2 we must have

^x___Jdlogb dx , \ , 9 Idlogb dx ~du~

As this equation must be satisfied identically, we must have either

c = 0, or V ^— = Vc. where V is a function of v alone and V

du

its derivative with respect to v. In the latter case the coordinates

of F., can be chosen of the form-^ - (Vx). Replacing Vx by x,

dv

dx

which does not change the form --of the coordinates of Fit we

du

have the result stated in the theorem.

34. Levy sequences of the first order 81

Furthermore, when the coordinates of FI and F% are of the

dx dx

form - - and — — , in the point equation of N we have c = 0.

ou ov

If we put 6 =8id0 where 00 is any function of u and v, in equation (1), the function 0X satisfies an equation of the type (1), and for this new equation c = 0 in case 00 is a solution of (1) and only in this case. As a result we have the corollary:

The homogeneous coordinates of the focal points of any con- gruence harmonic to a net N(x) are expressible in the form

(38)

in which case 6 is a solution of the point equation of N.

We have also the converse theorem:

If 6 is any solution of the point equation (1) of a net N, the points whose coordinates are of the form (38) are the focal points of a congruence harmonic to N.

For, as v varies the direction-parameters of the path of the first of these points are of the form

x d . a d lx\ d . b d I x

}()f? - • - I - 1 H

3

dvduei/ dv 36 due du 0 to

which evidently are the parameters of the line joining the two points; similarly when u varies.

We have also the theorem:

When 6 = £aiXl, where the a's are constants, the points of coordinates (38) lie in the hyperplane zlai xl = 0.

34. Levy sequences of the first order. If 6 is any solution of the point equation (1) of a net N, from (2) it follows that the functions

(39) ,_, «

_ )

du du dv dv

are solutions of the point equations of N- 1 and NI respectively. We call them the minus first and first Laplace transforms of d.

82 III- Sequences of Laplace

The points of coordinates (38) are the Levy transforms of N by means of 6 (cf. I, 53). In consequence of (2) and (39), we can take as homogeneous coordinates of these respective points

(40) x-i,i =

-, -, , .

0—1 v

Moreover, the net of coordinates x0,i is the first Laplace trans- form of the net of coordinates #_ 1,1.

By differentiation and reduction by means of (2), (4) and (39), we find that the point equations of these nets are denoted by (cf. § 31)

<«>

From the form of (40) it follows that N-iti and N0,i are Levy transforms of N-i and NI by means of 0-i and ^. More- over, from § 10 we have that the tangents to the curves of para- meter v of N0,i are harmonic to NI, and consequently this harmonic congruence Gt is determined by the solution 0i of the point equation of Nlf Its focal point of the first order generates a Levy trans- form NI,I of NI whose coordinates are given (analogously to (40)) by

(43) xit i = xs -- —• Xi ,

0i

where 02 is the second Laplace transform of 6. In like manner the function 0_i determines a congruence G-\ harmonic to N-\ whose focal nets are -AT- 1,1 and N-2,i, where

(44) X- 2, 1 = X-i- —X-2,

0—2

0_2 being the minus second Laplace transform of 0.

Continuing this process we obtain a sequence of Laplace whose focal nets are Levy transforms of the nets of the Laplace sequence arising from N. We call it the first Levy sequence of N determined by 0. The coordinates of the nets Nr,\ for positive and negative values of r are of the form

35. Levy sequences of higher order. Derived sequences 83

/..->. 0r+l

\_40J #p,l — #*•-)- 1 ~ ^ #7">

where 6r is the rih Laplace transform of 0.

35. Levy sequences of higher order. Derived se- quences. Let 0 and 0' be two solutions of the point equation (1) of N linearly independent of the coordinates of N. By means of 0 and 0' we determine two first Levy sequences Nr,i and N'r,i. These nets are the focal nets of two sequences of congruences Gr and Or harmonic to the nets Nr respectively of the Laplace sequence obtained from N. From § 11 it follows that the point of intersection of corresponding lines of Gr and O'r describe a derived net of Nr. Moreover, from the results of § 11 it follows that these derived nets form first Levy sequences of the sequences of Nr,i and Nr,i and consequently we call them Levy sequences of the second order. We shall obtain the analytical expressions for their coordinates.

We consider first the derived net of N by means of 0 and 0'. The functions 0_i,i and 00,i defined by

(46) 0-1,1=0'— -^- «li, 0o,i = 0I--y0',

where 01 1 and 0i are the minus first and first Laplace transforms of 0', are evidently corresponding solutions of N-i,i and NO,I, which as we have seen are Laplace transforms of one another. Hence the coordinates of the Levy transform of JV_i,i by means of 0-i,i are the form

/An\ ^0,1

(47) #-1,2 = #0,1 — -7 #-1,1.

0—1,1

In consequence of (40) and (46) this is equivalent to

0—1 0—1,1

0_! 6-1 X-\

0 0' x

0! 01 Xi

_i 0_

1,1

>_i 0'

if we make use of the following lemma concerning determinants:

6*

84

HI. Sequences of Laplace

If D = I aim 1 (l,m = 1, . . . . ri) is any determinant of the nth order, and we write

dr+l,s — - = Ar,s,

then

(48)

2,2

2,n

An— 1,2) • • • An— l,n

Hence the coordinates of ^-1,2 can be taken in the form (49) o;_i,2 = | 0-i 0' XL\.

