TRAN S FORMATI O N S 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 Printed by LijTCKE & WULFF, Hamburg. 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 pseudospherical 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 S and 82 are two transforms of S there can be found a fourth surface S' which is a transform of both S\ and S2. Later (footnote 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 Guichard (f. n. 100) announced two theorems concerning 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-parameter 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 Darboux 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 developables meet the surfaces on which the nets lie in these nets and corresponding points of the two nets divide harmonically the focal segment of the corresponding lines of the congruence; these transformations were first studied by Koenigs (f. n. 17) and are called transformations K. When two nets and the congruence of the joins of corresponding points are so related that the developables of the congruence 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 Ribaucour and transformations K 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 transform 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 n-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.......................2....... 28 14. Radial transformations.......................................... 31 Chapter II. Transformations F. 15. Fundam ental equations.......................................... 34 16. Inverse of a transformation F. Parallel transformations F.......... 37 17. Harmonic congruence of a transformation iF..................... 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 0.............. 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......................................I... 80 34. Levy sequences of the first order................................ 81 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 permutability...................................... 90 Chapter IV. Surfaces and congruences in 3-space. 40. N ets in 2-space............................................... 96 41. Tangential coordinates of a surface in 3-space................... 97 42. A sym ptotic lines.............................................. 98 43. Nets in 3-space. Point coordinates............................. 100 44. Ray congruence and ray curves........................... 104 45. N ets /....................................................... 106 46. W congruences................................................ 108 47. R surfaces.................................................... 111 48. R congruences. Transformations 1............................ 113 49. Reciprocally derived nets. Transformations WT of nets R........... 115 50. Theorem of perinutability of transformations TW................... 119 Chapter V. Transformations P2. W congruences. 51. Tangential coordinates of a net. Laplace transforms in tangential coordinates..................... 127 52. Transformations F in tangential coordinates...................... 129 53. Transformations 2 of nets with equal tangential invariants......... 134 54. Theorems of permutability of transformations 2 and of TW congruences 136 55. Nets permanent in deformation.................................. 138 56. Transformations S of permanent nets for which ( t 0 and ( i 0... 141 57. Transformations ~, of a sequence of permanent nets............... 143 58. Transformations 2 in point coordinates. Nets in relation 0........ 146 59. Transformations ~, and K of the focal surfaces of a WV congruence. 149 60. Nets with equal point invariants and equal tangential invariants... 150 Chapter VI. Orthogonal nets. 61. Nets 0 and p, 0. Congruences I and p, I...................... 156 (i2. Nets conjugate to congruences I and p, I........................ 159 63. Orthogonal determinants.................................... 160 64. Determination of 0 nets................................. 165 Contents VII Section Page 65. Congruences conjugate to 0 nets................................ 167 66. Transformations F of an O net for which the conjugate congruence is norm al 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. N ets 2, 0...................................... 188 Chapter VIL Transformations of Ribaucour. 77. Orthogonal determinants and 0 nets in 3-space................... 194 78. Transformations P 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 K and congruence K................................ 201 82. Transformations D,, of isothermic surfaces....................... 206 83. Theorem of permutability of transformations D................. 209 84. Special isothermic nets in 3-space............................... 211 85. Complementary transformations Dm of special isothermic nets...... 212 86. Transformations D,, of special isothermic nets.................... 214 87. Transformations D, of minimal surfaces......................... 215 88. Transformations E,, of 0 nets with isothermal spherical representation 215 89. N ets....................................................... 218 90. Transformations R of nets P.................................... 221 91. Theorem of permutability of the transformations of nets ~2........ 223 92. Surfaces of Guichard of the first kind........................... 224 Chapter VIII. Circles and spheres. 93. Coordinates of a sphere....................................... 233 94. Pentaspherical coordinates of a point............................ 235 95. Pentaspherical coordinates of an O net.......................... 237 96. Congruences of spheres....................................... 237 97. Derived congruences of spheres.......................... 240 98. Congruences of circles......................................... 241 99. Congruences of spheres and circles in cartesian coordinates........ 243 100. Congruences R of spheres...................................... 245 101. Plicker line coordinates........................................ 247 VITI Coiltents Section Page 102. The Lie line-sphere transformation.............................. 249 103. The Lie transformations of surfaces......................... 252 1.04. Congruences R of spheres. T 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 R........ 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 0 in 5-space................. 277 116. Cyclic congruences........................ 280 Chapter IX. Rolling surfaces. 117. Congruences harmonic to nets C............................. 287 118. Rolling surfaces......................................... 288 119. Special isothermic surfaces............................... 291 120. Rolling of a surface applicable to a surface of revolution......... 292 121. The fundamental theorems of Guichard................... 294 122. Deformable transformations R of the first type................... 296 123. Deformable transformations R1 of the second type................. 298 124. Deformable transformations of the second type of minimal surfaces. 304 125. Deformable transformations E............................... 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 Fi, of permanent nets on a central quadric........ 329 136. Transformations F7, of surfaces applicable to a central quadric... 334 137. Theorem of permutability of transformations F7i of surfaces applicable to a central quadric.............................. 339 Contents IX Section Page 138. Transformations B7, of surfaces applicable to a central quadric..... 340 139. Permutability of transformations F7k and Bk..................... 346 140. Theorem of permutability of trainsformations Bk.................. 351 141. 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 II of surfaces applicable to a quadric............. 365 147. Isothermal-conjugate nets on a quadric......................... 366 148. Transformations F and TV 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,.... x, 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 y,.... yn. The distance d between P(x) and P(y) is defined by the equation d2 = (yl- )2+... ( yn- xn)2 = _ (y- x)2. As thus used: indicates the sum of all terms of the type (yi —x), 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 y =x +uX, 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 determined only to within a factor. This equation represents each of the n equations yi = xi+uXi. 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-parameters X and Y are proportional are said to be parallel. The angle of inclination of two non-parallel lines of direction-parameters Xi and Yi is defined to be 27xy cos 0 X -- V'ZXz. Y2 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 + X- X vY, where u and v are parameters, is a plane. The locus of the points whose coordinates satisfy a relation of the form alx'+ a2x2+....- a'nxq+ an+l= 0, where the a's are constants, is called a hyperplane. For the sake of brevity we write the above equation in the formlax +an+ -= 0. In particular, xi - 0 is the equation of a coordinate hyperplane. Two hyperplanes n n Jaix+ a+ o, - bix + Fb?+1= i= —1 i=l are said to be parallel when the corresponding quantities a/ and b (i = 1,.... n) are proportional. The angle of inclination 0 of two non-parallel hyperplanes is defined by n Zaibi cos 0 / z(ai)2.(bi)2 When cos 0 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 dx whose direction-parameters are du'.e 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. 1) In this treatment the parameters may be 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 3 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 parameters of a surface are such that the coordinates x are solutions of the same equation of the Laplace form, (:1) a2 _ aloga e + alogb aO auav av au a u av' 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 ax2 F ax ax 2x 2 (2) E-^ G- ax (2) EZ au), au av' av it is readily found that [cf. ~ 63] aE aG aG aE a loga _ av F a a logb -_ a F v ~av 2rH2 ' au 2H12 The functions E, F, G are called the fundamental coefficients of the net. ax ax The functions -a and - are direction-parameters of the tanau av 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 a' and i' defined by ax _,, x q au ~? av 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. 1* 4 I. Conjugate nets and congruences p' and q' being functions of u and v. If these equations be differentiated with respect to v and u respectively, we have in consequence of (1) (4) D mlr + n', =m2 -t n, av au where a a q' logb i l -- log,, nl =, v -7 p" p' au ' a log a a b q'- av, 2 log-q, q/ av7 n au q The coordinates of points on the tangents to curves v - const. and u - const. have the respective forms x +rr^', x+t3'. By means of (4) we find that the derivatives of these functions with respect to v and u respectively are linear in a' and f'. Conversely, if a surface is referred to any system of parametric curves, a point on the tangent to v const. has coordinates 9x of the form x + t a When v varies the direction-parameters of the tangent are ax at ax a2x v av au az av Hence if this tangent is to be in the tangent plane to the surface, ax ax the preceding expression must be a linear combination of - and au av' 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 f' are also direction-parameters. We wish to consider now the particular parameters a and f, such that _x ax (5) ax aa', - b/ in which case equations (4) become (6) a n,, a ma, aU where m and n are functions of u and v given by ab aa (7) = a - - b Following Guichard we say that the a's and /'s are the normal parameters of the net. Conversely, if we have n pairs of functions a and fi 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 Iu- const. are parallel. A representation of all these nets is given by drawing through the origin lines whose direction-parameters are the a's and i's. We call this representation a point net. 3. Determination of nets on a surface. Consider the differential equation (8) A o a + a + a2 o a0 2ao -- 0o au 2 auav av2DE au,av where A, B, C, D and E are functions of u and v. If we change the independent variables, putting U! = 1 f(u,, V), V' = 92 (U, v), 6 6 ~~~~I. Conjugate nets and congruences the resulting equation is of the form (9) A' PO' + 2B' a20'av' -~2 Dfau' f1 where a ua av av B'-Aau' a' / au! a v' av' aU\ + an' av' au au B an -av+ au a av av' C~cAaVf~2Ba av' f+ (av' 2 \au lau av \av From these expressions it follows that if 9) and ~P2 are resolutions of equation (9) is. of the form (1). Since 9) and )2 must be functionally independent, they are obtained by solving the two differential equations of the first order which are factors of (10) Adv2 -~2Bdudv+Cdut2 - 0. Darboux3) has called (10) the differential equationof the characteristics of (8). There is only one such function cp when If we take it f or u', equation (9) is reducible to the f orm, a2o ___ ao Let S be a surface in 3-space whose cartesian coordinates x are functions of any two parameters u and v. We can find by differentiation anl equation of the form (8) satisfied by the three coordinates and by any function of them, say F~x1', x2, x3). Then 3) Lepons, 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 S 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 S, 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 ax' ax axf' ax (12) - au au' av av' where h and I are functions of u and v such that the conditions a a x'i_ a axt av \u ~au \a -v are satisfied. These equations show that the x's satisfy an equation of the form (1), where now a and b 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. 8 I. Conjugate nets and congruences (13) h ( — h) aloga, al (h -) alogb5) v av au aobau 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' 1 aloga aO' h alogb ao' (14) +_ auav h av au I au v ' which may be written in the form (1) a aloga' a' + alogb' aO' auav av azD au av' where (16) a'- ah, b'= bW. 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 0 ' ao ao0' ao (17) -at h - u ~ au, av av is a solution of (14); we call 0 and O' corresponding solutions of (1) and (14). The analytical problem of finding parallel nets may be given another form. If we define a function po by (18) h —l= p, equations (13) may be replaced by at logb aIt a_ (19) i log a Sp. -u ~ au av — ga 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 3 V2 aloga asp alogb a _ a2 (20) a2 + logab -0. auav av au u au auav 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, i-+-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 x' (15), we get alogb' a x' a ha acx alogb ax Ih log h au av - vh lg a u 3 4au av ax If the coefficient of is zero, the theorem is established. If it a zt 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 a2x Ax ax =a2 A - +B ax au2 au a~ v Expressing the condition of integrability of this equation and (1). we get an equation of the form a2x _ x ax av2- +D v' av2 au'1i av 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'- ). In consequence of (12) the derivatives of this expression are reducible to __ at a(1 + th — t) - ('- ), X (l+t -)+ a( -). av av Hence the points for which t has the respective values t 1- t2 1 1 1 —h' 1 —1 describe surfaces 2 and 22 such that the lines L are tangent to the curves v - const. on 2i and u const. on 4:2. The coordinates y and z of -2 and X2 are (21) y '-hx x'-l (21) Y- =1-h ' z=1-' 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 surface6). 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 11 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 u 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, F1 and F2, 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 2, the loci of F1 and F,2 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 a Dy _ x' -x ah 8y 1-h (/x x'-x Dloga\ 2u (1-h)2 au' av 1-h av 1-h av ' (22) s _a h-l ax x'-x alogb z _ a'-x ~u 1n-dl u -l Du ' v (1)2 Dv and ay _ a a/_a \ a8 a l -h ay J Dua av 1 —h au au au 1-h av (23) a h a a b ua Dv I~v u1 — a u 1-l abvh Du v - log-a 1) D_ + Da log( ) ~l__. From these equations it is seen that the parametric curves on 2x and ~2 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 surfaces 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 G 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 — X, z =x — X, such that as u and v vary respectively the corresponding point moves tangentially to the line. This is expressed analytically by y __, (25) -aX, a — = X, au av where a and T are determinate functions. Substituting the above values, we arrive at equations of the form a x _ a, - a a, (26) x -- iu + - _ +7e+ au au av av Expressing the condition of integrability of these equations, we find that the parameters X must satisfy an equation of the form (27) 0 alogA a 8alogB a 0 vu(27v v u + au av uav 3v au au av 5. Congruences conjugate to a net 13 Hence we have the theorem: The direction-parameters of a congruence referred to its developables 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 (28s) A -. a logB I logb a alogb and (29) alogA 1 A lo bga ), b v loga) 1( Dlga B-, log- 1* ~V av - - Du e e9 aDv If these two sets of values of the coefficients be equated, we get the following equations of condition: a 1+ e + loga/I 1 0 0av ~ - 9 'av (30) lo a( a I+v a]ogb 1)=, (-30) D v + -u a Y_ a 7). Dv A Du /k If the last of these equations be replaced by (v - logt Q _ logt (31) au av v i a 8ou D v ' where t is thus defined, the first two are reducible to a 8 t aDlo ga i t 0 (32) - t + log i t t\ =. v u, / u \ u1 7) We note that if we put., - =, - p - O, equations (26) and (30) reduce to (12) and (13). 14 I. Conjugate nets and congruences Moreover, equations (26) can be written ax i a a(Xt), x I ta (t). (33) (X t)a (X 0) a at au Xv t av 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 determine 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 whose 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-rx'. The first derivatives of this expression are of the form ax ar, ax ar (36) (1-rh) x, -(1-rl)- X. au au av av When r takes the values 1/h and 1/1 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 -~', z = x — x. From these results follows the theorem: Any congruence conjugate to a given net N can be obtained by drawing 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 conjugate 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 congruence conjugate to a net N is conjugate to an infinity of nets parallel to iN. 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 V and of a parallel net. 6. Focal Surfaces of a congruence. From (37) we have by differentiation and reduction by means of (12): ay __1 h., / l\ (a x x' loga\ au ~ h2 au av a v h av (38) a z / h\ /ax x' alogb\ xaz 1 la l au [ au al bu av av12 a and a2y a a ah) ay a (bh -) ay \ (39)ai o a u h ax a b ah a a(9 ____ av lo a h) + a log ) uav a ~ v a I au [a I av) v' 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 have, as follows from (37), (40) 1= -h. 16 I. Conjugate nets and congruences From (13) and (16) it follows that in all generality we may take (41) a — b (42) a' — -- ' = h. We recall that for an equation of the Laplace form a,_0 a0 a8 (43) a- a +b +-L c a u av au av the functions aa ab (44) IIT - a-+a b +c, K - b + ab + c, 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 01, where 01= 0-, 2 being any function of u and v. From (41) and (42) we have for n-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, 1 and ha, 11 of (13). The coordinates of the focal points of the congruences are of the respective forms (37) and X t Xtl (45) y1 = x- = x The coordinates of any point on the line joining the focal points of coordinates y and y1 are of the form y+t(y-y1). If we differentiate this expression with respect to u and v and make use 7. Laplace transforms 17 of (38) and similar equations for yg, we find that its first derivatives are proportional to the corresponding first derivatives of y when t - h1. Consequently the points of coordinates y and (46) y + hi (y- y) 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 yi. 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 =- 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 (47) 1 ax x- - x logb aut' aq In fact we have (48) a b ax-1 b a ag u ax av ab ov \ ab a u' au 1 ax -- X aloga av av /aa a1 X a a av ax au aa au log ab dav av The nets N, and N1- with the respective coordinates xI and x-1 are called the first Laplace transform of N and the minus first Laplace transform respectively. If the point equation of N-1 is written in the form (4) a28-0 a a (-1 + a a 1 _(49) a (loga-) a0-1 + a(logbl)-' a3iav av au a av we find that ab (50) a-_ — = ab, au ab a _ _ _ b_ _ b b-1 b - log a - a c ab alogb ' au The Laplace transforms of a net parallel to N are defined by equations of the form 2 18 I. Conjugate nets and con I _ ~ a x' (51) xi 1 a -) x1 x a log b' au ' au From these, (14) and (48) we have,gruences I- 1 ax' Xt —______ av (52) ax-1 a x- au au, I ax' h2H' ax Da lH an' ax-1_ lK' ax-1 av hK av axv - h ' au av' where H' and K' are the invariants of (15). Hence we have the theorem: If N and N' are parallel nets, their respective Laplace transforms are parallel. 8. Transformations of Levy. By means of (38) equations (37) can be written in the form 1 h ay X Y a 1 aQt' au h I ax xu -- i 1 3v' a 1 av av I It is readily shown that 1/h and 1/i are solutions of the respective equations (39). By a change of notation the second of these equations may be written 0 ax - a av' av 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 o ax - x-a a-O au 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 The 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 0 ax 0 ax (53) =- — x- a v' z= au av au where 0 is a solution of (1). Conversely, every solution of (1) gives two nets conjugate to the congruences of tangents. In order to prove the latter part of this theorem, we note that if 0 is a solution of (1), we have from (53): (54) ay - (- az alogA ( au alu av av ( where aO bo (55) A=a B - au av From (54) we have by differentiation 2- = a log A alogB) ay alogB ay a(uav av \ au au au av' a2 alogA 8 / logA az (alogA) az aua v au a 8u avog av Hence the points of coordinates (53) describe nets. We call these nets the Levy 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 0 is a solution of (1), the function 1 ao 0-I ' alogb a u au is a solution of (49), the point equation of the minus first Laplace transform of N. We call 0_- the minus first Laplace transform of 0. From these equations, (47), (48) and (53), we have 20 I. Conjugate nets and congruences 0-1 ax_ — 0 ax a-1 1 av - 0 a~ av a t Consequently the net of coordinates z in (53) is the Levy transform of the minus first Laplace transform of -N by means of 0-1. Similar results follow whenwe consider the net of coordinates y. Hence: The Levy transforms of a net determined by a solution 0 of the point equation of N are Levy transforms of the minus first and first Laplace transforms of NT by means of the corresponding Laplace transforms of 0. 9. Determination of congruences. We saw in ~ 5 that the direction-parameters X of a congruence are solutions of al equation of the form (27). If in (27) we put X - x'O, we obtain a2x' a lA\ax' a B\ ax',, (57) aulog + log C weeauav av v jo - u log a 0 av where (58) el 1 alogA a0 alogB a 0e a2o (58) C av a- - au av auav Hence a necessary and sufficient condition that the quantities x' are the cartesian coordinates of a net is that 0 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 directionparameters X, we find a solution 0 of the equation of Laplace satisfied by the X's; then the quantities (59) x' X 0 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 directionparameters 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 x', D loga' ax', a logb' -— x a ~ lo av aDv D u a e 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/. Hence: If the direction-parameters X of a congruence G are solutions of an equation (27), the functions aX a logA aX a logB (60) X X av av aau a are direction-parameters of the tangents to the 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 G, and G-1. As a corollary of this theorem we have: When two congruences are parallel, their focal nets of the same rank are parallel. Let M o(xo) be a generic point on the middle surface of a congruence 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- = o+- X, z - xo- -X. 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 axo aX a a ut a it a U -- = t -- Q —~ +-2 2 X-~- log BY p, 22 I. Conjugate nets and congruences and Q must satisfy the equation (63) a + a logA g a logB a ( a2 logAB-C)e O, au av av au au avt gau av 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 Xo 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 of a congruence is equivalent to the integration of the adjoint of the direction equation of the congruence. Io. 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 NV by means of the same solution 0 of the point equation of N are the focal points of the congruence of the joins of corresponding points of the transforms; that -is, the points defined by (53) U IWS(X) u are Laplace transforms of one another. The mutual arrangement of N 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 subsequent figures. A net and a congruence are said to be harmonic when the foci of the 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 This situation is illustrated by fig. 2. We have also the theorem: If N is conjugate to a congruence G, the osculating planes of the curves of Fa parameter u(v) of the first (second) focal net of G are determined by the lines of G -. and the tangents of the curves u(v) of N. Suppose that we have a net N v and a congruence whose lines lie in tan- \ gent planes of N and the developables of G correspond to the curves of AN, G: taken as parametric. As u varies, the Nx, corresponding focus of a line of the congruence must lie on the characteristic of F fg 2 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 developables 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)x ax (64) x - x —^r a', y x- t av Now az ax alogra x alogb av - u v av a -- ax ax Expressing the condition that this is proportional to r — t au av' we get (65) a 1 1 aloga 1 alogb 1 av r r av t au rt 8) This can be shown analytically by making use of the fact that the x's 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 ax ax be proportional to r- — t is u, av a 1 _ loga 1 alogb 1 au t rS av t a u tr Hence there must exist a function 0 such that 1 1 ao 1 1 ao r 0 a u' t 0 av Substituting these values in (65), we find that 0 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 be Levy transforms of N by means of the same solution 0 of the point equation of N. Since the direction-parameters of the harmonic congruence are of the form aO ax a0 ax au av Dv au' it follows that if a second harmonic congruence, determined by a function 01, is to be parallel to the given one, 0r must be a function of 0. Since both must satisfy (1), 01 is a linear function of 0 with constant coefficients. Hence we have: A necessary and sufficient condition that two congruences, harmonic to a given net N and determined by solutions 0 and 01 of the point equation of N, be parallel is that 01 be a linear function of 0 with constant coefficients. Consider now a congruence G and two nets NV and N2 harmonic to G. Corresponding tangents to N1 and NV meet in a point of a focal net of G, and the congruences of these tangents are conjugate to this focal net, by the theorem of Levy. These two congruences and the nets N1 and NV are in the relation discussed in the last theorem of ~ 6. Hence we have the theorem: 11. Derived nets. Derivant nets If two nets are harmonic to a congruence, the joins of corresponding points of the nets form a congruence conjugate to the nets. I. Derived nets. Derivant nets. Let G1 and G2 be congruences harmonic to a net N, determined by solutions 01 and 02 of the point equation (1) of N, it being understood that 02 is not a linear function of 0x. The function 01 a 02 y - 02. au is a solution of the second of equations (56) with 0 replaced by 0, that is, the point equation of the second focal net of G1. The coordinates of the Levy transform N(x) of this focal net by means of y conjugate to G1 are of the form -X_ a 8ga x z- - 8v which in consequence of (53) and (54) is reducible to ~_ (o% a 02j a ( 0a 02 (66) x ax —, 1 a au av av au The coordinates y2 and z2 of the focal points of G2 are given by (53) when 0 is replaced by 02. The point equation of the second focal surface of G2 admits the solution 01- 02 - /- 2, which determines a Levy transform of this surface conjugate to G2. The expressions for the A coordinates of this transform are reducible to (66). Hence: If twio congruences are harmonic to a ^I/, net 1N, the point of intersection of corre- \ \ sponding lines of the two congruences de- N^ \ scribes a net conjugate to the two congruences. u i This result is illustrated by fig. 3, where L1, and Liv are Levy transforms - I. Conjugate nets and congruences of N by means of 01 and L2,, 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 yl. If the expression (46) is differentiated with respect to v, the resulting expression is reducible by means of (12) -ay and (38) to l-, where av' -_., h 11-h1 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 +- h~(y-y1)-ly I-l which is reducible to (67) -x+ x' (l- h)-x"(l-h) (67) hx+- - hl I-hl, In like manner we find that the coordinates of the first focal point of the congruence conjugate to the nets of coordinates z and zi 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 ax 6av 02 av au ~a02 ao, ao2 ao1 au av a v au (68 a {a21 a0 a2D2 801 aDX aD201 aD02 8202 aelx - i 2X D Du2 aV au2 Dav Iu a-u2 au 8u2 ui aDv fa l0 a OD ae1l D au Dv Dv au 12. Determination of nets harmonic to a given congruence 27 a 01 a -02 a0 a x a aeul ae aeu a v a O2 0O DO2 D6l ~~(68b) a av a v au -" I a28r ae0 a2O a,\ a a2 a 201 aO2 e2 aD\ aDx x 2 \v a2 av aV2 av JaDu aDv2 u - v2 v u / av IQv2a DO, ae 1 a2 ae. i _ Du Dv Dv au We remark that if we replace 02 by 02 + const. the expressions in the parentheses are unaltered. Consequently: The ool derived nets' of N determined by 01 and 02 - c, where c is a parameter, are parallel to one another and conjugate to the congruence harmonic to N determined by 01. 12. Determination of nets harmonic to a given congruence. 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 -_. By (21) this 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-1 and N' 1. From these two theorems it follows that the problem of finding nets harmonic to a given congruence G is equivalent to the determination 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 G 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 x' X- 1 (69) = 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 consequence of (52). Hence: If G and G' are parallel congruences, and through the focal points of G lines are drawn parallel to the lines joining a fixed point to the corresponding focal points of G', these lines are tangent to the curves of a net harmonic to G. 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 (70) y a log (- y), a z logp (y_ ), a~u tu a v av where p and q are determinate functions. These equations are reducible to the normal form (6), if we put (71) a -- p, f - yq with ap aq (72) qn, vDq v au U 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) (73) ~~~Z~ a q-fipv. 13. Congruences harmonic to point nets 29 Conversely, suppose we have any point net of parameters c and fi, 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 congruence 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 given by (6). If p and q are a pair of solutions of (6), it follows from (5) that 0, given by the quadrature (74) u ap, bq, 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 ax 0a ax aO -- a b (a q -- /ip). au av v au a b q p). 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 be found by quadratures; when a congruence harmonic to any net is known, by a quadrature at most 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 G' parallel to the corresponding tangents to N, these lines are tangent to a net harmonic to G'. 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 congruence harmonic to T; its direction-parameters are of the Jbrm oa'q-f-'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), aZ aa _p AZ aq a8 au au au av av av _2Z aq 8a ap 8 8uav av au au av Hence the direction equation of the congruence is (75) alogq az alop aZ logp a log )g q (75) 3u- mnvzuav av au au av au av This is of the form (27), where now (76) q AU, p BV, m- alo a q U and V being functions of zt and v alone respectively. Hence: The direction-parameters of any congruence whose direction equation is (27) are expressible in the form (77) Z = A U- B TV, 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 0 be any solution of (1). From (57), and (58) it follows that the locus of the points of coordinates x given by x (78) x= -, is a net N, whose point equation is a20 - ae O a b ae (79) log- + - log b auav av - a u a l 0 av, 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 0. The tangents to the curves v = const. at corresponding points on N and N meet in the point whose coordinates are 1_ 0-1 ax _0 a a U 0 1a. au au 6\ e and the curves u = const. in the point 1 0-1 ax 11\ ax av av \e These points generate the Levy transforms of N by means of the function 0-1, and of N 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 xi and xI2 are expressible in the form (lxil-+- l12 x)/(l-1+ l2) 2. The coordinates of any point of a plane determined by three points of coordinates xl, Xi2 x3 are expressible in the form (l1 xil+ 12 x2+ 13 xI3)/(1L+ 12+ l ). 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 corresponding 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 20, 0 a logp 0 auav au au av' one of whose invariants is zero. In this case I = V+1/p. 6. When in equation (1) we have a = U and b = V, where U and V are a2o functions of u and v alone respectively, the point equation is = 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 Edtu2 + 2 Fdz d v +- G dv2 = 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 N'(x') has equal point invariants, that is b'= a', the equations ax _ 1 ax' ax 1 ax' au a'2 &u' 3v a'2 av 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-x'ja'2, x —x'/a'2. Consequently Nlies 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 (x0'- x'0)/(0' —0) describes a net conjugate to the congruence of the lines joining corresponding points on N and N'. 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 andN' 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. a0, iao a0 a 0 a 11. If 01 is a solution of (1), then 01 — a — and Oi —0 are av av a u a u solutions of the respective equations (56); and the former admits the latter as its minus first Laplace transform. 12. To each solution fo of the first of equations (56) there correspondends a solution 0O of (1) such that 5p =- 0-0 8 —/ v 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 transforms 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 N by means of solutions 01 and 06 of (1), the.. _ ix 80i ax A a\i o2 aoi ao2 o\1 quantities a = - - -a a- a a a are the coordinates au av a uv au au v av a u of a net parallel to N. 16. A necessary and sufficient condition that a net N and a parallel net N' defined by (12) be radial transforms of one another, to within a translation of either net, is that h - 1 = const., say c; then if N' is a radial transform, x' - cx. 17. A necessary and sufficient condition that two nets, N' and N", parallel to N and determined by pairs of solutions hi, 11, and h2, 12 of (13) be radial transforms of one another, to within a translation of either, is that = - const. h —h 11 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 Nis a radial transform of a net N by means of a solution 0 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 aO a ao ab au' aa av' 20. If Nis a radial transform of a net N by means of a solution 0 of the point equation of AT, and 01 is any other solution of this equation, then 01/0 is a solution of the point equation of N. Show that the Levy transforms of N and N by means of 01 and 01/0 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 II. Transformations F. 15. Fundamental equations. In this chapter we areconcerned with the determination of all nets N1 such that for a net N1 and a given net N the lines joining corresponding points form a congruence G conjugate to N and AT19). These transformations of N into nets N1 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 N1 are in relation F. We call G the conjugate congruence of the transformation10). 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 N1 are in relation F, the direction-parameters of the conjugate congruence of the transformation are proportional to the coordinates x' of a net T' parallel to N, and also to the coordinates x' of a net N,' parallel to Ni. Hence these coordinates must satisfy a relation of the form Xf (1) x'- a,, 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, 0' is necessarily a solution of the point equation of N'. The coordinates x1 of N, are necessarily of the form 0 (2) x -!-o where 0 is to be determined. The coordinates x' are given by equations of the form (cf. I, 12) ax' ax ax' ax (3) a h - Z b au,8 a v av Hence the first derivatives of xr are reducible to the forms ax1 of' \ Dax,,1 al ao 8a h a U h au au ' (4) (4) 8aX o' ax', 1 a8' a8\ av a( I a av ( I avi From these expressions it follows that N1 is parallel to NI, if, and only if, 0 and O' satisfy a ' a a0' a (5) -h -- ~ au au' av av' Expressing the condition of integrability of these equations, we find from (I, 13) that 0 is a solution of the point equation of AN namely (6) 8a20 aloga ao + alogb 8a auav av a t a u av 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 O' the corresponding solution of the point equation of N', then (2) defines an F transform of N. Hence: Any transformation F of a net N is determined by a net N' parallel to N, such that the joins of corresponding points of N and Nt are not concurrent, and by a solution of the point equation of N; and any such net N' and a solution determine an F transform. 36 II. Transformations F In consequence of (5) equations (4) may be written x) a ax ax~, o axI (7) au h au' v I av' where we have put (8) =ho-O_, -=16- 0'. By means of (1), (3), and (5) equations (7) are reducible to (9) ax1 / a., ax ax _ ' / ao ax -- _ - x' _ - -6, au 6 au au' a v 62av which, in consequence of (2), are equivalent to aX1 XI _ -,a 6 a x au o L(f' -x) 0 + 6 2x (10) a - [(x —X) a - + 0 ax av 6'1 av av From (9) it follows that the point equation of N1 is reducible to (1) a2O aloga, a 1 + alog b8 a 1 au v av au au v ' where r c (12) al = a, bl - b. This equation may also be put in the form a8201 / alog a h a6 ae 1 i a/logb I aoi aoe (13) auav- v a O' av) a '' au ' au) av In consequence of (I, 37) equations (2) can also be given the forms O hy a lz (14) x1: - +x+ -o ~ x - 6' 6' 6' 0 16. Inverse of a transformation lF Parallel transformations F 37 From (8) and (I, 18) follows (15) r — a - 0. Incidentally we observe that r and a satisfy aw a a' a 0 a= O - -logbp, a —= o -loga g b 5 8 ' v a V a (16) 8a 0 a O a a a= — up -- log - 0 -O log ac. 8u au b' av av Suppose we have any congruence G. There is a net N con-' jugate to G (~ 8), and a net N' parallel to T whose coordinates x' are direction-parameters of G (~ 5). Each radial transform of N', say NA, determines an F transform jN of N, and NVi is parallel to N'. Since there is an infinity of parallel nets Nt satisfying this condition (~ 5), we have the theorem: If the coordinates of a net are the direction-parameters oJ 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 oo' nets conjugate to the other. For, if N is a net conjugate to the first congruence, there is a net N', parallel to N, whose coordinates are direction-parameters of both congruences, and by the theorem there are o 1 nets conjugate to the second congruence and parallel to N'. i6. Inverse of a transformation F. Parallel transformations F. Evidently N can be looked upon as a transform of N, and now we seek the functions 0 —1 and (0')-1 giving this transformation. Since the roles of N' and N' are interchanged it follows from (1) that (17) (o')- - Hence if we make (2) conform to — 1 (18) X =( X /, 38 II. Transformations F we find that 0 (19) 0-1 = 0l It is readily verified that equations (7) are satisfied by these values of 0-~ and (0')-1. Hence 0-1 is a solution of (11). Moreover, we have r a (A) au (20) 1 a I (+) ^ Qv [e-1] <s v ' a V 0-1 a V e Hence equations (9) can be written au (0-1) au. ) (21) 1 / x - \ vv to-1} ~ v [e From these equations follows: If N and N1 are in the relation F determined by a solution 0 of the point equation of N, and 0-1 is the solution of the point equation of N1 likewise determining the transformation, the radial transforms of N and N1 by means of 0 and 0-1 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 N1 be parallel to N is that 0 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 X -- X- cCX where c is an arbitrary constant. In a general transformation F the function 0 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 N1 and N2, corresponding to the values 0 and 0 1- c, where c is a constant and to the same 0'. Now the coordinates of N1 are given by (2) and those of N2 by 0 + cf XZ2 — X ' ' Xs2-X-e, z From these follows 22 - 1 — CX1. 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 0' 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 N', parallel to 1V, be a radial transform of N it is necessary and sufficient that x' cx, where c is a constant. In this case corresponding solutions 0 and 0' of the point equations of N and A\ are in the relation ' = cO + d. Substituting in (2), we have xd X= — cO + d' that is N1 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 transformations. 17. HarmOnic congruence of a transformation F. From (20) and (21) we have 0-1 ax1 0 ax u au ' (22) ax 0-1 3x1 0 ax 0-1 a- ao av av a.v Hence the corresponding tangents to the curves u = const. of the nets N and N1 meet in points of a net which is a Levy transform 40 II. Transformations F of V by 0 and N1 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 congruence harmonic to the nets N and N1. Hence we have the following converse of the second theorem of ~ 15: WBien two nets N, N1 are in the relation of a transformation F, their corresponding tangent planes meet in a line generating the congruence harmonic to N determined by the function 0 and harmonic to N1 determined by 01. 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 G a congruence harmonic to it, the nets harmonic to G are obtainable from N by transformations F involving the same function 0, 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 N1 obtained from N by transformations F determined by the same function 0, and differing only in the additive constant of 0', 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 xi= 0 (cf. ~ 1) are of the form xi x- x. x. Since xi and xiare 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 NT is 1, as follows from (1). Consequently N' also is a net in a hyperplane. 18. Tansformations F and radial transformations. Let N(x) be a radial transform of N(x) by means of a solution o of the point equation (6) of N, so that x = x/o. From ~ 14 we have that the point equation of N is 820 a DO 8 b 8O (23) log- + log auav Dv -o Du Du - avv 18. Transformations F and radial transformations 41 If 0 is any other solution of (6), then 0 = /co is a solution of (23). Also it can be shown that if *r is a solution of the adjoint of (6), then y- o is a solution of the adjoint of (23). From (16) it follows that if r and a are functions of the transformation F of N by means of 0 and sp, these functions, i and a, serve also for the transformation F of N into a net N1 (1), determined by 0 and y. Hence similarly to (20) and (21) we have X24 (26) _T - function o01 by 0-11 - 0-1, equations (24) become (27) T_ ()- - a ( c- ) a2) O-' a- 0 a 'V 0-1 a- 0 Comparing these equations with (21), we note that o1 is a solution of the point equation of N1. Hence from (26) it follows that N1 is the radial transform of Ni by means of (o1. Thus by the quadrature (27) we obtain a net N, which is an F transform of N. Moreover, there are an infinity of such nets N,, since wo- co, + c 0-1, where c is an arbitrary constant, satisfies (27). Hence: If N and N are nets in the relation of a radial transformation, and NVi is an F transform of N, there can be found by a quadrature oo1 nets N1, which are F transforms of N and radial transforms of N1. When in particular 0-, w then 0 = 1 and consequently N and the nets N, are parallel, the functions co being co-~. Hence: 42 II. Transformations F' A transformation F is equivalent to the combination of a radial, a parallel and a radial transformation"). I9. Transformations F with a common conjugate congruence. Suppose we have two solutions 01 and 02 of equation (6). We seek the two nets obtained from N by transformations F determined by these respective functions and by a net N' parallel to N. We denote these nets by N1,1 and N2, 12). The point coordinates of Ni, and N2,1 are expressible in the forms 01 02 (28) 0 sl-z xi, 2,f X 02I, where 01, 06 and 02, 02 are pairs of corresponding solutions of the point equations of N and N', that is 01 and 02 are obtained from 01 and 02 respectively by quadratures of the form (5). We consider the functions 01 A. (29) 2- 02, of From (2) and (1) it follows that these functions are solutions of the point equations of N1V, and Ni,l respectively, and their derivatives are in relations analogous to (7). Hence a transformation of NA1, is given by (30) xl, 02 01-.- 0 By substitution we find that this expression is reducible to that of 2, 1, given by (28). Hence N2,1 is the transform of N1, 1 by means of N{,1, and the functions (29). It follows then from ~ 16 that N,,, is obtained from NiT,l by the functions 01- 02 0l/0 and 01/02. 20. Transformations F determined by the same function 0. Let T' and N" be two nets parallel to a given net N 1) 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 01 and 02 and the second to the subscript of common function, 9y,, determining the conjugate 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 a x' ax ax' ax a u au' av av' (31) X31) ax" ax ax' ax --- h1 - au a a av av' where hi, 11 and h2, 12 are pairs of solutions of equations (I, 13). The coordinates x' and x" are the direction-parameters of two congruences, G' and G" respectively, conjugate to N. By means of them and a solution 01 of the point equation of N we obtain two transforms NV,1 and N1,2 of N, whose point coordinates are of the respective forms (32) xI, -x- x, XI, 2 —,2-X- x It is our purpose to show that NV1, and N1,2 are in relation F. There is a net which is the F transform of N" by means of 0' and N'. Its coordinates are of the form (33) x =x- x'. 1 Differentiating and making use of (19), (31) and similar equations for 0f and 0l', we get ( a -x (h 1' - h2 1) X. 1, Jau ~ au (34)ax Olo) ax av o av0 Hence NI', as defined by (33) is parallel to NV1,113). We have seen that the solution of the point equation of N1,1 giving N by the inverse transformation is - Oi/01. When x1,1 in (34) is replaced by this value, we have by a quadrature a solution of the point equation of T,,'. By means of (20) we find that the 13) We have used the notation x"'" to mean that the net is parallel to YN1, and determines a congruence G"' conjugate to N1,. 44 II. Transformations F corresponding solution of (34) is -- 0'/01. Hence a transformation F of N1,1 is given by equations of the form (35) Xi, 1 - x11. xi By substituting the above values we find that this expression is reducible to the second of (32). Thus N1,2 is a transform of N1,i by means of the same function, -01/0, which gives the transformation of N1,1 into N. Hence we have the theorem: If a net N is transformed into two nets N1,1 and N1,2 by means of the same function 01, the latter two nets are in the relation of a transformation F; moreover, in the triad of nets N, N1,1 and N1,2, 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, N1, and N2 form a triad, and if N' and N" are the nets parallel to N determining the transformations from N to N1 and N2 respectively, the net Nl',' determining the congruence of the transformation from N1 to N2 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 01 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~ = 0 are in the relation of a transformation F. In general, we have: If a net N is conjugate to two congruences G' 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 N1 and N2, 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 N1 and N2 are parallel transforms of N. Let the coordinates of xl and x2 of Ni and N2 be given by 01 02 x 0 Xi = -- X2 =- - of. 1 2 21. The theorem of permutability of transformations F4 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 pbints, that is that there exist functions 11, 12, P1, P2 such that a[X-i + 4n (Xi - X2)] = thl (XI - XA) au a [Xi + 4 (x1- X2)] = PU2 (X1 - X2), av When the above values are substituted in these equations, we get equations of the form ax ax Ax'i+ Bix"+ C1 az =0, A2x' -- B2Xx" + C2 --- =0. au av Evidently we must have A = B1 = Ci = A2 = B2 = C2 = 0. These conditions are equivalent to ~1 + 4),r 02 +022 O1 + D 1 024L- Al 2 0 0 1 er2 au 2 j~ au 0, 1.r2 auo2 0202' 1 au Ite aA au au) 62 + (1' 2- 2 = 0 ao 62 ~ 02 02" 2-( fk= av av av av If VI 2 al = =0 -, then hi 11 coust. and h2 = 12 = const. (~ 4), and consequently G' and G" are not distinct. If 01 and 02 are constants, then N1 and N2 are parallel (~ 16). Excluding these cases, we find that the above equations necessitate 02/0, 1 const., that is N, Ni,.N2 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 N1,1. Making use of (1), (2) and (18) we can show that any solution of the point equation of N1,1 is expressible in the form (36) 012 0'.02 of 21 1 46 II. Transformations F where 02 is a solution of (6). Incidentally we remark that from (21) it follows that I om I 0)- m 8 i^ va 012/ )a 02 (37) This function 012 determines a transform of N1,1 such that its points lie on the lines joining corresponding points on N1,1 and N1,2, that is, G"' is the conjugate congruence. From (33) it is seen that the corresponding function 0"'i is given by (38) 02 02 ' 02. Hence the coordinates of the transform N12 are of the form 02 0i — O11 x2 (39) x~- xl,. o~ o ~ — o ' ou x1,1. The function 02 and the congruences G' and G" define two transforms of N, namely N2,1 and N2,2, whose coordinates are respectively of the forms (40) X2,1 X — 2, X2, - 2 Corresponding points of the nets N, N1,2 and N2,2 lie on a line,. and from (29), (32) and (40) we have that N2,2 is a transform of NVI,2 by means of the function (41) 02 -Al 02. In like manner it follows from (29) and (35) that N12 is obtainable from N1,2 by means of the function 012- 010i12t'/. But by means of (36) and (38) we show that this expression is reducible to (41). Hence N2,2 and V12, being transforms of N1,2 by means of the same solution of the latter's point equation are themselves in relation F. We wish to show further that N12 bears to NT2,2 and NV2,1 a relation analogous to that born to NI1, and N1,,z used to determine MNl2. 21. The theorem of permutability of transformations F 47 Since N2,2 and N2,1 are obtained from N by 02, they are in relation F. The corresponding net parallel to N2,2 is defined by equations of the form (cf.(33)) (42) rxit2 f- x1 14) X2,2 --- - off 02 A solution of the point equation of N2,2 is (43) 021 =- -o t 01' 2 and the corresponding function 02'" is given by (44) 21r' 1; -- o1' 62 The net N21,obtained by this transformation is defined by equations of the form 01 012 02 0,1 /"' (45) X21 = 2,2- of '- 60 2,. Ol U2 — 2 2 1L Making use of the above values, we find (o1' 02- 02' 1) x + (2 01 — o 02) Xl (46) X12-X X21-Xo, of -- 012 -0 —1 02 Hence the nets N12 and N2l coincide and the congruences G' and G"" are conjugate to N2,S15). In view of the above results we have that when two nets N' ^^V 2 and N" parallel to N are known, and two solutions 60 and 02 of equation (6) are given, the four MM^/M functions 0;, 01', o2 and 0(' (each / involving an additive constant of / fi integration) can be found by as many quadratures. When these are known, we have a group of 14) The mark."" indicates that the net is the parallel to N2.2 determining the conjugate congruence G"". 15) Cf. Jonas, Sitzungsberichte Berl. Math. Gesell., vol. 14 (1915), p. 103; also Transactions, vol. 18 (1917), p. 111. 48 II. Transformations F six nets pictured schematically in the fig. 4 by corresponding points of these nets. We say that any four nets as N, N1,1, N2,2, N12, such that the first and fourth are in relations F with each of the second and third form a quatern. We note that Ni,1 and N2,2 determine the additive constants in 0f and 0', but that the additive constants of 02 and 01' are arbitrary and consequently there are oo2 transforms N12 of a given N1,i and N2,2. However, it follows from (33) and (42) that when one of these constants is fixed, the oo1 transforms N12 are conjugate to the same congruence. We may gather together the foregoing results into the following fundamental theorem of permutability: If N1, 1 and N2,2 are two transforms of N by means of functions 01 and 02 and congruences G' and Gt", there exist 002 nets N12, each of which is an F transform of N1, and N2,2, their determination involving two quadratures; there are obtained incidentally two other nets N2,1 and N1,2 such that N2,1, N11,, N2,2, N12 is a quatern, and also N, N2,1, N1,2, N12, Moreover, the six nets can be associated into the triads N, N1,1, N1,2; N, N2,1, N2,2; N1,2, N2,2, N12; N2,l, N1,1, N12. Any set of corresponding points in the above configuration are the vertices of the complete quadrilateral formed by corresponding lines of the four conjugate congruences of the transformations. 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 conjugate 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', NN,1, N,1; \// _fjf ATa A7"' A7rrr A7"" A7rrr Ay7f "16\ N!, -1N2, -2,'2; 2,1,,12, 12; Ni2,1, N, 2,2t, a12 ). In consequence of (32), (36), (38), (40) and (43) equation (46) can be written (47) 012 X12 02 (0= X 1+ 012 2+ 021 2 2). \0 02 0 1 02 12X 16) In this notation NMa means that the radius vector is parallel to a line of G"' and the net is parallel to Ni,j. 21. The theorem of permutability of transformations F 49 From the expressions (36), (38), (43) and (44) we find (48) o1 012' 021 0 (02 021 + 01 012 -012 021) Hence the above equation may be written (49) 01 0io x12 (02 021 XI, 1 01 012 2,2 012 021 X). From (34) it follows that the functions hl2 and l12 of the parallel transformation determining GW"the congruence conjugate to N1,1 and N12, have the values h1 0' —h2 0o l1 06 —12 0o (50) 1'l12,=2 -- T! ^1 The functions T12,, d2, 2 of the transformations from IV1,1 into N12 are given by equations similar to (8) and (15), namely (51) T12 - h12 012- 012, 612 112 12 012- 02', r12 012 = 12 — 12. In consequence of the above values these functions have the expressions 1 01 0 02 02 01 121, T 12 -= 21 1 --- - - + i1 012+ 021). 1 02 h12 t T -- ('"2 — 1 +Si02' 02l) (-), 2 1 -- In an analogous manner the functions 7h21 and 121 of the transformation from N2,2 to NV12 are of the form 712^ 0a -hi of j 1 (53) hz2l - 221- -— 12 0 --- ——, and the values of the corresponding functions T2i1, 62 21, defined by (5l4) T2- h2l021-0l, 21 t1021 0 (221 21 T21 - '21, 4 50 II. Transformations F are reducible to 'r2r21 -= T1j12, 62021 = 61612, (55) 121 0.5 (2 61- l2+l2 0l o12). 9)21== -2 ( — n i ---ol+g^611). Consider in particular the case when N2,2 is a parallel transform of N. If we take 02 - =02' = — 1, 0 0, then 012= - ot1,2 and consequently N12 is a parallel transform of N1,1. From (43) we have 021 O — 01', and consequently from (49) (56) 01(x12,-2,2) (1Oi-O') (x,1-x). Since 0' involves an additive arbitrary constant, we have the theorem: If N1 is any transbfrm of a net N and NA2 is parallel to N, there are to transforms N12 of N1 and N2 which are parallel to N1; corresponding points of these nets lie on the line through M2 parallel to MMi. Consider also the case where 02 -= 0+ c, c being a constant. 'We have accordingly - o0+c', 0'-0 o '+ ec" (5 C —7)-C 021=-c+ 02 ', 01 c" — c o01 off " " 1 where c' and c" are constants. If c' =0 0, 12 is constant; it follows from ~ 16 that N12 is parallel to N1,i. Also N2,2 is parallel to N12 if c"- 0. In order to determine the effect of the additive arbitrary constants of 02 and 01', we replace them by 0-+ (c-1) and, 0'+ (e -1) in (33) and (36). If we denote the new functions by XeI and 0c2, we find in consequence of (1) and (19) (58) xe, — x 1-(e-)x, 0c2 = 012+ (-) 0\; 21. The theorem of permutability of transformations F 51 51 Hence the w onets N12 obtained by varying c and holding e fixed are conjuga~te to the same congruence conjugate to N1,1 also; similarly as e varies and c remains fixed the ooI nets NAT12 and N2,2 are conjugate to the same congruence. The foregoing formulas are interesting also in another connection. Thus if we look upon N and a net N12 as F transforms of N2,2, there are oo2 nets, including ANY,1, each of which forms a quatern with N2,2, NA and NV12. The above results lead to'a means of finding these nets N,,e. In fact we replace N1,1 by the transform of N by means of the functions (59) OC = 01 + (C 1 ) 02, X e 6;1 - (e x ) ", the quantities X(e) being taken as direction-parameters of the congruence of the transformation. Now we have a X(e) ax a x(e)_ ax (60) a he a le au ait' a av' where (61) he- hl+ (e1)h2 ler l + (e-i) 12 also (6)=e) Of+ (e- 1.) 0"' of - f 0' (c-i) off, (62) 22 2 O8e) - O' $ (c --— 1) Of +t (e — 1) 0" +r (c 1 (e - 1) 0".. When these values are substituted in the following expression which is the analogue of the right-haind member of (46), it is found that the result is reducible to the latter: (os' 02, 04' 0) a(e~ + (o(e) oc - 01e) 02)x" (63) C2 Hence: Let N1,1 and N21,2 be F transjbrms of N by means of finctions 01 and 02, x'I and x" being the directzon-plarameters of the congruences of the transformations; if-' NA, e is the- transform of N by means of 01 + (C -i1) 02 and a congruence qf direction-parameters x' + (e - i)x", c and e being constants, the C>O 2nets NV12 forming quaterns Ivith N, IV,1 and N2,2 frm qualter)~,ns also wcith X, Nc,e and NY2,2 whatever be c and e. 4* 0`2 II. Transformations F 22. Derived nets and transformations F. If N(x) is a net with the point equation (6), and 01 and 02 are solutions of (6), they determine a derived net A( 5) of N (cf. ~ 11), whose equations may be written ax ax (64) x au v where d qa 02v oa J av,av (65) 1 a02kk p ao2 ao1 ao02 ao01 u a u aau av av au Let 03 be another solution of (6), and N' a net parallel to N. An F transform N3 of N is given by 03 (66) The functions 03, and 032 defined by (67) 03i 0 i- 0. (i = 1, 2), where ao h aoi ao - a __ (68) au au' av av are solutions of the point equation of N3. They determine a derived net N3 of N3, whose coordinates are of the form a X:',+ 3aX (69) X3 -X33 at av where p; and q3 are given by (65) when 01 and 02 are replaced by 03, and 032 respectively. From (66) and (67) we have ax3 -03h-0 -, ao3 ax_ au, a, / e~0 2 X axu 03I /,t ao8 __ av 0~~~2 X0a 22. Derived nets and transformations F 53 a 03i _ Oh - 0' 8 a03 o a i 8u ~z ef a 3u au ' D- - (, ' ( D ___ a Os i 03 w 0' av Of2 a V 3 aV ) (i =1v 2) On substituting these expressions in (69), the resulting expression in reducible to (70) 01 02 03 01 0 03 01 0 O 03. - I i,.1J, 8^ 2 aeO+ ax x ax X3= +a, D D DO3 Dx 01 02 03- 01 02 03 3 At Du D Du Du Dv 01082 O3fa O1 DO2 DO3 O- 1 D 02 o83 a o1 e2 eO3 av av av av av v av au a az where a (2 aO3 aeO a0O + or i 03 ae a03 ae 0 \ v au au av 2\ v au au av +3 av au aDi av Since O0 and O' are solutions of the point equation of N', a derived net N' of N' is given by equations of the form a x' ~x ^xt-~~a ax, x '-X +'p -U +q av where p' and q' are obtained from (65) by replacing 01 and 02 by O and 0'. From equations analogous to (I, 68), we find Dx' x ax' -ax a.t u' v av' where a 02, a 01, 2 O, Oh a — a02 - a0. aD0 a 02 a, 0, O, a02 O1 av 02 Dv 1 au -02 a u Consequently N' is parallel to N. 54 II. Transformations F The functions 03 and 0O defined by - a a0 a03 aoa, ao 0= 03 + a av ' 03= 03 9+ au av are corresponding solutions of the point equations of N and N'. Hence quantities of the form - _,03 X - X of are the coordinates of an F transform of N. When the above expressions are substituted in this quantity, it is reducible to (70). Since 06 and 02, as given by (68) involve additive arbitrary constants, there are 0o2 nets N3. 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 o 2 nets N3, each of which is a derived net of N3 and an F transform of N. 23. Derivant net and derived net of two transformations F. We note that the corresponding points of six nets in the relation of the theorem of permulability are coplanar. From the second theorem of ~ 11 it follows that these planes envelope a derivant net N of 1V. Since the four congruences of the configuration are harmonic to IN, 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 ra 62 - 2 61 + 2 L A (71) -T' 0o of 1 0102.2 ~1 X o02- - +, x - q o From ~ 17 it follows that corresponding tangent planes of N, Ni,1 and N2, meet in the point which generates the derived net N of N determined by the solutions 01 and 02 of the point equation of N. We shall show that the corresponding tangent planes of the o2 nets N12 pass through this point. In fact from (I, 66) it follows that the coordinates x of this point may be given the form A i 02\ a J i 02\ ax _Xx + (1) a I 0 A1 'a A1 Lats \ - l02 av \6 v \o;J 1u o, 24. The extended theorem of permutability 55 Consider now the net N1,1 and look upon N and a net Nja2 as F transforms of it by means of 07' - 0j/0{ and 012. The corresponding formula for this case is _ _ _ _ _X _0 \ X I e (- In: a a (012 o1iax X1~~~~ ~~~ a U -01.u Bs v 1 au i Xj- 0 a e: a a e0, au 0 — -c 8 1 av 0 10 By means of (9), (20) and (37) this is reducible to the preceding form. Hence: Thle corresponding tangent planes of N 1, N:,2, 2 and the 2 nets N12 meet in the lpoint which generates the; derived net of N by means of the functions 01 and 02 Of the transformations of N into Nj,1 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 N,, N2, and, N3:be these transforms of NV by means of the functions Oi and 'wj (i 1, 2, 3), where of off0~j __ off/ ___ Of~l 2-,12 21 -02 011 ' 021' 01' 09.1 012 Oagl Applying the theorem of permutability to the three pairs of these nets, we get three families of nets N12, N23 and Npn, since N2) Ni. From (48) and (49) we have (72) Oi Oij wjj - wtj (Oj qjOi 4- Oi Oij - Oij Oj ), 1i Oij wij Xjj - 1,oj (0j Ojixi 4 0i Oij Xj Oj Ojx) (i 1 i). Since a net N12 and a net N1u are transforms of A',, there exist cc nets N for each of which N,, N12, NV1 and N form a quatern. It is our purpose to show that one of these nets N is such that Nj,2 N12, N28, N form a quatern; and likewise N8, N18, N~8 and N. We denote by Oij and ~iij the functions by means of which Nij is transformed into N; from their definition it follows that -ij -Opji and -ivj - wpj. According as we look upon N as belonging to the one or the other of the quaterns, N1, iV12, N18, N; IjV, NV12, N8, N, the coordinates Y of NI are given by the respective equations 56 5 I. Transformations F 0 012 012 W12X W18 (013 013X12 4 012 012 X13 - 012 013x1), ) 021211 W21 X W23 (023 023 X12 + 021 021 X23 - 021 023 X2), which are analogous to (72). When we equate these two values for x, the resulting equatiol is reducible by means of (72) to an equation of the form Ax, ~ Bx2, + Cx 0. Since A, B and C are necessarily equal to zero, we have the following equations of condition: 2012 (01W13 13- 033 012031) 0- 01W3 01302 Wx 2 021013 02 WV23 02 3 Oi W2 023 0 - ~021 032 - 029101 O _ 01 W12 O201.2 iV23023 8)- 02 W23 023 Oi W2 02 3~ 0-1 V13 01 2tV \2W23233 ~~~02301 - - 018 021 O1 l12 012 (Oi W3 032-02 u3 031)- 01 W13 013 02 0 + 02 W23 023 OiW2 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 f Oi 012 W12 012 - 102 (02 021 013 + 01 012 023- 03 012 021), (74) ~ 01 013 W13 013 w 3 (03 031 012 ~ O1 013 032- 02 013 031), 02 023 W23 023 W3 (01 032 021 -- 02 023 031- Oi 023 032). From (38) and (44) it follows that (75) 1V1 W12 01012 W2 W21O02 021, so that the above equations are consistent with the requirement that Oij O0ji. When equations (74) are compared with the second of (72), it is found that Oij is a solution of the point equation of.Nj, which is a necessary condition. The analogue of the first of (72) is 012 012 W012 - W13 (013 013 + 012 012 01.2 013). Substituting in the right-hand member the expressions for 012 and 60: from (741), we have 25. Transformations K 57 01 012?Vl2 012 W1 2,- 0 0=12 12 0-2 — 2 _- 01 (012 023 + 013 032- 032 023) W+ 02 (023 031 + 021 13 013 031) + 03 (031 012 + 032 021 - 021 01 2) — 012 023 031- 021013 032, which in consequence of (75) is consistent with the requirement that wiij= — wji. By means of these results equations (73) reduce to D =- x1 (02 023 031+ 03 032 021- 01 023 032) ~(76) 1 + ~- 0 (03 031 012- 01013 032 — 02 031 013) -+ X3 (01 012 023+ 02 021 013 - 03 012 021) — x (012 023 031+ 021 013 032). Since this expression is symmetrical in the functions involved, it follows that N defined by (76) forms a quatern with N2, N12 and V28 and also with N3, N13 and N23. 25. Transformations K. We inquire under what conditions a net N and a transform N1 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'- (h+ ) 0. When we require that this function satisfy (5), the resulting equations are reducible, in consequence of (I, 13, 18), to (8 a lg 0 0 logb 0 a log (78' log - = 2 ~,loga z T au av aP av From this it follows that the point equation of N is necessarily of the form (79) a2o a- og a a logl/V a aztav av au au av' where o is defined by (80) e -- 20. -58 II. Transformations F Then from (I, 18) and (77) we have (81) h 00 + ' 0 '. 