mmm Wf / I xSm*^i?c?^n A TREATISE ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES BY LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON CvA COPYRIGHT, 1909, BY LUTHER PFAHLER EJSENHART ALL RIGHTS RESERVED 89-8 gftc SUftengum GINN AND COMPANY PKO- PRILTORS BOSTON U.S.A. 6 A-.ATH.. STAT. LIBRARY PEEFACE This book is a development from courses which I have given in Princeton for a number of years. During this time I have come to feel that more would be accomplished by my students if they had an introductory treatise written in English and otherwise adapted to the use of men beginning their graduate work. Chapter I is devoted to the theory of twisted curves, the method in general being that which is usually followed in discussions of this subject. But in addition I have introduced the idea of moving axes, and have derived the formulas pertaining thereto from the previously obtained Frenet-Serret formulas. In this way the student is made familiar with a method which is similar to that used by Darboux in the first volume of his Lemons, and to that of Cesaro in his Geometria Intrinseca. This method is not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu able in developing geometrical thinking. The remainder of the book may be divided into three parts. The first, consisting of Chapters II-YI, deals with the geometry of a sur face in the neighborhood of a point and the developments therefrom, such as curves and systems of curves defined by differential equa tions. To a large extent the method is that of Gauss, by which the properties of a surface are derived from the discussion of two quad ratic differential forms. However, little or no space is given to the algebraic treatment of differential forms and their invariants. In addition, the method of moving axes, as defined in the first chapter, has been extended so as to be applicable to an investigation of the properties of surfaces and groups of surfaces. The extent of the theory concerning ordinary points is so great that no attempt has been made to consider the exceptional problems. For a discussion of such questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. In Chapters VII and VIII the theory previously developed is applied to several groups of surfaces, such as the quadrics, ruled surfaces, minimal surfaces, surfaces of constant total curvature, and surfaces with plane and spherical lines of curvature. iii iv PEEFACE The idea of applicability of surfaces is introduced in Chapter III as a particular case of conformal representation, and throughout the book attention is called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely with the recent method of Weingarten and its developments. The remaining four chapters are devoted to a discussion of infinitesimal deformation of surfaces, congruences of straight lines and of circles, and triply orthogonal systems of surfaces. It will be noticed that the book contains many examples, and the student will find that w r hereas certain of them are merely direct applications of the formulas, others constitute extensions of the theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as would enable the reader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish no such key, only to remark that the Encyklopadie der mathematisclicn Wissenschaften may be of assistance. And the same may be said about references to the sources of the subject-matter of the book. Many important citations have been made, but there has not been an attempt to give every reference. However, I desire to acknowledge my indebtedness to the treatises of Darboux, Bianchi, and Scheffers. But the difficulty is that for many years I have con sulted these authors so freely that now it is impossible for me to say, except in certain cases, what specific debts I owe to each. In its present form, the material of the first eight chapters has been given to beginning classes in each of the last two years; and the remainder of the book, with certain enlargements, has constituted an advanced course which has been followed several times. It is im possible for me to give suitable credit for the suggestions made and the assistance rendered by my students during these years, but I am conscious of helpful suggestions made by my colleagues, Professors Veblen, Maclnnes, and Swift, and by my former colleague, Professor Bliss of Chicago. I wish also to thank Mr. A. K. Krause for making the drawings for the figures. It remains for me to express my appreciation of the courtesy shown by Ginn and Company, and of the assistance given by them during the printing of this book. LUTHER PFAHLER E1SENHART CONTENTS CHAPTEE I CURVES IN SPACE SECTION PAGE 1. PARAMETRIC EQUATIONS OF A CURVE 1 2. OTHER FORMS OF THE EQUATIONS OF A CURVE 3 3. LINEAR ELEMENT 4 4. TANGENT TO A CURVE 6 5. ORDER OF CONTACT. NORMAL PLANK 8 6. CURVATURE. RADIUS OF FIRST CURVATURE 9 7. OSCULATING PLANE 10 8. PRINCIPAL NORMAL AND BINORMAL 12 9. OSCULATING CIRCLE. CENTER OF FIRST CURVATURE .... 14 10. TORSION. FRENET-SERRET FORMULAS 16 11. FORM OF CURVE IN THE NEIGHBORHOOD OF A POINT. THE SIGN OF TORSION 18 12. CYLINDRICAL HELICES 20 13. INTRINSIC EQUATIONS. FUNDAMENTAL THEOREM 22 14. RICCATI EQUATIONS 25 15. THE DETERMINATION OF THE COORDINATES OF A CURVE DEFINED BY ITS INTRINSIC EQUATIONS 27 16. MOVING TRIHEDRAL 30 17. ILLUSTRATIVE EXAMPLES .33 18. OSCULATING SPHERE 37 19. BERTRAND CURVES 39 20. TANGENT SURFACE OF A CURVE 41 21. INVOLUTES AND EVOLUTES OF A CURVE 43 22. MINIMAL CURVES . 47 CHAPTER II CURVILINEAR COORDINATES ON A SURFACE. ENVELOPES 23. PARAMETRIC EQUATIONS OF A SURFACE 52 24. PARAMETRIC CURVES 54 25. TANGENT PLANE 56 26. ONE-PARAMETER FAMILIES OF SURFACES. ENVELOPES .... 59 v vi CONTENTS SECTION PAGE 27. DEVELOPABLE SURFACES. RECTIFYING DEVELOPABLE .... 61 28. APPLICATIONS OF THE MOVING TRIHEDRAL 04 29. ENVELOPE OF SPHERES. CANAL SURFACES 66 CHAPTER III LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAME TERS. CONFORMAL REPRESENTATION 30. LINEAR ELEMENT 70 31. ISOTROPIC DEVELOPABLE 72 32. TRANSFORMATION OF COORDINATES 72 33. ANGLES BETWEEN CURVES. THE ELEMENT OF AREA .... 74 34. FAMILIES OF CURVES 78 35. MINIMAL CURVES ON A SURFACE 81 36. VARIATION OF A FUNCTION 82 37. DIFFERENTIAL PARAMETERS OF THE FIRST ORDER 84 38. DIFFERENTIAL PARAMETERS OF THE SECOND ORDER .... 88 39. SYMMETRIC COORDINATES 91 40. ISOTHERMIC AND ISOMETRIC PARAMETERS 93 41. ISOTHERMIC ORTHOGONAL SYSTEMS 95 42. CONFORMAL REPRESENTATION 98 43. ISOMETRIC REPRESENTATION. APPLICABLE SURFACES .... 100 44. CONFORMAL REPRESENTATION OF A SURFACE UPON ITSELF . . 101 45. CONFORMAL REPRESENTATION OF THE PLANE 104 46. SURFACES OF REVOLUTION 107 47. CONFORMAL REPRESENTATIONS OF THE SPHERE . 109 CHAPTER IV GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD OF A POINT 48. FUNDAMENTAL COEFFICIENTS OF THE SECOND ORDER .... 114 49. RADIUS OF NORMAL CURVATURE 117 50. PRINCIPAL RADII OF NORMAL CURVATURE 118 51. LINES OF CURVATURE. EQUATIONS OF RODRIGUES 121 52. TOTAL AND MEAN CURVATURE 123 53. EQUATION OF EULER. DUPIN INDICATRIX 124 54. CONJUGATE DIRECTIONS AT A POINT. CONJUGATE SYSTEMS . 126 55. ASYMPTOTIC LINES. CHARACTERISTIC LINES ....... 128 56. CORRESPONDING SYSTEMS ON Two SURFACES 130 57. GEODESIC CURVATURE. GEODESICS ... 131 58. FUNDAMENTAL FORMULAS . 133 CONTENTS vii SECTION PAGE 59. GEODESIC TORSION 137 GO. SPHERICAL REPRESENTATION 141 61. RELATIONS BETWEEN A SURFACE AND ITS SPHERICAL REPRE SENTATION 143 62. HELICOIDS 146 CHAPTEE V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63. ClIRISTOFFEL SYMBOLS 152 64. THE EQUATIONS OF GAUSS AND OF CODAZZI 153 65. FUNDAMENTAL THEOREM 157 66. FUNDAMENTAL EQUATIONS IN ANOTHER FORM 160 67. TANGENTIAL COORDINATES. MEAN EVOLUTE 162 68. THE MOVING TRIHEDRAL 166 69. FUNDAMENTAL EQUATIONS OF CONDITION 168 70. LINEAR ELEMENT. LINES OF CURVATURE 171 71. CONJUGATE DIRECTIONS AND ASYMPTOTIC DIRECTONS. SPHER ICAL REPRESENTATION 172 72. FUNDAMENTAL RELATIONS AND FORMULAS 174 73. PARALLEL SURFACES 177 74. SURFACES OF CENTER 179 75. FUNDAMENTAL QUANTITIES FOR SURFACES OF CENTER . . . 181 76. SURFACES COMPLEMENTARY TO A GIVEN SURFACE 184 CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. ASYMPTOTIC LINES 189 78. SPHERICAL REPRESENTATION OF ASYMPTOTIC LINES .... 191 79. FORMULAS OF LELIEUVRE. TANGENTIAL EQUATIONS .... 193 80. CONJUGATE SYSTEMS OF PARAMETRIC LINES. INVERSIONS . . 195 81. SURFACES OF TRANSLATION 197 82. ISOTHERMAL-CONJUGATE SYSTEMS 198 83. SPHERICAL REPRESENTATION OF CONJUGATE SYSTEMS . . . 200 84. TANGENTIAL COORDINATES. PROJECTIVE TRANSFORMATIONS . 201 85. EQUATIONS OF GEODESIC LINES 204 86. GEODESIC PARALLELS. GEODESIC PARAMETERS 206 87. GEODESIC POLAR COORDINATES 207 88. AREA OF A GEODESIC TRIANGLE 209 89. LINES OF SHORTEST LENGTH. GEODESIC CURVATURE .... 212 90. GEODESIC ELLIPSES AND HYPERBOLAS . . 213 viii CONTENTS SECTION PAGE 91. SURFACES OF LIOUVILLE 214 92. INTEGRATION OF THE EQUATION OF GEODESIC LINES . . . 215 93. GEODESICS ON SURFACES OF LIOUVILLE 218 94. LINES OF SHORTEST LENGTH. ENVELOPE OF GEODESICS 220 CHAPTEK VII QUADRICS. RULED SURFACES. MINIMAL SURFACES 95. CONFOCAL QUADRICS. ELLIPTIC COORDINATES 226 96. FUNDAMENTAL QUANTITIES FOR CENTRAL QUADRICS . . . 229 97. FUNDAMENTAL QUANTITIES FOR THE PARABOLOIDS .... 230 98. LINES OF CURVATURE AND ASYMPTOTIC LINES ox QUADRICS . 232 99. GEODESICS ON QUADRICS 231 100. GEODESICS THROUGH THE UMBILICAL POINTS 236 101. ELLIPSOID REFERRED TO A POLAR GEODESIC SYSTEM . . . 236 102. PROPERTIES OF QUADRICS 239 103. EQUATIONS OF A RULED SURFACE 241 104. LINE OF STRICTION. DEVELOPABLE SURFACES 242 105. CENTRAL PLANE. PARAMETER OF DISTRIBUTION 244 106. PARTICULAR FORM OF THE LINEAR ELEMENT 247 107. ASYMPTOTIC LINES. ORTHOGONAL PARAMETRIC SYSTEMS . . 248 108. MINIMAL SURFACES 250 109. LINES OF CURVATURE AND ASYMPTOTIC LINES. ADJOINT MINI MAL SURFACES 253 110. MINIMAL CURVES ON A MINIMAL SURFACE 254 111. DOUBLE MINIMAL SURFACES 258 112. ALGEBRAIC MINIMAL SURFACES 260 113. ASSOCIATE SURFACES 263 114. FORMULAS OF SCHWARZ 264 . r CHAPTER VIII SURFACES OF CONSTANT TOTAL CURVATURE. TF-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CUR VATURE 115. SPHERICAL SURFACES OF REVOLUTION 270 116. PSEUDOSPHERICAL SURFACES OF REVOLUTION 272 117. GEODESIC PARAMETRIC SYSTEMS. APPLICABILITY 275 118. TRANSFORMATION OF HAZZIDAKIS 278 119. TRANSFORMATION OF BIANCHI 280 120. TRANSFORMATION OF BACKLUND 284 121. THEOREM OF PERMUTABILITY 286 CONTENTS ix SECTION PAGE 122. TRANSFORMATION OF LIE . . 289 123. JF-SURFACES. FUNDAMENTAL QUANTITIES 291 124. EVOLUTE OF A TF-SuitFACE . 294 125. SURFACES OF CONSTANT MEAN CURVATURE ....... 296 126. RULED IF-SuRFACES 299 127. SPHERICAL REPRESENTATION OF SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS 300 128. SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS 302 129. SURFACES WITH PLANE LINES OF CURVATURE IN ONE SYSTEM. SURFACES OF MONGE 305 130. MOLDING SURFACES 307 131. SURFACES OF JOACHIMSTHAL 308 132. SURFACES WITH CIRCULAR LINES OF CURVATURE 310 133. CYCLIDES OF DUPIN 312 134. SURFACES WITH SPHERICAL LINES OF CURVATURE IN ONE SYSTEM 314 CHAPTER IX DEFORMATION OF SURFACES 135. PROBLEM OF MINDING. SURFACES OF CONSTANT CURVATURE . 321 136. SOLUTION OF THE PROBLEM OF MINDING 323 137. DEFORMATION OF MINIMAL SURFACES 327 138. SECOND GENERAL PROBLEM OF DEFORMATION 331 139. DEFORMATIONS WHICH CHANGE A CURVE ON THE SURFACE INTO A GIVEN CURVE IN SPACE 333 140. LINES OF CURVATURE IN CORRESPONDENCE 336 141. CONJUGATE SYSTEMS IN CORRESPONDENCE 338 142. ASYMPTOTIC LINES IN CORRESPONDENCE. DEFORMATION OF A RULED SURFACE 342 143. METHOD OF MINDING 344 144. PARTICULAR DEFORMATIONS OF RULED SURFACES 345 CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN 145. REDUCED FORM OF THE LINEAR ELEMENT 351 146. GENERAL FORMULAS 353 147. THE THEOREM OF WEINGARTEN 355 148. OTHER FORMS OF THE THEOREM OF WEINGARTEN .... 357 149. SURFACES APPLICABLE TO A SURFACE OF REVOLUTION . . 362 x CONTENTS SECTION PAGE 150. MINIMAL LINES ON THE SPHERE PARAMETRIC 364 151. SURFACES OF GOURSAT. SURFACES APPLICABLE TO CERTAIN PARABOLOIDS 366 CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152. GENERAL PROBLEM 373 153. CHARACTERISTIC FUNCTION .... 374 154. ASYMPTOTIC LINES PARAMETRIC 376 155. ASSOCIATE SURFACES 378 156. PARTICULAR PARAMETRIC CURVES 379 157. RELATIONS BETWEEN THREE SURFACES S, S v S 382 158. SURFACES RESULTING FROM AN INFINITESIMAL DEFORMATION 385 159. ISOTHERMIC SURFACES 387 CHAPTER XII RECTILINEAR CONGRUENCES 160. DEFINITION OF A CONGRUENCE. SPHERICAL REPRESENTATION 392 161. NORMAL CONGRUENCES. RULED SURFACES OF A CONGRUENCE 393 162. LIMIT POINTS. PRINCIPAL SURFACES 395 163. DEVELOPABLE SURFACES OF A CONGRUENCE. FOCAL SURFACES 398 164. ASSOCIATE NORMAL CONGRUENCES 401 165. DERIVED CONGRUENCES 403 166. FUNDAMENTAL EQUATIONS OF CONDITION ........ 406 167. SPHERICAL REPRESENTATION OF PRINCIPAL SURFACES AND OF DEVELOPABLES ...... 407 168. FUNDAMENTAL QUANTITIES FOR THE FOCAL SURFACES . . . 409 169. ISOTROPIC CONGRUENCES 412 170. CONGRUENCES OF GUICIIARD 414 171. PSEUDOSPHERICAL CONGRUENCES 415 172. IT-CONGRUENCES 417 173. CONGRUENCES OF RIBAUCOUR 420 CHAPTER XIII CYCLIC SYSTEMS 174. GENERAL EQUATIONS OF CYCLIC SYSTEMS 426 175. CYCLIC CONGRUENCES 431 176. SPHERICAL REPRESENTATION OF CYCLIC CONGRUENCES . 432 CONTENTS XI SECTION PAGE 177. SURFACES ORTHOGONAL TO A CYCLIC SYSTEM 436 178. NORMAL CYCLIC CONGRUENCES 437 179. CYCLIC SYSTEMS FOR WHICH THE ENVELOPE OF THE PLANES OF THE CIRCLES is A CURVE 439 180. CYCLIC SYSTEMS FOR WHICH THE PLANES OF THE CIRCLES PASS THROUGH A POINT 440 CHAPTEE XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181. TRIPLE SYSTEM OF SURFACES ASSOCIATED WITH A CYCLIC SYSTEM 446 182. GENERAL EQUATIONS. THEOREM OF DUPIN 447 183. EQUATIONS OF LAME 449 184. TRIPLE SYSTEMS CONTAINING ONE FAMILY OF SURFACES OF REVOLUTION 451 185. TRIPLE SYSTEMS OF BIANCHI AND OF WEINGARTEN .... 452 186. THEOREM OF RIBAUCOUR 457 187. THEOREMS OF DARBOUX 458 188. TRANSFORMATION OF COMBESCURE 461 INDEX . 467 DIFFERENTIAL GEOMETRY CHAPTER I CURVES IN SPACE 1. Parametric equations of a curve. Consider space referred to fixed rectangular axes, and let (x, y, z) denote as usual the coordi nates of a point with respect to these axes. In the plane 2 = draw a circle of radius r and center (a, b). The coordinates of a point P on the circle can be expressed in the form (1) x a -{- r cos u, y = b H- r sin u, 2 = 0, where u denotes the angle which the radius to P makes with the positive o>axis. As u varies from to 360, the point P describes the circle. The quantities a, 5, r determine the position and size of the circle, whereas u determines the position of a point upon it. In this sense it is a variable or parameter for the circle. And equations (1) are called parametric equations of the circle. A straight line in space is determined by a point on it, P Q (a, 6, c), and its direction-cosines a, /3, 7. The latter fix also the sense of the line. Let P be another point on the line, and let the distance P Q P be denoted by u, which is positive or negative. The rectangular coordinates of P are then expressible in the form (2) x = a + ua, y = b + u(B, z c + wy. To each value of u there corresponds a point on the line, and the coordinates of any point on the line are expressible as in (2). These equations are consequently parametric equations of the straight line. When, as in fig. 1, a line segment PD, of constant length , per pendicular to a line OZ at D, revolves uniformly about OZ as axis, FIG. 1 2 CURVES IN SPACE and at the same time D moves along it with uniform velocity, the locus of P is called a circular helix. If the line OZ be taken for the 2-axis, the initial position of PD for the positive a>axis, and the angle between the latter and a subsequent position of PD be denoted by u, the equations of the helix can be written in the parametric form (3) x = a cos u, y a sin u, z = bu, where the constant b is determined by the velocity of rotation of PD and of translation of D. Thus, as the line PD describes a radian, D moves the distance b along OZ. In all of the above equations u is the variable or parameter. Hence, with reference to the locus under consideration, the coordi nates are functions of u alone. We indicate this by writing these equations The functions / x , / 2 , / have definite forms when the locus is a circle, straight line or circular helix. But we proceed to the gen eral case and consider equations (4), when / r / 2 , / are any func tions whatever, analytic for all values of u, or at least for a certain domain.* The locus of the point whose coordinates are given by (4), as u takes all values in the domain considered, is a curve. Equa tions (4) are said to be the equations of the curve in the parametric form. When all the points of the curve do not lie in the same plane it is called a space curve or a twisted curve ; otherwise, a plane curve. It is evident that a necessary and sufficient condition that a curve, defined by equations (4), be plane, is that there exist a linear relation between the functions, such as (5) of i + 5f 2 + c/3 + d = 0, where a, b, c, d denote constants not all equal to zero. This con dition is satisfied by equations (1) and (2), but not by (3). If u in (4) be replaced by any function of v, say (6) *fc*^(*0, equations (4) assume a new form, * E.g. in case u is supposed to be real, it lies on a segment between two fixed values; when it is complex, it lies within a closed region in the plane of the complex variable. EQUATIONS OF A CURVE 3 It is evident that the values of x, y, z, given by (7) for a value of i , are equal to those given by (4) for the corresponding value of u obtained from (6). Consequently equations (4) and (7) define the same curve, u and v being the respective parameters. Since there is no restriction upon the function </>, except that it be ana lytic, it follows that a curve can be given parametric representation in an infinity of ways. 2. Other forms of the equations of a curve. If the first of equa tions (4) be solved for w, giving u $(#), then, in terms of x as parameter, equations (7) are (8) x = x, y = F 2 (x), z = F 8 (x). In this form the curve is really defined by the last two equations, or, if it be a plane curve in the o?y-plane, its equation is in the customary form (9) y =/(*) The points in space whose coordinates satisfy the equation y = F z (x) lie on the cylinder whose elements are parallel to the 2-axis and whose cross section by the xy-pl&ne is the curve y = F 2 (x). In like manner, the equation z = F 3 (x) defines a cylinder whose elements are parallel to the #-axis. Hence the curve with the equations (8) is the locus of points common to two cylinders with perpendicular axes. Conversely, if lines are drawn through the points of a space curve normal to two planes perpendicular to one another, we obtain two such cylinders whose intersection is the given curve. Hence equations (8) furnish a perfectly gen eral definition of a space curve. In general, the parameter u can be eliminated from equations (4) in such a way that there result two equations, each of which in volves all three rectangular coordinates. Thus, (10) Qfa y, z) = 0, <S> a (a;, y, z) = 0. Moreover, if two equations of this kind be solved for y and z as functions of x, we get equations of the form (8), and, in turn, of the form (4), by replacing x by an arbitrary function of u. Hence equations (10) also are the general equations of a curve. It will be seen later that each of these equations defines a surface. 