From the symmetry of this expression it follows that N—i,2 is also a Levy transform of N-^-L, which shows that it is the derived net of N for the functions 6 and 0'. Evidently the derived net of NL for 0! and 0{ and of N—i for 0_i and 0Li are given by

(50)

#0,2 = | 0 01

#-2,2= | 0-2 0-1 #(•

Since N— 1,2 and No,2 are the Levy transforms of JVo.i determined by 0o,i they are Laplace transforms of one another; similarly JV_2f2 and N-it2 are the Levy transforms of jV-i,i determined by 0-i,i. Hence the solutions 0 and 6' determine a Laplace sequence of nets Nr,2 which are derived nets of the nets Nr+i, and a first Levy sequence of the nets Nr, i. The coordinates of Nr, 2 are of the form

(t)]\ __ I n nl

for r positive and negative, and where A^ = N.

If 6" is another solution of (1) independent 0, 0' and the coordinates of N, the functions

e

)" nit nil

)-^~e -JH6-1'

solutions of the point equation of .#-1,1, determine a derived net of N-I,I. Analogously to (49) we have that the coordinates of this net are of the form

0-

36. Periodic Levy sequences 85

Applying the lemma to this case, we have that the coordinates of the net may be taken in the form

(52) Z_2)3= |0_2 0-1 0"#l|.

Hence the functions 0, 0' and 6" determine a sequence of Laplace whose coordinates are of the form

(53) Xr,8 = I Or O'r + 1 0"+2 Xr + 3\.

We remark that from the symmetry of (52) it follows that the net JV-2,3 is a derived net also of the Levy transforms N-i,i and N-i, i determined by 6' and 6" respectively.

In general m independent solutions 0, 0', ---- 0o»-« of (1) determine a Levy sequence of order m whose coordinates are of the form

(54) Xr,m = I Or 0r-f 1 "

for positive and negative values of r21).

36. Periodic Levy sequences. Suppose that we have a Laplace sequence of period p. We seek under what condition the Levy sequence of the first order determined by a solution 0 of (1) is also of period^?. It is necessary that xp,i = A #0,1, where A is at most a function of u and v. In consequence of (40) and (45) this condition is equivalent to

/r-i-N

(55)

From (19), (25) and (29) we have

7\ ft 7\

(56) Xp+i == mxi, Op+i = -~- --- — loga-0p.

Consequently from (55) it follows that

, 9 , O

l = m, — !<*

21) Cf. Hammond, 1. c., p. 252.

86 HI. Sequences of Laplace

In like manner the condition 2^-1,1 =- f*x—i,i is equivalent in consequence of (21) and (32) to

3 . dp P = m lo = 0.

Hence we must have

(57) dp = ml 0,

where wx is a constant. We have seen that equation (1) admits solutions of this type. If we have such a solution, the Laplace transforms of 6 satisfy equations (33) and (34) with m replaced by Wi. Hence we have

(58) Xp— $,1 = Xp—t+i -- ^p —Xp—i = mH . . . . Hp-i #— »,i.

Op— i

Therefore:

If N is a net of period p in (p — \}-space such that xp = mx and 0 is a solution of the point equation of N such that 6P = m^ 6, where rat is a constant, the Levy sequence determined by 6 is of period p; when m^— m, 0 is necessarily a linear function of the x's and the nets of the Levy sequence lie in (p — 2)-space.

The latter part of the theorem is a consequence of the last theorem of § 33.

If we take two solutions 6 and 6' of (1) satisfying the con- ditions Op=mld, dp—m20', the first Levy sequences determined by 6 and 6' are periodic. Also in consequence of (56) and (33) we have from (51)

Xp—i,2 =

Consequently the second Levy sequence determined by 6 and 0' is of period p.

Similar results hold for the Levy sequences of higher order. Consider in particular, the case of the Levy sequence of order p determined by the p independent solutions 0, 0', . . . . 9(p~1) of (1) for which 0£> = m10(»)(« = 0, ---- p — 1). From (54) we have

x0,p = \e e[ ev....e(/-Vxp\==(m—mi)\dei....e</-L»\x. Similar results hold for xr,p. Hence:

37. Transformations F in homogeneous coordinates 87

If N is a net of period p in (p — l)-space, such that xp = mx and 6, 6', 0", . . . . 0^—1) are p independent solutions of the point equation of N such that 0® = mi6(f)(i = o, . . . . p — 1), the pth Levy sequence coincides with the given sequence**).

37. Transformations F in homogeneous coordinates. When two nets N and N{ are in the relation F, the tangents to the curves v = const, at corresponding points M and M{ meet in the focal points of a congruence harmonic to both N and NI (§ 17). In accordance with the first theorem of § 33 the coordinates of N and .ZVi can be chosen so that we have

dXi dx dxi dx

(59) •S=* ==V

where r0 and cr0 are functions of u and v. Hence the equations of any transformation F can be given this form. As previously remarked, in this case c = 0 in (1) and likewise Ci = 0 in the point equation of NI.

When c ^ 0 in (1), the equation can be reduced to this special form by replacing x by xQ, where 6 is any solution of (1). Hence when the point equation of the net N has the general form (1), the equations of a transformation F are

/ /-*f\' 30\ 0 I OC \ 0 OC\ 0 i OC

'du ' ~ T du \ 0/' dv a dv \ 0

In order that the conditions of integrability of (60) be satisfied for any solutions x and 9 of (1), it is necessary and sufficient that T and a be solutions of

QT 9 . 0 80- 80

or in other form

8,ra a d . a 8 . ab

(62) ^ 10«

2J) Of. Hammond, 1. c., p. 256.

88 HI. Sequences of Laplace

If we put

(63) T — o=<fB,

we get equations (IE, 16), and find that y is a solution of the adjoint of (1), which is denoted by (cf. § 31)

Suppose we have any pair of solutions of (61), and the net NI whose coordinates x± are given by the corresponding equations (60). The points FI and Fz whose coordinates y and z are of the forms

/ s*r \ ™ ***

(65) y = Xi—T—, z = xl — ff—,

lie on the line joining corresponding points of the nets JVand NI. Moreover, as u, or v, varies the point F1} or F%, moves tangentially to this line. Hence Ft and F% are the focal points of the con- gruences of these lines, and N and NL are in relation F.