9Q 0 9 0 Now the equations (5) may by written a _ 1 ao a I ao (82) DD (0) 1 ao (6) 1 au \ 0 a u' av 0 a a v' From (8) it follows that 02 (83) r= — — o Hence equations (10) become ax _ 1 0 0 _ a auzt e ~ 6) au M + "a It (84) aDx, 1 0 aD 80 ax\ av e o' (x-) a- + ) av and the point equation of N1 is (85) D2e01 a log V1 1, a logV aO1 Duav aDv aD& r Di v ' where (86) -e o' We note that the invariants H and K 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 N1 meet the lines of the conjugate congruence in points harmonic to the focal points, it is necessary that both N and iV 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 transformation IK. 17) 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 associate 1). In all generality equations (84) are in this case ax - 1 ax axl 1 ax (87) au -- av au a It' a v Q av Suppose now that N has equal point invariants. The knowledge of a parallel net N' gives h and 1, and consequently a solution y* of the adjoint (I, 20) of the point equation which in this case is reducible to (88) a2 __ _auav^ ( )* Since equation (79) can be written (89) 2 0 \ av_ ( a2 / V \ a solution of this equation is given by (80). It is readily shown that 0' given by (77) satisfies (5), and thus we have: When N has equal point invariants, each parallel net N' determines without quadrature a transformation K into a net N1, the function 0 of the transformation having the value ' Qe/2, 0' being given by (77). 2 In particular, fp - is a solution of (88). In this case = 1 from (80) and consequently -- is associate to N. Suppose conversely that we have a solution 0 of the point equation. From (88) and (89) it follows that s =2 0/Q is a solution of the adjoint equation. From ~ 4 we know that by means of T( we find oo1 nets N' parallel to N, of the form x'+ kx, where x' are the coordinates of one of these nets and k is a constant. Accordingly we have the theorem: When N has equal point invariants, each solution of the point eqzuation 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 II. Transformations F such that corresponding lines of the conjuqate congruences of the transformations which pass through a point of N are coplanar. 26. Theorem of permutability of transformations K. If N1 and N2 are two nets, each in the relation of a transformation K with a given net N, we apply the theorem of permutability of general transformations F, and seek the nets N12, which are K transforms of N1 and N2. For this case we must have r12- 2, 21 -2 2121 Since (90) 1=- - =L= -, 62 these conditions are equivalent, in consequence of (52), to (91) 021 + 12 - 0. ~2 01 In consequence of (90) equations (37) for the case of transformations K are reducible to a e[ 1 a 0 a 02 (2| (012 00) 1 2 / 01 __ (92) au o o a u o- - ~a O[ _ 1. ^i a \ a2 v (012 0j) =-e(02 eV-01 ). aV o,, - a -v av From these equations and similar ones in 021 we find that the lefthand member of (91) is necessarily constant. Since each term of this expression is determined to within an additive constant, there are oo sets of solutions satisfying (91). In fact, in consequence of (36) and (43) we can put (91) in the form (93) ' - lo, a+ - ~o. 02 01 From (52), (86) and (93) it follows that 9ll2- 2 012- 2 021 9)12- -, 2q21i- 2 2. el e2 Hence each N12 for which (93) is satisfied is a K transform of N1 and N2. 27. Transformations F of applicable nets 61 Now equations (48) and (49) become 06X12 012 0102 —02 012,1+ 2 ( 2 0 1 f 0 By means of these results the coordinates (71) of the point of contact Mof the plane of the quatern with its envelope are reducible to A 0 21 X2,2-022 X1,1 _ 1201 X1201012 X - f f _ 1' 2 o 2 02 l 0 — 2102 12 01 — 0e012 Hence M is the intersection of the lines MM1 and M1Mf2; consequently the points M12 of the oo nets lie on a line, MM12. Therefore in consequence of the theorem of permutability of general transformations F (~~ 21, 23), we have the following theorem of permutability of transformations K: If NJ and N2 are K transforms of a net N with equal point invariants, there can be found by a quadrature ool nets N12 which are K transforms of N1 and N2; corresponding points M12 of these nets Ni2 lie on a line I through the corresponding point AM of N and in the plane rc determined by Ml and the corresponding points Mfi and Ml2 of Ni and N2; the plane Tr touches its envelope at the intersection of I and the line M1M2; the parametric lines on the envelope form a net to which are harmonic the congruences generated by the lines IMM1, MMI2, MiMi12, M2M12, and the tangents to this net are harmonic to I and M1M2. 27. Transformations F of applicable nets. We say that a net N in n-space and a net N in p-space are applicable, when their coordinates x and x satisfy the condition dx2 -d2 1 1 which is equivalent to a )2rax ( N ) 2 ax ax __ax ax 1au a u b au av \au av' G ax( D)2 (av) 62 II. Transformations 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 (I, 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 Nto a transformation F determined by a parallel net N' and a solution 0 of the point equation of N. From (9) it follows that the fundamental coefficients of the new net N, are given by El =,2 (X a. \ 2 a M au + of2 -= a )2 0x', a '-' au an au F, o (X aoao a,,tax oao,,ax aoO+o, F) 014 aua av an av av au f4 x'" 2 a av a We transform N by means of the net N', applicable to N', and the same function 0 used above. In order that N1 and N1 shall be applicable, the expressions for the fundamental coefficients for N1 must be equal to the above. Equating the corresponding expressions, we get three equations, which in fact are equivalent to the two,(y2^2), ao - 2( x a_)' o ', a _ ax _x ax (y 2 y/2) a - 2 (x' avx -1 a a' x ^ v ~ xv ax By integration we find, to within a negligible constant factor, (95) o-ix2 - ' 2 It is readily shown that this function is a solution of the common point equation of the applicable nets AN' and N'. Consequently the F transforms of NA and N by means of 0' and 0 given by the quadrature 28. Nets corresponding with orthogonality of linear elements 63 ae/ a x _, Dxx ae a _2\zx a, Be 8- 6 ) aV = 2 (x a -a >,xau ayi x Du Dk v a v ) are applicable. Hence19): If N and N are applicable nets, each net parallel to 1V determines a pair of applicable nets N2 and 1V which are respectively F transforms of 1V and N. We shall establish a theorem of permutability of these transformations. Let two applicable nets N and N be transformed into pairs of applicable nets N,, N1 and N2, N12 be means of parallel nets N', TN' and N", N" and functions 01 and 02 given by (96) [01 = 2 x- l2, 2 =zx 2 -1fy2 We apply the results of ~ 21 to this case and seek whether NT1 and N1V2 are applicable. To this end we take ro"f' B 2p42 oW' -'2 (" _ 2 /12 ^12 ^ ^ — ^ 1 J ^21 — 21 2 - X2 Substituting the values of these functions from ~~ 20, 21, we get (97) O' + o2 - 2 (x' " - ' x") = 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 Ot' and Of involve additive arbitrary constants. Hence they can be chosen in an infinity of ways to satisfy (97), and we have: Of the o2 transforms N2 and N12, 00 pairs are applicable, these cases arising when the constants in 0' and Of 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] (98 ao Dx a x Dx ax a x - Z ax-.x ) Transactios, vo. 19 (1918) p. 17v0. 19) Transactions, vol. 19 (1918) p. 170. 64 II. Transformations F Let (6) be the point equation of N(x) and let 82 Dloga ad aDlogb aD Duav Dv au au av 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) ah - h) loa n - (/) a log b (99) v ^r - v) Dv ( v a u - u the nets N'(x') and N'(Y'), whose coordinates are given by a x' ax ax' ax ax' ax x' a x (100) -h 1 - D D, =D D - av a u av a, av a are parallel to N and N respectively. Moreover, from (98) it follows that N' and N' are in relation 0. If we say that N' and N' are corresponding parallel nets of N and N, we have: If N and N are two nets in relation 0, correspondin.g parallel nets are in relation 0. We seek transformations F of nets N and N in relation 0 into nets NV and N, in relation 0. Let N' and N' be two corresponding parallel nets of N and N, and let 0 and O' be corresponding solutions of the point equations of N and N' respectively, that is ao' ao ao ' ao (101) =h ='t' vt au a ' a V v The equations of the transformations are (102) x = -V - xi x —ex' 28. Nets corresponding with orthogonality of linear elements 65 From the first we have by differentiation equations (9) and similar 8x1 8x1 expressions for - and --. 8 Iu av When we express the condition that these quantities satisfy equations of the form (98), the resulting equations give, to within a negligible constant factor, (103) a x' ' Since V' 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 N'(x') are two corresponding _parallel nets, the nets NV1 and N1 which are FE transforms of N and N respectively by means of the equations (104) x =x — - x, x -, - C^f — ' X ' ~X where 0 is given by 0o 1 a a 0 1 8 ' (105) I8'^ -h - - DvA m a u h au't av I av are in relation 0. We shall establish a theorem of permutability of these transformations. Let two nets N and NV in relation 0 be transformed into pairs of nets iN1, N1 and NT2, N1 by means of parallel nets N', N' and N", N" and the functions O' and 02 given by (106) 01 - '', 02 '- xT ". We apply the results of ~ 21 to this case and seek under what condition Nl2 and 1T2 are in relation O. For this to be the case we must have Or' 1x"'i~~", 01f - M"""-"" ^12 -- 1 1 1 21 — 2 X2: where x" and x2"' are given by (33) and (42) and,x" and x-" have analogous expressions. Substituting the values of 0'12 and 01" from ~ 21, we have (107) e+ 2e ---(x"x'-xx ' ") - 0o. 66 II. Transformations F In consequence of (106) we 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 0t' 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 Co2 F transfbrms N2 and N12 arising from the theorem of permuttability of any nets cand N in relation 0, there are o1 pairs in relation 0; they arise when the constants in 06 and 1' are chosen so that (107) is satisfied; all these oo1 pairs of transforms can be found by two quadratures, when N1, N1 and N2, N,2 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 N1 is the F transform defined by (2), and (N)1, (N1)1 and (N')1 are the first Laplace transforms of 1V, Ni and N' respectively, then (X) ( - (x)- (- ) where I a0 1 -o' (O) 0- l (0')% - 0'aloga av' a log a, av av av 4. If N1 is an F transform of a net NAT, any Laplace transform of N admits as F transform the corresponding Laplace transform of N1I, the equations of the transformation being (0C) c where (0)r, (0')r. and (x')r are the rth Laplace transforms of 0, 0' and x'. 5. If N and N1 are in relation F, and o and <p are corresponding solutions of their point equations, the Levy transforms of N and N1 on the tangents to the curves v = const. of N and N, are in relation F, and also the Levy transform 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 y conjugate to the congruence of the transformation from N into Nl. 6. By means of Ex. 5 and ~ 11 show that if N and N1 are nets in relation F, and,,1 and 0, (p are corresponding pairs df solutions of the point equations Exercises 67 of N and Nl, the derived net of N by g and,' is in relation F with the derived net of N1 by V1 and '1. 7. If NT and NA are nets in relation F, and AN2 and N12 are parallel to N and N1Y respectively in accordance with the second theorem of ~ 21, corresponding lines of the congruences of parameters x2 and Xz1 conjugate to N and N1 meet in a point describing a net forming a triad with N and N1. 8. Show that if in ~ 24 we take 03 = 1, 0 =- 0. 0, the nets Ns, N13, 823, N are parallel to N, NA, N2, Nl2 respectively. 9. If N, N, J, N12 are nets of a quatern, the respective functions 02, 02a, 02/012, - 012/012 determine radial transforms N, N2V, 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 I == - cO, where the c's are constants define a net Ni which is a transform of N, the congruence of the transformation consisting of parallel lines. The tangent planes to N and N1 meet in the harmonic congruence of N determined by 0. 11. If N and N are applicable nets, the equations x =- x - cO, ~x = -- ~c determine applicable nets N1 and Ni, if 0 2 (c X~ — -c)/(~c2 — c2). 12. When the functions a and b in equation (6) satisfy the condition a82a 2b a bi aa 8b 0) ca b -- _ a i) u av a -v \v b a a8v 8u' the equations 8log7 _p 1 b 8log iy/ _ 1 Da 8u a 8u 8v b 3 are consistent, and the functions (ii) h7,1 - -~ l b i 1 a a ' - ' ~ pb satisfy equations (I, 13). The two nets N and No parallel 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 i 1 0' a — a 80 solution 0' (a2 - b2). If we put 0 = - /p(a - b), we have -- a _ 80' b U 8- === -a~ - These values satisfy (81), and consequently 0 and 0' and the av a7; congruence conjugate to N whose direction-parameters are the coordinates x' of N' determine a K transform of N. 14. Let iM,, M2, M, M2 be corresponding points of four nets N, AN1, Ns, N12 of a quatern under transformations F. Show that a necessary and sufficient condition that another net Ns '.obtained by varying the additive constants of 0" and 02 be such that its points lie on the corresponding lines MM112 is that the transformations be JK. 68 II. Transformations F 15. If N is a net with equal point invariants, NIV, AN, NA are K transforms of N, and IV12, 113, N23 are the nets with equal point invariants which together with the respective groupsA, NV, N, N2; N, N1, NT3; IN, N1T, No form quaterns under transformations IK, then IN defined by (76) is a K transform of T12, N113, N23. Transactions, vol. 16 (1915), p. 296. 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 N with the point equation (6) and determined by solutions 0 and 01 of (6) be K transforms of one another, it is necessary that 00,1 b2VT, where V is a function of v alone. 18. If N and N1 are two nets in relation K, their respective associates can be so placed in space that they shall be in relation K. 19. If in (46) 0' and 0' are replaced by 0' + c and i' - c, where c is a parameter, the corresponding points of the ool nets N12 lie on a conic which passes through the corresponding points of N, N1 and N2; 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 x',... xn?, the n + 1 quantities y, of which yn"+'l 0, satisfying the conditions y+ - "1.. y + 1 are called homogeneous cooTrdinates 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) aly lt.... - - a - y'-1y 0. Now y — 0 ( 1,...., n) is the equation of a coordinate hyperplane. Also 1-n-l- 0 is taken as the equation of a hyperplane, namely the hyperplane at infiniity. This hyperplane likewise is a coordinate hyperplane in homogeneous coordinates. Thus we have n+ 1 hyperplanes forming a coordinate (n +l)-hedron. Moreover, a point all of whose homogeneous coordinates save one, say y4, are zero lies in all of these hyperplanes except the hyperplane y — O; it is a vertex of the coordinate (n+lj)-hedron. Suppose now that we have any n +1 hyperplanes, say,y+.... * +a l-yV-< 0 (i.., n i), subject to the single condition that they do not have a point in common, that is, the determinant of the a's is not equal to zero; thus a [: 0. If we put e^=,l --....+ a1-l'- y — (i 1...,,n + 1), 70 III. Sequences of Laplace where Q is a factor independent of i, the quantities zi 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 2's general homogeneous point coordinates, and as such they are distinguished from the particular system of y'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 (n + 1)-hedron, but is defined when it is required that a particular point, not on the faces of the new (n+ 1)-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 linear transformations Po by means of which from a general system of homogeneous coordinates z we pass to a system y so that yl/y7+1... yn/y.-. 1 are cartesian coordinates. Hereafter we 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 P1 (,i) and P~ (z2) are of the forv, and conversely. Consider the point with these coordinates and apply a transformation Po (referred to above). The resulting expressions will be of the form y1+-tPy2. Hence the cartesian coordinates of the point are ) i't i i i ' i'y +~yr. yll + 1 ^y,y *. + 1I which shows that the point lies on P (cf. I, Ex. 1). Evidently which shows that the point lies on P, Pg (cf. I, Ex. 1). Evidently the converse also is true. In like manner it can be shown (cf. I, Ex. 2) that: The homogeneous coordinates of any point on the plane through three points P1 (a'), P2 (z2) P3 (3) care of the Jorm 29. Homogeneous point coordinates 71 Z1 + 12 + I 3g2 and conversely. From ~ 1 the locus of a point whose homogeneous coordinates z are functions of a parameter ie is a curve. We wish to prove: The homogeneous coordinates of any point on the tangent to a curve z (t) are of the form d+ du and conversely. If the coordinates of a point are in this form, and we apply a transformation Po we get for the new coordinates of the point ~y + P -dY. Hence the cartesian coordinates of the point are dyu d x du 1 + ' dtu_ du i d^-i - i -[ - ^ -+ ^ l o g 0 ' t + As these are of the form x + d, the point lies on the tangent (~ 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 ciz in the form 2z + - d. Two functions Q and t are defined by du' dt eQ - 1, et=,,.t d 'a In terms of e and t the coordinates of the point are of the form e -- (tz). Hence we have: 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 formdu The osculating plane of a curve z(u) at a representative point is by definition the locus of points whose coordinates are of the form 72 III. Sequences of Laplace z + - -+ v -- 2z If we apply a transformation Po to the du du2 ' coordinates and proceed as above, we can show that the cartesian coordinates of any point of the osculating plane are of the form +K Cv Jt i aX ~X dx ~ ct2X cdum du"2 We observe that this is the result previously found for 3-space [~ 7]. 30. Laplace transformations. We have seen in ~ 2 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 N with n + 1 homogeneous coordinates x. The coordinates of any point P on the tangent to a curve v = const. ax are of the form xx +-, - * As v varies, the point moves in the tangent plane, provided - (x + q - x) is expressible linearly in ax ax terms of x, a- and v. Hence a net is characterised analytia u av cally by the condition that its homogeneous coordinates are solutions of the same equation of the form 1) a 2 0 aloga aO alogb ao aua v av a u aut av 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 transform of N, we must have a logb It O_ = 0. a u Since similar results follow for the first Laplace transform, we have: The coordinates of the minus first and first Laplace transjbrms of a net with the point equation (1) can be taken in the form(2)._ ax alogb 1_ ax aloga 1(2) I:-1- Iix x — - ~. (2)/ x1- am ~u av av 31. Sequences of Laplace 73 In consequence of the fourth theorem of ~ 29 the equations of a Laplace transformation can be put in the simple form indicated in the following theorem: The homogeneotus coordinates ~i and yi of the focal points of a congruence can be chosen so that a^ zv (3) - y, -a- ' z. a I. a v From (2) we have by differentiation (4) a l x _ a log a K a x a log b av av lau u X7 where H and K are the invariants of (1) namely H _ a2loga + aloga alogb +, ua a av Du (5) a _ 2log b oa loga alogb c ua u av D v a c. If K i 0, we get on integration x-1- 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 M-1 describes a curve, and not a net; we say that AT-1 is degenerate. Substituting in the first of (2) and integrating the resulting equation, we find (6) x (b V+j d Ub V being an arbitrary function of v alone. In like manner, if H= 0, the integral of (1) can be given the form (7) x a (Uia + dv). Hence we have the theorem: When either invariant of an equation of Lacqlace is equal to zero, the 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 III. 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 A', — ),.... -1, A, N....,.... 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) [o0; ai, bi, e,] the equation a 2i alogai a0i a alogbi ao0 a uv av au at a v (i 82 10. alogai alogb.+ aloga alogb + \ ]+ — Sav- log ei v -- - at Oi 't a 1 V v Qut a V a u Then from (5) it follows that (10) eH a log a alog b Hi logei_ + c, { u I v ai 8v a u (10) = a 2 log e o + aloga alogb + Kj + Clog.-t- +c, a u av b - a ' a u If in equation (1) we put (11) sx where i is a function of u and v, the Laplace equation satisfied by x' is denoted by (12) [0'; ac, b ], ]. 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 N1, we differentiate the second of (4) with respect to v. Making use of (2), we find that the Laplace equation of N1 is denoted by (13) [01; aH,,-] and that (14) a —2 log al ~-H. (14) H, H- K H=R. auav b Proceeding in a like manner with the first of (4), we find that the Laplace equation of N-i is denoted by (15) [B0i;a, bK,-] and that __ _____ bK a (16) MYTH-, W K-1 KJ- d a log The coordinates x2 of N1 are given by the equations (17) x ax --- - x log aH av dv analogous to (2), and the point of equation of N, is denoted by (18) [02; arHHl, b, a2] In general the coordinates of Nid are given by (19) X - a -x._ a log(aHH,.... H-2) Xr-l, dv dv and the point equation is (20) 0l.; aHH'.... H-l b,,aH-. -r- 2.. L~~~~~~~a H" - I -tL 1 M-2 - 76 76 ~~~~~III. Sequences of Laplace The equations analogous to (4) and (17) are ax X__ a logb (21)ax a a' alog (a H. H...H 1) Xr+,xr+1-L. The coordinates of N-r are given by (2 2) x~ - log (bKK1....K-r —2) X-rI,'!'1 and the point equation is (23) ar b K-K- i K-I-r~i L ~~~~~~~b K9 ij.K-r~2] Also we have (24) jax- a logct-+ -I-lX-,-j 32. Periodic sequences of Laplace. Ordinarily asequence of Laplace is unlimited i-n both directions. If H,, 'or K-,. is zero, the sequence terminates in the positive or negative sense (cf. III, Ex. 5). In the present section we are concerned with the case when the sequence is j)ecriodic, that is when a certain net NV~ coincides with IN. In this case we must have (25) -XJ litx, where m is at most a function of 'a and v which is the same for all n coordinates xi Since nt is at least equal to 3, the coefficients of. (12) and (20) for r- - must be equal. Hence we must have (26) -alo H.... HP1 __ a o i -0 (27) log V0. anta V aP HP HIP.. HI 32. Periodic sequences of Laplace 77 Diff erentiating the first of (26) with respect to ua and makingz use of the second, we find that where U and V are functions of ua and v respectively. If we change the independent variables in accord ance with the equations 'it- ~(U1), v V) we find that the invariants H', Hi',... H, of the transforme d equations of the nets N, N1. Np are given by H' p (Wa) ~P' (') H, H~ (W ~u) t/1' (v')HiL (i 1... 0 where cp' and tp' denote the derivates of (p and V.). Hence qp and f can be chosen so that (28) becomes Then from (26) it follows that rn in (25) is constant 20). Suppose now that (27) and (29) are satisfied for equation (1). Applying (1 0) to (20) for r - p-i1, we get, in consequence of (27), 82 a1lH H 2.....~~ HP - ~ L~vlog (30) 82 b In like manner, making use of (27), (29) and (30), we obtain V2_log a-2Hp-2.... H- -~~log 2 K- log -K1._ auav a 2altv a 20) Cf. Tzitzeica, Comiptes Rendus, vol. 157 (1913), p. 9108; also Hammond, Annals of Mathematics, ser. 2, vol. 22 (1921), p. 245. 78 III. Sequences of Laplace Hence in general we have (31) HP- i = —K-i+1. If we differentiate (25) with respect to u, and make use of (2) and (21), we can reduce the resulting equation to (32) p- p- - mx_ which gives the analytical form of the condition that the nets N-,_ and N-1 coincide. Again differentiating (32) with respect to 1u, we get, in consequence of (21), (24), (30) and (32), Hp-iH1p-2 Xp-2 - ntx-2. And in general because of (31) we have (33) Hp- Hlp-2.... Hp-i Xp- i M= x-i, which in consequence of (29) is equivalent to (34) xp-i = 3HE1 ^.... Hp_ i —1 i. Finally we have (35) x -- mx-p, showing that N and ~Tp 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 i by 1. In order to point out the significance of this observation, we note that the expressions (19) and (22) for x, and x-, are expressible in the forms arx ar-1x ax. r, s^-1 -1 + * ~ * * +r, o..Xr -- -t + ABr-,..... +-, o, +Br- r- +.i+ BroX 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,.... i + 1, which necessarily is (33) with i replaced by i —1. Suppose now that we consider a periodic net of odd order, and write - 2n + 1. If in (34) we put i = - and in (33) i -- + 1, we get IXn, + 1 - t m HH.... Hn X-n, (36) X-91n1 - H 'll H2ZH n-l.... HnxA. an-1x 3-: 1 an1x These two equations express - and - + 1linearly in terms of the 2 n + 1 quantities a(37x a x alx ax (37) 7 a '.....' at I a' 2v *' I' v ' ' I 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 mi there exist p independent solutions of (1), (33) and (34) including the condition (25); and not more than p independent solutions. When p is even, and we put p — 2 n, we have in place of (36), X2$, = mHHff.... Hn -l X-, X — — 1- H2n-l.. Hnz-1 Xn-1. Then all the derivatives of order n and higher are expressible linearly in terms of the p quantities anx Dx an-'x ax a t ' a' Duvn 1 ' v x 80 III. Sequences of Laplace Hence we have the theorem: WThen 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 homnoeneous coordinates of the focal points of any harmonic conax -ax gruence are - and - respectively. a dv a Let N(x) be a net with homogeneous coordinates x satisfying (1). Let F1 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 ax of F, are - (~ 29). Then the coordinates of F2 according as it is looked upon as on the tangent to the curves u const. at iMl or on the tangent to the curves u = const. at F1 are of the respective ax dx d2x forms )ix + ju -av a - - -—.a These forms must be proportional to one another in consequence of x being a solution of the corresponding equation (1). Expressing this requirement, we find that both of these must be proportional to - -gb x- + ex. la vg Evidently - lgb 0, otherwise F2 and M coincide. Looking upon d u F1 as the minus first Laplace transform of F2 we must have ax /alogb -x \ a /alogb adx + cx a tt au a v a ' an a v As this equation must be satisfied identically, we must have either c -O, or T' -g - Vc, where V is a function of v alone and V' its derivative with respect to v. In the latter case the coordinates of F2 can be chosen of the form (Vx). Replacing Vx by x, 8x which does not change the form of the coordinates of F, we have the result stated in the theorem. 34. Levy sequences of the first order 81 Furthermore, when the coordinates of F1 and F2 are of the ax ax form - and, in the point equation of N we have c = 0. au av' If we put 0 = -0 0 where 0o is any function of u and v, in equation (1), the function 01 satisfies an equation of the type (1), and for this new equation c = 0 in case 0o 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 congruence harmonic to a net N(x) are expressible in the form (38) 'u0 ~] v v in which case 0 is a solution of the point equation of N. We have also the converse theorem: If 0 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 -- -lo a * \\ _- + log b ' --- x\ aav au \0 av 0 au (e au O a v (e)' which evidently are the parameters of the line joining the two points; similarly when u varies. We have also the theorem: When 0 =-aixi, where the a's are constants, the points of coordinates (38) lie in the hyperplane aixi = 0. 34. Levy sequences of the first order. If 0 is any solution of the point equation (1) of a net N, from (2) it follows that the functions 0 ao a logb 0 ao a loga0 (39)au au ' 0'av av are solutions of the point equations of N-1 and N, respectively. We call them the minus first and first Laplace transforms of 0. 6 82 III. Sequences of Laplace The points of coordinates (38) are the Levy transforms of N by means of 0 (cf. I, 53). In consequence of (2) and (39), we can take as homogeneous coordinates of these respective points 0 01 (40) X-1,1= --- X-1, Xo,1 0= X1-X. Moreover, the net of coordinates Xo,i is the first Laplace transform of the net of coordinates x-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) (41) [0-,1; — 1 -- (42) [Oo,; a,0 b, From the form of (40) it follows that N-_,1 and No,1 are Levy transforms of N-1 and N1 by means of 0_- and 01. Moreover, from ~ 10 we have that the tangents to the curves of parameter v of No,i are harmonic to N1, and consequently this harmonic congruence Gi1 is determined by the solution 01 of the point equation of N1. Its focal point of the first order generates a Levy transform N1,1 of N1 whose coordinates are given (analogously to (40)) by (43) X1, 1 X -- 2 X, where 02 is the second Laplace transform of 0. In like manner the function 0_- determines a congruence G-1 harmonic to N-1 whose focal nets are N-1,1 and N-2,1, where 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,i for positive and negative values of r are of the form 35. Levy sequences of higher order. Derived sequences 83 (45) Xr,l = r+1 - r+ r Or where 0r is the rth Laplace transform of 0. 35. Levy sequences of higher order. Derived sequences. 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, 1. These nets are the focal nets of two sequences of congruences Gr and Glr 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 Gr 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,1 and consequently we call them Levy sequences of the second order. We shall obtain the analytical expressions for their coordinates. H We consider first the derived net of N by means of 0 and 0'. The functions 0-1,1 and o0,1 defined by (46) 0-1,1= I '-1 o0,I o- ', 0-1 0 where OL and 0a are the minus first and first Laplace transforms of 0', are evidently corresponding solutions of N-1,1 and No,l, which as we have seen are Laplace transforms of one another. Hence the coordinates of the Levy transform of N-1_, by means of 0-1,1 are the form 00,1 (47) X-1,2 o0, - - - X-,1. 0-1,1 In consequence of (40) and (46) this is equivalent to 0-1 OL'1 xX-1,2= 0 0 ' x 1 01 0' x 1, 0-1 0-1,1 0-1 0 —1,1 01 01 XI if we make use of the following lemma concerning determinants: 6* 84 III. Sequences of Laplace If D I al 11 (1, m,.... n) is any determinant of the nth order, and we write ar +,s ar+ ls- a, Ar, ar, 1 then A1,2 Al,3 8..... Al, n (48) D =all A2,2 A2,3..... A2,n (48) D- = a _ __ An-1,2,......... Anl, Hence the coordinates of N-1,2 can be taken in the form (49) x-1, 2- 0-1 o' x From the symmetry of this expression it follows that N_1,2 is also a Levy transform of N-x,1, which shows that it is the derived net of N for the functions 0 and 0'. Evidently the derived net of Ni for 01 and 01 and of VA- for 0- and '0L are given by (50). XO,2 0 01 x, X- 22 -= 0-2 0 -i x Since N-1,2 and No,2 are the Levy transforms of No,1 determined by 00,1 they are Laplace transforms of one another; similarly N-2,2 and N-1,2 are the Levy transforms of N-1,1 determined by 0-i,1, Hence the solutions 0 and 0' determine a Laplace sequence of nets Nr,2 which are derived nets of the nets Nr+1, and a first Levy sequence of the nets NW, i. The coordinates of Nr, 2 are of the form (51) Xr,2- | Or 0 - 1r +2 2^ for r positive and negative, and where No= N. If 0" is another solution of (1) independent 0, 0' and the coordinates of N, the functions of ~ ef-e -0 o/0 O'- 0'_ 1, O 0"- 0 { —1 ' —1 solutions of the point equation of N1-1,, determine a derived net of N-1,1. Analogously to (49) we have that the coordinates of this net are of the form I 0'-2,1 0"-1,1 XO 1. 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) X-2,3 10-2 t -i 0" x1 Hence the functions 0, 0' and 0" determine a sequence of Laplace whose coordinates are of the form (53) Xr,3 O= r Or+l 0r+2 Xr+31. We remark that from the symmetry of (52) it follows that the net N-2,8 is a derived net also of the Levy transforms NL'1, and N~ 1, 1 determined by 0' and 0" respectively. In general m independent solutions 0, 0',.... 0(w-1) of (1) determine a Levy sequence of order m whose coordinates are of the form (54) Xr,n -- |Or O0r+l 2.... or+.m-l1, Xr-m, for positive and negative values of r21). 36. Periodic Levy sequences. Suppose that we have a Laplace sequence of periodp. We seek under what condition the Levy sequence of the first order determined by a solution 0 of (1) is also of period p. It is necessary that xp, -= ixo,, where ) is at most a function of u and v. In consequence of (40) and (45) this condition is equivalent to (55) x+ x (x- x op \ u x From (19), (25) and (29) we have (56) X3p0- = mxi aOp a ---p log a p. Consequently from (55) it follows that Consequently from (55) it follows that;~- ~'a-;; -7 -o 21) Cf. Hammond, 1. c., p. 252. 86 III. Sequences of Laplace In like manner the condition xp-, 1= px-i,1 is equivalent in consequence of (21) and (32) to a Op ~ t=m, -log - 0. Hence we must have (57) Op ml 0, where mi is a constant. We have seen that equation (1) admits solutions of this type. If we have such a solution, the Laplace transforms of 0 satisfy equations (33) and (34) with m replaced by mi. Hence we have (58) Xp-i,i Xp —i+l —p- xp-i mH.... Hp- x-i, i. Op-i Therefore: If N is a net of period p in (p-l)-space such that xp = mx and 0 is a solution of the point equation of N such that Op - mi 0, where mn is a constant, the Levy sequence determined by 0 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 0 and 0' of (1) satisfying the conditions Op = m1 0, 0p= m2 0', the first Levy sequences determined by 0 and 0' are periodic. Also in consequence of (56) and (33) we have from (51) Xp,2 -= nmm t2 Xo, 2 Xp-i,2 mMl (HH1.... p-i-i1)3 He iHp-i+l X-i,2 Consequently the second Levy sequence determined by 0 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',.... 0P-1) of (1) for which (i) = m O ()(i - O,....p —l). From (54) we have Xop = 0 01 '..... (Pi - X) x = (m-m)l 0 0.. 0 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 - )-space, such that Xp mx and 0, 0', ",.... O(p-l) are p independent solutions of the point equation of N such that 0(i) = m1 0) (i - o,.... p-1), the pth Levy sequence coincides with the given sequence2r). 37. Transformations F in homogeneous coordinates. When two nets N and Nr are in the relation F, the tangents to the curves v- const. at corresponding points M and Mf1 meet in the focal points of a congruence harmonic to both N and N1 (~ 17). In accordance with the first theorem of ~ 33 the coordinates of N and N1 can be chosen so that we have ax,. ax ax _ ax (59) ~^ TO- 60 ---" au(9) a-T u' av av, where ro and ao 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 cl 0 in the point equation of N1. When c t 0 in (1), the equation can be reduced to this special form by replacing x by xO, where 0 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 (60) ax \I/ ax, a \au au av av 0 In order that the conditions of integrability of (60) be satisfied for any solutions x and 0 of (1), it is necessary and sufficient that r and a be solutions of ___ __, a 0 a8 a 0 (61) Da = (T- C) log a- = (ad-T) log b 6 av av a a au b or in other form 8 a. a a a a o8 b, a b (62) log -- alog - log. 22) Cf. Hammond, 1. c., p. 256a 21) Cf. Hammond, I. C., p. 256. 88 III. Sequences of Laplace If we put (63)- -, 0, we get equations (II, 16), and find that c is a solution of the adjoint of (1), which is denoted by (cf. ~ 31) (64) [P; a ' b' ab-] Suppose we have any pair of solutions of (61), and the net Nr whose coordinates x1 are given by the corresponding equations (60). The points F, and F2 whose coordinates y and z are of the forms X X (65) y = x1 -z- x — lie on the line joining corresponding points of the nets N and N1. Moreover, as u, or v, varies the point F1, or F2, moves tangentially to this line. Hence F1 and F2 are the focal points of the congruences of these lines, and N and N1 are in relation F. With the aid of (61) we show that the point equation of NA is (66) a2 0 - a log a a 01 a logab a auav av 0 au au 0 av When we put 0= -1 in (61), we get the conditions of integrability of (59), namely (67) o _ (-r) loga a, 0 - (- ao) a log b. As a consequence of these results we have: Whenever 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 ax ax whose focal points have the coordinates- and -a Hence in au U, aV 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 congruence. Triads of nets. Suppose we have a transformation in the form (59), so that the new net N' (x') is given by ax' ax ax' ax (68) — 0 -a = ^ (68) au au' av ~ av Now the point equation of N is necessarily of the form (1) with c 0. If 0 is any solution of this equation and O' is given by the quadratures a0' ao a ' a_ (69) TO --— '0 — '~6O au au' av av then the functions xI, defined by 0 (70) X x- tx', are the homogeneous coordinates of a net VN, since fa ( = (o' —ro)o -, (71) I lv (0 (' ) av ) From the form of (70) it follows that corresponding points M, M' ', M on the three nets N, N', N1 are collinear. Hence not only is N1 an F transform of N, but also of N'. In the latter respect it differs from the case of ~ 15 (cf.III, Ex. 24). As an application of the foregoing results we prove the theorem23): 23) This theorem for 3-space is due to Ribaucour, Comptes Rendus, vol. 74 (1872), p. 1491. 90 III. Sequences of Laplace If a net N lies on a hyperquadric laik(x)x(k) M 0, i,k 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 a (i) a X(k) X(i) a x(k)\ haiku 3'! + 0V aU } an tav av au since the coordinates x satisfy (1) with c 0. In consequence of this result it follows readily that 0 given by 0 = ai7 (xi) x'(C) + x(k)x'(i) is a solution of (1), and that O' where 0o faikx (i)x (k) satisfies (69). If N, denotes the corresponding transform of N with the equations (70), it is found that Ni lies on the quadric. If r, oa and rz, o2 are two sets of solutions of (61) for the same 0, by means of equations of the form (60), we get by quadratures two nets, N1 and N2, transforms of N. From their equations we find a x2 r2 ax1 ax2 _ 2 ax1 au au' av i av' Consequently N, 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 9i, 92 of the adjoint of (1), so that by quadratures Exercises 91 of the form (II, 16) two pairs of functions rl, 61; '2 o 2 are to be found. Transforms N1(x1) and N2 (x2) are given by aXi a X a xi a x (72) Ti ~~~ --- - - - I- 2) au - au oi)' av av a k,I (i Solutions of the point equations of N, and NV2 are given by (3 a oij a (oj) ao __ a ().(7 3) -~ - ri - au ) a a — 0 auau Oi av av Oi j- 1, 2j The coordinates of the transform N12(X12) are given by (it being understood that N12 and N21 are the same net) au4 fii au 0 ' avi a '~ (~12 i (74) ax - a (xi ~ - av Comparing these results with (60), (71) and with (II, 21, 37) we have from (II, 52, 49) ( 02 021r1 01 0122 0 01 02 (75) 1 12 2 21 - 02 021 61 + 01012o2 -012 021, 01 02 t ~~~~02 Yp12 0 (021 02 Tl + 61 2 - 2fi.), 02 61'Cl and (76) 01 012X2 X 02 021x1 -4- 1 012 X2 - 012 021x 24). Exercises. 1. The homogeneous point coordinates of any point of the tangent plane to a surface are expressible in the form ax ax au av 2. Show that equatiois (3) hold for the special homogeneous coordinates such that yi/yn+l and zi/znfl 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 xiOVO1, 1ijO~0i, X12 012o'/012 of Chapter II must be replaced by - xi, - Oij, - X12 respectively in order to conform to the usages of the present chapter. 92 III. Sequences of Laplace =X -- yi _ z yn+l zn+l we have a2X a ogy"+l aX alogz+'l ax auav av au au av 3. The invariants of the Laplace transforms N1 and 1V-_ of a net N are given by H = 2H-K- a2log H v - a- a og, K=i; H-1 = K, K-, = 2K- - a logK. Darboux, Le-ons, vol. 2, p. 28. 4. The invariants of the Laplace transforms Nr and N-r of a net N are given by Hr, = Hr-I + H- K - a _ logH.... Hrl, Kr = r-1; 82 H-r = _T-r+l, -r- = -r+ + K- H — - logK.... K-r+l. au av Darboux, Lecons, vol. 2, p. 30. 5. If Hi = 0 for a Laplace sequence, then =A (U+SV dv) +A( U'+SV a dv) +....+A U(i) +S v dv), where fi, A,.... Ai are determinate functions; U and V are functions of u and v respectively, and U(i) is the ith derivative of U with respect to u. Darboux, Legons, vol. 2, p. 33. 6. When an equation of Laplace admits a solution of the form x =AU+ A U'.... - Ai U(i), where the A's are functions of u and v and U(i is the ith derivative with respect to u of a function U of u, then -H = 0. Darboux, Lecons, vol. 2, p. 35. 7. If Hi - 0 and K- = 0 for a sequence of Laplace, the point equation for N admits solutions of the form x = A U +.... + Ai U() + BV+.... + BjV(), where the A's and B's are determinate functions; U and V are arbitrary functions of u and v respectively, and U() and V( denote ih derivatives with respect to u and v. Darboux, Legons, 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 transforms are -- and- respectively is that the point equation of the net have au av equal invariants. 9. If the parametric curves on a surface S form a net N, the tangents to the curves defined by A du2+ B dv2 = 0 meet the lines joining the points M_, and M1 of the Laplace transforms of N in points harmonic to M_1 and M1. Exercises 93 10. Show that if a point equation (1) satisfies the conditions (27) and (29), it satisfies also the conditions 82 ap KK t.....K-,+1 1, log - 2 -... +1- u log KKI _.... K_.+ 2 11. Show that if equation (1) admits solutions of periodp, so also does its adjoint equation [cf. ~ 37]. Hammond, Annals, vol. 22 (1921), p. 260. 12. The point equation of N1, 1 defined by (43) is denoted by 02 b2 - [01,1; aH-0l, b,- l]. This may be obtained from (13) by multiplying the second term by 02/01, retaining the third term, and multiplying together the third and fourth terms of (13) and dividing by 01. Show that the same method applied to (1), denoted by [0; a, b, 1] gives (42). 13. Apply the method of the preceding exercise to (20) and obtain for the point equation of N,,1 the following Fl 07+167+11 [07,1; aHH1....H_1 0, b, aHIr-1 r- 2....H i. [8,Q,~ aIH....ar,_,H,.. b —tHr-_, ' Verify this result directly for r = 2. Hammond, 1. c. p. 249. 14. The point equation of N 2,1 is denoted by [ 2,1; a 2, b 0_, 2 This may be obtained from (15), by multiplying the second term of the latter by 0_1/0_-2 retaining the third term and taking for the fourth term the product of the second and fourth of (15) divided by 0_2. Show that (41) is obtained from (1) by the same method. 15. Apply the method of the preceding exercise to (23) and obtain for the point equation of N_(r+ 1) 1 0_(r+1),1, 0_.-l _1 r br Kr-1.... _lK r+20-r- Hammond, 1. c. 16. Show that for the derived net Ns for m >s the coordinates are expressible in the form _ rs8 as-1e () m-s-10(m-1) am-s X-sS,'- Laus, us-1 ' "" ' s - av-_ -1 ' a ---- Tzitzeica, Comptes Rendus, vol. 156 (1913), p.375. 17. If two nets N and N1 are in relation F so also are the nets resulting from a projective transformation of N and Ni. 94 III. Sequences of Laplace 18. Show that equation (66) is denoted by [ at b.a I] 01; 0 ' 0 ' 19. From (65) we have Dy - y z az au —x 1, av (x)l; - =()- X;l where (x)l and (x)_l denote coordinates of the first and minus first Laplace transforms of N, and l and n_l are first and minus first Laplace transforms of yi with respect to (64). Show that the coordinates of the first Laplace transform of F, and the minus first of F2 are of the forms (x)W +y v log a o_1, (x)_l y -z - log b. 20. If N and N1 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 hets N and N1 are in relation F, so also are their Laplace transforms of the same orders; the equations of these transformations are a (xl>, = 9r r!a i (X), ^l= (X (x~)3 - - (xi), = l where Tr-Tr-1+ '' 1 1.... r r-. for r positive. Hammond, 1. c., p. 260, 22. If the coordinates of a net N of periodp satisfy (25), and 0 is a solution of its point equation (1) such that 0 = m -O and o is a solution of the adjoint (64) of (1) such that -p = n p_, where ml and n are constants, the F transform of N by means of 0 and i is a periodic net of the same period as N. Hammond, 1. c., p. 261. 23. If N(x) and N (xl) are nets in relation F on the hyperquadric arix()x~() =-o, then,aik(x(()x() + x+(k)x?) - k, where k is a constant. When i, O ik ki -0, the congruence of the transformation consists of generators of the hyperquadric [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 = b =-"p the equations of a transformation K (~ 25) are xl 02 8 (x Dx 0 2 8 l x a ~u p a, 0r' av p av \ 26. Let N be a net in 3-space and N1 an F transform of N given by (60). Let corresponding points M and Ml of these nets and the two focal points F, Exercises 95 and F2 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), M1(0,0,0,1), Fi(1,00,0,), 'F2(0,0,1,0). Show that the pencil of conies tangent to the lines MFE and MF2 at the points (M)-1 and (M)1 of the Laplace transforms (N)-1 and (N)1 of N are given by 2Axz-( log x 9+- log- z,) aul b 0 av a where A is a parameter; also that according as A is K, or H, the conic osculates the curve u - const. of (N)-1 at (M)-_ or v = const. of (N)1 at (M)1. Annals, vol. 18 (1916), p. 11. 27. A necessary and sufficient condition that a conic of the pencil of Ex. 2G osculate the curve u - const. at (M)-1 and v = const. at (M)1 is that N have equal point invariants. Darboux, Lecons, vol. 4, p. 38. 28. Show that the pencil of conics tangent to the lines M1iF and M1iF2 at the points of the Laplace transforms of N1 are defined, in the coordinates of Ex. 26, by x a b za a 2xzz =1 ( log + ~~log~ + 2 -- W -1og- q- log q-; - a 8 6, 8, 0 02 that this pencil and the pencil of Ex. 26 determine involutions on the line F1 F2; 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 N1 are two nets in relation K, any two conies of the two pencils of Exs. 26 and 28, meeting on the line F1 F2 determine a pencil of quadrics which cut the line MM1 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 MM, at these focal points. Annals, 1. c. p. 15. 30. If N and N1 are nets in relation K, the doubly osculating conies of the pencils of Exs. 26 and 28 meet on the line F F2, and the vertices of the two cones in the pencil of quadrics determined by these conies are the corresponding focal points of the conjugate congruence of the transformation. Tzitzeica, Comptes Rendus, vol. 147 (1908), p. 1036; also Annals, 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 ~(1) 20 aloga Do Dlogb a -c auav Dv ~ u Du av 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 ___ DO b1 + c( 10 a t2= au a_ _ +al 62 (2) a20 a 0 a 0 aV2- a2 a + b2 av + C20. 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: a a 2O \ 2 2e D 0 aD lDay4 av \au ' a auav av DauavD' D v \V" Reducing the resulting expressions by means of (1) and (2), we get two equations of the form (3) A aD +B a + Ce=O, A, a- +B2 aDO -C = D u av Du av where Ai, B1.... C2 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: 41. Tangential coordinates of a surface in 3-space 97 aa2b + a,_ aloga alogb c log a ~a- -- = - -T- ~ u + c +-M --- aav av ai a u a)v a logb a b, al oga, /a log/ b 2 aogb al,~ t + bb 2 + cl + b- c + b, + 2 a1, ab 8u ag a- au aua ac bc ac2 - aloga a log b ac acb cl +. c1~ + cq a v a u au 9 ab +- b, aloga alogb alog b au av au 8+uv au ' a log a a2 a logb aloga2 loga b2 ~ a a2~ C2+ - +?2 av au au av av2 a 82 alogb aloga ac bc+a2c1+. c+ + cb2u 8Cu av av When these conditions are satisfied, the system (1) and (2) is completely integrable and there are at most three linearly independent solutions. For, the derivatives of the second and higher orders of 0 are linearly expressible in terms of ao a, and 0. au' av' 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 projective transforms of one another. 4I. Tangential coordinates of a surface in 3-space. Let x, y, z, w be the homogeneous point coordinates of a surface S 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 ax ax t, x + Pi a, ' Qx X- t v al~1 ZC +^2^? 7 98 IV. Surfaces and congruences 'in 3-space for any values of At, 1,:2 and 2. Hence there exist four functions, X, Y, Z, TW, of u and v, the tangential coordinates of the surface, [cf. ~ 67] satisfying identically the three conditions (5) I Xx + Yy + Zz + Ww - 0, x ax.u, ax av where. indicates the sum of four terms obtained by replacing x and X by y and Y; z and Z; w and TV respectively. In consequence of the last two we have, on differentiating the first, the two equations (6) S Dax -0, au V aX 0. Xrav 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 [~ 55]. Hence along an dv asymptotic line - must equal such a function of u and v that du the equation of the tangent plane is satisfied by i x +, dx + v d2x, for all values of i,, and v [cf. ~ 30]. Hence we must have XXdx = 0. Eliminating X, Y, Z, and W from this equation and (5), we have (7) where Ldu2 + 2M dudv +Ndv2- 0, (8) ax ax a2x au aDv au2 Dy Dy D.y Y at a U2 Y u av Du2 az az a2z 2 -a u av au2 Dw DW a2W au av aDi2 ax ax a2x x- au av auav ax ax a2x au av av2 M1 Z 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 -au-al D +bl a+c o, (9) a2o ao a - 2 a + b2- + 26. 8 v a it a v 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: a2 8a20 82 / a20 a v2 \a~u2- av v2 When the above expressions are substituted, the resulting equation is reducible. by means of (9) to an equation of the form a2~ ao aO A a --- a +B +C a +De -0, autav au ar 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 ( 8a, a b2 av 8u a2a2 a a1 a a a, ba b ) a c, 2a2 _ ~2av + a (al a) + b2 a2 (a ) + a2 0, au2 au~ 2 au av av av aau a2bl a2b2 + a (bb b)b a a2 a a c1 av2 an2 av a2u. an au av a _c2 a2c 2 a a ac1 a c2 ab, a c2 ac a u- a2v2 2 an +a2 a 1 a - c2 av av av 7* 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 ax ax a- x in terms of x, a, - and aa, there are four linearly indeul av au a Iv' 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 projective transforms of one another. When the surface S is subjected to a polar transformation with respect to the quadric (11) x2+ y2 2+ w2' — 0, 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 S' [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 projective transforms of one another. 43. Nets in 3-space. Point coordinates. Consider a surface S 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 -ax i 8 ax \ xrz( 4 du+ - dv x -x+ xt\ au d + v d. A necessary and sufficient condition (~ 2) that two families of dv 6v curves determined by and form a net is that, as M moves du 6u along the curve of the second family through it, P moves in the tangent plane to the surface at f. Hence the point whose coordinates are of the form x+- 86x -- u- u — a6v must lie in the tangent plane. This gives the equation of condition X6'x+ Yay+ Zaz+ S va- w = 0. 43. Nets in 3-space. Point coordinates 101 Combining this equation with the identities (5), we get (12) Lduzu- +M(du6u+dudv)+ATdvdv 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 [~ 56] we have: A necessary and sufficient condition that the curves defined by R d2 - 2 Sdu dv + Tdv2= 0 form a conjugate system is (13) RN+ TL - 2SM = 0. 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) du2+ r dv2 z O, where r NIL is a function of u and v. Comparing this equation with (7), we have in consequence of (8), ax ax a2x 82x cx au Ta r a2 y -0. w 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) =r 2 + a +' +c'o. v -- aru2 au a 1 - 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 av uav u \av' When the expression from (1) and (15) are substituted, the resulting equation is reducible to a38 - a2_ a0 a0 (16) a3 A B a 0 + 1a where I log a b a 11 a2a a b a loga A& — ulog, B --- v+i - a+ - log7 - v-' — a r r B r a uv a a' v ) r c E1+( 2 a2ogb a b') r(c a au u av au' where K is one of the invariants of (1) [cf. I, 44]. From (1) we have also by differentiation a3O a 2 0 ao ao u --- 3u, - - B2 —u + C2 — { D2 O, (18) a2a aa v aau B2 au0 2 av where (19) _ a aboga Oaloga 1 a2b A --, B2=- H+2 -' b D - c log bc. av ' auav' bYau2' au Also from 5) we obtain and expressed linearly in Also from (15) we obtain a v and - expressed linearly in terms of 0, a a- and a2 There remains the condition au, av (a )t2' 8 / 830 a- a30 ' au a 2ai/ a8 au 83/ 43. Nets in 3-space. Point coordinates 103 By means of (16) and (18) this condition is reducible to the form a20 D0 D0 (20) P Q + R + S-, where P, Q, R and S 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 higher derivatives in 0 are linearly expressible in terms of 0, - Do a20 v'a 2D 2 Since all further conditions of integrability are satisfied, 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 projective with the given net. When the expressions P, Q, R and S in (20) are calculated, it is found that equations (21) reduce to ( DA,1 D, ~+ Dva Cir = A2 +B2, av au aBi +A1B2' aB2 aloga 2 Dv Du Dv (22)| aC a log b C+A= 2C Ca logb Dv D D+A 2 a+B,c+Cc'= - a+A2+C2. — +A1D2+Bl c+01c' D +A2D,+C2c. Dv 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"/D. Consequently by the elimination 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, 221 r 111 b 2 J r 2 -1 2 0. 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 consequence of the preceding theorem, we have: An isothermal-conjugate net is transformed into an isothermalconjugate net by a projective transformation. 44. Ray congruence and ray curves. Consider a net N in n-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 Ix-i_+ -px,. Hence it must be possible for u and v to vary in such a way that d(xx-i+-x,) is a linear function of x-_ and x1. From (III, 2, 4) we have 3ax _ a2x 1 92b alogb ax-i aloga....UY ' X-X — l, X-i + Kx, au au~2 b a ax a = a v av 3ax alogb ax a2x 1 a2 a a log a ~ ~ u a x~i a~x i +U H x a V2 X x k a I, Xa1 1Hx, a v~~ a ~av 2 v 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: x-x b a ux dU-+xKdv] ++ pxxdu- + (x x aa \d au2v a Uv ax dv (, I ax b x 8l aloga X 0, \u au a \v av _ where v and a are to be determined. 44. Ray congruence and ray curves 105 92x a2x ax aX Equation (24) is linear in au, v2- au' - andx. If n 3, this equation must be satisfied by five or more linearly independent functions x, which is possible only when the coefficients of the a2x quantities 3...., x are zero. This gives 2 du = dv = 0, which from (24) is seen to be impossible. When n — 3, by means of (15), (24) is reducible to an exa2x ax ax pression linear in- a -2 a x. The coefficients of the latter expression must then be zero, which gives the four equations of condition ( du+-Irdv0-, la'dv+-v 0, pb'dv +a-0, (26) Hdu2+ Rdudv- rKdv2= 0, where r 2b 1 aa alog b a (27) = b a -+ + a' - '. Pb " a a 2u au av Hence the system of lines forms a congruence. FollowingMTilczynski25) 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 ina aplane has an envelope, the developables of such a congruence for nets in 2-space haveno 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), ax- _ i(A a logb ax_ (B C) ax ax (28)a Da U 1 au av a A 1C a u a 11 au au2 (28)4 + (AlC-.32 ~+ ) x-+ tAi a - logb 21og ba x__ aloga ax- + ax a K a2loga ~Vr ~~ + X-1^+ x- X'.~av2 - av av av av av. ax If we replace x and a by the linear expressions for them obtained au ax from (III, 2, 4) and then eliminate x from these two equations, we av obtain for the net N-1 an equation of the form (15), where r_- has the value K/C1. From this result and (14) it follows that a necessary and sufficient condition that the asymptotic lines correspond on the two surfaces is that C7r K, which by (17) is equivalent to a2logb a b' (29) 2 a- 0. auav au In like manner a necessary and sufficient condition that the tangents to the curves u = const. of N form a WTcongruence is (a 2log a 3 a'. ( ~30) 2 auav av r 45. Nets R 107 Tzitzeica"7) 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 a2 to - log r 0. Hence we have the theorem of Tzitzeica: auav An R net is isothermal-conjugate. By means of the first of equations (22) we establish the converse theorem due to Demoulin28): If the tangents to the curves in either family of an isothermalconjugate 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 b 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 log6b aloga b' --- 2 a' a- 2 av au Hence: The two differential equations satisfied by the homogeneous point coordinates of an B net are reducible to the form a20 aloga a9 + alogb a -co auav av au au av (31) a2 + aua2 au au a v av We return to the consideration of the net N-1. From (23) we have by differentiation f a2x- _ aloga ax-_ a logKb axauav av au au av (32) 2log a a logb alogK aloga uav au av a +cjx-l 27) 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 C — -K. Making use of (III, 2, 4) and the third of (22), we find from (28) ( a2 a _ 2 a v. aq - a (33) (33) 2 aloga ax -1 a logKJb aX1 x- + c-....-'+2 - lX —I au - au av av where c'-1 is a determinate function. Comparing equations (31) and (33), we see that N-1 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 W29). 46. W congruences. Let S be a surface referred to its asymptotic lines a = const., f = const. The cartesian coordinates, (1), x(2) x(), of S are given by the Lelieuvre formulas of the form [~ 79] 8 V2 'V3 82 V3 (34) a c av2 av -- av2 av3 a" a~ aa aa af a8, where v, v2, v3 are proportional to the direction-cosines of the normal to S, such that the total curvature of S is given by 1 (35) - ( + ^ )2 The functions vi are solutions of an equation of the form a20 (36) = M where M is a function of a and f. 29) Cf. Demoulin, 1. c., p. 591. 46. W congruences 109 If 01 is any solution of (36), the equations [cf. ~ 172] a(o1v) e a (,) vi (37) a 1 a avi, a a o ai a aa ad a # are consistent, and the functions v- are solutions of (38) 0a9o 06 a {38) ^- a aaga 1 ad f (01) The equations of the form (39) X(1)= X(1)+ 2v3 — Y3V2 define the coordinates, x(i), of a surface 8, such that S and S are the focal surfaces of the congruence of lines joining corresponding points on S and S; this is a W congruence, since the asymptotic lines are parametric on S. Moreover, any W congruence with S 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 9a0 a~ ao +ob 0a aa 0 a0 ao a a2 aaaa +b AdI a A0aat a0 a,8 Differentiating (34), we obtain (41) 9 = a2a 2 ^3, x2=- aV3 2x2 a.3 a vaa2 aD2 a= 2 a82 The functions Yi 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 a 2V a9) a i b avi +5b a9) ab\ a a2 aa aa aiad " aig a la a 2 V, a,~~yteai a2~k i aa aT av' ~(a al + a+ 2~~ ~ a~2a a) aV a9 aa4) We write a Vi vi (43) 01 Vz= kyj + 1~ + m ac and seek the conditions which k, I and m must satisfy in order that this expression may satisfy equations (37) for i 1, 2, 3. By means of (42) we find.at am (44) a alm =0, aa -bl - 0, 1 a a a9) a_9 k -- +- J+l +m l 2 aa ad aa a (45) I a am a SO ac 0, + I m a 9i a cc a,3 aa and a k a 01- +1 b a~ + a b + m 2 7 a aib 0 (46){~ o~a 9~ ) a2+~)0 ~ ak +a 01 +i~ acg) + a, b + m( a2Ca + a a, 0 a~ afi (i9 a+ aa Conversely, if 1 and m are any pair of solutions of (44), the functions k and 01, defined by (45) satisfy (46). Furthermore, if equations (46) are differentiated with respect to # and a respectively, and the resulting equations are substracted, we find that 01. 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). Hence 30): 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 11 and b are the functions appearing in the equations (40) of the surface; when a pair of solutions is known, the coordinates x(i) of the other focal surface of the congruence are given by ax(i) ax(i) ( 7 ) (i) — x i) _ 2 - aa (47) x -x am al a ay ao ~^_ a8 _ I m 8a, 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 X surface when it contains an R net. In this section we establish an analytical characterization of R surfaces. Let S be a surface referred to its asymptotic lines a = const., f — const. Its cartesian coordinates satisfy equations of the form (40). Any isothermal-conjugate system of curves on S is defined by [cf. ~ 82] (48) u = (a) + B (), v=-A (a)-B (,), when A and B are arbitrary functions of a and,f respectively. If we effect the transformation of variables (48), equations (40) become a2 + a2 (K+L) ao + (M+ a) 8 au2+ v (K L) au ~( N) v a2O 1 ao 1 aO uav s (K-L) u + (M-N) auav 2 au 2 av' where _- 1 1 A'- - Al+ ' A", L- -2B a,4'+ — B'-B", 2 2 ' /a A'2bB'- A" ' M — T 2 A1'2 1 A 2B'2 a B- +B 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 ha (a Nn- a (M+ ), av (K+L) au (K- L) av (M-N) au 112 IV. Surfaces and congruences in 3-space or in terms a and 3 1 DK 1 L o 1 am 1 a3N Y -a'd-Y aaO-.4 ' -- 0. Substituting the above expressions for K, L, M, N, we are brought to the single condition (49) B' +B'Bt b = A' a 4-A'A"al3). Since the quadrics are characterized by the condition that a= -b = 0, we have the theorem32): 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 A1, B1, it is satisfied also by A, B, where A2 A'2+ kA12, B'2-B'+ kB12, k being a constant. Hence we have the theorem of Demoulin33): If a surface is an R surface in two ways, it is an R surface in an infinity of ways; that is an infinity of nets R lie upon it. If the condition (49) is satisfied by a surface, the parameters of the asymptotic lines can be chosen so that A c, B = t. Consequently: A necessary and sufficient condition that a surface be an R surface is that its cartesian coordinates satisfy equations of the form ( 0 a 3q0 o l a3o.(50) I aa2 aa aa + aace d82 ad a d ai o When the conditions (10) are applied to (50) we obtain the equations to be satisfied by 9y and p, whose solution gives the complete determination of R surfaces. 31) Cf. Demoulin, Comptes Rendus, vol. 153 (1911), p. 799. 32) Cf. Bianchi, Rendiconti dei Lincei, ser. 5, vol. 222, (1913), p. 5. 33) L. c. 48. 1? congruences. Transformations O1.13 113 For this case equations (44) become a I ' - a__ - d mn of which a solution is I m equation (43) becomes in this casi a ce a a - ed'. Making use of (45), the K (f+'tP) + dV (51) raaa __ I (~~+'~P)+ (~v+ -2 48.. R Congruences. Transformations 01. By definition the W congruence of tangents to the curves of either family of an R net is an -R contgruence. In this section we show that when an B surface is known, a congruence BR can be f ound by quadratures, whose linies are parallel to the normals to the given surface BR34).,Equations (50) canl be written __ a\ - atp ao a / ao_ atp ao (52) a 0 ad a a'V ~ad C ad3 a", Hence if we put2e (53) 2a a W -+~i- 2eax we have a ~~~a dp, a() adq ( Diff erenitiating these equations with respect to id and a respectively, we find that (55) a2 ~ ae~ a2 cc a _ e aca ~ Cac~d'' a~da~d 34) eoln Comptes IRendlus, vol. 153 (1911), p. 798; also, Jonas, i. c., pp. 67 et seq. 114 IV. Surfaces and congruences in 3-space Since equations (54) may be written in the form (37) with 60 replaced by e-~' and v,,i by_ %, i, it follows that the surfaces 2 and 2 of coordinates (i) and (i) given by equations of the form (34) in /i and analogously to (39) by equations of the form (56) ( 1) ~(t) A- 23 —3~2 are the focal surfaces of a W congruence r. In consequence of (53) we have a () a Xy.C V (@(_yi)-f U - (dz)_ $) _ a - 0) Consequently the lines of the W congruence are parallel to the normals to the given R surface. From (53) we have (57) '1 (a ax - a x)1\ aa ad ' If we differentiate this equation with respect to c and d and add the resulting equations, we find in consequence of (50) and (53) (58) -^ + a- + 2, Hi (^) +a(+)] 2 8a a [1 (fi) _ a ) 2 KI[ (+ )- ( a(+P ) Comparing this result with an equation analogous to (43) written ac. ad. (59) -gri — D e + — we see that -, and hence from equations analogous to (44) and (49) it follows that Z is an X surface, and consequently the congruence r is B. Moreover, from (53) it is seen that the normals to Z and -2 are parallel to the tangents to the curves of the R net a ---f -- const., a +-f - const. on S. The equations for 2 analogous to (50) for S are obtained from (50) by replacing p and zp by functions j and fij, where 2) -= e", 49. Reciprocally derived nets. Transformations TV of nets R 115 as follows from equations analogous to (44). From (58) and equations similar to (51), we find that gi can be chosen so that y+ + + VIP 0. Also from (54) we have that e- ~ Hence equations (53) can be written a. (i).,X/yW i (i+ ) =- eSlO1-i1 ax e(-') ' Suppose conversely that we have an R congruence for which one of the focal surfaces is S whose point coordinates x) satisfy (50), and for which the functions Vy are given by (51). In consequence of (37) and (51), the equations - = 601 (Vi+vi) e-(o, = 1 (V-Vi ) e-( are consistent, and (i) are solutions of a20 a ao alog 0 a0 8. 1 -0 (log 8 — a) a a8' 8a20 aloge1 ao a a -a a8- a —+ - (log 01- y) 8 Hence $) are the coordinates of a surface?R. Accordingly we have the theorem of Demoulin: WVhen an R surface is known, an associated R congruence can be found by quadratures, and conversely; the lines of the congruence are parallel to the normals of the associated R surface, and the normals to the Jocal surfaces of the congruence are parallel to the tangents to the P net on the R surface. Demoulin has called this reciprocal relation a tranformation 01. 49. Reciprocally derived nets. Transformations W of nets R. In ~ 11 we remarked that if 0, and 02 are solutions of the point equation (60)a82o 8loga a0o alogb aO af av v a z au av 8* 116 IV. Surfaces and congruences in 3-space of the cartesian coordinates of a net N(x), they determine a derived net N(x) of N, whose coordinates are given by equations of the form a x ax (61) x + x+- a where ( P) (a013 -02 a v)' q +(02 a 01 a- 0 3m 3v 3v a3 m Z/ d a02 aol_ ao2 ael a u av a v. a u ' From these equations we have a o0 a o0 302+ a 0. (63) O- +-p w q- q A 0, o 02+1p +q a 0. Ordinarily IN is not a derived net of N. When it is, we say that N and N are reciprocally derived nets35). If N and N are reciprocally derived nets, the tangents planes at corresponding points, Mi and M, must pass through 1Mi and I respectively. Hence the surfaces, S and 8, on which N and N lie are the focal surfaces of the congruence G of lines MM. Since the lines of G are not tangents to the curves of N or N, there exist upon 8 and S corresponding nets to which these lines are tangent, namely the focal nets. Consequently S and S are surfaces in 3-space (~ 3), and since two nets on S correspond to two nets on 8, the asymptotic lines correspond on S and S. Hence G is a W congruence. Since 01 and 02 are solutions of (60), we have from (62) ' _ 1_ I 9 loga + l20 _ 12 a8u -q av ' 3 \ J 32 av a32 av ' ap a log a 22 a-2i \ (2 a20 - a2a 3 -- 23 v \ 8' ~ a32 3! a2 3v a3 a3v-2 ' 3m)-q 3a v '-P + a2 a \a^ av 23m aa 2 1' a-u a a1( e2 au2" a, " J av a t av au2 3q alogb q /8202 301 320 0 \ 'a - 1. 