4 CURVES IN SPACE It should be remarked, however, that when a curve is defined as the intersection of two cylinders (8), or of two surfaces (10), it may happen that these curves of intersection consist of several parts, so that the new equations define more than the original ones. For example, the curve defined by the parametric equations (i) x = w, y = w 2 , z = w 3 , is a twisted cubic, for every plane meets the curve in three points. Thus, the plane ax + by -f cz + d = meets the curve in the three points whose parametric values are the roots of the e( l uation CM* + &n" + an + d = 0. This cubic lies upon the three cylinders y = x 2 , z = x 3 , y 3 = z 2 . The intersection of the first and second cylinders is a curve of the sixth degree, of the first and third it is of the sixth degree, whereas the last two intersect in a curve of the ninth degree. Hence in every case the given cubic is only a part of the curve of intersection that part which lies on all three cylinders. Again, we may eliminate u from equations (i), thus (ii) xy = z, y* = xz, of which the first defines a hyperbolic paraboloid and the second a hyperbolic- parabolic cone. The straight line y = 0, z = lies on both of these surfaces, but not on the cylinder y = x 2 . Hence the intersection of the surfaces (ii) consists of this line and the cubic. The generators of the paraboloid are defined by x = a, z = ay ; y 6, z = bx ; for all values of the constants a and 6. From (i) we see that the cubic meets each generator of the first family in one point and of the second family in two points. 3. Linear element. By definition the length of an arc of a curve is the limit, when it exists, toward which the perimeter of an inscribed polygon tends as the number of sides increases and their lengths uniformly approach zero. Curves for which such a limit does not exist will be excluded from the subsequent discussion. Consider the arc of a curve whose end points m , m a , are deter mined by the parametric values U Q and # , and let m v m 2 , , be intermediate points with parametric values u^ w 2 , . The length l k of the chord m k m k+l is =V2,r/;.^, L1 )-/v(oi 2 . = i, 2, 3 LINEAR ELEMENT 5 By the mean value theorem of the differential calculus this is equal to where f t . = w t + 0<( w *+i ~ %) < ^ < * and the primes indicate differentiation. , As denned, the length of the arc m m a is the limit of 2Z 4 , as the lengths in k m k+l tend to zero. From the definition of a definite integral this limit is equal to r a n Hence, if s denotes the length of the arc from a fixed point (u ) to a variable point (u), we have This equation gives s as a function of w. We write it (12) =(w), and from (11) it follows that which we may write in the form (14) ds 2 =dx i As thus expressed ds is called the element of length, or linear element, of the curve. In the preceding discussion we have tacitly assumed that u is real. When it is complex we take equation (11) as the definition of the length of the arc. If equation (12) be solved for u in terms of s, and the result be substituted in (4), the resulting equations also define the curve, and s is the parameter. From (11) follows the theorem : A necessary and sufficient condition that the parameter u be the arc measured from the point U = U Q is (15) /, 2 +/r+/s 2 = l- An exceptional case should be noted here, namely, /r+/ 2 "+/ a 2 =o. 6 CURVES IN SPACE Unless//, / 2 , / 3 be zero and the curve reduce to a point, at least one of the coordinates must be imaginary. For this case s is zero. Hence these imaginary curves are called curves of length zero, or minimal curves. For the present they will be excluded from the discussion. Let the arc be the parameter of a given curve and s and s + e its values for two points M(x, y, z) and M^(x^ y^ z^. By Taylor s theorem we have (17) ^ = z -f z e where an accent indicates differentiation with respect to s. Unless x 1 , y , z ! are all zero, that is, unless the locus is a point and not a curve, one at least of the lengths x l x, y^y, z l z is of the order of magnitude of e. If these lengths be denoted by &u, %, Sz, and e by 8, then we have where 1 2 denotes the aggregate of terms of the second and higher orders in 8s. Hence, as M l approaches M the ratio of the lengths of the chord and the arc MM l approaches unity ; and in the limit we have ds 2 = dx 2 + dy 2 + dz 2 . 4. Tangent to a curve. The tangent to a curve at a point M is the limiting position of the secant through M and a point M l of the curve as the latter approaches M as a limit. In order to find the equation of the tangent we take s for par ameter and write the expressions for the coordinates of M 1 in the form (17). The equations of the secant through M and M l are If each member of these equations be multiplied by e and the denominators be replaced by their values from (17), we have in the limit as M 1 approaches M y TANGENT TO A CURVE 7 If #, /3, 7 denote the direction-cosines of the tangent in conse quence of (15), we may take When the parameter u is any whatever, these equations are * ft f! ft (20) a= / Jl =, 0= . /2 = > y = -= 3 * 9 / nt *i ^ * o . /wo / /*o . /*->. /!*) * // They may also be written thus : /oi\ dx dy dz 21) a = :T = / V = T" ds ds ds From these equations it follows that, if the convention be made that the positive direction on the curve is that in which the par ameter increases, the positive direction upon the tangent is the same as upon the curve. A fundamental property of the tangent is discovered by con sidering the expression for the distance from the point M^ with the coordinates (17), to any line through M. We write the equa tion of such a line in the form (22) *= = !=* = =, a b c where a, 5, c are the direction-cosines. The distance from M l to this line is equal to (23) {[(bx>- ay )e + \(bx"- ay")e* + ] 2 bz )e + -] 2 + [(az 1 - cx )e + . -] 2 }*. Hence, if MM l be considered an infinitesimal of the first order, this distance also is of the first order unless in which case it is of the second order at least. But when these equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent is nearest to the curve. * Whenever the functions x , y , z appear in a formula it is understood that the arc s is the parameter ; otherwise we use /{, / 2 , / 3 , indicating by accents derivatives with respect to the argument u. 8 CURVES IN SPACE 5. Order of contact. Normal plane. When the curve is such that there are points for which (24) ^=4 x y z the distance from M l to the tangent is of the third order at least. In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in gen eral, the tangent to a curve has contact of the wth order at a point, if the following conditions are satisfied for n = 2, , n 1, and n : x w i(0 jyV V 5>V"y < 25 ) ^=^=^rr When the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations fl ~jTf // -f(.n-\) f (rt-1) f(n-l) Jl J% J% J\ J 2, Ja The plane normal to the tangent to a curve at the point of contact is called the normal plane at the point. Its equation is (26) (X x) a + (Y y} ft + (Z z) 7 = 0, where a, /3, 7 have the values (20). EXAMPLES 1. Put the equations of the circular helix (3) in the form (8). 2. Express the equations of the circular helix in terms of the arc measured from a point of the curve, and show that the tangents to the curve meet the elements of the circular cylinder under constant angle. 3. Show that if at every point of a curve the tangency is of the second order, the curve is a straight line. 4. Prove that a necessary and sufficient condition that at the point (x , 2/o) of the plane curve y f(x) the tangent has contact of the nth order is/"(x ) = / "(BO) = . . . r=/()(z ) = ; also, that according as n is even or odd the tangent crosses the curve at the point or does not. 5. Prove the following properties of the twisted cubic : (a) Of all the planes through a point of the cubic one and only one meets the cubic in three coincident points ; its equation is 3 u*x - 3 uy + z - w 3 = 0. (6) There are no double points, but the orthogonal projection on a plane has a double point. (c) Four planes determined by a variable chord of the cubic and by each of four fixed points of the curve are in constant cross-ratio. FIKST CUKVATUKE 9 6. Curvature. Radius of first curvature. Let Jf, M l be two points of a curve, As the length of the arc between these points, and A0 the angle between the tangents. The limiting value of A0/As as M l approaches Jf, namely dd/ds, measures the rate of change of the direction of the tangent at M as the point of con tact moves along the curve. This limiting value is called the first curvature of the curve at M, and its reciprocal the radius of first curvature ; the latter will be denoted by p. In order to find an expression for p in terms of the quantities defining the curve, we introduce the idea of spherical representa tion as follows. We take the sphere * of unit radius with center at the origin and draw radii parallel to the positive directions of- the tangents to the curve, or such a portion of it that no two tangents are parallel. The locus of the extremities is a curve upon the sphere, which is in one-to-one correspondence with the given curve. In this sense we have a spherical representation, or spherical indicatrix, of the curve. The angle A# between the tangents to the curve at the points M, M^ is measured by the arc of the great circle between their representative points m, m l on the sphere. If ACT denotes the length of the arc of the spherical indicatrix between m and m^ then by the result at the close of 3, dO v A<9 = lim = 1. da Ao- Hence we have (27) ! = , p ds where da- is the linear element of the spherical indicatrix. The coordinates of m are the direction-cosines a, /3, 7 of the tangent at M\ consequently When the arc s is the parameter, this formula becomes (28) ^.jji+yw+gw * Hereafter we refer to this as the unit sphere. 10 CURVES IN SPACE However, when the parameter is any whatever, u, we have from (12), (13), (20), and W-fifl+JSJf*flfr Hence we find by substitution (30) which sometimes is written thus : \ _(d*x}* f- The sign of p is not determined by these formulas. We make the convention that it is always positive and thus fix the sense of a displacement on the spherical indicatrix. 7. Osculating plane. Consider the plane through the tangent to a curve at a point M and through a point M^ of the curve. The limiting position of this plane as M l approaches M is called the osculating plane at M. In deriving its equation and thus establish ing its existence we assume that the arc s is the parameter, and take the coordinates of M l in the form (17). The equation of a plane through M (x, y, z) is of the form (32) (X- x)a + (Y- y)l + (Z-z)c = Q, X, Y", Z being the current coordinates. When the plane passes through the tangent at Jf, the coefficients a, >, c are such that (33) x a + y b + z c = 0. If the values (17) for a; , y^ z^ be substituted in (32) for X, F, Z, e* and the resulting equation be divided by , we get where 77 represents the aggregate of the terms of first and higher orders in e. As M 1 approaches Jf, 77 approaches zero, and in the limit we have (34) x"a + y"b + z"c = 0. OSCULATING PLANE 11 Eliminating a, >, c from equations (32), (33), (34) we obtain, as the equation of the osculating plane, (35) X-x x 1 x" Y-y y y" Z-z = 0. From this we find that when the curve is defined by equations (4) in terms of a general parameter w, the equation of the osculating plane is x _ x Y (36) y z _ z // /," fi The plane defined by either of these equations is unique except when the tangent at the point has contact of an order higher than the first. In the latter case equations (33), (34) are not independent, as follows from (24); and if the contact of the tangent is of the wth order, the equations ^ + ^ + ^ G = 0? for all values of r up to and including n are not independent of one another. But for r = n + ~\., this equation and (33) are inde pendent, and we have as the equation of the osculating plane at this singular point, X-x x Y- y Z-z = 0. When a curve is plane, and its plane is taken for the rry-plane, the equation (35) reduces to Z = 0. Hence the osculating plane of a plane curve is the plane of the latter, and consequently is the same for all points of the curve. Conversely, when the osculating plane of a curve is the same for all its points, the curve is plane, for all the points of the curve lie in the fixed osculating plane. The equation of the osculating plane of the twisted cubic (2) is readily reducible to where JT, F, Z are current coordinates. From the definition of the osculating plane and the fact that the curve is a cubic, it follows that the osculating plane meets the curve only at the point of osculation. As equation (i) is a cubic in w, it follows that through a point (o, 2/o, ZQ) not on tne cnrve there pass three planes which osculate the cubic. Let MI, w 2 , u 3 denote the parameter values of these points. Then from (i) we have = 3 XG, 3 ?/o, 2t i \ n 12 CURVES IN SPACE By means of these relations the equation of the plane through the corresponding three points on the cubic is reducible to (X - XQ) 3 7/0 - ( Y - y ) 3 x + (Z - z ) = 0. This plane passes through the point (x , 2/0, ZQ) 5 hence we have the theorems : The points of contact of the three osculating planes of a twisted cubic through a point not on the curve lie in a plane through the point. The osculating planes at three points of a twisted cubic meet in a point which lies in the plane of the three points. By means of these theorems we can establish a dual relation in space by mak ing a point correspond to the plane through the points of osculation of the three osculating planes through the point, and a plane to the point of intersection of the three planes which osculate the cubic at the points where it is met by the plane. In particular, to a point on the cubic corresponds the osculating plane at the point, and vice versa. 8. Principal normal and binormal. Evidently there are an in finity of normals to a curve at a point. Two of these are of par ticular interest : the normal, which lies in the osculating plane at the point, called the principal normal; and the normal, which is perpendicular to this plane, called the binormal. If the direction-cosines of the binormal be denoted by X, />t, z>, we have from (35) X : / : v = (y z"- z y") : (z x"~ z z") : (* / - y x"). In consequence of the identity the value of the common ratio is reducible by means of (19) and (28) to p.* We take the positive direction of the binormal to be such that this ratio shall" be -f- p ; then (37) \ = p(y f z"-z y"), ^ = P (z x"~ x z"), v = p(x y When the parameter u is general, these formulas are (38) x= 282 or in other form : /oof dycPz dzcfy dzd*xdxd?z dxd y dyd^x ~ P ~~df~ ^~ P ~ ds 3 ~ P ds* *For SV / = 0, as is seen by differentiating 2x"*= 1 with respect to s. PRINCIPAL NORMAL AND BINORMAL 13 By definition the principal normal is perpendicular to both the tangent and binormal. We make the convention that its positive direction is such that the positive directions of the tangent, prin cipal normal and binormal at a point have the same mutual ori entation as the positive directions of the x-, y-, z-axes respectively. These directions are represented in fig. 2 by the lines MT, MC, MB. Hence, if Z, m, n, denote the direc tion-cosines of the principal normal, we have* (39) m /JL =4-1, FIG. 2 from which it follows that I a = mv nu, ft n\ Iv, 7 = Z/i m\, I = fjij vft^ m = va \7, n = Xp yuo:, \ = ftn 777*, /Ji = yl #n, i^ = am ftl. , Substituting the values of #, /3, 7; X, /u-, i^ from (19) and (37) in the expressions for , m, w, the resulting equations are reducible to Hence, when the parameter u is general, we have (42) l=-(W- or in other form, _ 2 2 dzd 2 s In consequence of (29) equations (42) may be written: da dft dj ds ds ds or by means of (27), da dft dy (43) m do- da- Hence the tangent to the spherical indicatrix of a curve is parallel to the principal normal to the curve and has the same sense. *C. Smith, Solid Geometry, llth ed., p. 31. 