With the aid of (61) we show that the point equation of JVi is

(66)

dudv

When we put 0 = 1 in (61), we get the conditions of inte- grability of (59), namely

As a consequence of these results we have:

Wh&never the homogeneous coordinates of two nets are in either of the relations (59) or (60), the nets are in relation F.

From the manner in which equations (59) were obtained it is evident that the transformations F obtained by taking all possible solutions of (67) possess the property that all the corresponding tangent planes of the nets pass through the line of the congruence

dx dx

whose focal points have the coordinates - - and — •. Hence in

du dv

38. Transformations F with the same conjugate congruence 89

order to obtain general transformations F, especially in dealing with two such transformations, it is desirable to take the equations in the form (60). However, when a solution of (1) is known, we can write (60) in the form (59), and then the analysis of the transformations F in homogeneous coordinates is the same as that of parallel nets in cartesian coordinates.

38. Transformations F with the same conjugate con- gruence. Triads of nets. Suppose we have a transformation in the form (59), so that the new net N' (x') is given by

(fiQ\ dx> jg. 9a?> - dx

\ / o _, 0 o ) o 0 Q A. *

Now the point equation of N is necessarily of the form (1) with e = 0. If 6 is any solution of this equation and 8' is given by the quadratures

80' 80 30' 80

^r: = r°^ ~w ~-a°^>

then the functions xl} defined by i

a

(70) xl = x—-jp-x',

are the homogeneous coordinates of a net NI, since

du (71) { ,

dv

From the form of (70) it follows that corresponding points M, M', M^ on the three nets N, N', N: are collinear. Hence not only is NI an F transform of N, but also of Nr. In the latter respect it differs from the case of § 15 (cf.ni, Ex. 24).

As an application of the foregoing results we prove the theorem23):

23) This theorem for 3-space is due to Ribaucour, Comptes Eendus, vol. 74 (1872), p. 1491.

90 HI. Sequences of Laplace

If a net N lies on a hyperquadric

each congruence conjugate to N meets the hyperquadric again in a net, which consequently is an F transform of N.

We assume that the given conjugate congruence is conjugate to a net N' whose coordinates are given by (68). From the above equation we have by differentiation

dot® .

since the coordinates x satisfy (!•) with c = 0. In consequence of this result it follows readily that 0 given by

is a solution of (1), and that 0' where

0' =

satisfies (69). If NI denotes the corresponding transform of N with the equations (70), it is found that NI lies on the quadric. If TI, o"i and ra, <r2 are two sets of solutions of (61) for the same 6, by means of equations of the form (60), we get by quadratures two nets, N! and Ni} transforms of N. From their equations we find

du ti du' dv ' GI dv '

Consequently NI and N2 are in relation F, and form with N a triad (§ 20).

39. Theorem of permutability. In view of the remarks of the preceding section it follows that the results of §§ 20, 21 can be translated at once into analogous forms for transformations F in homogeneous coordinates.

Suppose then that we have two solutions 01} 02 of (1) and two solutions <pi} g>2 of the adjoint of (1), so that by quadratures

Exercises 91

of the form (II, 16) two pairs of functions rlt a^ r2, <rg are to be found. Transforms N^x^ and N2(x2) are given by

/y» /_. _.\ V *A/», vy I IAS \ v tA/» v i t/x

(72)

Solutions of the point equations of NI and Ns are given by

The coordinates of the transform ^12 (#12) are given by (it being understood that JV12 and N21 are the same net)

9 v ^ a v \e j = 1, 2,

Comparing these results with (60), (71) and with (II, 21, 37) we have from (II, 52, 49)

(75)

ft 1 ft ft • T

02

and

(76) 0i 0X2^12 =l= 02 021^1 + 01 012^2 — 012 021 X 24).

Exercises.

1. The homogeneous point coordinates of any point of the tangent plane to a surface are expressible in the form

dx . dx

2. Show that equations (3) hold for the special homogeneous coordinates such that yllyn+i and z1'/2"*1 are cartesian coordinates of the foci; also that if the direction-parameters of the congruence are taken in the form

24) In making this comparison it must be noted that the quantities a^flj/fy, QijByd^ £Ci20'i'»'/0i2 of Chapter II must be replaced by — a?f, — 0,7i — on respec- tively in order to conform to the usages of the present chapter.

92 III. Sequences of Laplace

yn+i we have

cx = 0

dudv dv du du dv

3. The invariants of the Laplace transforms NI and N—i of a net N are given by

Darboux, Lemons, vol. 2, p. 28.

4. The invariants of the Laplace transforms Nr and N—r of a net N are given by

92

Sr = Hr—1 -\~ H — -BT -- x - ^ — log jET . . . . Hr—i, Kr ==L Hr—1 ',

ouov

H-r = JT-r+1, #-r = -K-r+1 + JBT- B"— -^— log ^ . . . . 1T_,+1.

Darboux, Lemons, vol. 2, p. 30.

5. If Hi = 0 for a Laplace sequence, then

where /?, >!,.... w4{ are determinate functions; U and V are functions of w and v respectively, and U® is the ith derivative of U with respect to w.

Darboux, Legons, vol. 2, p. 33.

6. When an equation of Laplace admits a solution of the form

x = AU + AiU'+....+At U®,

where the As are functions of u and v and U® is the ith derivative with respect to u of a function U of u, then Hi = 0. Darboux, Legons, vol. 2, p. 35.