3m t- ' z aj\ 2 3a a-2 a _' av -Jr- aZ/ a dv2 a 't a V2 a 35) Cf. Tzitzeica, Comptes Rendus, vol. 156 (1913), p. 666. 49. Reciprocally derived nets. Transformations W of nets R 117 With the aid of these expressions we show that the point equation of N is a20 8a ao a ao (65) v = a logap + logbq a.zi a v av a +t 1~gbq& a v' If N is to be a derived net of N7, we must have (66) x - q _ and the point equation of NT is, analogously to (65), a2o a ao a 8 9o (67) 2 -.ala ogap a - + -j logbq q. --- Comparing this equation with (60), we have (68) pT-U, qq=-; where U and V are functions of u and v alone. From (61) and (66) it is seen that the parameters can be chosen so that (69) PP -- -- C where c is a constant. When the expression (61) for x is substituted in (66), the result is reducible to (0x D2x ax ax (70) a- v-ct aa + b' where j -a', -cp+1 a logp q a log a 1 + I _ a - -a +-+ + + av q a I., (71) 1 alogqp + p q a logb +1 aq -b ' -cq+q-++ -v -+ 9 p J - By means of (64) equations (71) are reducible to /2., - L8 c '8 _ 8^ a 8 e -, 02 a-8 v2..,8 0, a { az/l ~.. -- ' a vC a 02 a82o1 1 a 2 f 8aaO1 a v 21 a22- a622 a,, -a2 18201 801 \ -01 /29 822 f 802 C\ aT an2 av a" an 1 a Z a t2 + aV2 b av 118 118 I~~~V. Surfaces and congruences in 3-space fromt which it f ollows that 01, and 02 are solutions of (72) a20 a20 C a'0 + b' a0~+c O. a2 2 m a In order that equations (60) and (70) shall admit three independent solutions, we must have equations (22) satisfied with c - 0. When we make use of these results in obtaining equation (20) for the system (60) and (72), we find that P- Q -B- 0 and consequently S - 0. But this is thfe las t of (22) and it reduces to -2 a' logb -0 Then from the first of (22) we have a a' 2a2log a at a ua V From the results of ~ 45 it f ollows that NY is an It' net, whose cartesian coordinates satisfy, equations which can be written in. the f orms a 2 0 a20 aloga ao alogb ao a u2 0 V- at au t a; av' (73) j 27 a loga ao + a logb ao0 au av atV a u a atV Now equation (72), to be satisfied by 01 and 02, is a2o a 2o __ aloga ao lob ao (74) a2~a2 2 a a+2 alg +C0. Since (60) and (74) satisfy the corresponding equations (22), they admit four independent solutions. If ini (61) the expression (66) for x is substituted, wie finid in consequence of (69) and (71) that -XY is a solution of (7) a2o a 2o0- a lgao)a0 a loblao (7) am2 +a2 V2. ~ oau, ad +2t alv a ilence N is an B net whose equations are (6 5) and (75). 50. Theorem of permutability of transformations TV 119 In ~ 50 it is shown that, if 03 and 0- are two solutions of (60) and (74) independent of 0t and 02, the functions 03 and 04, defined by (76) OiO Oi a - a oI a q 0 (i = 3, 4) are solutions of (65), and that, moreover, the functions p and q in (66) are expressible in terms of 03 and 0, in the form analogous to (62). Hence N is a derived net of xN and consequently is reciprocally derived. We say that N is obtained from N by a transformation Kec. Since any solution of (60) and (74) is expressible linearly in terms of four independent solutions, it follows from the form of (62) that there are oo~ transformations WJT for each value of c, different from zero, in (74). Hence: Any It net admits of 0o -transformations 1ITt, into R nets, for each value of the constant c different from zero6). 50. Theorem of permutability of transformations W. Let 03 and 80 be two solutions of the equation (60) and — 0 a20 aloga aO 2 logb a ( co -+! - + --- ____ 2 a a Dy aV2 au at- ' aDv av and consider the functions (76). Analogously to (I, 68) we have a -i 290/ i a- i,a 2 a201 a0 a )02 a G | a i, a a? O. at a 2D a O; a l2 a Oi a, 802 01 aO a2021l, av a Da s92 Ia D JJ (77) a __ O JD20 + 1 DOi aDO2 D21_ DO1 D02 (7) D v { I D2 J [ \ D av Dyi Dv |v d D aa a2 a2a att aVa aV2a ~I;'" d ze av a Vu" av a v" a \O a a^v2 ae, a2 jl' -p D-{- ai+ ('-c) i 36) Annals of Mathematics, vol. 22 (1921), p. 170; these transformations are given in a different form by Demoulin, Bulletin de la Classe des Sciences, Academie Royale de Belgique, 1920, p. 226, and by Jonas, Jahresbericht, vol. 29 (1920), p. 58. 120 IV. Surfaces and congruences in 3-space Differentiating the first of these equations with respect to n, we get a a i a [ alog a2b + 1 0, a92 0 a01 202 a?G S a 8 a 9' a V 8at2 aV a2rt ~ (, - a ) i a logb + a i a O^ a u) a?? e a U& av1| Making use of the expression (64) for -- and (76), we reduce this to a n a2o a [ a 2 2 1 + 2alog al I 0 a o 8?2 elot 'Ia b -+ - b p C-C an - = l a logbOa ~ i -+ c O Also we find that a20i a a Oi a a Oi a-^ __ a log ap +- logbq. auav av a it a 1b a- ' From the second of (77) we get a_0 _ q aloga ao8 allogq q alogbq+c Qi ~/,a a O8 a v2 p a v a u a av p a a a With the aid of (71) and (77) we obtain a20i a20o a logp a) a 2 log bq a Oi a tt2 av2 2 a u a u a v av v (; Hence the functions Oi determine a We' transformation of N. The coordinates of the transform NV are of the form (78) X ^ a v' -- u a+ _ ___ ___ 50. Theorem of permutability of transformations WV 121 Where (cf. I, 68) 8t ( ja2t 1 [8 /a a-20 8aO9 a2a ax 1 axt2 a S2 0v Qu. 1 [at u2 - Z a a, av tf2 av acu, ax / a 02 a2 0 a O a 2021 a? a t a zF atu a2~] a x 1 aIx I aSx ax\1 av 1- aU ait +u av I' and lI a - a - a 03 -- a80 P =- 5703 - ) = j(-04 03 (79) __ 5 0a 03 a 04 a 03 \i u a v 9v aU ' From (77) and (63) we have _pq{ fC\0z a 1 a 04 a l a804 2 803 Z [ L\au av av aS a \ 2 Sa m/ -04t +i i -e,_) a )~ ( a V a201, a 0 a4 2 a Sz a 0\ v azSm Sr SM2/ / a 04 a a 0 /a O2 a 202 0 a201 (80) 03- ^ 04 Sv aS St2 aSm S2!] [I SO' S 520 520\ LS a r av av aJ:]1 2 02 8a2 ___g2 SO1 SOi/af4 S20 S0 S201 + (1 a 02 a auv at2 v au^2 o- a 02 ) a,~ a Ol a2O3 a 03 a 204 02 aSOi SaO4 5 203 Sa 03 a21 \01 a 02r \a Stt2 aS St2!IV 122 IV. Surfaces and congruences in 3-space By means of the above formulas equation (78) is reducible to C C fl a 04 ao -3+ (, -- o o1 xz x qi{( +-t I 3 ) 04 +(0304 403) aU c/-Ci 9xS, ax\ 0 8 3 / 204 a2o3 ~ a —o a v /[a u2 a 62 ) - al0,, 0 - 0 0 2, ( i -03 04- 01 03 02 au2 D a au a i a 2a0 x a 02, a0 l e03 a204 a2 9) oi. ao0. 0 j 013 - 3 D ] and the pairs of functions 0, 06 and 0, 0. Hence the net can be obtained also by applying to VN the We' transformation determined by 0s and 01, and then to the resulting net iV the transformation determined by the functions 803, 80D 80i 1 0, 304\ a0.I - __ ~ 04~ ~0 _ 0- 03 / _ — av a3v t av a3 ^_^, r4"^ ""3!^""^4 i"13^,0 and 0 3.'y'- z 1 i^^03i a4 ae8 4 't- which are analogous to (76). If c'= c, the net N coincides with N. Hence if a net N is a bic transform of N by means of functions 0t and 0o, solutions of (60) and (72), and 0s and 04 are two other solutions linearly in endependent of and 0, the functions (76) determine N as a IT', transform of KN. When c' c, we have that 1N is a IWe transform of N and a WTCe transform of N. Hence: Exercises 123 If N is an R. net, and Nj and N2 are obtained fromn N by tralnsfobrmnations W,, and IV42,, there can be found directly an R net N-A2 which is a J/1'V, transform of N1 and, a I'1"'c, transformn of NV287). Exercises. 1. If xi (i = 1, 2, 3, 4) satisfy equations of the formn (9), from the first of (10) and the first and fourth of (4) it follows that x1, x2, x3 satisfy an equation of the form (1) with equal invariants. ' Hence the projection from a point upon any plane of the asymptotic lines of a surface is a plane net with equal point invariants. Koenigs, Comptes liendus, vol. 114 (1892), p. 55. 2. If xi, x, x:3 are the coordinates of a planar net satisfying equations (2) and a2o alogo ao + alogo ao altav al la auit a v the equations y- 1 (ax3 13 ax)i, ay1 1 a X2 au - a2 -j2331 - a av - X2 a2 - X3ai are consistent. In like manner we obtain by quadratures two functions?2 and y3 by permuting the subscripts of the x's cyclicly. Show that the function X= -XIy1+X2y2+ X3/ satisfies equations (2). Hence xi (i = 1, 2, 3, 4) are the coordinates of a surface S referred to its asymptotic lines. Annals, vol. 18 (1917), p. 224. 3. The functions yi, Y 2, Y.3, - 1 in Ex. 2 are the homogeneous point coordinates of the asymptotic lines of a surface which is the polar reciprocal. of S with respect to the quadric x2 + x2 + X2 + ~2 --— 7 0. Annals, 1. c., p. 225. 4. A necessary and sufficient condition that the surface S whose coordinates satisfy (9) be a ruled surface is that a2 0 or bi - 0; in this case the curves,u = const. or v- = const. are straight lines. a log2 a log 2 5. If in accordance with the first of (10) we put a, - 2 a? o b2- 2 ao then the coordinates -XI x/l)2 satisfy equations of the formn a2 b a+ a2o ao zzb- +ci avi02an $-c2 a,?,t2 av a V2 -ast called by Wilczynski the canonical fo2'm of (9). Trans., vol. 8 (1907), p. 247. 6. Let S be a ruled surface for which u = const. are the straight lines and the equations are in the canonical form (Ex. 5). NOW 0Ct2 = 0, and from (10) we a 2 -have a-. The tangents to the curved asymptotic lines at points of a generator it const. form a ruled surface R,, whose coordinates are given by ax Y = tvx -+ 37) Annals, 1. c., p. 172. 124 IV. Surfaces and congruences in 3-space It is readily found that the y's satisfy the equations &2 ca ~ry -0. v2 '- 2 a, 2 Hence Rl is a quadric, which osculates S along the given generator. 7. If two surfaces S and S in 3-space are so related that each net on S is a radial transform of a net on S, then -,(i) __ ____ _ 2 a, xa()+d ' where as, d are constants. 8. If two parallel nets in 3-space have areas preserved, the total curvature is the same at corresponding points. Guichard, Comptes Rendus, vol. 136, p. 151. 9. The focal points of a ray congruence are givenl by ARx_1+px,, where K)2+ R1?)-rH/2" — O; they are the intersections of the lines of the congruence and the tangents to the curves Kdu C2+ R du tv - -rHdv2 - 0. 10. A necessary and sufficient condition that the tangents to the ray curves of a net N pass through the corresponding focal points of the ray congruence is that N have equal point invariants. Green, Amer. Journ., vol. 38 (1916), p. 313. 11. Let N be a net with equal points invariants on a surface S and N7o its associate net on a surface So (~ 25); also let 2 and 2o be the surfaces corresponding with orthogonality of linear elements to S and So as determined by No and lN respectively [~ 157]. If 1rT, is a K transform of Nr, it is possible to place the associate, N1o, of Ni so that INlo and N1o are in relation K (II. Ex. 18). These nets determine surfaces 'Z anld -2o corresponding to S1 and Slo by orthogonality of linear elements. Show that the pairs of surfaces 2, 2; 2, Zo; 2lo, Zo and clo 21 are the focal surfaces of W congruences; that is the K transformation from N1 into NL1 determines a quatern of TV congruences. Trans., vol. 15 (1914), p. 415. 12. Four nets 1NT, AT,, N2, X12 forming a quatern under transformations Ki determine twelve TV congruences, forming six quaterns. Trans. 1. c., p. 416. 13. The focal nets of the congruence of normals to a suiface of constant curvature are R nets. Tzitzeica, Comptes Rendus, vol. 152 (1911), p. 1078. 14. The lines of curvature of a surface of constant curvature and the normals to the surface are in relation 01. Demoulin, Comptes Rendus, vol. 153 (1911), p. 798. 15. An isothermal orthogonal system on a sphere and the normals to the minimal surface, whose lines of curvature admit this system for spherical ripresentation, are in relation 0,. Demoulin, 1. c. 16. An R net N admits oo4 parallel nets determining congruences G of transformations F of N into R nets NITN; when such a parallel net is known, each solution 0 of equations (60) and (74) defines a transform N~1. Annals, vol. 22 (1921), p. 176. Exercises 125 17. If N is an P, net, and NTi and NX are two F transforms of N by means of functions 01 and 02 which are solutions of equations (60) and (74) for the same constant c, all of the oo2 nets AN12 which are F transforms of N1A and NV (~ 21) are R nets and their determination requires two quadratures; when the constant c in (74) is different for 0t and 02, there is a unique net N112 which is an R net and it can be found without quadratures. Annals, vol. 22 (1921), p. 178. 18. If N is a W transform of an R net N by means of solutions 01 and 02 of equations (60) and (74), and N3 is an R net, which is an F transform of N by means of a solution 03 of (60) and (74) with c replaced by c', there can be found directly a unique net NA which is a W transform of Ns and an F transform of N; when c = c', there are oo2 such nets NT obtained by two quadratures. Annals, vol. 22 (1921), p. 181. 19. From the equation (15) of a net N(x) on a surface S it follows that a2x ax the point of coordinates z -- + - lies on the intersection I of the osculating planes of the curves of N at the corresponding point. This line generates a congruence whose focal points are determined by functions and y, such that d(Ax-,ltz) is a linear functions of x and z. Proceeding as in ~ 44, we show that the developables of the congruence are determined by the equation (i) rC ciu d2+ td Ldt cv - r (B + a(- d i v= 0, where P is given by (27). The curves on S defined by (i) are called the axis curves and the congruence the axis congruence of N. By means of (22) equation (i) may be written (ii) rCt du2+ tdudv —r H (~ — rC1- a — a logr) dv — 0. Wilczynski, Trans., vol. 16 (1915), p. 316; Green, Amer. Journ., vol. 38 (1916), p. 308. 20. A necessary and sufficient condition that the axis curves form a conjugate system is..H - r - log 0. Green, 1. c., p. 311. 21. When a net has equal point invariants and is isothermal-conjugate, the ray curves and the axis curves form nets; and conversely. Green, 1. c., p. 321. 22. When a net is subjected to a polar reciprocation the axis and ray congruences aie interchanged. Wilczynski, 1. c., p. 317. 23. Show that at any point on a surface S the tangents to the two systems of curves defined by the equations ai dul"-2 2 bi d t d v +- c1 dv2 - 0, Ca du12 -2 2 b,,l dI v + c2 Cdv2 - 0, 126 IV. Surfaces and congruences in 3-space are separated harmonically by the tangents to the curves of the system defined by al du + bi dv, bj du + ci dv a2 d + b2dv, b2 du + c2dv 24. If the parametric curves on a surface S form a net N with equations (1) and (15), the curves defined by du -- r dv2. 0 form a net, whose tangents at a point separate harmonically the tangents of N and the asymptotic tangents. 25. A necessary and sufficient condition that a net N(x) on a surface be isothermal-conjugate, that is -a- log r = 0, is that the tangents to the axis curves, the curves defined by Ex. 24 and the curves defined by Hidz2 — Rdu dv rKdv2 = 0 be pairs of the same involution, provided that the double lines of the involution are not the tangents of the net. Green, Amer. Journ., vol. 38 (1916), p. 323. Wilczynski, Amer. Journ., vol. 42 (1920), p. 216. Chapter V. Transformations Q. W Congruences. 51. Tangential coordinates of a net. Laplace transforms in tangential coordinates. In [~ 84] we found that a necessary and sufficient condition that the parametric curves on a surface in 3-space form a net is that the tangential coordinates X, Y, Z, W satisfy a Laplace equation a 82 a logc a) alog/D a uav av an a u av From [22, ~ 66] and [34, ~ 67] it follows that the tangential coordinates satisfy also the equation a 28 a8) aX 8 a), (2) + + r'X, 2)a v =r au2 a at 2+ where in terms of the spherical representation of the net D", J22 ill' 221' J11 ' (3) r-D ='11'- ^lP - 2 r2,:Conversely, if two equations of the form (1) and (2) admit four linearly independent solutions, the latter may be taken as the tangential coordinates of a net N, whose homogeneous point coordinates x, y, z, w are given directly by the equations (4) XXx - O aX, zx- -O. We have also the following theorem analogous to the third theorem of ~ 43: 128 V. Transformations V. WV congruences Any four linearly independent solutions of equations (1) and (2) are the tangential coordinates of a net vwhich is a projective transform of the given net N, We call (1) the tangential equation of the net rN. If this equation has equal invariants, we say that the net has equal tangential invariants. The homogeneous point coordinates x, y, z, w and the tangential coordinates X, Y, Z, W of a net in 3-space, satisfy respectively equations (IV, 1) and (1), and also the relations (IV, 5). If the second and third of the latter be differentiated with respect to u and v, we have accordingly V 8 2^ + a xx a (5) 0auxV ~ zyX 0a o, From the first and last of (5) we have in consequence of (IV, 5) 8u av av a ~t Z(6) x _Xx X ax _ From the first and last of (5) we have in consequence of (IV, 5) (68) l~a -- - rls 0. av av t u au~ When the second and third of (5) are differentiated with respect to v and u respectively, the resulting equations are reducible by means of (1) and (5) to /ax alog \ a2 _0 (7) azt a og av2 o v ~V 8at2 0. The minus first and first Laplace transforms of equation (1) are given by a_ -_log E a X a loga (8) X-l- aX X1 — X.a a u au aV av 52. Transformations F in tangential coordinates 129 From ~ 30, (5) and (7) we have X 1 x1 0, x i? X-i' 0, ax, axI D' u 0 2 X 1- -u 0, ~x~l-1~a, Dx10x-1 0. Dv av Hence we have the theorem: -When the Laplace transformations are applied to the tangential coordinates of a net N, the resulting functions are the tangential coordinates of the Laplace transforms of N, but in opposite sense. 52. Transformations F in tangential coordinates. Since the analytical processes of ~ 37 are independent of their geometrical interpretation, it follows that if X is a solution of (1) and jip a solution of its adjoint, namely ____~ a loga Da D log/3 a2 __ (9 Dcuav Dv au au av +( auavav loga/3), there exist two functions 7t and F, defined to within additive constants by ar a aI7 a l IZl~log~ifi, 2~l log-a auLD au av av a (10) - DklD ~j -~log- = -X log ru a. ate au Dv av Consider now a net N with tangential coordinates X, Y, Z, IF satisfying (1). Then the functions X1, sY1, Z1, TY1, given by quadratures of the form (11) aDu u D \V ' a u auav d 2,\1j (which are consistent in consequence of (10)) satisfy the equation _________ a _a l IC a _ a__ (12) + -10-g - log - 07 auav 'c av a au or aD o a = 9 130 V. Transformations Q. W congruences and consequently are tangential coordinates of a net NV. It is our purpose to show that N and Ni are in the relation F, and furthermore to find 2 in terms of the functions defining the transformation F, when Nand N1 are given in terms of homogeneous point coordinates, x and x1. On the supposition that N and JVt are in the relation F, their homogeneous point coordinates are related as in (III, 65). Necessarily we must have (13) 1 aaaX';-0, lax ~v0, which in consequence of (III, 65) and (11) lead to ax alogZ __ _Y au au ~ IyX ' al ax a log1 _ Y av av yX From (III, 60, 65) it follows that f -ay- 0, a u x ay - 0. av Hence, to within a constant factor, the integral equations is i-=zyX. of the above In consequence of (5) we have by differentiation a_ y ax a yax aa a 2 alu ae ' av av ' aav - Y uaav ' so that 1 is a solution of (1). From (III, 65) it follows that (14) i =WyX=- IzX. The analogues of equations (III, 65) are (15) X- = Xi- X I -X X1-.(~5~~~~~):q-x ---7,x-x -- 52. Transformations F in tangential coordinates These functions Xi and X2 satisfy the equations 131 1 6) a 9^ a a - -X11l loges, ~au a U (16) ax al e l h log We must have also a8Zi __Xi, a log,X a X V* aVlogav~-~ /~ xax 3a 0 1 8u, - 0 ax1 0 av which by means of (III, 60) and (15) are reducible to - ax a log <- _ xi a - a it -Xt7 - atx a logo - xi av av ZxXv From these we find that we may take (18) 0 =zX1x =-X2x. Moreover, this expression for 0 satisfies equation (III, 1). As a consequence of (13) and (17) we have that IXlx is a constant. However, we wish it to be equal to zero. Substituting from (III, 65) and (15), we find with the aid of (14) and (18) that this is accomplished, if the additive constants of integration of T and Wf are chosen so that il- 9> J.+ ---=yX - 0, (19) + -2yNC 0. I+ r-2IxY2- o. Thus we have established that the net Ni whose tangential coordinates are given by (11) is in the relation F with N. Moreover, when the transformation is given in point coordinates, the functions X and pt follow from (14), (11) and (10); and conversely, when the transformation is defined in terms of tangential coordinates. In consequence of (III, 65) and (15) the expressions for i and 0 can be given also the forms (20) i= iz,, e-=] xx. 132 V. Transformations ~2. W congruences By making use of the results of ~ 39, we can obtain the equations of the theorem of permutability of transformations F from the standpoint of tangential coordinates. The functions X12 and 921 must satisfy aij - a j ay __ a / Ii a - l~i an-V-K41' a ar ~i)i (ij - 1, 2, i j). The functions 612,, 6 12, T2 21 are given by -- -- 2 ^"i21t i a112T22 i (tI 12 T2 T21 -- -- + - 1221 (21) -- - _ - _ 2 2 21 t i i 112 02_ z 61 112 = 62621 - t -I 122 12 -211 and the tangential coordinates X12 of N12 are of the form (22) aI 2 X12 -X "2 21 XI i+ I "21 i12 A2 -,- 2 X21 X.If equations similar to (20) are to be satisfied, we must have (23)X x12 =z1 xa- 21, 21 -ZX2 Xl 2 =,X2 xl - 12. When these equations are differentiated, we find that the resulting equations are satisfied in virtue of the preceding formulas. Also we find that x12, given by (III; 76), and X12 given by (22) satisfy the condition 2Xl1 x2 = 0. Hence we may take X12 and X21 as just given. Equations similar to (III, 65) and (15) are ijj j - Xj j wx Y i i _x d -i j ( = 1, 2, ( ij Xyij, xj- iJ X — yj Xi, Aij i From these equations, (20) and (23) we obtain jJ?jx-'ijij -- nj T~J 0, ZX2, ij ij - ij -dij = 0. 52. Transformations F in tangential coordinates 133 Consequently when 412 and 421 have the values (23), the expressions (22) are the tangential coordinates of N12, whose point coordinates are given by (III, 76). From the form of (22) we are led at once to the theorem of ~ 23, namely: When N, N1, N2 aud Ny2 form a quatern under' transformations F four corresponding tangent planes meet in a point. When the point coordinates are cartesian, we make use of the preceding results by taking w -1, and assuming that X Y, Z are direction-cosines. Then we have (24) Xx+ yy + z = —Z so that W is the distance of the tangent plane from the origin. From (III, 60) and (II, 20) we have in this case vi = 0'/0 and xl in the formulas of Chapter 3 must be replaced by - x1'/O, so as to obtain formulas of Chapter 2. From these results, (II, 2) and (20) we have X =zXx'38), so that i) is the distance from the origin to the tangent plane to the net iV' determining the conjugate congruence of the transformation. If we call it w and denote by w-1 the corresponding tangential coordinate of NV, parallel to N1, and determining the same congruence, we have (25) = Xx'Z-, = 1 t-1, where X1, Y1, Z1 are the direction-cosines of the normal to NT. If TVW denotes the other tangential coordinate of N1, we can write (11) in the form a (Xi -a L a _ a (26) au (w' ai \W a' av w-1 au. ~ -- / ~' ~ G v au \ Vw-1 a uv 1) In this case equations (15) become Xi x~ _x (27) Xi -- T —, o r-1T o. W 1 IV W~1 w 38) When we are dealing with cartesian coordinates, the symbol ( denotes the sum of the terms in x, y and z. 134 V. Transformations Q. W congruences From these expressions it follows that if E denotes the angle between the tangent planes of N and N1, a necessary and sufficient condition that the harmonic congruence of the transformation be normal is that (28) (w-'-1)2 ~ -wv- o-1 COS (+ 0) + w2 = 0. 53. Transformations a of nets with equal tangential invariants. From (15) it follows that a necessary and sufficient condition that the focal planes of the harmonic congruence of the transformation be harmonic with respect to the tangent planes to N and N1 is that (cf. V, Ex. 8) -j-o - o. In this case we have from (10) a l a a X a log 2 llog =f, log 2 log a. a u i au a v P a v Recalling that a and f are determined only to within factors which are respectively functions of u alone and v alone, we see that in all generality the tangential equation (1) can in this case be written (29) a2 + alogVle i alog +, -O. aua/ u^va a au u av If we put = — 2 w, w being the solution of (29) determining the transformation, equations (10) can be integrated in the form (30) - — ew" Now the tangential equation of N1 is (31) a2l a a l a - aua v alv ea. u auz av Thus the tangential equations of both N and N1 have equal invariants. We say that A and N1 are in the relation of a transformation 2 39). 39) Rendiconti di Palermo, vol.39 (1915), p. 161; cf. also Demoulin, Bull. Acad. de Belgique, 1919, p. 273. 53. Transformations 9 of nets with equal tangential invariants 135 If we put '9' - Q, 1/r W equations (29) and (31) are transformed into a2 1 a 2 a24 2 8 ze av i as~e,), au a jy,, t. 1u/v \j u a v auav au av tv These equations are satisfied respectively by (32) IVI- _Vx, ew' V2J-I/QY, - _ 1 V. 2 p/ 7 Q W~ew Z, P _- z -[/ fQ-w In terms of these functions equations (11) are reducible to a (J a __C a4). (3 Sn ' u Sn (33) a (i 4) - a i V~ a s\ a 2 a V ia V 7 (i - 1, 2, 3). Since N has equal tangential invariants, its spherical representation is the spherical representation also of the asymptotic lines on a surface 2 whose point coordinates, ~, i, 5, are given by the Lelieuvre formulas [~ 79], namely a it an U an' a__ a v a v2 av av Ivsav Similar equations in the functions vi give the point coordinates ~,1 p611, 41, of a surface Z, with the same spherical representation of its asymptotic lines as the curves of the net N1. Moreover, equations (33) express the condition that _1 and Z are the focal surfaces of a TW congruence [~ 172]. The surfaces F and I are associate to the respective nets N and N1, [~ 155]. 136 V. Transformations Q. W congruences From the theory of W congruences [~ 172] it follows that if X, Y, Z and XY, Y1, Z1 are direction-cosines of the normals to 2 and Fl, then (34) v - -X, v-l= / Xi, where - 1/q2 and- 1/e are the Gaussian total curvatures of 2 and:1 respectively. From these expressions, (26) and (32), we have (35) w-1 -. V 0Q w Hence equations (26) become a a IV a V -a V ((36) VaXa a T ) ^(Ve^lW~) -- — ew2 Q W - w (36) _ VQQ1wT1) -ew_ 2 / ( eX),l = eTv,av (X)2 ld we (avtWIetV1) -QWo rv and we have the theorem: Each solution w of the tangential equation of a net with equal tangential invariants leads by quadratures (36) to the determination of a net N1 which is an 1Q transform of N. From [72, ~ 172] we have that the coordinates of the surface 2 associate to N2 are given by equations of the form (37) = + leel (YZ - ZY). In [~ 172] it is shown that direction-parameters of the normals to the focal surfaces of any W congruence satisfy equations of the form (33). Since these are of the form (11) with - O. 0, we have If two nets N and NV1 with equal. tangential invariants are in relation F, and if the surfaces 2 and 2 with the same spherical representation of their asymptotic lines as the curves of N and N1 respectively can be so placed in space that they are the focal surfaces of a W congruence, then N and N1 are in relation 9. 54. Theorems of permutability of transformations.Q and of W congruences. Let N be a net with equal tangential invariants, and NV1 and N2 be Q transforms of N by means of solutions wl and w2 of equation (1). 54. Theorem of permutability of transformations Q1 137 The functions W12 and wV21 defined by the quadratures a a j (38) aw (VI~ i) Q~~/J (ilj -1, 2 i tj) ar a v "(yeTe — wi Iv~j) Q tvt a v -i are evidently solutions of the tangential equations of N1 and N2 respectively. It is clear that wij as thus defined is determined to within the additive function -. The constants ci can be JeQ Qwi chosen so that there are ool pairs of functions W12 and w21 satisfying the condition (39) J 1? VIs+V1 2 w 2w2 040). By applying the results of ~ 52 we shall show that each pair of functions determines a net N12 which is an Q. transform of N1 and N2. In fact, the tangential coordinates X12, K12, Z12, W12 of N12 are given by the following equations analogous to (22): (40) x 1 2XUz ~- i (40) V~Q2 X12 Q~x W12 W21 where / 2 W1 2 W2 W1 WI W_ Wi -2~~ cose-2 coe22 1W (41) 1022 e ____2 + 2 2 ___ _ _- 2 COS____+ 2 _ W1o 2 721 WV12 W21 W12 W21 It is readily shown that, in consequence of (36) and (39), the functions (40) satisfy the equations a 1/u i JQj W i X12) - i -Qw~j aW u (42) a a (K) (ij 1,2, i tj) av a VTi Hence we have established the theorem: If N has equal tangential invariants, and N1 and N2 are two S transforms of N, by a quadrature we can find co1 nets NA12, which are f.Q transforms of N1 and N2. 40) The reason for this choice is to be found in (21). 138 V. Transformations Q2. W congruences If 2, 21 and 22 are the surfaces associate to N, N1 and Ns so that 2 and 21 are the focal surfaces of a TV congruence, and also 2 and 22, the associate 22 of a net N12 can be so placed in space that 21 and 212 are focal surfaces of a Wcongruence, and likewise -2 and 22. From equations analogous to (34) and (37) we have for the coordinates, $12, ',12;12, of 212 (43) s12= - - - (Y12 -I2 YZt). 7/)21 Hence we have the theorem of Bianchil4): If 2 and 21 are focal surfaces of a TV congruence and Z and 22 of a second W congruence, there can be found by quadratures an infinity of surfaces 2i2 such that Z1 and 212 are focal surfaces of a W congruence, and likewise 21 and 212. 55. Nets permanent in deformation. When the harmonic congruence of a transformation S is normal, the focal planes of the congruence are perpendicular, and consequently the angles between the tangent planes of the nets N and AT, are bisected by these focal planes. Now from (28), (30) and (35) we have Q = Q - Hence: A necessary and sufficient condition that the harmonic congruence of a transformation P. be normal is that the associate surfaces 2 and 21 of N and N1 have the same total curvature at corresponding points; moreover, the focal planes of the congruence bisect the angles between the tangent planes of N and Ai. If e =, we have from (34) (44) vi= cos, vl = 1 Multiplying the first of (33) by vi and summing; and also by 3i and summing, we get _ a__ ae e(cos ) a log9 au 2 at au) _ av i ia a log+ au 2 a8u, anu 41) Lezioni, vol. 2, pp. 71-74. 55. Nets permanent in deformation 139 Adding, we have a v - _a 8 V~_ aQ riy i -- au a Differentiating the first of (44), we get a a logQ> (45) a cos E = - (t + cos ) log au aa In like manner from the second of (33) we get (46) cos E - (1 cose) alog av av Expressing the condition of integrability of these two equations, we find that (47) Q u- v, where U and V are functions of u and v alone respectively. Now the integral of equations (45) and (46) is (48) tan -- V+ c' where c is an arbitrary constant. From [~ 141] it follows that in this case both N and N1 are nets which admit an infinity of applicable nets, which we shall call nets permanent in deformation. If D and D" are the second fundamental coefficients of N, the coefficients Dk and Dk' of the nets Nk applicable to N are given by [cf. ~ 141]. (49) Dk tanh 9 D, Dk' = coth D", where p is defined by au f a log-n, av 9 ag othg. (50) - — tanh, cothan. au au av av 140 V. Transformations 2.- -W congruences Moreover, a log e 12a( ( aloge 12v (51) 2{11 the Christoffel symbols being formed with respect to the linear element of the spherical representation of N, namely (52) da2== OdU+2 Rdudv+Odv2. From the formulas of [~ 83], namely GD 2 FDD" ED"2 H H ~ Hit follows that the coefficients (7, ak, O3k of the linear element of the spherical representation of Nk7 are given by (53) (97 = tanh 2SO i=,ak = RI 05 = coth2SO. The integral of equations (50) for Q given by (47) is (54) tanh 9) where k is a constant, such that when k 0 we have the net N, that is No0 N. If we form the Christoffel symbols 1 k and 1121 lu/ a 2je with respect to (53), the equations analogous to (51) are a log Qk - 12 2 U1 1+kV 1 u 2 k U+V 1-kU' (55) a log el _ 212t V' 1-kU ~av UV1(kU~V1+kVP from which it follows that we may take (6 UV 11 ( U)~k 1kU 1+kV (56) e7c ~-k)( -7V 56. Transformations ~2 of permanent nets 141 56. Transformations S~ of permanent nets for which Cito and to. We write the linear element of the sphere in the form (57) do'2zz U2%2du2+21/V cos 2 w du dv +~ dv2, where 2co is the angle between the parametric curves. We denote by X', 7', Z'; X", i1,.3Z" the direction-cosines of the bisectors of the angles between the parametric curves on the sphere, so that we have ax Ia V/C(sin wCX'+ cosco X"), au (58) ax a 1 (- sin wcX'+ cos co X"'). av If these equations be differentiated with respect to v and u respectively and we make use of [(22) ~ 66], we find ax I/ a XI I aX AX -V(Ysin w X, B X"+ V'asinco X (59) aX// ax11 a A X' —JfVcos CX, - -BX'-V 1/4 cos oX, au av where ( a co 1/' a logl~ A -- au v sin2c a2 v (60) (60) ~~~~a co f ~ alog v-Q B iav n au The direction-cosines, X1, Y1, Z1, of the normal to an.Q transform N1 may be written in the form (61) X1 =-X cos e + sine (sin aX'- cos a A"). Since Q,, equations (36) become 6 \ a _ _ a _ _ (a (i)2 k2) u;X_ Q'aU? (QWvXi) Q W VI When the expressions (61) are substituted in (62), we find the following equations: 142 V. Transformations Q. W congruences a log aau+ MVsin 2 w Q + y sin (a + -o) cot-E =, (3 |/ q~ a1ol V-sin (a o)tan -=0, (63) a V av au 2 a 10gw loge, |l -og _ cot 2 V cos (a +- o) +l- I os e 0a a logw,, ^i7t i 1 loge 0. + -tan Ve3cos(a- w )+ --- — =0. av. 2 1 + cose av The condition of integrability of the first two equations is satisfied, and by means of them we show that the last two are consistent, and that w satisfies the tangential equation of N, namely (29) with =V@@cos2. Hence: If Nis a permanent net with the spherical representation (57), each pair of functions a and w satisfying (63) leads by quadratures to an P transform N1 which is a permanent net; and the associates surfaces c and 2' of N and N1 have the same total curvature at corresponding points and can be so placed in space that they are the focal surfaces of an W congruence42). From (37) it follows that the coordinates 1, t*, 5X of r are of the form (64) = - + sine (cos a X'+ sin a X"). Since w in (63) is not a general solution of (29), we say that the permanent nets whose tangential coordinates are X, Y, Z and a function w defined by (63) is a special permanent net. Suppose that N, and N2 are two 2Q transforms of Nby means of functions ai, wi, ei (i -1, 2). In order that one of the ooc P transforms N12 shall be a permanent net, it is necessary that 012 - e. From (39), (41) and (61) we have in this case zw12 (coS e1- cos E2) - w2 [1 - cos cos E2- sinei sin e cos (al- 2)]. It is readily shown that this function w,2 satisfies the corresponding equations (38), namely 42) The existence of such W congruences was established by Bianchi Lezioni, vol. 2, pp. 74-80. 57. Transformations Q of a sequence of permanent nets 143 a __ a /,2 a a w2 2\ (65) - (QW1 W12) - 1 at a1 (Q2W1 W12)= ) a u a 1u t aV \w/ Hence we have the following theorem of permutability: If N is a permanent net and NT, and V2 are permanent nets which are P transforms of N by means of functions a,, wl, e1 and,2y, w e2 (e2 1 e1), of the oo1 transforms N12 one is a permanent net and it can be found without quadratures. Incidentally we have established the theorem of Bianchi43): If 2 and ~- are focal surfaces of a W congruence and.' and '2 are focal surfaces of a second W congruence such that the total curvatures of -, 21 and 22 are the same, there can be found without quadratures a surface 212 of the same total curvature such that 1 and 212 are focal surfaces of a W congruence, and likewise 22 and 2-2. When in particular Q const., N is the net of geodesics on a surface of Voss [~ 141]. In this case E is a constant. Each set of solutions of the corresponding equations (63) determines an Q transform N1 which is a net of geodesics on a surface of Voss. Moreover, these transformations admit a theorem of permutability. The associated W congruence is pseudospherical, and the preceding theorem is the theorem of permutability of transformations of Backlund [~ 121]. 57. Transformations S2 of a sequence of permanent nets. In ~ 56 we saw that in order to obtain an - transformation of a permanent net N into a permanent net NV, it is necessary and sufficient to take for the net N' parallel to N one of the special permanent nets determined by a pair of functions satisfying (63). Suppose now that we consider one of the deforms Nk of N (~ 55). The equations for the LQ transformations of Nk analogous to (63) are (aak _ aw / - sin 2 a log i - + k k sin (a+ C) cot 2 ak + a _ ~7 sin 2 C a log 1V, -V/ sin (cac- 0) tan 0 av +_v V Sa 2o a w — cot 1(k cos (ak,+ )o) + I -.- _ 0O a - 2 1 — cos u an a o tan- V k COS (a —C) + I E - os = 0 av 2 +pp. 80-C82 43) Lezioni, vol. 2, pp. 80-82. V. Transformations Q. W congruences 144 where ek, Ok and Qk are given by (53) and (56), and analogously to (48) we have l___ +_ kck (67) tan-k l _ _ 1 When the expressions for Ok, Sk and Qk are substituted in the first two of (66), we find that ak-= a is a solution, if coth * tan = tan `2 2' which is consistent with (67), if we take kck (1 -ck) =1. On this hypothesis the last two of (66) are reducible to alog k a log ogw 1 U'k (au ait 2 1 - kU' (68) alogwk a logw 1 V'k av v 2 1+kV' of which the integral is wk = awV(l1 -kU) (1 + kV), where a is a constant. We shall show that when a= (1-kc)-1 the corresponding net Nk' parallel to Nk is applicable to N'. To this end we remark that the second fundamental coefficients (Dk)' and (D1')' of the net with the spherical representation (53) and determined by wk are given by [cf. ~ 67] ( ) &2WJk + )11' aW, 111 ' ak - au{- wll Ik ~u. ^ + (69) ( D) a=2w J22 ak + J22' aWk w D) a 2 i1 k au 2fk av where the symbols /rs are formed with respect to (53). When we put k 0 in these expressions, we obtain the expressions for the coefficients (D)' and (D")' for N' by definition. 57. Transformations Q of a sequence of permanent nets 145 When the epressions (66) are substituted in (69), we deduce the relations (D)' f1-kU (D)' (Dr)' I1+ kV (D")' WVk 1 -cc w ' 'Wk 1-icc W From (49) and (54) it follows that we must have (ox)'= l/+kV I-kU IfY. (Dk)' =I~f ~ (D)', (Dk)' = V - (D")'. Hence (70) W 7" - T/-(1 - kU) (1+ kV)T_) 1-Icc In ~ 27 we saw that if N(x) and Nkk(Xk) are applicable nets, and N' (x') and N1(x) are a pair of applicable nets parallel to N and Nk respectively, the F transforms, N1 and Nk,1 of Nand Nk determined by the nets N' and Nk' and the function 0' = F- _X' are applicable to one another. In order to apply this result to the present case we calculate the expression for 0'. From [~ 67] we have 2_2 ____a~k 2 aW aWk ak 2 u4+~ [=k~a )2? au av av 12 2 1/a \ aw2 aw 2, Ix~8 W 2 + 1 2, a 2 a E a w- aw + (Y i" au)2 at tav av Substituting the values from the preceding equations, we find that k -!kC W2sin 2 c [(U+V)i V ot-j1-ksin( - 2o) U'+ V tanjsina2)c] l~aoot 2 2 We note that ic appears in this- expression only in the factor k/l(l-icc), and consequently it does not appear in the equation XI-x - -x -x', since 0 also involves this factor. Hence as ic 10 146 V. Transformations Q. W congruences varies and consequently Nk, we find that all the transforms -A,i are applicable to N144). 58. Transformations Q in point coordinates. Nets in relation 0. In ~ 28 we saw that if N(x) and N(x) are two nets corresponding with orthogonality of linear elements, that is are in relation 0, they have the same point equation, say (71) a20 _ aloga aD alogb ao auav Dv au Da Dv' Moreover, if h and I are any solutions of equations (I, 13), the corresponding parallel nets N'(x') and N'(x) are in relation 0, and 0', given by (72) 0' - x' x', is a solution of the common point equation of N' and N' (cf. ~ 28). Furthermore, if 0 is the solution of (71) corresponding to 0', then the nets N (xj) and NA (1) defined by (73) xl x- -X, x- == — ' are in relation 0 and are respective F transforms of N and N. Suppose now that N and N are nets in 3-space. Then, as follows from [~ 157] they have equal tangential invariants; similarly N, and Nj. It is our purpose to show that N and N. are in relation S, and likewise N and N1, and thus obtain the equations of transformations S2 in terms of the point coordinates. If N(x) is any net with equal tangential invariants, and N(x) is in relation 0 with N, we have [cf. ~ 157] ax _ y _ a x -_ y _ az (74) -- - oYo o -Yo au(74) au - ' D av av av' and ax ay as ax ay az (75 u Yo Dit' Dv v ao Dv' 44) Cf. Transactions, vol. 19 (1918), p. 179. The existence of transformationsQ of permanent nets for which e -= 0, (:= 0 is established also; the restriction in the theorem of page 183 is not necessary, since the second of equations (61) should include the term - V'/(1 + cos a) V. 58. Transformations ~2 in point coordinates. Nets in relation 0 147 where xoYo, zo nd xo, o, Zo are the point coordinates of the surfaces 2 and 2 associate to N and N, and the asymptotic lines on 2' and 2 are parametric. From (74) and (75) we have also (76) (77) xo au 0, VX~ ax -n ^a u^0 v- ax v I ax XvO 0. a2Ov Consequently Xo, Yo, o and Xo, yo, Zo are direction-parameters of the normals to N and N respectively. Furthermore, if the expressions for the derivatives of y and z as given by (74) are substituted in (75), we have in consequence of (77) (78) Xoo + Yoo o+ oo+ 1 - 0. Hence 2 and E are polar reciprocal with respect to the imaginary sphere45) (79) x 2+ + 2+ z t 1 0. If N' and N' are nets parallel to N and N by means of the same pair of solutions h and I of (I, 13), we have ax) _ ay - a D' (80) zo -D Yo Dv' a U a u ax' _ ay' _ av' av ~0 Yov av For the F transforms NIV and Ni1 defined by (73) we have aDX1 _ - D,- __i (81) aD Do a - Yoi aD, (82) a - y aDy Da DU a y u' Dxl - ay, - aZ1 - 01 o -- o av av01 -v a a i a Yo a ' av g b av Yoi aD ' where xoi, yoi, Zo1 and Xol, Yo0, zo0 are the point coordinates of surfaces 21 and 21 associate to NiN and Ni with their asymptotic 45) Darboux, Legons, vol. 4, p. 67; also [Ex. 15, p. 391]. 10* 148 148 ~~~V. Transf ormations ~~. W congruences lines parametric, and they are direction-parameters of the normals to N1 and N1 respectively. Moreover, we have analogously to (78) (83) xoixo+ Yoi1 -I+ ';'0o1l + -0. From (81) we have (84) a1a1 a1a ~ ~ a,~ aut av av au 201\ au ar av aul On diff erentiating (73) we obtain au 01 a 0 -O ' av 0X2_7 f avl o av)' au 012 au au 'ar 02 O\, ao' __ au_ 120 au' 0 a u ) aV{ 012( -j av avk When these expressions and analogous ones in y7' an d -Z. are substituted in (84) and 0' is given the value (72), the resulting equation is reducible, by means of (80) and the f act that x0, yo, zo are directionparameters of the normal to N', to the f orm (85) "1- x Y Similarly we have (86) X z From (85) and (86) we have (87) x0 X-1~+yo 7o1~+ JO1A-1 0, xXO 1+yy1+zozo 0. Consider now the surfaces -Y and 21. The quantities X01 y0, Z0 and Y0l, y-0i oi, are direction-parameters of the normals to these respective surfaces. Moreover, from (78), (83) and (87) it follows that -X3 (x01 - x) = 0, ~X501 (x01-xo) = 0. Consequently 2F and XI are the focal surfaces of the congruence of lines joining corresponding points on these surfaces, and it is 59. Transformations 2 and K of the focal nets of a W congruence 149 W congruence, since the asymptotic lines are parametric on 2 and 21. Hence from the last theorem of ~ 53 it follows that N and N1 are in relation Q, and also N and N1. When a net N with equal tangential invariants is known, the associate surface 2 can be found by quadratures [~ 78]. Then by equations (74) we obtain a net N, with equal tangential invariants, which is in relation 0 to N. Each net N' parallel to Nleads by a quadrature to an z2 transformation of N and N. Hence: - Each net parallel to a net N with equal tangential invariants determines an P2 transformation of N, and the determination of the point coordinates of the transform requires only quadratures46). 59. Transformations Q and K of the focal nets of a W congruence. We have seen that the surface 2 of the preceding section is associate to the surface S on which 1V lies. Since this relation is reciprocal, there exists a surface Sin relation 0 to 2; its coordinates, z, y, z, are given by quadratures of the form (analogous to (75)) ax _ ~ Yo az ao ayo y a o au ai u - a U' v av aY v' From these equations and (75) we see that S can be so placed in space that (88) X = x+ zyo-yZo From [~ 157] it follows that the asymptotic lines correspond on S and S the surface on which N lies. Also since XQ, yo, o and x, y, z are direction-parameters of the normals to S and S respectively, it follows from (88) that S and S are the focal surfaces of a W congruence. Since S and 2 are in relation 0, there can be found by means of equations analogous to (74), the coordinates Xo, yo, o of a surface 2 associate to S; moreover, from [~ 156] it follows that the parametric curves on S and 2 form nets with equal point invariants. Furthermore, since 2 and S bear to S and 2 relations analogous to the relations of - and - to S and 8, it follows that 2 and S are polar reciprocal with respect to (79). 46) Cf. the first theorem of ~ 53. 150 V. Transformations S. W congruences When the net N is subjected to a transformation Q, in accordance with the preceding section, we get surfaces Si, St, 2-, S1 and A2 related to one another as in the preceding paragraph. Since S and 2, and S1 and 21, are polar reciprocal with respect to (79), it follows that 2 and 21 are in the relation of a transformation K (V, Ex. 11). Moreover, S and S1 being associate to - and r respectively are also in relation K (II, Ex. 18). Furthermore, Si and S1 are the focal surfaces of a W congruence. From [~ 172] it follows that if S and S are the focal surfaces of any WI congruence, it is possible to find by quadratures two surfaces S and 2 associate to one another and in relation 0 with S and S respectively. If 2 is referred to its asymptotic lines, the parametric curves on S and S form nets with equal tangential and point invariants respectively. Hence: A W congruence admits of transformations into W congruences such that one pair of corresponding focal nets are, in relation 5 and the other pair in relation K47). 60. Nets with equal point invariants and equal tangential invariants. In consequence of [(38) ~ 83] equations [(36) ~ 83] can be written in the form -logD log a, logD"- log b,. av av au au Hence: When a net has two of the following properties, it has the third also; equal point invariants, equal tangential invariants, isothermalconjugate. We consider such a net, and put J12t a logV'l ) 12} alogV D=-D" /11= — 8 'av 12 au From [(3) ~ 63] and [(13') ~ 64] we have jlt a H ill D l = u log 12 av log v 422) a D l 22l i 8 a H 47) Cf. e ulin 1. c.. 276. ") Cf. Demoulin, 1. c., p. 276. 60. Nets with equal point invariants and equal tangential invariants 151 Consequently the point coordinates of the net satisfy the equations D20 D20 D logr DO alog'r DO ( D2+ D Du Du + v Dv' a20 aDlogV D O a logVYe a D DUD) aDv Du Du Dv' where lID From ~ 25 we have: ITf N(x) is a net with equal point and equtal tangential invariants, a parallel net of the same kind is given by the quadratures Dx 1 ax Dx 1 ax Du e Dii av - avD In order to establish transformations F of a net N of this type into nets of the same kind, we consider N expressed in terms of homogeneous coordinates, the point and tangential equations being PO2O a+ DogI-'- ao a logV aD O auav v au aDu a+ (90) / (9) 2 a logV aD a log DX a _ D Uav Dv Du i aDv It is readily shown that the equations Du Q\Du 1xX1 Di XI _lx xX x ~1 a Dx (D x Dx av (91) XI a x ixlX27 x a ~u alX x~, e r an Dv XD a v X 1 X — 2,7x1 DXi and similar equations in y1, zi, w1, Y,1, Z1 and W1 form a completely integrable system. For each set of solutions the function2x1x, is constant. Consequently each set of solutions, for which 152 V. Transformations 2. W congruences (92) z2 - 0, defines a net Nj for which these functions are the point and tangential coordinates. Furthermore, for a set of solutions the functions (93) 0 -ZXZ x, i = Xxi are solution of the respective equations (90), so that equations (91) are equivalent to (III, Ex. 25) and (36). Consequently N1 is both a K and Q transform of N, and therefore has equal point and equal tangential invariants. Hence, since (92) is bilinear we have: A net with equal point and equal tangential invariants admits 0o6 transformations K and QP into nets of the same kind. If N1 and N2 are two of these transforms of N, from (III, 73, 75) it follows that for a net N2 to be a K transform of N1 and N2, we must have 012 + 021 = 0. In like manner for N12 to be an 1Q transform we have from the equations preceding (21) that 12 +~ 21 0= O Hence from (23) we have X1 x2a+X2x1 = O. It is readily shown that the left-hand member of this expression is constant. Only when it is zero is there a net N12 which is both a K and an Q transform of N1 and N2, and then there are an infinity of such nets, since 012 is determined only to within an additive constant, but 021, 412 and 421 are then completely determined. Exercises. 1. If (1), x(2) x() are the cartesian coordinates of a net N in 3-space x(1) x(2) ~x(3) and 0 is any solution of the point equation of N, then a -, -^-, X-, ao 3 are tangential coordinates of a net. 2. A necessary and sufficient condition that the curves v = const. of a net be plane is that the minus first Laplace transform be a developable surface; the analytical condition is that the invariant H of the tangential equation of the net be equal to zero. 3. A necessary and sufficient condition that the curves of a net be plane is that the tangential coordinates can be taken in the form X=Ui+V1, Y== U2+ V2, Z- Us+ V3, W-= U4+ V4, where the U's and V's are functions of u and v respectively. Exercises 153 4. When two nets N and N1 in relation F are subjected to a polar reciprocal transformation, the resulting nets are in relation F; and the conjugate and harmonic congruences of the given relation F are transformed into the harmonic and conjugate congruences of the resulting relation F. 5. In a transformation F defined by (11) the functions A,.. a (. ) and (-),..., a ( are tangential coordinates of the first and second focal surfaces of the conjugate congruence of the transformation. 6. A necessary and sufficient condition that the transformation defined by (11) be radial is that =- aX+ b Y-+ cZ+ dW, where a, b, c, d are constants. 7. When in (26) we take w =aX- +bY+cZ, where a, b and c are constants, the conjugate congruence of the transformation consists of parallel lines. 8. Show that the functions Xi and X2 given by (15) are tangential coordinates of the focal surfaces of the harmonic congruence of the corresponding transformation F. 9. Each solution of the adjoint of the tangential equation of a net determines a congruence harmonic to the net (cf. Ex. 8). What is the dual of this theorem? 10. If the direction equation of a congruence G is the same as the tangential equation of a net N, the determination of nets harmonic to G and congruences harmonic to N are equivalent problems, namely the solution of the adjoint of the given equation. 11. When a polar reciprocal transformation is applied to two nets in relation S2, the resulting nets are in relation K, and conversely. 12. A necessary and sufficient condition that the axis curves of a net N form a net is that N have equal tangential invariants (cf. IV, Ex. 20, 22). 13. If four nets N, Nl, NT2, N12 form a quatern under transformations 2 determined by wl, w2, i12 and w21, the four nets parallel to them determined by w1, 1/V'ppi wl, w2i and 11/p2 P12 W21 form a quatern; likewise the four nets determined by w2, 1/1/po2 w2, W12 and 1/ pl plz wi2. 14. Let N(x) be a net with equal tangential invariants and N(c) the net of the same kind in relation 0 to N, and N'(x'), N" (x") and N' ('), N" (-") corresponding parallel nets to N and N. If we put 0 =- x'x ', 0/' = - x` Y", then the nets Ni (xi) and N2(x2), where 1 1, 602 XI X -- -X, X2= X — A fXf are 2 transforms of N. If we draw through points of Ni and N12 lines with direction-parameters of the respective forms (cf. ~ 21) Ot! wt1 -: ad a sO > ' - 2 N", N' c where 0'' and 06 are solutions of the point equations of N" and N' corresponding to 01 and 02, the point M12 of intersection of these corresponding lines generates a net N12. Show that this net N12 will be an S2 transform of N1 and N2 if 0f' + 0' -- 2x"x' + 2 x' ". 154 V. Transformations S2. W congruences 15. Show that all the conditions of the preceding example are satisfied, if we replace 06l and 0, by 0''+ c and 0- c, where c is an arbitrary constant, and consequently there are ool nets NU2 in relation 9 with N1 and N2; that corresponding points of these nets N12 lie on a conic through the corresponding points of N, Nt and N2; also that the tangents planes to these nets N12 at corresponding points form a pencil (cf. ~ 26 and II, Ex. 19). 16. Let N be a net with equal point invariants, N1 and N2 be K transforms of N, and AN12 one of the K transforms of N1 and N2 in accordance with the theorem of ~ 26; show that the tangent planes to the coo nets N12 at corresponding points envelop a quadric cone (cf. Ex. 15). 17. Prove directly that the surfaces S and Si of ~ 59 are in relation K. 18. Two nets N and N with equal tangential invariants which are in relation O determine twelve surfaces forming a closed system of Darboux [Ex. 19, p. 391]; similarly two nets N1 and Nt which are 2 transforms of N and N determine a second system; find the relations between the two systems of surfaces other than those treated in ~ 59. 19. If S and S two surfaces in relation 0 are referred to their common nets, N(x) and N(Z), the equations of the form x 4- ex, where e is an infinitesimal, define a surface S which is an infinitesimal deform of S, and the parametric curves form a net N [cf. ~ 158]. If N'(x') and N'(x') are corresponding parallel nets to N and N respectively, the equations of the form x' - x'+ ex' define a net N' which is an infinitestimal deform of N'. Then equations of the form Xi= — x~, where 0'= - x'x', define a net Nt, which is an F transform of N and an infinitestimal deform of Ni whose coordinates are given by (73). 20. When the spherical representation of a permanent net is such that e i 0, ( * 0, the parameters can be chosen so that either (i) p = a, a constant; (ii) p v; (iii) p= u + v. In these respective cases the coefficients of the spherical representation can be given the respective forms (i) = S =1, = = cos 2; i i a ( _, U v/ au, v (iii) (u v) a =-D av; (iii) e== UA(.i+ )-1, a -, v (.+v)-. Determine the conditions which ow and 0. must satisfy in each case. Exercises 155 21. When the curves v - const. of a permanent net are represented on the gaussian sphere by isotropic generators, the coefficients of (52) are reducible to the form 2 2V' + f-GV)2 T ~Vv(1 + Uv) where, V and V1 are functions of v alone; also the tangential coordinates of such a net are given by ___ai__ a2 a3 Xz1-+u +bi y=v1+ —uv +b2 Z- 1+ nv I V1+Uuv where ro and b are arbitrary functions of u and v respectively, and the a's and b's are functions of v alone subject to the conditions: 4 ___ 4 _.i'a~K~zO ia~2 - v a VV 2V b ai+- ai (i 1=, 2, 3), the primes indicating differentiation. Drach, Ann. de Toulouse, ser. 2, vol. 10 (1908), p. 135. Chapter VI. Orthogonal nets. 6I. Nets 0 and p, O. Congruences I and p, I. A net N(x), whose cartesian coordinates x satisfy an equation (1) 260 Dloga 0O Dlogb 0O auav V Du au av' is called an orthogonal net, or for the sake of brevity an 0 net48), if (s) 2 ( x o, F) ax ax 0 a) 0 (2) E = i 0, F O au au av av The second of these conditions is equivalent to the requirement that o, defined by (3) 2 o=zx 2 is a solution of (1). We say that a net N is p, 0, if its point equation (1) admitsp — 1 solutions, y,..., yp-i linearly independent of the x's, such that ay 2D aox\2 ay (4) au au av av ax ax ay ay 0. Du Dv au av We call the y's the complementary functions49). This means that Zx2 y2 is a solution of (1). It is understood that none of 48) The results of ~~ 61-65 are due to Guichard, Annales de L'Ecole Norm. Sup., ser. 3, vol. 14 (1897), pp. 467-516; vol. 15 (1898), pp. 179-227. 49) Note that if each y be replaced by iy, where i2 -1, a net p, O is the projection in n-space of an 0 net in (n +p —1)-space. It is this point of view that Guichard takes in considering these systems. 61. Nets 0 and p, O. Congruences I and p, I. 157 the y's are constant. Hence if we have a net so that the x's and p-1 functions y satisfy x2 —_ y2 = const., we say that the net is p, 0 and not p + 1, 0. We say that the equations (5). yi = ilyl+... +. ip-lyp-l_ (i = 1... p —l), define an orthogonal substitution, when the constants aij satisfy the conditions p-1 p-1 (6) Zjj = 1 i Soaj aiuk 0=, (j + k). When such a substitution is effected upon the complementary functions y of a net p, O, we have y ~2 =f y2 and consequently the functions y serve equally well as complementary functions. When there is a linear relation between the y's, such as (7) al y +.. + ap-1 yp ---= a, where I at + 0, that is, when the relation is non-isotropic, we can assume that C at = 1, so that in the new variables y, we may take yp-_= a where a is a constant. Hence the net is p-1, O. If, however, a at 0, that is, if the relation is isotropic, we can choose the orthogonal substitution so that (7) is reducible to ap-2 p-2+ ap-i yp-i= a, where now ~2p-2 +-~p- 0. Hence yP-1 = - i yp-2+ b, where b is a constant. Since any complementary function may be replaced by itself with an additive constant, we have y2 + -....+y + +pHence the net is p-2, 0, and we have: When for a net apparently p, 0, there is an isotropic linear relation between the complementary functions, the net is in fact 158 VI. Orthogonal nets p —2,0; when there is a non-isotropic linear relation, the net is p-1, O. We say that a congruence is I, when its direction-parameters X satisfy the conditions X = 0, (8) a x o0 2 Since the X's satisfy an equation of the form () 82a0 alogA ao alogB ao au av - av au aue av it follows from the first of (8) that (10) a ax =0. au av It is readily seen that these conditions are satisfied also when each X is replaced by AX, where ~ is any function of u and v. A congruence is p, I when there are p-1 functions Y satisfying (9) such that -X2 Yz o, 0 ai 2 a Y\2/y aX 2 aY 2 i (M) z g + ~' ( 8 )-z ( a + ~ The functions Y are called the complementary functions of the congruence p, I. We have also (12) aX aX ay a. ' au av au av When a congruence is 2, I, we may choose for the parameters the X's divided by the single complementary function. In this case we have (13) JX2-=, a a au av 62. Nets conjugate to congruences I and p, I 159 so that a congruence 2, I is the generalization of the normal congruence in 3-space. 62. Nets conjugate to congruences I and p, I. In ~ 5 we saw that if 0 is a solution of (9), the functions x' defined by (14) '_ x 0 are the cartesian coordinates of a net N' whose radii vectores from the origin are parallel to the lines of a congruence G of directionparameters X. If G is a congruence I, we have from (8) (15) and I 2 ___ I ax \2 1 ly aX)2 2 ax' ax' au av ( av) o,' Hence N' is a net 0. In consequence of the second theorem of ~ 5 and the last of (16) we have: All nets conjugate to a congruence I are 0. If G is a congruence p, I, the p -1 functions y' defined by Y (17) y are solutions of the point equation of N'. In consequence of (11) and (12) we have (18) a a Y ax' ax' au av av ay + lay, ay' _ au av Hence in general N' is a net p, 0. When, however, 0 = aY1+... + ap-1Yip-, (a? t+ 0) 160 VI. Orthogonal nets there is a non-isotropic linear relation between the functions y', and N' is p —, 0 (cf. ~ 61). When in (19), (20) = 0, there is an isotropic linear relation between the functions y', and N' is p- 2, 0. We have shown that the nets conjugate to the given congruence are parallel to nets of the type (14), and are given by equations of the form (I, 12). By means of the same equations we obtain p-1 functions y from the functions y'. It is readily seen that these functions satisfy equations of the same form' as (18). Hence the nets conjugate to the congruence are of the same type as the parallel nets N'. In accordance with the fifth theorem of ~ 5 there are olI of these nets parallel to each net N'. If we call them a parallel family, we have: Of the nets conjugate to a congruence p, I, there are 00o-3 parallel families of nets p -2, 0, o0p-2 parallel families of nets p- 1, 0, the others are p, O. In particular, we have: Of the nets conjugate to a congruence 2, I there is one parallel family of nets 0, and all the others are 2, 0. The nets 0 arise from the case when 0 in (17) is equal to Y. For a congruence 3, J, when 0 is equal to Y,+ i Y2 or Y —i Y2 condition (20) is satisfied, and only in this case. Hence: Of the nets conjugate to a congruence 3, I there are two parallel families of nets 0, ool parallel families of nets 2, 0, all the others are 3, 0. 63. Orthogonal determinants. Since F= 0 for an 0 net, it follows from (I, 3) that a and b in (1) may be chosen so that (21) a=- E, b= G, in which case the normal parameters i and 'i of the net, given by (22) au = BVE, av au av 63. Orthogonal determinants. 161 are the direction-cosines and are in the relations In23) thi cs (, 6, 7) beco 0e Tn this case (1, 6, 7) become (24) and (25) av VW a u frl, iava I a VE Consider now the orthogonal determinant zI~2 X I 1 X2 V7 2 A2~~~~~ (26) c1 2.2 X XV, 2 n-2- - - 2 ~1 ~....1 I 1 2 n I "I 1..... the X's being functions of u and v satisfying the conditions 12 i=1 1 1xi xi o 0, JgliX~z, i=1 i=1 zexi 0. Therefore we may look upon these functions as the directionparameters of n mutually perpendicular lines. Since the first derivatives of the X's of any row are direction-parameters of some direction in the space, they are expressible as linear homogeneous functions of the other terms of the same column. It is our purpose to show that there is a set of X's such that the latter expressions take the simple form (28) - ak, -and (29) aC _ aU k a k bz, av.4akXk -Mn, __ - ~Ib Xk —It, 11 162 VI. Orthogonal nets the }, i and the X's in each equation having the same superscript, and aic, bk, m and n being the same for all values of the superscripts of }, / and the X's. From (24), (28), and (29) we get a2$ an a ak bk7 7rnbkXk7 ) t am 8 m a ( -ab kX b + w -bkX) -i - + An D +n aDuav \v 6v a2Xk aak ab a- M8U= a tk - + + an - k + bow auav - ar a+ukny~7-fl3 Since there are n equations of each of these types, obtained by letting the superscript i take values 1, 2,....n, the following 2n + 3 relations must be satisfied: aak bk - = mbk, -a iak av a u (30) av - a -m a+ n + akbk = 0. av au k In consequence of (25) the last of these is equivalent to a~iafE a/i aVG (31) av 1/ a O auv 1 - +akb =0 Whenever we have a set of functions satisfying these equations, the system (24), (28) and (29) is completely integrable. Moreover, from the form of the equations of the system it follows that each set of solutions satisfies the conditions (32) tz 2 X2+ j2+ - const., (32) f 2 Xi + Mi cj m_ if _ = const. k K k 63. Orthogonal determinants 163 Hence by a suitable choice of the constants of integration the functions will satisfy (27). It can be shown, as in the case of three dimensions [cf. ~ 65], that the most general solution can be obtained from a particular solution by effecting an orthogonal substitution with constant coefficients on the elements of the determinant 4; thus n n a (33) XJX, -ai f Xi t C j=1 j=1 j=1 A determinant of the type (26) satisfying the conditions (24), (28), (29) and (30) is called an orthogonal determinant in space of n dimensions. Suppose we have 2n functions 5 and g satisfying conditions of the form (23) and (24). We inquire whether there exists a corresponding orthogonal determinant J for which these functions are the elements of the last two rows. By purely algebraic processes we can find n(n -2) functions YT, (i 1,....n; j-, =.... n-2) such that.Y1..... Y^n WI an is the determinant of an orthogonal substitution. Since ilk o, n r i-1 i=1 we have equations of the form aYi (34) I |a Hi i - -, A d - 11* 164 VI. Orthogonal nets where i - 1,... n; j, k - 1,... n- 2 (j t k). Expressing the condition of integrability of the first two equations, we find that the functions P, Q, A, B must satisfy the following equations: i -~-_P~a p a _ p P (35) a B- = PjkBk+ Ajn, aV _ zQjkAkc+Bjm. We are interested, however, in seeing whether there is a set of Y's for which the first two of (34) shall assume the form (28). Suppose we put 2....... Xk - Y1 + Y' Y2 + Y 1-2 Yn-2 1, 2 \k 2 ' where the y's are the elements of an orthogonal substitution of order in-2. These functions X' serve with the d's and N's to make / (26) the determinant of an orthogonal substitution of order n. We have accordingly in the general case equations of the form (34). If, however, we wish the first two to be of the form (28), the functions y must satisfy the equations (36 _ -2 a yk n —2 (36) V Yc + I + P -O + Q 0. au J71 av 1j=, The conditions of integrability of these equations reduce to the first equation (35). But all of the latter are satisfied, and hence there exist solutions of (36). Hence we have the theorem: fthen 2 n functions a and y are known satisfying (23) and (24), there exist functions Xi which with ~ and q form an orthogonal determinant S. From the form of (36) it follows that the functions y are determined to within an orthogonal substitution with constants coefficients. The effect of such a substitution on the y's is equivalent to an orthogonal substitution of the form 64. Determination of 0 nets 165 a -2 Cej- x X -- X j~1 on the elements of the columns of the matrix of the first n-2 rows of J. Hence the last two rows of an orthogonal determinant J determine d to within such an orthogonal substitution with constant coefficients effected upon the matrix of the first n- 2 rows of dz. When an 0 net is known, the functions ) and " can be found at once, and the further problem of putting its determinant in the canonical form requires the solution of 2 (n- 2) completely integrable equations of the form (36). Guichard50) calls this a problem of the order n-2. In particular for a net in 5-space this is equivalent to finding the direction-cosines of an 0 net in 3-space when the rotations of its trihedral are known. In [~ 65] we saw that this requires the solution of a Riccati equation. 