14 CURVES IN SPACE 9. Osculating circle. Center of first curvature. We have defined the osculating plane to a curve at a point M to be the limiting position of the plane determined by the tangent at M and by a point M l of the curve, as the latter approaches M along the curve. We consider now the circle in this plane which has the same tan gent at M as the curve, and passes through M { . The limiting posi tion of this circle, as M l approaches 7I/, is called the osculating circle to the curve at M. It is evident that its center C is on the prin cipal normal at M. Hence, with reference to any fixed axes in space, the coordinates of (7 , denoted by X Q , F , Z Q , are of the form X =x + rl, Y y -f rm, Z^=z + rn, where the absolute value of r is the radius of the osculating circle. In order to find the value of r, we return to the consideration of the circle, when M l does not have its limiting position, and we let X, F, Z\ Zj, m^ n^ r^ denote respectively coordinates of the cen ter of the circle, the direction-cosines of the diameter through M and the radius. If x v y v z l be the coordinates of M^ they have the values (17), and since M l is on the circle, we have rl = 2(A - xtf = 2(7-^- ex - i e*x". -) 2 . If we notice that ^x ^ = 0, and after reducing the above equation divide through by e 2 , we have 1-r^Z/ +*? = (), where 77 involves terms of the first and higher orders in e. In the limi-t r l becomes r, ^x\ becomes 2z 7, that is - , and this equation reduces to so that r is equal to the radius of curvature. On this account the osculating circle is called the circle of curvature and its center the center of first curvature for the point. Since r is positive the center of curvature is on the positive half of the principal normal, and consequently its coordinates are (44) X =x + pl, Y =y + pm, Z Q = z + pn. CENTER OF CURVATURE 15 The line normal to the osculating plane at the center of curva ture is called the polar line or polar of the curve for the corre sponding point. Its equations are /45\ X-x-pl = Y-y-pm = Zz pn ^ \ JJL v In fig. 2 C represents the center of curvature and CP the polar line for M. A curve may be looked upon as the path of a point moving under the action of a system of forces. From this point of view it is convenient to take for parameter the time which has elapsed since the point passed a given position. Let t denote this parameter. As t is a function of 8, we have dx _dx ds _ ds dy _ ds dz _ ds ~ ds~dt~ C *dt dt~ ~dt ~dt~ y ~dt Hence the rate of change of the position of the point with the time, or its velocity, may be represented by the length laid off on the tangent to the curve. In like dt manner, by means of (41), we have d^ _ d?s n /ds\ 2 ~ 7 From this it is seen that the rate of change of the velocity at a point, or the acceleration, may be represented by a vector in the osculating plane at the point, through the latter and whose components on the tangent and principal normal d*s , 1 /dY - and - I ) df* P \dtj EXAMPLES 1. Prove that the curvature of a plane curve defined by the equation M (x, y)dx cy ex p (J/ 2 + N 2. Show that the normal planes to the curve, x a sin 2 it, y = a sin u cos w, z = a cos M, pass through the origin, and find the spherical indicatrix of the curve. 3. The straight line is the only real curve of zero curvature at every point. 4. Derive the following properties of the twisted cubic : (a) In any plane there is one line, and only one, through which two osculating planes can be drawn. (6) Four fixed osculating planes are cut by the line of intersection of any two osculating planes in four points whose cross-ratio is constant. (c) Four planes through a variable tangent and four fixed points of the curve are in constant cross-ratio. (d) What is the dual of (c) by the results of 7 ? 16 CURVES IN SPACE 5. Determine the form of the function so that the principal normals to the curve x = w, y = sin w, z - <f> (u) are parallel to the yz-plane. 6. Find the osculating plane and radius of first curvature of x a cos u -f 6 sin w, y = a sin u + 6 cos w, z = c sin 2 u. 10. Torsion. Frenet-Serret formulas. It has been seen that, un less a curve be plane, the osculating plane varies as the point moves along the curve. The change in the direction depends evidently upon the form of the curve. The ratio of the angle A^ between the binormals at two points of the curve and their curvi linear distance As expresses our idea of the mean change in the direction of the osculating plane. And so we take the limit of this ratio, as one point approaches the other, as the measure of the rate of this change at the latter point. This limit is called the second curvature, or torsion, of the curve, and its inverse the radius of second curvature, or the radius of torsion. The latter will be denoted by r. In order to establish the existence of this limit and to find an expression for it in terms of the functions defining the curve, we draw radii of the unit sphere parallel to the positive binormals of the curve and take the locus of the end points of these radii as a second spherical representation of the curve. The coordinates of points of this representative curve on the sphere are X, /*, v. Pro ceeding in a manner similar to that in 6, we obtain the equation (46) i_ r 2 ds* where dcr l is the linear element of the spherical indicatrix of the binormals. In order that a real curve have zero torsion at every point, the cosines X, /*, v must be constant. By a change of the fixed axes, which evidently has no effect upon the form of the curve, the cosines can be given the values X = 1, /* = v = 0. It follows from (40) that a = 0, and consequently x const. Hence a necessary and sufficient condition that the torsion of a real curve be zero at every point is that the curve be plane. In the subsequent discussion we shall need the derivatives with respect to s of the direction-cosines a, & 7; I, m, w; X, p, v. We deduce them now. From (41) we have (4T) a =i, /3 =, y-. FRENET-SERRET FORMULAS 17 In order to find the values of X , /* , i/, we differentiate with respect to s the identities, X 2 +At2+z , 2 = 1? a\ + /A + 7* = 0, and, in consequence of (47), obtain XX + fjLfj, 1 + vv 1 = 0, \ + /V + yv r = 0. From these, by (40), follows the proportion X : fjL f :v r =l:m:n, and the factor of proportionality is 1/r, as is seen from (46). The algebraic sign of r is not determined by the latter equation. We fix its sign by writing the above proportion thus : (48) V ,: - - A = > l*> = t V = T T T If the identity I = ^7 vfi be differentiated with respect to s the result is reducible by (40), (47), and (48) to (49) I Similar expressions can be found for m and n . Gathering to gether these results, we have the following formulas fundamental in the theory of twisted curves, and called the Frenet-Serret formulas : (50) *-i, / *-5, y = W p p /> l =-(- +-Y = -(-- ^ \p T 7 V T % , I ,_m / n X = j / " T T T Y ^-/l+2), v \^ v As an example, we derive another expression for the torsion. If the equation \ = p(y z " z y") be differentiated with respect to s, the result may be written If this equation and similar ones for WI/T, rz/r be multiplied by ?, wz, n respectively and added, we have, in consequence of (50) and (41), x (51) y y" z z" x " y " z " 18 CUKVES IN SPACE The last three of equations (50) give the rate of change of the direction-cosines of the osculating plane of a curve as the point of osculation moves along the curve. From these equations it follows that a necessary and sufficient condition that this rate of change at a point be zero is that the values of s for the point make the determinant in equation (51) vanish. At such a point the osculat ing plane is said to be stationary. 11. Form of curve in the neighborhood of a point. The sign of torsion. We have made the convention that the positive directions of the tangent, principal normal, and binormal shall have the same relative orientation as the fixed x-, y-, 2-axes respectively. When we take these lines at a point M Q for axes, the equations of the curve can be put in a very convenient form. If the coordinates be ex pressed in terms of the arc measured from M^ we have from (19) and (41) that for s = P When the values of I and X from (41) and (37) are substituted in the fourth of equations (50), we obtain (5 2) x " = ---- (y z" - z y") - - x". p r p From this and similar expressions for y " and z" 1 we find that for s = -i f -i P* p 2 P T Hence, by Maclauriri s theorem, the coordinates #, y, z can be ex pressed in the form / -i (^%\ < v = s 3 -f---, \<JO I v Ct J 2 2 p b p z s -f- , 6 pr where p and r are the radii of first and second curvature at the point s = 0, and the unwritten terms are of the fourth and higher powers in s. From the last of these equations it is seen that for sufficiently small values of 8 the sign of z changes with the sign of s unless THE SIGN OF TORSION 19 I/T = at M . Hence, unless the osculating plane is stationary at a point, the curve crosses the plane at the point.* Furthermore, when a point moves along a curve in the positive direction, it passes from the positive to the negative side of the osculating plane at a point, or vice versa, according as the torsion at the latter is positive or negative. In the former case the curve is said to be sinistrorsum, in the latter dextrorsum. As another consequence of this equation, we remark that as a variable point M on the curve approaches Jf , the distance from M to the osculating plane at M Q is of the third order of magnitude in comparison with MM Q . By means of the other equations (53) we find that the distance to any other plane through M is of the second order at most. Hence we have the theorem : The osculating plane to a twisted curve at an ordinary point is crossed by the curve, and of all the planes through the point it lies nearest to the curve. From the second of (53) it is seen that y is positive for suffi ciently small values of , positive or negative. Hence, in the neighborhood of an ordinary point, the curve lies entirely on one side of the plane determined by the tangent and binomial on the side of the positive direction of the principal normal. These properties of a twisted curve are discovered, likewise, from a consideration of the projections upon the coordinate planes of the approximate curve, whose equations consist of the first terms in (53). The projection on the osculating plane is the parabola x = *, y = s 2 /2 p, whose axis is the principal normal to the curve. On the plane of the tangent and binomial it is the cubic x = s, z = s 8 /6 pr, which has the tangent to the curve for an inflectional tangent. And the curve projects upon the plane of the binormal and principal normal into the semi- cubical parabola y = s 2 /2 /o, z= s 3 /Q pr, with the latter for cuspidal tangent. These results are represented by the following figures, which picture the pro jection of the curve upon the osculating plane, normal plane, and the plane of the tangent and binormal. In the third figure the heavy line corresponds to the case where r is positive and the dotted line to the case where r is negative. *This result can be derived readily by geometrical considerations. 20 CUEVES IN SPACE The preceding results serve also to give a means of determining; ue variation in the osculating plane as the point moves along the curve. By r ns of (50) the direction-cosines X, /u,, v can be given the form where the subscript null indicates the value of a function for s = and the un written terms are of the second and higher terms in s. If the coordinate axes are those which lead to (53), the values of X, p, v for the point of parameter 5s are X = =-, v = \ TO to within terms of higher order, and consequently the equation of this osculating plane at this point MI is . Y- + Z = 0. TO If we put Y = po, we get the z-coordinate of the point in which this plane is cut by the polar line for the point s = ; it is po5s/T . Hence, according as T O is positive or negative at Jf, the osculating plane at the near-by point MI cuts the polar line for M on the negative or positive side of the osculating plane at M . 12. Cylindrical helices. As another example of the use of formulas (50) we derive several properties of cylindrical helices. By definition, a cylindrical helix is a curve which lies upon a cylinder and cuts the elements of the cylinder under constant angle. If the axis of z be taken parallel to the elements of the cylinder, we have 7 = const. Hence, from (50), from which it follows that the cylindrical helices have the following properties : The principal normal is perpendicular to the element of the cylinder at the point, and consequently coincides with the normal to the cylinder at the point ( 22). The radii of first and second curvature are in constant ratio. Bertrand has established the converse theorem : Every curve whose radii of first and second curvature are in constant ratio is a cylindrical helix. In order to prove it, we put T = icp, and remark from (50) that dv da_d\ ds ds dp _ dfj. dy _ ds~ ds ds ds from which we get a K\ + a, /3 = */* + 6, 7 = KV + c, where a, 6, c are constants. From these equations we find a 2 + 52 + C 2 _ i + K ^ aa + bp + cy = CYLINDRICAL HELICES 21 Hence the ta -ents to the curve make the constant angle cos- * =; with the lines whose lin 3n-cosines are V * Consequently the curve is a cylindrical Vl + K 2 helix, and t .u e ts of the helix have the above direction. EXAMPLES 1. Find the length of the curve x = a (u sin w), y = a cos w, between the points for which u has the values TT and IT ; show that the locus of the center of curva ture is of the same form as the given curve. 2. Find the coordinates of the center of curvature of x = a cos M, y = a sin w, z a cos 2 u. 3. Find the radii of curvature and torsion of x = a (u sinw), y = a (1 cosw), z = bu. 4. If the principal normals of a curve are parallel to a fixed plane, the curve is a cylindrical helix. 5. Show that the curve x e u , y = er M , z V% u is a cylindrical helix and that the right section of the cylinder is a catenary ; also that the curve lies upon a cylin der whose right section is an equilateral hyperbola. Express the coordinates in terms of the arc and find the radii of first and second curvature. 6. Show that if 6 and denote the angles which the tangent and binormal to a sin 6 dd r curve make with a fixed line in space, then - = - - sin </> d<p p 7. When two curves are symmetric with respect to the origin, their radii of first curvature are equal and their radii of torsion differ only in sign. 8. The osculating circle at an ordinary point of a curve has contact of the sec ond order with the latter ; and all other circles which lie in the osculating plane and are tangent to the curve at the point have contact of the first order. 9. A necessary and sufficient condition that the osculating circle at a point have contact of the third order is that p = and I/T = at the point ; at such a point the circle is said to superosculate the curve. 10. Show that any twisted curve may be defined by equations of the form where p and r are the radii of first and second curvature at the point s 0. 11. When the equations of a curve are in the form (4), the torsion is given by /I /2 /3 f fff f/f f where has the significance of equation (12). 22 CURVES IN SPACE 12. The locus of the centers of curvature of a twisted curve of constant first curvature is a curve of the same kind. 13. When all the osculating planes of a curve pass through a fixed point, the curve is plane. 14. Determine f(u) so that the curve x = a cosw, y = a sin w, z =f(u) shall be plane. What is the form of the curve ? 13. Intrinsic equations. Fundamental theorem. Let C^ and C z be two curves defined in terms of their respective arcs s, and let points upon each with the same values of s correspond. We assume, furthermore, that at corresponding points the radii of first curva ture have the same value, and also the radii of second curvature. We shall show that C l and C z are congruent. By a motion in space the points of the two curves for which s = can be made to coincide in such a way that the tangents, principal normals, and binomials to them at the point coincide also. Hence if we use the notation of the preceding sections and indicate by subscripts 1 and 2 the functions of C l and C 2 , we have, when s = 0, (54) x l = x z , a l = a z , ^ = Z 2 , \ = \ z , and other similar equations. The Frenet-Serret formulas for the two curves are ds ds r ds = = > = ( -| -- I -=- = -> ds p ds \p T/ ds r the functions without subscripts being the same for both curves. If the equations of the first row be multiplied by 2 , Z 2 , X 2 respec tively, and of the second row by a^ l^ X : , and all added, we have (55) ^(,.H*,+ x i\)=0. and consequently a^ + IJ 2 + \\ = const. This constant is equal to unity for s = 0, as is seen from (54), and hence for all values of s we have INTRINSIC EQUATIONS 23 Combining this equation with the identities we obtain (a t - a 2 ) 2 + ft - >) 2 + (X, - X 2 ) 2 = 0. Hence a l = a# ^ = Z 2 , X t = X 2 . Moreover, since in like manner & = & 7i = 7 2 we nave 1(^-^=0, ^-^=0, I <*,-,) = o. Consequently the differences 2^ # 2 , y^y^ z l z 2 are constant. But for s = they are zero, and so we have the theorem : Two curves whose radii of first and second curvature are the same functions of the arc are congruent. From this it follows that a curve is determined, to within its position in space, by the expressions for the radii of first and second curvature in terms of the arc. And so the equations of a curve may be written in the form (56) />=/,<), T =/,(.). They are called its intrinsic equations. We inquire, conversely, whether two equations (56), in which f^ and/ 2 are any functions whatever of a parameter s, are intrinsic equations of a curve for which s is the length of arc. In answering this question we show, in the first place, that the equations /trrv du v dv /u w\ dw v (pi) - = -, = /--| , __ ds p ds \p T/ ds r admit of three sets of solutions, namely : (58) u = a, v = 1, w = \; u = fi, v = m, w = /JL ; u y^ v = n, w = v; which are such that for each value of s the quantities a, fi, 7; /, 7?z, n ; X, /-i, v are the direction-cosines of three mutually perpen dicular lines. In fact, we know * that a system (57) admits of a unique set of solutions whose values for s = are given arbitra rily. Consequently these equations admit of three sets of solutions * Picard, Tralte d Analyse, Vol. II, p. 313; Goursat, Cours d Analyse Mathematique, Vol. II, p. 356. 