7. If Hi = 0 and K—j = 0 for a sequence of Laplace, the point equation for N admits solutions of the form

x = AU+.... + AtU(t> + BV+ ....+ BjVW,

where the A's and J5's are determinate functions; U and V are arbitrary functions of u and v respectively, and Z7" and V ^ denote ito derivatives with respect to u and v.

Darboux, Lemons, vol. 2, p. 38.

8. A necessary and sufficient condition that the point coordinates of a net can be chosen so that the coordinates of the minus first and first Laplace trans-

OX OX

forms are -=— and -~— respectively is that the point equation of the net have

equal invariants.

9. If the parametric curves on a surface S form a net N, the tangents to the curves defined by Adu2-\-B dv*= 0 meet the lines joining the points M_^ and MI of the Laplace transforms of N in points harmonic to M_± and MI.

Exercises 93

10. Show that if a point equation (1) satisfies the conditions (27) and (29), it satisfies also the conditions

11. Show that if equation (1) admits solutions of period^), so also does its adjoint equation [cf. § 37]. Hammond, Annals, vol. 22 (1921), p. 260.

12. The point equation of Nt x defined by (43) is denoted by

This may be obtained from (13) by multiplying the second term by 0»/0i, retaining the third term, and multiplying together the third and fourth terms of (13) and dividing by 61. Show that the same method applied to (1), denoted by [6; a, 6, 1] gives (42).

13. Apply the method of the preceding exercise to (20) and obtain for the point equation of Nr>l the following

r er+1 _ 6^

[0r>1;a fii.....!^ -£-,&, arH-iHr-* Hr_t

Verify this result directly for r = 2. Hammond, 1. c. p. 249.

14. The point equation of N_2 ^ is denoted by

This may be obtained from (15), by multiplying the second term of the latter by 8_1/0_2, retaining the third term and taking for the fourth term the product of the second and fourth of (15) divided by 6_2. Show that (41) is obtained from (1) by the same method.

1 5. Apply the method of the preceding exercise to (23) and obtain for the point equation of N_(r,l) x

0 — - _l...._r+v

-r—l OK

_!

---- _

Hammond, 1. c. 16. Show that for the derived net N_am for m>s the coordinates are

expressible in the form

0 ds~le' (s)

»' *-i''

Tzitzeica, Comptes Rendus, vol. 156 (1913), p. 375.

17. If two nets N and Ni are in relation F so also are the nets resulting from a protective transformation of N and Ni.

94 IH. Sequences of Laplace

18. Show that equation (66) is denoted by [a aT bff ]1

e ' e ' 0-r

19. From (65) we have

3y 9y , . dz . . dz

-ft=-**-v ^= -<*>'" -ft = v>-i* 8^ = **'

where (x)i and (af)_1 denote coordinates of the first and minus first Laplace trans- forms of N, and <p l and y>_l are first and minus first Laplace transforms of <p with respect to (64). Show that the coordinates of the first Laplace transform of F\ and the minus first of Ft are of the forms

(aOiP + y -g^-logap-i, (*)_!? — *-g^- log 6 ?!•

20. If N and ^ are nets in relation F, the lines joining corresponding points of their first Laplace transforms (minus first Laplace transforms) meet the corresponding lines of the conjugate congruence of the transformation in the focal points of the first (second) order (cf. Ex. 19).

21 . When two nets N and N\ are in relation F, so also are their Laplace transforms of the same orders; the equations of these transformations are

where

TV = TV-i-f — , <Tr = Tr-l,

for r positive. Hammond, 1. c., p. 260.

22. If the coordinates of a net N of period p satisfy (25), and 6 is a solution of its point equation (1) such that 6P — mrf and f> is & solution of the adjoint (64) of (1) such that <p = n<f>—p, where mi and n are constants, the F transform of N by means of 8 and <p is a periodic net of the same period as N.

Hammond, 1. c., p. 261.

23. If N(x) and NI(XT) are nets in relation F on the hyperquadric

5of*ac®x^ = o, then ^anc(x(i)x(^ + x(k)x^) = k, where fc is a constant. When i,* <,*

k — 0, the congruence of the transformation consists of generators of the hyper- quadric [cf. § 38].

24. Show that if corresponding points of three nets in relation F are collinear the relation between their coordinates may be put in the form (70).

25. Show that for a net with the point equation (1) with a = 6 =V~/p the equations of a transformation K (§ 25) are

8x, 62 9 / x \ 9*, 02 8 / x

du p du \er dv p dv \e

26. Let N be a net in 3-space and NI an F transform of N given by (60). Let corresponding points M and MI of these nets and the two focal points FI

Exercises 95

and FI of the harmonic congruence of the transformation be taken as the vertices of a tetrahedron of reference of homogeneous coordinates x, y, z, w such

M (0,1, 0,0), 3fx (0, 0, 0, 1), .F, (1, 0. 0, 0), Ft (0,0, 1,0).

Show that the pencil of conies tangent to the lines M Fl and MFt at the points (M)-i and (M)i of the Laplace transforms (2V)- 1 and (2V)j of J\T are given by

where ^ is a parameter; also that according as ^ is -BT, or H, the conic osculates the curve u = const, of (2V)_i at (M)-i or v = const, of (2V)i at (M)i.

Annals, vol. 18 (1916), p. 11.

27. A necessary and sufficient condition that a conic of the pencil of Ex. 26 osculate the curve u = const, at (M )— i and v =• const, at (M )t is that N have equal point invariants. Darboux, Lecons, vol. 4, p. 38.