64. Determination of 0 nets. We consider now the determination of the 0 nets corresponding to a given orthogonal determinant. Evidently the coordinates x of any net NA are expressible in the form n-2 (37) x - X + q +ry, k-i 1 the superscripts of x, X, f, and V being the same, and py, q and r being functions of u and v independent of the superscript of x. We have now to find the conditions to be satisfied by Pk, q and r in order that (37) be the equation of the net. By differentiation of (37) we find au - a~ X k-a ax nP~- 2 ~\Q ) aaq,kS a a a a2q +mir)+t7 M h-i k=1 \zk aU ax -2 a-P, a,, q var a: vt (a bk r Xk+ k-( -a-vvrn) -I b7 p+ a i +l q' Equating these expressions to those of (22), we see that we must have 50) L. c., p. 500. 166 VI. Orthogonal nets 8 — ak, Dp -bk, (38) | = ak sP mr+ ~ = r, r r -- v bk pl - n q + I/G, where m and n are given by (25). By differentiating these equations, we find that they form a completely integrable system, and consequently their solution involves n arbitrary constants. We find also that a function ps and the n functions Xk are solutions of the equation (39) a28k Dlog ak a k alogabk DOk (39) - a Duv Dv au a u v Suppose now that we have a solution of any one of these n —2 equations, say pi. From the equations a-p1 -pi cl aP -blr a DU al q, Dv- bi r 8u 3v we find two functions q and r. Then from the other n -3 equations of this type we find by quadratures the functions pk. When these values are substituted in (37), we have an 0 net corresponding to the given orthogonal determinant. Hence: The determination of the 0 nets corresponding to a given orthogonal determinant is equivalent to the integration of any one of the n-2 equations of Laplace (39). We shall speak of pkc and Xk as the tangential coordinates of the 0 net defined by (37). From (3) and (37) we have (40) 2 o - 2+ q+ '+. Making use of (38), we find (41) u - Gq, D G. 65. Congruences conjugate to 0 nets 167 As an immediate consequence of these equations we have the theorem: A necessary and sufficient condition that an 0 net corresponding to a given orthogonal determinant be such that IX2 is constant is that its coordinates be given by n -2 (42) x = ek Xk, k-1 in 'which the e's are constant. When we have a solution 0 of the point equation (1) of an 0 net N, the functions q and r given by (43) fa Eq, v o Gr satisfy the fourth and fifth of equations (38). These functions and the functions pk (k 1,...., n-2) obtained by the quadratures apk alpk (44) a - a, bv~= br, determine an 0 net N', whose coordinates are of the form (45) x' - pck Xc + q~ -+ r1, and which corresponds to the same orthogonal determinant as N. Since the p's are determined only to within additive constants, there are con-2 nets V' corresponding to a solution 0 of (1). From (41) and (43) it follows that when 0 -=, AN' coincides with N. When 0 is a constant, we have the case of the preceding theorem. Hence we have: Each solution of the point equation of an 0 net N other than o given by (3) leads by quadratures alone to co',-2 nets N' corresponding to the same orthogonal determinant as ~. By means of (40) and (41) we establish the converse theorem: Each net parallel to an 0 net N gives by quadratures a solution of the point equation of N. 65. Congruences conjugate to 0 nets. From ~ 5 it follows that the direction-parameters of any congruence conjugate to an 0 net N can be chosen so that they are the cartesian coordinates 168 VI. Orthogonal nets of a net N' parallel to V, and therefore corresponding to the same orthogonal determinant. These nets N' are of three kinds. 1~. When the coordinates of N' are of the form (42) with the e's in the isotropic relation (46) l e 0, the congruence is I. Evidently there are oo0-4 such congruences. 2~. When in (42) the constants satisfy (47) e2 = the congruence is 2, I; there are oon-3 such congruences. 3~. For the others, if we put Xi —tn Y x/ +1) Y", (X-2_ 1)' we have:x 2 y= -,y22 PXVY /8aY1\\ 2 Y a 22x' 2 ( \2 a (afl2 ( a l ax )2 ~h" [j\ ~v [ ~ ~jv J2 [^v + Hence all the other congruences are 3,, and we have: Of the congruences conjugate to an 0 net in n-space, no- 4 are I, oon-3 are 2,, and all the rest are 3, I. In view of the second theorem of ~ 64 and the above results we remark: 1~. In 3-space of the congruences conjugate to an Onet, one is 2, I and the others are 3, I. 2~. In. 4-space of the congruences conjugate to an 0 net, two are I, their parameters being Xi + iX2 and X1-iX2; co1 are 2, 1 and the others are 3, I. 66. Transformations F of an 0 net for which the conjugate congruence is normal to the net. A congruence of direction-parameters X is said to be normal to a net N(x), if (48) X a=-0o, a0 av (48)a m a 66. Transformations F of an 0 net 169 When we take for the direction-parameters the cartesian coordinates x' of a suitably chosen net N' parallel to N, these equations of condition may be replaced by au a,' ~ v a ~ 0 a ze av from which follow (49) Ix'a2 k uax' av au av ax Dx - 0 where k is a constant. Hence the congruence is I or 2, I, according as k is zero or not, provided that 2(ax) o D u _o, ( av _ 0, which we assume to be the case. Furthermore from (49) we see that N and N' are 0 nets. Hence: A congruence conjugate to a net for which (50) holds can be normal to N only in case N is an 0 net; then the congruence is I or 2,1. Conversely: A congruence I or 2, I is normal to the 0 nets conjugate to it. If a congruence G conjugate to an 0 net N is normal to SN, it follows from the second theorem of ~ 64 that the coordinates 5: of the parallel net N determining the congruence are of the form (51) n -2 - lek x, k=-1 where the e's are constants. By differentiation sequence of (22) and (28) we have in con (52) where (53) ax A_ x u -h au' ^= ee ak 7k= 1 E ax _ -ax av av' n-2 v= ek b@ I - = V 'G. kc== y o 170 VI. Orthogonal nets An F transform No(.xo) of N for which G is the conjugate congruence is given by (~ 15) (54) Gtt -- X -, X, xo~~~ where 0 is a solution of the point equation of V, and (55) a a _ - ao Sut ' & o _ ao -lav If p7c are the solutions of (43) and (44) for this function 0, we find that these equations are satisfied by (56) n1-2 0 -= ekc1k. c=1 We assume that it is this value determine No (cf. ~ 17). From (II, 9) we have (57) a =^ X - a I 02 a QU u of 0 which is used in (54) to axO a ^ a O ax\ av- K2 -a av - av o av av' and consequently (58 aX0 2a aN -Xo ao -aIx ^ ax 2 a ^\ au av o \ an av an u av aV av. When the congruence is I, that is when (46) holds, then a X a Xo - 0, whatever be 0. Hence all the F transforms are au av 0 nets (cf. ~ 62). When G is 2, I, that is when (47) holds, we have from (58) (59) ax0 axO a 0o a 0o 0 3 au av au av where 0o, given by (60) 00 6 67. Transformations F of 0 nets into 0 nets 171 is the solution of the point equation of No which determines N as an F transform of No (~ 16). Since (59) is the condition that x o — 0 be a solution of the point equation of No, we have that No is a net 2, 0, 00 being the complementary function. Furthermore from (54) and (60) we have (61) 0(x- o)2 - 0. From. (44), (56), (59) and (60) it follows that, when 0 in (43) is constant, No is an 0 net and is parallel to iV (~ 16). Hence: The F transforms of ai 0 net for which the congruences of the transformations are normal to N are 0 nets and 2, 0 nets; when the transform is 2, 0, the complementary function is equal to the distance between corresponding points on N and the transform. When in particular 0 = w, we have from (52) xx = a, where a is the corresponding solution of (55). Hence Jx2 _- 0o that is, the hyperspheres with centers on No and radii given by 00 pass through the origin. Each of the o0o-3 congruences normal to an 0 net is conjugate to a net of this kind51). 67. Transformations F of 0 nets into 0 nets. In ~ 65 we saw that in n-space the congruences conjugate to an 0 net are I, 2,I or 3,I. In the preceding section we discussed the transformations F of 0 nets for which the congruence of the transformation is I or 2, I. We consider now the case where the congruence is 3, I. We recall (~ 62) that in addition to N and its parallels there is one other family of parallel 0 nets conjugate to the congruence. We shall make a study of the transformation F of N(x) into one of the nets of the second family, say N1(xi). We recall from ~ 15 that the coordinates of N1 are given by 0 (62) xl=- -- x Tx, where a x' ax ax' ax (63) a - au' a, l av' 51) Similar results follow when 0- = — (x —a)2, where the a's are constant. In this case the spheres pass through the point whose coordinates are the a's. We shall refer to this as the case where 0. wo. 172 VI. Orthogonal nets and (64) Also we have a D' aD au aWu' a e' a "z" - 7_ (65) Dx 2 ( 8o'0, ' Du at 'h x u 9ui' ax' v a I,a ' _,x'\ Dv o'21 X Dv DvI T and a having the values (66) r = hO-0, - 10- 0'. a necessary and sufficient condition From (65) it follows that that NA be an O net is that (7 l a? a 0ea a X o a o I ax, (67) D D0O'n0 i 2 f d y x D Du av Du D afv Dv 0 This equation is satisfied by (68) 0' The coordinates x' of the net N' parallel to N which are the direction-parameters of the congruence of the transformation are of the form (69) -- 2 k Xk+ q+r where pk, q and r satisfy equations (38) with E and G replaced by E' and G', the first fundamental coefficients of N'. Hence equation (68) may be written (70) o' = 2 2 + (2 ~+ 2). 2 2 k The corresponding function 0 of the transformation is given by the quadratures (64), which are reducible by means of (38) to (71) o -E a t ao = VG-. 9v 68. Transformations R 173 From these equations and the results of ~ 64 it follows that 0 is the solution of the point equation of N which determines N'. Hence 0 is given by a quadrature and the additive arbitrary constant leads to a family of parallel nets NV (~ 16). Accordingly: Each net N' parallel to a given 0 net determines ool parallel 0 nets each of which is an F transform of N; they are obtained by a quadrature. Conversely, if 0 is any solution of the point equation of N other than co, and N' is any one of the oon-2 nets parallel to Ndetermined by 0 (~ 64), the function 0' defined by (70) satisfies the corresponding equation (64). Hence: Each solution of the point equation of N other than o leads to oon-2 0 netsNj (x) which are F transforms of N; the coordinates xi are given by Ox' (72) xi - x — 2 x, where -2 I D e O0 (73) x'-, 1 pkX 1 aZ + - -_+ - a v X Pk=1 YE au~ VG av the functions Pk being obtained by the quadratures Dpk s a PCk bk aD (74) k Da - -/E au' av V1G Dv Moreover, all of these nets N (xi) are harmonic to the congruence harmonic to N determined by 0. 68. Transformations R. The nets No and N1 defined by (54) and (62) respectively form with Y a triad (~ 20). From (51), (69) and (56) we have (75) Ex' -ek p = 0. Consequently we obtain from (65) and (60) ( DxO) -XU _ OI -z ) aX (xu D v ~(xI - XO)2 (0)2 O0. 174 VI. Orthogonal nets From these equations and (61) we remark that the hyperspheres with centers at points of No and radii determined by 00 are tangent to N and N1. Since there are oo-3 congruences 2, I conjugate and normal to N, there are oon-3 nets No forming such a triad with Nand No. Hence: If N and N1 are two 0 nets in the relation of a transformation F, for which the congruence is 3,1, there are oo"-3 two-parameter families of hyperspheres tangent to N and N1; the locus of the centers of the hyperspheres of any family is a net No which is 2, 0, the complementary function being the radius of the hypersphere; the nets N, At and No form a triad under transformations F. In view of this result we call the transformation from N into N1 a generalized transformation of Ribaucour52), or simply a transformation R. We call the corresponding net No a central net of the transformation. From (63), (64) and (68) we have (ad) v;tax o, ax ae ax ao (76) D x IDu Du' Dv av By means of these equations, (57) and (75) we find that ax O=o, y x~ -o. Du av Hence: The tangent planes of No are normal bisectors of the joins of corresponding points of N and NI. Also it follows from ~ 17 that: Corresponding tangent planes to N, Nt and No meet in a line generating a congruence harmonic to each of these nets. By means of N' we can obtain a transformation R of any net parallel to N (~ 67). From the above results we obtain the theorem: When a transformation R of a net Nis known, a transformation R of any net N parallel to N can be found by a quadrature; the transforms of N and N are parallel, as are also corresponding central nets of the transformations. 52) In his study of cyclic systems Ribaucour considered two-parameter families of spheres in 3-space upon the sheets of whose envelope the lines of curvature correspond. The relation between the two sheets has been called a transformation of Ribaucour. 69. Transformations R in another form 175 When in particular N is N', the transform of Nis the origin, as follows from (62). This is the case treated at the end of ~ 66. From (II, 1) and (68) we have 2x' Hence: When 1N and NT, are Onets in the relation of a transformation R, the nets 1N' and NI whose coordinates are direction-parameters of the congruence of the transformation may be obtained from one other by inversion. In ~ 75 we shall show that only for certain types of 0 nets do there exist transformations F into non-parallel 0 nets which are not transformations R. Hence: In general an inversion is the only radial transformation of an O net into an 0 net. The function wo, where (77) 2 co= x;, is a solution of the point equation of Ni. We wish to show, furthermore, that co, is the transform of o by an equation analogous to (62), namely 0 (78) oo -- Co- 0- ~t, where o/ is defined by (79) aw h I au a t' v a v When the expression (62) for x1 is substituted in (78), the latter reduces to (80) o + ' — xx' 0. By differentiating the left-hand member of this equation we find that it is constant. Hence the additive constant of co' can be chosen so that (80), and consequently (78), is satisfied. 69. Transformations R in another form. From (65) and (68) we have (81) E,-2d ( \-)E G/ = (ax)2 ' ~~~~~~~~~\-~~ ]u — g 0' 176 VI. Orthogonal nets If we choose the algebraic signs of V1-E and VGh, so that (82) f = jfE' G' we have from (66) (83) I - 1+ El GI 88 '\m '-F If we define a function w by (84) 0' o ow, equation (70) becomes n-2 (85) p2 + q2+ r2- 2w - 0. k-=l The equations (62) of a transformation 1R may be written i (86) xi - w T 7 Xk+ 2 + r, where the functions involved satisfy (85) and -= VEq, - = V-,G, au av apk apk ^P k k aq - n- 1 — a.- r~ K E) I au 7zla7GPck V G av r+w(VE+ El (87) a q 1 a G ar 1 a-'VE av ~/E a U at 1/G av ar -2bk aVG alogw -_ q a logw 53). au av a log w an log w 53) The expressions for ag and agw follow from (64), (71), (83) d (84), and are obtav cand (84), and those for and are obtained by differentiating (85). 6u ov a 69. Transformations 1? in another form17 177 The integrability conditions of this system of equations are satisfied provided that 6 is a solution of the point equation of N and that aVE1 Va1 1_ av - y7G av E (88) ___ VE~~1 ( ( - VV+,/) 4) as can be verified readily with the aid of (21), (30) and (31). Moreover, any set of functions satisfying (87) make the left-hand member of (85) constant, as can be shown by differentiation. If we write (89) ax -I / 1. aX1 __ TIl it follows from (65) and (82) that (90), _C q, - Si 6' q1' If we define functions X1, k by (91) 19n (ic - 1... ) the functions ~, and Xi, 7, satisfy equations of the form (27). By differentiation we have (92) where av -ni,. ax1 k,,k - J an 1lcu. anj aXl,k - bi, k fi, av~~~ Ml M r(V, V I6 1 ~~~0 (93)j PkJ t a, ait - Cak- k E( /+ 16IE,), 6 Hence (~ 63) the functions ~1, ~1, X1, k are the elements of the orthogonal determinant corresponding to _N1. 12 178 VI. Orthogonal nets 70. Inverse of a transformation R. Let p~1, q~1, r-l, w-1 denote the functions determining the inverse transformation by means of which N is obtained from Ni, so that X =1 x(-,, PT,1, c-1 1+,-~1). When this expression for (x —x) is equated to that given by (86), and Xl,k, at and Ia are replaced by their above values it is readily found that pA1 Pk= q-1 - 1 - W. jpr- ~Qqck, q =Qq, w -- where Q is to be determined. When we require that these functions and 0-1 satisfy an equation analogous to (85), we find that 0-1- -- 0. Comparing this with (II, 19), we have -= 1/0'1/Ow. Hence ___ P-1- q -Ph p 0 q Ow' (94) ___ I 1 | 1- I -1-=_ 0-1 =_ Ow 0W7 ~ 0 TV-' By means of (93) we show that these functions satisfy a system of equations of the form (87) for N1. From (51), (54), (56), (60) and (91) we find (95) x- Xo= - o, ek Xi,, which shows that the parameters of the congruences conjugate to Ni and No and to N and No are the same linear functions of the quantities X1,k and X7k respectively. 71. Transformations R in tangential coordinates. Let Pk, Q and R denote the tangential coordinates of an 0-net N(x). They satisfy (cf. ~ 64) a Pk ac Q Pk bk P, =ak Q, bkRI au av (96) a a -/, ao _ VGR, aue av 72. Theorem of permutability of transformations 1 179 where co is given by (97) Also we have (98) a 2 &) == jPk2+ Q2 + R2. f kP k Q + RI Equation (80) may be written (99) (0 =I PPk pk+Qq+ Rr-, where p,, q, r and 6 determine a transformation P of N into an 0-net N1 (xi), whose coordinates are of the form (100) xI,-,I PI, k XI, k + Q1 5I + -RR III the functions P1,,,, Q, and R1 being the tangential coordinates of N,. In consequence of (86), (90) and (91) we have (101) Pi,k z 'pk.- P Q C oV 0~~~z~qQ f R-n r -B. 6IE These formulas and (91) define the given transformation 1B in terms of the tangential coordinates of N and N1. 72. Theorem of permutability of transformationsR. We apply the theorem of permutability of transformations F to the case of transformations R, and assume that IV, and N2 are two R transforms of N determined by functions et and 62 respectively, and nets N' (x') and N" (x"). Now the analogues of (68) are (102) x'2 = 2 6{, fx/2 2- i. Making use of (I, 31), we obtain a D1- -2:, Dxi a UD au' DO02 - i Dx aDU Ix ar a D1 =-Ix ax Dv Dv' 8 8 a V Id DO2 2" aDx Dv Dv, 12* 180 VI. Orthogonal nets By means of these results we establish by differentiation that x' xf- xO'- O' is constant. From the above and (II, 33) we have (104) Z1(x) 2 0'-2 (zx x.- Ofi). The functions 01' and 02 are determined to within additive constants. Hence there are sol cases for which (105) X' X" — '- - o. Then, as follows from (II,38), (106) 0t- 2 Hence, by ~ 67, N12 is an R transform of N1. As (105) is symmetrical in functions of the two transformations, N 2 is an R transform of N2 also. Consequently: If N1 and N2 are R transforms of an 0 net N, there are ool 0 nets N12 which are R transforms of N1 and N2, and they can be found by quadratures54). The transformations from N into iVi and N2 are given by equations of the form (107) Xix- X- (ZpX1,x k+ q +r ) (i =1 2), Wi where the functionsp, q and r are solutions of (87). If we write the coordinates of Ni" (xi"), which are the directionparameters of the transformation of N1 into a net N1,, in the form (108) xi" = p 2, X1, + q12 t1 -+ r2 l, k 54) This theorem is the generalization for n-space of the similar theorem for 3-space established by Bianchi by applying the Lie line-sphere transformation to the theorem of permutability of W congruences; cf. Rendiconti dei Lincei, ser. 5, vol. 131 (1904), p. 361. 72. Theorem of permutability of transformnations? 181 it follows from (II, 33), (90) and (91) that (109) P12,7c k -pl, k-P2,k, q.1= o- qI-2, r-l2 O- ri-r2 These quantities satisfy the relation P12,1c + q12 ~ r12 = 20.1,.k and also because of (II, 37) the equations 3 012 V__ -0l. a 12 -/ o rl, D2-VE^ - V~i~q1 DEl2 ~G, r12, atu av aO12, k a,q aDpl2, k b al, k ^21 -- hi, ] ~'12 au av The tangential coordinates of T12 are given by the following formulas which are analogous to (91) and (101): i2, k XI - X1, k, P12, k - fPl12, k -1,c, q12 q12 (110), t ft Q12 === ffs 2- Ql? -Rl2 Off ^f 11-2 q12 q12 where 012 is given by (II, 38) and 0j' (111) j"-t -co"- I'_ When these expressions and those from (101) and (109) are substituted in (0' 2 x')pI, i7 + (' XI - 'O x")p2, k XJ12, k - k+ _J 2 - -: 2, (k 1 12 c 2 2)1 7 + 192, Q2 q1 + 0l1 q2 (112) (P1olf 1, Pfc ' Q12 - Q+_ q1 + 2 1 12 O1 q12 R2z -- + -q2-r + P2-r2 I1 12 where (113) Q-1 O '- Ow", z 02o" -2w' we find that they are satisfied. 1.82 VI. Orthogonal nets Since the first of (112) involves the direction-cosines of N, the coordinates of N' and N", and solutions of the point equations of N' and N", we have the theorem: If N,1 N2, N12 bfrm a quatern under transformations B, N is any net parallel to N, and Nj and N2 are R transforms of N parallel to N1 and N2 respectively, lines drawn through points of N1 and N2 parallel to the joins of N1 and N2 respectively with N12 meet in points of a net N12 parallel to 12. 73. Cyclic congruences. Let N(x) be any net and N1 (G,) an F transform defined by (62). The tangent planes to N and Nl meet in the lines of the congruence harmonic to N determined by 0; the coordinates y and z of the focal points F1 and F2 of the congruence are of the forms (II, 22) 0 Dx 0 Dx (114) y~x- O -- Z - - — v D — 0 av a30 ae' Dv au If M and Ml denote corresponding points of N and N1, a necessary and sufficient condition that F M- F-F M1 is that Z x D -2 x' 0 =-0. av av Hence in order that F1 Mi= F1 Mi and F2 M1= F M1, it is necessary and sufficient that 'r cCYx ofr 0 2= 0, where c is a constant different from zero. In this case N' is an 0 net, and N and NV are 0 nets. Consequently (~~ 66, 67): Wheni N and N1 are in relation F a necessary and sufficient condition that each of the focal points of the harmonic congruence of the transformation be equidistant from the corresponding points of N and NA is that N and N1 be O nets in relation R, or conjugate to a congruence I. We consider the case for transformations R. In consequence of (22) and (87) the expressions (114) are reducible to the forms (115) - y -= x-2 1, = - x-R, 73. Cyclic congruences 183 where 0 8 (116) R1= -, J=. From these equations and the preceding theorem it follows that the hyperspheres, S1 and S2, with centers at the focal points and of radii PR and R2 pass through corresponding points of N and of each of the oo11-2 R transforms of N by means of 0 (~ 67). Moreover, since (117) a (y- ) 2+e,, S1 and S2 meet orthogonally. From the fifth theorem of ~ 10 and the second of ~ 67 it follows that the hyperspheres with centers at the focal points of any congruence harmonic to an 0 net N and passing through points of N meet orthogonally and pass through the oo'1-2 R transforms of N determined by the solution 0 of the point equation of N which determines the harmonic congruence. We call them focal hyperspheres of the congruence. In 3-space S and S2 are spheres meeting in circles orthogonal to oo,' R transforms of IV, that is in circles of a cyclic system. Conversely, in [~177] it was seen that the 0 nets orthogonal to the circles of a cyclic system are harmonic to the congruence of axes of the circles. Hence we say that for space of any order any congruence harmonic to an 0 net is a cyclic congruence. From the last theorem of ~ 13 we have the theorem: If the direction-2patrameters X of a cyclic congruence are solutions of the equation a o a alog ao a logB +a -CO (119) X2=-= U2 2A2+ VT B2, vwhere U and V are functions of u and v alone respectively. In ~ 16 we show that the condition (119) is characteristic of cyclic congruences. If we put X — y —, equation (117) is a special form of (119). 184 VI. Orthogonal nets In fact, from (I, 54, 55) we have (120) ~y a log2 a a logR1 a Ut a u (Y a V a If we express the condition that X satisfy ( 18), we have (121) RI4 AU, A R2 BV, where U and V are functions of u and v alone respectively. 74. Multiply cyclic congruences. Let N -and Ni be two o nets in relation F and let G denote the harmonic congruence of the transformation F. If N and N1 are R transforms of one another, the focal hyperspheres with centers on the focal nets of G and passing through points of N,' also pass through the corresponding points of N1. If, however, Ni is not an P transform of N, then there are at least two pairs of focal hyperspheres associated with G. If we denote by P, and R2 the radii of the focal hyperspheres passing through points of NAT1, it follows from (121) thatR, - U1 RI and -R2 = VI -R2, where U1 and V, are functions of u and v alone. From (111) it follows that I' + 11 - 11 -- R~. From these two conditions we find that the functions R must be of the forms (122) 11= Uk 2 12= V2; R, U2 L12=+ 2 c, where c is a constant and ~ is to be determined. From (114), (115) and (122) we have ao YE o ao _ VG o (123). -r Since 0 must satisfy the point equation of N, namely a2e alog YE ao a logVG a& (t24) - + auav a anU ant a ' we must have a o __ v 1 aYE a o U 1 a UG (125) log a loga ze 3, 231 ~~G av av V / a tt 75. Transformations F of O nets into O nets 185 The condition of integrability of these equations is (126) a (U a V a u (V i aV x 8 U VoG av a V Y-E Sua u From this equation and (31) we have (U2+ V2) a G av v71 an f ~ a VE |= U 1,'- a -vvI U2akbk, ~U V_1 1 Y av lak, (127) ( + )( 1 = ---UU' a V + VI a VE V2 kbk. y aj an/ y av When a net N satisfies conditions (127), the function 0/2 is found from (125) by a quadrature and then 0 from (123) by another quadrature. Since these conditions do not involve the constant c in (122), it follows that: IT7hen a cyclic congruence admits more than one pair of focal hyperspheres cutting orthogonally it admits an infinity of pairs. In this case we say that congruence is multiply cyclic. The constant factor of integration of (125) can be taken equal to unity in all generality, but the additive constant in (123) gives a family of parallel multiply cyclic congruences. Hence: When an 0 net Nsatisfies the conditions (127) there is a unique family of parallel multiply cyclic congruences harmonic to it, which can be Jbund by quadratures. 75. Transformations F of O nets into O nets which are not transformations R. We return to the consideration of the 0 nets N(x) and N1 (xa) harmonic to a multiply cyclic congruence, and such that the focal hyperspheres of radii R, and R2 given by (122) pass through points of N1. If we put (128) E = Z a1 V2 186 VI. Orthogonal nets then from (115) and analogous equations for iV1 we have R2 _ x R_2 ax _1 R x R a Dxi (129) - D~ B2- -- Xv Dx, x -- X (1/ av af vE a z -u/7 atD Substituting in these equations from (62), (122) and (65), we get (130) VE, = -C V-ju cV, /G - T v c. 0' U 0' VV When these expressions and those given by (65) are substituted in (128), we get, in consequence of (63), (64) and (123), 3 a i -,\ VE )., (131) aa ( zx'2) = C E, aD ( x2) - ci- / T If we put ' (132) (132) a u U 0' Dv V 0' we find that these equations are consistent, and that e so defined is a solution of (124). Consequently Q', defined by DQ' hQ DQ' 1e (133) ' -- hv' <u a v 3a ' is a solution of the point equation of the net N'. Hence the integral of (131) is (134)- x'2 -cC o ', where Q' is determined only to within an additive constant. From (129) and (62) we have R2 aX 0, R2 ax1 R ax 0, _ 0 I ax, 1VG/ av D' GI/7 aD V ' E a/ D t V&E1 a tD Squaring these equations and summing for-the x's, we have, in consequence of (122) and (134), (135) 2x':~ = - ( —.),x' 9 = e q + - ), v U ~ ~~ (o, 75. Transformations F of 0 nets into 0 nets 187 The coordinates x' of N' are expressible in the form (69), where functions p, q, r are subject to the conditions (cf. 35): the apk a au it ak q) a- b r, av (136) ag _ 1 aVE - - +h a 1 aVG au - akpk - Vr+ h ' a - y- au Vu kIV a G av k V ar 135 and (6r b) we ha it l-= (-15 ad ( ' av = P-VE a From (135) and (69) we have (137) 2q C- Q + ), 2r -V ( i c(e [' Substituting these expressions in the last four of equations (136), we obtain th V -c) = U kp- U + V) av-r (138) k V -a l V1G(V2+c) =+ V2 yk- VVr + (U2 + V2) 1' a - q l,VGii(V'+ c) == 'Flp,- ibp.-VV'ra U q In consequence of (30), (127) and (136) these expressions for h and 1 satisfy the equations (139) (1- h) alog al (h-) alogVG a It a U Hence equations (123), (125), and the following in which q, r, h and I have the expressions (137) and (138) form a completely integrable system: (140) au ae' 1/E2 - h aitt U o' au U 7' apk av ae' 1-G i - — 7 av V O' ao' Vl-. av V 0 188 VI. Orthogonal nets In addition to c the complete integral of the system involves n + 1 constants of integration, since for each set of solutions the left-hand member of (141) 2+ + r2 co 0' is constant, and in order that (134) be satisfied, we must choose solutions satisfying (141). Therefore we have the theorem: If an 0 net N satisfies the condition (127), it admits co"'2 transformations F into 0 nets, which are not R transforms of N;V the harmonic congruences of these transformations are multiply cyclic. 76. Nets 2, 0. Let N(x) be a net 2, O in n-space, the complementary function being y. Then x(,...., (n), iy are the coordinates of an 0 net N in n + 1 space. From ~ 65 it follows that there are oon-8 nets N' parallel to N for which (142) x2 y2. We say that such a net N' is a special net 2, 0. Hence there are xon-3 special nets 2, 0 parallel to N. If N' is a special net, we effect the transformation F of N by means of the solution y of its equation and the conjugate congruence of direction-parameters x', so that the coordinates x1 of the transform N1 are of the form (143) x1 x- -x' y In consequence of (142) we have (144) I(x _-x)"2 y Also in consequence of (142) and the equations ( a145x ax' = x a ' x a y Dy' ay (145) = h ' =, - =h -= a u au av av, au 8W av av we have from (143) by differentiation 1 -x1 ax- = 0, that is NA Du v is an 0 net. 76. Nets 2, 0 189 Consider two of these transforms N1 and N2 by means of two parallel special nets N'(x') and N"(x") satisfying (142) and x"/2 y"2. In consequence of (145) and analogous equations we find that (146) x'/" — y'y" — i=c, where c is a constant. Since NV and N2 are transforms of N by means of y, the -nets 1V, N1 and N2 form a triad (~ 20), and N2 is an F transform of N by means of - y/y', and the corresponding solution -y"/y' of the point equation of the net NA", parallel to Ai1, whose coordinates are given by (cf. II, 33) (147) x" X -- x". In consequence of (146) and (142), we have I(xi 2 - 2c Y Hence if c = 0, the conjugate congruence of the transformation from N1 into N2 is I. If c 4 0, N1 and N2 are in relation R (~ 67). Hence in consequence of (144): If N is a net 2, 0 in n-space, the conmplementary function being y, the net N admits oon-3 F transforms Ni which are 0 nets and the corresponding points of these nets lie on the hypersphere of radius y and center at the corresponding point of N; moreover, any two nets N1 and N2 are R transforms, unless the congruence of lines joining corresponding points of N1 and N2 is I. When, in particular, N is a net 2, 0 in 3-space, it follows from the last remark of ~ 65 that there are two special nets N' parallel to 1V. Moreover, there are no congruences I conjugate to an 0 net in 3-space (~ 65). Hence: If N is a non-special 2, 0 net in 3-space, the complementary function being y, on the envelope of the spheres of radius y and centers on N the parametric curves form 0 nets in relation R with one another and in relation F with N. We return to the consideration of a net N in n-space which is 2, 0, the complementary function being y. Let N' be a parallel 190 VI. Orthogonal nets net; then equations (145) hold. If 0 and 0' are corresponding solutions of the point equations of N and N', the equations of the form (62) define an F transform N1 of N, and the point equation of NV1 admits the solution 0, Y1 =y-,y From these equations we have (148) zx- -yl = xy2-2-0 ( x-yy ) + 2 (X22- y'). Since,x2_-y2 is a solution of the point equation of N, the function a' defined by au au a' ar (149) Gil h'L( y_) ^ =1J (f y 2) is a solution of the point equation of N'. If N' is a special net, a solution of (149) is a' -2(lxx'-yy'), and equation (148) becomes amp -y2 _ Z2_82_ 0 at (150) -= x-y- 0. Consequently x~ — y is the solution of the point equation of N1 corresponding to the solution x2- y2 of the point equation of N whatever be 0. If N' is not a special net, a solution of its point equation is f (xI'_-y'2), and the corresponding solution of the point equation of Nis given by ao ax,y ai =,ix ax yay au a a u' av v a av' In this case a solution of (149) is '- 2(zxx'-yy'-o). By means of this function equation (148) is reducible to (150). Hence: Exercises 191 If N is a net 2, 0 and V' is a parallel net which is not special, an F transform Nj of N which is 2, 0 can be found by a quadrature; if N' is special, each solution of the point equation of N determines an F transform which is 2, 0. Exercises. 1. A net N' parallel to a net N which is p, 0 is p, 0, the complementary functions of N' being solutions of its point equation corresponding to the complementary functions of N. 2. Of the 0 nets corresponding to a given orthogonal determinant, those defined by n-2 x- Z ekXk, ke k k=1 where the e's are constants, lie on the hypercone x2- =0, and any such 0 net is so determined; for such a net E = ' ek ak, f-G- ek bk. k 3. If N is an 0 net on the hypercone x2 0, and G is any congruence conjugate to N, the developables of G meet the hypercone in a net which is an R transform of N; for this transformation 0 E e^p, pk w 0. 4. If NL and N2 are II transforms of a net N and all of these nets lie on the hypercone 2x2 = 0, so also do the nets N12 which are R transforms of Ni1 and N2, in accordance with ~ 72. 5. A congruence parallel to a cyclic congruence is cyclic, and for oo1 of the parallel cyclic congruences the circles of the cyclic system pass through a point (cf. ~ 13). 6. The equation of Laplace satisfied by the direction-parameters of a multiply cyclic congruence has equal invariants. 7. If N is an 0 net harmonic to a multiply cyclic congruence G, any net N' parallel to N possesses the same property, and the multiply cyclic congruence harmonic to N' is determined by the solution of the point equation of N' corresponding to the solution determining G. 8. Any congruence parallel to a multiply cyclic congruence is multiply cyclic. 9. If N is a net 2, 0 and N1 is any 0 net conjugate to a congruence 2, I conjugate to N, the distance between corresponding points of N and N1 is the complementary function of N. 192 VI. Orthogonal nets 10. In order that the first and minus first Laplace transforms of a net N with point equation (1) be 0 nets it is necessary that a and b can be chosen so that (i) (~8)ax\2 /8ab2 (8ax)2 /(a)2 (i) Z\au =- \a/ ' From (I, 3) it follows that we must have a1b'8 2(\aa'2 ( x/ a8 x 2 ab 8a 82b a2a lax ax). a)/ av a a-u =-8 au av aua av 88 8a avV b(11) a,- 8b aa a2a 82b \ ax ax) - t a8 8v auav 8ayv au u av The consistency of the equations (ii) necessitates the condition b (a A b a2a -b 82b (za av Vb -a) aua v u av When this condition is satisfied, the equations (iv) 8log a_ 1 8b log _ 1 aa au a au' av b av are consistent (Cf. II, Ex. 12) and (ii) may be replaced by v8 aQx 82 (v) a = ab 2logo. au av auav When (i), (iii) and (v) are satisfied, the net N possesses the desired property. We call it a G net. 11. The point equation of a G net N admits the solutions 01 2x2 - a, 02 =:x2 - b2. The first Laplace transform of 0i is X l and the minus first Laplace transform of 02 is a.21, where x1 and x-1 are the coordinates of the first and minus first Laplace transforms of N. 12. Any net parallel to a G net is a G net. From (II, Ex. 12) and (iii) of Ex. 10 it follows that two of these parallel nets have point equations with equal invariants, and are associates of one another. If the point equation of one of them is written 820 l ogo ao 8 alog~ a0 (i) a uv av a v' auav v u + u av' in place of (i) and (v) of Ex. 10, we have Exercises 193 V/axx2 ( \a 2 /aV \2 a9y\2 aL a 9 a2 <"I) Aid D J = {^) a 2, ^; E3 = (; a J. L D a _ v-" 8Flog a. A net G of this type we call a net Go. 13. The point equation of a net Go admits the solution 0o = ZX2 — 2. The first and minus first Laplace transforms of Oo are.2x and 2x2,. 14. Show that the radial transform of a net Go by means of the f auction 0o = -X2-2 is a net Go.: 15. If a G net N is subjected to a transformation F for which the directionparameters of the conjugate congruence of the transformation are the coordinates x' of one of the nets Go parallel to N and the function 0' of the transformation is _x 2-_ 2, the transform is a G net, and its Laplace transforms are R transforms of the corresponding Laplace transforms of N. 18 Chapter VII. Transformations of Ribaucour. 77..Orthogonal determinants and 0 nets in 3-space. In the case of 3-space an orthogonal determinant is of the form (1) xl1.. x3. ' /= Z'.... 3 1l... 3 Now (2) 1. ax, ax ___ - - aX-mvq, a z fl-' -1 a u-,av whereas it follows from [~ 651 that _ D _ _E _1 aV-1-E m'l J/' av b D1- - __-1 -_ iaV~~ - J/ G 0a (3) Ia a b a b a v a u nct where el and e2 are the radii of principal curvature of an 0 net corresponding to J1. When an orthogona~l determinant is known, the coordinates of a corresponding net are of the form (cf. VI, 37) 78. Transformations R in 3-space (4) x PX+ Q +- R1, where, as follows from (VI, 38), P, Q and R satisfy 195 aP D aP D" au -YE 8vQ' v / D P 1 8YER+IE aQ 1 _/, 1~E V av a av afE au 1 a VE Q _ R D" 8G Q V~G av a v V-G E au aR a 78. Transformations R in 3-space. In 3-space the equations of a transformation R of an Onet N(x) are of the form (6) 1 xi =: x- — px + q _ +t), W where, as follows from (VI, 87, 88), the functions satisfy the equations I ao _ VE q, 8zf6 Qi a Y = — G r, v ' a - -v Q r 8q /YE 1 q 1 E - 8 1 8u- = — — aT E+v E ) I -/ v au r au el ' / a v a -V ~ au' ar 1 aVE au /G ' av ' ar V 1V aV; (G av Q2 V a 8u a log w _- q a — ^vE, a logw _ = — r 8av 0 v av E aV/ Vu[ I E V<V ( ~V GI> I 8n' YE 8 u 196 VII. Transformations of Ribaucour and also from (VI, 85) (8) p2+q q+r 2 w 0=0. From (VI, 93) we have that the principal radii of curvature of the net VN, denoted by qe and y2; are given by (9),1 - VE — - +,E), =V-+ — -(-G- y oil(V) V 2 Q2 6 From (VI, 54, 56) we have that the coordinates Xo, Yo, Zo, of the central net and the radius R of the spheres, are given by. 0 (10) xO x — --, = -. We recall from ~ 68 the following theorem: 4When a transformation R of a net N is known, a transformation R of any parallel net N can be found by a quadrature; the transforms of N and N are parallel, as are also the central nets of the transformations. The equations of the transformation of N are x e — -- X+ ( q + - tO), where w- satisfies equations of the form (7). Since (8) must be satisfied, we have 6 w — 06 w. Also we have the theorem: When N and N1 are in relation R, the nets N' and N1, whose coordinates are direction-parameters' of the congruence of the transformation, may be obtained from one another by an inversion. Conversely: If two 0 nets N' and Nt are related by an inversion each net N parallel to N' admits an R transform N1 parallel to N[ which can be found by quadratures. 79. The cyclic system associated with a transformation R 197 From (VI, 91) we have that the direction-cosines of the normals to a net N and to an R transform NA are in the relation (11) X = -X-, a; a' = to a. From (VI, 28, 74) we have Dx' -VIE' DX ax' VG' aX au a a u' av b v ' Do ' E' / ap o ' V ' a p au a a az ' av b v ' where E' and G' are the first fundamental coefficients of N'. Hence (11) is of the form (II, 2), and we have: TWhen N and NV are in relation R, their spherical representations are in relation F. Conversely, we have the theorem: If N(x) is an orthogonal net on the unit sphere and N'(x') is an 0 net with this spherical representation, the equations Ix' X (12) X =I - +2-, x IX12 define an orthogonal net N, (x1) on the sphere such that N1 and iV, are in relation F. From (11) and (VI, 99, 101) it follows that the distance from a point of any 0 net N to the corresponding tangent plane to an R transform iVN is given by (13) (13) 2XY1(x - x) - PI +P — - xx - - S 79. The cyclic system associated with a transformation R. If N and NV are two 0 nets in relation R, the circles orthogonal to N and Ni at corresponding points have for axes the lines of the harmonic congruence G of the transformation, that is the harmonic congruence determined by the solution 0 of the point equation of N (~ 73). In ~ 64 we saw that there are 198 VII. Transformations of Ribaucour oo1 nets N' parallel to N determined by 0. By the last theorem ~ 67 each of these nets N' determines an R transform N1 of N and all of these transforms are harmonic to G. Consequently the above circles are orthogonal to all of these nets N. and therefore form a cyclic system [~ 174]. Incidentally we have established the theorem: If N is an 0 net and G any congruence harmonic to it, the circles with lines of G for axes and passing through corresponding points of N form a cyclic system. We call the planes of the circles orthogonal to two nets N and N1 in relation R the circle-planes of the transformation. From the second theorem of ~ 11 it follows that the circle-planes envelop a net r. Since the circle-plane at a point M of N is determined by the normal to N at M and the line joining corresponding points of N and N, it follows that any point on the circle-plane has coordinates of the form (14) x x-+jx'+ kY, and the direction-cosines of the plane are proportional to r —qi. In order that the point M(x) be the point of contact of the plane with its envelope, the functions j and k must be such that (i0.-qr) | - o, ( -q0a ) a 0. By differentiation we find X V-E I 7 \ ) + X D + A.,k Hence j and 7 are determined by (16) 1 +jh — O 1 +jl —= and we have \ a / U it a' a- a, + a' (17) OX- -_ +k uau au u' Dv Dv v' 79. The cyclic system associated with a transformation R 199 If equations (16) are differentiated with respect to v and u respectively, we get (18) haj 1 ak 0 l a ak 0. 8v Qa v a u 2 a8z Making use of these relations, we find from (17) by differentiation (i9) _8!^ -// —^82, J 82k (1 9) auav x a a v au av' Hence j and k are solutions of the point equation of N. We shall show that a, defined by (20), = j -pk + 0, also is a solution. In fact, we find that Q,ac ac 8u at 8 av v avav k a -0))8ko8 aw a 8k a 8Z8u 8au8v au 88. v' Moreover, the linear element of N may be written (22) cds = dk +- 2 dj dow. If we put (23) j - m - in, 2w - - in, equation (22) becomes (24) ds" dk2 - dnm2 + cn2. Since k, in and n are solutions of the point equation of N, they are the coordinates of a net applicable to NV. This result is in accord with [~~ 141, 176] where it was shown that the planes of the circles of a cyclic system envelope a net corresponding to the developables of the congruence of axes. Moreover, it was shown that the applicable net is known intrinsically when the cyclic system is given, and conversely when the appli 200 VII. Transformations of Ribaucour cable net is known a cyclic system can be found directly. For the sake of brevity we say with Guichard that a net is C when it admits an applicable net. We have just seen that N is a net C. It is harmonic both to the congruence normal to N and to the congruence G of the transformation from N into N1 (~ 11). The coordinates of the focal points of first rank of these two congruences are of the respective forms x+-eX and x-x'/h (I, 37). By means of the preceding formulas we prove the theorem: The normal congruence and the congruence G are the harmonic congruences of N determined by the solutions m + in and k of the point equation of N. Moreover from ~ 11 it follows that the derived net of N by means of m + in and k + const. is parallel to N. Conversely, any net harmonic to a normal congruence Go is enveloped by the planes determined by Go and by any congruence conjugate to a net N normal to Go. But in ~ 65 we saw that any other congruence conjugate to N is 3, I, and consequently leads to a transformation R of N. Hence the resulting net harmonic to Go is a net C. Since the determination of congruences conjugate to N is equivalent to finding nets parallel to N, we have: The nets harmonic to a normal congruence are nets C; their determination is equivalent to finding nets parallel to a net orthogonal to the congruence: when such a parallel net is known, the cornesponding harmonic net follows directly and the coordinates of the applicable net can be found by quadratures55). 80. When the circle-planes of a transformation R pass through a point. Consider in particular the case for which the circle-planes pass through a point. From (17) it follows that j and k are constant. If the point is taken as the origin, we have from (14) that in all generality we may take (25) p= - dP, q Q, r=R, d being a constant. 55) In ~ 117 it will be shown that every net C is harmonic to oo families of parallel normal congruences. 81. The circles K and congruence K 201 From (VI, 96) and (7) it follows that 0 = o-c, where c is a constant. From these results and (10) we have, = - 2 d- 0+2 c. 0 20 J) P Hence the spheres cut the fixed sphere with center at the origin and radius ro under the constant angle e, where r 2 2c, o0 cose - d. Conversely, it can be shown that when the spheres of a transformation R meet under constant angle a fixed sphere with center at the origin, the functions of the transformation are of the form (25). It is evident that any 0 net admits such transformations R. In consequence of (25) the coordinates of any point on the line of the congruence G of the transformation R are of the form (1 + -) x'+ d * X. When; = -1, the point lies on the sphere of radius d with center at the origin and describes the spherical representation of N. These lines meet the same sphere again in the point of coordinates d X- I x', which describes the spherical representation of 1V (cf. 11). 8I. The circles K and congruence K. Let Vi (xl) and 7A2 (x2) be two R transforms of an 0 net N(x), and write the equations of the transformations in the general forms 6 x x —^y' 6j= 2 e~v ' z 016 1,2,, (26) X -,02 X1 o" = X 2 0 where N' (x') and N" (x") are the nets parallel to N determining the conjugate congruences of the transformations56). If 0" and e, are solutions of the point equations of N" and N' respectively corre56) Cf. Jonas, Sitz. Berl. Math. Gesell., vol. 14 (1915), p. 109. 202 VII. Transformations of Ribaucour sponding to the solution 01 and 02 of the point equation of N, and the additive constants are chosen so that (27) o'+ of = x'x", then as shown in ~ 72 the ool nets As1, defined by (II, 46), namely +(Of 02- - 01) x'+ (02- 1 - ai' 02) " (28) X12. - x + o o — of of —) are 0 nets in relations R with Nj and N2. Consider the circle K through three corresponding points, Mi, Mi and M2 of N, IV and N2. The coordinates of its center are of the form (29) X + ia X +P, x, where X and It are determined by the condition that the lines joining the center to the mid-points of the segments 1MMll and IMM are perpendicular to these segments. These conditions are reducible by means of (26) to (30) 2 A 0i + p x" x+ 01 = o, /% O.x'" + 2 L 0' + 02. In like manner the condition that the line joining the center to the mid-point of the line joining 11 to the corresponding point of one of the nets N12 is perpendicular to this line is 012 X\) t O. ZJx + X' + Ix" —X i XI + 77,,,. (\~~ 0^~~12 2 By means of (II, 33, 36), (VI, 106), and (27) we find that this condition is satisfied when (30) hold, and consequently the corresponding points of each of the nets Vi2 lie on K. Hence: If N, NV, N2, IV2 are four Onets in the relation of a quiatern under transformations R, four corresponding points Mi1, 1Ai, M2, 12/ lie on a circle K; the four corresponding points on any four of the possible ool nets N12, forming a quactern with, N1, N, Vs are in constant cross-ratio57). 57) The last part of this theorem is due to Demoulin, Comptes Rendus, vol. 150 (1910), p. 156, and is left as an exercise. 81. The circles K and congruence K 203 If we put (~ 21) (31) 0-= 01+ (c-) 02 (c X x'+ (C- 1)x", then (32) 4 -= 2X- J6(C) (C) x" - c' + O(C) (32) C 2 I " 2 Hence the'transformation F determined by Oc and the congruence of direction-parameters x(c) is a transformation R. Moreover, if Nc is the transform, each of the co1 O nets N12 are R transforms of Nc, and consequently corresponding points of NT, N1, N2, each of the ool nets Nc and each of the oo1 nets Nx2 lie on a circle K. Hence: If 01 and 02 are solutions of the point equation of an 0 net N determining two R transforms, corresponding points of N and of the oo1 R transforms Nc of N by means of 1 t- (c- 1)02 lie on a circle K, upon which lie also the corresponding points of the ool 0 nets N12 which are B transforms of all the nets lNc. Let Nc, and Nc, be any two of these transforms of N, and let VNoI and N02 be the central nets of the corresponding B transforms, both being conjugate to the congruence normal to N. From the third theorem of ~ 68 it follows that the tangent planes to N01 and N02 meet in the axis of the circle K, and consequently these axes form a congruence KI harmonic to the nets No, and No02. Since the congruence K is determined by the circles K, we have that it is harmonic to the central net of the transformation R of N into any net Nc. Since N and any net N6l2 may be looked upon as R transforms of a net Nc, it follows that the central net of the transformation from Nc into any net N12 is harmonic to K. From ~ 23 it follows that corresponding tangent planes to N, to all the nets 1c and to all the nets N12 meet in a point generating a net N, the derived net of N determined by 01 and 02. Since the tangent plane of a central net passes through the intersection of the tangent planes to N and a transform Nc, a point of N is on the corresponding tangent plane of each central net of all the transformations R. Consequently the congruence K is conjugate to N. Hence: 204 VII. Transformations of Ribaucour Corresponding tangent planes of V, the so' nets Nc and the 0o nets Ni2 meet in a point generating a net N conjugate to the congruence K of the axes of the circles K through corresponding points of these nets58). Since the congruence K is harmonic to the nets Nos, and No02 and the tangents to the curves of these nets at corresponding points lie in the principal planes of N at the corresponding point, the focal points of K lie in these principal planes. In order to obtain the coordinates of these focal points, we remark (~ 78) that the coordinates of NVol and No, are of the forms P1 P2 Xo = x —X, Xo02 x — X. The point equation of ATo admits the solution 01o 02 —01p2/p1, and from ~ 19 it follows that No2 is the F transform of No,, by means of o01. Hence from (II, 22) and (7) we have for the coordinates of the focal nets F1 (yi), F2(y2) of the congruence K, that is the intersections of the corresponding tangent planes to No, and Vo2,, expressions of the forms 01 ax01 (01p2 - 02p) + (02 q, - 01 q2) X I~1 ^ t -, t 0ot e O u -X p q2 — p2 q (33) u I(3 3) 1 a _ (01p2 -0 22p) + (02 r - 01 r) )X | Ys Xo~ 300, V v pa rs —2s pr From the preceding theorem it follows that the normals to all the nets Nc and to all the nets N12 at points of a circle K are tangent to the sphere S with center at the corresponding point on N and passing through K. Moreover, the normal to any net Nc meets the normal to any net N12, in the center of the sphere tangent to Nc and 2V1. Hence these normals lie on a hyperboloid of revolution to which S is tangent along K, unless all the normals lie in the plane of K. We consider the latter possibility. 58) Cf. Demoulin, 1. c., p. 310. 81. The circles K and congruence K 205 If a, f, denote the direction-cosines of the congruence K, we have in consequence of (26) and (27) if, — Y if IX If I I Xl li 1ffy (34) a/ d, X -z x,y x x g j4 O8 lt -2 (Oj' + 6 )2] The direction-cosines of the normals to N6, and N12 are of the forms, by- (11), (31) and (VI, 112), _1) + (C ~ -)x -X _ZYC(C (Xf + (C 1)XII _LY) C XI 2 X+ _ ___2 2 oI~f off I - Y+ (- off)of In consequence of (VI, 69) we have qJ, r2 qI e~$-r,"] ~~XC a - a q1 ~2 1 2l Hence either all the normals lie in the plane of the circle K, or none do. In order that they do, we must have (35) q1,r2-q21 0. From ~ 79 it follows that this is a necessary and sufficient condition that the circle-planes of the transformations of N into N1 and NV2 coincide. If we replace (35) by q2 = q, r2 h 4, from the sixth and seventh of (7) it follows that 2 is a constant, which may be taken equal to unity. Then we have (36) q2 = q1, r2= r1,, P2 =` P1i +d, 02 = 01 + e, where d and e are constants. When these expressions are substituted in (33), the latter become e e+)X j W t~ d I q1 d C, e 206 VII. Transformations of Ribaucour Hence the congruence K is harmonic to the 0 net of coordinates x-eX/d, that is K is. a cyclic congruence (~ 79), and the net is orthogonal to the circles of the corresponding cyclic system. Accordingly we have the theorem of Demoulin59): If Nis any 0 net and N1 and N2 are two R transforms of iV, the circles K determined by corresponding points of N, N1 and N2 are of two kinds; 1~, if the circle-planes of the two R transforms are coincident, the normals to the ool nets Nc and ool nets N12 lie in the planes of the circles K, and the congruence K is cyclic, the circles of the cyclic system being concentric with the circles KI; 2~, i the circle-planes of the two transformations R are distinct, the normals to the nets Nc at points of a circle K form one set of generators of a hyperboloid of revolution, and the normals to the nets NT1 the other family of generators60). 82. Transformations Dm of isothermic surfaces. From (VI, 81) it follows that a necessary and sufficient condition that the correspondence between the two surfaces of a transformation P be conformal is that T2 = a2. From (II, 8) and ~ 4 it is seen that T and a cannot be equal. Also from ~ 25 it follows that when + a -- 0 the transformation is K and the 0 nets N and iVN have equal point invariants, that is they are isothermic. Hence we have the theorem of Cosserat61): In order that the correspondence between two surfaces in relation iR be conformal it is necessary and sufficient that both surfaces be isothermic. We proceed to the consideration of these transformations and put (37) YE= f= e', where t, is a function thus defined. Now the point equation of N is (38) 20 a P8+ a t a u a v Dv 8u 9u Dv' 59) L. c. 60) Cf. ~ 108. 61) Annales de la Faculte des Sciences de Toulouse, vol. 8 (1894), p. E. 13. 82. Transformations D, of isothermic surfaces 207 Comparing this equation with (II, 79), we have (39) - -= we-~12'' (39) - M e-Q2, q where m is a constant. From [~ 65] it follows that ro wQi \Q2 211 v' from which it follows that from whnich it f ollows that the Codazzi equations of N are a (1) (1 1) _a \ — t 1 at, 88 \ee2 \Q QJ I u (40) aKo _e_2, aL a ue au' Ko e2 -2 av av' where we have put (41) o = 1+ I 1 Q2 L__ e { 1 )l q2 el Expressing the condition of integrability of (40), we find that L is a solution of (38). The net conjugate to the congruence normal to N which is in relation K to N, that is the harmonic of N with respect to the centers of curvature of N, is given by (42) x+ 2 X. KQ From this result, (40) and (II, 82) it follows that the functions 0 and 0 of this transformation K are (43) L 0 Ko 2mn' 0. 2 We return to the consideration of transformations R of N and introduce the function v by means of the equation (44) o == e- -=0 mv. (44) w == m v. 0 1208.VII. Transformnations of:Ribaucour We have from (II, 82) (45) ar e_-2V' a e-q, - e-e 0r. Comparing these equations and (7), we get (46) VE1 —V1 -In terms of these functions equations (7) become a -e0, ao __ a — q, a-P ei a v (47) aqe' 1' -/ aqatpev e~* a v, el a n(&PraeV a), a r a_ ap ar et) a_ am ~~au q, V aind (6) assumes the form (48) XI zx- (p X q +rj) It is readily found that equations (47) form a completely integrabe system, and that for' any set of solutions the left-hand member of the following equation is constant: (49) p2 iq2 + r2 -2 m0) -O.. The complete integration of (47) involves five constants in additionto rn. When these constants a-re chosen so that (49) is satisfied, three of these constants a-re essentially arbitrary. Hence we have the theorem of Darboux 12): 62) Annales de LtIcole Norm. Sup., ser. 3, vol. 16 (1899), p. 503. 83. Theorem of permutability of- transformations Dm 20 209 An isotherinic net admits oc' transform7,ations ft into isothe-rmic nets. These transformations have been called DR. by Bianchi 63). 83. Theore m of permutability of transformations D,1 We consider two isothermic nets, NI and NAT, transforms of an isothermic net N by means of two 'Sets of solutions, %p Vp p9i qp, r?, m (i — 1, 2) of equations (47) and (49), and apply the results of.~ 2 1, 7 2 to obtain a theorem of permutability of transf ormations D1.. From (VI, 105) it follows that the additive constants of iinteration of O' and 01' must be chosen so that (50) ~~~~Pp2 + qlq2 + 'l9r2 & 11 01' Since in the two transf ormations of Nwe ha-ve taken o - 2 -equations (II, 92) become at (012 01) -in,e- 1 (u 1) From these and analogous equations in 02, we find that the lefthand member of the following equation is constant; we consider in Particular the case when (5,1) 021 )2 +012 V1 0. From this equation and (II, 36, 48) we have.(52) 011 OWf~ = 2 m012 (Ml 12 01! 021 in ) 'from which follows, when 021 and 01"' are replaced by their values (II,~ 38, 43), (53) Ml M2 (V2 019 VI 02) - in2 O 2+ In11 0 1'. 63) Annali, ser. 3, vol. 11 (1905), pp. 93-158. 14 210 VII. Transformations of Ribaucour From (VI, 83) it follows that for any transformation R | rnU = AllrY1 (V ~VI E) q2, b= MI V1 (V/W+J (h)r,, | = ' v, (V - E+Vi~-E2) ql 1 = r,2 a2 (VG + 1/G) r1. a Ve at When equations (50) and (53) are solved for 02 and i', and the results are substituted in these equations, the latter are satisfied. Since 0' and 02 are completely determined there is only one transform iVNI, which is isothermic. Making use of (54), we find a 012 eVe,{O 2 012 2 O 1 / 1 2 i 0 \ a o \ M 2 \1 2 \U1 (55) \ oi '012) = av.012 In consequence of (VI, 109) the expressions for 12 and 12 analogous to the last two of (47) differ from the right-hand members of (55) only by the factor m2. Consequently (56) 012 = ' 12 012 V12, and the transformation from NV into N12 is Dn2. In like manner the transformation from N2 into N12 is Dl,. By means of (51) and (52) the expressions (II, 46) for the coordinates of N12 are reducible to (57) X12 -- x +-n2 2 0 x'- m 01 X (5'i) 21M = X+ (01" `2 02 VIY Making use of (50) and (53), we reduce this equation to i2m2-mrn in2 02 x't- rn 01' (58) x12 h- X-2 —1 t2m02X-tl 0-.. m( mt pI 2 + qi q2 -i r2 -+ i 1 1v2- 22 02 V1 From this it is seen that we must have m nMi. Hence we have the following theorem of Bianchi64): 64) L. c., p. 120. 84. Special isothermic nets in 3-space 211 If N is an isothermic 0 net, and N1 and N2 are two isothermic 0 nets obtained from N by transformations D., and D,,, there can be found without quadrature an isothermic 0 net N12 such that N12 is a D,,, transform of Nt and Dm, transform of N2. 84. Special isothermic nets in 3-space. In ~ 81 it is shown that in order that two transformations R have the same circle-planes it is necessary that (59) 02 = +j, +p2 -p1+ h 2 qk1, = r, where j and k are constants. In ~ 119 we shall show that in the tangent planes of a deform of a quadric, meeting the circle at infinity in four distinct points, there are eight points which describe isothermic surfaces, any one of which is in the relation of a transformation R with three others, the corresponding cyclic systems admitting the tangent planes to the surface for circle-planes. We apply the preceding results to the determination of all isothermic surfaces admitting two transformations D. for which the circleplanes are the same. From the last of (47) it follows that v2 = - + n, where n is a constant. Moreover, since (49) must be satisfied by both transformations, we must have also (60) it k1 +- = m (01 +j) (vl + n)- 01 h mi. When this equation is differentiated with respect to u and v, the two resulting equations are equivalent, by means of (47), to 2(m2-m )v - +2m = - Ko 2m -k 2(ma -m) 0 - 2m2j k- L, where Ko and L are defined by (41). These equations may be replaced by (61) 01 L+e, 1 = -Ko+f, where e and fare constants. From the last two of (47) we find that (62) 1 -- e' - Ko 14* 212 VII. Transformations of Ribaucour In consequence of (40) these functions satisfy (47), if we take (63) p -- - LKo g, where g is a constant. In order that (49) hold we must have (64) ( I[o -t l +) 1- +^ o+ (64) e2'AL( ~)+( ~ )i+ 4 K L2+AK0oLK2BKo+2CL+D-0, where (65) A - 2 mi-g, B= -m e, C =- mlf, D - g2- 2 m ef. Eliminating e, j and g from these equations, we have that mn must satisfy (66) (A — 2 m)2 - D?,-L 2BC 0. If then a net N satisfies the condition (64) where A, B, C, D are constants, there are in general three solutions n of (66). When these are substituted in (65), we have three sets of values of e, f and g and the corresponding functions (61), (62) and (63) determine transformations of NA into three isothermic nets such that the circle-planes of the three transformations are the same. The foregoing results are due to Darboux65). Bianchi66) also has considered these surfaces, and in order to put in evidence the essential constants appearing in (64), he refers to such a surface as a special isothermic surface of class (A, B, C, D). 85. Complementary transformations D,, of special isothermic nets. The three nets obtained from N by the transformations determined by the values (61), (62), (63) have been called the complementary nets by Bianchi. We shall show that they are special nets of class (A, -B, - C, D). In fact, in consequence of (9) and (46) we find that the functions Ko,1 and L, for one of these nets N1, defined by equations analogous to (41), have the values -(67) Ko, L +- 22p1, O1 -o + 2p, (67) Lo, - i = 01 VI 65) L. c., 5. 507. 66) L. c., p. 130. 85. Complementary transformations D. of special isotheunnic nets 213 From these and the foregoing forimulas we find LI-e ef+2g Ko,14-f ef + 2g V/1 ' 01 -'K iL + g = (ef+2 2g) 2 From these expressions and (VI, 94) it follows that YAT may be obtained from N1 by a transformation Din, determined by the functions 0y1= L1.-e,;1 — (Ko+f, If - o 1L - Since the effect of changing the signs of e and f is to change the signs of B and C, but not to affect (66) we have that -AT is a special net (A, -B, -C, D). Now we show, as Darhoux has done 68), that the circle-planes of a complementary transformation envelope a surface applicable to a general quadric. To this end we make use of the results of ~ 79. From (VI, 83) and the preceding formulars, we find that for the present case h= mi (v + 0e2"'), 2 = V i (v- e- 2 e - 0 20 2g + 2 ef-ve rn (Of- e v)' Of- er' B Of ev From (23) it follows that if we put x-lc, y +jrz -j, y-iz 2 -0B, the point (x, y, z) describes a surface. applicable to the envelope of the circle-planes. When the above' values of j, k and o are substituted, and 0 and v are eliminated, we find that, is the qiiadric (Y - iz) [x + 2 mn (y + iz)] - 2 gX2 + 2 ex~+ m in (2g + ef) x (y + iz). 67) If 2g -- ef 0, N1 has constant mean curvature and the corresponding equation (64) is _~j[02 0+ A -T0 IL,- BK0 1- 2 CL, D = which is satisfied because of (65). 68) L. c., p. 508. 214 VII. Transformations of Ribaucour 86. Transformations Dm of special isothermic nets. Let N be a special isothermic net of class (A, B, C, D) and N, a complementary transform by means of the functions given by (61), (62) and (63). We shall show that it is possible to find solutions 021, V, p2, q2 and r2 of equations (47) such that the net NV is a special isothermic net. This is done by finding under what conditions a net NXl2, arising from N1 and N2 in accordance with the theorem of permutability of ~ 83 is a complementary transform of N2. From (II, 43), (VI, 109) and (52) we have Ojf Off 021= 01- of 02, p)21 =ofoPf2 1 I - 0 221 _ P -Pi, n 21 -" 1 oei Also from (50) and (53) we obtain (w2 - M) 0 m = p2 2 ( 2 + 1 q2 +l ri 2 - nV ^ 01 - m1 ^1 02). The analogues of (67) are v2 L + 2P2 02K0+ 2p2 Ao,2- = - - The equations 1 (68) 021 — L2-e, v21 -=Ko, 2 — f, 21= -K,2L2 —, are consistent with the above results, provided that (69) pIp2 + q2 q2 + rl r2- 2m (,201 + V102) + (m2-m1) (v2 e - 02f+ 2p)2) 0 -By differentiation we find that the left-hand member of this equation is constant for each set of solutions 02, v2, p2, q2 and r2 of (47). Hence if the constants entering in the latter are chosen so that (69) is satisfied, the net N12 is a complementary transform of N2. The latter is a special isothermic net of class (A, -B, -C, D). Hence we have the theorem of Bianchi69): Of the oo4 isothermic nets obtained from a special isothermic net of class (A, B, C, D) by transformations Din, oo3 are nets of class (A, - B, - C, D). 69) L. c., p. 141. We note that Bianchi showed that N2 is of the same class as N. This is due to his choice of directions in the trihedron of N2. 