24 CURVES IN SPACE whose values for s = are 1, 0, ; 0, 1, ; 0, 0, 1 respectively. By an argument similar to that applied to equation (55) we prove that for all values of s the solutions (58) satisfy the conditions (59) aft + Im + \p = 0, 7 + mn + pv = 0, ya + nl + v\ = 0. In like manner, since it follows from (57) that du dv dw A u-r+v + w = (), as as as we prove that these solutions satisfy the conditions ( 60) o a +Z a +X a = l, /3 2 + m 2 +/x 2 = l, y+n +i/^l. But the conditions (59), (60) are equivalent to (40), and conse quently the three sets of functions a, & 7; Z, TH, w; X, /z, v are the direction-cosines of three mutually perpendicular lines for all values of s. Suppose we have such a set of solutions. For the curve (61) x I ads, y= I fids, z=* I yd*, the functions a, /3, 7 are the direction-cosines of the tangent, and since ds* = dx 2 + dy* + dz 2 , s measures the arc of the curve. From (61) and the first of (57) we get d?x_l_ ofy^m d^z^n. /d*x\* /^>\ 2 /^\ 2 = 1 ds*~p ds 2 ~ p* df~~p W/ W/ W/ p* Hence if p be positive for all values of s, it is the radius of curva ture of the curve (61), and Z, m, n are the direction-cosines of the principal normal in the positive sense. In consequence of (40) the functions X, yu., v are the direction-cosines of the binomial; hence from (50) and the third of (57) it follows that r is the radius of torsion of the curve. Therefore we have the following theorem fundamental in the theory of curves : Given any two analytic functions, f^s), f 2 (s), of which the former is positive for all values of s within a certain domain ; there exists a curve for which p =/j(s), r =/ 2 (), and s is the arc, for values of s in the given domain. The determination of the curve reduces to the find ing of three sets of solutions of equations (57), satisfying the conditions (59), (60), and to quadratures. KICCATI EQUATIONS 25 We proceed now to the integration of equations (57). Since each set of integrals of the desired kind must satisfy the relation (62) u 2 we introduce with Darboux * two functions cr and &), defined by (63) 1 w u iv u iv 1 + w \ w u + iv ft) It is evident that the functions cr and - are conjugate imaginaries. Solving for u, v, w, we get 1 o-ft) ,1 + o-ft) cr + co (64) u = 1 v = i i w = If these values be substituted in equations (57), it is found that the functions cr and co are solutions of the equation (65) Miieip. ds 2 r p 2 r And conversely, any two different solutions of (65), when substi tuted in (64), lead to a set of solutions of equations (57) satisfying the relation (62). Our problem reduces then to the integration of equation (65). 14. Riccati equations. Equation (65) may be written (66) ^ = L + 2 MO + NP, where L, M, N are functions of s. This equation is a generalized form of an equation first studied by Riccati, f and consequently is named for him. As Riccati equations occur frequently in the theory of curves and surfaces, we shall establish several of their properties. Theorem. When a particular integral of a Riccati equation is known, the general integral can be obtained by two quadratures. * Lemons sur la Thdorie Generate des Surfaces, Vol. I, p. 22. We shall refer to this treatise frequently, and for brevity give our references the form Darboux, I, 22. t Cf . Forsyth, Differential Equations, chap, v ; also Cohen, Differential Equations, pp. 173-177. 26 CURVES IN SPACE Let O l be a particular integral of (66). If we put = l/c -f 0^ the equation for the determination of </> is (67) - + 2(M+2WJ<l>+N**Q. As this equation is linear and of the first order, it can be solved by two quadratures. Since the general integral of (67) is of the form <=/ 1 (s)+ 0/2(3), where a denotes the constant of integration, the general integral of equation (66) is of the form -Sri- where P, (), R, S are functions of s. Theorem. When two particular integrals of a Riccati equation are known, the general integral can be found by one quadrature. Let l and 2 be two solutions of equation (66). If we effect the substitution 6 -- \-6 , the equation in ^ is If this equation and (67) be multiplied by 1/^r and !/< respec tively, and subtracted, the resulting equation is reducible to ty/<l>)=N(0 1 2 )^/<t>. Consequently the general integral of cts (66) is given by 00, -v|r fv(0 t -0c/ < 69 > where a is the constant of integration. Since equation (68) may be looked upon as a linear fractional substitution upon a, four particular solutions V 2 , # 3 , 4 , corre sponding to four values a v a z , a 3 , a of a, are in the same cross-ratio as these constants. Hence we have the theorem : The cross-ratio of any four particular integrals of a Riccati equation is constant. From this it follows that if three particular integrals are known, the general integral can be obtained without quadrature. DETERMINATION OF COORDINATES 27 15. The determination of the coordinates of a curve defined by its intrinsic equations. We return to the consideration of equation (65) and indicate by a.P+O bP+Q (70) o\ = -^ -^i (o i = T> -v (i = l,2,3) six particular integrals of this equation. From these we obtain three sets of solutions of equations (57), namely (71) *1 I and similar expressions in cr 2 , &> 2 ; <7 3 , &> 3 respectively for /8, m, ft; 7, n, ZA These expressions satisfy the conditions (60). In order that (59) also may be satisfied we must have CT ft),, ft), ft), which is reducible to (72) = -1. z. _ 1 9 q * ^ A/ JL ^ ^ O 5 X ; 2 , J 2 ; a 3 , Hence each two of the three pairs of constants form a harmonic range. When the values (70) for <T., &) t . are substituted in the expressions for a, /3, 7, it is found that (73) 7 = ,1+a, a where, for the sake of brevity, we have put (74) RS PQ PS-QR 28 CURVES IN SPACE The coefficients of U, F, and W in (73) are of the same form as the expressions (71) for cr, Z, X ; , ra, /* ; 7, n, z>. Moreover, the equations of condition (59) are equivalent to (72). Hence these coefficients are the direction-cosines of three fixed directions in space mutually perpendicular to one another. If lines through the origin of coordinates parallel to these three lines be taken for a new set of axes, the expressions for #, /3, 7 with reference to these axes reduce to U, V, ^respectively.* These results may be stated thus : If the general solution of equation (65) be (68) " = STV aii -f- o the curve whose radii of first and second curvature are p and T respectively is given by f\ T)v I/ w-K ) /~r> ci RS-- PS- QR It must be remarked that the new axes of coordinates are not necessarily real, so that when it is important to know whether the curves are real it will be advisable to consider the general formulas (73). An example of this will be given later. We shall apply the preceding results to several problems. When the curve is plane the torsion is zero, and conversely. For this case equa tion (65) reduces to = - of which the general integral is ds p -if = ae 1 J P = ae~ i<r , where a is an arbitrary constant, and by (27) <r is the measure of the arc of the spherical indicatrix of the tangent. This solution is of the form (08), with O -I Therefore the coordinates are given by (75) x=Ccos<rds, y=Csir\<rds, 2 = Hence the coordinates of any plane curve can be put in this form. * This is the same thing as taking ai &i = l, c*2= b^ i, 3 oo, &3=0. t Scheffers, Anwendung der Differential und Integral Rechnung auf Geometrie, Vol. I, p. 219. Leipsic, 1902. DETERMINATION OF COORDINATES 29 We have seon that cylindrical helices are characterized by the property that the radii of first und second curvature are in constant ratio. If we put T = pc, equation (65) may bj written ^ = _l(l_2c0-02). ds 2 T V Two particular integrals are the roots of the equation 2 + 2 cd - 1 = 0. These roots are real and unequal if c is real ; we consider only this case, and put (76) e l = -c -Vc 2 + 1, 2 = - c + Vc2 + l, 01&2 = - 1. From (69) it follows that the general solution of the above equation is (77) where we have put (78) Since <r and -- in (63) are conjugate imaginary, if we take then a and 6 must be such that ae it _ i & oe - my _ 0j where 6 denotes the conjugate imaginary of 6. This reduces, in consequence of (76) to n 4- M (7 9 , ; , * = -|=-* One solution of this is given by taking <x> and for a and 6; we put a s = GO, 6 3 = 0. If these values be substituted in (72), we get a 4- k - 0, where i = 1, 2. So that equation (79) becomes && = 0^, where i = 1, 2. The solutions of this equation are 61 = 1? 6 2 = - i0i- From (77) P = e0 2 , Q = - 0i, R = e r , S = - 1, so that W==- Vc 2 + l When the foregoing values are substituted in (73), and the resulting values of a, /3, 7 in (61), we get (80) x- C - Ccoslds, y= - fstafdt, g = = Vc 2 + 1 J Vc 2 + 1 J Vc 2 + 1 From the la ie expressions we find that the tangent to the curve makes a constant ang c -v\ the z-axis the direction of the elements of the cylinder. And the cross-section of the cylinder is defined by Xi = fcos t dsi, y\ = J sin t dsi, where Si denotes the arc of this section measured from a point of it. If pi denotes the radius of curvature of the right section, we find that pc 2 = pi(c 2 + !) 30 CURVES IN SPACE EXAMPLES 1. Find the coordinates of the cylindrical helix whose intrinsic equations are p = T = S. 2. Show that the helix whose intrinsic equations are p = T = (s 2 + 4)/V2 lies upon a cylinder whose cross-section is a catenary. 3. Establish the following properties for the curve with the intrinsic equations p = as, r = 6s, where a and b are constants : (a) the Cartesian coordinates are reducible to x=Ae ht cos, y = Ae ht sm t, z Be hi , where J_, B, h are functions of a and 6 ; (6) the curve lies upon a circular cone whose axis coincides with the z-axis and cuts the elements of the cone under constant angle. 16. Moving trihedral. In 11 we took for fixed axes of refer ence the tangent, principal normal, and binormal to a curve at a point M Q of it, and expressed the coordinates of any other point of the curve with respect to these axes as power series in the arc s of the curve between the two points. Since M is any point of the curve, there is a set of such axes for each of its points. Hence, instead of considering only the points whose locus is the curve, we may look upon the moving point as the intersection of three mutually perpendicular lines which move along with the point, the whole figure rotating so that in each position the lines coin cide with the tangent, principal normal, and binormal at the point. We shall refer to such a configuration as the moving trihedral. In the solution of certain problems it is of advantage to refer the curve to this moving trihedral as axes. We proceed to the con sideration of this idea. With reference to the trihedral at a point Jf, the direction- cosines of the tangent, principal normal, and binormal at M have the values a=l, /3 = 7 = 0; I = 0, m=l, n = 0; X = p = 0, i/ = l. As the trihedral begins to move, the rates of change of these functions with s are found from the Frenet formulas (50) to have the values da _ ft d{$ _ 1 dy _ ft dl _ 1 dm _ ~ = o = , o j - 5 u, ds ds p ds ds p ds dn 1 d\ n dfJL 1 dv ft = j r~ = "i ~T~ = ~ 7~ = U * ds r ds ds r ds MOVING TRIHEDRAL 31 Let f, 77, f denote coordinates referring to the axes at Jf, and f, 77 , f those with reference to the axes at Jf , and let JfJf = As (see fig. 4). Since the rate of change of a is zero and a =1 at Jf, the cosine of the angle between the - and f -axes is 1 to within terms of higher than the first order in As. Likewise the cosine of the angle between the f- and Tj -axes is As/p. We calculate the cosines of the angles between all the axes, and the results may be tabulated as follows: f v S (81) As ., _ As FIG. 4 Let P be a point whose coordinates with respect to the trihedral at M are f, 77, f. Suppose that as Jf describes the given curve (7, P describes a path T. It may happen that in this motion P is fixed relatively to the moving trihedral, but in general the change in the position of P will be due not only to the motion of the trihedral but also to a motion relative to it. In the latter general case, if P denotes the point on T corresponding to M on (7, the coordinates of P relative to the axes at M and M may be written ?4Af Tj+A^, f+A^; f + A 2 f, 17 +A^, f+A 2 ?. Thus A 2 # indicates the variation of a function relative to the moving trihedral, and A^ the variation due to the latter and to the motion of the trihedral. To within terms of higher order the coordinates of M are (As, 0, 0) with respect to the axes at Jf, and with the aid of (81) the equations of the transformation of coordinates with respect to the two axes are expressible thus : . 32 CURVES IN SPACE These reduce to As As o As As . . . H As As /a T In the limit as Jtf approaches M these equations become ds c?s p ds ds p T ds ds T 80 d0 thus -T- denotes the absolute rate of change of 0, and -=- that relative to the trihedral.* If t denotes the distance between P and a point P^ (f^ rj^ fj), that is ^ 2 = (? 1 -f) 2 +(7? 1 -7;) 2 + (? 1 -?) 2 , we find by means of the formulas (82) that If a, 5, <? denote the direction-cosines of PP^ with respect to the axes at Jf, then When we express the condition that f 1? 77 L , f t as well as f, ?/, f satisfy equations (82), we are brought to the following fundamental relations between the variations of a, 6, c: &a da b &b db a c $c __ dc b (83) ~T~ ~ ~7 --- T" == ~T I --- 1 * ~r == "T * c?s ds p ds ds p T ds ds r If the point P remains fixed in space as M moves along the curve, the left-hand members of equations (82) are zero and the equations reduce to (84) --l, = ds p ds \p T/ c?s T Moreover, the direction-cosines of a line fixed in space satisfy the equations or . - da b db (a c\ dc b (85) - = - = (- + - )i _ = _. c?s /) c?s \p T) ds r * Cf. Cesaro, Lezioni di Geometria Intrinseca, pp. 122-128. Naples, 1896. MOVING TRIHEDRAL 33 These are the Frenet-Serret formulas, as might have been expected. We shall show that the solution of these equations carries with it the solution of (84). Suppose we have three sets of solutions of (85), a, I, X ; & m, /JL ; 7, n, v, whose values for s = are (86) 1, 0, 0; 0, 1, 0; 0, 0, 1. They are the direction-cosines, with respect to the moving trihedral with vertex M, of three fixed directions in space mutually perpen dicular to one another. Let be a fixed point, and through it draw the lines with the directions just found. Take these lines for coordinate axes and let #, y, z denote the coordinates of M with respect to them. If f, ??, f denote the coordinates of with respect to the moving trihedral, then f , ?;, f are the coordinates of M with respect to the trihedral with vertex at and edges parallel to the corresponding edges of the trihedral at M. Consequently we have f = - (ax -f &y + 7*)> (87) rj=-(lx+my + m), If these values be substituted in (84) and we take account of (50) and (85), we find that the equations are identically satisfied. If fo ^o ?o Denote the values of f, ?;, for s = 0, it follows from (86) and (87) that they differ only in sign from the initial values of x, y, z. Hence if we write, in conformity with (21), (88) x and substitute these values in (87), they become the general solu tion of equations (84). We have seen that the solution of equa tions (85) reduces to the integration of the Riccati equation (65). 17. Illustrative examples. As an example of the foregoing method we consider the curve which is the locus of a point on the tangent to a twisted curve C at a constant distance a from the point of contact. The coordinates of the point MI of the curve with reference to the axes at M are a, 0, 0. In this case equations (82) reduce to (i) ^-l^- a ^- ds~ ds~ P ds~ 34 CURVES IN SPACE Hence if Si denotes the length of arc of C\ from the point corresponding to s = on C, we have and the direction-cosines of the tangent to Ci with reference to the moving axes are iven b are given by 7 a 2 + p 2 Va 2 4- p 2 Hence the tangent to Ci is parallel to the osculating plane at the corresponding point of C. point By means of (83) we find d / p \_ a Sa:\ d / ds ds y V(i 2 + p 2 p Va 2 + p 2 ( 2 + P 2 ) P Va 2 + p 2 Proceeding in like manner with 0i and 71, and making use of (ii), we have d(*i & 2 pp d 5/3i ftp 2 p p _ (a 2 + p 2 ) 2 a 2 + p 2 5Si ~ (a 2 + p 2 ) 2 a 2 + p 2 871 _ ap From these expressions and (21 } we obtain the following expression for the square of the first curvature of C\ : app/ -II -t Pi 2 a 2 + p 2 \a 2 + The direction-cosines of the principal normal of C\ are 5/?i 571 By means of (40) we derive the following expressions for the direction-cosines of the binormal : r (a 2 + p 2 )^ r (a 2 + p 2 )^ > ft2 + ? Va 2 + p 2 In order to find the expression for TI, the radius of torsion of Ci, we have only to substitute the above values in the equation _ S\i _ p /d\i _ MI\ ~ 5i Va 2 + p 2 \ ds P We leave this calculation to the reader and proceed to an application of the preceding results. We inquire whether there is a curve C such that Ci is a straight line. The necessary and sufficient condition is that I/pi, be zero (Ex. 3, p. 15). From (iii) it follows that we must have ILLUSTRATIVE EXAMPLES 35 From the second of these equations it follows that C must be plane, and from the former we get, by integration, log (a* + ,*) = + , where c is a constant of integration. If the point s = be chosen so that we may take c = log a 2 , this equation reduces to = a \e - 1. P If 6 denotes the angle which the line C\ makes with the -axis, we have, from (i), 8rj a 1 tan 6 = = - = e - 1 Differentiating this equation with respect to s, we can put the result in the form dd__ 1 ds p consequently (89) When these values are substituted in equations (75), we obtain the coordinates of C in the form x = i \ 1 e as, y = ae or, in terms of 0, (90) x = a log tan - + cos 6 , y = a sin 0. The curve, with these equations, is called the tractrix. As just seen, it possesses the property that there is associated with it a straight line such that the segments of the tangents between the points of tangency and points of intersection with the given line are of constant length. Theorem. The orthogonal trajectories of the osculating plane of a twisted curve can be found by quadratures. With reference to the moving axes the coordinates of a point in the osculating plane are (, 77, 0). The necessary and sufficient condition that this point describe an orthogonal trajectory of the osculating plane as M moves along the given curve is that and in (82) be zero. Hence we have for the determination of and 77 ds ds the equations ?*_+ ^ + = 0, da- da- where a is given by (89). Eliminating , we have d?-t] ^ + " = - Hence 77 can be found by quadratures as a function of <r, and consequently of S, and then is given directly. 