28. Show that the pencil of conies tangent to the lines M\FV and MiFz at the points of the Laplace transforms of NI are defined, in the coordinates of Ex. 26, by

,/x9. b z 9 a w \2

2 X Z ^^ 'u I — "5 — log ~7\ H 5 — 'lofif "Tr *^ 5" I '

• <T uu 0 T 9v w 0 '

that this pencil and the pencil of Ex. 26 determine involutions on the line FI Ft; and that a necessary and sufficient condition that the two involutions be identical is that the transformation F be K. Annals, 1. c., p. 12.

29. If N and NI are two nets in relation K, any two conies of the two pencils of Exs. 26 and 28, meeting on the line FI Ft determine a pencil of quadrics which cut the line M MI in an involution whose double points are the focal points of this line for the congruence of these lines, and the two cones of the pencil are the quadrics tangent to M MI at these focal points.

Annals, 1. c. p. 15.

30. If N and NI are nets in relation K, the doubly osculating conies of the pencils of Exs. 26 and 28 meet on the line FI F%, and the vertices of the two cones in the pencil of quadrics determined by these conies are the corre- sponding focal points of the conjugate congruence of the transformation.

Tzitzeica, Comptes Rendus, vol. 147 (1908), p. 1036; also AnnaJs, 1. c., p. 16.

Chapter IV.

Surfaces and congruences in 3 -space.

40. Nets in 2-space. It is evident that any three functions of two parameters, u and v, satisfy an equation of the form

m 920 _ 8 log a 90 . 9 log 6 90

dudv dv du du dv

Consequently any two families of curves in 2-space form a net. It is likewise true that we can find two equations of the form

(2)

820 __ 80 80

9 u9 du dv

i!l_ ,li + 6l*+r0

9*2 ~~a29u+S<; +C2^'

which the three given functions satisfy.

Conversely, we seek the conditions which must be satisfied by the coefficients in (1) and (2) in order that they admit three linearly independent solutions. To this end the following conditions of integrability must be satisfied:

820\ _ _8_ / 820 \ _8_ / 820 \

dv \9w2/ du\dudvj' dv\dudvj du\dv*

Reducing the resulting expressions by means of (1) and (2), we get two equations of the form

where AI, Si . . . . (72 are determinate functions, which must vanish, if equations (2) are to hold for three linearly independent solutions. Putting them equal to zero, we get the following conditions:

(4)

41. Tangential coordinates of a surface in 3-space 97

, deb __ 9 log a 3 log 6 92loga

-" ~""~~ ~

9loga 9log& 9c

= * Ci + — r^- c + —

9logq 9log6 „ , 9*log6 -~~ + < ~

9loga 9«2 __ 9 log 6 /9loga\2. 92loga

~^r ^+c*+-^r ^r~a2+l"^rj+~8v~

9c2 9 log 6 9 log a 9c

-— -- -- - -—

. du du dv dv

When these conditions are satisfied, the system (1) and (2) is completely integrable and there are at most three linearly in- dependent solutions. For, the derivatives of the second and higher

3 f) 9 0

orders of 0 are linearly expressible in terms of - — , — — , and 6.

Hence all the integrals are expressible as linear functions, with constant coefficients, of three solutions. Therefore we have the theorem:

When the homogeneous coordinates of a net satisfy equations of the form (2), the net is in 2-space; and all nets whose homogeneous coordinates satisfy the same equations (1) and (2) are protective transforms of one another.

41. Tangential coordinates of a surface in 3-space. Let Xj y, z, w be the homogeneous point coordinates of a surface 8 in 3-space referred to any system of parametric curves u = const., v = const. Since the tangents to the parameteric curves at a point of the surface lie in the tangent plane to the surface at that point, the equation of the tangent plane must be satisfied not only by x, y, z, and w, but also by the coordinates of any point on these tangents, that is by expressions of the form

dx , dx

! x

98

IV. Surfaces and congruences in 3-space

for any values of &i, 1*1, ^2 and /*2. Hence there exist four functions, X, Y, Z, W, of u and v, the tangential coordinates of the surface, |cf. § 67] satisfying identically the three conditions

(5)

where £ indicates the sum of four terms obtained by replacing x and X by y and F; z and Z\ w and W respectively.

In consequence of the last two we have, on differentiating the first, the two equations

(6)

dX

dX

42. Asymptotic lines. An asymptotic line on a surface is characterized by the property that its osculating plane at a point is tangent to the surface at that point [§ 551. Hence along an

dv asymptotic line -=— must equal such a function of u and v that

the equation of the tangent plane is satisfied by kx + pdx + vcPx, for all values of A, /*, and v [cf. § 30]. Hence we must have

2Xd*x = 0. Eliminating X, Y, Z, and W from this equation and (5), we have

(7) where (8) L =	L dx dx d*x	du*-\-2 ,M =	Mdudv-\- Ndv dx dx d*x	2-o, ,N =	dx dx d*x
du dv du* dy "dy tfy	X du dv dudv	X du dv dv*
y du dv dus dz dz d*Z	y	y
" du dv du2 dw dwd*w	B
'V du dv du2	W	w

42. Asymptotic lines 99

This is the equation of the asymptotic lines on S. We have immediately the theorem [cf. § 77]:

A necessary and sufficient condition that four functions x, y, z, w be the homogeneous point coordinates of a surface referred to its asymptotic lines is that x, y, z, w be four linearly independent solutions of two partial differential equations of the form

(9)

dd . . dd

9ws

820 _ 80

c, 2 ' o

dv du dv

We seek now the conditions upon the coefficients of equations (9) so that two equations (9) shall have four linearly independent solutions. It is necessary that the following condition be satisfied:

820\ 82 /320

dv* \8tt2/ du* \dv

When the above expressions are substituted, the resulting equation is reducible by means of (9) to an equation of the form

A-£k+Bi£+c%+Dt=0'

where A, B, C and D are determinate functions of the coefficients of (9) and their derivatives. These functions must be equal to zero, otherwise we can have at most three linearly independent solutions of (9) [cf. § 40]. Putting them equal to zero, we obtain the four equations of condition

dai 862

dv du

dv* du dv dv dv du

9 &2 i 9,77x1 8 &2 ,8#2 9 / jx I o 8 Cj _

2 o_-\12/llo-- J-o_. o - . V *• 1- / I o__

du* dv ^ l du du du

d*ci , _ 3 a

dv* ' du *du L du "dv * dv

7*

fc^L . fc dci_ A

°1 o T °2~^ V.