87. Transformations Dm of minimal surfaces 215 87. Transformations Dm of minimal surfaces. A minimal surface is isothermic, and the spherical representation of its lines of curvature is such that we may take [~ 109] (70) G = e- * el e2 where VI/E= G —e. From (9) and (46) it follows that the, transform is minimal, if v p, and only in this case. The equations of such a transformation are: a 0 _ eq, 8 = er, _ _ a o _ _ _ b -- e - q, - 'r (71) aa a q _ a a P a_ l a -e_ v r+m (c p+ e- ), - a a a n a8 q, e a - + m (ea'p e- a6). t au av q' av 1 q These equations form a completely integrable system. Since p2-_q2++ - 2-2nmpO 0 must be satisfied, we have: A minimal sirfacte admits ooS transformations Dr into minimal surfaces 70). From (56), (52) and (VI, 109) we have ^12 -p2,, and therefore: If a minimal surface is transformed into minimal surfaces S_ and 82 by transformations Dint and DM,, there can be found without quadrature a minimal surface which is a Dn, transform of St and a Din, transform of S271). 88. Transformations Em of O nets with isothermal spherical representation. Since the spherical representation of the 0 net on a minimal surface is isothermal, each transformation Dm of a minimal surface into a minimal surface leads by a quadrature to a transformation R of an 0 net with isothermal spherical representation into an 0 net of the same type, in accor70) Bianchi, Rendiconti dei Lincei, ser. 5, vol. 81 (1899), p. 151. 71) Cf. Bianchi, 1. c. 216 VIL. Tralisformations of iRibaucour dance with the first theorem of ~ 78. These transformations were studied by the author 72) and later by Bianichi 73), who called them transformations Em. In this section we show that these transformations are the most general transformations Pi of 0 nets with isothermal spherical representation into nets of the same kind. The tangential coordinates of N satisfy (V, 29) with o, e andy 0. From (VI, 94) we have that the functions 10 and wn' of ~ 52 are p) and p)/O' respectively. Hence- if a transformation PR of NT is to be a transformation A~ that is if N1I is to have isothermal spherical representation, its tangential equation must be of the form a2X + D D) Ip O._ Dumav v Du Dv, aDv From (V, 35) it follows that iJ_1, can be chosen so that (72) if we take (73) 0' rOl V0 If in accordance with (9) we punt (7 4) Vl-EhI+ Vt-y ie )+ Pe- V p we find thlat the last two of equations (7) are satisfied, and consequently the following system of equations is completely integrable: an a __-/ -G I a V aDu a V a q - 1~ a zP m In~e~+O e'1), Dq _ alp am - a a v au' (75) a av- q, Dr -D-ll ) P / Or 0 V e - j a v -- a a pe' ja r O V a-f r } v - r 0p V V ()) 72) Transactions Amner. Mlath. Soc., vol..9 (1908), pp. 149-177. 73 ) Rendiconti (lei Lincei, ser. 5, vol. 24 (1915), p. 371. 88. Transformations Em of 0 nets 217 From these equations we have (76) ( -, ellq, Bu \ Iv \ I and consequently O1/p is a solution 0 of the point equation of the minimal surface N with the same spherical representation as N. In terms of this function the above expressions for the derivatives of q and r assume the same form as (71). Consequently the transformation of N by means of p, q, r, 0 is a D,, into a minimal surface. Hence we have the theorem: When an 0 net N with isothermal spherical representation admits a transformation R into a net N of the same kind, the minimal surface parallel to N is transformed into the minimal surface parallel to iNT\ by a transformation Dn. Conversely, if N and NV are 0 nets on minimal surfaces in relation D,, and N is a net parallel to N, the set of functions p, q, r of this transformation and 0 and v, where 0 v= 0p determine transformations Em of N into oo nets N1 parallel to 1V. For 0 is determined to within an additive arbitrary constant by the first two of (75) and then v is fixed. Hence: An 0 net with isothermal spherical representation admits O 2 transformations Em, for each value of m, into ox parallel 0 nets of the same kind. From these results and the fifth theorem of ~ 78 we have: An isothermic net on a sphere admits oo transformations F into isothermic nets on the sphere. Consider a quatern of minimal surfaces in accordance with the last theorem of ~ 87, and four nets NV, N1, N2, 2 parallel to the 0 nets N, NT, N2, NV of the minimal surfaces in accordance with the last theorem of ~ 72. From (56) it follows that 012 - -mt2 01 2p2, 021 - -M1 021P21,P where 012 and 02 determine the transformations from N1 and Ni into NV2. From equations analogous to (76) and (49) we have 01212 v -— 2 012, 218 VII. Transformations of Ribaucour Hence, since 0'2 = 0'2 and 6o1" == Of7, 12 = m- 212 12 2' mJ vt21 12 M 22 012 V127 v2t"f M I 021 2 1. and consequently we have the theorem: If N1 and N2 are obtained from an 0 net N with isothermal spherical representation by transformations E,,,, and E,, there can be found without quadratures a net N12 of the same kind, such that N/12 is an Em2 transform of N1 and an Em, transform of N2. 89. Nets P. By definition an O net N is a net 52 if there exists a net No with equal point invariants conjugate to the congruence G normal to N. Let N be a net S. and let 0o be the solution of the point equation of N, namely (a0 alogVE a0 _ alogKGl/ a0 (77) auav av au au v' by means of which Ao (xO) is obtained from N by a transformation F. The equations of the transformation are of the form 0o (78) Xo = --, 0o where 60 is defined by (79) au a 0 a0 __ av {i)a zu ' av v 8^ and (80) h=_ 1 A 1 ei' e2 since [~ 51] aX_ 1 ax ax 1 ax (81) a~~ ae l au' av e2 av If the point equation of No is written in the form a2 a - ao a -ao (82) a,logV 0 + log VO a7 u a v av a it auo 89. Nets 9 219 we have from (77) and (II, 12) (83),~o,VE o 1VG eQo = ^ 0 -- U, 00 00 where U and V are functions of u and v respectively, and from (II, 8) (84) TO= - + 0 To ~ - 00 ) =_0 - (:0 When these expressions are substituted in (83), we obtain (o85) _ Q2 UVE —e VG (85) ~0 ~ e0l2 (vV- UVE)' / — uJV E(e —e2) UVEe2-VVGQ1_ The Codazzi equations for N are [~ 65] (86) _a ( av el I_ aVE 2 av a Ce( au lQf 1 alG~ Qi a 'V '-e~l au When we require that the function (85) satisfy consequence of (86), (79), we find, in a log 0o au (87) { a log 0o av UJ/ e Q 2I a 1 a vVG e-e au j aulog(UVlE-V-), V/G ele2 a (I)+ og(U-/E -V1_) UVE e2 — el av ea av The condition of integrability of (87) is (88) a (UVE ee a (1)\ a vfG Q1esl 2 1 a IV\ y i - 1-2 au \e/l au / UVE Qe2 =v \ av\ V a V -B a \2 220 VII. Transformations of Ribaucour When this condition is satisfied, we find that 0o given by (87) satisfies (77), and consequently N is a net Q. Hence we have the theorem of Demoulin74): A necessary and sufficient condition that an 0 net be a net - is that (88) be satisfied. In consequence of the second theorem of ~ 25 we have that when N is a net.2, there is a second net N1X with equal point invariants conjugate to the congruence G normal to N, and that the points of No and No on a line of G are harmonic to the focal points of G on this line. Since the coordinates of these focal points are of the forms, x-+e1X and x -e2 X, we have from (78) and the analogous equations for No, namely xc ~= x -- -- XA 0, the equation (89) O, Oct) UVE+QVVG eThe equations a s t (U187) Vo ro y The equations analogous to (87) are obtained from them by replacing V by -V. From these equations we have (90),o00 = U2E-V2G, v E= v EG(2- 1) Since No and N, are conjugate to the normal congruence to AN they are nets 2, 0, the respective complementary functions being 0o/0o and 0,/0, (~ 66). Conversely, if No is a net 2, 0 with equal invariants, the two sheets of the envelope of spheres with centers on No and radii given by the complementary function of No are surfaces Q in relation R. The foregoing results may be stated as follows: When a net N is a net S, the nets with equal invariants conjugate to the congruence normal to N can be found without quadratures; these nets are 2, 0. Conversely, when a 2, 0 net No with equal invariants is known, there can be found without quadratures two nets 52, in relation R, whose normal congruences are conjugate to No. 74) Comptes Rendus, vol. 153 (1911), pp. 590, 705, 927. 90. Transformations R of nets s2 221 From (II, 87) it follows that the net NV (xo), where ax _ 1 ax ax o_ 1 axo a8, eo a' av,- a v has equal point invariants. From ~ 76 we have that NV is 2, 0, the complementary function t' being given by a quadrature, and that the sheets of the envelope of the spheres with centers on N; and radius t' are parallel to the corresponding sheets of the envelope of the above congruence R of spheres with centers on NTo. Hence: When a net S2 is known, a parallel net D can be Jbund by quadratures. 90. Transformations R of nets 2. Let N be a net!and NT an R transform of N by means of a set of solutions O, p, q, ) and w of equations (7) and (8). As in the preceding section, we denote by No one of the nets with equal point invariants conjugate to the congruence normal to N; it is an F transform of N by means of 0o and 00. In accordance with the theorem of permutability of transformations F (~ 21) there exist oo2 nets NVo, which are F transforms of No and N1. From (11) and ~ 21 it follows that if we take 0 p we obtain oo nets NA1o conjugate to the congruence normal to ANT, as the additive constant of 0f varies. We seek the conditions that one of these nets be a K transform of NT, and thus has equal point invariants, in which case NV is a net -Q. From (II, 43, 44) we have that the functions 001 and 0'.' of the transformation from No into a net NVo are given by 0o o o' (91) o1- 0 0-^P 0ol-0 ^-.p, 00o 00 and from (II, 53) and (80) we have (92) hotl --- PT, 1 - I P2 to -,o 222 VII. Transformations of Ribaucomir with (93) 01 ho0a1 a' a A necessary and sufficient condition that No, be a K transform of No is that (II, 81) M001 01 moo, 01 (94). hol== m1 Oo 1 r0 o{ Q0 O 01 Q0 001 where m is a constant. From (II, 36, 38) we have that the functions 0lo and 0OW of the transformation from N1 into No, have the form (95) 910 0- 0 010 0- where 6' m mv0. When we equate the expressions (92) and (94) for hol and lt,; the resulting equations are reducible by means of (VI, 83), (91) and (95) 010 v00o 0 001 (96) ~611/f~l= — V1/-~ II-3- 01 0 0V 10 I v J -vv 010 + Oo Q~jf;0 0 -Bo 0VVo 0o 001 From (91) we have by differentiation a 00, 4 a 00, au — 0 00, -OA 8 1/ a 00 eav -~r~~/ vi By means of these equations and the last two of (97), which are obtained by differentiating (95), we find that the expressions (96) satisfy the last two of (7). Hence the following system of equations in which El and G, have the form (96) is completely integrable: 91. Theorem of permutability of the transformations of nets P. 223 a86 - av - ap _ _ E a_ 1/VG - a- ~ 1n qe ' av Q2 8a Ve 1K avE, + mV(E,+V, ), 1 a 8 Q J 1/r a8' a' }/E a8 ar 1 aVE ar V. 1 a VF a q- -- — G av a" — a -q — -- -Em q + m) ( V G + V /), a8 El o0 av 0 a _ =- -r_ [ r (Oo -00) +- a ]' lav '-[( -10 )+ - avS When we have a set of solutions 0, v, p, q, r, 010 satisfying the quadratic relation (98) 2+ q2 + r2 - 2 mv 0, the R transform N1 is a net Q, and the coordinates of IV, are given by equations of the form (48). The same functions multiplied by the same constant determine the same transformation. Hence in addition to m there are four significant constants of integration, and consequently: A net SP admits oo5 transformations R into nets Q 75). From (II, 86) (91) and (95) we find that the function Qo for Nio has the expression (99) o 0 __ 10 _ o - g — Veo ]" ' e /l mi on~ O, o 91. Theorem of permutability of the transformations of nets P2. Let N be a net Q, and NM and N2 two R transforms of Nlbymeans of functions oi, pi, qi, ri, tvi, Oio, i (i 1, 2), solutions 75) Cf., Transactions of the Amer. Math. Soc., vol. 16 (1915), pp. 275-310. C )'/ 2,24 224 ~~~VJJ. Transformations of Ribaucour of (97) and (98). We wish to. show that of the 00' 0 nets NV12, which are Pb transf orms of N1 and N2 in accordance with ~ 72,,one is a net S~.. Consider the nets N10 and NV2 which are K transforms. of NTO by means of the pairs of functions 0ol, wol and 002, W02. From 24 we have that there is a net N120 which is an F transform of each of the niets NT1, N10 and N20, and the functions 0j0 and 0Of of the respective transformations from NVI and NY20 are given by 00 001 W0,Volo00 Wl (Go 001 012 + 01 010 002 - 02 01o 001), 00 00 2 W0 2190' 2(00 00260214+ 02 020 0o1- 01 020 002). Moreover, from (II, 91) it follows that N120 will be a K transform of INi0 and IN~,, and consequently have equal point invariants, if mn1 0r2tO2 + W12 Ow01W ~ 0. Substituting the above values in this equation, we reduce the resulting equation to 2 __ ni Ml MJ [01 010 002 0 O' U n o,'4O - 02 (010Oj (100) Oo 001 01(02 if this equto is differentiated with respect to u anP.d v, we find that the resulting equations are satisfied identically. Since 09f and Off are completely determined by (50).anld.(100), only one of the nets AN12 is an S2 net. Hence we have the theorem: if 'N1 and N2 are SQ nets which are B transforms of an.!~ net N, there exists a unique S~ net NV12 which is an 1? transform of N1 and N2; moreover, iV12 can be found directly, as soon as the coordinates -of Ni an d _AT are known. 92. Surfaces of Guichard of the first kind. In order that the 0 net NT on a surf ace S be defined by VE $r esinh a V1zzz e$ osha, (101) 11 t - = e(coth a +h), e e tanh a +h), 92. Surfaces of Guiehard of the first kind 2255 it is necessary and sufficient that h,, and a satisfy the equations ah aS ah h_ a (coth a+ —h) a (tanh a + h) a2s" __ __ __ as aae a~ aa I a - a -I- a -+ coth a + tanh a a + a2a ca av au av av auc (102) acc aa2 ~ av2 aa aS a U2 aV coth a r + tanh cc esch a- - aa a% + sech2 a CZ a + (cosh a -- h sinh a) (sinh a + h cosh a) - 0, av a v as follows from the Codazzi equations -(86) and the Gauss equation [~ 64]. These equations are satisfied also by, a, h, where I-h (103) (Ih2), ea e 14h 1 +-~h' An 0 net N parallel to IN is defined by {V e sinh — a e cosh a, (104)1 e1 e-(coth a + h), 1(tanh + h). e2 By means of (101) and (104) we have Q1 e2+2e1l -2. Following Calapso 76) we call S a surface of Guichcrd of the first kind, and the parallel surface S its -associate (cf. VII, Ex. 28). When the expressions for J7 V&, el and Q2 from (101) are substituted in (88), this condition is satisfied, if we take U = V L. ience we have the theorem of Demoulin77): The surfaces of Guichard of the first kind are surfaces S. From (87) and (85) we have (105) 00=e, e 0 = 1-h, y'o e __________ 0 eo h - i1' 76) Annali, ser. 3, vol. 11 (1905), p. 211. 77) L. c. 15 226 226 ~~~VII. Transformations of Ribaucour We apply the results of ~ 90 to establish transformations 1? of surfaces of Guichard of the first kind. Analogously to (105) we have From (95) it follows that (106) h, + Ae zh + e$ t, where t is thus defined. Now 00 1 0 e-p e/(1 -h), and consequently (96) are reducible to V~~l + -(11 (107) { 1 eP w;here p=cosh + tsinh a, iP zsinh a +tcosh a. Since 9p)-_V2 1- t2, when we require that G1 -E - 2 $2 we get (108) e ~ (t_ V If we put V A e4+al/rn (h1- 1) in (99), we obtain (109) ea, ea t t+1* Hence we find VE/-El _ k 4'sinh a, /Gl- -e cosh ai. Furthermore, equations (9) are satisfied by _VL' _ _ _ _ _-l -(cosh a, + h1 sinh cc,), — snhacc,+ h1 cosh a,. 92. Surf aces of Guicliard of the first kind27 1227 Hence the surf ace S1 is a surface of Guichard of the first kind. By means of (108) the expressions (107) are reducible to V111Ezr - - ~e$ ~Qt +i i, V V// 7 - -0e YJt+9) V Consequently the functions determining a transformation B of a surface of Guichard of the first, kind into surface of this type are solutions of the completely integrable system: au0 - e~sinh aq, 8-u - - (cosh cc + h sinh a) q, auI a e ceosh a r ap= -- (sinhae+ h cosh a) r, ~ — =-(co sh e+ h sinh a).P- anha cc-+ m,[-6Oe- (s t + IP) + ve~sinh a], 10) a q (!- cotha a ~+ accr a v t au ___ ah c I + ~a a +m[-6e - QP t + 9) + ve~cosh a], a V -(Tt I zp)q, u=: - e -F arV =- e-(qpt+ )r. The equations of the transf ormation are of the f orm. (48) in these functions. Accordingly we have the theorem 781): A surface of 6Guichard of the first kind admits oo I transformations B.,~ for each value of m t 0, into surfaces of the same kind. When S is transformed into S1, the associate surface 8 is transformed into a surface S,,, parallel to S,, by means of the functions p, q, r, 0 and jT, where in consequence of (49) O 0: Op. The equations in 0 are e$ sinh -aq, a o - - -e~coshar, 78) Annali, ser. 3, vol. 22 (1914), p. 205; also Transactions of Amer. Math. Soc. vol. 17 (1916), p. 68. 15* 228 VII. Transformations of Ribaucour It is readily found that _ p2 v62 0v — -,. 0' y ep2' The associate of S1 is determined by equations analogous to (103), namely ehrAz e' (1-h) 2p t + 2 e-(1 —t2), a -h-1-h ct+ e- e-a ea 1+h1 t —1 1+- t-e hi =h i. If this associate is to be the R transform of S by means of 0, a, p, q, r, then analogously to (106), (108) and (109) we must have h1+ -- e =- h + e — t, o 0 e== 0 (-1), t1e-at1 It is readily shown that these expressions are equivalent to the preceding. Hence: When a surface of Guichard of the first kind is transformed into a surface of the same kind, the associate surfaces are in relation R 79). Exercises. 1. If N is a net 2, O the complementary function being y, the lines of curvature on the sheets of the envelope of the spheres of radius y and centers on N are represented on the unit sphere by the central projections on this sphere of the two special nets 2, O parallel to N; the coordinates of the spherical representations are x'ly' and x"/y". 2. Corresponding tangent planes to two pairs of 0 nets in relation R whose central nets are parallel are parallel. 3. A necessary and sufficient condition that the spheres, tangent to a surface S and with centers at the mid-points of the segments of the normals to S 79) Annali, 1. c. p. 209. Exercises 229 included between the centers of principal curvature of S, determine an R transform of Sis that the lines of curvature of S have isothermic spherical representation (cf. ~ 6 and I, Ex. 8). 4. A necessary and sufficient condition that the spheres tangent to a surface S with centers at points harmonic to the points on S with respect to the centers of principal curvature determine an R transform of S is that S be an isothermic surface (cf. ~ 25). Darboux, Annales Norm., ser. 3, vol. 16 (1899) p. 504. 5. Show that the first three theorems of ~ 81 are equally true for the transformations R of nets 0 in n-space. 6. For a system of circles K of the second kind (~ 81) the focal points of the congruence K are harmonic to the foci of the meridian sections of the hyperboloid of revolution whose generators are the normals to the nets Nc and N12. Demoulin, Comptes Rendus, vol. 150 (1910), p. 159. 7. If N is an isothermic O net, the point equation of N admits the solution O0 = 2i (1 - -,1 and the corresponding parallel 0 nets No(xo) are defined Pi P2 by equations of the form So= 1 2 ( oo+oi\X+ e1 a(Ko a 3o \) Xo= - 0o KO +- C X + e ' -- ~ where c is an arbitrary constant and Ko - +. Pi P2 8. If N is an isothermic 0 net defined by (37), the Christoffel transform N [~ 159] is defined by VEW- -Jf, ep 1 P- e p2 — P2 P2 Show that if N1 is a Dm transform of N, a D. transform N1 of N is given by = P, q q, r — r, 0= 0=0,, m-m, and that N1 is the Christoffel transform of N1. Also if 1, 321 and M1, 1M1 denote corresponding. points on these nets, the product of the segments M 11/ and M Mt is equal to 2/w. Bianchi, 1. c., p. 105. 9. If 3M, 111, -M2, 3112 are corresponding points of four nets satisfying the theorem of permutability of transformations Dm (~ 83), the cross-ratio (M M12, All M2) is equal to m2/mi. Demoulin, 1. c., p. 157. 10. Show that the O net of a minimal surface may be considered of class (0, 0, 0, 0) (~ 84) and obtain the results of ~ 87 from those of ~ 86. Bianchi, 1. c., p. 149. 1 1. From [~ 125] we have for a surface of mean curvature Ko 1, -= e- sinh k, - e- cosho. PI P2 In this case from (47) it follows that 0- r + 2p =-a, where a is a constant. Show that when a = 0, the mean curvature of N1 is — 1. 230 VII. Transformations of Ribaucour 12. Show that the 0 net of a surface of mean curvature Ko - 1 may be considered a special isothermic net of class (-, 0, 0, I), and apply the results of ~ 86 to obtain those of Ex. 11. Bianchi, 1. c., p. 151. 13. The Christoffel transform of a special isothermic surface of class (A, B, 0, D) is a special surface of class (A, - C, - B, D). Bianchi, 1. c., p. 131. 14. If from a special isothermic net N of class (A, B, C, D), we obtain two nets of Ni, N2, of class (A, -B,- C, D) by transformations D,, and Da,, the surface N12 is of class (A, B, C, D). Bianchi, 1. c., p. 146. 15. Determine the special isothermic surfaces for which the circle-planes of the complementary transformations pass through a point. 16. Show that the nets N' of transformations Dm of isothermic nets and of transformations Em of 0 nets with isothermal spherical representation have the property E'- G' - 2 m ('2 + y'2 + z'2). Calapso, Annali, ser. 3, vol. 26 (1917), p. 168. 17. If N1 is a D. transform of an isothermic 0 net N, an isothermic 0 net N is defined by 1 0 1 0 lE= _G —= -, + — - -= p + -. 0' pi pi p2 p2 Bianchi, Annali, ser. 3, vol. 12 (1906) p. 22. 18. If N and N1 are two 0 nets in relation Em, the minimal surfaces -5 and Z1, whose asymptotic lines have the same spherical representation as the curves of N and N1 respectively, can be so placed in space that they are the focal surfaces of a Wcongruence (~ 53). 19. When two 0 nets N and N1 are in relation Em, the spherical representation of these nets are isothermal orthogonal nets on the unit sphere in relation F, and these two nets are related conformally. The equations of the transformation are of the form XI = -X+ P (pXX+q$+r), mOp where 0, v, p, q and r are solutions of (75). 20. A necessary and sufficient condition that the spherical representations of two 0 nets in relation R be conformal is that the transformation be Es. Annals, Ser. 2, vol. 17 (1915) p. 69. 21. If four nets N, NT, N2, N12, form a quatern under transformations Em, and Ns is an E,3 transform, then the nets 33, 3,, N23, N' of ~ 24 form a similar quatern. Transactions, vol. 9 (1908), p. 170. 22. If N is an 0 net with isothermal spherical representation, the point equation of N admits the solution Oo = E- G, and the corresponding parallel 0 nets No are defined by equations of the form.O = [2e (E + ) + cl X+ a - aG ( taE G where c is a constant. where c is a constant. Exercises 231 23. If in (88) we put 1/~E= 1G, U = - = 1, this condition is satisfied, that is isothermic surfaces are surfaces 52. In this case 00 = e2t (t _) P_ P+ P. p2 Pi I 2 pip2 In order that the transformations determined by (97) be Dm, we must have e 0,o V o+ 0o 0o01 = 0. 24. Determine the character of the surfaces 2 which are R transforms of an isothermic surface other than Dm transforms. 25. If in (88) we put -PE l = Gp2 e-Y, U = V== 1, the equation is satisfied in consequence of (86); that is a surfase with isothermal representation of its lines of curvature is a surface P2. In this case 20o =- -o (p +-p2); consequently the mid-point of the segment between centers of principal curvature of the surface describes a net No with equal point invariants; also Nc7 is at infinity. When we express the condition//-/pil =V1 Gl/p we get 0 e- ( 01~ 00i 2e- +(/G-VY)) 1- +2 =0. By means of this relation the expressions (74) are obtainable from (96), and these transformations Rm are Es. 26. Determine the character of the surfaces ~2 which are R transforms of a surface with isothermal spherical representation of its lines of curvature other than the Em transforms. 27. If Si and S2 are surfaces of Guichard of the first kind obtained from such a surface S by transformation RB, and BR, there can be found without quadratures a surface S12 of the same kind which is in relations Rn, and R, with Si and S2. Annali, ser. 3, vol. 22 (1914), p. 212. 28. In order that the 0 net N on a surface S be defined by 1 - a i/i= eze sina, /G= e'cosa, - = e (cota-h), - - e (-tana+h), P2 it is necessary and sufficient that h, t and a satisfy =h (h- h+cot a) =a, ah = ( - tana) 8a, 82E a8 a aa 8: a-a 8a -= - cot a -- - tan a a8ala aq 8v ai a8 a8 av 82a 82a 82c 2 2 a a 8a 2 8a 8a 2~ ~ Q2- +cot a- + tana csc a 0 — + sec a —~ a (2 aV 2 a a s2 v osu au av av - (cos a + h sin a) (sin a - It cos a) = 0. 232 VII. Transformations of Ribaucour These equations are satisfied by -, a and h defined by e= e- (l+ h2), 1 sn sin - [sina (1 - h2) -2 hcos a], cosa = + [cos a (1 -h2) + 2 h sin a]. An O net N parallel to N is defined by JV' e sin a, T =-e cosa, - - e -E(cot a+h), - e (-tana+-h) P1 P2 and P1 P2+P2 P1 -2. The surface S is called a surface of Guichard of the second kind, and S its associate. Calapso, Annali, ser. 3, vol. 11 (1905), p. 216. 29. Show that a surface of Guichard of the second kind is a surface Q and determine the transformations R of such a surface into surfaces of the same kind. Annali, ser. 3, vol. 22 (1914), p. 205. 30. A necessary and sufficient condition that a surface of Guicharcl of the first kind be a spherical surface [~115] is that e= a, h 0, where a is a constant; then p, = a tanh a, P2 = a coth a. In this case the associate surface is homothetic to the given one. 31. If S is a spherical surface (cf. Ex. 30), equations (106), (108) and (110) are consistent when v = (a2p2 02)/Oa2. Thus a spherical surface admits oo2 transformations R,, for each value of m $: 0, into spherical surfaces. Annali, 1. c. p. 230. 32. For the circle-plane of a transformationoR, of a spherical surface (Ex. 31), the functions j, k and w of ~ 79 have the values j = a2/2mn0, k = pa2/0, - = (2-_p2 a2)/20. Consequently the planes of these circles envelope a surface applicable to the quadric of revolution x2 - 2 m(y2 z2) = a2. Annali, 1. c. p. 235. 33. If S is a surface of Guichard of the first kind and Si is an Rm transform of the same kind, a surface of Guichard of the first kind is defined by e~ e' P1 2 - - _ — =Ce (1 al-2m) (-tanha+ise. 3, l. Calapso, Annali, ser. 3, vol. 29 (1920), p. 84. Chapter VIII. Circles and spheres. 93. Coordinates of a sphere. The equation of a sphere in 3-space can be written in the form (1) 2 cac + y 2 Y2 + 2 as Y + ia4 (yl +y y3+ A + 1) + 85 (Y2 + Y +-l) = 0, where ac,...., a5 are constants, and y1, Y22 y3 are current coordinates. We call the five constants a the coordinates of the sphere. If Xl, x2, xs are the coordinates of the center of the sphere and R its radius, we have (c acl- hxl, a-2 - 1 hx2, (2)124 i(X1+~X+X-R2+1), a5= L 2 a3s hX3, _-h h(X+_ X+x A 1) 2 2 3 t where h is a factor of proportionality. These equations can be written also thus I - _ _ _ _ _ 2 JXil —., o2 (3)1t Cc + i C-4' a + i 4 ' 1 X + 3X — -- + i4' a3 - f X3 - a5 + ia4' I = - (as + ia4). An exception arises when (4) a. + ia4 -, which is a necessary and sufficient condition that (1) defines a plane. From (3) we have (5) R,' a5 + ia4 234 VIII. Circles and spheres where we assume the sign of the radical chosen so that R is positive for a real sphere. It follows that a necessary and sufficient condition that a sphere defined by (1) be a point-, or null-sphere, is (6) a- + Cae +a +a +a I 0. By definition the power of a point with respect to a sphere is the product of the distances to any two points of the sphere collinear with it, that is the square of the tangent to the sphere when the point is outside. Hence the left-hand member of (1) divided by ia4-+ a is the power of the point (y, y2, y3) with respect to the sphere (1). Consider the sphere (1) and another of center (x1, xa, x]) and radius R' defined by f 2 y + 2 cy2+ 2 c2s3 +- (y + y + y + 1) (+ a (y' + 8y + y - 1) = 0. The angle under which they cut is given by 3 R2 + Rf2 — (i- )2 (8) cos 2 — 11' With the aid of (3) and (5) one shows that this is expressible in terms of the coordinates a and a' of the two spheres in the form Zi at' (9) cos 0. In certain discussions it is advantageous to introduce a sixth coordinate s defined by (10) iac = R (as. + i.,). Comparing this equation with (5), we note that (1 1) CC2.. - + a2 '. 1 '''I 94. Pentaspherical coordinates of a point 235 Now equation (9) becomes 5 (12) cosO - 1 a.= ~6 6 94. Pentaspherical coordinates of a point. When the sphere (1) is not a null-sphere, its coordinates a may be chosen so that aO2 = 1. These coordinates a are direction-cosines of a line in 5-space and from (9) it follows that the angle between two spheres in 3-space is equal to the angle between two lines in 5-space whose direction-cosines are the corresponding a's and a's. Accordingly we can choose a set of five mutually orthogonal spheres in 3-space whose coordinates aij satisfy the conditions 5 5k With this choice we consider the functions 3 (14) 2 )j= 2, aiji i i ( + y + y+' 1) + (y +y +y ). i=1 The right-hand member is the power of the point (yl, y2, ys) with respect to the sphere Sj, divided by its radius Rj = 1/(a5j+ ia4j). When (15) ia4j +a5 = 0, the sphere Sj is a plane and zj is the distance from the point (yi, Y2, Y ) to the plane. The quantities j(j =1,.... 5) are called the pentaspherical coordinates of the point, i being a parameter. In consequence of (13) we have the fundamental relation 5 (16) zj - 0 j=1 From equations (14) we have 5 5 5 Y1y aliXj zj, y= - 2jj, Ys=,a sj zj, j=1 j=1 j=1 1 2 (asj-+i a4j) z --- O, yA-+y -~y -- Z (Casy —iaaj)Zj. j=I j = 1 236 2III. Circles and spheres Hence when five quantities satisfying (16) are given, they are the pentaspherical coordinates of a point, whose cartesian coordinates are given by (17)80). As an immediate consequence of these results, we have: The equation of a general sphere is of the form (18) 6a1 Z1 +a t +.. -Ja.5 5 0, where zi are the pentaspherical coordinates of a point on the sphere. and the a's are constants. If zj and zji are the pentaspherical coordinates of two points. JUL and M1', whose respective cartesian coordinates are yi and y~, the distance between them is given by 3 5 5 (19) _)fjj7y2,_ = -2 t'2;sj zj j=1 ___ in consequence of (17), where, becatse of (5) and (14) 5~~~~~ j=JRj j=1Rj Rj being the radius of the sphere of coordinates cj. If M' approaches 111 along a curve, the linear element is given by 5 5 2;(d Zj) 2 2;(dzj)2 (20) ds2zzz ~' _____j=_ j=1 (~!L)~ [~ZJ (asj + ia4j)j j=1 _Rj =1~(.rir j When in (17) we take aii= 1 (i = 1. 5), aij= 0(i t A these equations reduce to (21) Y z1, 12 Z2, Ys Z ~1+ n (Z+3 iZ4)>r y +y + y Y~ (+ Y' Z5 80)~ An exception is afforded by Zj asj + i apj in which case the point- is at infinity (cf. VIII, Ex. 1). 95. Pentaspherical coordinates of an 0 net 237 From (4) it follows that the fundamental spheres of coordinates ai, aC2, ai2 3 are planes. In this case we say that the z's are special pentaspherical coordinates. For this case equation (20) reduces to (22) ds2- (z i4)' 95. Pentaspherical coordinates of an O net. If y1, Y2, YS are the cartesian coordinates of a point on a surface referred to its lines of curvature, the functions 2zj satisfy the point equation of the surface, as follows from (14); and consequently, if A involves u and v, the z's satisfy an equation of the form (23) a0 a_ loga ao + alogb 80 +o. av a av av a u av Conversely, if we have five solutions of this equation satisfying (16), say zj, and an independent solution 1/2, then the functions z4, where zj3 = zj, satisfy an equation of this form with c - 0, and consequently yi, y2,y, y,+y,+y, 1, given by (17) or (21), are solutions of the latter equation. Hence: If five particular solutions of an equation of the form (23) satisfy the relation (24) 2 + + +.... + =, they are the pentaspherical coordinates of a suTface referred to its lines of curvature. If the invariants of (23) are equal, the 0 net is isothermic. In this case the equation is reducible to the Moutard form as0 (25)= Mo1. Hence: If an equation of the Moutard form admits five particular solutions zj satisfying the relations jz]_ O0 the quantities are the pentaspherical coordinates of an isothermic net. 96. Congruences of spheres. When the quantities a in (1) are functions of two parameters ul and vi, we say that (1) defines 238 VIII. Circles and spheres a congruence of spheres. When the a's are independent, we can find by differentiation alone an equation of the form o a90 a20 a 0 a. D + 2E +F + +K0==O, v12u+ G1~ + v1~ qKO ~ O, O u+ a + G a u, av au, a v+ of which the five functions a are solutions. If we take as independent variables u and v, defined by u= y- (u, vi), v = ip (1, v1) such that sp and ip satisfy the differential equation D(a +)2E l av+F l=0 the above equation is transformed into the Laplace form (26) ao alogA ao alogB ao (26)._+ - + co. auav aV au an The determination of u and v requires the integration of the differential equation (27) Fdu -2Edux dv +D dvl =Dd 0. The curves on the envelope of the spheres uniquely defined by this equation are called the principal curves. We assume hereafter that the parameters of a congruence of spheres are such that the spherical coordinates a satisfy an equation of the Laplace form (26). Then the principal curves on the envelope are parametric. The equation of the congruence is of the form 5 (28) ii = 0. i=-1 We consider the congruence of spheres defined by (28). As i varies a sphere touches a tubular envelope along the circle C-1, given by (29) a=-O, - -Z u zaa O0 au - 96. Congruences of spheres 239 and as v varies, along the circle CU7 given by (30) i = z0, Z i a- O. Dv The intersections A, and A2 of these circles are the points of tangency of the sphere (28) with the envelope of the congruence of spheres, that is, the points defined by (31) zasi=, -, ai O. au av The circles (29) and (30) are called the focal circles of the first and second rank respectively of the congruence of spheres. Differentiating the first two of (31) with respect to v and making use of the third, we get ~ ajiDZ z CZ0i D- -- Oz ev cu v 0 Hence as v varies the points A, and A2 begin to move along the circle C_1. Similarly as u varies, these points begin to move along C1. But as u and v vary, the points Al and A, trace out the principal curves on the envelope of the sphere of coordinates a. Hence we have the theorem of Darbouxs1): On the envelope of a congruence of spheres, when a displacement is made along one of the principal curves, the four points of contact of two infinitely near spheres with the envelope lie on a circle which is a focal circle of the congruence. We have also from the above that the line A1 A2 and the line As Al, joining nearby points, meet in the plane of C-_ or C0 as v or u varies. Hence these planes are the focal planes of the congruence of lines A1 A2. Accordingly in view of the preceding theorem we have the theorem of Ribaucour: The chords of contact of the spheres of a congruence with its envelope generate a congruence whose focal planes are the planes of the circles.of contact u = const., v = const., and the developables of the congruence are parametric. 81) Lemons, 2nd ed., vol. 2, p. 335. 240 VII. Circles and spheres 97. Derived congruences of spheres. Since the five functions a satisfy equation (26), they are the direction-parameters of a congruence G of lines in 5-space. Conversely, the direction-parameters of such a con/ \- gruence determine a congruence of spheres in 3-space. From (I, 60) it follows that the directionparameters of the congruence Gi and G-1 of tangents to the curves of parameter v on the &\ G first focal surface of G and of parameter u on the second focal surface may be given the respective forms nFig5 (32) ai log, a logB These congruences G1 and G_- in 5-space correspond to the congruences of spheres in 3-space with the respective equations (33) a v log a * i = 0, i log Zi - = 0. Hence we speak of these congruences of spheres as the first and minus first derived congruences of the congruence defined by (28). Evidently the circles (29) are the intersections of corresponding spheres of the given congruence and its minus first derived congruence, and similarly the circles (30) of spheres of the given congruence and its first derived congruence. Consider now the congruence of spheres defined by the first of (33). As u varies, a sphere describes a tubular surface, to which the sphere is tangent along the circle defined by (33) and a 10 O_ a u av logA which is readily seen to be the circle (30). In like manner for the congruence of spheres defined by the second of (33)'as v varies a sphere describes a tubular surface to which the sphere is tangent along the circle (29). 98. Congruences of circles 241 From (3) it follows that at, a2, ae, A,,', C -- (C5 + i a4) are homogeneous coor-, 7Ai dinates of the center M of the sphere. &A/ >,- /?/ Since they satisfy equation (26), M1, \:," —it,describes a net N. When we compare '" —2 ' " > (32) with (III, 2) we remark that the centers lMf and M-1 of these spheres y:6 describe nets which are the first and minus first Laplace transforms of V. Since the lines MM_1 and MMi are necessarily the axes of the circles (29) and (30) respectively, we have the theorem: If G is a congruence of spheres, and G-1 and Gh its minus first and first derived cong?-uences, as u or v varies a sphere describes a tubular surface tangent to the sphere along a circle zwhich is the intersection of the sphere and 'the corresponding sphere of the minus first or first derived congruence respectively. The centers.1, MA-1 and Mi of corresponding spheres describe nets in 3-space forming a Laplate sequence, and the joins MM-1 and IMM are the axes of the circles of intersection. Consider further the circle (29). We saw in ~ 96 that its plane is the focal plane of the congruence Al A2 tangent to the second focal surface of this congruence. Hence on the envelope of the plane the parametric curves form a net N. This circle lies also on the corresponding sphere given by the second of (33). If B1 and B2 are the points where this sphere touches its envelope, this plane is a focal plane of the congruence of lines B. BA and N is the first focal net of the congruence. Hence: The planes of the focal circles of either rank of a congruence of spheres envelope a net whose tangents are the chords of contact of the spheres and the corresponding derived congruence of spheres Zvith their envelopes. 98. Congruences of circles. By definition the circles of intersection of corresponding spheres of two congruences of spheres determined by the congruences of tangents to'a net in 5-space form a congruence of circles, for which the spheres are the focal spheres. The centers of the spheres are the focal points of the congruence of axes of the circles (~ 97). Moreover, as we have just seen, the planes of the circles envelop a net, whose tangents 16 242 VIII. Circles and spheres meet the circles in the points where the focal spheres touch their envelopes. Equations (29) define a congruence of circles, as do also equations (30). Moreover, any congruence of circles may be so defined. When for the congruences of circles C-1 as defined by (29) the parameter v varies, the circles C- (u, v) and C-1 (u, v + dv) meet in the points, A, and A2, where the sphere of parameter cc touches its envelope; they are defined by (31). When the parameter u varies, the circles C-1 (u, v) and C-1 (u + cdu, v) meet in the points, A', A', of intersection of the spheres of parameters a a2a (34) i au' a z Hence as u and v varies the circle C_- meets the infinitely near circle in two points. Consider, conversely, a two parameter family of circles such that as either parameter, u or v, varies, two infinitely near circles meet in two points. Hence as u or v varies the axes of the circles generate developables. Let Fi and F2 be the focal points of this congruence. The spheres S1 with centers F1 and containing the circles of the system envelope, as u varies, a tubular surface tangent along the circles. If the equation of the spheres S, is fizi 0, the circles are defined by this equation and -hi z = 0. In like manner, the spheres 82 with centers F2 and passing through the circles envelop, as v varies, a tubular surface to which they are tangent along the circles. Thus the circles are defined also by bizi =0, z ---=- 0, v i where the first equation defines the spheres S2. Since the circles are the same in both cases, we must have a I _li }+ nmi, D nai + r di. But from ~ 2 it follows that the a's and f's are direction-parameters of a net in 5-space. Combining this result with the second theorem of the preceding section, we have: 99. Congruences of spheres and circles in cartesian coordinates 243 A congruence of circles is characterized by the property that as u or v varies a circle meets the infinitely near circle in two points, which are the points of intersection of the circle and the tangents to the net enveloped by the plane of the circle. We call these points the focal points of the congruence of circles. 99. Congruences of spheres and circles in cartesian coordinates. From (3) it follows that the cartesian coordinates xi, x2, X3 of the center of the sphere, and also the function (35) 2 0 = x+ x+ x R2 are solutions of the equation (36) 9 20 a loga ao + alogb o0, auav v au au av' where as follows from (3) and (26) A B (37) a - b h' Conversely, if x1, x2, x3 and 0 are four solutions of an equation (36), it follows that the quantities a defined by (2) are solutions of an equation of the form (26). Hence: If 0 is a solution of the point equation of a net N, the spheres with centers on N and radii given by (38) R x~ —2+x2+X20 (38) 2 = x+ + - 2 0 form a congruence upon whose envelope the parametric curves are the principal curves. As a corollary we have: The points of an 0 net in 3-space form a congruence of null spheres. When the equation of a congruence of spheres is written in the form (39) o(y-x)-R2 0, 16~ 244 VIII. Circles and spheres the y's being current coordinates, the equations of the focal circles C-1 and C1 of the congruence are given by (39) and the respective equations (40) (y,-x) - 0, ( - x) aX aR a ~ v av which shows that the axes of the circles are tangent to the curves of the net of centers. Since equations (39) and (40) are equivalent to (31), we have: Equations (40) define the focal planes of the congruence of lines joining corresponding points on the envelope of the spheres (39), the developables being parametric. By means of (38) equations (40) may be written (41) y ax a 0 ly ax - O au 8u av av If we write b a a a0 0-1-, ~i — 0' __b au' = aa av au 3v in consequence of (I, 48), the preceding equations are equivalent to ya-1 _ a -1-0 0 Y aX1 a01 -a v av au au Hence: If 0 is a solution of the point equation of a net N, the spheres with centers on the minus first and first Laplace transforms of N and with radii given by the respective equations R2e- te s e -i- 2 0-1, Rl =z b-2o1 meet the spheres with centers on N and radius given by (38) in congruences of circles. When the central net N(x) of a congruence of spheres is an O net, the congruence of lines joining the corresponding points of contact on the envelopes of the spheres is normal, since the focal 100. Congruences R of spheres 245 planes are perpendicular. A surface N normal to the congruence is given by equations of the form [~ 67] - aX+ W 1 aX aw 1 ax X- + + -a a u atu t Dv a Dv' where the X's are the direction-cosines of the congruence and also of the normal to the given net N. The function W is to be determined by the condition that the point of N lies on the line whose equations are (40). Making use of (41), we find that this condition reduces to aDW ao aw ao (42) -e u= Du a2 av'2 where el and 92 are the principal radii of curvature of N. Hence W is found by a quadrature. Conversely, if a net N parallel to N is known, we find by the quadrature (42) the radius of the corresponding sphere. Hence we have the following theorem of Ribaucour82): IfS and S are two surfaces with the same spherical representation of their lines of curvature, the normals of either are the chords of contact with their envelope of a family of spheres having their centers on the other surface; the radii of the spheres are found by a quadrature. Ioo. Congruences R of spheres. We are in a position now to establish the theorem: If the principal curves form a net on each of the sheets of the envelope of a congruence of spheres, they are 0 nets in relation R, provided that the radii of the spheres are not constant. If the quantities y in (39) are the coordinates of the point of contact of the spheres with one sheet of the envelope, we have (3 Dy Dy 0. (43) -2 (y ( -)a =0 a u Dv If equation (41) be differentiated with respect to v and u, respectively we obtain ay ax = O, ay ax 0. av au a u Dv s2) Comptes Rendus, vol. 67 (1868), p. 1334. 246 VIII. Circles and spheres Differentiating the first of (43) with respect to v or the second with respect to u, we get E(-y ) -a 2Y +2ay ay 0. auav an av Hence if the y's are to satisfy 'an equation of the form (36), we must have J 3Y ay= 0, which is the condition that the net au av N(y) be an 0 net. In consequence of this result the central net is 2, 0, and by the last theorem of ~ 66 and the second theorem of ~ 76, the theorem is established. By definition a congruence of spheres is a congruence R if the central net No is 2, 0 and the complementary function of No is the radius of a representative sphere. From ~ 68 we have: The spheres of a transformation R form a congruence R. When two congruences R of spheres have parallel central nets we say that the congruences of spheres are parallel. Hence from the last theorem of ~ 76 we have: If G and G' are parallel congruences R of spheres and the spheres G' do not pass through a point, there can be found by a quadrature a congruence R of spheres UG such that the central nets of G and G. are in relation F. In this sense the congruences G and G1 are in relation F. The spheres with centers on any surface S and passing through a point form a congruence R, the central net 2, 0 on S being determined by the equation of the lines of curvature of the envelope of the spheres; it is the net on S determined by the function V^X2+y2+z2 (~ 3). In like manner the spheres with centers on a surface S and tangent to a plane A, or a fixed sphere, form a congruence R, the central 2, 0 nO t on S corresponding to the lines of curvature on the other sheet of the envelope. The spheres tangent to two fixed spheres form a congruence R with centers on a central quadric of revolution, whose foci are the centers of the fixed spheres. When one of the fixed spheres is replaced by a plane, the surface of centers is a paraboloid of revolution. 101. Pliicker line coordinates 247' If the centers of the fixed spheres are the points (a,, 0) and ( a,, 0) and the radii are r1 and r2, we have ( - a)2 + y + 2 ( + 1) 2, (x + a)2 + y2 + z2 (R + 2 r^ )2 where — = -1 and 2- 4- 1. From these equations it follows that x2+ y2+ 2- R= bx +c, 1 dx+e, where b, c, d and e are constants. Hence these two functions are solutions of the point equation of any net on the quadric. Similar results hold for a paraboloid of revolution. Consequently: On a quadric of revolution any net is 2, 0. Conversely, if every net on a surface S is 2, 0, the functions x2 + y22 2 2 and R1 must be linear functions of the coordinates. Consequently: If every net on a surface is 2, 0, the surface is a quadric of revolution. IoI. Pliicker line coordinates. If xI, x2, x3, x4 and yl, y2, ys, y4 are the homogeneous point coordinates of two points P and Q, the numbers 12, pl3, P14, p23, P34, p42 defined by (44) pij = i x (j 1 2, 3, 4, i are called the Piicker coordinates of the line PQ. Evidently the ratios only of these coordinates are so determined. Since these ratios are unaltered, if x1, x2, x3, xS, are replaced by axx1+-ty,....,,ix4+ y, any two points of the line determine the same set of line coordinates. The six coordinates satisfy the equation (45) p12 P34 +4p13 P2 +P4 p23 =- 0, as may be seen by expanding in terms of two-rowed minors the identity Y1 Y2 Y3 Y4 2 3 4 xi x2 X3 x8 248 VIII. Circles and spheres Conversely, if we have any six numbers pij satisfying (45), we can find two points, (x,, x2, x3, 0) and (yl, 0, y8, y4) satisfying (44), and consequently there is a line with these coordinates. Consider two lines p and p' meeting in the point (xi) and determined by Q(yi) and Q'(y~). Since XI X2 X3 X4 YL Y2 y3 Y4 0==, XI X2 Xs X4 1 yl y'2 Y8 Y we have (46) p12 P4 +P13a p2 +tIl4 p12 +p841 p2 -- p42 pS -p283 P14 0, where pj = (xi yj -xj ye). Conversely, if two sets of line coordinates satisfy this condition, any two points on one line are coplanar with two points on the other. The line coordinates of the line joining P and the point (Qyi+-tky) are of the form pij + tpp j. Hence: A flat pencil of lines consists of the lines whose coordinates are Apij + lpj, if p and p' are lines of the pencil. Consider the lines of space whose coordinates satisfy a linear relation (47) as4 pl2 + a42Pl3 + a23P14 + a12p34 + a13p42 + a14p23 = 0, where the a's are constants. If (b1, b2, bs, b4) is any point of space, the coordinates (x1, x2, x8, x4) of any point of the lines through the given point for which (47) holds, satisfy a linear equation whose coefficients involve the a's and the b's. Hence all the lines through the given point lie in a plane. Consequently the lines satisfying (47) form a linear complex. The plane in which all the lines of the complex through a point lie is called the null-plane of the point. When the a's in (47) satisfy (45), they are the coordinates of a line which all the lines of the complex meet. In this case the complex is called special. 102. The Lie line-sphere transformation 249 o12. The Lie line-sphere transformation. Consider the relation between two 3-spaces r and R whose cartesian coordinates xi and Xi are in the relations (48) (Xi+iX)+ x3i+xz = 0, (X - i) X)x-x2-Xs =0 3) To each point of either space corresponds a line of the other. If (xi) and (x~) are two points of a line of r corresponding to a point (Xi) of B, we find that equations (48) and similar equations in x\ are consistent, provided that (49) p12 -34 ==0; here pj = xi x' —x xi and x, x= = 1. Hence to the points of R correspond lines of the linear complex (49), which is called the complex of the transformation. If we have any line of the complex and substitute the coordinates of any two of its points in (48), we find the corresponding point in R. In like manner the coordinates Xi and Xi of two points in R whose line corresponds to a point in r, satisfy the condition,(Xi-y_)2 = o, that is to a point in r corresponds a minimal line in R. To the points of a line a of the complex (49) correspond minimal lines in R passing through the point A which corresponds to a. Also to the points of a minimal line M in R correspond the lines of the complex through the point of r corresponding to Ml. Moreover, since the transformation is continuous, to the tangents of a minimal curve in R correspond the points in r of a curve whose tangents are lines of the complex (49). A general line in r is defined by x 1 PS4 + X3 P41 + p13 = 0, (50) X2 p34 + X3 42 +p823 = 0. 83) Lie, Geometrie der Berihrungstransformationen, p. 453 et seq. 250 250 ~~~~~~VII. Circle's and spheres Eliminating the x's from these equations an d (48), we get (51) 2a, X1+2a2X2+ 2a3 X3 +i a4 (XI'+X'+ X'+ 1) + a5 (XZyiFX+ X3'-1) 0, where (5) P12 as isP1 al+1, P14 -a5+i' tps4zz a3-ia6, 142~ al-r, P23- 514 Since the _p's satisfy (45), the a's satisfy (11), and consequently: To a general line in r correspond a sphere in 1B. From these results and (10) it follows that to a line of complex (49) of the transformation corresponds a null-sphere in B. From (46) and (52) it follows that to two intersecting lines of coordinates pij and _pj correspond two spheres S and S' whose coordinates a, and a' satisfy the condition (53M) al ala2 C2~....a6a6' -=0, that is, as follows from (12), the spheres are tangent. From (52) it. follows that the line of coordinates _p~j, given by (54) J P12 a3-ia6, i 1), 4 V3'4 ag -J- ia"6, JJ42 _P42, jA23 1)23,~ corresponds -to the sphere (51). The equations of this line are (55) ~~X1 P12 + Xg 1)41+-i Pl1c - 0 X192+X3 P42 +P123 0. The lines of the- complex (49) meeting the line (50) form a congruence. The line coordinates _j of a line of this congruence satisfy (49) and _P34 P12 +1)42 P13 +192,3 P14 +_P12 P34 +1)13 _P42 +P1-114 P23 0. It is readily seen that all of these lines meet the line (55). Hence the lines (50) and (55) are the directrices of the congruence. We refer to two such lines as con4jugate with respect to the complex (49). Hence: 102. The Lie line-sphere transformation 251 To a sphere in R corresponds two lines in r conjugate with respect to the complex of the transformation. From the foregoing discussion it follows that to the points on the conjugate lines correspond the minimal lines of the sphere and to the lines of the complex meeting the conjugate lines correspond the points of the sphere. Consider two intersecting lines in r of coordinates pij and pzj, not of the complex (49), and the two tangent spheres corresponding to them. The homogeneous point coordinates of the centers, C and C', of the spheres are (a2, a2, as, -a -ia) and (c, oa, ca, -a -i a), and the point of tangency of the spheres is the intersection of the line CC' with either sphere. To a line of coordinates ).pij + ppij corresponds a sphere of coordinates oai + c4. Hence: To a flat pencil of lines in r corresponds a pencil of spheres tangent to one another at the same point; moreover, the centers of the spheres form a linear pencil of points projective with the given pencil of lines. From (4) it follows that the sphere of the pencil which is the tangent plane to all the spheres of the pencil is determined by -= - (a - ia )4/(a5 + ia4). Suppose that this is the sphere of coordinates a. From (52) and (54) it follows that to a plane in R corresponds in r two lines of the special complex P14 0 conjugate with respect to the complex (49). All the lines of this special complex meet the line I at infinity in the plane x- 0, that is the line x 0, x4 = 0; this line is in the complex (49). Hence: To a plane in R corresponds in r two lines of the special complex p4-= 0 conjugate with respect to the complex of the transformation, and to the points of these lines the two pencils of minimal lines in the given plane. Let g and g' be the conjugate lines corresponding to a sphere S. A line a of the complex (49) meeting g and g' corresponds to a point A of S, and to the minimal generators of S through A correspond the points m and m' where, a meets g and g'. Since g and g' are conjugate with respect to the complex, all the lines of the complex through m' meet g: that is, m' and g determine the null-plane of m'. In like manner m and g' determine the nullplane of m. 252 VIII. Circles and spheres If S1 is a sphere tangent to S at A, the conjugate lines g# and gl corresponding to S1 pass through m and m' respectively. Hence: To all the spheres tangent to S at A correspond a pencil of lines through m lying in the null-plane of m' and a pencil of lines through n' lying in the null-plane of m. 103. The Lie transformations of surfaces. A point and a plane through it constitute a surface element of which the point is the center. We consider the surface elements consisting of the points of a surface.' in B and the corresponding tangent planes. By the Lie transformation the minimal curves of 2 correspond to two one parameter families of curves c and c' in r admitting lines of the complex (49) for tangents; these curves lie on two surfaces ac and a'. To a point A of 2 corresponds a line a of the complex tangent to a curve c and to a curve c' at points m and n' corresponding to the respective minimal tangents to 2 at A. Hence oa and a' are focal surfaces of a congruence of lines of the complex. Two nearby tangents to c determine a plane tangent to a', that is the null-plane of rn in the tangent plane to a' at n'. In like manner the null-plane at n' is the tangent plane to o at nm. The spheres tangent to ' at A correspond to two pencils of lines with centers at in and n', as shown in the preceding section. Hence to the surface elements of 2 correspond two sets of surface elements whose centers are the points of a and a', and whose planes are the tangent planes to a and a' at corresponding points. Therefore: Then a Lie transformation is applied to a surface 2 in R, the two corresponding surfaces a and a' in r are the focal surfaces of a congruence of lines of the complex of the transformation; corresponding directions in the tangent planes to a and a' correspond to spheres tangent to 2; and the surface elements of 2 correspond to the surface elements of a and a'. Thus the correspondence is a contact transformation. As a point i moves along a curve y on a, the tangent planes intersect in directions conjugate to y. Hence only in case y is an asymtotic line do nearby surface elements have a corresponding direction in common. Two nearby surface elements of 2 are tangent to the same sphere only in case the center of the element moves along a line of curvature of 2. Consequently we have the following important property of the Lie transformation: 104. Congruences R of spheres. WV congruences of lines 253 The spheres with centers at the centers of principal curvature of Z correspond to the tangents to the asymptotic lines on a and a'; the lines of curvature of 2 correspond to the asymptotic lines on a and a', which are the focal surfaces of a TW congruence consisting of lines of the complex of the transformation. Since the tangents to a at a point m are projective with the centers of the spheres tangent to 2Y (102), we have: The centers of spheres corresponding to conjugate directions at m are harmonic with respect to the centers of principal curvature of 2. o04. Congruences R of spheres. W congruences of lines. From ~ 100 it follows that a necessary and sufficient condition that a congruence of spheres be R is that the radius R be a solution of (36), which from (10) is equivalent to the condition that a6 be a solution of (26). Hence: A necessary and sufficient condition that a congruence of spheres be R is that the six coordinates of the congruence satisfying (11) be solutions of the same equation of Laplace. If ai and ao (i = 1,....6) are the coordinates of two congruences R of spheres tangent to the same surface, we have a2 2 0, =, a o, a ac' 0. Hence the quantities car and ad are homogeneous coordinates of two nets, N(a) and N'(a'), in 5-space lying on the hyperquadric 2x2 0, and in relation F for which the lines of the congruence of the transformation are generators of the hyperquadric (cf. III, Ex. 23). As an immediate consequence of the results of the preceding section we have the theorem: If two surfaces ' and Si are in relation B, and o, a' and a,, ao are the transforms of Z and Zi respectively by a Lie line-sphere transformation, the surfaces a and a6 are the focal surfaces of a W congruence and likewise a' and a' moreover, a and a' are focal surfaces of a W congruence consisting of lines of the complex of the transformation and likewise oa and ai. Conversely if a and a1 are the focal surfaces of a W congruence, and ' and Z1 are transforms of C and ar by a Lie transformation 254 VIII. Circles and spheres then 2 and 1 are in relation R. Hence we have the following theorem of Darboux84) as a consequence of the preceding theorems: A necessary and sufficient condition that a congruence be a W congruence is that the Plicker coordinates of the lines be solutions of an equation of Laplace. Since a W congruence is transformed into a W-congruence by a projective transformation, we have: A net N(x) in 5-space lying on a quadratic variety of four dimensions whose equation is (56) 0aijx(ix()- =0 represents a W congruence. From the results of ~ 38 we have: Each transform F of a net representing a W congruence leads directly to a net representing another W congruence. Consider in particular a surface a upon which there lies an R net N (~ 45), that is a net for which the tangents to the curves of the net in each system form W congruences. When a Lie transformation is applied to a, to the tangents to either family of curves of N correspond spheres tangent to 2 with centers describing nets 2, 0 conjugate to the congruence of normals to 2. Moreover, corresponding points of these central nets, C and C', are harmonic to the centers of principal curvature of. and the nets have equal invariants (~ 25), that is Z is a surface P (~ 89). Conversely, if a Lie transformation is applied to a surface 5, there result two nets R. Consequently we have the theorem of Demoulin85): A net R is transformed into an 0 net D2 by a Lie transformation and conversely. Since the nets C and C' are in the relation of a transformation K (~ 25), the coordinates ai and ax (i 1,.... 6), of the spheres with centers on C and C' tangent to 2 satisfy equations of Laplace with equal invariants. Conversely, if the coordinates ao of a sphere are solutions of such an equation, the central net C has equal invariants, and consequently (~ 89) the sheets of the envelope are surfaces 2. Hence: 84) LeMons, 2nd ed., vol. 2, p. 358. 85) Comptes Rendus, vol. 153 (1911), p. 590. 105. Harmonic congruences of spheres and circles 255 When the Laplace equation of a congruence R of spheres has equal invariants, the sheets of the envelope are surfaces!Q, and conversely. As a corollary we have: A necessary and sufficient condition the a net be I is that the Plicker line coordinates of the tangents to either family of curves of the net are solutions of a Laplace equation with equal invariants. 105. Harmonic congruences of spheres and circles. Suppose we have a congruence of circles C1, whose axes are tangent to the curves u - const. of a net N, the circles being determined by a solution 0 of the point equation of N, as discussed in ~ 99. Any net N conjugate to this congruence of axes is defined by equations of the form -0 ax (57) x x-01 av av where 01 is another solution of the point equation (36) of N (~ 8). a& I ao 1 Evidently 0 given by 0 =6 0 — a — 1 -is a solution of the point equation of N. Hence the spheres S with centers on points of N and radii determined by Yx-2R2 = 20 form a congruence. It is readily found that the circles CG lie on the corresponding spheres of this congruence. We say that a congruence of spheres and a congruence of circles are harmonic, when the centers of the spheres describe a net conjugate to the axes of the circles, and the latter lie on the corresponding spheres86): We may state the preceding results as follows: If a congruence of circles is determined by a solution 0 of the point equation of the net N whose curves u = const. are tangent to the axes of the circles, and 01 is the solution determining a net N conjugate to the axes, the congruence of spheres harmonic to the circles and with centers on N is determined by O1 aO (58) - — ao av av 55) Cf. Guichard, Annales L'Ecole Norm. Sup., ser. 3, vol. 20 (1903), p. 197. 256 VIII. Circles and spheres Consider the converse problem when the spheres are given with centers on a net N and radii determined by a solution 0 of its point equation. A congruence G conjugate to N is obtained by drawing through points of N lines parallel to the lines joining the origin to points of a parallel net N' whose coordinates x' are given by (59) a _ ax ax' _ ax (59) ~ h -~ 1? au a u av av A function 0' is given by the quadratures a o' a o a o' a 0 (60) - h, -1 au, au' a v av From (I, 37) we have that the focal point generating a net N whose curves u = const. are tangent to the congruence G is given by equations of the form (61) xzx -- x-x Moreover, 1/1 is the solution of the point equation of N determining N as a Levy transform of N7 (~ 8), that is the function 0, referred to in the above theorem. Hence it is readily seen that the function (62) 0 - o - of enables us to express 0 in the form (58). Consequently the spheres with centers on the net N and radii determined by 0 meet the corresponding spheres of the given congruence in circles harmonic to the latter. Accordingly we have: The determination of congruences of circles harmonic to a congruence of spheres is the problem of finding congruences conjugate to the net of centers of the spheres and a quadrature. Since 0' is determined by (60) to within an additive constant, there are co 1 congruences of circles harmonic to the given congruence of spheres such that corresponding circles have the same axis. This result may be obtained in another manner which brings to light the role of transformations F in this theory. Let N1 (i) 105. Harmonic congruences of spheres and circles 257 be an F transform of N by means of a solution 0 of the point equation of N, the congruence G of the transformation having parameters '. Thus 0, Xi1 - X -- - X From (II, 7) we have axl 8x a51_ ax, a U a8 av av where hi = - e' - = h7 1 ' -10 Consequently from (61) we have X -i l 11 We call the functions 6 0 _ _ 0 _ _' o 01 -- -0', O, the solutions of the point equations of N1 and Nj corresponding to the solution 0 of the point equation of N. We find from (62) that - 1 0' 0 -- 0 1.-..... Consequently we have: If N and Ni are F transforms of one another, and 0 and 61 are correspondinq solutions of the point equations of N and N1 respectively, the congruences of spheres S and S1 with centers on N and N, and radii. determined by 0 and 01 respectively, intersect in a congruence of circles having the lines of congruence of the transformation F for axes. When, in particular, 0 =, the function 0- == 0, and the spheres 81 pass through the origin. Hence: If the central net N of a congruence of spheres S whose radii are determined by a solution 0 of the point equation of N is subjected to a transformation F determined by 0, the spheres 81 zith centers on the transform N1 and passing through the origin meet 17 258 VIII. Circles and spheres the spheres S in a congruence of circles harmonic to S; moreover, the o 1 congruences of circles harmonic to the spheres S dnd with the same axes are obtained in this way by varying the additive constant of integration in 0'. From this result and (38) it follows that the circles are the intersections of the spheres xSz - 2:xo X + 2 ~= 0, 2 O. x - 2 Z0o x1 = -0 where 0 (63). - _ x and the XO's are current coordinates. Moreover, in consequence of (63) the equation of the planes of the circles is (64) o' — - =. Consequently the coordinates xo of the net No enveloped by these planes are found by solving (64) and (65) xo 8 8 ao ox ax _ au ae u av Ov in consequence of (59) and (60). From (41) it follows that (65) are the equations of the chord of contact of the spheres S with their envelope. Consequently No is conjugate to the congruence of these chords of contact and we have the theorem of Guichard87): The congruence of lines joining the points of contact of a congruence of spheres with their envelope is conjugate to the net enveloped by the planes of any congruence of circles harmonic to the spheres. The quantities x' and 0' are tangential coordinates of No. The tangential coordinates of the net N1o enveloped by the planes of any other congruence of circles harmonic to the spheres are given by a h 'uX ' a" 'v a8' iaiu h 8u' av I av ' 87) L. c., p. 197. 