36 CURVES IK SPACE Problem. Find a necessary and sufficient condition that a curve lie upon a sphere. If , ??, f denote the coordinates of the center, and R the radius of the sphere, we have 2 -f if* + f 2 = R 2 . Since the center is fixed, the derivatives of , 17, f are given by (84). Consequently, when we differentiate the above equation, the result ing equation reduces to = 0, which shows that the normal plane to the curve at each point passes through the center of the sphere. If this equation be differen tiated, we get 77 = p ; hence the center of the sphere is on the polar line for each point. Another differentiation gives, together with the preceding, the following coordinates of the center of the sphere : When the last of these equations is differentiated we obtain the desired condition (92) - -f (rp Y = 0. Conversely, when this condition is satisfied, the point with the coordinates (91) is fixed in space and at constant distance from points of the curve. A curve which lies upon a sphere is called a spherical curve. Hence equation (92) is a necessary and sufficient condition that a curve be spherical. EXAMPLES 1. Let C be a plane curve and Ci an orthogonal trajectory of the normals to C. Show that the segments of these normals between C and Ci are of the same length. 2. Let C and Ci be two curves in the same plane, and say that the points corre spond in which the curves are met by a line through a fixed point P. Show that if the tangents at corresponding points are parallel, the two curves are similar and P is the center of similitude. 3. The locus of the point of projection of a fixed point P upon the tangent to a curve C is called the pedal curve of C with respect to P. Show that if r is the distance from P to a point M on (7, and 6 the angle which the line PM makes with the tangent to C at M, the arc Si and radius of curvature pi of the pedal curve are given by where s and p are the arc and the radius of curvature of C. 4. Find the intrinsic and parametric equations of a plane curve which is such that the segment on any tangent between the point of contact and the projection of a fixed point is of constant length. 5. Find the intrinsic equation of the plane curve which meets under constant angle all the lines passing through a fixed point. 6. The plane curve which is such that the locus of the mid-point of the seg ment of the normal between a point of the curve and the center of curvature is a straight line is the cycloid whose intrinsic equation is p 2 -f- s 2 = a 2 . 7. Investigate the curve which is the locus of the point on the principal normal of a given curve and at constant distance from the latter. OSCULATING SPHERE 37 18. Osculating sphere. Consider any curve whatever referred to its moving trihedral. The point whose coordinates have the values (91) lies on the normal to the osculating plane at the center of curvature, that is, on the polar line. Consequently the moving sphere whose center is at this point, and whose radius is V p 2 -f- r 2 // 2 , cuts the osculating plane in the osculating circle. This sphere is called the osculating sphere to the curve at the point. We shall derive the property of this sphere which accounts for its name. When the tangent to a curve at a point M is tangent likewise to a sphere at this point, the center of the sphere lies in the normal plane to the curve at M. If R denotes its radius and the curve is referred to the trihedral at M, the coordinates of the center C of the sphere are of the form (0, y v z t ) and yl + z* = ^ 2 . Let P(x, y, z) be a point of the curve near M, and Q the point in which the line CP cuts the sphere. If PQ be denoted by 8, we have, from (53), 6/r which reduces to Hence 8 is of the second order, in comparison with JMTP, unless ?/ 1 =/3, that is, unless the center is on the polar line; then it is of the third order unless z l = p r, in which case the sphere is the osculating sphere. Hence we have the theorem : The osculating sphere to a curve at a point has contact with the curve of the third order ; oilier spheres with their centers on the polar line, and tangent to the curve, have contact with the curve of the second order ; all other spheres tangent to the curve at a point have contact of the first order. The radius of the osculating sphere is given by (93) JS* =,! + TV, and the coordinates of the center, referred to fixed axes in space, are (94) x l = x + pi p f T\, y^ = y + pm p rfji, z l = z + pn p rv. 88 CURVES IN SPACE Hence when p is constant the centers of the osculating sphere and the osculating circle coincide. Then the radius of the sphere is necessarily constant. Conversely, it follows from the equation (93) IP that a necessary and sufficient condition that E be con stant is [P, that is, either the curvature is constant, or the curve is spherical. If equations (94) be differentiated with respect to s, we get (96)- #1 = From these expres sions it is seen that the center of the osculating sphere is fixed only in case of spherical curves. Also, the tangent to the locus of the cen ter is parallel to the binormal. Combin ing this result with a previous one, we have the theorem: The polar line for a point on a curve is tangent to the locus of the center of the osculating sphere to the curve at the corresponding point. This result is represented in fig. 5, in which the curve is the locus of the points M\ the points (7, C^ C 2 , are the correspond ing centers of curvature ; the planes MCN, M^C^N^ are normal to the curve ; the lines CP, C^P^ are the polar lines ; and the points P, Pj, P 2 , are the centers of the osculating spheres. FIG. 5 BEETEAND CUKVES 39 19. Bertrand curves. Bertrand proposed the following problem : To determine the curves whose principal normals are the principal normals of another curve. In solving this problem we make use of the moving trihedral. We must find the necessary and sufficient condition that the point M l ( = 0, TJ = k, ? = 0) generate a curve C^ whose principal normal coincides with the ?;-axis of the moving trihedral. Since the point M 1 remains on the moving ?/-axis, we have d% = d% = 0. And since M l tends to move at right angles to this axis, Brj = 0. Now equations (82) reduce to (96) }Ll-, vO, 5--*. ds p ds > ds r From the second we see that k is a constant. Moreover, if co denotes the angle which the tangent at M l makes with the tangent at M, we have, from the first and third of these equations, 8? kp tan co = -^r = or Sf T (k p) sin co cos co sin co (97) k We have seen ( 11) that according as r is positive or negative, the osculating plane to a curve at a point M near M cuts the polar line for M below or above the osculating plane at M. From these considerations it follows that when r > 0, co is in the third, fourth, or first quadrants according as k > ^, < k < p, or k < ; and when r < 0, co is in the second, first, or fourth quadrant, accordingly. It is readily found that these results are consistent with equation (97). By means of (97) it is found from (96) that the negative sign being taken so that the left-hand member may be positive. Thus far we have expressed only the condition that the locus of M^ cut the moving T^-axis orthogonally, but not that this axis shall be the principal normal to the curve C l also. For this we consider the moving trihedral for C l and let a x , b^ c^ denote the 40 CUKVES IN SPACE direction-cosines with respect to it of a fixed direction in space, as M^D in fig. 6. They satisfy equations similar to (85), namely M (99 ) l If a, 6, c are the direction-cosines of the same direction, with respect to the mov ing trihedral at M, we must have a l a cos &> -f- c sin &), b l = 6, ^ = a sin eo + <? cos a>, for all possible cases, as enumerated above. When these values are sub stituted in the above equations, we get, by means of (98), P sin &) sin tw _l_ _ T p COS ft) sm cos ft) a sin &)) - = 0, as [T sin G) cos <w sin &)1 Tsin to sin &) cos &)1 _ ~ - I \d -\- I - I ( I c u, */> PI T I J L * PI T I J [sin &) cos &) k ~| , , , . x dco ~ . 6 -h (<? sm &) + a cos &))- = 0. /3 T r t r sin &)J as Since these equations must be true for every fixed line, the coeffi cients of a, 6, c in each of these equations must be zero. The resulting equations of condition reduce to &) = const., (100) sm &) cos &) sm <w ~ 1 ;; = " Since &) is a constant, equation (97) is a linear relation between the first and second curvatures of the curve C. And the last of equations (100) shows that a similar relation holds for the curve C r Conversely, given a curve C whose first and second curvatures satisfy the relation (ioi) p + 7 = c> where -4, B, C are constants different from zero ; if we take A B k = , COt ft) = ;> TANGENT SURFACE OF A CURVE 41 and for p l and r l the values given by (100), equations (99) are sat isfied identically, and the point (0, k, 0) on the principal normal generates the curve C v conjugate to C. We gather these results about the curves of Bertrand into the following theorem: A necessary and sufficient condition that the principal normals of one curve be the principal normals of a second is that a linear relation exist between the first and second curvatures; the distance between corresponding points of the two curves is constant, the oscu lating planes at these points cut under constant angle, and the torsions of the two curves have the same sign. We consider, finally, several particular cases, which we have excluded in the consideration of equation (101). When C = and A=Q, the ratio of p and T is constant. Hence the curve is a helix and its conjugate is at infinity. When A = 0, that is, when the curve has constant torsion, the conjugate curve coincides with the original. When A = C = 0, k is indeterminate ; hence plane curves admit of an infinity of conjugates, they are the curves parallel to the given curve. The only other curve which has more than one conjugate is a circular helix, for since p and T are constant, A/C can be given any value whatever ; both the given helix and the circular helices conjugate to it are traced on circular cylinders with the same axis. 20. Tangent surface of a curve. For the further discussion of the properties of curves it is necessary to introduce certain curves and surfaces which can be associated with them. However, in con sidering these surfaces we limit our discussion to those properties which have to do with the associated curves, and leave other con siderations to their proper places in later chapters. The totality of all the points on the tangents to a twisted curve C constitute the tangent surface of the curve. As thus defined, the sur face consists of an infinity of straight lines, which are called the generators of the surface. Any point P on this surface lies on one of these lines, and is determined by this line and the distance t from P to the point M where the line touches the curve, as is shown in fig. 7. If the coordinates x, y, z of M are expressed in terms of the arc , the coordinates of P are given by (102) f 42 CURVES IN SPACE where the accents denote differentiation with respect to s. When the equations of the curve have the general form the coordinates of P can be expressed thus : (103) =./+/, ?=/,() +./, f -/,() where v = -- From this it is seen that v is equal to the distance MP only when s is the parameter. As given by equations (102) or (103), the coordinates of a point on the tangent surface are functions of two parameters. A rela tion between these parameters, such as (104) f(s, t) = 0, defines a curve which lies upon the surface. For, when this FIG. 7 equation is solved for t in terms of s and the resulting expres sion is substituted in (102), the coordinates f, ?;, f are functions of a single parameter, and consequently the locus of the point (f, 77, f) is a curve (1). By definition, the element of arc of this curve is given by da- 2 = di; 2 -f drf + c?f 2 . This is expressible by means of (102) and (41) in the form (105) d<r z = l + - 2 ds 2 + 2dsdt + dt\ where t is supposed to be the expression in s obtained from (104), and p is the radius of curvature of the curve (7, of which the sur face is the tangent surface. This result is true whatever be the relation (104). Hence equation (105) gives the element of length of any curve on the surface, and do- is called the linear element of the surface. According as t in equations (102) has a positive or negative value, the point lies on the portion of the tangent drawn in the TANGENT SUBFACE OF A CUKVE 43 positive direction from the curve or in the opposite direction. It is now our purpose to get an idea of the form of the surface in the neighborhood of the curve. In consequence of (53) equations (102) can be written 1 \ L 1 -^r+.-.u, 6 pr The plane f = cuts the surface in a curve F. The point M Q of (7, at which s = 0, is also a point of F. From the above expression for f it is seen that for points of F near M Q the parameters s and t differ only in sign. Hence, neglecting powers of s and t of higher orders, the equations of F in the neighborhood of J/ are f=0 , ,=-., r=_ < 2/o 3 pr By eliminating t from the last two equa tions, we find that in the neighborhood of M Q the curve F has the form of a semi- cubical parabola with the T^-axis, that is the principal normal to (7, for cuspidal tangent. Since any point of the curve C can be taken for Jf , we have the theorem : The tangent surface of a curve consists of two sheets, corresponding respectively to positive and negative values of t, which are tangent to one another along the curve, and thus form a sharp edge. On this account the curve is called the edge of regression of the surface. An idea of the form of the surface may be had from fig. 8. 21. Involutes and evolutes of a curve. When the tangents of a curve C are normal to a curve C v the latter is called an involute of (7, and C is called an evolute of C r As thus defined, the involutes of a twisted curve lie upon its tangent surface, and those of a 44 CURVES IN SPACE plane curve in its plane. The latter is only a particular case of the former, so that the problem of finding the involutes of a curve is that of finding the curves upon the tangent surface which cut the generators orthogonally. We write the equations of the tangent surface in the form Assuming that s is the parameter of the curve, the problem reduces to the determination of a relation between t and s such that By means of (50) this reduces to dt + ds = 0, so that t c s, where c is an arbitrary constant. Hence the coordinates x^ y v z l of an involute are expressible in the form (106) 2^= x + a(c s), #! = # + (<?), z x = z + ?(<? s). Corresponding to each value of c there is an involute ; consequently a curve has an infinity of involutes. If two involutes correspond to values c^ and c 2 of c, the segment of each tangent between the curves is of length c l c 2 . Hence the involutes are said to form a system of parallel curves on the tangent surface. When s is known the involutes are given directly by equations (106). Hence the complete de termination of the involutes of a given curve requires one quad rature at most. From the definition of t and its above value, an involute can be generated mechanically in the following manner, as represented in fig. 9. Take a string of length c and bring it into coincidence with the curve, with one end at the point s = ; call the other end A. If the former point be fixed and the string be unwound gradually from the curve beginning at A, this point will trace out an involute on the tangent surface. By differentiating equations (106), we get 7 I (c s} j , m (c s) j , n (c s) , dx l = ds, dy l = ds, dz l - ds. FIG. 9 INVOLUTES AND EVOLUTES 45 Hence the tangent to an involute is parallel to the principal nor mal of the curve at the corresponding point, and consequently the tangents at these points are perpendicular to one another. As an example of the foregoing theory, we determine the involutes of the cir cular helix, whose equations are x = a cos te, y = a sin M, z = au cot 0, where a is the radius of the cylinder and 6 the constant angle which the tangent to the curve makes with axis of the cylinder. Now sin u, cos M, cot s a cosec 6 - u, a, /3, 7 = - : -- cosec Hence the equations of the involutes are Xi = a cos u + (au c sin 0)sin M, yi = a sin u (aw c sin 0) cos u, zi = c cos 0. From the last of these equations it follows that the involutes are plane curves whose planes are normal to the axis of the cylinder, and from the expressions for x\ and yi it is seen that these curves are the involutes of the circular sections of the cylinder. We proceed to the inverse problem : Given a curve C, to find its evolutes. The problem reduces to the determination of a succession of normals to C which are tangent to a curve G . If M Q be the point on C corresponding to M on (7, it lies in the normal plane to C at If, and consequently its coordinates are of the form where p and q are the distances from M Q to the binomial and prin cipal normal respectively. These quantities p and q mi^t be such that the line M Q M is tangent to the locus of Jf at tiis point, that is, we must have ,, where /c denotes a factor of proportionality. "When the above values are substituted in these equations, we et P and two other equations obtained by replacing a, Z, X by /3, m, ft and 7, n, v. Hence the expressions in parentheses vanish. From 46 CURVES IN SPACE the first it follows that p is equal to /o; consequently M Q lies on the polar line of C at M. The other equations of condition can be written , , dp q da p J: + 1 + p a, * _ + 0> ds r ds T Eliminating /c, we get p For the sake of convenience we put <o = I - > and obtain by integration - = tan (o> + c), P where c is the constant of integration. As c is arbitrary, there is an infinity of evolutes of the curve C\ they are defined by the following equations, in which c is constant for an evolute but changes with it: x Q =x+lp + \p tan(o) + c), y Q =y + mp + fip tan(o> -f c), Z Q = z -f np + vp tan (o> -f- c). From the definition of q it follows that q/p is equal to the tangent of the angle which MM Q makes with the principal normal to C at M. Calling this angle 0, we have 6 = &> + c. The foregoing results give the following theorem : A curve C admits of an infinity of evolutes; when each of the normals f" (7, which are tangent to one of its evolutes, is turned through the sa^g angle in the corresponding normal plane to C, these new normals are tangent to another evolute of C. In fig. 5 the locus of the points E is an evolute of the given curve. Each system o; normals to C which are tangent to an evolute C constitute a tang en t surface of which C is the edge of regression. Hence the evolves of C are the edges of regression of an infinity of tangent svff aces? a ll o f which pass through C.. From tb e definition of w it follows that w is constant only when the curve C is plane. j n this case we may take w equal to zero. Then when c we have the evol\te C in the plane of the curve. The other evolutes lie upon the right MINIMAL CUEVES 47 cylinder formed of the normals to the plane at points of Co, and cut the elements of the cylinder under the constant angle 00 c, and consequently are helices. Hence we have the theorem : The evolutes of a plane curve are the helices traced on the right cylinder whose base is the plane evolute. Conversely, every cylindrical helix is the evolute of an infinity of plane curves. EXAMPLES 1. Find the coordinates of the center of the osculating sphere of the twisted cubic. 