100 IV. Surfaces and congruences in 3-space

When these conditions are satisfied, the system is completely integrable, and as all the higher derivatives are expressible linearly

dx dx d^x

in terms of x, - — , - - and - — , there are four linearly inde- ' du dv dudv

pendent solutions, and only four. Hence we have the theorem: A necessary and sufficient condition that a system (9) admit

four linearly independent solutions is that the coefficients satisfy (10).

All surfaces whose four point coordinates satisfy the same system (9)

are protective transforms of one another.

When the surface S is subjected to a polar transformation with

respect to the quadric

(•\1\ r2_i_?.2_|_«2 I M;2__ A

v11/ ^ i y \ 6 \ w u?

the point and tangential coordinates of S are tangential and point coordinates respectively of the transform S'. Since asymptotic lines are transformed into asymptotic lines on 8' [cf. § 84], we have the theorem:

Any four linearly independent solutions of the system (9) are tangential coordinates of a surface referred to its asymptotic lines; all surfaces whose tangential coordinates satisfy the same system of equations (9) are protective transforms of one another.

43. Nets in 3-space. Point coordinates. Consider a sur- face 8 referred to any system of curves, u = const., v = const., and upon it a net, or conjugate system. Any point P on a tangent at M to a curve of a family of the net has homogeneous point coordinates of the form

. Idx 7 . dx X = **X + t*(d^du + -Jv'1

A necessary and sufficient condition (§ 2) that two families of

J J£

curves determined by -r— and -r- form a net is that, as M moves du ou

along the curve of the second family through it, P moves in the tangent plane to the surface at M. Hence the point whose

coordinates are of the form x-\-8x = x-\-- - 8u-\-- - 8v must

ou ov

lie in the tangent plane. This gives the equation of condition

= 0.

43. Nets in 3-space. Point coordinates

101

Combining this equation with the identities (5), we get

(12)

Ldudu-}- M(du6u-\-dudv}-\- Ndvdv = 0,

where L, M and N are given by (8). This is in keeping with (7) which defines the asymptotic or self-conjugate directions, and could have been inferred directly from (7), since these differential equations in the parameters are independent of the point coordinates and consequently should be equivalent to the similar equations found when cartesian rectangular coordinates are used [cf. §§ 54, 55]. As an immediate consequence of these observations and the results of [§ 56J we have:

A necessary and sufficient condition that the curves defined by Rdus-\-2 8dudv-\- Tdv2 = 0 form a conjugate system is

(13)

RN+TL — 2SM=Q.

From this result, and from (12) also, it follows that a necessary and sufficient condition that the parametric curves form a net is that M = 0. But from (8) this means merely that x, y, z and w are linearly independent solutions of an equation of the form (1). In this case the equation of the asymptotic lines is of the form

(14)

where r = NIL is a function of u and v. Comparing this equation with (7), we have in consequence of (8),

dx dx

du dv

9 it'

= 0.

Hence we have the theorem:

The homogeneous point coordinates of a net in 3-space are simultaneous solutions of two equations of the form (1) and

(15)

3*0 _ 320 ,30 ,30

c'e.

102

IV. Surfaces and congruences in 3-space

Conversely, we shall show that two equations of the form (1) and (15) admit at most four linearly independent solutions. In the first place in order that they admit a common solution it is necessary that they satisfy the condition of integrability

820 \ _ _8_ /8'0

dv dudv

du

When the expression from (1) and (15) are substituted, the resulting equation is reducible to

(16) where

830

9*8

(17) <

1 I 82a

, 8

-- K\ 2

l"A

dudv

where K is one of the invariants of (1) [cf. I, 44]. From (1) we have also by differentiation

(18) where

du*dv

80 du

n 80 ^ dv'

1 826

8

—

Also from (15) we obtain -

f terms of 0,

98

dv

, and

8 du

820 8w2'

83

and - r expressed linearly in

dv

There remains the condition

0 \ _ 8 / 830\ 9v/ " " dv \ dus}'

43. Nets in 3-space. Point coordinates 103

By means of (16) and (18) this condition is reducible to the form

where P, Q, R and 8 are determinate functions. If the coefficients in (1) and (15) are not such that

(21) p=Q = R = S = 0,

we have a system to be satisfied similar to (1) and (2), which, as we saw in § 40, admits at most three linearly independent solutions. Hence we must have (21) satisfied, in which case the third and

O f\

higher derivatives in 6 are linearly expressible in terms of d, — — ,

0 It

ri ft 3^ $

-=-. Since all further conditions of integrability are satis-

ov on

fled, we see that there are at most four linearly independent solutions of a completely integrable system of equations of the type (1) and (15). Accordingly we have the theorem:

The homogeneous point coordinates of a net in 3-space satisfy a system of equations of the form (1) and (15); conversely } a net whose coordinates satisfy such a system lies in 3-space. Any four linearly independent solutions of the same system of equations (1) and (15) are the homogeneous point coordinates of a net protective with the given net.