106. Representation in 5-space 259 in consequence of (59) and similar equations defining the other congruence of circles. But from these equations it follows that No and N1o are in relation F or are parallel. Hence: When two congruences of circles are harmonic to the same congruence of spheres, the nets enveloped by the planes of the circles are inr relation F or are parallel. Conversely since the tangential coordinates of any two nets in relation F can be chosen so that they satisfy equations of the form (66), it follows that the tangential coordinates of any other net conjugate to the lines joining corresponding points of No and Nio satisfy equations of the form (66). Hence: If S is a sphere describing a congruence and G is the congruence of lines joining the points of contact of S with its envelope, the tangent planes of any net conjugate to G meet the spheres S in a congruence of circles harmonic to the spheress8). io6. Representation in 5-space. When we interpret the results of ~ 105 in terms of nets and congruences in 5-space, we find that a net representing the congruence of circles is harmonic to a congruence representing the congruence of spheres. We shall establish this result. From (2), (32), (37) and (38) it follows that the parameters of the spheres giving rise to the circles C1 are of the form a4 = hxi (i = 1, 2, 3), ac = i h 0( - ); hi ( ) a -v ao v X) (i =1, 2, 3), 18~-= ioga\-~v q- a loga D 1 D loga v 2 v-(+ a ' v Dv2 )- ag) Since the functions a' and fi' are the direction-parameters of the tangents to a net, they are solutions of a system of equations of the form (1,4). Any other solution 01 of (36) gives two other solutions of this system, namely p'zrzh01i, d qIa /a(loga 01) 88) Cf. Guichard, 1. c. p. 197. 17* 260 VIII. Circles and spheres We make use of this fact to obtain the parameters of the congruence in 5-space representing the congruence of spheres S harmonic to the circles C1 in ~ 105. In fact, from (57) and (58) it follows that the parameters -~ of the congruence are given by oaj = Xhzi= h -ao (cq'-dAp') (i 1, 2, 3), TV av a,- q! 7h -= ae- (a q'-a 5 p'). eer;-( ~r Hence (~ 13) the congruence and net in 5-space corresponding to the congruences of spheres and circles are harmonic. In order to prove conversely that congruences of circles and spheres represented by a net and a congruence harmonic to it in 5-space are harmonic to one another, we take the direction-parameters, a and /, of the net in the normal form, so that (67) a= u, = m. The parameters of any harmonic net are of the form (68) a- = q-fip, where p and q are a pair of solutions of (67). From these expressions it follows that the spheres S of coordinates o pass through the circles of intersection of the corresponding spheres of coordinates a and a. Hence their net of centers is conjugate to the congruence of the axes of these circles, since the curves of the net and the developables of the congruence are parametric. From (68) we have by differentiation Q 8 a 3p B a, = a-2 3 P (69) a -q c aQ v a aq a U u a u av a aapav 107. Conjugate congruences of spheres and circles 261 When u varies the characteristics of the spheres S are circles, Fi, the intersection of the spheres of coordinates a and, which in consequence of (67), (68) and (69) pass through the corresponding points, Al and A2, where the spheres of coordinates a touch their envelope. Similarly, when v varies the characteristics of S are circles F2 passing through the corresponding points B, and B2, where the spheres of coordinates f touch their envelope. We have seen that the lines A1 A2 and B1 B2 are the tangents to the parametric net N on the envelope of the circle C1 (~ 97). Moreover, we know that the lines of intersection of the planes of the circles Fr and T2, that is the joins of points of contact of the spheres S with their envelope, form a congruence with the developables parametric. Hence we have the fifth theorem of ~ 105. 107. Conjugate congruences of spheres and circles. We return to the consideration of the congruence of spheres S whose centers describe a net N and whose radius is determined by a solution 0 of (36), the point equation of N. By means of a solution 01 of (36) we determine Levy transforms N-1 and N1 of N, such that MM-i and i11M1i are tangent to the curves of parameter uI and v respectively of N. By means of the functions ft -ft 01 30 ft - fO 07 ~ (70) C'-1= 0- l T1 =8 (70) 0 - 0 0 a ae ~u: 3v we determine spheres S-1 and S1 with centers on N-_ and N1 respectively. The spheres S meet the spheres S-1 and 81 in congruences of circles C-1 and C] respectively, harmonic to S-1 and S1. Moreover, corresponding circles C_- and C1 meet in the points A1 and A2 where S touches its envelope (cf. fig. 6). Since the nets N-1 and N1 are Laplace transforms of one another, the same is true of the functions 0_- and 01 (I, Ex. 11). Consequently by the fourth theorem of ~ 99 the circles C of intersection of the spheres S-1 and 81 form a congruence. Since the spheres 8-1 and 81 pass through A1 and A2, so also does their intersection C. From ~ 97 it follows that the points of contact, B-1, B'L and B1, B' respectively, of the spheres -S1 and S8 with their 262 VIII. Circles and spheres envelopes lie on C, and that the lines AQ.^ oB- B1BL1 and B1B[ are the tangents to thenet N enveloped by the planes ( ^^^^?-E, of C. Since S-1 is harmonic to C-i, I l \M > ^ it follows from ~ 105 that the line ^ /s> B-1BL1 passes through the net enve"7S >/ loped by the plane of C-_, that is ^^A?42 the second focal net of A1A2. SimiFir. 7 larly the line B1 B passes through the first focal net of A_ A2 (cf. fig. 7). Consequently the congruence A A2 is harmonic to N. Hence89): If A1 and A2 are the points of contact with its envelope of a congruences of spheres S with centers at poirts of a net N and G is any congruence harmonic to N, the circles with the lines of G for axes and passing through the corresponding points A1 and A2 form a congruence. Moreover, the congruence A, A2 is harmonic to the net enveloped by the planes of the circles. Consider now the representation in 5-space. To the spheres S corresponds a congruence G, to C, and C_- the focal nets F1 and F2 of G, to S8 and S-1 congruences harmonic to Fi and F2, which are tangents to a net P representing the congruence of circles C. Hence the net P is conjugate to G. Conversely, if we have a congruence G and a conjugate net P in 5-space, AX G Qto G there corresponds in 3-space a con7;, / ~ gruence of spheres S with centers at points,,^ g / ~ of a net N. It touches its envelope at the points, Ai and A2, of intersection of ~/ ^^ ~ C, and C_-, the characteristics of S when Ah\ ~ v and u vary respectively. Since the Fz tangents T1 and T2 to the curves of parameter v and u of P are harmonic to the focal nets F1 and F2 of G, they generate congruences represented in 3-space by congruences of spheres harmonic to the circles C1 and C_- respectively, the centers of the spheres being on' nets N1 and N-1 conjugate to the tangents to 89) Cf. Guichard, 1. c., p. 198. 108. Congruences of circles K determined by transformations R 263 the curves of parameter v and u respectively of N. However, since T1 and T2 are tangents to the net P, the two congruences of spheres with centers on NA and NV-_ meet in a congruence of circles, and corresponding points of IVN and N_1 are the foci of the axes of the circles (~ 97). Hence (~ 10) the congruence of axes is harmonic to N. In view of these results we say, with Guichard, that congruences of spheres and circles related as in the preceding theorem are conjugate to one another90). In order to obtain a congruence of circles conjugate to a congruence of spheres, we note that the coordinates of VN are of the form (57) where 01 is a solution of the point equation of N, the net of centers of the spheres. The function 01 determining the radius of S1 is given by (70). Hence the circles are defined by the equations (71),(y2- 2y )+-20 - 0, ZY 2- a = ', where the y's are current coordinates. Conversely, if we have a congruence of circles, any net harmonic to the axes of the circles is given by equations of the form (61) and the radius of the spheres of the conjugate congruence is determined by (62). The converse may be stated in geometrical form, if we observe that it follows from the representation in 5-spaqe that there is a reciprocal correspondence between the congruence harmonic to N and the nets harmonic to the congruence of axes of the circles. In view of this fact and the preceding results we have: If G is a congruence harmonic to the net N enveloped by the planes of the circles of a congruence, and A1 and A2 are the points of meeting of a line of G and the corresponding circle of the congruence, the plane determined by the mid-point of the segment A1 A2 and the axis of the circle envelopes a net N; the spheres with centers on N and passing through AI and A2 form a congruence conjugate to the given congruence of circles 9). Io8. Congruences of circles K determined by transformations R. In ~ 81 we found that if N1 and N2 are R trans0) L. c., p. 197. 91) Cf. Guichard, 1. c., p. 198. 264 VIII. Circles and spheres forms of an 0 net N, there can be found by quadratures alone a set of oo nets Nc, of which N1 is a member, which are? transforms of a family of oo 1 nets N12, of which N is a member; corresponding points of these oo2 nets lie on a circle K; the tangent planes to the nets at points of K pass through a point Ma which describes a net Nd conjugate to the congruence K of the axes of the circles K; this congruence being harmonic to the central nets of the transformation of N into Nc; the normals to the nets N12 and Nc at points of K are tangent to the sphere S with center at Md and passing through K. We wish to show that the circles K form a congruence of circles and the spheres S a congruence of spheres harmonic to the former. Since w -- x2 is a solution of the point equation of N, 2 the function (cf. ~ 81 and VI, 96) l00 w - - P Pi1 is a solution of the point equation of No0, and consequently =00 a O1-l = 02 2 I - + (02 1+ 01 q2)P o oi0 a 01 - (01 p12-02 p) + (0 i —o r' )P ad W 01 a 0 iV 1)1 )pO 2Pr1 are solutions of the point equations of the focal nets F1 and F2 of the congruence K, and either is a Laplace transform of the other. From (VII, 33) it follows that the radii of the spheres with centers on F1 and F2 and passing through the corresponding circles K are given by R - (1p2- 02 p1)2+ (02 ql — 1 q2)2 y~1Yi ~ (pi q2-2p2 q1) (2 (12 0-2 p1)2p + ((2 ri- 01 r2)2 m2- ~ (pi r2-p2 rl)2 109. Orthogonal congruences of spheres and circles 265 From these equations and (VII, 33) we find that y; -Ryt = 2 w, = 2 Ye Jo. Hence (~ 99) the circles K form a congruence, and the spheres S form a congruence harmonic to these circles (~ 105). From the second theorem of ~ 11 it follows that the lines joining the focal points of the same rank of corresponding lines of the congruences of the transformations R of N into two nets of the class Nc meet in a point M which describes the net N enveloped by the planes of the circles K. If Al and A2 are the points of contact of the sphere S with its envelope, the lines Al A2 form a congruence G which is conjugate to the net N (~ 105). Since normals to a pair of nets N and N1 are tangent to the corresponding sphere S, the circles orthogonal to N and N1 form a cyclic system (~ 79) and lie on the spheres S. In ~ 113 it is shown that a cyclic system is a congruence, and consequently this cyclic system is harmonic to the sphere. From the results of ~ 105 it follows that the axes of these circles form a congruence conjugate to Nd. Io9. Orthogonal congruences of spheres and circles. Consider the congruence of spheres S with centers on N(x) and radii determined by a solution 0 of the point equation of N, and also the harmonic circles whose planes are defined by (64). The spheres S with centers on the net N enveloped by these planes and with radii determined by a solution y of the point equation of N are orthogonal to the corresponding spheres 8, when, and only when, (72) x-6. In consequence of (65) we have r7^ - ^ ax ^ - a ax (73) x u A X au au, av av, Because of these equations it follows that the spheres S are orthogonal also to the first and minus first derived spheres of S (~ 99). Moreover, equations (73), being of the form (41), show 266 VIII. Circles and spheres that the line joining the points where S touches its envelope passes through the corresponding point of N and hence is the axis of the circle. Conversely, let p be any solution of the point equation of N, and let S denote the spheres with centers on N, and radii determined by (. Let N(s) be any net conjugate to the line joining the points of. contact of S with its envelope, then equations (73) hold. By differentiating them with respect to v and u respectively, we get ax ax - a=ox ax _0. av av = au av In consequence of these equations 0 given by (72) is a solution of the point equation of V, and therefore the spheres S with centers on N and radii determined by 0 meet S and the two derived congruences of spheres orthogonally. We say that the congruence of spheres S are orthogonal to the congruence of circles in which the spheres S are cut by the tangent planes of V. By definition the poles of a circle are the two points on its axis, equidistant from the plane of the circle, which are the centers of the null spheres passing through the circle; any sphere with center in the plane of a circle and orthogonal to a sphere through the circle passes through the poles of the circle 9). For the present case the points of contact of S with its envelope are the poles of the circles, since the spheres S are orthogonal to S and its derived spheres. Hence: The circles in the tangent planes of the surface of centers of a congruence of spheres which have for poles the points of contact of the spheres with their envelope form a congruence of circles whose focal parameters are the same as for the congruence of spheres. Also: If C describes a congruence of circles, the spheres with centers on the net enveloped by the plane of C and passing through the poles of Cform a congruence orthogonal to the congruence of circles93). We have seen that the spheres S orthogonal to a congruence of circles C are orthogonal to any congruence of spheres S har92) Note that when the circles are real the poles are imaginary and vice-versa. 93) Cf. Guichard, 1. c., p. 195. 110. Transformations F of congruences of circles 267 monic to C, as are also the first and minus first derived spheres St and S-i of S. Hence if the equation of S is iz = 0, and the equations of C are as = 0, t1-Z = 0, where the a's and l's satisfy (67), it follows from (68) that Ya =o, 27f=0o, O' au a -o, a = o, av =0. Consequently the spheres S, SI and S-1 are represented in 5-space by a congruence and its first and minus first derived congruences whose lines are perpendicular to the tangents to the net representing C. Conversely, if we have such a configuration in 5-space, the corresponding spheres S, S, S-1, in 3-space are orthogonal to the congruences of spheres harmonic to the circles C represented by the net. Consequently the points of contact of S with its envelope are the poles of C. Since the centers of the spheres lie in the planes of C, and these planes are the tangent planes of the central net of 8, the spheres are orthogonal to the circles. Hence: A necessary and sufficient condition that a congruence of spheres and a congruence of circles be orthogonal is that the congruence of spheres and its first and minus first derived congruences be represented in 5-space by congruences of lines perpendicular to the tangents to the curves of the net representing the congruence of circles. io. Transformations F of congruences of circles. Consider the congruence S of spheres and the circles C harmonic to them, as defined in the preceding section, and in addition a second congruence of circles Ci harmonic to S. Let Ni(xi) be the net enveloped by the planes of these circles and i be the solution of the point equation of N1 determining the congruence of spheres with centers on N1 and orthogonal to S. Then (74) x -, and (75)- ^ ~ x I Du ~ a? u ~ au~ a v at; 268 VIII. Circles and spheres From the fifth theorem of ~ 105 it follows that N1 is an F transform of N, or is parallel to N, the nets N and N1 being conjugate to the congruence G of lines joining the points where S touches its envelope. Consider first the case where N and N1 are not parallel, then (76) x1 x- XI', where 0 is a solution of the point equation of N, the direction-parameters x' of G and 0' being in the relations ax' ax a x' a ax a8' Do a 0' - 0 (77) hh a ' o { 4 au aS v ai' S a? ~vs' av av Since G is normal to the tangent plane to N, we have (78) Z a' a 0, Z a' =, and consequently y' defined by (79) = Z ' satisfies the equations (80) hf -9 t au,' av av' From (72), (74), (76) and (79) we have (81) = - that is S1 is the transform of (r. Conversely, let N and N1 be two nets in relation F, and p and Sol be corresponding solutions of their point equations in the relation (81). Let C and C1 be congruences of circles in the planes of N and N1 determined by the functions S and Spo respectively. The equations of the axes of these circles are 110. Transformations F of congruences of circles 269 (o2) \ y aax as ax aQx SO (82) 2 - =0, Y- -^- 0, ut au av av and (83) Z _y XI 0 y _ - -0. au ua av av These two lines are found to intersect in consequence of (76) and (81). If we denote by x the coordinates of the point of intersection M, we find that they satisfy (82) and (79). From (79) and (80) equations (78) follow. In view of this result and the fact that x' and )' are solutions of the point equation of N', equation (79) is the tangential equation of a net, and therefore M describes a net conjugate to both axes. From the preceding section it follows that the spheres S with centers M11 and radius determined by 0 given by (72) are harmonic to the circles C. But by means of (76), (79) and (81) we find that (74) is a consequence of (72). Hence the spheres S are likewise harmonic to (1. We say that the circles C and Ci whose planes envelope nets in relation F and which are determined by solutions of the point equations of these nets in the relation (81) are F transforms of one another, or simply in relation F. Hence: Two congruences of circles zhich are F transforms of one another are harmonic to a congruence of spheres, and two congruences of circles harmonic to a congruence of spheres are F transforms of one another unless the corresponding circles lie in parallel planes. Incidentally we have: When two congruences of circles are in relation F,- corresponding circles meet in the two points where each circle cuts the corresponding -line of the harmonic congruence of the transformation F of the nets enveloped by the planes of the circles. If N, Ni, N2 and N12 are four nets of a quatern in relations F (~ 21) and the planes of congruences of circles C, C1 and C2 envelope the respective nets N, N1 and N2 and are determined by functions ), y1 and (y2 respectively, where 61 602 f q IT 2v - o9 ) ( — y q ' 270 VIII. Circles and spheres the function )12, defined by (84) 6 612 2 12= 2 — (02 021 1 4+- 01 012 2 0-12 021 T), is the corresponding solution of the point equation of N12, whose coordinates are given by (II, 49). Hence the circles C12 whose planes envelope N12 and which are determined by 912 are F transforms of C( and C2. We shall give another geometric interpretation of transformations F of congruences of circles. Take two congruences of circles conjugate to the same congruence of spheres. Since their axes are harmonic to the net of centers of the spheres, the intersections of corresponding axes form a net, and consequently the spheres with centers at points of this net and containing the circles form a congruence of spheres harmonic to the two congruences of circles. Hence: Two congruences of circles conjugate to a congruence of spheres are in the relation of a transformation F. As a corollary we have: The nets enveloped by the planes of the circles of two congruences of circles conjugate to a congruence of spheres are in the relation of a transformation F. In consequence of the last theorem of ~ 107 and the second of ~ 110 we have: Two congruences of circles in the relation of a transformation F are conjugate to a unique congruence of spheres. From the foregoing theorems follows the theorem: Any two congruences of circles harmonic to a.congruence of spheres are conjugate to a congruence of spheres, and conversely. Since two congruences of circles in relation F are conjugate to a congruence of spheres, it follows from that the nets in 5-space representing the congruences of circles are conjugate to the congruence of lines representing the spheres. Hence: The nets in 5-space representing congruences of circles in relation F are F transbfrms of one another; and any two nets in 5-space in relation F correspond to congruences of circles in relation F. The theorem preceding this one is a consequence of the latter and the fact that two nets in relation F are harmonic to a congruence. 111. Parallel transformations of congruences of circles 271 I I. Parallel transformations of congruences of circles. We consider the exceptional case where the corresponding planes of two congruences of circles harmonic to a congruence of spheres are parallel. Now xa xI a x ax I a1 (85) ahI a9 au au' av av' and consequently (75) is satisfied by (86) A = h a avz au,' av av Suppose, conversely, that we have two congruences of circles, C and. C1, whose planes envelope parallel nets N(x) and N1 (xi), and the circles are determined by corresponding functions T and yi, that is in the relation (86), we say they are parallel transforms of one another. We consider the point M of coordinates x defined by (73) and (87) Hx (xi-x) = (pi —p. By differentiating this equation and making use of (73), (85) and (86), we find zax ( -) -- 0. ax (X(i-x) = o. au av Since x —x and pi —y satisfy an equation of the Laplace type, they are the tangential coordinates of a net, the locus of M. From ~ 109 and (87) it follows that the spheres of center M and radius determined by 0 given by (72) are harmonic to the circles C and C1. Hence: Two congruences of circles which are parallel transforms of one another are harmonic to a sphere; and congruences of circles harmonic to a sphere such that corresponding planes are parallel, are parallel transforms of one another. II2. Congruences of spheres with applicable central nets. Let N(x) and N(x) be two applicable nets in 3-space. Since they have the same point equation, the function 272 VIII. Circles and spheres (88) 2 00 -_ xZ -Z2 is a solution of this equation. If 0 is a solution of this equation, the spheres S with centers on N and radii given by (89) R2 x-222 0 form a congruence. Since this may be written (90) R2 z - 2 (o + Oo) the spheres S of radius R and centers on N also form a congruence. Moreover, corresponding spheres of these two congruences touch their envelopes at the same points when N and N are applied to one another. This result follows from the following theorem of Beltrami which is a consequence of the last theorem of [~ 164]: When the surface of centers of a two parameter family of spheres is deformed, the points of contact of the spheres with their envelope (which itself changes in the deformations) are fixed points with respect to the spheres. When 0 = 0 in (89), the spheres S pass through the origin. The envelope of these spheres is the origin 0 and the locus of the point 0' symmetric to 0 with respect to the tangent planes to N. In accordance with the above theorem, when N is applied to N, the loci of 0 and O' are the sheets of the envelope of the spheres. Since 0 and 0' are the points of contact, we have by ~ 109 that the null spheres with centers at 0 and O' meet the tangent planes to Nin a congruence of circles orthogonal to the spheres S. Moreover, as N may be displaced in space the null sphere at the origin may be replaced by any fixed null sphere. Hence: If N and N are applicable nets, the circles, in which the tangent planes of N are met by a fixed null sphere, become a congruence of circles when N is applied to N. II 3. Generation of cyclic systems. It is our purpose now to show that the congruences of circles referred to in the last theorem are cyclic systems (~ 79) and that all cyclic systems can be obtained in this manner. 113. Generation~ of cyclic systems 273 Suppose we have a surf ace S referred to a general moving trihedral. From [~ 69, (51)] it follows that if (x0, zo,~) is a point Afxed in, space, it is necessary and sufficient that ax0 + ~_+q oro 0, ax 'c+ 1tqj.zo- r, yo 0,O aua a0 ~+ rxo-l~z0 - 0, ~ ~+rixo-jvjzo 0, a~~~~~~~~~~~ ___ az0 ~,+ PJLyo- qiXo0O. If we put au ___ auO Bz a ~+ irxo, B, - ag + ijrxo, the firstE four of the above equations may be written (9 1) qz~0 —A, q1 z0 -Al, 1pzo B, p~0Bj and the last two au -(XO A+ yo B), ao ~-(xo Al+yo B1). It follows from (91) that (pq1-pvlq)z-(AB -AB) =0. Comparing these results with equations [~ 174, (12), (13)], we have that the circles of radius izo and center (x0, yo) lying in the plane z -- 0 of the moving trihedral form a cyclic system. Evidently these are the'circles in which these planes are met by the -fixed null sphere I(X -xo) 2 -0. Conversely suppose we have any cyclic system consisting of circles, of center (x0, yo) and radius 1?, lying in the plane z ~~0; then xO, yo, I? must satisfy [~ 174, (12), (13)]. Since these equations 18 274 VIII. Circles and spheres involve only the first fundamental coefficients of S, an infinity of cyclic systems is obtained by the deformation of the envelope of the planes of the circles. As the values of p, q, pi, q1 given by (91) satisfy [~ 69, (48)] one of the deforms of S is determined by these values. The point whose coordinates with respect to its moving trihedral are Xo, yo, — i is fixed in space and consequently we have the theorem of Darboux94): A general cyclic system of circles situated in the tangent planes to a surface S is obtained by taking a deform of S, say S, constructing the circles C of intersection of the tangent planes of S by a fixed null sphere and then taking the positions of C when S is deformed into S. If the center of the fixed null sphere is (a,, a2, a3), 2 0 in (89) is 2 ax —' a2, and consequently we have: If N(x) and N(x) are applicable nets, the circles in the tangent planes of N(x) determined by the solution Ix2- (x-a)2 of its point equation form a cyclic system. In other words we have: If N(x) and N(x) are applicable nets, the circles in the tangent planes of N which have for poles the points of contact with their envelope of spheres with centers on AN and radii given by B2= (X a)2 form a cyclic system. 114. Transformations F of cyclic systems. In ~ 110 we established transformations F of congruences of circles. Now we wish to show that certain of these transformations transform a cyclic system into cyclic systems95). Let N(x) be the net enveloped by the planes of the cyclic system C, N(x) the applicable net and ~ given by (92) 2, = 2_X- 2 the function determining the cyclic system, the point equation of N and N being (36). 94) Lecons, vol. 3, p. 354. 95) Proc. Nat. Acad. Sci., vol. 5 (1919), p. 555. 114. Transformations F of cyclic systems 275 In ~ 27 we showed that if h and I is any pair of solutions of the system (93) -h - h) a loga al_ (h ) a log Dv Dv a' u u the nets A' (x') and A' (x'), whose coordinates are given by a x' x a x' ' x ax x' ax -94), 1 8;_ =D -8 =88 au a z' au v av a au' au D v ' are applicable, and parallel to N and N respectively. Also, if 0' is the solution of the point equations of N' and N' defined by (95) o' = Sx'2- '2, and 0 is given by D ' DO Do' D (96) 0 =h Dv 0 IDv a?,t a~u' av avI the nets N1V(x1) and V1 (1), whose coordinates are defined by (0, - (97) xi=x- x, x =x — x, are applicable, and are F transforms of N and N respectively. The circles C1 lying in the tangent planes of NV, and determined by the function ip, given by (81), form a congruence which is an F transform of C (~ 110). These circles form a cyclic system, if Substituting the values from (81) and (97), we find p' = X ax' -IX 2-. As this function satisfies (80), we have: If N(x) and N(x) are applicable nets and N'(x') and iV'(x') are applicable nets parallel to them, and NV is the Ftransform of 18* 276 VIII. Circles and spheres N by means of N' and the corresponding functions 0 and 01' -Ix -X'2, the circles in the tangent planes of N, determined by g1, where (98) 2 IZx2-2 2j x_ xform a cyclic system which is an F transform of the cyclic system of circles in the planes of N determined by 2y x _-,- 2. We remark that similar results hold for the cyclic systems of circles in the tangent planes of N and N, determined by y = -- and y - -- respectively. We shall show that these transformations admit a theorem of permutability. To this end we take two nets N'(x') and AN"(x") parallel to N and the nets N' and N" parallel to N and applicable to N' and N" respectively. By means of these we determine two F transforms, Ni and N2, of NV, and N1 and N2 of N such that N1 and N1 are applicable, and also N2 and N2. If we choose the additive constants in the functions 0' and 0, so that (cf. II, 97) 0i' + — -2 (x' x" —i 'x")= 0, there are ot pairs of applicable nets N12 and N12 such that A, N1, NT, N12 and N, N1, N2, N12 are quaterns under transformations F. In accordance with the preceding theorem, we set up a cyclic system in the planes of N1 with circles determined by Sp given by (98), and a cyclic system in the planes of N2 with circles determined by '2 = )- 02 (-of f 2 ) Since p1 and q2 are transforms of, by (81) and an analogous equation, it follows that a solution q12 of the point equation of V12 is given by (84). In order that q12 shall define a cyclic system for N12, we must have, in accordance with (98), (9P12 8 = 012 I xfttf - 1tft 0122 9t2=T1-o777 Xdi o -I diis i - I12. It is readily found that this condition is satisfied identically. Hence: 115. Cyclic systems in 3-space and nets 0 in 5-space 277 If C1 and C, are two cyclic systems which are F transforms of a cyclic system C, there can be found by quadratures ooC cyclic system C12 which are F transforms of Ci and C2. I 5. Cyclic systems in 3-space and nets 0 in 5-space. From the theorem of [~ 177] it follows that the spheres focal to a cyclic system are orthogonal. Consequently a cyclic system is represented in 5-space by an 0 net (~ 98). We consider the converse problem, taking an 0 net N in 5-space corresponding to the determinant xI xI.... x5 X2 X/ X...... (99) d- X.... 1,............ We recall that av ' au ak bk - mbk, - nakc. av a;( Let Hi and Z2 denote the spheres of coordinates ~ and q respectively. If a are the coordinates of the congruence of spheres S orthogonal to the circles C represented by N, we have (101) J r= O, ao =0 O. Since the two derived spheres of S must also be orthogonal to Z and Z2, we must have also (~ 97) (102) aaU a=0, J a $-, a a r _ 0, 0 a - auZ av - u av In order that (101) be satisfied, we must have 8 (103) a -iXk k=l1 278 VIII. Circles and spheres where x1, x2, x are determined by (102). Making use of (100), we find that they must satisfy (104) Xk ak = 0, A k bk 0, and consequently they are determined to within a factor, say t. Now au au av av (105) k= S as 0, 2A a Ab 6-~. -v au From these equations it is readily shown that the x's satisfy the same Laplace equation as the a's. If we choose the factor t so that ca, a2, a3 are the coordinates x, x2, xs of the net N of centers of S, this equation assumes the form (36), and consequently x1, x2, xS are the coordinates of a net N. Moreover, from (5), (104) and (105) we have R2= ca2 x, Clda2=z dx'2 dX2. Hence N and N are applicable, and the conditions of the second theorem of ~ 113 are satisfied, so that we have: A necessary and sufficient condition that a congruence of circles be a cyclic system is that it be represented in 5-space by an 0 net. From this result and the last theorem of ~ 110 we have: The problem of transformations F of cyclic systems is the same as of transformations F of 0 nets into 0 nets in 5-space. The centers of the spheres Zx and Z2 are the focal points of the congruence G of axes of the cyclic system. Their homogeneous coordinates are 2, 2, -(,,-(-+ i a n -(+ 4). Since the homogeneous coordinates of the net Nk of the centers of the sphere Sk of coordinates Xk are XC, xX, xy_, — (Xk+-iXk), it follows from (100) and ~ 33 that the nets N'Tc are harmonic to G. 115. Cyclic systems in 3-space and nets O in 5-space 279 Moreover, the cartesian coordinates x of NI satisfy an equation of the form (36), where ak b bk Xa: +i?X4 X +iX As Rk= 1/(X1 +iXk) and (X7- iXZ)/(Xk-+ iXZ) are solutions, so also is k — RB. Hence the nets Nk are 2, 0. The equation of the sphere S, is xlz = o. The pentaspherical coordinates z and zi of the points where it touches its envelope are common solutions of this equation, and (106) z O =, l = 0. Since they must satisfy z2 = 0 also, we have that they may be taken as (107) -- X+ iXs, z - X2-iX3. These points lie on the circle of intersection of the spheres.. and Z2, namely (106), and since their pentaspherical coordinates satisfy an equation of the Laplace type (26), the points describe O nets N and N, (~ 95). As in the case of Nk, these nets are harmonic to the congruence G, and consequently the net is orthogonal to the circles. Similar results follow for the spheres S2 and S3, which proves again that the circles of intersection of the spheres IF and Z2 form a cyclic system. In the same manner it is shown that if al, a2, ac are any constants satisfying the isotropic relation la2 - 0, the quantities OakXa are pentaspherical coordinates of an 0 net orthogonal to the circles. Hence: The last two rows of an orthogonal determinant of the fifth order determine a cyclic system; any isotropic linear functions of the terms of the first three rows are pentaspherical coordinates of an 0 net orthogonal to the cyclic system. 280 VIII. Circles and spheres 1 6. Cyclic congruences. In ~ 73 we found that a necessary condition that a congruence be cyclic is that the direction-parameters satisfy the relation (108) JX2 =- A2 U2+ B2 V2, where U and V are functions of u and v alone respectively, and A and B are the functions which appear in the direction-equation of the congruence (26). Guichard96) has shown that this condition is also sufficient for space of any order. We adapt his proof to the case of 3-space. In ~ 30 we saw that the homogeneous coordinates a and yg, of the foci of a congruence can be chosen so that ay (109) a -my, a nz, au av in which case the direction-parameters yi zi x - -4 - (i =1,2, 3) satisfy an equation (26) with 1 1 A= B — YA- 4 - ' Since A and B are determined to within respective factors U and V, functions of u and v alone, the condition (108) may be written (110) ZX2= 24 +2 If we define functions _ and g by the equations -- = yi si = i (i = 1, 2, 3), -(+ + i )-, -- (5+ i44) z= S 5 =96) Aales Lcle Nrm. Sup., ser.3, ol. 15 188), p. 203. 96) Annales L']Ecole Norm. Sup., ser. 3, vol. 15 (1898), p. 203. Exercises 281 the condition (110) reduces to 5 i-1 __l = O. Moreover, from (109) it follows that the functions 5 and ij satisfy the equations (100). Hence the i's and I's are the elements of the last two rows of a determinant z/ of the fifth order. In consequence of the preceding section, we have that the congruence is harmonic to 0 nets. Hence: A necessary and sufficient condition that the direction-parameters of a congruence in 3-space be cyclic, that is harmonic to a net 0, is that its parameters satisfy a condition of the form (108). Exercises. 1. Show that Zi-= are the pentaspherical coordinates of any point of the plane at infinity not on the circle at infinity; and that a point on the circle of infinity has an infinite set of coordinates of the form zi = -z + h/l, where h is any constant and z,2 = 0, zZ/R = 0. Darboux, Lecons, 2nd ed., vol. 1, p. 270. 2. When two spheres are orthogonal, the center of either is the pole of their radical plane with respect to the other; consequently any one of the five spheres of ~ 94 is conjugate with respect to the tetrahedron whose vertices are the centers of the other four spheres. Darboux, Principes de Geom. Anal. p. 384. 3. If a sphere S of center (a, b, c) and radius R is subjected to an inversion with respect to the sphere So of center (0, 0, 0) and radius k, the center (a', b', c') and radius R' of the transform S' are given by a' b c' R' k_ a -b c -R a2+b2+c2 —RB2 When S is orthogonal to So, S' coincides with S. Darboux, 1. c., p. 386. 4. When in Ex. 3 we replace S by So, we have RP' - B. Consequently the pentaspherical coordinates zk of a point P and its inverse P' with respect to the fundamental sphere Sk differ in sign, but the other four coordinates of P and P' are respectively equal. Hence five successive inversions with respect to the five fundamental spheres transform a point into itself. Darboux, 1. c., p. 387. 5. From (19) it follows that the equation in pentaspherical coordinates z of a sphere of radius p and center of pentaspherical coordinates zi is 2 zi2i zi pR 0. ~: B, 282 VIII. Circles and spheres Hence the pentaspherical coordinates of the center and the radius of the sphere 2aizi - 0 are given by zi - a 2-V - 21? a 2 i I a' a BL i Bi R Darboux, 1. c., P. 390. 6. If in (20) we replace dzj by dzy + kdzj and equate the coefficients of k on both sides of the equation, we get ds 8s cos (ds, 8s) = for the angle between two displacements. Darboux, i. c., p. 389. 7. A homogeneous equation Vo (zj. z5) 0 defines a surface. For ' dr z O=. If wepu )= a displacement on the surface w p az we have Z'xi 6xi = ~ ~ ~_ -Z aZi h a v e i 5z= Az 07 since ep is homogeneous. From Ex. 6 it follows a zi that the displacement Ozi is nornal to the surface. Darboux, 1. c., p. 403. 8. If!p (zi,.. z5) 0 and iS (zi,. 05 = 0 define two surfaces, the angle 0 between their tangent planes at any point is given by (cf. Exs. 6, 7) cos z= Darboux, 1. c., p. 404. 9. The equation 22O a O a 2(uI -v) auv + + =o0 admits the solutions 0 = A 1,f2u - a) (v - a), where A and a are arbitrary constants. If we put f(w) = 2(w - aj) where ai (i = 1,... 5) are constants, the five functions Zi ~ U)(as-v) (as-h) satisfy (24) and for each value of the constant h -are the pentaspherical coordinates of a surface referred to its lines of curvature. The equation of this one parameter family of surfaces is 0 0 (i) a-h O which, in consequence of (21), is expressible in cartesian coordinates as follows: 2 i) 2 _ y- (y2+y2+y+1)2 (y2+y+ 1)2 Yt ++- Y2 +2 ai-h a2-h a3-h 4 (a,. - lb) 4 (a5 - h) Exercises 283 These surfaces of the fourth degree admitting the circle at infinity as a double line are called cyclides. Darboux, Legons, 2nd ed., vol. 1, p. 258. 10. Show that z2 z2 z2 (h - h') I ^____ = - 2 - (ah- -) (ai -h') ai-h z a h' and consequently two cyclides meet orthogonally along their line of intersection. Show that three cyclides of the family (ii) Ex. 9 pass through a point, and that this equation defines a triply orthogonal system of surfaces. Darboux, Principes de Geom. Anal., p. 478. 11. A cyclide defined by (i) Ex. 9 is transformed into itself by an inversion with respect to any of the five coordinate spheres (Ex. 4). Darboux, 1. c., p. 421. 12. Show that transformations R of an 0 net defined in pentaspherical coordinates is the same problem as finding congruences conjugate to a net in 5-space on the hypercone (24) (cf. VI, Ex. 3). 13. A necessary and sufficient condition that the spheres of a congruence be orthogonal to a fixed sphere or pass through a point is that 0 in (38) be a linear function of the x's. Show that in the former case either sheet of the envelope is obtainable from the other by an inversion with respect to the fixed sphere. 14. If the spheres S of a congruence cut a fixed sphere ~ under constant angle, different from 0 and 7r, by the addition of a constant to the radius of each sphere S the new spheres SI are orthogonal to a fixed sphere 2i concentric with Z. Hence the sheets of the envelope of the spheres S being parallel to the sheets of the envelope of the spheres S are in relation R. Darboux, Legons, 2nd ed., vol. 1, p. 310. 15. When the equations (50) of a line in r are written in the form x - rz + p, y = sz-+, the equation of the corresponding sphere in R is (X- P+s) (Y-i s _ V Z- - 1 r) 2r 2r 2r 2r where - = sp-ro. Lie, 1. c. 16. To the points in R correspond in r the lines defined by the Pfaffian equation xdy-y'dx+-dz =-0. Lie, 1. c. 17. The surface elements of a surface a (~ 103) are defined by x, y, z, p, q. FromEx.16 it follows that the null-plane of the point (x, y, z) is yxa'-xy'+z-z'=- 0, where x',, z' are current coordinates. Since this is the tangent plane to a', the corresponding surface element of the latter is defined by x' q y'p, z' - z-xp-yq, p', q' -x. Lie, 1. c., p. 467. 18. The lines through m (x, y, z) of a lying in the tangent plane are defined by (x'- x) = (z' — ), 284 VIII. Circles and spheres where x', y', y ' are current coordinates, and r and s are such that pr+ q-s 1I To these lines correspond the spheres (i) r(X2+Y2+Z2) (s +x-rz)X-i(s-x-+ rz)Y+(1l-sx+ry)Z+y-sz — 0 Lie, 1. c. 19. The tangent plane to ~ is defined by (i) Ex. 18 when r = 0, qs 1, that is (xq+l)X-i(q-)Y(x1) (x-q)Z+z-qy - 0. This plane is tangent to each of the spheres (i) at the point (X, Y, Z), where X+Y= z+ px+q - qY, y y px -+q yg X q- iY --- — z x X —i Y — Z.~ - - q+x 'q+x' q+x and the other functions, P, Q, of the surface elements of 2 are xq+ 1.xq-1 -L Q -i q-x q-x These five equations determine the surface element of Z corresponding to a surface element of a. Lie, 1. c. 20. Show that when the equations of Ex. 19 are applied to the surface a" we get the same result as for a. Lie, 1. c. 21. Let a line d generate a congruence G, its focal points being A and B; and let di and d_l be the corresponding lines of the first and minus first derived congruences of G. When this configuration is subjected to a Lie transformation' spheres S, Si and S_1 correspond to d, di and dl; the spheres S and S, are tangent at a point Mt of one sheet 2t of the envelope of the spheres S, and S and S_1 at a point M_1 of the other sheet _l; moreover, the centers C and C, of S and Si are harmonic to the centers of principal curvature of 1i at M1 and the centers C and C_1 are harmonic to the centers of principal curvature of 2_at MX1. Demoulin, Comptes Rendus, vol. 153 (1911), p. 590. 22. If in the preceding exercise the lines d belong to the linear complex of the transformation, the spheres S are points M, the two surfaces X, and 2-t coincide and are the locus 2 of these points; the spheres Si and S-1 coincide, are tangent to 2 and their centers are harmonic to points of S with respect to the centers of principal curvature of E; and the lines di and d_, are conjugate with respect to the complex. Demoulin, 1. c. 23. To a surface with isothermal spherical representation of its lines of curvature corresponds by a Lie transformation a surface upon which there is an R net for which the tangents to all the curves is one family meet a fixed line (cf. VII, Ex. 25). Demoulin, 1. c. 24. To an isothermic surface corresponds by a Lie transformation a surface upon which there is an R net for which the tangents to all the curves in one family are lines of the complex of the transformation. I)emoulin, 1. c. 25. When a congruence in 5-space is I, the corresponding congruence of spheres in 3-space is formed by point-spheres, whose centers describe a surface upon which the parametric net N consists of its lines of curvature. The orthogonal Exercises 285 congruence of circles consists of circle points with centers on N and in the tangent plane of N. Guichard, 1. c., p. 199. 26. The spheres of a transformation R in 3-space are represented in 5-space by a congruence 2, I, such that if the parameters of the congruence are given by (2) with h = 1, the complementary function is R. Conversely any congruence 2, I of spheres gives a transformation of Ribaucour of the sheets of its envelope. 27. A necessary and sufficient condition that on an envelope of spheres the lines of curvature in both systems be spherical is that the six coordinates of the sphere be of the form a == Ui + Vi, where Ui and Vi are functions of u and v respectively subject to the condition (Ui + V)2I= 0. Darboux, Lecons, 2nd ed., vol. 2, p. 357. 28. A necessary and sufficient condition that on an envelope of spheres the lines of curvature in one system be spherical is that the six coordinates ae of the spheres satisfy an equation of Laplace for which one of the invariants is equal to zero. Darboux, 1. c. 29. When the envelope S of the planes of a congruence of circles is referred to a moving trihedral, and the coordinates of the center are (a, b) and the radius is R, the coordinates of the poles of the circles are (a, b, iR) and (a, b, - iR). The radii of the spheres with centers on S orthogonal to the circles are given by /'a2+ b2-R2. In this case equations (40) assume the form [~ 174, (14)]. 30. By means of the preceding exercise and [~ 175, 20] show that a necessary and sufficient condition that the circles form a cyclic system is that the focal points of the congruence of lines joining corresponding points on the envelope of spheres are harmonic with respect to the points of contact. Ribaucour, Journ. de Math., ser. 4, vol. 7 (1891), p. 260. 31. From the preceding exercise and ~ 82 it follows that a necessary and sufficient condition that the circles orthogonal to the spheres of a transformation B form a cyclic system is that it be a transformation Dm of isothermal 0 nets. 32. If N and N1 are two 0 nets in relation R, the circles orthogonal to N and N1 form a cyclic system (~ 79); the poles Po and PI of the circles have coordinates of the form Xo= — xi 0 -- O i q+ir 1' q-ir' the points Po and P' describe the sheets of the envelope of the spheres through Po and P' with centers on the net N(s-) enveloped by the plane of the circles (cf. ~ 79); the radius of the spheres is given by R2 = 2j(jO'+ kp-+)+k = 2j= +k2; also 2I — R2 -= x2+- 2jw' + 2 Pk, where w' is given by (VI, 80). Show that the right-hand member of this equation is a solution of the point equation of N. Show also that Po and P' are harmonic with respect to the focal points of the congruence of axes of the circles. 286 VIII. Circles and spheres 33. A necessary and sufficient condition that the poles Po and P' of Ex. 32 describe 0 nets in relation R is that the point equation of N admit as solution -B =- 2j- +k2. This condition is reducible to (i) 02 +u(pl +2) +pl p [0 (h + ) - q2 r2] 0. Show that the two sheets of the envelope are conformally represented upon one another and the transformation R is Dn. 34. If S and Si are two surfaces of Guichard of the first kind in the relation of a transformation Rm (~ 92), the condition (i) of Ex. 33 is satisfied when m =. The same is true for the associate surfaces S and S1. The points Po and Po describe isothermic nets No and No which are Christoffel transforms of one another; likewise P' and P'; and the product of the segments Po P' and Po P' isequal to 4. The nets No and N' are in relation Di and likewise No and N'. Calapso, Annali, ser. 3, vol. 24 (1915), p. 25. 35. If two applicable nets N and N are 0, the cyclic congruences determined by the method of ~ 112 are normal, and parallel to congruences of normals to a spherical surface [cf. ~ 178]. 36. The problem of finding cyclic systems whose planes envelope a sphere is equivalent to the determination of surfaces S applicable to the sphere. If Xi, x2, x3 are the coordinates of a surface applicable to a sphere of radius a, the circles in the tangent planes of the sphere determined by 2o =- a2- -2x form such a cyclic system. 37. Let S be a surface applicable to a sphere So, and xi (i 1, 2, 3) the coordinates of S; then xa, X2, xs, 2x are the tangential coordinates of a surface whose lines of curvature correspond to the lines of curvature of S. Ribaucour, 1. c., p. 55. 38. Consider a Go net N in 3-space as defined by (VI, Ex. 12), and take the congruence of spheres of radius a with centers on Ny. Show that corresponding points of the Laplace transforms of N lie on the corresponding sphere; that the developables of the congruence G of lines joining corresponding points of contact of the sphere with its envelope are parametric and the focal planes of G are tangent to the sphere at corresponding points of the Laplace transforms, Ni and N-1 of N; that the focal point of the first rank of G is the center of the osculating sphere of the curve v = const. of N1 and the center of normal curvature of the curve u = const. of N-1, and similarly for the focal point of second rank; and that the curves u = const. of N-1 and v = const. of N1 have common osculating circles at corresponding points. Stetson, Annals, vol. 19 (1917), p. 123. 39. If the curves of two nets N1 and N2 correspond and the osculating planes of the curves v = const. of N1 osculate the curves u = const. of N2 at corresponding points, then Ni and N2 are 0 nets, and Nt is the first Laplace transform and N2 the minus first transform of a net N, which is a net Go. Stetson, Annals, vol 19 (1917), p. 123. Chapter IX. Rolling surfaces. 117. Congruences harmonic to nets C. Let N (x) be a net C in 3-space and N (x) the applicable net. If 0 is a solution of their common point equation, the congruences G and G harmonic to these respective nets and determined by 0 have directionparameters of the form axi ao axi ae axi ao axi ao z. - x Q_6__xi _0 Y __ au a~ a~ v au' a av auv a These functions satisfy the same direction-equation and are in the relation since N and N are applicable. If 0 = x., then X3 = 0; if 0 1+ ix2, then X1+X = O. In the latter case G is 2,I, that is a normal congruence. The intersections of corresponding lines of G and of the harmonic congruence G' determined by 0 = x3 generate a net Nd conjugate to G and G', being a derived net of N (~ 11). From (I, Ex. 15) it follows that Nd is parallel to the net of coordinates Xi _i aYao ao a 8ao at auv av au and the complementary function y', defined by (VI, 17) is equal to 1. Hence this net and consequently Nd, is an 0 net. Therefore the X 1 0 nets normal to G are determined by the congruences harmonic to N for which 0 = x3 + const. Then by applying an orthogonal substitution to the x's, or what is the same thing displacing N in space, we have: 288 IX. Rolling surfaces If N(x) is a net C and N(x) the applicable net, of the congruences harmonic to N: o00 families of parallel congruences are 2, I; 00 fjamilies are 3, I and the others are 4,I; in the first and second cases 0 - ac+ d with I a2 O 0and a2 4 0 respectively. The 0 nets normal to a congruence 2, are generated by its points of intersection with the parallel harmonic congruences determined by 0 = b x + e, whereZ a b = O, zao b = 0, ao being the conjugate imaginary of a. If now we consider the congruence G harmonic to N determined by 0 = + i+i 2, we find that the lines of this congruence lie in the plane x,-+ix-O 0. Moreover the points of these lines corresponding to the points on the congruence G which generate the 0 nets normal to G are the intersections of these lines in the plane xL-+i2 -- and the planes x3 const. In the general case when 0 = ax- d, with I a2 0, the lines of the congruence G lie in the isotropic plane I ax -+d 0, and the points corresponding to the points on G describing the 0 nets are the intersections of these lines and the ol' isotropic lines lying in this plane. I8. Rolling surfaces. Let S and S be two applicable surfaces, and M and M corresponding points. Imagine S held fixed and S moved in space so that M comes into coincidence with 1M, and the tangent planes at M and M come into coincidence, and likewise corresponding linear elements arising from M and l. For each pair of points M and M we have a different position of S. Consequently S assumes o 2 different positions, unless S and S are ruled surfaces with generators in correspondence [cf. 142], in which case there are on' different positions. In this particular case the two surfaces do not have a net in common; it will be assumed that this case is excluded in what follows, unless mention is made of it. As S undergoes this rigid motion of two parameters it is said to roll on S. The common tangent plane at the point of coincidence of S and S is called the plane of contact. The results of the last paragraph of the preceding section may be stated as follows97): 97) Darboux, Annales L'Ecole Norm. Sup., ser. 3, vol. 16 (1899), p. 467. 118. Roiling surfaces 289 Whlen a surface S rolls over an applicable surface 8, a plane r invariably fixed to S, cuts the plane of contact of S and S in a line d which generates a congruence, which is harmonic to the net common to S and S. WVhen ir is isotropic, the congruence of,lines d is normal to a family of surfaces whose lines of curvature correspond to the net common to S and S; moreover, every isotropic line in? meets d in the point which describes one of these normal surfaces. Any line fixed to S may be looked upon.as the intersection of two planes fixed to S. Since furthermore, there is a unique isotropic plane through an isotropic line, we have: When a surface S rolls over an applicable surface 8, a line I invariably fixed to S meets the plane of contact in a point which generates a derived net of the net common to S and 8; if I is an isotropic line this derived net is an 0 net, the normals to which are the positions of the intersections of the tancgent planes to S and the nnique isotropic plane through 1; all derived 0 nets are so determined. As a corollary we have: If two parallel isotropic lines are invariably bound to a surface S.ns it rolls over S, the two 0 nets generated by the points of meeting of these lines and the plane. of contact have the same spherical representation. We may now restate the first theorem of ~ 113 as follows: If a surface S rolls over an applicable surface S and Q is a point invariably fixed to S, the isotropic lines through Q (null sphere) meet the plane of contact in points P, on a circle C, which generate the 0 nets orthogonal to the cyclic system of the circles C. Also: If S and S are applicable surfaces and lines I are drawn through a fixed point normal to the, tangent planes to 8, as S rolls over S the lines I generate a cyclic congruence whose developables -correspond to the net common to S and S. If N(x) and Nl (xi) are nets in relation F by means of a net N/ (x') and corresponding solutions 0 and 0' of the point equations of N and N' respectively, and 5p, c',,fi and s, /', P1 are sets of corresponding solutions of the point equations of N, 7N' and NT so that 0, 0, 0 8X'IX — XI',q'-1 ' 19 290 IX. Rolling surfaces the derived nets of N and NV by means of the respective pairs of functions y,- and yiJ, ' are in relation F (II, Ex. 6). Suppose now that N, N' and N1T are nets C, and that the applicable nets are AT N' and NV so that _ _ 0 —, Xi -- -- X the functions 0 and 0' being chosen in accordance with ~ 27. If we take a, = acx - biy - cz + di, Cy' = a1x'+ b1y'+ cl ', ~p- a~x~-b~-c~-d~, ~'~ a~'~-b~' -c2Y', = 0a2x + b2y + c32+ d2, /' = a2x+ b2y'+ c2Z, where a, b1,.... d2 are arbitrary constants, then yi a = 1+b11+c1+d1, 1 X1 - a 2 Y1 += C2 bi+2 (+. In this case the derived nets of N and Ar are the loci of the points of meeting of the line a1x+ b y + c1+d =- 0, a2 x + b2y+c2z+ d = 0 with the respective planes of contact as N rolls over AT and NJ over 'ANT. Hence: If N and N are applicable nets, and N1 and N1 are F transforms of N and N respectively, by means of 0' x' —_x'2, stuch that they are applicable, and I is a line invariably bouzd to N and N1, as NT rolls on N1 the point where I meets the plane of contact of Ni and N1 generates a net which is an F transform of the net generated by the point where I meets the plane of contact when N rolls on N. If the line I is isotropic, the two nets thus found are 0 nets in the relation R. Hence: If N and N are applicable nets and N, and NV are F transforms by means of 0'W - x'2 - '2, the cyclic systems in which a point sphere invariably bound to N and NAT meets the common tangent planes as N rolls on N and as N2 rolls on N2 are in relation F; moreover, the two surfaces orthogonal to these cyclic systems determined by each generator of the null sphere are in relation B. 119. Special isothermic surfaces 291 I 9. Special isothermic surfaces. We apply these results to the case where S is a quadric Q which meets the circle at infinity in four distinct points Pi. Through each of these points pass two isotropic generators, di, d. The twelve points of intersection of lines di and dC, where i t kc, are the umbilical points of Q. If mi and mn are the points where di and dX meet the plane of contact of Q and S, and 2i and Si denote the loci of mi and mt as Q rolls over.S, on the eight surfaces 2Z and 2' the lines of curvature correspond to the nets N and N common to Q and S. Two surfaces 2i and 2k (k 4 i) are normal to the circles of intersection of the planes of contact with the point sphere having its center at the umbilical point of intersection Pik of di and dX. Two surfaces 2i and X have the same spherical representation of their lines of curvature, by the third theorem of ~ 118. The 0 nets on hi and s. are conjugate to the congruence of the lines of intersection of the plane of contact and the plane of the lines di and d~. Likewise, the 0 nets on Zi and ' are conjugate to the congruence of lines of intersection of the plane of contact and the plane of the lines di and dk, that is the tangent plane to Q at Pi7. The congruence of lines mi ma is harmonic to the net N on S, and its focal points divide harmonically the segment mi mW, since the points mi and mi lie on the generators of Q. Similar results hold for the congruence of lines mi mi. Hence in and CZ are isothermic surfaces in the relation of a Christoffel transformation [~ 159]. Moreover Zi and Hi are in the relation of a transformation K (~ 25) which is a transformation Dm since the nets are 0 nets. Hence we have the theorem of Darboux98): When a quadric Q which meets the circle at infinity in four distinct points rolls on an applicable surface 8, the eight surfaces generated by the points of intersection of the isotropic generators and the plane of contact are isothermic, their lines of curvature corresponding to the net common to Q and S; two of these surfaces determined by generators through a point at infinity are Christoffel transforms of one another: two determined by generators through an umbilical point are in the relation of a transformation D,. In ~ 84 we considered these isothermic surfaces from another 98) L. c., p. 492. 19* 292 IX. Rolling surfaces point of view and found that they are the type called special by Bianchi. 120. Rolling of a surface applicable to a surface of revolution. Let S be a surface of revolution defined by x r cosv, y - r sinv, z- = (r). Its linear element is (1) dS2 - (1 + (12) dr2, + r,,2 dv. Let S be a surface applicable to S. If we put (2) du - ~1+ ' dr, U r, the linear element assumes the form ds2 du2+ U2 dv2 The tangents to the deforms of the meridians of 8, that is the curves v - const. form a normal congruence G for which the other focal surface, S, called the complementary surface to S, is given by equations of the form [cf. ~ 76] /(3) X1 _ X-U ax (3) l x-U au' We recall from [~ 124] that this surface S is applicable to a surface of revolution, whose linear element is (4) ds? = U2 ()du'~+,2 dw, that the normal congruence G is a TV congruence and that any normal W congruence may be obtained in this manner. When we apply equations (3) to 8, we obtain xl=Y1 =0, 1= - -rp r which is the point of intersection of the tangent to the meridian at a given point and the axis of revolution of S. Hence: 120. Rolling of a surface applicable to a surface of revolution 293 When a surface of revolution S rolls on an applicable surface 8, the point in which the axis of S meets the plane of contact generates the complementary surface S8 of S, and the line joining this point to the corresponding point of S generates a normal W congruence of which S and St are the focal surfaces; moreover, any normal W congruence may be generated in this manner. If we put (5) rid " k 1/ 1 -2, kVx, where k is any constant, by means of (2) equation (4) is reducible to 2 (6) ds' -d dr- +r' dvd. If S is the quadric of revolution '2 y2 (7) - -1, we have (8) 1 +,p', ((a- ) r2+ b b (b -- r2) Substituting this expression in the first of (5), solving for r2 and putting k2- b2/(b -a), we have ~ (a- b) r+b2 k2 b(b - 1) Hence (6) assumes the same form as (1) for the expression (8), that is S1 is applicable to S. Similarly, if S is the paraboloid of revolution (9) 2 z -- r2, we have 1+ '2 1 + m2r2 and by taking k" - 1/m2, we find that S_ is applicable to S. Hence: When a non-developable quadric of revolution S rolls on an applicable surface 8, the point in which the axis of S meets the IX. Rolling surfaces 294 common tangent plane generates the complementary surface 81 of 8, and S1 is applicable to S. As a consequence of the results of ~ 100 we have the theorem: When a non-developable quadric of revolution S rolls on an applicable surface 8, the spheres with centers on S passing through either focus are tangent to two surfaces in relation R. 121. The fundamental theorems of Guichard. Let S be a surface applicable to the quadric of revolution S which is defined by (7). Denote by A and A' the extremities of the axis of revolution of 8, and let S be in contact with S at a point M. Denote by d, d] and d', d' the isotropic generators of S through A and A' respectively. These lines meet the common tangent plane 2T in the respective points a, a,, a', al. The lines a a and a' a are the generators of S through M. The lines a a' and a, al are the intersections with the plane ir of the isotropic planes through AA' and the pairs of parallel generators d, d' and di, d. _ From the theorems of ~ 118 it follows that as S rolls on X, the lines a ' and a1a[ generate normal congruences, and the points a, a' and al, a[ generate O nets normal to these congruences. Moreover, these congruences are harmonic to the net N common to S and S. Since the lines a a and a'a, are the generators of S and therefore form a harmonic pencil with the tangents to any net at Ml it follows that the focal points of the lines aa' and a1adivide harmonically the segments a a' and a1la both of which are of the constant length AA'. It can be shown by the methods of [~ 73], that the surfaces generated by a, a', a1, ac, have constant mean curvature AA4- - AA' The lines a a' and a al meet in the point IM' in which the axis of S meets the common tangent plane. Hence M' describes a derived net N' of N, and as shown in the preceding section N' lies on a surface applicable to S. From the fourth theorem of ~ 118 it follows that a and a, describe two of the O nets orthogonal to the cyclic system of circles in which the null sphere at A meets the common tangent plane. Hence these O nets are in relation 1, and the net N' is the central net. Similarly a' and al describe 0 nets in relation R with the same central net. 121. The fundamental theorems of Guichardl 295 Hence we have the theorem99): Wfhen a central quadric of revolution S rolls on an applicable surface S, the points a, a and a', a' in which the isotropic generators through the extremities, A, A', of the axis of S meet the common tangent plane, generate surfaces of constant mean curvature, such that (a) and (al), and (a') and (aW), are pairs in relation R; and (a) and (a'), and (a1) and (a'), are pairs of parallel surfaces. From the theorem of Bonnet [~ 73] it follows that the mean points of the segments a a' and ia aa generate surfaces of constant total curvature in r'elation B. Since the tangent planes to S and its complementary surface S1 are normal to one another, when S rolls on 8, the tangent planes:I to S1 pass through the axis of S, an infinity of them corresponding to one plane. When S1 is applied to S the planes 7r1 become tangent to S and the lines which coincided with the axis of S when S was applied to S become a family of generators of 8, since an infinity of the planes r,1 pass through each line. The isotropic planes through the axis become the isotropic tangent planes to S which have the property of passing through the foci of S. Consequently when 81 is applied to S either the lines a a' (or a' a) coincide with the axis of S and a and a' (or a'and al) become the foci of S. Accordingly as S rolls on 81 the surfaces generated by a, a, and by a' aj are the sheets of the envelope of spheres with centers on S and passing through one and the other focus of S respectively. Since a and a1 are symmetric with respect to the common tangent plane, we have the above results in the following form as given by Guichard100): If- is a central quadric of revolution whose axis is of length 2 a, Fi and F2 its foci, and qP1 and cp2 the symmetric points of F1 and F2 with respect to the tangent planes of S, then as S rolls over an applicable surface the points F1, F2, qYi, 2 describe surfttces of constant mean curvature +- l/a. When S is the paraboloid (9), two of the points a' and a' are at infinity, and consequently the points a and a1 bisect the focal segments of the normal congruences generated by the lines in which 99) Darboux, 1. c., p. 477. 