2. The angle between the radius of the osculating sphere for any curve and the locus of the center of the sphere is equal to the angle between the radius of the osculating circle and the locus of the center of curvature. 3. The locus of the center of curvature of a curve is an evolute only when the curve is plane. 4. Find the radii of first and second curvature of the curve x = a sin u cos w, y = a cos 2 it, z = asinw. Show that the curve is spherical, and give a geometrical construction. Find its evolutes. 5. Derive the properties of Bertrand curves ( 10) without the use of the moving trihedral. 6. Find the involutes and evolutes of the twisted cubic. 7. Determine whether there is a curve whose bmormals are the binormals of a second curve. 8. Derive the results of 21 by means of the moving trihedral. 22. Minimal curves. In the preceding discussion we have made exception of the curves, defined by z =/! (w), y =/ a (u), z =/ 8 (u), when these functions satisfy the condition As these imaginary curves are of interest in certain parts of the theory of surfaces, we devote this closing section to their discussion. The equation of condition may be written in the form _ f f __ if J3 Jl V2 where v is a constant or a function of u. These equations are equivalent to the following : (108) ^ : ^lz* SI .<l+*>:.. 48 CURVES IN SPACE At most, the common ratio is a function of M, say f(u). And so if we disregard additive constants of integration, as they can be removed by a translation of the curve in space, we can replace the above equations by (1 09) x =.f(u)du, y = if( n )du, z = We consider first the case when v is constant and call it a. If we change the parameter of the curve by replacing I f(u) du by a new parameter which we call w, we have, without loss of generality, 1-a 2 .1+a 2 (110) x = ^u t y = i^--u, z = au. For each value of a these are the equations of an imaginary straight line through the origin. Eliminating #, we find that the envelope of these lines is the imaginary cone, with vertex at the origin, whose equation is (111) z 2 +2/ 2 +z 2 =0. Every point on the cone is at zero distance from the vertex, and from the equations of the lines it is seen that the distance between any two points on a line is zero. We call these generators of the cone minimal straight lines. Through any point in space there are an infinity of them ; their direction-cosines are proportional to > where a is arbitrary. The locus of these lines is the cone whose vertex is at the point and whose generators pass through the circle at infinity. For, the equation in homogeneous coordinates of the sphere of unit radius and center at the origin is 3? + y 2 + z 2 = w 2 , so that the equations of the circle at infinity are Hence the cone (111) passes through the circle at infinity. We consider now the case where v in equations (109) is a function of u. If we take this function of u for a new parameter, and for convenience call it it, equations (109) may be written in the form (112) g _ll~p >()<**, y = i^^F(u)du, z= CuF(u)du, MINIMAL CUEVES 49 where, as is seen from (108), F(u) can be any function of u different from zero. If we replace F(u) by the third derivative of a function f(u), thus F(u)=f "(u], equations (112) can be integrated by parts and put in the form uf (u)-f(u), (113) 1y-4 Since F must be different from zero, f(u) can have any form other than c l u*+ c 2 u -f c 8 , where ^, c 2 , c 3 are arbitrary constants. EXAMPLES 1. Show that the tangents to a minimal curve are minimal lines, and that a curve whose tangents are minimal lines is minimal. 2. Show that the osculating plane of a minimal curve can be written (X x) A + (Y-y)B + (Z-z)C = Q, where A 2 + B 2 + C 2 = 0. A plane whose equation is of this sort is called an isotropic plane. 3. Show that through each point of a plane two minimal straight lines pass which lie in the latter. 4. Determine the order of the minimal curves for which the function /in (113) satisfies the condition 4/ "/ v - 5/ iv2 = 0. 5. Show that the equations of a minimal curve, for which /in (113) satisfies the condition 4/ "/ v 5/ iv2 = a/ ///3 , where a is a constant, can be put in the form 8 8 . 8i, x = - cos , y - sin , z = t. GENERAL EXAMPLES 1. Show that the equations of any plane curve can be put in the form x=J*cos0/(0)d0, y - J sin 0/(0) d0, and determine the geometrical significance of 0. 2. Prove that the necessary and sufficient condition that the parameter u in the equations x =fi(u), y =f 2 (u) have the significance of in Ex. 1 is 3. Prove that the general projective transformation transforms an osculating plane of a curve into an osculating plane of the transform. 4. The principal normal to a curve is normal to the locus of the centers of curvature at the points where p is a maximum or minimum. 50 CURVES IN SPACE 5 . A certain plane curve possesses the property that if C be its center of curva ture for a point P, Q the projection of P on the x-axis, and T the point where the tangent at P meets this axis, the area of the triangle CQT is constant. Find the equations of the curve in terms of the angle which the tangent forms with the x-axis. 6. The binormal at a point Mot a curve is the limiting position of the common perpendicular to the tangents at M and M , as N approaches M. 7. The tangents to the spherical indicatrices of the tangent and binormal of a twisted curve at corresponding points are parallel. 8. Any curve upon the unit sphere serves for the spherical indicatrix of the binormal of a curve of constant torsion. Find the coordinates of the curve. 9. The equations r Idk - kdl r hdl - Idh r kdh ~ hdk x a I - i y a I z d I J #2 + 2 + 12 J h Z + fc2 + 1-2 J h 2 + where a is constant and h, k, I are functions of a single parameter, define a curve whose radius of torsion is a. 10. If, in Ex. 9, we have k = sm/i0 + -%sinX0 I = 2 - cos \/ 2 where X and /x are constants whose ratio is commensurable, the integrands are expressible as linear homogeneous functions of sines and cosines of multiples of 0, and consequently the curve is algebraic. 11. Equations (1) define a family of circles, if a, &, r are functions of a parameter t. Show that the determination of their orthogonal trajectories requires the solution of the Riccati equation, *! = l*?,__L* ( i-"), dt r dt 8rdr where 0=tanw/2. 12. Find the vector representing the rate of change of the acceleration of a moving point. 13. When a curve is spherical, the center of curvature for the point is the foot of the perpendicular upon the osculating plane at the point from the center of the sphere. 14. The radii of first and second curvature of a curve which lies upon a sphere and cuts the meridians under constant angle are in the relation 1 + ar + fy> 2 r = 0, where a and b are constants. 15. An epitrochoidal curve is generated by a point in the plane of a circle which rolls, without slipping, on another circle, whose plane meets the plane of the first circle under constant angle. Find its equations and show that it is a spherical curve. 16. If two curves are in a one-to-one correspondence with the tangents at corresponding points parallel, the principal normals at these points are parallel and likewise the binormals ; two curves so related are said to be deducible from one another by a transformation of Combescure. 17. If two curves are in a one-to-one correspondence and the osculating planes at corresponding points are parallel, either curve can be obtained from the other by a transformation of Combescure. GENERAL EXAMPLES 51 18. Show that the radius of the osculating sphere of a curve is given by E 2 T 2 p 4 [x"" 2 + y" /z + z ///2 ] r 2 , where the prime denotes differentiation with respect to the arc. 19. At corresponding points of a twisted curve and the locus of the center of its osculating sphere the principal normals are parallel, and the tangent to one curve is parallel to the binorinal to the other ; also the product? of the radii of torsion of the two curves is equal to the product of the radii of first curvature, or to within the sign, according as the positive directions of the principal normals are the same or different. 20. Determine the twisted curves which are such that the centers of the spheres osculating the curve of centers of the osculating spheres of the given curve are points of the latter. 21. Show that the binormals to a curve do not constitute the tangent surface of another curve. 22. Determine the directions of the principal normal and binormal to an involute of a given curve. 23. Show that the equations x = a C<f> (u) sin u du, y = a \ $ (u) cos u du, z a f 4>(u)\l/(u)du, where (u) (1 4- ^ 2 4- ^ /2 )- (1 4- ^ 2 ) * and \f/ (u) is any function whatever, define a curve of constant curvature. 24. Prove that when ^ (u) = tan w, in example 23, the curve is algebraic. 25. Prove that in order that the principal normals of a curve be the binor- mals of another, the relation a I h ) = - must hold, where a and 6 are con stants. Show that such curves are defined by equations of example 23 when (1 4. ^2 _|_ ^/2)3 _|_ (1 _|_ ^2)3(^" _j_ ^,\2 = . (1 4. i// 2 )^(l 4- 1// 2 4- 1// 2 )^ 26. Let \i, /ii, *i be the coordinates of a point on the unit sphere expressed as functions of the arc <TI of the curve. Show that the equations x = ek I \idffi k cot w / (MI^I y = ek j mdai k cot w \ (v\\{ v{\\] d<?i, z = ek I vida-i k cot w | (\i/4 where k and w are constant, e = 1, and the primes indicate differentiation with respect to o-i, define a Bertrand curve for which p and T satisfy the relation (97) ; show also that X 1? /t 1? v\ are the direction-cosines of the binormal to the conjugate curve. CHAPTER II CURVILINEAR COORDINATES ON A SURFACE* ENVELOPES 23. Parametric equations of a surface. In the preceding chapter we have seen that the coordinates of a point on the tangent surface of a curve are expressible in the form (1) x where f l (u), ?=/(*), are the equations of the curve, and v is proportional to the distance between the points (f , 77, f ), (x, y, z) on the same generator. Since the coordinates of the surface are expressed by (1) as functions of two independent parameters w, v, the equations of the surface may be written Consider also a sphere of radius a whose center is at the origin (fig. 10). If v denotes the angle, measured in the positive sense, which the plane through the z-axis and a point M of the sphere makes with the #z-plane, and u denotes the angle between the radius OM and the positive z-axis, the coordinates of M may be written FIG. 10 (3) x = a sin u cos v, y = a sin u sin v, z = a cos u. Here, again, the coordinates of any point on the sphere are ex pressible as functions of two parameter^, and the equations of the sphere are of the form (2)*. * Notice that in this case /^ is a function of u alone. PARAMETRIC EQUATIONS OF A SURFACE 53 In the two preceding cases the functions f v / 2 , / 3 have par ticular forms. We consider the general case where / 1? / 2 , / 3 are any functions of two independent parameters w, v, analytic for all values of u and v, or at least for values within a certain domain. The locus of the point whose coordinates are given by (2) for all values of u and v in the domain is called a surface. And equa tions (2) are called parametric equations of the surface. It is to be understood that one or more of the functions / may involve a single parameter. For instance, any cylinder may be defined by equations of the form x =fi M y =/ 2 M z =/a ( u ^ v )- If we replace u and v in (2) by independent functions of two other parameters u v v v thus (4) u = F l (u v v,), v = F z (u l9 vj, the resulting equations may be written (5) x = fa (u^ VJ, y = fa K, vj, z = fa (u t , vj. If particular values of ^ and v l be substituted in (4) and the result ing values of u and v be substituted in (2), we obtain the values of #, y, z given by (5), when u^ and t^ have been given the par ticular values. Hence equations (2) and (5) define the same sur face, provided that F l and F 2 are of such a form that fa, fa, fa satisfy the general conditions imposed upon the F s. Hence the equations of a surface may be expressed in parametric form in the number of ways of the generality of two arbitrary functions. Suppose the first two of equations (2) solved for u and v in terms of x and y, and let u = F t (x, y), v = F 2 (x, y) be a set of solutions. When these equations are taken as equations (4), equations (5) become x = x, y = y, z =f(x, y}, which may be replaced by the single relation, (6) 2 =/(*, y). If there is only one set of solutions of the first two of equations (2), equation (6) defines the surface as completely as (2). If, however, there are n sets of solutions, the surface would be defined by n equations, z =f t (x^ y). 54 CURVILINEAR COORDINATES ON A SURFACE It may be said that equation (6) is obtained from equations (2) by eliminating u and v. This is a particular form of elimination, the more general giving an implicit relation between x, y, z, as (7) F(x,y,z)=0. If we have a locus of points whose coordinates satisfy a relation of the form (6), it is a surface in the above sense. For, if we take x and y equal to any analytic functions of u and v, namely f^ and / 2 , and substitute in (6), we obtain z =/ 8 (w, v). In like manner equation (7) may be solved for z, and one or more equations of the form (6) obtained, unless z does not appear in (7). In the latter case there is a relation between x and y alone, so that the surface is a cylinder whose elements are parallel to the z-axis, and its parametric equations are of the form x =/i W y =/ 2 M 2 =/ 3 (w, v). Hence a surface can be denned analytically by equations (2), (6), or (7). Of these forms the last is the oldest. It was used exclusively until the time of Monge, who proposed the form (6); the latter has the advantage that many of the equations, which define properties of the surface, are simpler in form than when equation (7) is used. The parametric method of definition is due to Gauss. In many respects it is superior to both of the other methods. It will be used almost entirely in the following treatment. 24. Parametric curves. When the parameter u in equations (2) is put equal to a constant, the resulting equations define a curve on the surface for which v is the parameter. If we let u vary continu ously, we get a continuous array of curves whose totality consti tutes the surface. Hence a surface may be considered as generated by the motion of a curve. Thus the tangent surface of a curve is described by the tangent as the point of contact moves along the curve ; and a sphere results from the revolution of a circle about a diameter. We have just seen that upon a surface (2) there lie an infinity of curves whose equations are given by equations (2), when u is constant, each constant value of u determining a curve. We call them the curves u = const, on the surface. In a similar way, PARAMETRIC CURVES 55 there is an infinite family of curves v = const.* The curves of these two families are called the parametric curves for the given equations of the surface, and u and v are the curvilinear coordinates of a point upon the surface.f We say that the positive direction of a parametric curve is that in which the parameter increases. If we replace v in equations (2) by a function of w, say (8) v = <t>(u), the coordinates #, y, z are functions of a single parameter w, and consequently the locus of the point (#, y, z) is a curve. Hence equation (8) defines a curve on the surface (2). For example, the equation v = au defines a helix on the cylinder x = a cos w, y o> sin u, z = v. Frequently equation (8) is written in the implicit form, (9) F(u, v) = 0. Conversely, any curve upon the surface is defined by an equation of this form. For, if t be the parameter of the curve, both u and v in equations (2) are functions of t\ thus w = ^ 1 (Q, v = (j> z (t). Elimi nating t between these equations, we get a relation such as (9). We return to the consideration of the change of parameters, defined by equations (4). To a pair of values of u^ and v l there correspond unique values of u and v. On the contrary, it may happen that another pair of values of u^ and v l give the same values of u and v. But the values of x, y, z given by (5) will be the same in both cases ; this follows from the manner in which these equations were derived. On this account when equations (4) are solved for u^ and v l in terms of u and v, and there is more than one set of solutions, we must specify which solution will be used. We write the solution (10) u^ = <$>! (w, v), v^ = 4> 2 (u, v). In terms of the original parameters, the parametric lines u^= const. and v l = const, have the equations, * On the sphere defined by equations (3) the curves v const, are meridians and u const, parallels. t When a plane is referred to rectangular coordinates, the parametric lines are the two families of straight lines parallel to the coordinate axes. 56 CURVILINEAR COORDINATES ON A SURFACE where a and b denote constants. Unless u or v is absent from either of these equations the curves are necessarily distinct from the parametric curves u = const, and v = const. Suppose, now, that v does not appear in ^j then u^ is constant when u is constant, and vice versa. Consequently a curve u^ = const, is a member of the family of curves u = const. Hence, when a transformation of parameters is made by means of equations of the form or ^=(2,, ^(M), the two systems of parametric curves are the same, the difference being in the value of the parameter which is constant along a curve. EXAMPLES 1 . A surface which is the locus of a family of straight lines, which meet another straight line orthogonally and are arranged according to a given law, is called a right conoid ; its equations are of the form x = u cos v, y = u sin u, z = <j> (v). Show that when (v) = a cot v + b the conoid is a hyperbolic paraboloid. 