When the expressions P, Q, R and S in (20) are calculated, it is found that equations (21) reduce to

(22)

dv du

:

dv du du

104 IV- Surfaces and congruences in 3-space

When the point coordinates of a net N are cartesian, we have from (14) and [(40) § 55] that r = D"ID. Consequently by the elimi- nation of X from the first and third of the Gauss equations [(7) § 64] we find that the cartesian coordinates of N satisfy an equation of the form (15) with

22) Jill , (221 Jill

-r --- c =

From (14) and [§ 82] it follows that a necessary and sufficient condition that N be isothermal-conjugate is that r = U/V, where U and V are functions of u and v alone respectively. As a con- sequence of the preceding theorem, we have:

An isothermal-conjugate net is transformed into an isothermal- conjugate net by a, protective transformation.

44. Ray congruence and ray curves. Consider a net N in w-space, and the system of lines joining corresponding points of the first and minus first Laplace transforms of N. If this system of lines is to form a congruence, there must be two points of a line generating curves to which the line is tangent. The coordinates of any point are of the form Ax_i + /*#!. Hence it must be possible for u and v to vary in such a way that d(hx—i-\-ftXi) is a linear function of x-\ and x±. From (HI, 2, 4) we have

)x—i d2x 1 32& 9log6 dx—i 8loga

' •—- , ^__ „ sy* *-* /y» ^ . ^ *-*.._ /y» [_,

du ' ~ du* b du2 du dv dv

dXi d log b dxi d*x I 92a 9 log a

Z_ ^ t-J /y r-\— I— I SY* _ -~—_ — ^— . /y* S-? /y»

du du dv dv* a dv2 dv

where H and K are the invariants of (1). Expressing the above condition, we get equations of the following form to be satisfied by the coordinates x:

I -./8* x I -

v\du du xr (dv dv x~

where v and ff are to be determined.

44. Eay congruence and ray curves 105

. d*x d2x dx dx Equation (24) is linear m ——5-, — -y, — — , — — and x. If n >• 3,

O i/- (/ 1? 0 % C7 ^

this equation must be satisfied by five or more linearly independent functions x, which is possible only when the coefficients of the

dzx quantities— —g-, . . . . , x are zero. This gives A du — pdv — Q, which

0 If/

from (24) is seen to be impossible.

When n = 3, by means of (15), (24) is reducible to an ex- pression linear in— -=-, -—,-—, x. The coefficients of the latter du*' du' dv'

expression must then be zero, which gives the four equations of condition

(25)

v — 0, pafdv-\- v = 0, pb' dv -\- 0 = Q,

b du*

[ v |

du dv

Eliminating A, p, v and tf from these equations, we obtain

(26) where

^ T? r d*b 1 d*a /, ,9log& , ,./ 8 log a

(27) It = — — -3 ^~5- + c + a — ;-^- — HO .

& 3w2 a 9y2 9?< 9y

Hence the system of lines forms a congruence. Folio wing Wilczynski25) we call it the ray congruence of the net N, and the curves on the surface of the net defined by (26) the ray curves. Since any one parameter family of lines in a plane has an envelope, the developables of such a congruence for nets in 2-space have no significance. Hence we have the theorem:

The lines joining corresponding points of the first and minus first Laplace transforms of a net form a congruence only when the net lies in 3-space.

From (13), (14) and (26) we have the theorem of Wilczynski26) :

25) Transactions of the American Mathematical Society, vol. 16 (1915), p. 318.

26) L. c. p. 319.

106 IV. Surfaces and congruences in 3-space

A necessary and sufficient condition that the ray curves for a net N form a net is that N have equal point invariants.

In consequence of the fourth theorem of § 10 we have:

The ray congruence of a net with equal point invariants is harmonic to the corresponding net of ray curves.

45. Nets R. We determine the condition that the tangents to the curves v = const, of a net N form a W congruence, that is the asymptotic lines on the surface of N and its minus first Laplace transform correspond [§ 172].

From (23) we have by differentiation, making use of (16),

d*x-i I t aloffZA dx-i , x dx , „ dx

(28)

du

<-l,

d*x~i 9loga dx-i.^dx.dK , 98loga ~ + -A-~TT + "^"^H 0...8

dv dv dv

dx If we replace x and — — by the linear expressions for them obtained

dx from (HI, 2, 4) and then eliminate — — from these two equations, we

dv

obtain for the net N-i an equation of the form (15), where r-i has the value K/C^.

From this result and (14) it follows that a necessary and sufficient condition that the asymptotic lines correspond on the two surfaces is that Cir= K, which by (17) is equivalent to

du

In like manner a necessary and sufficient condition that the tangents to the curves u = const, of N form a W congruence is

45. Nets R 107

Tzitzeica27) has defined an R net to be one for which the tangents to* the curves of both families form W congruences. Equations (29) and (30) are the analytical condition that N be an R net. When these conditions are satisfied, the first of (22) reduces

a8 to — - log r = 0. Hence we have the theorem of Tzitzeica :

dudv

An R net is isothermal-conjugate.

By means of the first of equations (22) we establish the con- verse theorem due to Demoulin28):

If the tangents to the curves in either family of an isothermal- conjugate net form a W congruence, it is an R net.

Since an R net is isothermal-conjugate, the parameters can be chosen so that r = — 1. Since a and & in (1) are determined only to within factors, which are functions of u and v respectively, these can be chosen so that (29) and (30) may be replaced by

, 3 log 6 ,_09loga

o — £> — - — , a — & — - - . dv du

Hence :

Tlie two differential, equations satisfied by the homogeneous point coordinates of an R net are reducible to the form

(31)

8 log a 90 . dlogb 86 .