100) Comptes Rendus, vol. 128 (1899), p. 232; cf. also Darboux, 1. c., p. 477. 296 IX. Rolling surfaces the isotropic planes through the axis of S and the isotropic generators through the vertex of S meet the common tangent planes of S and S. Hence: When a paraboloid of revolution S rolls on an applicable surface S, the points in which the two isotropic generators through the vertex meet the common tangent plane generate two minimal surfaces in relation R, and the central net of the transformation R is described by the point in which the axis of S meets the common tangent planes. In ~ 120 we showed that this central net lies on a surface also applicable to S. By reasoning similar to that for a central quadric of revolution, we obtain the theorem of Guichard101): When a paraboloid of revolution rolls on a surface applicable to the paraboloid, the focus and the point symmetric to it with respect to the common tangent plane generate minimal surfaces. In ~ 127 we give analytical proofs of these theorems of Guichard. 122. Deformable transformations R of the first type. Let So be the surface of centers of a congruence of spheres of radius R. Corresponding points on the sheets, S and S1, of the envelope are symmetric with respect to the tangent plane to So at the corresponding point M. Let So be referred to a moving trihedral whose plane y=-0 is the plan ]1IMo M, and let the parametric lines on So be chosen so that the x and y axes are tangent to the curves v = const. and u - const. respectively. If a denotes the angle which the line 1 MMo makes with the plane z - 0, the coordinates of M1 are R cos a, 0, R sin a. The displacements of a point P of coordinates t cosa, 0, t sin are given by [~ 69] J6x = d (t cos a) + du - (qdu + ql dv) t sin a, (10) 6y ' = 1 dv + (r du + r dv) t cos a —(p du +pi dv) t sin a, 6z = d (t sin a) -(q du- q dv) t cos a. When we replace t by R and require that cos a 6x sin a z 0 o01) L. c. 122. Deformable transformations R of the first type 297 for all displacements of Mo, we find aR+ cos -- ~ cos 6 - 0, au aR Dv 0. av In order that M describe a line of curvature on S, it is necessary that there be a point P such that 6x a_ y _ 6 cos a 0 sin a These conditions are equivalent to I(1 ddv + [(r du + -r dv) cos d- (p du +-p dv) sin o] t 0, ( sina du [qdzu+ q dv- da] t 0. Eliminating t from these equations, we obtain as the equation of the lines of curvature on S sin a (p sino'-r cos () dua2+ i (qi -~a dva ~-+ I q( — a o +- sina(pisino-ri coso') dudvZ- O. The equation of the lines of curvature on 81 is obtained from (13) by replacing a by -oa. In order that the lines of curvature on S and S1 correspond, that is that the congruence of spheres be E, it is necessary and sufficient that r cos8 _ Dv p sin a q Du '1 -iy + sino' cos o r1 fi q + 'P1 sin2 o We recall that, in accordance with the theorem of Beltrami (~ 112), if So is deformed, the spheres touch their envelope at the same points of the spheres as for So. We investigate under what conditions a congruence R of spheres remains a congruence R in all deformations of the central surface So. Evidently a remains unaltered, as do also the functions ~, i, r, ri, since they depend 298 IX. Rolling surfaces only on the linear element of So [~ 72]. However, the functions p, q, pi, q1 vary in the deformation. Consequently if equations (14) are to hold for all deformations, we must have [cf. ~ 72, (75)] _ __a a _. (15) 1 a -- 0, a= 0, ' q-+sin 'oso a =0 1 auv a v ' au a Hence a must be a function of u alone. Then from the last of (15) it follows that the parameter v can be chosen so that Ad is a function of u alone. Moreover, from the first of (15) we find that ' is a function of u alone. Hence So is applicable to a surface of revolution. If we take the linear element of So in the form (16) dso U2 du+ -i udv2, from the last of (15) we obtain (17) tan-or = where c is a constant. Then from (11) we have o Uudu (18) R -J Vu+ c - const. Hence we have the theorem of Bianchi102): In order that a congruence R of spheres remain a congruence R in all deformations of the central surface So, it is necessary and sufficient that So be applicable to a surface of revolution and that the radius of the spheres be given by (18). I23. Deformable transformations R of the second type. In this section we determine every congruence R of spheres whose central net No admits an applicable net NTo which is 2, 0 and consequently is the central net of another congruence R of spheres108). To this end we make use of the equations of a transformation R as given in ~ 78. 102) Lezioni, vol. 2, p. 117. 103) Cf. Trans. Amer. Math. Soc. vol. 17 (1916), pp. 437 —458; also, Calapso, Annali, vol. 26 (1917), pp. 151-190. 123. Deformable transformations R of the second type 299 From (VII, 7) we have (19) a~ L)= L2 au p I p? av p Lp where (20) L1= -VE( - e ), L=VG (l+I-. Since the radius of the spheres of the transformation is O/p, and this radius must be the same for the spheres of centers on No applicable to No, by the theorem of Beltrami (~ 112), we must have for the functions 0 and p of the corresponding transformation (21) - e, p-Qp, where e is a factor of proportionality to be determined. From (VII, 10) we find that the first fundamental coefficients E0, Fo, Go of No are (22) Eo = L [1+, Fo L L,' Go L = [l+ ()J If we denote by S and S1 the sheets of the envelope of spheres of radius /lp with centers on No and by q, r, w the functions of the R transformation from S into S1, the latter functions must satisfy equations of the form (VII, 7). From (19), (20) and the analogous equations for the transformation of 8, we get I1nqQ(1~+ -i Eqv(I+), (23) I + (ll l Gre(l l+ l 0 -=1/7(+ 1 0 ( 1 2 P I 2 1P From these equations and the ones obtained by expressing the equality of the first fundamental coefficients of So and So as given by equations of the form (22), we have (24) q - M r, T 20 W- $ Y 300 IX. Rolling surfaces by making a suitable choice of signs of VE and VG, and requiring that the quadratic relation (VII, 8) be satisfied also by the functions p, q, r and w. When the above expressions for q and r are substituted in the equations analogous to (VII, 7), we obtain (25) lc - E- g - ( - G), u z 0 ~' v 0 - 1 V 1 DYE _ r(V-VE) { F av IG aT v r (26) 1 aVG: 1v + a V_ vE (V U, 1/E a 1/E a e (28) {21E-VE K --- -- Vo. The Gauss and Codazzi equations for S are [~ 65]. (V )1 af u (_ _.f 7 0 a E 1 a E t1 a a _V av, i 2 av a 2 e l a Similar equations for S are satisfied by the functions.E, G, Go and and - given by (23) and (26), provided (28) is satisfied. Hence if (23), (26) and (28) hold, the conditions of the problem are satisfied. The central nets No and No of the two transformations R are 2, 0, the complementary function being the radius R of the spheres for each net. In ~ 27 it was shown that any net No parallel to to is applicable to a net No determined by a quadrature, which is parallel to No. From (VI, Ex. 1) it follows that No and No are nets 2, 0, and that they have the same complementary function R' which is 123. Deformable transformations R of the second type 301 obtained by a quadrature. Moreover, from ~ 76 it follows that two of the nets No parallel to No are special, that is '-2= R'2. Hence if the central nets No and No of two transformations R are applicable, there exist two transformations R whose central nets No and No are parallel to No and No, such that the spheres of the transformation with centers on No pass through the origin. We apply the preceding formulas to this case. If ill (27) we put VEi = 1'f = 0, we get (30) VE' = 1/E'-fEI, K', = V -VG. Now equation (28) reduces to V-re FG V+1 E'= o, that is N' and NA are conformal. But as shown in ~ 82 this is possible only in case N' and N[ are isothermic nets in the relation of a transformation Dm. If we make use of the results of ~ 82 and put 0' (31) = V'-e, V/~El -V- G - e-e, in (30) we find that (26) is satisfied. Hence we have the theorem of Bianchi 104): The central net No of a transformation Dm of an isothermic net N' into an isothermic net N[ admits an applicable net No such that as N" rolls on No the spheres of the transformation pass through a fixed point 0. This result may be looked as follows: When No rolls on No, the point 0 invariably fixed to No generates the surface 81. In the terminology of Bianchi~05) S1 is a surface of rolling, being generated by a point fixed to one surface as the latter rolls over an applicable surface. From the foregoing results we have also the converse theorem 106): "'4) Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 303. 10') Rendiconti dei Lincei, ser. 5, vol. 23' (1914), p. 4. "to) Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 349. 302 IX. Rolling surfaces Transformations Dn are the only transformations R for which one of the two sheets of the envelope of the spheres is a surface of rolling as the central net rolls on its applicable net. In ~ 78 we saw if No is any net parallel to No and R is the complementary function, the sheets of the envelope of the spheres of centers on No and radius R are parallel to the isothermic nets N' and NM. Hence we have the theorem: A necessary and sufficient condition that an 0 net N admits an R transform N. such that the central net of the transformation is applicable to a net is that N have the same spherical representation as an isothermic net. 1 a / 7 From (VII, 3) it follows that the functions - a -- and VG Dv 1A - have the same values for all parallel 0 nets. Consequently we must have 1 aVE a I aI8l a8p (32) (32) ~G av av' YE 8 - au From (31) it follows that 1 a VE - a lof e \ VG av -av ' e (33) 1l alau au log e(. -YE~ Du -- D-u We remark that p, q, r of the R transformation of N into N1 and N' into N[ are the same (~ 78). Hence if in (VII, 8) we put w - m v, where m is a constant, it follows from this equation and (VII, 49) written p2 + q2 + r2 = 2 m 0' v' that (34) 0 v 0f V. Consequently from (33), (VII, 47) and the last two of equations (VII, 7) we get (35) VE1+VE =e-, e- -, V/~ +V~QI = e1- ^,-e - 0' V 0 V 123. Deformable transformations R of the second type The equations of the R transformation from N into N1 are 1 X1 -- -- - (pX+q+-'Jrr), where the functions satisfy the system 303 3 0 -a 'u a-u QO8 ~ Q, au ~ Dx a 30 /, - =VYGr, av ap ' -- = ^e r, av e2 (36) a q au a — r + m (t I + e- t 0) el 3av4 q a, av au ar a 3 a zt av a 4 - v 34 \ 0 ), -u q+m el' of v e-'O a~ Q ai aty 0 +e-~, auv 0 [e - av -r [ e~' - [ (o —G)- - -'] Hence: Any 0 net having the same spherical reprensentation as an isothermic net admits 0 BI P transformations for which the central net No is applicable to a net No which is 2, 0. For the case of N' and N' equations (25) can be integrated with the result Q = 1/v'. Since p, q, r are the same for the transformation from N' into NI and N into Nr, it follows that Q has this value in general. Hence the transformation functions from AN' into N- are (37) pq -_ r -- a(37) fo' ' t = 1, and for N into V, (38) p, q, r, 0 -,, - 2-'~ 304 IX. Rolling surfaces.From (30), (31) and (23) we find V VG~ e&'+ -,e1 ~~ -~ V where O', Q' and _j, _' are the principal radii of the isothermic net N' and the net N' respectively. Since the left-hand members of (26) are the same for N', AT' and N, N, we have from this equation, (39) and (34) EVE —e-. V (40) o E VE _~~V e V J/P _ p O e 2 Q 6 Also from (27), (35) and (VII, 9)' 6.(41) Y _ _ Qi El 6''+&Po - GI / +O19 ei 6202o Since the nets NV' and N are defined intrinsically, the determination. of the cartesian coordinates of these nets requires the solution of a Riccati equati on. 1124. Deformable transformations of the second type of minimal surfaces. We apply the results of the preceding -section to the case when the transformation Dm, is the one considered in ~ 87 which transforms the 0 net N' of a minimal sur-.face into the 0 net Ni of a minimal surface. Then v' p and from (39) and (VII, 70) we have /1 V~ - -, ~z0, 125. Deformable transformations E,, 305 that is V' is a plane net. We have-seen in the preceding section that N~ is a point net. Hence the spheres of the transformation pass through a point and are tangent to a plane; consequently the central net lies on a paraboloid of revolution. In order to obtain the equation of the paraboloid, we find the distance from the point iNI to the plane N'. From (VII, 48) and the condition v' p we find that this distance is 1/m. Hence the equation of the paraboloid is mr2z 2a. Therefore we have the theorem: The surface of centers of a transformation Dm of a minimal sltfcace into a Tminimal su'facCe is applicable to the paraboloid of revolution m (x2 + y2) = 2 z 07). 125. Deformable transformations E,,. From ~ 88 and (32) it is seen that a transformation Em of a surface with isothermal representation of its lines of curvature into a surface of the same kind is a deformable transformation P of the second type. We apply the results of ~ 88 to this case. Comparing equations (VII, 76) and (36), we find that the function O' in (36) is given by (42) O - From (40) it follows that IE/~Q = -/G/l2 0, that is N is a planar net. This result may be stated as follows: As So rolls over So, a plane invariably fixed to So has S for its envelope. Bianchi108) calls S an envelope of rolling in this case. Hence we have the theorem of Bianchi109): A surface S with isothermal representation of its lines of curvature is an envelope of rolling, when the central net No of any Eo, transformation of S rolls on. its applicable net. We shall prove the converse theorem110): Transformations E,, are the only transformations P for which the given surface is an envelope of rollingc as the central net ANo rolls on its applicable net. 107) Cf. Bianchi, Memoire dei Lincei, ser. 5, vol. 12 (1918), p. 532. 108) Rendiconti dei Lincei, ser. 5, vol. 231 (1914), p. 3. 109) Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 367. 11) Annals of Mathematics, ser. 2, vol. 17 (1915), p. 64. 20 306 IX. Rolling surfaces If S is to be a plane, we must have from (40) YE +. --- V p. ol ~2 0t 0 r From the first it follows that the spherical representation of S is isothermal. Replacing the first by (VII, 70), we have from the second that (42) must hold, and consequently the transformation is E")ll) I26. Transformations F of deformable transformations R of the second type. Consider a deformable congruence R of spheres of the second type G, and denote by G' the parallel deformable congruence R which gives a transformation Dn_ (~ 123). Let No denote the central net of G and No of the deform G; likewise No and No. Then, as we have seen (43) =Zo2 R= From ~ 76 it follows that if we take the Ftransform of No by means of 0' - (x o2- - R'2), the coordinates x' being the direction-parameters of the conjugate congruence of the transformation, we get a net Vx10 which is 2, 0, the complementary function being 0 R -= R o. Moreover, in consequence of (43) we have o'-f2 1-zo2). 21 r Hence from ~ 27 it follows that the F transform of No by means of 0', the coordinates xo being the direction-parameters of the conjugate congruence of the transformation, is applicable to N1o. Then 2x-0 2 is a solution of the common point equation il) Similar results follow, if we require that Si be a plane. 127. Converses of the theorems of Guichard 307 of lio and ISAo. Since X1o - BR is a solution of this equation (~ 76), so also is x20 — R2, and consequently N1o is 2, 0. Hence: TYhen a deformable congruence R of spheres of the second type is known, another congruence of the same type can be found by quadratures 112) 127. Converses of the theorems of Guichard. In this section we determine under what conditions the two sheets of the envelope of a congruence of spheres have the same constant mean curvature for all deformations of the central surface So. We consider first the case when the sheets are minimal surfaces. If el and Q2 denote the principal radii of curvature of 8, then e1 - -t, e2 = R-t2 where ti and t2 are the roots of the equation obtained by eliminating du and dv from (12). In consequence of (15) this equation is reducible to f~r ^. i / oau'\J. 1u It (pqfi —pqi ndCrl cosa ( —j-uu-19sina-~uu (44) 1(4) +t[n (q-2 u sn2 +)-isinp] lsina 0. If S is to be a minimal surface, we must have 2R t1 + t2, that is sin2 apl-i l(- 2 ) 2R. (p q.-p9 q) sin a + ri cos -- a + sin~ a — If Si also is to be minimal, this equation must be satisfied also when a is replaced by -. This gives the two equations p [(q -pi q)sin — r cos< R _ a (45) (2 Rricos + r) q + (2 R sin- -s -- sin2 0. t12) Rendiconti dei Lincei, ser. 5, vol.302 (1921). 20* 308 IX. Rolling surfaces The first of these equations involves only functions which do not vary as So is deformed. In order that the second hold for all deformations of S8, we must have ao (46) 2 Rr cos + - 0, 2R -E sin a= 0. Eliminating R, we get the third of (15). From (16) we have =, u, r = 1/U. In consequence of (17) the first of (46) becomes 2 = - U Vc2+- u. When we substitute this expression in (11), we obtain U= c,/c2+ u2, where cl is an arbitrary constant. Hence by a suitable choice of the parameters the linear element of So is reducible to the form (47) ds2 = (1 + mu2) du2 + +udv2. This is the linear element of the paraboloid of revolution Q, defined by (9). Also we find (48) R2 (1- 2). 2 'a7 These expressions satisfy the first of (45) and thus all the conditions of the problem are satisfied. Moreover, it can be shown that R equals the distance from a point of Q to its focus on the axis of revolution. Hence we have the theorem 13): The spheres with centers on a paraboloid of revolution and passing through the focus constitute the only congruences R of spheres the sheets of whose envelope are minimal surfaces in all deformations of the central surface. We consider next the case when the two sheets have the same constant mean curvature 1/a for all deformations of So. In this case R2-2a R+ (a-BR) (t1+ t2) + t t2 0. Substituting the expressions for t1 + t2 and t, t2 from (44) and requiring that this condition hold for both S and Si, we get 113) Cf. Bianchi, Memoire dei Lincei, ser. 5, vol. 12 (1918), p. 479. 127. Converses of the theorems of Guichard 309 aa(_R2 2a-R) (_pqi2-.piq Z~) sinat rjc ssa au-0 f~ (B2 -2 aB~~p-,) [1 q p q i a- r1 cos ci1 [(B-2 aR)rcos o'-(a-R) ~1]q I +[B2 -2aB) a or +.(a-R) sinciu sino =: 0. I.~~~~~~a Since these equations must be true for all deformations of S0, the coefficients of q and p11 in the second must be equal to zero. In consequence of the third of (15) -this is equivalent to the single condition (50) (?2 - 2 2aR) cos aor+ Uu (R - a) = 0. On differentiating this equation with respect to u and making use of (11) and (15), we obtain (51) (B - a) (Ucos2 o-u_ I U') + u U2cos a = 0. Eliminating R from (50) and (51) a11nd substituting for cos a its expression from (17), we find (52) )~ 2+) 0.?t a O~~(z" t C8) When (50) and (52) are satisfied, so also is the first of (49). The general integral of (52) is where 1 is an arbitrary constant. Hence the linear element of So is (53) a2(u2 c d t2 +ztt2dV2. (-_ C2) (u22 + 1) This is the linear element also of the quadric Q of revolution of the conic x2 2 (54) X y a2 1 310 IX. Rolling surfaces about the x-axis, provided that (55) c2 12 (55) C2- l 6a2+ Hence to each choice of I there corresponds a value of c. Equation (51) reduces to R-a+ VaI + 1u2+ 0. Furthermore since a2 enters only in (52), we have that a in (50) and (51) may be replaced by - a. Accordingly we have also R+ a+va a2+l VU~2+l- 0. Hence the spheres with centers on the quadric Q may pass through either of the foci on the x-axis, and we have the theorem114): The spheres with centers on a central quadric of revolution and passing through either focus on the axis of revolution constitute the only congruences R of spheres the sheets of whose envelope have the same constant mean curvature for all deformations of the central surface. I28. Theorems of Ribaucour and Bianchi. In accordance with the theorem of Beltrami (~ 112) the tangent planes to either sheet of the envelope of a congruence of spheres envelop a surface for all deformations of the surface of centers, if these planes are understood to be carried along in the deformation. We seek the general solution of the problem: To determine all cases for which the surface elements of a surface S1 associated with a surface S continue to be the surface elements of a surface in any deformation of S115). Let S be referred to any system of parametric lines. Between the coordinates, x, y, a, of S and x1, yl, v1, of S~ we have relations of the form 14) Cf. Bianchi, 1. c., p. 481. 115) Cf. Bianchi, Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 3. 128. Theorems of Ribaucour and Bianchi 311 ax ax (56) xI = x+ - a + m ax + nX, where 1, m and n are the same for yi and zi. Differentiating this equation and making use of [~ 64, (7), (8)], we obtain ax1 L/- FD'-GD \ax (L+n 12 a,i~+ ~ /, FD-ED' x a n8w,,,,., \y + M+n -- )+ + + D1' m X, (57) 'I _a X FD"-GD'\ ax +v ( a 11 u + (Q+nFD H D) -a + (a + D' l+D"m )X, where II L - ai P J 12,+ (221 a hf 12! n am +LJ^^P-L22) -- l (58) -au + {2}1+{12 Q= a~V+ 12m+2w+ 1, the Christoffel symbols being formed with respect to the linear element of S. If X1, Yi and Z, denote the direction-cosines of the normal to Si, we have (59) X I ax + ax au av and similar equations for YI and Z1, where I,,, must be such that a x1Xo, axXI 0o. aznk, av Substituting from the above equations, we get an ( IA-v ) D+ (nl^-vm)D' A (EL+ FM) +p (FL+ GM) + r - (60) D (an (nA 1- l) D'+- (ni - vrm) D"- ) (EP+ FQ) +~, (FP+- G Q)+ v an av 312 IX. Rolling surfaces If S is deformed in any manner carrying the surface-elements of S1 invaribly bound to it, for a deform S of S we have XI - -x, I-xn — +nX, (61) _ X=:x- + a x. + -X, where, m, n, i, Al, v are unaltered by the deformation. This fact is of prime importance. The right-hand members of (60) involve only quantities unaltered by the deformation. Consequently we must have (62) ni = il, ntV = yvm. There are two cases to be considered, 1~. When v $t 0, then n x -1 x — -XYI, which expresses the fact that 8S is a sheet of the envelope of spheres with centers on S, that is the theorem of Beltrami. 2~. When v = 0, then n = 0. In this case the points of S, lie in the corresponding tangent planes to S, and since A XX1 = 0, corresponding tangent planes to S and S& are perpendicular. The existence of this case was established by Ribaucourl16). Hence we have the theorem of Bianchil17): A necessary and sufficient condition that c 2 suTface-elements invaribly associated with a surface S continue to be the surface-elements of a surface in all deformations of S is that they are the elements of a sheet of the envelope of a congruence of spheres with centers on 8, or of a surface S8 such that corresponding tangent planes to S and S1 are perpendicular and points of S1 lie in the corresponding tangent planes to S. We remark that if equations (62) hold for one deformation of 8, they hold for every deformation. 116) Journ. de Math., ser. 4, vol. 7 (1891), p. 92. 17) L. c., p. 4. 129. The surface S as S rolls on an applicable surface S 313 I29. The surface generated by a point in the tangent plane to a surface S as S rolls on an applicable surface S. The formulas of the. preceding section can be used to find the surface generated by a point P associated with S as S rolls on an applicable surface S. When in particular P lies in the tangent plane to S, we have n = 0. We consider this case. If we put (63) {' — L2E+2LMJF+ M2G, Q2= LPE+-(LQ+-1MP)F+MQG, Q -Q P1E3 - 2 PQF+ Q2G, the first fundamental coefficients of S are found from (57) to be (64) JE1 =2 + (ID +mD')2, F- = -2+ (ID + mD') (D' mD"), l{ ~QG -G1 sQ3+ (1lD' + mD")2. Also the coefficients of S8 are (65) {E =-l+ (IlD + mD')2, Fl =-24 (ID+ (D mD) (1D' + rnD"), GI - 3+(1D' +mD)2, where D, D', D" are the second fundamental coefficients of S. Since the functions 2Q, Q2, 'l3 are the same for both surfaces, we have from (64) and (65) ) ds2 - dS = [(/D + nD') du + (ID'+- mD") dv]2 -() [(ID+ mD') dut + (ID'+ m-D") d v]2. As an application of this result we consider the case when S is a ruled surface and S is not ruled, and we take for the curves v = const. the generators of 8, that is D =. We take the point P determined by the condition that it lies on a generator of S and is such that as S rolls on S it describes a line of length zero on S. Then we have m = 0, d- = 0. Since DD" -D'2 -- D'2, we have from (66) (67) ds, = 12 D(Ddu2+ 2 D'd ddv- +D"dv2). Hence we have the theorem of Darbouxl18): 118) Annales de l'Ecole Norm. Sup., ser. 3, vol. 16 (1899), p. 497. 314 IX. Rolling surfaces When a ruled surface S rolls on a non-ruled ap2plicable surface S, the points where the different lines of length zero of S meet the generators of 8 describe surfaces which are conformably represented on one another; and their lines of length zero correspond to the asymptotic lines on S. The surfaces described by the points a, a,, a', ai in ~ 121 are examples of this theorem. I30. Kinematically conjugate directions on rolling surfaces. If on a surface S we take a curve C of the family determined by an equation of the form dv Mdu, and at each point of C draw the tangents to the curves of another family determined by an equation dv N6u, the coordinates of this ruled surface R are given by (61), when we put t6u, m = t6v, n = 0, t being the parameter of a point on the line. As S rolls on an applicable surface 8, along C, these lines generate a ruled surface R tangent to S. From (66) it follows that a necessary and sufficient condition that R and R be applicable is that (6;8) (D + D)dutdu+ (D' +D')(du v + dv u) -- (D"f+ D")dv v 0, or (69) (D- D) d6u ut+ (D'-D')(d 6v + dv u) + (D': —D") cv 6v 0. If we desire the condition to be satisfied for all deformations of S and in particular when S and S are congruent, we must take (69). In fact (68) is obtained from (69), if S is replaced by its symmetric with respect to the origin. Equation (69) coordinates with every family of curves on S a second family, and the relation is involutoric. Beltrami119) has called two such families kinematically conjugate. The preceding results may be stated as follows: If S and S are applicable surfaces, and C and C are any two corresponding curves, the ruled surfaces, R and Rl consisting of the tangents to S and S in directions kinematically conjugate to C and C are applicable. As S rolls on S and C rolls on C, coincident lines of R and R form the instantaneous axis of rotation of the rolling. Hence: 119) Giornale di Battagline, vol. 10 (1872), p. 103. 130. Kinematically conjugate directions on rolling surfaces 315 When a surface S rolls on an applicable surface 8, the instantaneous axis of rotation lies in the plane of contact and its direction is kinematically conjugate to the direction of motion of the instantaneous center. Since the common conjugate system of curves on S and S satisfies the equations D du 6u +D' (du dv+dv du)+ Ddv 6v = 0, Ddu 6u+D'(du 6v+dv u)+D"dv dv O0, we have the theorem: The common conjugate system on two applicable surfaces is kinematically conjugate. When S rolls along a curve of either family of the common conjugate system, the surfaces R and R are developables whose edges of regression correspond, since the points of these edges are Laplace transforms of points of C and C. The equation (70) (D —D) du2+- 2 (D'-D') du dv+(D"-D") dv2 = 0 defines two families of curves on S and S which are kinematically auto-conjugate, that is as S rolls on S along one of these curves the instantaneous axis is tangent to the curve itself. In this case also the surfaces B and R are developable. Since S and S are applicable, the curves C and C have the same geodesic curvature, at corresponding points. Also as follows from (70) and [~ 49] their normal curvatures are equal, and consequently their first curvatures. As an example of the foregoing we consider two applicable surfaces S and 8, and assume that the curves u const. are the asymptotic lines on S. Then D2 D _ -DD. In consequence of this relation we have that the equation of the asymptotic lines on S may be written [(D' —D') du + D"dv] [(D' +D') du+- D"dv] = 0. From this equation it follows that the curves kinematically conjugate to u- const. on S correspond to one family of asymptotic lines 316 IX. Rolling surfaces on S. Bianchi has called them virtual asymptotic lines on S. Moreover, the other family of virtual asymptotic lines on S satisfy (68). Hence we have the theorem120): If S and S are applicable surfaces, the ruled surface R consistiny' of the tangents to the asymptotic lines in one family where they meet a virtual asymptotic line of S is applicable to the ruled surface R of tangents to the corresponding virtual asymptotic lines of S where they are met by its corresponding asymptotic line. If S is a ruled surface, we may take it for R, and then R consists of the tangents to the corresponding virtual asymptotic lines on 8, which are geodesics. Hence as a corollary of the above we have the theorem of Chieffi121): If S is applicable to a ruled surfajce R, the ruled su/'face consisting of tangents to the geodesic virtual asymptotic lines of S at points of meeting of any asymptotic line a of S is applicable to S with a rigid in the applicability. 13 I. Congruences of rolling. When a non-ruled surface S rolls on an applicable surface S, a line I invaribly fixed to S generates a congruence G called by Bianchi122) a congruence of rolling. In order to find the focal points of G, we note that there are two instantaneous axes of rotation in the plane of contact Tr at a point M for which I and a nearby position I' meet, namely the direction li joining M and the point Pi in which I meets Ar, and the direction lS normal to the plane through I perpendicular to wr. When It is the axis of rotation, the point P1 generates one of the focal surfaces. When 12 is the axis, the lines I and I" meet in the foot P2 of the common perpendicular of I and 12, that is the foot of the perpendicular from M to 1; and thus P2 generates the second focal surface of G. The plane of the lines I and I" is the tangent plane to the locus of P, and is perpendicular to rr. The plane of I and I' is the tangent plane to the locus of P2 and is normal to the line MP2. Hence we have the theorem of Bianchi 23): For any congruence of rolling one focal su.face is generated 120) Cf. Bianchi, Memoire dei Lincei, ser. 5, vol. 12 (1918), p. 445. 121) Giornale di Battagline, vol. 43 (1905), p. 9. 122) Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 15. 123) L. c. These results have been established by him by analytical processes in the Rendiconti di Palermo, vol. 39 (1915), p. 187. Exercises 317 by the point P1 in which I meets the plane of contact mz, and the other by the foot P2 of the perpendicular from the point of contact M to 1; the tangent planes to these respective surfaces are the plane through I normal to Yf and the plane through P2 perpendicular to MlP2. From the above treatment it is seen that the focal surfaces of G are generated by the same surface-elements attached to S whatever be the surface S on which S rolls. From the results of ~ 130 and the above considerations we have that the developables of a congruence of rolling correspond to the curves kinematically conjugate to the curves on S whose tangents are the lines l1 and 12 for a point of contact. The latter are the curves corresponding to the curves in which S is met by the pencil of planes through I and by the parallel pencil of planes normal to 1. Bianchi has called them the meridian profiles and the curves of level respectively. Hence: For a congruence of rolling the developables correspond to the curves on S kinematically conjugate to the deforms of the meridian profiles and the curves of level on S. Since a right conoid is the only surface posessing a family of meridian profiles which are also curves of level, we have the theorem of Bianchi: The congruences of rolling with coincident developables are those, and only those, generated by the axis of a right conoid as it rolls on an applicable surface. From the preceding considerations it is evident that the developables of a congruence of rolling are real. We remark also that the focal surfaces of a congruence of rolling afford examples of the theorems of Beltrami and Ribaucour (~ 128). Exercises. 1. Show that the theorem of Bonnet [~ 73] may be interpreted as follows: When a sphere rolls on an applicable surface, its center describes a surface of constant mean curvature. 2. When a quadric S tangent to the circle at infinity at a single point P rolls on an applicable surface S, the two isotropic generators of S through P meet the common tangent plane in two points, a and a', which describe two parallel surfaces of constant mean curvature + I/aa'. Darboux, Annales L'Ecole Norm. Sup., ser. 3, vol. 16 (1899), p. 468. 318 IX. Rolling surfaces 3. If S and S are applicable surfaces and the spheres with centers on S and passing through the origin become tangent to the plane z = 0 as S rolls on S, it is necessary that 2+ 2 + 72 z2 If we put [cf. ~ 47] _ a _ a+ f - a _ a -1 ax- z l Y a-p+ '1 z a+ --- x+iy -- d X-iy = o, then since S and S are applicable 4z2dad - dd (a +1)2 -. Hence the projection of S on the unit sphere and of S on the plane z 0 are conformal. The general solution is given by ( = f(a), Vo =fo (, z= (l + ) /'() ft (a), where f is arbitrary and fo is the conjugate function. Calo, Annali, ser. 3, vol. 4 (1900), pp. 123-130. 4. If S and S are applicable surfaces and the spheres with centers on S and passing through the origin become tangents to a sphere of radius a with center at the origin as S rolls on S, is is necessary that 2 + y2_2 + = R2, 2+y2+ z2 (R a)2. If we put p —a -- R K 4,a y isl=, z-iR z i= a Arty+ a B 1 a + 1 +- - ( Ba) aa+ Zx (R ~ —i) Y =i(R+ a) a z= (B - (a) af - a_+l I__l+1 af+1' since S and S are applicable, we have R2< da-d _ dad# (i) R2didl =(R + a)2 dadf (I +a-')2 a(1 qaf)2' Hence the projections of S and S on the unit sphere are conformal. The general solution is given by a =-f(a), = fo(,), where f is an arbitrary function and fo is the conjugate function. Then R can be found directly from (i). Calo, 1. c. 5. When the Christoffel transform (cf. VII, Ex. 8) of N' in ~ 123 is taken as N, the corresponding net N is a point, and consequently the central net of this transformation is the other special net 2, 0 parallel to No. 6. Show by means of (VII, 3) that a necessary and sufficient condition that an 0 net have the same spherical representation as an isothermic 0 net is that au a/ I1 = Au Trans. Amer. Math. Soc., vol. 17 (1916), p. 447. Exercises 319 7. A necessary and sufficient condition that an 0 net on the unit sphere be the spherical representation of an isothermic net is a I _ 8__ 8 /a 1 a8f\ 8at% a V as v au Ya ~ 8. Show that the net N' of ~ 123 is parallel to two isothermic nets (N)' and (N)" for which -V-f' ()'=_ = e-, v(ET = (), et that a transformation D of (N)' is determined by the functions p, q, r of (37) and (0)' - - 1/', ()' = - ', (-m)' = — m; and that the resulting transform is the net obtained from the given transformation of N' referred to in (VII, Ex. 17). 9. Show that there exists an 0 net, with the same spherical representation as an isothermic net, for which the tangential coordinate P is equal to e --. Calapso, Annali, ser. 3, vol. 26 (1917), p. 171. 10. If we have an Onet N of the kind in Ex. 9 and put V-r= ] -/ = P -cosh Y, Yf = yP sinh o, from (VII, 3,5) we have ap 8a 1 aQ. a8 8a 1 R sinhp, a cbsh So. alt aul 2 ]F av av 2 1/Y From (VII, 9) and (35) we have for an R transform of this 0 net 2 P2 =P+2= e'-fP P +~ + ' +4 (2 In order that this expression be equal to Pi, namely (VI, 101) P - P- m (Pp+Qq +Rr — ), we must have Pp+Qq+Rr-0+21/ Pmv (e'- o+ e-+ +4 mp =0. The left-hand member is found by differentiation to be a constant. Hence there are oo4 transformations R of N into nets of the same kind. Calapso, 1. c., p. 152. 11. If N' and N" are isothermic O nets in the relation of Christoffel (cf. VII, Ex. 8), the mid-point of the join of corresponding points on N' and N" describes an O net N parallel to N' and N" for whichV/E-= cosh i, V1-=G sinh 4. When these expressions are substituted in (36), the function 22- 0'- v0/0' is constant. Show that there are oo4 transformations R for which this constant is equal to 320 IX. Rolling surfaces zero, and that for each of these transformations E — G1 - 1; also that NlTl is the locus of the mid-point of the joins of corresponding points of N1 and N'. Calapso, Annali, ser. 3, vol. 26 (1917), p. 152. 12. A necessary and sufficient condition that the parametric curves of two applicable surfaces S and S' be kinematically conjugate is that D'- D'. 13. A necessary and sufficient condition that the kinematically auto-conjugate curves for two applicable surfaces, S, S, reduce to a single family is that S and S be ruled surfaces applicable with generators corresponding. Bianchi, Memoirie dei Lincei, ser. 5, vol. 12 (1918), p. 441. 14. When two applicable surfaces are referred to their kinematically auto2a 12 conjugate-system, D D '- D' = -D'D "- D, and -a 1t = 8 12t the Christoffel symbols being formed with respect to their linear element. Bianchi, 1. c., p. 445. 15. Given two quadratic differential forms (1) all du2 —2a2 a d dv+ a22 dv2, bll du2+-2 b2 du dv- b2 dv2. When the Jacobian of these forms is equated to zero, the resulting equation, namely aul du -al2 dv, al2 du + a22 dv v bti du-+ bi'2 dv, b12 d u+ b22 dv defines two functions it == - (u, v), ~- ~ (u, v) in terms of which the forms (1) become ail d~-2 - a2- dv2, b1 du-i+ -b2 dv2. Bianchi, Lezioni, vol. 1, p. 82 16. If S and S are applicable surfaces, the curves defined by equating to zero the Jacobian of the forms Edu2+ 2Fdudv+ Gdv2, (D - D) du + 2 (D'- D') du dv + (D"-D") dv form an orthogonal kinematically conjugate system. Bianchi, Lezioni, vol. 2, p. 38. 17. On two applicable surfaces of constant mean curvature + 1/a whose lines of curvature correspond and for which oi and p2 are the radii of principal curvature of one surface and -?p2, -pl of the other [cf. ~ 125], every orthogonal system is kinematically conjugate; moreover, this is the only case where every orthogonal system is kinematically conjugate. Bianchi, 1. c., p. 39. 18. The lines of curvature on a surface of rolling described by a point O fixed with respect to a rolling surface S correspond to that kinematically conjugate system on S which projects into an orthogonal system on a sphere with center at 0, the projection being from 0 as center. Bianchi, Memorie dei Lincei, ser. 5, vol. 12 (1918), p. 448. 19. The lines of curvature on an envelope of rolling described by a plane wT fixed with respect to a rolling surface S correspond to that kinematically conjugate system on S which projects orthogonally into an orthogonal system on ir. Bianchi, 1. c., p. 448. Exercises 321 20. When a surface S rolls over an applicable surface S, any two parallel lines fixed with respect to S generate congruences for which one family of developables correspond, namely, those corresponding to the curves kinematically conjugate to the transforms of the curves of level of S. Bianchi, Rendiconti dei Lincei, ser. 5, vol. 24 (1915), p. 16. 21. When a surface of revolution S rolls on an applicable surface S, the axis of S generates a normal congruence, since the focal planes are perpendicular. One of the focal surfaces is the complementary surface Si of S corresponding to the deforms of the meridians of S. Bianchi, Rendiconti di Palermo, vol. 39 (1915), p. 205. 22. For the surface S defined by X = rrcos0, y = r sin, ~ = clogr+f(0), where f is an arbitrary function, the distance cut off on the z-axis by the perpendicular from any point P of the surface to the axis and the tangent plane at P is equal to c. Hence when S rolls on al applicable surface, the axis generates a congruence of rolling for which the focal distance is constant. Bianchi, 1. c., p. 207. 23. When the surface S defined by a = rcosO, y = r sin 0, z = clogr+mnz rolls on an applicable surface S, the z-axis generates a congruence G of rolling for which the distance between the focal points is c and the angle between the focal planes is sin'- c//c2 + m2. Hence G is a pseudospherical congruence whose focal surfaces have the same gaussian curvature - 1/(c2 + m2) [cf. ~ 171]. Bianchi, 1. c., p. 208. 21 Chapter X. Surfaces applicable to a quadric. 132. Transformations F of nets on a quadric. Consider a net N on the general quadric Q, whose equation is (1) ex2fY2 +gz2+2ayz+2bsxx 2cxy+2rx+2sy+2tz+,w zO. Since the coordinates are solutions of an equation of the form (II, 6), we have on differentiating (1) with respect to u and v XeaX +fay ay g i az ~a ay a'+ ay az au, at au at au av a It a V av aduI +hj az ax a ax ay.+ av aY ant av av au au av a_ a_ Any net N'(x') parallel to N is given by equations of the form (II, 3). Consequently we have an equation of the form (2) in which x, y, z, are replaced by x', y', z'. From this it follows that the function (3) 0' ex' 2+fyf'2+g I 2 2(ay'zb' w b 2~'x + c 'xy') is a solution of the point of equation of N' 124). It is readily found that 0' and 0, given by (4) - 2 [e xx'~fyy'+ g e z'+ a (y z'+ y'z) + b (x'+ z'x) ' c (x y1+ xy 'y) + r x'+ s y'+ tz'] satisfy the equations ao' h ao ao' ao au ait' av, av, 124) The function 0' t 0, since N' cannot lie on a cone. 133. Permanent nets on a quadric 323 When these values are substituted in (II, 2), it is found that the F transform N, (x1) lies on Q. It has been shown in ~ 5 that any congruence conjugate to a net N cati be obtained by drawing through points of N lines whose direction-parameters are the coordinates of some net parallel to N. Hence we have the theorem of Ribaucour: Any congruence conjugate to a net on a quadric meets the quzadric again in a net to which it is conjugate. We apply the results of ~ 21 to the particular case when N is on the quadric Q, and also N1 and N2, that is when 01 and 02 are of the form (4). In order that 012 and 0' be of the form (3) and (4) with x' y' z'; x y z replaced by xj", y', z"; X1, yi, ai respectively, we must have 01 +02 -2 [ex'x" +-fy'y"+ g '"+ a(y z'+ y"') (6) l Jr+ b (z' x"+ z" x') + c (x' y"+ -x" y')] = 0. By differentiation it is found that the left-hand member of this equation is constant, and consequently the additive constants in 06 and 02 can be chosen in oo ways so that (6) shall hold. Hence: If N, and N.c are F transforms of N and all three nets lie on Q(t), there are ool other nets N12 on Q which are F transforms of JN and N2; they can be found by a quadrature. I33. Permanent nets on a quadric. Let Q be a quadric referred to its asymptotic lines, and write its linear element.in the form (7) ds' = Eo da 2 Foda d +- Go d/2, and its second quadratic form (8) q= 2 Do dadO. If S is an applicable surface, its linear element is (7) and its second quadratic form is (9) =Dd -- Dod a2 - 2 d a d ad Df+ d f2. Since the asymptotic lines on Q are straight lines, we must have [~ 85] 21* 324 X. Surfaces applicable to a quadric (10) 11V 2 2 1 0o the Christoffel symbols {t;s being formed with respect to (7). If we put (11) Ho= VEO-F r2 = Fo = K — Ho from [~~ 63, 77] we have J1 _ aoga J12 alog 11 /o - a/~ ' t o~ — ' (12) I 1= ia, 22}= logH. I) a f 2 o A Hence the Codazzi equations [~ 64] for S are reducible to aDo aD o alogagDo a log Do _0, aO at, o, ao aa (13) 8 D ao 8 Do a log a a log D __ a- "" + D-l-og - D ogDr ~ 0. aa ad a a Since Q and S are applicable, Do D' — = — Do2 -- Ho. If we put -Bo (14) z -D07, I" Do, '= j)J, we have (15) 'V-d" - DoIV'"-l. Then (13) may be replaced by (16) a-a =Do = D aa. The equation of the common conjugate system of Q and S is [~ 56] z d a — z"d /a 0. 133. Permanent nets on a quadric 325 By means of the above formulas we find that the Gaussian curvature of the left-hand member of this equation is zero, and consequently it can be given the form dudv. Hence we liave the theorem of Servant'25): If S is a surface applicable to a quadric Q and the latter is referred to its generators, the common conjugate system can befound by quadratures. If we put (17) e J(V da-1f i7'd~) du, e'o (Vjda + V dd ) dv, we find, in consequence of (16), (18) e2co zdf+V 1/fI2 1, e-2cok = jfJ/ 12_1J From equations (17) we have (19) { 2Vzda eO du + e- dv, 2VJ" d3 -e'mduFe (Odv. In terms of u and v the second quadratic forms of Q and B7are necessarily of the form a) - DdU24-D" dv2, D du2+- D" ddV2. In consequence of (8), (9) and (19) we have 1.4f (20) D +I d -1 I D11 2s - - ~~~- 1 (21) D 2 -D/' From these follow the theorems 126): If a net upon a quadric admits an alpplicable nJet, then (22) D+D" =126) Bull. Soc. Math. de France, vol. 30 (1902), p. 19. i'C) Cf. Servant, 1. c., p. 20. 326 X. Surfaces applicable to a quadric The permanent net on a deform of a quadric is isothermalconjugate. From (19) we have e( aa e- @ ade eo a e-@" (23) (3 u 21/' v - 21a '2 at 21/i"' av 21/f' In consequence of (15) and (18) we have from these equations a aA aa aS a4u av av au (24) a./ _ a _ a_ 1= au a av av 2~ D' Conversely, if we have a solution of this system, and define functions c, d and z" by (23) and z' by (15), we find that co and a' satisfy (18). Since the corresponding equations (17) are consistent, we obtain (16). Then a surface S applicable to Q is defined intrinsically by (14), and we have: The determination of permanent nets on a quadric is equivalent to the solution of equations (24). I34. The permanent net on a deform of a quadric. When a quadric Q and an applicable surface S are referred to the permanent nets N and N, we have, in consequence of (21), DD" D2 (25) 4= — H 2 EG- F, where a' = -K, K being the total curvature of Q and S, and the linear element of Q and S is (26) ds2 = E du2- 2 Fdu dv+ G dv2. If we define two functions a and b by means of the equations (27) D = - a, D"= o b2, we have from (22) (28) a2 —b2 1 134. The permanent net on a deform of a quadric32 017.. 341 In. consequence of (21) and (25), we may take (29) -D== D"~ a oab, Ha - a b. The Codazzi equations for NV and NV are [ef.- ~ 64]1 aD D ( 2 2 _ ~ {1 ) a D" { 22 D+ 2, an u 1 J aD ({221 the Christoffel symbols 1 being formed with respect to (26). From these equations, in which D, D" and D are replaced by their expressions from (27) and (29), and the identities [cf. ~ 63] (31) al g H = {11 + av i l2lf I f' we obtain, in consequence of (28), P - an log a p1 1- a: G~ {12 a 1 a log a, )12 av 12 f an log b, J22 -b 2 alob 221 a lgb {2 ~ av lo -j If _,X j denote the cartesian coordinates of S, from the Gauss equations [~ 64, (7)] for S we have that _, -j T are solutions of the equa tions: a20 o a2o -2 lo a o 2 alogb ao (3 3_ _ _ _ _ _ _ alo g a + 2 a V (33) anav -2 aloga ae + alogb ao Hence from ~ 45 we have: The permanent net on a deform of a qutadric is an I? net12) 127) Cf. Tzitzeica, Comptes Rendus, vol. 152 (1911), p. 1077; also Bianchi, Rendiconti dei Lincei, ser. 5, vol. 22 (1913), p. 3. 328 X. Surfaces applicable to a quadric We desire to prove the converse theorem: When the functions a and b in the equations (33) of an R net satisfy the condition (28), the net is applicable to a net on a quadric. From (31) and (33) we have for any R net N j11 g H 12}a log a j221 a, 2ab log log H' {11 a ab2 J12 a logb 1221 /a E W2f log H2= — log. When these values are substituted in the second set of (30), we find that the second fundamental coefficients of an R net satisfying (33) are of the form -D = D"- asbl2/H. Then since DD"= -o4H2, we may take Hao ab, in which case we have (29) and the above expressions for the Christoffel symbols are reducible to (32), in consequence of (28). As a result we have that (27) furnishes a solution of the first set of (30) in which the symbols have the values (32); that is N admits an applicable net N, for which the second fundamental coefficients are given by (27). If a and f, are the parameters of the asymptotic lines of the surface S on which N lies, we have id dca a du-b dv, t dfA = a du-+ b dv, where the integrating factors i, t are subject to the conditions a /a\ a lb a a a \ as \i/ au \a? v a W az In consequence of (27), (29) and [~ 64, (7)] the coordinates of S satisfy the equations a2 ae20 a a0 a0 ao 2 + a2 2 - b lg a b - =a2 loga-+ a b8zt2 8 u2 8u au v v & v a2 _ aloga ao, alogb a0 auav av au au av 135. Transformations Fk of permanent nets on a central quadric 329 If we express these equations in terms of a and,, we obtain 1 632 1 ] ab a1 0 1 ao r a 1a f~1 +-b 1 aa 2i2 a ta + 2,I2 a; - 2 aa [au, a au av / 1 9o a 1 +a 36 -- 1 1\ 1 8 + 2 a[ u + av ( 1 au av = ~' ab\~ J2 a +c P2 a +a cc av ( + T a~8u 36 0 3 1 01a I abb +a[ Dv (-) Adding and substracting these equations, the resulting equations are reducible in consequence of the above conditions on a and p to the form 20 3ae 320 3a -. AB8a'2 A a c' a -2 a' and consequently S is a quadric. I35. Transformations Fk of permanent nets on a central quadric. For the central quadric Q, whose equation is (34) ex -fy2 - g!2 = 1, we have (35) -, Y -- ex fy, gz and the Gaussian curvature is given by128) (36) K -- cVex2, C4 ( e2x2)2' o where c4 = - 1/efg. If Q is referred to a net N whose point equation is the"second of (33), we have a (3 a3 I t= a lo[ a l og b a, (37) a-v logp.( e) ~ - a, log- e -^p ^T~ r ^0v - 128) Cf. C. Smith, Solid Geometry, 9th edition, p. 223. 330 X. Surfaces applicable to a quadric and also (38) Hence (39) x ax Dz av aXe(Dx\z a2 U I(a)x \D/ Dva where U and V are functions of u and v alone. In consequence of (27), (32), and the Gauss equations for Q we find a a 2 x\ a a2 ax /Qx2 ~.V -^ax a e ]=a log2 e +a-2ca e2JeX ax(40 f3 a [ (36) a log h a av From (35) and (36)we have From (35) and (36) we have (41) ZeX x-=- - -u ( D u c Du a I' av c Dv o' When the expressions (39) are substituted in (40), the result is reducible by means of (41) to 1 a U I 8 alog a 2 a3u C a u ' 1 aV 1 aloga 2 v V+-1v 'a Also on differentiating equation (38) with respect to u and v we get a logI+(+ i)olog (U+ V) log a (V+ o) a 0, Dv / Dv a lo b +(U I) alog (U+V) ogb + (U _ 0. From these two sets of equations it follows that U= - = l/c, and consequently (42). axz2 a2 aea C ' laX 2- b2 Ie Dxv c'b Dav c 135. Transformations Fe of permanent nets on a central quadric 331 Hence from (28) and (36) we have the theorem: For any permanent net on a central quadric (34) the coordinates satisfy the condition (43) + ele ( ) where 4 = -1/efg129). This equation may be written (44) a2- b2 - c2 e2xz. Suppose, conversely, we have a net N on the quadric (34) satisfying the condition (43). Since equations (39) hold for any net on the quadric, the functions a and b in the point equation of N can be chosen so that we have (42). If these equations are differentiated with respect to u and v respectively, and the Gauss equations [~ 64, (7)] are used, the resulting equations are reducible by means of (41) to + D(a D-aaa 0 (45) 1 au {222 a. (1a b {aD"- b 0. lb12 a v() bav From (38) and (42) we have a2b2 Dx Dy Dx D_ 2 2_ yH2 2 c2 l1'ef Z= c2 2 c2 ~ e f au av av au c2 Hence, if we take Ha a b, from (31) we obtain (6 D a 22 b (46) log a log In consequence of these expressions equations (45) give D = - -a2", D"= a b2. Substituting these expressions in the first 129) Of. Calapso, Annali, ser. 3, vol. 19 (1912), p. 62. 332 X. Surfaces applicable to a quadric set of equations (30) and making use of (28), which is a consequence of (42) and (43), we get the expressions (32). When these values are substituted in the second set of (30), a solution is -D = D'"= aab. Hence there exists a net N applicable to N. Therefore: When a net on a central quadric (34) satisfies the condition (43), it is a permanent net. Let N be a permanent net on the quadric Q (34). From (3)and (4) it follows that if in the equations of the form (47) x =x- x, we put (48) = 2(exx'+fyy'+gzs') - 2ezx', 6' =Zex'2, the F transform N1 of N lies on Q. In order that N, be a permanent net it is sufficient that &rx12 - e(x) 2 c 2e (49) Ze( a ) +Ze( al ) _ C e From (47) we have by differentiation (50) X' = a ( a O a-x a —,- a0 t where (51) = h e- ', a= - 0'. Substituting these expressions in (49), we can reduce the resulting equation by means of (42) to (52) a2- 2 _ 2b2 b= c2 2e2 x2. In consequence of (II, 16) we have from (51) 8 _ a- log 8 a ha a8o (v3 aV a avlga- 6avvf2 8 a\ - a a h IT a Quate -d [f ~z Q Q a 8ulog au 135. Transformations Fk of permanent nets on a central quadric 333 Differentiating (52) and making use of (53), the resulting equations are reducible to " 1 a o _ b__2("1 -a I + 1 o 2 [ —,0 XI 1x(XI -x) au 0 auu 0 (+ c, ee -- (X - ]O, where gs = a,-b- -0+' czeS2xxl. By means of (44), (51), (52) and (54) these equations are reducible to (55) -+ a0 (h -2) = 0 a, + 3a0 (o +u 0 (u, e av+ 0v - o, which can be integrated in the form kca (56) O'-= kc- 02 2 where k is an arbitrary constant. When this value for s is substituted in (54), this equation and (52) are equivalent, in consequence of (47) and (51), to (57) h a2 -1b2 = C2J (e- k e) x2, h a2- b = C2 (e2-ke) xx. Differentiating the second of these equations, we obtain a h log ah b2 alogb c2 x j + - h- a2 — 3 (ea-ke) x' a au bL a a3 ^ a o, al + logb a2 aloga (-ke) x' x, 3v 3v "3v b av in consequence of (44) and 334 X. Surfaces applicable to a quadric i aa (i-h) 1 loga al (h- ) 1og b a v a' u au; a X(5 ax a x aX a y' ay y' -ay (5h - -- 1;. h au au zu' av 'v a u a u' av aV v, —, a. It is readily found that equations (58) and (59) form a completely integrable system, in consequence of (44). Moreover, for every set of solutions of this system equations (57) are satisfied to within additive constants, as is found by differentiation. Hence each set of solutions satisfying (57) determines an F transform which is a permanent net. From (47), (48) and (59) it is seen that if x', y', z', h and I are multiplied by the same constant, the transform N1 is unaltered. Hence when k is any constant different from e, f and g, there are a 2 sets of solutions satisfying (57) and giving distinct transforms. When k e, there are o 1 sets of solution s, y', z', h and I of (58) and (59) satisfying (57). Thenx' is given by a quadrature (59) and involves an additive constant, say m. In this case each set of solution s, y', a', h and I determines oo transformations, such that the corresponding points of the oo1 transforms lie on a conic, the section of quadric by a plane parallel to the lines from the origin to the points (x'+mn, y', ') as m varies. Similar results hold when k -f or k = g. Hence: A permanent net on a central quadric ex2+fy2+ -gz2 1 admits o 2 transformations Fk into permanent nets on the quadric for each value of the constant k; when k is equal to e, f or g, the transforms N1 may be grouped into o 1 families of oo transforms each such that corresponding points of the nets of a family lie on a conic130). 136. Transformations Fk of surfaces applicable to a central quadric. It is our purpose to show that each transformation F of a permanent net N on a quadric Q into a permanent net N1 on Q leads directly to a transformation F of the net N applicable to N into the net N1 applicable to NT,. In fact, we 130) Cf. Journ. de Math., ser. 8, vol. 4 (1921), pp. 37-66. 136. Transformations F7, of surfaces applicable to a central quadric 335 shall show that it is possible to find without quadratures a net IN' parallel to N such that 0' given by (48) can be put in the form (cf. II, 95) (60) 0'= k (IXf2_fx 2), and then the desired transform is defined by - 0-, Equating these expressions for 0', we have (62) (e - k) x'2 (f-k)y'2 (g-k).'+ k f2 0. Differentiating this expression and assuming that equations of the form (63) a h -a au aul av av' hold, we obtain,ax az 1t -f ax (e -lc) x +(f-k)y +(g-k)z' +k x au 0, au au au a (64) ____ f ax ay axza (e-k)x +(f-k)y ay (g —k)z z k ' a =0. av av av av If these equations are differentiated with respect to u and v and in the reduction use is made of equations [~ 64, (7)] for N and N, two of the resulting equations are satisfied identically in consequence of (64) and the other two are reducible to hle (Dx)2~D j(e-k)x'X4-kDj 'X= 0, (65) \aU2 (5 l e _ D "l (e- k) xf X+ k DU ' X 0. In consequence of (27), (28), (29), (35), (42) and the second of (57) these two equations are equivalent to (66) (h-i) aab+ck I'X = 0. 336 X. Surfaces applicable to a quadric Solving equations (64) and (66) for x', y', ', we have expressions of the form crab r, 1 f~v - al x' aFx ax - a b (l-h)x+ - I d-k)' (Fd G d (67)1 C ax C (e- ( -k)X d ~ DX (k ( - F E Dv Du Dv If we differentiate these expressions, we find that x', y', ~ satisfy equations of the form (63), by making use of equations of ~~ 134, 135 and of [~ 631. Also from (67) we have, with the aid of [~ 48, (11)1, a2 b2 (68) k2 2= x - 2 ( h)2 + (e -) - (e-k) xX)2. Substituting this expression in (62), we find that it is satisfied in virtue of (57). From these results and the last theorem of ~ 135 we have, when k is not equal to 0, e, f or g, the theorem: If N is a net applicable to a net N on a central quadric ex2 -fy2 + gZ2 = 1, for each value of k different from e, f, g and zero, there exist W 2 transformations Fk of N into nets NV applicable to the quadric; these transforms are conjugate to o2 congruences G; their determination requires the finding of solutions of a completely inteqrable system of equations subject to a quadratic and a linear homogeneous relations 13). When k -- e, the function x' is determined to within an additive constant m, as seen in ~ 135. There are only oo sets of solutions y', z', h and 1, and in consequence from (67) it follows that there are only soo congruences G of the o2 transformations. As m varies we obtain w1 transforms N1 conjugate to the same congruence. They are defined by (61) with 0 = 2 exx'+ 2mex, O'= e(x'+ m)2+fy'2+gz'2. From this expression for 0 and the results of ~ 23 it follows that the tangent planes at corresponding points of these nets N, enve131) Journ. de Math. 1. c. p. 53. 136. Transformations yiY of surfaces applicable to a.' central quaciric 337 lope a cone. If ~,~~ are the coordinates of the vertex, the equation of the tangent plane is where ~,',~ are current coordinates, and XY adZ ar direction-parameters of the normal to N,. When their expressions are calculated, it is found that they involve m. to the second degree, and consequently the cone is a, quadric. When k e, w' does not appear in (62) and (64). Solvinig the latter for y' and z', and substituting in (62), xve obtain a homogeneous quadric equation in V', -',Z' Hence the lines of the conigruences G through a point of N form a quadric cone. Since similar results hold. when k is equal to f or g, we have in consequence of the last theorem of ~ 135 the theorem: If NA is a nJet applicable to a net IV on the centr-al quadricQ (34), there are 2sets of solutionsoi qain (57), (58), (9 wi'ith. A: equal to e, f or g; there are thus three families of0 transforms N, of NA. which are applicable to Q:- the tr ansformvs of each famnily a-re conjugate to oc' congruences G, there beinq oc transformns con)ju.gate to each GC; the lines of' the (congruences G thro~ugh apoint of -N form a quadric cone; the tangent _planes at p~oints qfaline of C qf' the nets N1 co~' aate to it envelop a quadric cone,__and the _points on Q correspQjonding to these p~oints of the nets NV, on a line of' C lie onl a conic'312). When Q is the quadric of revolution e(x 2 +y2) +gs2 1 the transformations of the type described in the first of the above theorems exist. There is, however, only one family of transf ormations of the type described in the second theorem; they are Fg. When 1;- e, equation (62) may be written 132) The two types of transformation set forth in this section were, discovered by Guichard in a diff erent manner in his M1'itmoire sur la deformation des quadrics, AR md'ires ih L'Acad~nmie des Sciences de France, ser. 2, vol. 24 (1909). His method dlid not reveal the relation between the nets NA and N1, on the quadric in either case, nor (lid he show that the nets N1 and NA1 are in relation F when k i c, f or g. These results were announced by the anther in 1919, Trans. Amcr. Math. Soc., vol. 20 (1919), pp. 3 23-338. 22 338 X. Surfaces applicable to a quadric From the results of ~ 100 it follows that N is a net 2, 0 and consequently N is 2, 0. From (69) it follows that N' is one of the two special nets 2, 0 parallal to V, the complementary function being (70) - -g'. e Suppose conversely that we take for N' one of these two special nets 2, 0 and define z' by (70). Equations (64) and (65) hold for k =e f. Adding equations (65), we have in consequence of (27), (28), (29), (35) and (42), the second of (57). From these follow (66), (67), (68), and then the first of (57), as in the general case. Thus h and 1 determined by N' satisfy the equations of ~ 135, z' is given directly by (70) and x' and y' by the quadratures (59), and thus involve additive arbitrary constants. Each of the two parallel special 2, 0 nets N' determines a normal congruence conjugate to N (~ 61). In accordance with the theorem of Beltrami (~ 112) and the results of ~ 100, when the net N is applied to its applicable net N on Q, the lines of these two normal congruences coincide with the lines joining points of NA to the foci on the axis of revolution of Q. Hence we have the theorem: Let N be a net applicable to a net N on a central quadric of revolution Q; the lines joining points of N to the foci of Q on the axis of revolution become lines of two normal congruences G1 and G2, conjugate to N when N is applied to N; there can be found by two quadratures c 2 nets -AT conjugate to G0 and 002 nets N2 conjugate G-2 which are applicable to o 2 nets Ni and c 2 nets N2 on Q; the nets N1, or N2, can be grouped into co1 families of c1 nfets such that their tangent planes at points on the same line of the congruence form a quadric cone and the corresponding points of the applicable nets on Q lie on a conic13). When Q is a sphere, real or imaginary, with the equation e(x2+y2 + 2) - 1, an applicable net N consists of the lines of curvature on a surface of constant gaussian curvature, since every 133) In order to obtain one of these families we hold fixed the additive constant in x' and let the additive constant of y' vary, and vice-versa; cf. Trans. Amer. Math. Soc., 1. c., p. 337. 137. Theorem of permutability of transformations Fk 339 net on Q is orthogonal. If we take k e, it follows from (62) that V' is a cone; hence there are no transformations of the type of the second theorem. When k + e, we have from (60) and (62) ek ^ ~ ~x that is the transformation is R (~ 67). Hence: k-e A suSface of constant gaussian curvature admits o 3 transformations R into surfaces of the same curvature. I37. Theorem of permutability of transformations Fk of surfaces applicable to a central quadric. Let N1 (xi) and 1N2(42) be obtained from a net N(x) applicable to a net N(x) on Q by means of transformations FZ1 and F7,, and let N1 (xl) and I2 (x2) be the nets on Q to which N1 and N2 are applicable. Let 0, and 02 be the functions of these transformations, where (71) 01 - k (=2_ 1i2) (z2 _ z X, x"; x, x" being obtained from x and x by the quadratures 'ax a _ x' ax ax" ax a" ax h- -- Zl~; h2 12- l a l- ua av v au a z av av (72) ) x' ax ax' ax a x" x a x" a x — ~, ~, ~-; h2, - 12. a at av av an a'u av ava We seek under what conditions the net N12 whose coordinates are of the form 012 ^T'" (73) x 12 - X - -01 X ~12 where (cf. ~~ 20, 21) Off off "', |xl"= -01 x " - = o 2 — o12 — 0T, 0o = 0 -- 0 — (74)a a, a a, a a aot h2 ao 8 ao' aoi a8 h a8 ' av8 8av a? 2 a2 an' av av anu a~u' a-v a v is applicable to a net N12 on Q, defined by 012 f 01 (75) X-1 2 f XxzO If we put (76) 012 - 2 A(231-Z Xl )2 22* 340 40X. Surfaces applicable to a quadric we find that this expression is equivalent to that of (74) if (77) k2 Oi'+ k,1 6- 2 k k2, (x'x"- '") 0. When the left-hand member of this equation is differentiated, it is found to be constant in consequence of (71), (72) and (74). Hence the additive constants entering in 0" and O' can be chosen so that (77) holds. From (6) it follows that nI2 defined by (75) is on Q, if (78) O" + '- 2 (ex'x" /1-fy'+y" cj ' ") - 0. Solving equations (77) and (78) for Oj' and O', we find expressions which satisfy (74). Hence: If N1 and NAT2 are transforms of a net NT applicable to a net N on a central quadric Q by mneans of transjborations F1i, andFi,, (k2 t k1), there canl be found directly a net N82, appicable to a n Iet N on Q, which is in relations F'. and FZ. with N1X and NAT2 respectively. This theorem holds when k1 or k2, or both, is equal to e, f or g, but k1 + h2. It holds equally when Q is a central quadric of revolution, or a sphere, real or imaginary134). 138. Transformations Bk of surfaces applicable to a central quadric. Let N(s~) be a net applicable to a net IV(x) on the central quadric (34). We consider a transformation F /: of N' and NA' as treated in ~ 136. We have (79) 0 - 2 exx'. From (35) and (36) we have (80) lex'X 6 ca~e2x a. By differentiation of (79) we have (81) ao-21ex ax ao _a azt, au ~~a r a 134) Trans. Amner. Math. Soc., I. c., pp. 324, 335. 