2. Find the equations of the right conoid whose axis is the axis of z, and which V 2 z 2 passes through the ellipse x a, " -\ -- 1. 3. When a sphere of radius a is defined by (3), find the relation between u and v along the curve of intersection of the sphere and the surface x 4 -f y* + z 4 = a 4 . Show that the curves of intersection are four great circles. 4. Upon the surface x v w 2 + -J- cos t>, y M a 4- sin v, z = w, determine the curves whose tangents make with the z-axis the angle tan- 1 \/2. Show that two of these curves pass through every point, and find their radii of first and second curvature. 25. Tangent plane. A tangent line to a curve upon a surface is called a tangent line to the surface at the point of contact. It is evident that there are an infinity of tangent lines to a surface at a point. We shall show that all of these lines lie in a plane, which is called the tangent plane to the surface at the point. To this end we consider a curve C upon a surface and let M(XJ y, z) be the point at which the tangent is drawn. The equations of the tangent are ( 4) f-s = t)-y _ ?-g = ^ dx dy dz ds ds ds TANGENT PLANE 57 where f, 77, f are the coordinates of a point on the line, depending for their values upon the parameter X. If the equation in curvi linear coordinates of the curve C is v = <f>(u), the above equations may be written . * \ j ( ^ , dx\ du , dvl ds ^ ^^ = \ -f- 4> )-r \du cv/ ds where the prime indicates differentiation. In order to obtain the locus of these tangent lines, we eliminate $ arid X from these equations. This gives (U) dx dy dz du du du dx dy dz dv ~dv dv = 0, which evidently is the equation of a plane through the point M. The normal to this plane at the point of contact is called the normal to the surface at the point. As an example, we find the equation of the tangent plane to the tangent surface of a curve at any point. If the values from (1) be substituted in equation (11), the resulting equation is reducible to (12) /i fi fi fi fi fs Hence the equation of the tangent plane is independent of u, and depends only upon u. In consequence of (I, 36) * we have the theorem : The tangent plane to the tangent surface of a curve is the same at all points of a generator; it is the osculating plane of the curve at the point where the generator touches the curve. When the surface is defined by an equation of the form F(x, y, z) = 0, we imagine that x, y. z are functions of u and v, and differentiate with respect to the latter. This gives Hx du dy du dz du ~ dx dv dy dv dz dv * In references of this sort the Roman numerals refer to the chapter. 58 CURVILINEAR COORDINATES ON A SURFACE By means of these equations the equation (11) of the tangent plane can be given the form (l-^+fo-jO^+tf-*)?^. ex cy cz When the Monge form of the equation of a surface, namely z =/(x, y\ t is used, it is customary to put dz cz ( 14 r- = -P = 9- dx cy Consequently the equation of the tangent plane is (15) (* - x)p + (77 - y)q -(f-z) = 0. In the first chapter we found that a curve is defined by two equations of the form (16) F l (x,y,z) = J F 2 (x, y, z) = 0. Hence a curve is the locus of the points common to two surfaces. The equa tions of the tangent to the curve are g-X^q -y _- Z ^ dx dy dz where cfcc, dy, dz satisfy the relations 5*1* + ^dy + ?*& = 0, ?**, + ^dy + ?** = 0. dx cy cz dx cy dz Consequently the equations of the tangent can be put in the form - z 77 - y $ - z (17) dy . dz dz dy dz dx dx dz dx dy dy dx Comparing this result with (13), we see that the tangent line to a curve at a point M is the intersection of the tangent planes at M to two surfaces which intersect along the curve. *" EXAMPLES 1. Show that the volume of the tetrahedron formed by the coordinate planes and the tangent plane at any point of the surface x = w, y = u, z = a s /uv is constant. 2. Show that the sum of the squares of the intercepts of the axes by the tan gent plane to the surface z = w 3 sin 3 u, y = M 3 cos 3 v, z = (a 2 - it 2 )*, at any point is constant. 3. Given the right conoid for which 0(u) = a sin 2 u. Show that any tangent plane to the surface cuts it in an ellipse, and that if perpendiculars be drawn to the generators from any point the feet of the perpendiculars lie in a plane ellipse. ENVELOPES 59 4. Show that the tangent planes, at points of a generator, to the right conoid for which (u) = a Vtan u, meet the plane z in parallel lines. 5. Find the equations of the tangent to the curve whose equations are ax 2 + by* + cz 2 = 1, 6x 2 + cy 2 + az 2 = 1. 6. Find the equations of the tangent to the curve whose equations are z(x + z)(x a) = a 3 , z(y + z)(y a) = a 3 , and show that the curve is plane. 7. The distance from a point M of a surface to the tangent plane at a near-by point M is of the second order when MM is of the first order ; and for other planes through M the distance from M is ordinarily of the first order. 26. One-parameter families of surfaces. Envelopes. An equation of the form (18) F(x, y, z,a) = Q defines an infinity of surfaces, each surface being determined by a value of the parameter a. Such a system is called a one-parameter family of surfaces. For example, the tangent planes to the tangent surface of a twisted curve form such a family. The two surfaces corresponding to values a and a of the param eter meet in a curve whose equations may be written > * a) = o. a a As a 1 approaches a, this curve approaches a limiting form whose equations are (19) ^( W ,)=0, *(**.)-(). The curve thus defined is called the characteristic of the surface of parameter a. As a varies we have a family of these characteristics, and their locus, called the envelope of the family of surfaces, is a surface whose equation is obtained by eliminating a from the two equations (19). This elimination may be accomplished by solving the second of (19) for a, thus: a = $ (#, y, z), and substituting in the first with the result 60 ENVELOPES The equation of the tangent plane to this surface is For a particular value of a, say a , equations (19) define the curve in which the surface F(x, y, z, a ) = meets the envelope ; and from the second of (19) it follows that at all points of this curve equa tion (20) of the tangent plane to the envelope reduces to This, however, is the equation of the tangent plane to the surface F(x, y, z, a ) = 0. If we say that two surfaces with the same tan gent plane at a common point are tangent to one another, we have : The envelope of a family of surfaces of one parameter is tangent to each surface along the characteristic of the latter. The equations of the characteristic of the surface of parameter a l are (21) This characteristic meets the characteristic (19) in the point whose coordinates satisfy (19) and (21), or, what is the same thing, equa tions (19) and F(x,y, z, a l )-F(x,^z, a) As a l approaches 0, this point of intersection approaches a limiting position whose coordinates satisfy the three equations (22) F-0, ^=0, !" 0. da da 2 If these equations be solved for a;, y, z, we have (23) * = /, y=/,(a), *=/.(a). These are parametric equations of a curve, which is called the edge of regression of the envelope. DEVELOPABLE SURFACES 61 The direction-cosines of the tangent to the edge of regression are proportional to - -^, -. If we imagine that x, y, z in (19) da da da are replaced by the values (23), and we differentiate these equa tions with respect to a, we get, in consequence of (22), dx da dy da dz da tfF dy *r ** , - - - | da d# c?# da dy da da dz da dz == " From these we obtain dF dF dF dx da dy dz da da dx dy d 2 F ~dz d 2 F dadx dady dadz But from (17) it follows that the minors of the right-hand mem ber are proportional to the direction-cosines of the tangent to the curve (19). Hence we have the theorem: The characteristics of a family of surfaces of one parameter are tangent to the edge of regression. 27. Developable surfaces. Rectifying developable. A simple ex ample of a family of surfaces of one parameter is afforded by a family of planes of one parameter. Their envelope is called a developable surface ; the full significance of this term will be shown later (43). The characteristics are straight lines which are tangent to a curve, the edge of regression. When the edge of regression is a point, the surface is a cone or cylinder, according as the point is at a finite or infinite distance. We exclude this case for the present and assume that the coordinates x, y, z of a point on the edge of regression are expressed in terms of the arc s. We may write the equation of the plane (24) (X- x)a + (Y- where , 5, c also are functions of s. The characteristics are defined by this equation and its derivative with respect to s, namely : (25) (X- x)a +(Y- y)V+(Z- z)c - ax - by - cz = 0. 62 ENVELOPES Since these equations define the tangent to the curve, they must be equivalent to the equations X-x _Y-y = Zz x y z Hence we must have (26) ax + % + cz r = 0, a x + Vy + c z = 0. If the first of these equations be differentiated with respect to s, the resulting equation is reducible, in consequence of the second of (26), to if E jf , IT A ax" + by" + cz"= 0. From this equation and (26) we find a : b : c = (y z"- z y") : (z x"- x z") : (x y"- y x"}. Hence by ( 7) we have the theorem: On the envelope of a one-parameter family of planes the planes osculate the edge of regression. We leave it to the reader to prove that the edge of regres sion of the osculating planes of a twisted curve is the curve itself. The envelope of the plane normal to the principal normal to a curve at a point of the curve is called the rectifying develop able of the latter. We shall find the equations of its edge of regression. The equation of this plane is (27) (X- x) I + (Y- y) m + (Z - z) n = 0. If we differentiate this equation with respect to the arc of the curve, and make use of the Frenet formulas (I, 50), we obtain (28) (I - + From these equations we derive the equations of the character istic in the form RECTIFYING DEVELOPABLE 63 t being the parameter of points on the characteristic. In order to find the value of t corresponding to the point where the character istic touches the edge of regression, we combine these equations with the derivative of (28) with respect to s, namely : and obtain (jL--J\t + - s *Q. VP PT/ P Hence the coordinates of the edge of regression of the rectifying developable are (29 ) t = x , p p r TP pr T p pr Problem. Under what conditions does the equation F(x, y, z) = define a devel opable surface ? We assume that x, y, z are functions of two parameters w, u, such that the curves u = const, are the generators, and v = const, are any other lines. The equation of the tangent plane is This equation should involve u and be independent of u. Its characteristic is given by (i) and where we have put, for the sake of brevity, .t.^.^w+y, _ ax 2 dxdy dxdz dx Since equation (i) is independent of u, we have (iii) A* + B* + c = 0. dv dv at) Comparing equations (ii) and (iii) with (13), we see that ^-X^=0, B-\^=0, C-X^ 3x dy oz 64 ENVELOPES where X denotes a factor of proportionality. If we eliminate X x, Y y, Z z, and X from these equations and (i), we obtain the desired condition 2 F d 2 F d^F dF z 2 dxdy dxdz dx 2 F d 2 F d z F dF dx dy dy 2 dx dz dy dz d_F_ aF dx dy dy dz dy ? az 2 " ~fa ** dz = 0. EXAMPLES 1. Find the envelope and edge of regression of the family of planes normal to a given curve. 2. Find the rectifying developable of a cylindrical helix. 3. Prove that the rectifying developable of a curve is the polar developable of its involutes, and conversely. 4. Find the edge of regression of the envelope of the planes x sin u y cos u -f z au = 0. 5. Determine the envelope of a one-parameter family of planes parallel to a given line. 6. Given a one-parameter family of planes which cut the xy-plane under con stant angle ; the intersections of these planes with the latter plane envelop a curve C. Show that the edge of regression of the envelope of the planes is an evolute of C. 7. When a plane curve lies on a developable surface its plane meets the tangent planes to the surface in the tangent lines to the curve. Determine the developable surface which passes through a parabola and the circle, described in a perpendicular plane, on the latus rectum for diameter, and show that it 4s a cone. 8. Determine the developable surface which passes through the two parabolas y 2 = 4 ox, z = 0; x 2 = 4 ay, z = 6, and show that its edge of regression lies on the surface y*z = x 3 (6 z). 28. Applications of the moving trihedral. Problems concern ing the envelope of a family of surfaces are sometimes more readily solved when the surfaces are referred to the moving trihedral of a curve, which is associated in some manner with the family of surfaces, the parameter of points on the curve being the parameter of the family. Let (30) F(& 77, , *) = APPLICATIONS OF THE MOVING TRIHEDRAL 65 define such a family of surfaces. Since f, 77, f are functions of *, the equations of the characteristics are (30) and ^-^^ + ^^ + ^^4.^=0 ds ~ d% ds dr) ds 0f ds ds ~ But the characteristics being fixed in space, we have (I, 84) Hence the equations of the characteristics are (32) ,_ , /i If, for the sake of brevity, we let $(, ??, f, *) = denote the second of these equations, the edge of regression is defined by (32) and < 8S > !( For example, the family of osculating planes of a curve is defined with refer ence to the moving trihedral by f = 0. In this case the second of (32) is rj = 0, and (33) is - 4- - = 0. Hence the tangents are the characteristics, and the edge of regres sion is the curve ; for, we have = ?? = f = 0. In like manner the family of normal planes is defined by = 0. Now the second of .(32) is 17 p=0 ; consequently the polar lines are the characteristics. Equation (33) reduces to f -f p r = ; hence the locus of the centers of the osculating spheres is the edge of regression (cf. 18). The envelope is called the polar developable. The osculating spheres of a twisted curve constitute a family of surfaces which is readily studied by the foregoing methods. From ( 18) it follows that the equation of these spheres is The second of equations (32) for this case is which, since spherical curves are not considered, reduces to = 0. And equation (33) is ?? = 0, so that the coordinates of the edge of regression are f = 77 = f = 0. Hence : The osculating circles of a curve are the characteristics of its oscu lating spheres ; and the curve itself is the edge of regression of the envelope of the spheres. 66 ENVELOPES 29. Envelope of spheres. Canal surfaces. We consider now any family of spheres of one parameter. Referred to the moving tri hedral of the curve of centers, the equation of the spheres is By means of (32) we find that a characteristic is the circle in which a sphere is cut by the plane The radius of this circle is equal to rVl r n . Hence the char acteristic is imaginary when r n > 1, reduces to a point when r = s + const., and is real for r f * < 1. By means of (33) we find that the coordinates of the edge of regression are given by (34) f = -n- , , = [l-(rr ) ]p, r Hence the edge of regression consists of two parts with corre sponding points symmetrically placed with respect to the oscu lating plane of the curve of centers (7, unless When this condition is satisfied the edge is a single curve, and its points lie in the osculating planes of C. We have seen that this is the case with the osculating spheres of a curve. We shall show that when the above condition is satisfied the spheres osculate their edge of regression <7 r We write the above equation in the form ^ (35) p[l~-(rr ) ] = er^l-r \ where e is + 1 or 1, so that p may be positive. We have seen ( 16) that the absolute and relative rates of change with s of the coordinates f, ?;, f of a point on C t are in the relations M = ^_?? + i, ^ = ^Z + l + f, ^ == ^_!?. Ss ds p &s ds p T Ss ds T When the values (34) are substituted in the right-hand members of these equations, we obtain, in consequence of (35), ENVELOPE OF SPHEEES 67 Hence the linear element Ss^ of C l is given by cs 1 = and (36) Uo, Since these are the direction-cosines of the tangent to C^ we see that this tangent is normal to the osculating plane to the curve of centers C. Moreover, these direction-cosines must satisfy (cf. I, 83) the equations /37\ 8a _da b 8b _ db a c 8c do b 8s ds p 8s ds p T 8s ds r Hence we have from which it follows that the radius of curvature p l of C l is (38) Pl =ee rVT^, where e r is + 1 or 1 , so that /o 1 may be positive. Since, now, the direction-cosines of the principal normal have the values it follows that the principal normals to C and C l are parallel. Furthermore, since these quantities must satisfy equations (37), we have g 3 g 3 , ^^ -. where p[ denotes the derivative of p l with respect to s r By means of (I, 51) we find that the radius of torsion r l of C l is given by .From (38) we find p[= - so that the radius R^ of the oscu lating sphere of C^ is given by R* p? + p[ 2 TI = r 2 , and consequently the osculating spheres of C l are of the same radius as the given spheres. 