'

dudv dv du du dv

920 . 920 3loga 90 , 9 9log& 90

9v2 9w2 3w 9w 9v 3y

We return to the consideration of the net N-I. From (23) we have by differentiation

-i 9 „, 9ic_i

i-f ~logJT6^t

\ouov ov ou

(32)

2T) Comptes Rendus, vol. 152 (1911), p. 1077. 28) Comptes Rendus, vol. 153 (1911), p. 592.

108

IV. Surfaces and congruences in 3-space

Since r = — 1, it follows from (17) that Ci= — K. Making use of (in, 2, 4) and the third of (22), we find from (28)

(33)

9 v!

9loga

— *

du du

3 u*

dv

--f c-ia;_]

where c'-i is a determinate function. Comparing equations (31) and (33), we see that N-i is an R net. Similar results hold for the first Laplace transform of N. Consequently:

The Laplace transforms of an R net are R nets.

Making use of the terminology of [§ 165], we have:

If either of the first derived congruences of a W congruence is a W congruence, then all of the derived congruences are TF29).

46. W congruences. Let 8 be a surface referred to its asymptotic lines a — const., fi = const. The cartesian coordinates, xa\ #(2), x®\ of 8 are given by the Lelieuvre formulas of the form [§ 79]

(34)

dec

da da

where v1} v2, vs are proportional to the direction-cosines of the normal to S, such that the total curvature of S is given by

(35)

K= —

The functions Vi are solutions of an equation of the form

(36)

= M6,

where M is a function of a and 8.

Cf. Demoulin, 1. c., p. 591.

46. W congruences If 0! is any solution of (36), the equations [cf. § 172]

109

(37)

da

0!

90!

da da

9/5

0i

90t

d/3 9/5

are consistent, and the functions vi are solutions of

92 0 (38) — — T-T- = 0]

The equations of the form

(%Q\ ^T(l) -y.(D _

^<Ji/y JU JU

define the coordinates, x(i} , of a surface S, such that 8 and 6' are the focal surfaces_of the congruence of lines joining corresponding points on S and 8; this is a W congruence, since the asymptotic lines are parametric on S. Moreover, any W congruence with 8 as a focal surface may be obtained in this way. We shall give this result another form.

From (9) and the first of equations (10) it follows that the coordinates x(i) satisfy equations of the form

(40)

980 9«2

90 . 90

920

da da 9/5' dp

90

— (Jh ~ ~\~ da

90

9/5'

Differentiating (34), we obtain

(41)

9 a8 9 «2

The functions Vi satisfy three equations of the form (1) and (2). From (36), (40), (41) and the first, second and fifth of equations (4), it follows that these equations are

110

IV. Surfaces and congruences in 3-space

(42) We wr US\	d*Vi dy> dfi	7i * 1	fft89>4- 8Mv
da* da da	&9/5" fM)^, , dy d*i	i 3/5 3/5/ 1 /x. ^^ 1 ^ttl
dadft \dadft
ite fl. i7. —	3/5 3/5 fri/..- -U 7——	3 Vi —1- <WJ

and seek the conditions which k, I and m must satisfy in order that this expression may satisfy equations (37) for i=l, 2, 3. By means of (42) we find

dl dm

(44)

(45)

and

(46)

da

— bl = Q,

k =

1_ 2

1 / dl

dk 30x

da da

I o ft

3a

3?

dm

dm ~W

dl

, 9f/ , - 4-m

da

3y

"sT

m

dfl

m

m

+01^=0,

+ -^- =0.

3« '

Conversely, if I and m are any pair of solutions of (44), the functions k and 61} defined by (45) satisfy (46). Furthermore, if equations (46) are differentiated with respect to ft and a respectively, and the resulting equations are substracted, we find that 0i is a solution of the second of equations (42), in consequence of (46) and the equations for (42) analogous to the third and last of (4). Hence30):

The determination of the W congruences with a given focal surface S is equivalent to the solution of equations (44), in which a:

30) Cf. Jonas, Jahresbericht der deutschen Mathematiker- Vereinigung, vol. 29 (1920), p. 52.

47. R surfaces

and b are the functions appearing in the equations (40) of the sur- face; when a pair of solutions is known, the coordinates x(i) of the other focal surface of the congruence are given by

~ _

8«

This expression follows from (39), (43), (45) and (34).

47. R surfaces. In § 45 we established the conditions to be satisfied by a net N in order that it be an R net. A surface is said to be an R surface when it contains an R net. In this section we establish an analytical characterization of R surfaces.

Let 8 be a surface referred to its asymptotic lines « = const., 0= const. Its cartesian coordinates satisfy equations of the form (40). Any isothermal-conjugate system of curves on Sis defined by [cf. § 82]

(48) u=A(a) + B(fl, v'=A(a)—B(fl,

when A and B are arbitrary functions of a and ft respectively. If we effect the transformation of variables (48), equations (40) become

920 d»0

3*0 1 . dO 1 36

where

the primes indicating differentiation.

From (31) it follows that a necessary and sufficient condition that the net of parameters u and v be an R net is that

dv ^ du v dv

IV. Surfaces and congruences in 3-space or in terms a and ft

_l_aJST_ J_M.- I dM . 1 dN

B' dft ~ A' da~~ U' B' dft "h A' da ''

Substituting the above expressions for K, L, M, N, we are brought to the single condition

(49) £'2 +B'B" b = A'

Since the quadrics are characterized by the condition that Oi = b = 0, we have the theorem 82) :

Every isothermal-conjugate net on a quadric is an R net.

If the condition (49) is satisfied for_ two sets of functions A, B and AI, Blf it is satisfied also by A, B,