138. Transformations BA of surfaces applicable to a central quadric 341 and with the aid of (27), (28), (42) and (57) we find _ _ J11 e80 a1Ji 2 a ( a 2 a'b2) - =0= 3i u a1 +12f1 ae (- i(62ta(-) Because of (28) and (32) we have (83) a2 20 alog +2 ob a 0 (83)..3t.. c 3. a^2 auaC22 a v 1 Cav? From the results ~ 49 it follows that if we take two transformations F, of N(k + e,; g) and write (84) 0, 1 2'exx, 0= 2 exx, the net 1\T(), defined by _ 3 a'x a 3x (85) -- x t+q a,' where 1t- /t (0 e a 0, 82 3 ) / o ( aa o a )\ (86)/ 302 0z 3 01 3 02 {'', 6s _ ~ i. _ u a8v au av' is a TW transform of IN, that is these two nets are on the focal surfaces of a Wcongruence. On substituting the expressions for the derivatives of 0. and 0s analogous to (81) and (82) in (I, 68), we reduce the resulting expressions to 9_ - - 2ca2' A2 8 0a2 aax 3a02) (b2(lt 2)_c27ik la ' t zt u a- x a2 a2 02 a'x a1(a( h,2) cekO ( av av? )a (123-x2) ) I0)/,1 (a'x aO-t ,342 342 ~~~~X. Surfaces applicable to a quadric By means of the same functions 0, and 02 We obtain a derived net.!K(.,) of NA Its equations are __ ax +qax (88) -X X+p +q a I av' where p and q are given by (86). On substituting the expressions for 01 and 02, as given by (84) in the expression (86) f or ii, we find, in consequence of (35) and (cf. (29)) (89) Ha - ab, that4 11jxay ax ay (90 zI) efxyxY\an an an an! (9) 4HjeJ(x, ty'I- x'y") z 4 ca befgfx (y z'T-y'z" Hence the expression f or s~ is reducible to (91) From this we have (92) ~ex 1. The equations analogous to (57) are (93) 1 1 h~2-l 2zz lh, a- 11 b2 - 2~( k e) xx', h2, a- 12 b - ~ e-ke) xx". By differentiation it can be shown that the left-hand member of the following equation is constant for any two transformations Fi, We choose the nets N' and N" so that (94) is satisfied. By means of these relations we show that f Jjf 2 - 2 - k g) (y`s'z"-y11z')2 -2(e 2-ke)x x2 I~'- ke) x"2 (95) - ~~(1(e 2~k e) x'x 1) 2 - a b2 (11h212h)2, 138. Transformations Bk of surfaces applicable to a central quadric 343 also that (e2- k e ) f _ -f ) (y - k g) [ x (y"-y z "l)2 (96) ka2b2 - 12 ). C4 (t h2 - 12i)2. In consequence of these identities we have (97) 5 -1 = 1, e k that is N(X) lies on a quadric confocal with Q. The equations for N analogous to (87) are obtained by removing the bars from the quantities x, X, D and D". Substituting from (81) the expressions for the derivatives of 01 and 02 and making use of (35), (36), (84), (89) and (90), we reduce the resulting equations to a _x 4 2a3 bpq { c2(e2. e)2x. x(y,, k(y yz) (a_ __ 4aJabIqfq {c2(e2 ke)x.x(y'z"-y"') +c'k(y'&" y"z') ( |+ a2 [(zy"- yz") (i - h1) — (y'- yz') (12 -A2)]} ~~~~t + 2 -[(Zy'l Y"^) (11 h)-(y- y/ ) ( 2 - 2)]} From (93) we have (zy" — Yz") (h a2 — 1 b2) (zy'- yz') (ha2 — a2 12 2) (99) I c2 4(e2- ke) xZ x (y'z"- y"z') - ( 2 ) (z'y"- Adding the left-hand member of this expression to the expressions in parentheses in (98) and subtracting the right-hand member, we get in consequence of (28) (IH -I 4 bjfg [h1 (/zy" —yz") —h2 (y'-yz') + (z"y' —z y")], (100) 4aqfg ( y )- ( '- )+(2/ — )] [l~ (zY"- Pyz")- 12 (z y-yz') q - (z"y' —z'y" )1 344 344 ~~~~X. Surfaces applicable to a quadric For -N- the expressionis analogous to Ie (I)and I e (x2 for ZATareY ae 2 d Mk/D ~ aking use of wellk-c Dul D- -e8v! known theorems on determinants, we find ultimately that -ka 2, 2 (a2) kb2 q2, (101) k k u a V From (87) and the analogous equations for N~ we have, in consequence of (27) and (29), [y x (D~~ 2 (D2.-.D2) - 2 (102) DuD u 0, ax 2 j tq2 (D2D2) b2q2. From (101) and (102) we have _ _ _ __k _ _x a __ ' k Da D~ D ~ k-e u't aDU aV~ - Dut Dv' (103){ k-e Dv) Hence if we put the net N()is applicable to the net N0o (,,o), which in. consequence of (97) lies on. Q. The equations (104) define the relation of Ivory between a quadric and a confocal quadric; the point of coordinates-, x0,O y0, io is the intersection with Q of the orthogonal trajectory of the family of conifocal quadrics which passes through the 'point of coordinates 7, ~ of (97)185). 13)Bianchi, Lezionli, vol. 3, p. 59. 138. Transformations Bk. of surfaces applicable to a central quadric 345' The functions of a transf oritiation WV are x', y, I, x", /~X/~y",~'? hl, l1, h2 and 12. They satisfy a completely integrable system of the formi (58) and (59). Moreover, the five conditions (93) and (94) must be satisfied. ilowever, these equations are satisfied also by the functions where ce, 3,; and 1 are constants. In this case, as follows from (85), (86) and (88) we get the same nets NX and NA7. Consequently. for each value of k there are, ' transf ormations of the kind sought. If NJA (xj) and _AT, (x 2) denote the Ek, transforms of NAT w'~e have XI _ _ 02,,0 x1~~~~~~~~x —b~~~~~~~~~~~~~x, ~~~~~~ X2 -X 0,,X In consequence of (91) and (92) we have ex1-1 ~ e;X2 1 Hence the point of coordinates, X, ~, s the pole of the plane of the corresponding -points on N, AT, an d N2. ~We may state the foregoing results as follows. Jf N, is a p~ex;manent net on a central qu adric Q, there arc set's ofjrnsrmain Fk of' NA jnto nets N, and 1N, so that the (condition (94) is satisfied ftor each value of ki different from zer-o; the locus of the p~ole 1ii1, of the p~lane ill Mj N1j wvith, resp~ect toQ Its a net N2~ onl, a.quadr-ic confocal to Q; as N rolls on its applicable net N!, the _point i111h describes a net N,~ such that NA and NI~ ar.;e the./bcal nets of a _14 cong ruence, and NIA is apl)Vicable to the net onQ which is the Ivory tramsformn of' the net N~T These are the transformations Bk found by Bianchi by entirely different processes"'6). Let NA be a permanent net on the quadric (34) and NA,,an F1, transform of NA by means of the functions x', y', Y', h, and i, satisfying (57), (58) and (59). From the form of equations (93) and (94) and the observations concerning (105) it follows that I3~Lezioiii, vol. 3, Ghaps. 1, 2, 3; cf. also Journ. de Math., 1. C., p. 61. 346 346 ~~~~X. Surfaces applicable to a quadric there exist only two sets of functions x", y, ifs",l h2, 12 and. x"', Yfi, s"',f h,3, 1.3 each satisfying (93) and (94) with x', y', z' h,, l, and determining different Bk transforms of NIA, and also of the net N applicable to N. The corresponding transforms N2~.9 and N3~ of NV are determined by 01,~ 02 and 01, 0-3 where 01 and 02 are given by (84) and 03 ~ lxx"'l. Consequently these transforms are conjugate to the congruence harmonic to N1 determined by 01... The same is true of' the transf orms N27 an 1ofN dtrmndb 01, 02 and 01_, 03. The congruence 'harmonic to NA determined by 01 consists of the linies of intersection of the corresponding tangent planes to NV and its F1 7 transf orm N1, by means of x', y', z', h, and 1,. In view of the preceding results we have the theorem: If NA is a permanent net on a central quadric Q and IV1 is ane FI, tr-ansj'or~z of' A1, and P1- and P2 are the points of inte~rsection?,with the confocal quaadric Qk of the tine of intersection of- corresp~ond'u~q tangyent planes of AT and -AT, then as N, rolls onl its applicable net NAI, the pQints Pi and P2 generate nets appIlicable to Q, each of which is a Bk transform of N. __139. Permutability to transformations FL. and Bk. Let N (-) be the permanent net on a deform of the quadric Q (34) and NA(3') its Bk, transform by means of functions x', y', z'; x", y" z"; h1, li, it2, 12. Let x"'1, y",s' h3, i.3, be a set of solutions of (5`8) a,-nd (59) with k replaced by k' satisfying the conditions (16 -i~a-l b-c~e-1ex", ita- 13 b2 c (e -kf;e) xx"'f, so that the Fk'c transf orm NV3 of NAT defined by equations of the form (107), __J 03 - where (108) 03- 2 ex"' 0" ex", is Onl Q. From (11, 33) it follows that the equations of the forms (109) x-X- ~"X, X,3zX- X 139. Perinuitability to transformations Flk and Bk34 347 define nOts parallel to N.3; the corresponding functions h3l1,13, and,3 2, 13 2 are given by (cf. II, 50) /h___ hff -h 0ff" A3- 3 03 -2 03Pf The functions a3 and b3, appearing of the form (cf. II, 12) In order that these functions to (93) and. (94), namely b - C 13 03-01 Off 132 13 O.'~l 03f 1303-3 in the point equation of NT3 are ba = b (13 03 I may satisfy equations analogous a3h31 A32 -b2 131 132 c~(e ~ Ae) Ax3 ", it is necessary 'and sufficient that (1 10) C2 a2h2 3 123_ 2(e (k - k) 03 0 Differentiating these equations with respect to u and v, and making use of (58), (59) and analogous equations, we find that the lefthand members are constant. If k': k, the functions 03' and 0"' are uniquely determined by (110), and consequently x' and x3' are uniquely determined by (1 09). Then by means of the functions (111) 031 21eX34 032 2jex3 X' we obtain a Bk~ transf orm N~T o f NAT which is app"licable to Q. 348 X. Surfaces applicable to a quadric It is readily found by differentiation that the left-hand'members of the equations (112) 0 + Ot-'-2 exe'X"' = 0, 01' + ft' - 2 ex"x"' = 0 are constants. If we take 0." and 0.' as given by (112), we find that (113) 0 3 0" 031 ---:1 ---1 ~7 0:1 032- - 02 -- 7 0. These are the conditions that NAT and N73 are in relation F (cf. ~ 22). We desire to show that this transformation is in fact a transformation Fk, as defined in ~ 136. From ~ 22 we have that the functions 0a and 0~" of the transformation F from N1? into iN are given by a 3 oa,,,,f,, I of' q,, 3 0f" (114) 03= 0,+p +q,, = 03 + + q" a q av 030' a" I- avu where p and q are given by (86) and -t,_ 1 0 1 ', a 0 " I 2 t 1 /0,i,,, ao ". a,"2 a2" ao," a '" au av av a and (115) a 9," a0i h- - 3 a O au -, 3 a (i- 1, 2, 3). av av 2 Moreover, the direction-parameters of the conjugate congruence of this transformation are the coordinates "', y"', Z f"' of a net AT,', parallel to Nr~, where (116),x -.x + ap 8 +q av From ~ 22 corresponding to (85) and (1.16), we obtain (117) a,'^ ^ a i a&=" ' __ ' a a V av 139. Permutability of transformations Fki and B4 349 where (118) I=- {(061,,D21 -- 2, —, a, I a 2 0 a- 0 pxl D v ' v!' q: DuD au I' Substituting the expressions (108) for 03 and 0." in (114), we get, in consequence of (88) and (115) (119) 03 =2 Q e.X x: and,,,-, 1, +^| 1ae aDO 2 D 3 _ ael a D3 a- O3 (120) a a v a u a a v au aD, av (120o) - e"r +-4efgab [0", b 1(y"' "- y","')- " z: (y'"s' -y'z)1; this second expression for 0"' is derived from the first with the aid of (29), (35) and (36). From (110) and (112) we get -t2hl A - 2h1 = c2 (eS- 'e) x'x"'+ (k'-,1) ', (121) a 2/1,fh - b 1213 = c (ez-k'e)x"x"'+ (:'-,) o". From equations (121), (100), (106) and (90) we deduce |O" (y"z — y i'") -- 0". (y'z- -y z') - 2 C2 (k' —k) a bfq (122) ~ B + h3 a.;+ l as + e (e- k2)] By means of (122) and analogous equations equation (120) is reducible to t,,, 1 L,2 7 - in2 2h3 i eXI' ax 10,' [^ [ (2 e 2- e- e) x" 'd 2 h (123) (- 21, z/ aDS' + 2;e 1,1 350 X. Surfaces applicable to a quadric Equations of the form a "' -— f" a ax" define a net N"' parallel to N. This expression is reducible to ~'"_ i 1 \n IsO |a 02 aX aO2 ax\ ao, ax aoe ax' (124){ a a u a&u a 1 v av u v a]',,+ 2fgabc e (y"z-yz")- ' (yz -yz')] which in consequence of (122) becomes (125) 1 ' [(e-k)x"+ --- h3 a l a+X] 1l-' q cp aa cq av Since N is applicable to Vo, defined by (104), it follows from the results of ~ 136 that a transformation Fk' of N is given by the functions 3=- 2U e-oV"= 2 k _-_xx, (126) 6 s12 ek. e 031f- exo.. x provided that analogously to (57) and because of (101) (127) jX2(t2p"2b2q2 - y (e-k 'e) "o- C ~-Z e-k', k e —k (127) tC ar Cb qe2-k Ah jfa2 22 _ i b2q2 3=(e2 —k'e) xoo"-= cZ — 'e x k e k x. When the expression (125) for."' is substituted in (126), the resulting equations are reducible by means of (97), (101) and (106) to (2 )03 k —k' -k(128) 2 [ (2 e2-ke-k'e) x"'2 2 h3 exll - 2 - ex" ax~ ^ cp Ia eq an 140. Theorem of permutability of transformations B73 351 In consequence of (118) we can write the first of (124) in the form ~/~' —x"' ax i- ax (129) a"'_x "'+a a2 From this equation, (38) and (42) we obtain e (ti' — X"')2= 1 (h2 2p-2_ 12 b2 q2). c In order that this equation be consistent with the first of (127), we must have (k-k') z e- " 2+ 2j e/ x"x"'-zl eX"'2I= 0. e-k - Because of (101) and (106) this condition is satisfied identically when the expression (125) for 5"' is substituted. Again from (129) and (88) we have e (x"' -x"') (xS-x) = (ha2p2- ib2q2). It is readily found that this equation is consistent with the second of (127). Moreover, when we compare the functions (128) giving an Fk' transform of with the functions (119) and (123) determining a derived net of Ns, we find that they differ by the same constant factor. Hence: Let N be a permanent net on the central quadric (34) and N the applicable net; if N is a Bk transform of N and NT is an Fk' transform of N, there can befound directly a net Ns zwhich is a Bk transform of NV and an Fk' transform of -AT 140. Theorem of permutability of transformations Bk. Let N be a net applicable to a net N on the central quadric Q (34), and a, and NV3 nets applicable to nets on Q and obtained from N by transformations Bk and Bk' respectively, determined by the respective pairs of functions 0,, 02 and 03, 04, where (130) o9 = 2je xx', 9 2= 21exx I? 03 - 2exax"', 04- 2exx"". 352 X. Surfaces applicable to a quadric From ~ 50 it follows that the functions Op and 04, defined by [a ol a (6,)36 a ___i a_ 6 0aok\ aua~ av av av au a (131)1 Ta h a Oj a Ok -1 a z at: a v; aur for i - 3, 4; j - 1, k 2, determine a W transform of NT which is also the TV transform of NV3 defined by the two functions given by (131) for i 1, 2; j - 3, kh- 4. From (127) it follows that conditions analogous to (93) are satisfied. The analogue of (94) is (1 -- k If hh4a p2-i3hb~q c which can be shown to be satisfied identically, by the use of methods similar to those used at the end of ~ 139. Consequently the above TVtransform of -T, is in fact a Bk' transform. Hence we have: If IV, a n d IN2 are obtained fronm the permanent net NV of a deforms of a central quadric Q by transformations Bk and Bk' (k' t k), there can be found directly a net NVl2 applicable to a net on Q such that N1 and N,12, and JrN, and iVP2 are in relations B'< and BL respectively 137). 14-1. Transformations Fk of permanents nets on a paraboloid and of surfaces applicable to a paraboloid. For the paraboloid P, (132) ex 2 +fy2 2e - 0, we have (133) i'y, I- _ exfy, 1 Vle2X 2~f2y2+1 and 1 ~a/e"1 2_2+ffy2 (134) X7 e (134 K _ - 4 (2x2+f2y2~ 1)V ~ -cVs2fy+1 c(e X -f""i i" where c' - - 1/4 ef. 137) Cf., Bianclii, i. c., chap. 4. 141. Transformations FL; of permanents nets 353 If P is referred to a net N whose point equation is the second of (33), we have ax ax ay ay e' A-f 0 anI ar au av and by processes analogous to those used in ~ 135 we find (135) ( ax\2 fay a2 IaxW I ay\2 __ e - aU c eaAf Ca and we prove the theorem: A necessary and siffficienI condition that a net IN on a pau aboloid (132) be pe)-manent is that the parameters can be chosen so that (1 36) e ) f(?) e ( x)a-f( a Y) c(eex2A-f2y2+ 1). (136)ej~~ia u av av In order to obtain Ek transforms of N, which are permanent nets on P, we take in place of (48) (137) 6 - 2 (exx'A-fyy'A-s,;'), O' - ex '12+fy1'2. Each set of functions satisfying the completely integrable system of equations (59) and (58) with g - 0, and the conditions (138) fh a2-1 b2= c2 [Ce -ke) Af2-kf)Y'], 1a. a2-1b2 c2[(e2-k e)xx'+f2 —kf) yy'-ks'] determine a transformation F1 of NV into a net N1 on P. When k - e or f we have two special types of transformations, as described in the last theorem of ~ 135. Consider now the net N applicable to iM From (60) and (137) we have in place of (62) (139) ~(e - k) - k " IX 0. Proceeding as in the case of (62), we find equations obtained from (64), (65), (67) and (68) 'by putting g - 0. Hence there exists for transformations Fi7 of N theorems similar to the first two theorems 23 354 X. Surfaces applicable to a quadric of ~ 136, but there are only two types of transformations as described. in the second theorem; they are Fe and Fy. When e- f; P is a paraboloid of revolution and. the transformations Fuk (k 1: e) are described by the first theorem of ~ 136. When k =e, equation (139) reduces to If'2 —z'2= 0. Hence N' is a special net 2, 0, the complementary function being o'. In this case, as for the central quadric of revolution, the transforms of N are conjugate to the two normal congruences conjugate to N. When N is applied to P, the lines of one of these congruences pass through the focal point of P on the axis of revolution, and the lines of other congruence are normal to the tangent plane to P at its vertex (cf. ~ 100). Hence: Let N be a net applicable to a net N Vn a paraboloid of revolution P; the lines joining points of N to the focus of P and the lines parallel to the axis of P become two normal congruences, G, and G2, conjtugate to NA when N is applied to N; there can be found by two quadratures o 2 nets N1, conjugate to Gi and o 2 nets N2 conjugate to G2 which are applicable to oo 2 nets 1V, and o 2 nets N1T2 on P; the nets N7 or N2 can be grouped into cr families of oo1 nets such that their corresponding tangent planes envelope a quadric cone, and the corresponding points on P lie on a conic138). I42. Transformations Bk of surfaces applicable to a paraboloid. In this section we establish for surfaces applicable to P(132) transformations analogous to those treated in ~ 138. Equations (82) and (83) hold in this case also. In place of (84) we have (140) 0 - 2 (exx'-fyy' — '), 2 - 2 (exx"-+f yy"- "). In place of (90) and (91) we have 4 efa b c (x" y'- x' y"), and 1 y"'i'-y z" 1 z" 1x'- x"/ e x"y —x y f xy-x "' I Y (x" ' —x'") X +- x(y'"- y"z') S- "'r t'ixy'-x y Now exx — JY Ffy- y z- ~ Z = 0. 138) Trans. Amer. Math. Soc., 1. c., p. 338. 142. Transformations Bk of surfaces applicable to a paraboloid 355 The first two of equations (93) with g 0 hold and in place of the second two we have (141) hi a2- 1 b2- c2 [(e2-ke)xx'+ (f2 —kf)yy'- k'], h2 a2- 12 b2 c[(e2 ke)xx" (f -kf) yy" — kz"] and (94) with g - 0 holds. By making use of the expressions for x(y'y"- y"') and y(z'x"-z"x') which are obtainable from (141), we find that x, y and z satisfy the condition j2 y2 1 3 + Y +2i+- o, I ~- 1 01 e k f k that is the net N lies on a quadric confocal with P. In place of (100) we have a8 4pa3bf a(y 4q ab3). (yl h -y'^h2), q (Y, -y/1). a u z av From these and analogous equations we obtain, by means of equations similar to (95), ek /8x\ 2 fkt 2 __ 2, k- e \a k -f k a a22 ek ax x fk ay ay k-e au av — f au av ek 2 fk ag 2 k-e ( [v ) k-f ( aY) _ b2q2 From these equations and (102) we have y a\2 _ l8ao\)2 ax a a _ axo axO \ au \ au}' 9au av au av ' 2 ( 2 -av) ( av 28* 356 X. Surfaces applicable to a quadric Here the symbol 2 refers to the three variables, where X '0 X - 7,cS1 i k__ k -J 2Ik' These are the equations of the transformation of Ivory for P'39). The other observations for transformations of a central quadric hold also for the case of the paraboloid, and consequently we have the analogous theorem: If AT is a permanent net on a paraboloid P, there are w pairs of transformations Fc of AT into nets N-T and A&2 so that the condition hl h22 a2 -112 b 2 C2e2-he)xx"+ (f2-Ikf)y'y] holds; the locus of the pole 11 of the plane ilIMj ]iI2 with respect to P is a net NT on a paraboloid conA-cal to P; as AT rolls on its applicable net N the point 1I describes a net N such that N and N are the focal nets of a T4 cong) —ttence, and IAT is applicable to the net on P which is the ivory transform of the net N. These are the transformations B11 of surfaces applicable to a paraboloid, as found by Bianchi in another maniner 10). 143. Determination of the asymptotic lines on a surface. When a surface S is referred to a general system of parametric lines, the translations and rotations of a general moving trihedral1 satisfy the conditions [cf. ~ 69] P ap,1 qri- qr a 8V art av auz (142)i aIq Iq a-, 'Y Pi - p I - 1 av, a it avl azi ___ - ar1 -pq1-p ~jp-np1+ q- q 0. ' &v a I q The equation of asymptotic lines on S is [~ 71, (68)] (pn q -q) d t2 + (p'qr-q11 _q +p1 ' - q1 ~) dudv + (prq1-q1+ )dv2 = 0. 139) Cf. Bianchi, i. c., p. 30. 110) Lezioni, vol. 3, chap. 1. 143. Determination of the asymptotic lines on a surface 357 If we replace this equation by the two J p u +pi dv = z (Q du + i v), (143) J pd u+pidv du+idv), 1q cdu + q1 dv -= d ( du + -i dvr), the function 2 is determined by 22 -- 1 k2, k -- k 1 Q2 as is seen by the elimination of du and dv from (143) and [~ 70]. If a and, are the parameters of the asymptotic lines on S, we must have P? +1 - k + 'C I it a V ^- 1^ ba ~^ a ^ - Da ~a Dv, + Dv )\ (144) a a Dvp _ k 1 at Dv If these equations are solved for p, q, pi, q1 and the results are substituted in (142), we obtain the equations which u and v must satisfy. This calculation is simplified, if we note that the first of (142) is equivalent to D u av aQ r tu av a a it av \a a cc a a a cc a ak, I av\ a I t/a avv a D av\ ( k aV 2 a Da a a a3 av On replacing the expressions in parentheses by their values from (144), we obtain a2tt a2v a u a 1t ll \ 8 a a,8 a cc ad a8a a3 7 a r ] + av av /1 ak8j \ T- a- -a1 a + 2 a a a 8 ai8 8a ( 8k +v k a it 1 ) 358 X. Surfaces applicable to a quadric This relation may be replaced by two equations because the translations 1, 5,, 1 are not completely determined. In fact, they are subject only to the conditions E - 2+, +2 2 51+r1 G 2= + W If we equate a and, to zero successively, we obtain the following equations due to Darboux141): ( a2U + a8l a 8alog7l J1i aa'8 aa a~v au - \ 1 ( a8 aV a8 a8V 1 alogk 1I21\ av v avJ221 aT acg a as a4 12 aogk l -a a a/l J-~ (145) a 2. av a ov ( alog, _ 22/ aa8 a a 8a ( av \ 2 / a8? v.8at av\ l alogk 121\ a, 8a Jil1 a va 8aB a8 a+ c 2 a, 1 2 J a1 '3 1 2 J {r tB am j where the symbols ts are formed with respect to the linear element of 8, 't, and v being parametric. Since the coefficients in (145) involve only E, F, G and their derivatives, each pair of solutions enables us to transform the linear element, so that it becomes the linear element of a surface referred to its asymptotic lines; that is, the complete solution of (145) gives not only the asymptotic lines on S but each family of curves which correspond to the asymptotic lines on a surface applicable to S. Bianchi calls these curves the virtual asymptotic lines on S. From the results of [~ 77] it follows that when a system of virtual asymptotic lines are known on 8, the function D' of the surface 8, applicable to S and upon which the parametric lines are the asymptotic lines, can be found by a quadrature. Hence we have: The deformation of a sucface and the determination of its virtual asymptotic lines are equivalent problems. 144. Deformations of paraboloids and central quadrics of revolution. We apply the results of the preceding section to the case when S is a quadric referred to its generators. If ao and AB 141) Leqons; vol. 3, p. 290. 144. Deformations of paraboloids and central quadrics of revolution 359 are the parameters of a family of virtual asymptotic lines on S, in consequence of (10), (11) and (12), equations (145) become, since k - 2 a~ a, 0o:8 a a 2, - 8, a (146) a lga a 8 Do lo- - logDo a' -o=0. aa1 ) af0 ado ao80 ac aao o acao aM aid It is readily found that the first integral of these equations is (147) 142) aao aao 8- a8i o rDo' In accordance with these equations we introduce a function o, thus (148) ce - eli (148) aao ao' a,?_o aao Differentiating these equations with respect to rio and ao respectively, and substituting in the first of (146), we get in consequence of (147) id8 a c a a 1 a2B a to 3l/ a I (149) ~ a — _ - a 0. ta9)- - a8o a 8O D a ' ca, ao 8 a8o a a cdDo Expressing the consistency of these equations and the second of (146), we obtain ____ a2 11 \ a2o-c (150) 8 e - e,3- - coa-o Mo D' D-0-/b For the paraboloid ex2-2-fy2-+ 2, 0, we have x a- J y "+ ' --.' - e '?/ fy ==2c From these expressions we find H EoG O-F [/(+ )2 + + e( - )2 +1]. ef Also 4 _ 4 1 ej' e o " ef -- If [(CC- [( ) -+ e (a- )2+ 1]. ) Ser, ll. Sc. t. Fance, vol. 99), p. 23 12) Servant, Bull. Soc. Math. France, vol. 29 (1901), p. 232. 360 X. Surfaces applicable to a quadric a2 1 \ 2 1 \ Hence a-a (,) and -a (-D) are equal to the same constant and equation (150) is reducible to the equation of surfaces of constant curvature [~~ 118, 119]. For each solution of (150) the above equations for the determination of a and,f as functions of ao and?o are completely integrable. Hence: The determination of surfaces applicable to a paraboloid and to a sphere, real or imaginary, are equivalent problems. We remark that when the paraboloid is a surface of revolution (-e f), equation (150) can be integrated directly. In like manner for the central quadric ex +-fy2-g2 - 1 we have 11 +a- a 1 a-f _ 1 af X e a+ ' / If a + 1, + - q (1 — a)2- +- e(1 + a )2 +f(a_-) 4 f 0 ^ -{ -ftY ~ e g e f g (a + e)6 (a + ) g e- Ho Wefg (. + )Ho Do- 1 [g [(1 —a)2+ e(l + qaf)2-+f(a-_/)2]. Do -a 22 Vefg When g - -e, the coefficients of eo and e-" in (150) are equal to the same constant and we have: The determination of surfaces applicable to a central quadric of revolution and to the sphere, real or imaginary, are equivalent problems. 145. Surfaces conjugate in deformation. Following Bianchi143), we say that two non-developable surfaces, S and S, are conjugate in deformation, when the asymptotic lines correspond on S and S and to every system of virtual asymptotic lines on S correspond a system of virtual asymptotic lines on 8, and vice-versa. In order that the latter condition be satisfied, the coefficients of the equations (145) must be equal to the corresponding coefficients of the similar equations for S. These conditions may be written 143) Lezioni, vol. 3, p. 201. 145. Surfaces conjugate in defornmation 361 a D log _h12[ 12i3 2 1 l1 ji ( "^r -2 [12 r~ m ) i I (a log1) { 2 l2t _2 2 j121 ] 22\ (221 III jil _221 )221 (152) 12 _-12 ) 2 ~121, where the symlbols rs( are formed with respect to the linear element of 8, and (153) k k' From [~ 85] we have that the equation of the geodesic lines on S is d(v J22j 2dv-, { + ({ 22 1 ( dv d154 du 2 du/ + (2 )1221 { lij~)d +{21ll 0. Hence from (151) and (152) it follows that the geodesics on S and S correspond. Suppose, conversely, that the geodesics on two surfaces S and S correspond and also the asymptotic lines. Then from (154) we have (152) and (155) 2 2 1 22l 2 121 IJll2 2 2 J 2 {2 and also I) D D' D' D" D"' (156) = L = H H' H H' fH H' where 2 has the value (153). The Codazzi equations for S are [~ 64] av (i )D ) ffa 2 i 2 { H +'l2 j H a (D)\ a ', j221 D i 12t O' l21t D" _\j — \ }+ l J E-~ 2A l J- -+\- 1 J" 362 X. Surfaces applicable to a quadric and similar equations for S. Substituting in the latter from (156), we find a log; _ - Jill a logi, _ 122\ _22t on condition that S is not developable. Equations (151) are satisfied and we have the theorem of Servantl14) and Bianchil45): A necessary and sfjficient condition that two non-developable surfaces be conjuqate in dejormation is that the geodesic lines and the asymptotic lines respectively correspond on the surfaces. When two surfaces S and 8 are conjugate in deformation, each set of solutions of equations (145) leads to a pair of surfaces S' and S', applicable to S and S respectively, upon which the asymptotic lines correspond, and since conditions (151) and (152) hold for these surfaces, they too are conjugate in deformation. Hence: If two surfaces are conjuvate in deformation, each set of solutions of the correspondizng equations (145) determine two other surfaces conjugate in deformation and applicable to the respective given surfaces. Dini146) solved the problem of finding pairs of surfaces upon which the geodesics correspond. We reproduce his results. We assume that the surfaces are referred to their common orthogonal system. Then equations (155) can be integrated, thus E E_ 1 G G 1 G~ G~ V E2 V' E E2 2 U3' where U and V are arbitrary functions of u and v respectively. From these we have ( 157) 7E V- u = By the substitution of the above values of E and G in (152), we get (- (U-V) -- V'E, (U-V) - U'G. t14) Comptes Rendus, vol. 136 (1903), p. 1239. 144) L. c., p. 206. 140) Annali, ser. 3, vol. 3 (1869), p. 269. 145. Surfaces conjugate in deformation 363 Neglecting the case where U - V - const., that is when S and S are homothetic, we have on integration Ef U2 (U_-V), C Vo(U-TV) where UI and VI are arbitrary functions of m and v respectively. If we choose the parameters so that U = u + h, V- v+/h, wAhere h is a constant, we find that the tw o linear elements are d s'= (u~- v) (U' du2" + VT dv2), (158) ____ ___ (___ ______t t 2l~ze ~ 2; — 2 h both of the Liouville for~m [cf. ~ 93]. Since ht is arbitrary, there are 0 1 surfaces S corresponding to S with geodesics in correspondence. We inquire under what condition two such surfaces are conjugate in deformation147), We exclude the case where either U or V is constant, that is when S is applicable to a surface of revolution (cf. X, Ex. 6). Substituting the values from (158) in (151), we find k_ 1 2 - (u h) 2 + h3-j)'l where a is a constant. From [g 64, (12)] we have __ a j K 1 V i) 11,C E r a lit VF, a u av 2~ av and similarly for k2. Calculating the expressions for k and k, and substituting in the above equation, the resulting equation is reducible to, on dropping the subscripts of U1 and V1, (t$7+) U2 (JZ j + U ii') 2(u h)2 U2 -.(159) 1 1 V - 0. 8 v +h U 2(v + h)2 V2 147) Cf. Servant, 1. c.; also Bianchi, Rend. dei Liacei, ser. 5, vol. 11 (1902), _p. 265. 364 364 ~~~~X. Surfaces: applicable to a:"qiadric Differentiating with respect to u and v, we find -ultimately that U and V must be, such that (160) where a, 4 and;' are constants which are arbitrary, since these expressions satisfy (159). When a + 0, in (1 60), the functions U2 and 172 are of the f orm I2 (ui- +a) (u+ a-b) (u-h a-c)' (161)-A V (v +a) (v-h a-b))(v+ ca-c)' where A, b and c are constants different front zero, and the linear element of S is j ds2 ~u~v)(u + a) (u-F a- b) (ut a- c) (162) VCV (v-ha)(v-ha-b) (v +a-c) When the expressions (161) are substituted in. (160), we find that h - a. If we put (163) ui-I — VIm 1 a uI-+a' a v-ha' the linear element of S is reducible to (164) d2 A Ct (u1- VI)[ t ln vI 1 dV, 146. Transformations H of surfaces applicable to a quadric 365 Comparing (162) and (164) with [~ 96, (14)] and taking A - 1, we note that S and S are applicable to the quadrics Q and Q whose respective equations are 2 2 z2 a a-b a-c (165) b2 c2 2 bc ) -a c — a aC2' From these equations it follows that when Q is an ellipsoid, Q is an hyperboloid of two sheets, and vice-versa; when Q is a hyperboloid of one sheet, so also is Q. Moreover, if Q is a surface of revolution (b = c), so also is Q. We remark that Q and Q are transformable into one another by the projectivity (166) x =J I - V Y | - zV a x ax' a x If in (161) we take c - 0, and proceed as in the general case, we obtain (162) with c 0 and in place of (164) we have -2 (V Aa [ uC du 1 vl dvt __ds = Lt-t | ) (^ ~ -- b-) (U + a V ) ( + a b a b a a Hence, as follows from [~ 97, (27)], the quadric Q is a paraboloid. If in (160) we take a - 0, we note that U2 is of the form A u/(u + a) (u — a - b) that is Q is a paraboloid. Proceeding as above, we find that Q is a central quadric of revolution. These results show, as in ~ 144, that the deformations of paraboloids and central quadrics of revolution are equivalent problems. The foregoing results lead to the theorem of Servant: Aside from surfaces capplicable to a surface of revolution, the only surfaces admitting surfaces conjugate in deformation are surfaces applicable to a quadric, and any such surface has this propoety. 146. Transformations H of surfaces applicable to a quadric. Let Q and Q be two quadrics conjugate in deformation. If a surface S applicable to Q is known, we have a set of virtual 366 X. Surfaces applicable to a quadric asymptotic lines on Q, that is a solution of equations (145), and consequently a deform S of Q is thereby determined intrinsically. Bianchis48) calls the relation between S and S a transformation H. Suppose that S is a ruled deform of Q. Since the transformation H of S into S preserves geodesics and asymptotic lines it follows that S also is ruled. Hence: The transformation H changes every ruled deform of Q into a ruled deform of Q. Since asymptotic lines correspond on S and S, to each conjugate system on S corresponds a conjugate system on S; the same is true for Q and Q. Hence to the conjugate system on S permanent in its deformation from Q corresponds the permanent conjugate system on S. Therefore: The transformation H chanfles the permanent net on S into the lermanent net on S. Let N and NT denote these permanent nets on S and S, and No and No the nets on Q and Q to which they are applicable. Let NATl denote an F;k transform of No, and ATN the corresponding Fk transform of N in accordance with the results of ~~ 136, 141. Since No0 admits the applicable net N1V, the net No, on Q corresponding to NOI admits an applicable net N1 conjugate in deformation to li. But' No, is obtained from NAo, by a projectivity, so that No0 is an Fk transform of NVo, since any transformation F is transformed into a transformation F by a projectivity. Consequently N1, applicable to N1Ao, can be so placed in space that it is an Fi, transform of TA Hence: If N and N are nets in relation H, and V1 is an Fk transform of AT there exists a net N1 which is an Fk transform of N and an H transform of NT. Thus we have established the permutability of the transformations Fe, and H (cf. X, Ex. 12). I47. Isothermal-conjugate nets on a quadric. From (12) it follows that the coordinates of a quadric Q referred to its asymptotic lines satisfy the two equations _ _ _ _ a _ _ _ 9 o8 a _ _ (167) a 2 loga 0 2 log a 148) Lezioni, vol. 3, p. 214. 147. Isothermal-conjugate nets on a quadric 367 where a2 - H a. If we put (168) u=a+, v-=a-i3, the parametric lines on Q form an isothermal-conjugate system, and every system of this kind is obtained by replacing a and B by arbitrary functions of a and f respectively. In terms of u and v equations (167) are equivalent to a2O a20 a aO a a (169) a v2 2 log a — 2 a log a a — 2(169) auz a at av a2 V aloga aO aloga a au a v av a a au av These equations are of the form (IV, 73). Hence every isothermalconjugate net on a quadric is a net? with equal point invariants. When we apply to (169) the conditions (IV, 22) that (169) admit three independent solutions, we find that a2 must satisfy the equation of Liouville a2logy a2log aUn2 av'2 where k is a constant. The general solution of this equation is known [~ 151]. When a solution is substituted in (169), the resulting equations are reducible to (167) by (168). Since equations (169) are the most general of the form (IV, 73) for which a= b, we have: Every isothermal-conjugate net on a quaadric is a net R with equal point invariants, and these are the only nets R with equal point invariants. When a quadric is transformed into a sphere by a projective transformation, each net with equal point invariants of the quadric becomes an isothermal-orthogonal net on the sphere; this net is also isothermal-conjugate. Since an isothermal-conjugate system is transformed by a projectivity into an isothermal-conjugate system [~~ 82, 84], we have the converse theorem: Every net with equal point invariants on a quadric is isothermalconjugate. 368 X. Surfaces applicable to a quadric From the first theorem of ~ 60 and the preceding considerations we have the theorem: Every isothermal- conj'agate system on a quadric has equal tangential invariants; and every net on a quadric with equal tangential invariants is isothermal-conjugate and has equal point invariants. In ~ 88 we saw that an isothermal net on a sphere admits c 8 transformations F into nets of the same kind on the sphere. Hence the same is true of nets with equal point invariants on any quadric, in view of the above remarks (cf. ~ 148). 148. Transformations F and W of isothermal-conjugate nets on a central quadric. Let N be an isothermal-conjugate net on the central quadric Q (34). An F transform iV also on Q is given by (47) and (48). From (II, 81) and (169) we have that Ni1 is a K transform, that is NAr has equal point invariants and is isothermal-conjugate, if _ 0 0 ' k 0 0' (170) h -- 4 + --- +-, where k is a constant. In consequence of (48) these equations are equivalent to (171) a2(h-1)-Cexx' - 0, a2(h2 - 2)-kZ e x2 0. Differentiating these equations and making use of (59), we have lh k fax faex,x+ (gI 1) a log a au a a a o a g (172) a (av a av -a av These equations and (59) form a completely integrable system. For each set of solutions of these equations the left-hand members of (171) are constants. Since all of the equations are homogeneous, there are o 2 sets of solutions satisfying (171) for each value of k. Then the conditions (170) are satisfied and we have the theorem: An isothermal-conjugate net on a central quadric admits o03 transformations K into isothermal-conjugate nets on the quadric149). 149) Cf. Tzitzeica, Bull. Sciences Math., ser. 2, vol. 36 (1912), pp. 151-164. 148. Transformations F and IV of -isothermal-conjugate nets36 369 IFrom equations (169) anid(31) we have the following expressions for the Christoffel symbols formed with respect to the linear element (26) of the net NV: il { a. H (173) 1 D ~ a3 = -log H j12t Dloga )12 a log a Du {22 aDlog a (2 Du H1 ~22 Dv log a'0 The Codazzi equations for. N can, be written as the second set of (30); then we have by integration D - -D" a-a/H, since a is determined by (1 69) only to within a constant factor. Hence if. a4z — K, we have (174) Hoa - a', D -D" — a' In consequence of (173) and (17,4). we show by methods similar to those used in ~ 135 that Ix2_ a2 (1-75) le a) c l eDx ax -_1 au av 2 le a X 2,- - a a V C where c4 -l1efy. Suppose now that we have two sets of solutions of (59) and (172), namely x', y', s', h1, i~1; x",~ y",~ z", h2, 12, satisfying the conditions (176) {IL 0 1 0 ~~ 0, a~~~~2 1h2l ) k e x" 2 _r 0, and also (177) a2 (hh2 —l1l 2)- -k ex'x1zz 0. By means of (173), (174), (175) and (1 76) we show as in 138 that the functions 01- 2 lexx', 02 2 1 exx"1 are so~lutionis of the equation D20 D 20 _ D -loga DO D2alga a0 4 Di 2 y Du D u Da D v C 24 370 X. Surfaces applicable to a quadric and the second of (169). Consequently (~ 49) 01 and 02 determine a W transform NT of N. Its coordinates a, y, z are given by (88), which in turn are reducible to (91). From (176) and (177) we have It f -y^ 2 e^-2 f_ 92 2 Zf g (y"z'- y' Z")y = ex'. e "2 ( e X = (hl -h 1 1) efg [ x (yYZ'- y'IZ)] = - k (hi 1- h2 t)2. In consequence of these identities we have that N lies on the given quadric (34). Hence the congruence of lines joining corresponding points on N and N17 degenerates into one set of generators of the quadric. From (34), (92), (88), (175) and exj2- 1, we have (P2a-2 b2) b z e x P + 2 )z e(-)8 = 0. Hence (cf.IV, 75) N is a net with equal point invariants. As at the close of ~ 138, we remark that there are only two sets of solutions x", y", y", h, s2; x"', y", z'o, ho, 13 each of which gives with x', y', z', hi, 11, a net V, say nets XN and N3. These nets are conjugate to the congruence of the lines of intersection of the tangents planes to N and its F transform NAT by means of x', y', z', h,, 1~; and their points are the intersections of the lines of this congruence with the generators of the quadric. Accordingly we have the theorem: If N and NZ are two isothermal-conjugate nets on a central quadric in relation F, the points in which the quadric is met by the intersections of corresponding tangent planes of N and N1 generate two isothermal-conjugate nets in relation F150). Exercises 1. A necessary and sufficient condition that N whose coordinate 5 satisfy (33) lie on a ruled deform of Q is that a/b be a function of u + v or u -v. 2. A necessary and sufficient condition that the nets permanent in deformation on every surface applicable to a surface S be isothermal-conjugate is that S be a quadric. Terracini, Annali, vol. 30 (1921), p. 145. 150) Tzitzeica, 1. c., has given a geometrical proof of this theorem. Exercises 371 3. When k'= k in ~ 139, the solutions of (106) and (110) are hs = ah-i+-h2, 13 - alll il+2, x'-= ax'+ x", y"'- ay'+-ty", z'= a z'+ P." In this case the transforms N and NT coincide. 4. Show that the transformations Ec and B1C of a deform of a paraboloid are permutable. 5. If N1 and N2 are obtained from the permanent net of a deform of a paraboloid P by transformations Blk and Bk' (k' k7), there can be found directly a net AT12 applicable to a net on P, such that J~i and Jir2, and N2 and Nl2 are in relations Bc. and B7 respectively. Bianchi, Lezioni, vol. 3, chap. 4. 6. The surfaces of revolution admitting a conjugate in deformation have the linear element (Ids2 - h[ -c2 (ar2 +1)] dr2+ q 9v2 a = 2 -- - where a, c and h are arbitrary constants. The linear element of the conjugate surface is _ h [1 - c2 (a r2+1)] dr2 1 v2. d s2 =- - - c+ - V2 (a'2 +- 1)3 ar2 +_ 1 Bianchi, Rend. dei Lincei, ser. 5, vol. 11 (1902), p. 272. 7. If two surfaces applicable to a surface of revolution are conjugate in deformation, so also are their complementary surfaces [~ 761. Bianchi, 1. c., p. 273. 8. If two quadrics, Q and Q, correspond in a projectivity which transforms the quadrics confocal with Q into the quadrics confocal with Q, then Q and Q are conjugate in deformation. Bianchi, Lezioni, vol. 3, p. 208. 9. If S and S are conjugate in deformation and S is deformed so that one of its asymptotic lines remains rigid, then S admits a deformation in which the corresponding asymptotic line is rigid. Bianchi, 1. c., p. 215. 10. If at points of two corresponding asymptotic lines on two surfaces conjugate in deformation tangents are drawn to the geodesics which are the deforms of generators of the applicable quadrics, Q and Q, the two ruled surfaces formed by these tangents are applicable to Q and Q, and are conjugate in deformation. Bianchi, 1. c., p. 215. 11. When the first of equations (165) is written in the form (34) and the second t72+f-722+ 2 - 1, the equations of the projectivity (166) are ee ef X' e g also /- e2 -C4 _ l - -e - c; and the point equation of a net on Q is 20_ a a aO a b ao _ log t + log iufv -- v gx of ' o nS x Sav if the equation of the corresponding net on Q is the second of (33). 372 X. Surfaces applicable to a quadric 12. If a permanent net on Q admits a transformation Fk by means of a set of functions x', y', z', h, 1, the corresponding net on Q (cf. Ex. 11) admits a transformation Fi determined by -/ i Y, - '-I\ x ' 1 - x' - - Y-,, == I r-9 \z'- XX Vee X ef X eg X A~-As~ x', - _, - _ ~e (e -,) h -- - x', 1l — x - x', I- - X ( ---. 13. If AT is a B, transform of a net N applicable to Q by means of functions X y,, z'; x", y", z"; h1, ii; 12, 12, the functions analogous to those of Ex. 12 determine a BT transform of N applicable to Q, and this transform and N are in relation H. Bianchi, 1. c., p. 231. 14. Show by means of [~ 127] that the pencils of planes whose axes are polar with respect to. a quadric meet the latter in an isothermal-conjugate net. Bianchi, 1. c., p. 244. 15. When a quadric Q is referred to an isothermal-conjugate net, so also is a quadric conjugate in deformation to Q, and Ho —3 = IH oa3c, where c is a constant. 16. In order that a net N on a quadric be permanent in more than one deformation of the quadric, it is necessary that N be isothermal-conjugate. Servant, Bull. Soc. de France, vol. 30 (1902), p. 21. 17. If NAT1 and N2 are isothermal-conjugate nets on a central quadric Q (34) obtained from an isothermal-conjugate net on Q by transformations KI7 and K7, (k' t k) of ~ 148, there can be found directly a net N12 on Q, which is a Kk' transform of Nt and a Kk transform of NT'; it is determined by the functions 0' and 06 given by (78) and I10+t - k' 2= 0 c01 ' + ek' - 0.2 01 18. Derive the equations of transformations AKl of isothermal-conjugate nets on a paraboloid similar to those for a central quadric ~ 148. 19. If a ruled surface S is subjected to an infinitesimal deformation which leaves the surface ruled and through each point of S and in the corresponding tangent plane a line is,drawn perpendicular to the direction of the deformation, these lines form a WT congruence for which the other focal surface is ruled; this construction gives the most general TVcongruence with ruled focal surfaces and generators corresponding. Bianchi, Comptes Rendus, vol. 143 (1906), p. 635. 20. If the focal surfaces, S1 and S2, of a W congruence are ruled, and if the curved asymptotic lines of S2 correspond to the generators of S1, then Si is a quadric. Segre, Atti di Torino, Dec. 28, 1913; Tortorici, Rend. di Napoli, vol. 28 (1922), Jan. 21. Index The numbers refer to pages. Adjoint equation, 9, 22, 88, 93. Angle, of two lines 1; of two hyperplanes, 2; of two spheres, 234; of two surfaces in pentaspherical coordinates, 282. Applicable nets, definition, 61; parallel to given pair, 62; transformations F of, 61, 275, 290, 325, 334; and congruences of circles, 272; and cyclic systems, 274; which are 0, 286; which are 2, 0, 298. See Nets C and Nets permanent in deformation. Asymptotic lines, definition, 98; parametric, 99, 123; in Lie line-sphere transformation, 252; determination of, 356. Axis curves, definition, 125; property of, 126; form a net, 125, 153. Axis congruence, definition, 125. Beltrami, 272, 299, 310, 312, 314. Bianchi, 112, 138, 142, 143, 180, 209, 210, 212, 214, 215, 216, 229, 230, 292, 298, 301, 305, 308, 310, 312, 316, 317, 320, 327, 344, 345, 352, 356, 360, 362, 363, 366, 371, 372. Bonnet, 295, 317. Calapso, 225, 230, 232, 286, 298, 319, 320, 331. Calo, 318. Chieffi, 316. Christoffel transformations of isothermic surfaces, 229, 286; and transformations D,,, 229; nets determined by, 319; of special isothermic nets, 230, 291. Circles K, definition, 202; normals to nets at points of, 204; of the first kind, 206; of the second kind, 206, 229; form a congruence, 263. Circle - planes of a transformation R, definition, 198; envelope of, 199, 213, 285, 286; pass through a point, 200, 230; coincident for two transformations, 206, 211. Congruence, definition, 11; developables of, 11; focal points of, 11; focal sur*faces of, 11; conjugate to a net, 12, 15, 34; direction equation of 13, 153; parallel, 15; middle surface, 16, 21, 32; harmonic to a net, 22, 24, 25, 29, 80, 81, 153; harmonic to a point net, 28; conjugate, of a transformation F, 34; harmonic, of a transformation F, 40, 153; conjugate to 0 nets, 167; normal to a net, 168. Congruences I and p, I, definition, 158; 2,, normal, 159,294; nets conjugate to, 159, 168. Congruence I, definition, 203; cyclic, 206; focal points of, 229. Congruence R, definition, 113; focal surfaces, 115. Congruences of circles, definition, 241; characteristic property, 243; focal spheres of, 241; focal points of, 243; in cartesian coordinates, 243; transformations F of, 269, 270; parallel transformations of, 272; a cyclic system, 272, 274, 278, 285; envelope of planes of, 258, 259, 269, 270, 271, 274. Congruence of rolling, 316; focal surfaces of, 317; developables of, 317; with coincident developables, 317; parallel, 321; normal, 321; with constant focal segment, 321; pseudospherical, 321. 374 Index Congruences of spheres, definition, 238; principal curves of, 238; focal circles of, 239, 241; chords of contact of, 239; derived, 240, 243; in cartesian coordinates, 243; null-spheres, 243, 284; with applicable central nets, 271; cutting a fixed sphere under constant angle, 283. Congruences R of spheres, definition, 246; parallel, 246; transformations F of, 246; direction-equation with equal invariants, 255; representation in 5-space, 285; permanent in deformation, 298, 308, 310. Congruences of spheres and circles, harmonic, 255-259, 264, 265, 267, 269-271; conjugate, 261-263, 270; orthogonal, 266, 267, 285. Conics, pencil of, 95. Coordinates, cartesian, 1; homogeneous point, 69; tangential, 97, 166, 286; of a sphere, 233; pentaspherical, see Pentaspherical. Cosserat, 206. Curve, definition, 2, 71; parametric, 2. Cyclic congruences, definition, 183; parallel, 191; characteristic property, 280, 285; generation of, 289; multiply, 185, 188, 191. Cyclic systems, definition, 183; circles pass through a point, 191; of a transformation RI, 197, 206, 285; generation of, 272-274, 285, 289; transformations F of, 275, 277, 278, 290; representation in 5-space, 277-279; envelope of planes of, 199, 213, 285, 286. Cyclides, 283. Darboux, 6, 92, 147, 154, 208, 212, 213, 239, 254, 274, 281, 282, 283, 285, 288, 291, 295, 313, 317, 358. Deform of a quadric, which meets the circle at infinity in 4 points, 291; of revolution, 293-296, 305, 360; tangent to the circle at infinity, 317; permanent nets on, 326-329, 353; R nets on, 327, 328; transformations Fk of, 334-340, 346, 353, 354; transformations Bi of, 340-352, 354-356; transformations I- of, 366, 371, 372. Demoulin, 107, 108, 112, 113, 115, 119, 124, 134, 150, 202, 204, 220, 225, 229, 254, 284. Derivant net, definition, 25; of two transformations F, 54; of two transformations K, 61; of two transformations R, 265. Derived congruences, definition, 21; of a 1V congruence, 108; of spheres, 240, 241. Derived net, definition, 25; Laplace transforms of a, 32, 33; parallel, 33, 53; transformations F of, 54, 66; of two transformations ', 54; reciprocally, 116; which is 0, 289. Developable surface, definition, 10; of a congruence, 11. Dini, 362. Direction-parameters, of a line, 1; of a congruence, 12, 14, 20, 30; of a cyclic congruence, 183, 281. Drach, 155. Envelope of rolling, definition, 305; lines of curvature, 320. Envelopes of spheres, with O nets corresponding, 174, 189, 229; conformal mapping of the two sheets, 206, 286; principal curves on, 238, 239, 243; congruence of chords of contact, 239, 241, 242, 244, 245, 258, 259, 262, 285; with applicable central nets, 272, 274, 298, 302, 307, 308, 310; cutting a fixed sphere under constant angle, 283; with spherical lines of curvature, 285. Equation, point, of a net, 3, 72; direction, of a congruence, 13, 153; tangential, of a net, 128, 153. Focal circles of a congruence of spheres, 239, 241. Focal nets of a congruence, 11, 15; of parallel congruences, 21; radial transforms, 33; in relation 'F, 34. Index 375 Focal points, of first and second rank, 11; coordinates of, 10, 14, 29; of a ray congruence, 124; of a congruence of circles, 245. Focal spheres of a congruence of circles, 241.; meeting orthogonally, 183. Focal surfaces of a congruence, 11, 15. Green, 124, 125, 126. Guichard, 5, 12, 26,. 124, 156, 165, 200, 255, 258, 262, 263, 266, 280, 285, 295, 296, 337. Hammond, 77, 85, 87, 93, 94. Homogeneous point coordinates, 69; of a net in 2-space, 96; of a net in 3-space, 101, 103; of an R net, 107. Hyperplane, definition 2; at infinity, 69. Hyperquadric, net on, 90, 254; transformations F of nets on, 90, 94. Hyperspheres, in transformations R, 171, 174, 189; focal, 183, 185. Invariants of an equation of Laplace, definition, 16; equal to zero, 73, 92; in a sequence of Laplace, 75, 92. Inversion, in transformations R, 175; of a sphere, 281; in pentaspherical coordinates, 281; of a cyclide, 283. Isothermal-conjugate nets, definition, 104; preserved by projective transformation, 10; R nets are, 107; on a quadric, 112,367,372; and nets with equal point invariants, 125, 150; characteristic property, 126; on a deform of a quadric, 326. Isothermic O nets, definition, 206; special, 212-214,291; are nets ~2, 231; in pentaspherical coordinates, 237; Lie transformation of, 284; nets with the same spherical representation as, 302-304, 318,319; sphericalrepresentation of, 319. See Transformations D,, and Christoffel transformations. Isotropic, relation, 157; line, 289, 291,296; plane, 289, 296. Jonas, 42, 47, 110, 113, 119, 201. Kinematically conjugate directions, definition, 314; axes of rotation, 315; auto-conjugate, 315, 320; for congruences of rolling, 317, 321; parametric, 320; orthogonal, 320. Koenigs, 58, 123. Laplace equations, 3, 12; invariants of, 16; sequence of, 74; Moutard form, 237. Laplace transforms, definition, 17; in cartesian coordinates, 17; of parallel nets, 18; of a function, 19, 32; of a derived net, 32, 33; of a radial transform, 33; transformations F of, 66, 94; in homogeneous coordinates, 72; of a net R, 108; in tangential coordinates, 129; which are Onets, 192, 193, 286. See Sequences of Laplace and Levy sequences. Levy sequences, of the first order, 81, 93; of higher order, 83-85, 93; periodic, 85, 86. Levy transforms, definition, 19; of parallel nets, 32; of radial transforms, 33; and transformations F, 66; in homogeneous coordinates, 81; in a hyperplane, 81. Lie, 249, 283, 284. Lie line-sphere transformation, equations of, 249, 283; linear complex of, 249, 284. Lie transformations of surfaces, definition, 252; equations of, 283, 284; W congruence of, 253. Linear complex, definition, 248; null-plane of, 248, 251, 252; special, 248; lines conjugate with respect to, 250, 251, 284; tangents to a curve lines of a, 284. Martin, 32. Minimal 'surfaces, transformations Dn of, 215, 217; 0net of, 229; focal sheets of a W congruence, 230; in relation R, 296, 308; deformable transformations R of, 304, 305. Nets, definition, 3; characteristic property, 4; tangent planes of, 4; point, 5, 28, 30; in 3-space, 6, 100-103, 105; parallel, 7-12, 124; planar, 9, 96, 123; conjugate to a congruence, 12, 14, 19, 32, 159; harmonic to a congruence, 22, 24, 27, 30, 34, 200; of translation, 32, 376 Index 66; minimal, 32, 66; in relation 0, 63, 146, 154; in homogeneous coordinates, 72; periodic, 76, 94; in tangential coordinates, 127, 128; of plane curves, 152. See Applicable nets, Derivant nets, Derived nets, Isothermal-conjugate nets. Nets corresponding with orthogonality of linear elements, 63, 146; parallel, 64; transformations F of, 64-66. Nets C, definition, 200; harmonic to a normal congruence, 200; on the envelope of planes of a cyclic system, 200; congruences harmonic to, 287, 288. See Applicable nets. Nets G, definition, 192; parallel, 192; Laplace transforms of, 192, 286; radial transforms of, 193; transformations F of, 193. Nets O, definition, 156, 194; conjugate to congruences I, 2,I and 3,I, 160; determination of, 165-167; parallel, 167; on a hypersphere, 167; congruences conjugate to, 168; transformations F of, 168-175, 185-188; radial transformations of, 175; inversion of, 175; transformations R of, 174-182, 195-197; on a hypercone, 191, 283; with isothermal spherical representation, 215-218, 229; in pentaspherical coordinates, 237, 279; in 5-space and cyclic systems, 277; generated by a rolling surface, 289, 291, 294; derived, 289. See Isothermic O nets. Nets 2, 0, definition, 156; as central nets of transformations R, 171, 189; special, 188, 228; transformations F of, 189-191; on quadrics of revolution, 247; associated with cyclic systems, 279; applicable, 298. Nets p, 0, definition, 156; conjugate to congruences, p, I, 160; parallel, 191. Nets permanent in deformation, definition, 139; fundamental coefficients, 139-140; transformations S2 of, 141-146; special, 142; spherical representation of, 154, 155. Nets R, definition, 107; equations, 107; Laplace transforms of, 108; on a quadric, 112, 367; reciprocally derived, 118; transformations W of, 115-123, 125; transformations F of, 124, 125; and nets Q2, 254; in line coordinates, 255; with tangents in. one family meeting a fixed line, 284; on deforms of a quadric, 327, 328. Nets 2, definition, 218; equations of, 219, 220; parallel, 221; transformations R of, 221-224; isothermic, 231; with isothermal sphericalrepresentation, 231; on surfaces of Guichard, 225, 232; and nets R, 254. Nets with equal point invariants, 16, 32, 67, 68, 95, 106, 125; characteristic property, 124; associate, 59, 67; planar, 123; and equal tangential invariants, 150; which are O nets, 206; on a quadric, 366-368. See Transformations K. Nets with equal tangential invariants, definition, 128; which are permanent in deformation, 139; and equal point invariants, 150; characteristic property, 153; on a quadric, 366-368. See Transformations 2. Nets with same spherical representation as isothermic 0 nets, 302, 303; transformations R of, 303, 319, 320. Normal parameters of a net, 5. Osculating planes of a net, 11; conjugate to a congruence, 23; and Laplace transforms of a derived net, 32; in homogeneous coordinates, 71; and nets Go, 286. Orthogonal determinant, definition, 163; in 5-space, 165, 279; in 3-space, 194. Parallel, lines, 1; hyperplanes, 2; nets, 7-12, 32; congruences, 15, 28. Pentaspherical coordinates, of a point, 235; fundamental spheres, 235, 281; equation of a sphere in, 236, 281; Index 377 special, 237; of an 0 net, 237, 279; of an isothermic net, 237; of a point of the plane at infinity, 281; angle between displacements in, 282; equation of a surface in, 283; transformations P in, 283. Perpendicular, lines, 1; hyperplanes, 2. Plane, in cartesian coordinates, 2; tangent, of a surface, 3; tangent, of a net, 4; in homogeneous coordinates, 71. Pliicker line coordinates, 247. Point equation of a net, in cartesian coordinates, 3; in homogeneous coordinates, 72. Point nets, definition, 5; congruences harmonic to, 28, 29; harmonic to a congruence, 30. Polar transformation, 100, 125, 153. Poles of a circle, definition, 266; points on an envelope of spheres, 266, 274; of a cyclic system, 285, 286. Principal curves on an envelope of spheres, 238, 239, 243; form a net, 245. Problem of order n —2, 165. Projective transformations, of space, 70; of nets in relations F, 93; of nets in 2-space, 97; of asymptotic lines, 100; of nets in 3-space, 103, 128; of isothermal-conjugate nets, 104. Quadrics, isothermal-conjugate nets on, 112, 366-370, 372; nets R on, 112, 367; osculating, 124; of revolution, 247, 293-296, 305, 337, 360; deformation of, 291, 293, 305, 326, 360; rolling, 291-296, 317; tangent to the circle at infinity, 317; transformations F of, 322, 329-334, 352, 353, 368; permanent nets on, 323-326, 329-334, 353, 370, 372; referred to generators, 323-325; conjugate in deformation, 365, 371; nets with equal point invariants on, 367, 368; nets with equal tangential invariants on, 367, 368; transformations TV of, 368-370, 372. Quatern of transformationsF, definition, 48. Radial transformations, definition, 31; harmonic congruences of, 31; of Laplace transforms, 33; of Levy transforms, 33; and transformations F, 40-42, 67, 153; of every net on a surface, 124; of nets 0, 175; of nets Go, 193. Ray congruence, definitisn, 105; focal points, 124. Ray curves, definition, 105; property of, 124. Relation, F, 34; K, 58; 0, 63, 146; R, 174; isotropic, 157. Ribaucour, 16, 89, 174, 239, 245, 285, 286, 312, 323. Rolling surfaces, definition, 288; plane of contact, 288; and cyclic systems, 289; on a quadric, 291-296, 317; on a surface of revolution, 293-298, 321; kinematically conjugate directions on, 314, 320; ruled, 314,' 315; axes of rotation, 315; on a sphere, 317; spheres associated with, 318. Ruled surfaces, referred to asymptotic lines, 123; osculating quadrics of, 124; rolling, 314; applicable to a non-ruled surface, 314, 316; applicable, 320; infinitesimal deformation of, 372; V congruences whose focal surfaces are, 372. Segre, 372. Sequences of Laplace, definition, 74; equations of, 75, 76, 92; invariants of, 75, 76, 92; periodic, 76-80, 93, 94; derived nets of, 83-85, 93; transformations F of, 94. Servant, 325, 359, 362, 363, 365, 372. Spheres, coordinates of, 233; point-, 234; null-, 234, 289, 294; power with respect to, 234; equations in pentapherical coordinates, 236; inversion of, 281; rolling, 317. See Congruences of spheres. Stetson, 286. Surface, definition, 2; tangent planes, 3; determination of nets on, 5-7; B, 111; of Voss, 143; determination of asymptotic lines on, 356-358; element, 252. 378 Index Surfaces applicable to a quadric, see Deforms of a quadric. Surfaces applicable to a surface of revolution, 293-298, 321; complementary, 292, 294-296, 321; conjugate in deformation, 371. Surfaces conjugate in deformation, definition, 360; characteristic property, 362; determination of, 362-365, 371; quadrics, 365, 372. Surfaces of constant mean curvature, transformations R of, 229, 295, 310; special isothermic -nets on, 230; generation of, 294, 317; kinematically conjugate lines on, 320. Surfaces of constant total curvature, equations of, 232; transformations R of, 232, 295, 339; cyclic systems associated with, 286; determination of 360. Surfaces of Guichard, definition, 225, 232; associated surfaces, 225, 228, 232; are surfaces Q, 225, 232; transformations R of, 227, 231, 232, 286; isothermic surfaces associated with, 286. Surface of rolling, definition, 301; one sheet of envelope of congruence R a, 302; equations of, 313; lines of curvature on, 320. Tangential coordinates, of a surface, definition, 98; referred to asymptotic lines, 100; of a net, 127; Laplace transforms in, 129; transformations F in, 129-134, 153; of a net 0, 166; transformations B in, 178-182. Terracini, 370. Theorem of permutability,- of transformations F, 45-51, 90, 132; for parallel nets, 50, 67; extended, 55-57, 68, 224, 230; of Tr congruences, 124, 138, 143. For references to the theorem of permutability of any type of transformation see this transformation. Tortorici, 372. Transformations B., of surfaces applicable to a quadric, 340-346, 354-356; permutability with transformations iFk, 346, 371; theorem of permutability of, 352, 371. Transformations D,, 208, 209, 231; theorem of permutability of, 209-211, 215, 230; of special isothermic surfaces, 212-214, 230, 291; of minimal surfaces, 215, 304; isothermic nets determined by, 230; parallel nets of, 230; characteristic property, 285; induced, 286, 319; generation of, 291; and surfaces of rolling, 301, 302; deformable, 304. Transformations E,, 215-217, 231; theorem of permutability of, 218, 230; characteristic property, 230; deformable, 305; and envelopes of rolling, 305. Transformations F, equations in cartesian coordinates, 34-37; conjugate congruence of, 34, 42, 89; inverse of, 37; parallel, 38, 50, 67; harmonic congruence of, 40, 182; and radial transformations, 40-42, 67, 153; triads of, 44, 90; theorem of permutability of, 45-51, 55-57, 67, 90, 132; of derived nets, 54, 66; of applicable nets, 61-63; of nets in relation 0, 64, 146, 154; of Laplace transforms, 66, 94; of Levy transforms, 66; in homogeneous coordinates, 87-91; of nets on a hyperquadric, 90, 94; projective transformations of, 93; of periodic Laplace sequences, 94; of nets R, 124, 125; in tangential coordinates, 129-134, 153; and polar transformations, 153; of nets 0, 168-175, 185-188; of nets 2, 0, 189 —191; of nets G, 193; of nets on a sphere, 197, 217; of congruences R of spheres, 246; of congruences of circles, 269-271; of cyclic systems, 274-278, 290; of deformable transformations R, 306; of nets on a quadric, 322, 323; of isothermal-conjugate nets on a quadric, 368, 372. References to particular types of transformations F are given under these transformations. Index 379 Transformations F,~, of nets on a quadric, 329-334, 353; of surfaces applicable to a quadric, 334-339, 354; theorem of permutability of, 323, 339, 340. Transformations H of surfaces applicable to a quadric, definition 366; of ruled deforms, 366; permutability with transformations Fk, 366, 372; permutability with transformations Bk, 372. Transformation, 01, definition, 113; of surfaces of constant curvature, 124; of isothermal systems on a sphere, 124. Transformations If, definition, 58; in cartesian coordinates, 57-59, 67; theorem of permutability of, 60, 61, 67, 68, 154, 372; of associate surfaces, 68, 124; of Levy transforms, 68; in homogeneous coordinates, 94, 95; characteristic property, 95; of nets with equal point and equal tangential invariants, 150-152; polar transformations of, 153; with normal conjugate congruences, 206,221; of nets on a quadric, 368-370, 372. See Transformations DI,. Transformations R, in cartesian coordinates, 173-177, 195; central nets, 174, 189, 196, 228; of parallel nets 0, 174, 196, 228; inverse of, 178; in tangential coordinates, 178, 181; theorem of permutability of, 180-182, 202-206; of Laplace transforms of a net G, 193; spherical representation of, 197; cyclic systems of, 197, 206, 285; circle-planes of, see Circle-planes; circles K of, see Circles K; congruences K of, see Congruences K; conformal, 206; of isothermic 0 nets, see Transformations D.; of minimal surfaces, 215, 296, 304, 305; of nets Q, 221-224; of surfaces of Giuchard, 227, 231, 232, 286; with conformal spherical representation, 215, 230, 306; of surfaces of constant mean curvature, 229, 295, 310; of surfaces of constant total curvature, 232, 295, 339; in pentaspherical coordinates, 283; deformable of the first type, 296, 308, 310; deformable of the second type, 298-306; with applicable central nets, 297, 302; of nets with the same spherical representation as isothermic 0 nets, 303, 319; and surfaces of rolling, 302, 318; and envelopes of rolling, 305; transformations F of, 306. Transformations W, of nets R, 115-123, 125; of isothermal-conjugate nets on a quadric, 370; of ruled surfaces, 372. See Transformations Bk. Transformations 52, in tangential coordinates 134-136; theorem of permutability of, 137, 138, 153, 154; of nets permanent in deformation, 141-146; in point coordinates 146-149; of a focal surface of a W congruence, 149, 150; of nets with equal point and equal tangential invariants, 150-152; polar transformation of, 153. Tzitzeica, 33, 77, 93, 95, 107, 116, 124, 327, 368, 370. Umbilical points of a quadric, 291. Virtual asymptotic lines, definition, 316; determination of, 358. TV congruences, definition, 106; of tangents to a net R, 107, 113; derived, 108; and reciprocally derived nets, 116; quatern of, 124, 138, 143; with focal surfaces of equal curvature, 138, 142, 143; transformations Q and IK of, 150; with minimal focal surfaces, 230; whose lines belong to a linear complex, 252, 253; in Lie transformations of surfaces, 252, 253; representation in 5-space, 254; in Plicker line-coordinates, 254; normal, 292, 293; with ruled focal surfaces, 372. Wilczynski, 105, 123, 125, 126. Erratum ~ 59. Change "focal nets" to "focal surfaces".