68 ENVELOPES The direction-cosines of the tangent, principal normal, and binor mal to C l are found from (36) and (39) to be Hence the coordinates (I, 94) of the center of the osculating sphere of C l are reducible, in consequence of (34), to + l iPi - P( T I\ = *? + m iPi ~ friPi = > ? + W 1 p 1 - XT^ = 0. Therefore we have the theorem : When the edge of regression of a family of spheres of one param eter has only one branch, the spheres osculate the edge. Since r does not appear in equation (35), it follows that when r is given as a function of s, the intrinsic equations of the curve of are where the function/(s) is arbitrary. Moreover, any curve will serve for the curve of centers of such an envelope of spheres. The deter mination of r requires the solution of equation (35) and consequently involves two arbitrary constants. When all the spheres of a family have the same radius, the envelope is called a canal surface. From (34) it is seen that in this case a characteristic is a great circle. Moreover, equation (35) reduces to p = r. Hence a necessary and sufficient condition that the edge of regression of a canal surface consist of a single curve is that the curve of centers be of constant curvature and the radius of the sphere equal to the radius of first curvature of the curve. GENERAL EXAMPLES 1. Let MN be a generator of the right conoid x = u cos u, y = u sin i>, z = 2 k cosec 2 D, M being the point in which it meets the z-axis. Show that the tangent plane at N meets the surface in a hyperbola which passes through M, and that as N moves along the generator the tangent at M to the hyperbola describes a plane. 2. A point moves on an ellipsoid -- h H -- = 1, so that the direction of its a 2 ft 2 c 2 motion always passes through the perpendicular from the center on the tangent plane at the point. Show that the path of the point is the curve in which the ellipsoid is cut by the surface x l y m z n = const. , where 1: m : n --- : -- : --- GENERAL EXAMPLES 69 3. If each of the generators of a developable surface be revolved through the same angle about the tangent to an orthogonal trajectory of the generators at the point of intersection, the locus of these lines is a developable surface whose edge of regression is an evolute of the given trajectory. 4. Show that the edge of regression of the family of planes (1 - w 2 )z + i(l + u*)y + 2uz +f(u) = is a minimal curve. 5. The developable surface which passes through the circles x 2 -f y* = a 2 , z = 0; x 2 -f z 2 6 2 , y = meets the plane x = in an equilateral hyperbola. 6. Find the edge of regression of the developable surface which envelopes the surface az = xy along the curve in which the latter is cut by the cylinder x 2 = by. 7. Find the envelope of the planes which pass through the center of an ellipsoid and cut it in sections of equal area. 8. The first and second curvatures of the edge of regression of the family of planes ax + /3y + yz p, where <r, /3, 7, p are functions of a single parameter u and a 2 + /3 2 -f y 2 = 1, are given by 1 A 3 A 2 where A = a a a" p p p p O f off -r\ oc. a a" a " P P P , D= ft , p, p,, y Y y" y " 9. Derive the equations of the edge of regression of the rectifying developable by the method of 28. 10. Derive the results of 29 without the aid of the moving trihedral. 1 1 . Find the envelope of the spheres whose diameters are the chords of a circle through a point of the latter. 12. Find the envelope and edge of regression of the spheres which pass through a fixed point and whose centers lie on a given curve. 13. Find the envelope and edge of regression of the spheres which have for diametral planes one family of circular sections of an ellipsoid. 14. Find the envelope and edge of regression of the family of ellipsoids (3^2 <j/2\ -j2 1 H - = 1, where a is the parameter. 15. Find the envelope of the family of spheres whose diameters are parallel chords of an ellipse. 16. Find the equations of the canal surface whose curve of centers is a circular helix and whose edge of regression has one branch. Determine the latter. 17. Find the envelope of the family of cones (ax + x + y + z - 1) (ay + z) - ax (x + y + z - 1) = 0, where a is the parameter. CHAPTER III LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAMETERS. CONFORMAL REPRESENTATION 30. Linear element. Upon a surface , defined by equations in the parametric form (1) x =fi (i*, v), y =/ a (w, v), 2 =/, (M, v), we select any curve and write its equations $ (u, v) 0. From 3 we have that the linear element of the curve is given by (2) d? fa j t foj j dij , t dy , , dz , , dz where ax = du -\ -- av, a y = au + -^ v f 2 = aw H --- ^M ^v ^w 9 du dv the differentials <^w, dv satisfying the condition 2$ , a^ , c?w + dv = 0. ^w dv ifweput du dv cu cv du dv or, in abbreviated form, equation (2) becomes (4) oV 2 = Edu? + 2 Fdudv + G dv 2 . The functions E, F, G thus denned were first used by Gauss.* When the surface is real, and likewise the curvilinear coordinates * Disquisitiones generates circa superficies curvas (English translation by Morehead and Hiltebeitel), p. 18. Princeton, 1902. Unless otherwise stated, all references to Gauss are to this translation. 70 LINEAK ELEMENT 71 w, v, the functions Vj, # are real. We shall understand also that the latter are positive. There is, however, an important excep tional case, namely when both E and G are zero (cf. 35). For any other curve equation (4) will have the same form, but the relation between du and dv will depend upon the curve. Consequently the value of c7s, given by (4), is the element of arc of any curve upon the surface. It is called the linear element of the surface (cf. 20). However, in order to avoid circumlocution, we shall frequently call the expression for ds 2 the linear element, that is, the right-hand member of equation (4), which is also called the first fundamental quadratic form. The coefficients of the lat ter, namely E, F, G, are called the fundamental quantities of the first order. If, for the sake of brevity, we put (5) d(u, v) du du dz d(u, v) d(u, v) dv dv it follows from (3) and (5) that (6) EG - F 2 = A 2 + B 2 + C 2 . Hence when the surface is real and likewise the parameters, the quantity EGF 2 is different from zero unless J, B, and C are zero. But if A, B, and C are zero, it follows from (5) that u and v are not independent, and consequently equations (1) define a curve and not a surface. However, it may happen that for certain values of u and v all the quantities J, B, C vanish. The corresponding points are called singular points of the sur face. These points may be isolated or constitute one or more curves upon the surface ; such curves are called singular lines. In the following discussion only ordinary points will be con sidered. From the preceding remarks it follows that for real surfaces, referred to real coordinate lines, the function H defined by (?) 1 is real, and it is positive by hypothesis. 72 LINEAR ELEMENT OF A SURFACE 31. Isotropic developable. The exceptional case, where the surface is imaginary and H is zero, is afforded by the tangent surface of a minimal curve. The equa tions of such a surface are (cf . 22) /I w 2 1 w 2 (u) du + (M) u, = ju(f>(u) du + u<j> (u) v, where 0(w) is a function of u different from zero. It is readily found that J=v 2 2 (w), F = G 0, and consequently EG F 2 = 0. This equation is likewise the sufficient condition that the surface be of the kind sought. For, when itjs satisfied, the equa tion of the linear element can be_written ds 2 = (Vjdw + V(?dw) 2 . If X denote an integrating factor of \^Edu -f- V(?du, and a function MI be defined by the equation \(\fEdu + VGdv) dwi, the above equation becomes ds 2 = duf. Hence, if we A"** take for parametric curves u\ const, and any other system for vi = const. , we have FI = 0, GI = 0. In other form these equations are 3v In accordance with the last equation we put 01?! 2 8i?i where A; is undetermined. By integration we have r X, M, v being functions of HI alone. When these values are substituted in the first of the above equations of condition, we get to be satisfied by X, /A, and v. The equation of the tangent plane to the surface (i) is reducible to (1 - W]l 2 ) (X - x) + i (1 + w- 2 ) ( Y - y) + 2 Ml (Z - z) = 0. Hence the surface is developable. Since its edge of regression is a minimal curve (Ex. 4, p. 69), the theorem is proved. The surface is called an isotropic developable. 32. Transformation of coordinates. It is readily found that the functions E, F, G are unaltered in value by any change of the rectangular axes. But now we shall show that these functions change their values when there is a change of the curvilinear coordinates. TRANSFORMATION OF COORDINATES 73 Let the transformation of coordinates be defined by the equations (8) u = u(u^ Vj), v = v(u l ,v l ); then we have dx _ dx du dx dv dx _ dx du dx dv du, du du, dv du. dv. du dv, dv dv i i i i 11 we find the relations (9) E du dv fa dv du dv dv. dv, du, 1 11 d^ dv l Hence the fundamental quantities of the first order assume new forms when there is a change of curvilinear coordinates. From (8) we have, by differentiation, du du dv dv du = du l -\ aVj, dv du. + - dv.. du. dv. du, dv, 11 11 Solving these equations for du v dv^ we get l/dv , du , \ 1 / dv , . du where (10) Hence we have d(u, v) du 8 dv l dv (du \ "du so that (12) 1 ^ From (9) we find the relation LINEAR ELEMENT OF A SURFACE By means of this equation and the relations (11), we can transform equations (9) into the following : (13) ^^\^2F^ l + G( 1 du EG I* E fa 1 to 1 __ F (faito 1 < i- 0V, du cu^ ct\ cu cu r^ 1 EG- cv CM ,T /" Jj<2 Tfi S~1 WA 1^1 1 JG/Cr .T 33. Angles between curves. The element of area. Upon a para metric line v = const, we take for positive sense the direction in which the parameter u increases, and likewise upon a curve u = const, the direction in which v increases. If efe,, and e?s M denote the elements of arc of curves v const, and u = const, respec tively, we find, from (4), (14) ds v = ^Edu, ds u = ^Gdv. Hence, if a w , #, y v and M , /9 M , y u denote the direction-cosines of the tangents to these curves respectively, we have fa du 1 dx cu "*</ E d u cy < 1 dz We have seen that through an ordinary point of a surface there passes one curve of parameter u and one of parameter v. If, as in fig. 11, &) denotes the angle, between and 180, formed by the positive directions of the tangents to these curves at the point, we have (15) and (16) cos ft) = aa + , + 77 W = -7== sin &) = VJSQ--F* H ANGLES BETWEEN CUKVES 75 When two families of curves upon a surface are such that through any point a curve of each family, and but one, passes, and when, moreover, the tangents at a point to the two curves through it are perpendicular, the curves are said to form an orthogonal system. From (15) we have the theorem: A necessary and sufficient condition that the parametric lines upon a surface form an orthogonal system is that F = 0. Consider the small quadrilateral (fig. 11) whose vertices are the points with the curvilinear coordinates (u, v), (u -f du, v), (u, v + dv), (u + du, v + dv). To within terms of higher order the opposite sides of the figure are equal. Consequently it is approxi mately a parallelogram whose sides are of length v E du and \ G dv and the included angle is o>. The area of this parallelogram is called the element of area of the surface. Its expression is (17) d^ = sin CD V EG dudv = H dudv. /(u+du.v) FIG. 11 If C is any curve on a surface, the direction-cosines a, its tangent at a point have the form dy /dy du dy dv /-/ __ _. __ ^ ,-^-- I ^4^ n , _ __1 g< _ ds \cu ds dv ds 7 of dx /dx du dx dv _ _ I _ _ _ _ I _ _ _ _ ds cu ds dv ds dz _ /dz du cz dv J ~ ds~\^uds dv ~ds If we put dv/du = X and replace ds by the positive square root of the right-hand member of (4), the above expressions can be written dv (18) du dv 7 = du dv 76 LINEAR ELEMENT OF A SURFACE From these results it is seen that the direction-cosines depend not upon the absolute values of du and dv, but upon their ratio X. The value of X is obtained by differentiation from the equation of (7, namely (19) Let C l be a second curve meeting C at a point M, and let the direction-cosines of the tangent to C l at M be a v /3 V y r They are given by " l du 8s dvW and similar expressions for /3 t and 7^ where 8 indicates variation in the direction of C r If 6 denotes the angle between the positive directions to C and C l at M, we have, from (18) and (20), Eduu + F(du 8v -f dv 8u] +Gdv8v (21) cos = #tf x + ppj + 77j = i j-^ and sin 9 = Vl - cos = H (8u dv 8v du 8s ds 8s ds This ambiguity of sign is due to the fact that 6 as denned is one of two angles which together are equal to 360. We take the upper sign, thus determining 6. This gives /nft . . Q Tr /8udv 8v du\ (22) sm6 = H -_ -__. \09 d8 08 ds/ The significance of the above choice will be pointed out shortly. When in particular C l is the curve v = const, through M, we have 8v = and 8s = V E 8u, so that /00 . ~ 1 { -^du T,dv\ . a If dv (23) cos<9 = = j + .F U 81X100=-= ds ds/ From these equations we obtain (24) tan* = Edu+Fdv The angle co between the positive half tangents to the para metric curves has been uniquely denned. Hence there is, in ANGLES BETWEEN CURVES 77 general, only one sense in which the tangent to the curve v = const, can be brought into coincidence with the tangent to the curve u = const, by a rotation of amount co. We say that rotations in this direction are positive, in the opposite sense negative. From (23) it is seen that is the angle described in the positive sense when the positive half tangent to the curve v = const, is rotated into coincidence with the half tangent to C. And so in the general case 6, defined by (22), is the angle described in the positive rota tion from the second curve to the first. From equations (15), (16), and (23) we find (26) These equations follow also directly from (20) and (21) by consid ering the curve u = const, as the second line. As an immediate consequence of equation (21) we have the theorem : A necessary and sufficient condition that the tangents to two curves upon a surface at a point of meeting be perpendicular is (26) E du Su + F(du Sv + dv 8u) + G dv $v = 0. EXAMPLES 1. Show that when the equation of a surface is of the form z =/(, y), its linear element can be written ds 2 = (1 + p 2 ) dx 2 + 2pqdxdy + (l + q 2 ) dy 2 , where p = dz/dx, and q = cz/dy. Under what conditions do the lines x = const. , y = const, form an orthogonal system ? 2. Show that the parametric curves on the sphere x = a sin u cos v, y = a sin u sin u, z = a cos u form an orthogonal system. Determine the two families of curves which meet the curves v = const, under the angles ir/4 and 3 7r/4. Find the linear element of the surface when these new curves are parametric. 3. Find the equation of a curve on the paraboloid of revolution x = wcosu, y = itsinu, z = w 2 /2, which meets the curves v = const, under constant angle a and passes through two points (M O , i>o), (MI, i). Determine a as a function of 4. Find the differential equation of the curves upon the tangent surface of a curve which cut the generators under constant angle a. 78 LINEAR ELEMENT OF A SURFACE . 5. Show that the equations of a curve which lies upon a right cone and cuts all the generators under the same angle are of the form x = ce " cosu, y cesinw, 2 = 6e", where a, 6, and c are constants. What is the projection of the curve upon a plane perpendicular to the axis of the cone ? Find the radius of curvature of the curve. 6. Find the equations of the curves which bisect the angles between the para metric curves of the paraboloid in Ex. 3. 34. Families of curves. An equation of the form (27) <(w, v)=c, where c is an arbitrary constant, defines an infinity of curves, or a family of curves, upon the surface. Through any point of the sur face there passes a curve of the family. For, given the curvilinear coordinates of a point, when these values are substituted in (27) we obtain a value of c, say C Q ; then evidently the curve $ = c passes through the point. We inquire whether this family of curves can be defined by another equation. Suppose it is possible, and let the equation be (28) ^(U,V) = K. Since c and K are constant along any curve and vary in passing from one curve to another, each is necessarily a function of the other. Hence i|r is a function of fa Moreover, if ifr is any function of fa equations (27) and (28) define the same family of curves. From equation (24) it is seen that the direction, at any point, of the curve of the family through the point is determined by the value of dv/du. We obtain the latter from the equation 36 , d<f> _ (29 ) g*H.2*.i.<* which is derived from (27) by differentiation. Let $ (u, v) = cbe an integral of an ordinary differential equation of the first order and first degree, such as (30) M(u, v) du 4- N(u, v) dv = 0. The curves defined by the former equation are called integral curves of equation (30). From the integral equation we get equation (29) by differentiation. It must be possible then to obtain equation (30) FAMILIES OF CURVES 79 from the integral equation and (29). But c does not appear in (29), consequently the latter equation differs from (30) by a factor at most. Hence M N = 0. Suppose, now, that we have dv du o . o , another integral of (30), as ^Hw, v) = e. Then M -^- N -^- = 0. dv cu The elimination of M and N from these equations gives ^ ^ = ; d(u,v) from which it follows that ty is a function of </>. Moreover, ^r can by any function of <. But we have seen that if ^ is a function of <, the families of curves </> = const, and -fy = const, are the same. Hence all integrals of equation (30) of the form <f>=c or ^ = e define the same family of curves. However, equation (30) may admit of an integral in which the constant of integration enters implicitly, as F(u, t>, c) = 0. But if this be solved for <?, we obtain one or more integrals of the form (27). Hence an equation of the form (30) defines one family of curves on a surface. Although the determination of the curves when thus defined requires the integration of the equation, the direction of any curve at a point is given directly by means of (24). If at each point of intersection of a curve C l with the curves of a family the tangents to the two curves are perpendicular to one another, C l is called an orthogonal trajectory of the curves. Sup pose that the family of curves is defined by equation (30). The 7 r\ relation between the ratios and > which determine the direc- du ou tions of the tangents to the two curves at the point of intersection, r\ ~~\/T is given by equation (26). If we replace by , we obtain cu A (31) (EN- FM) du + (FN GM) dv = 0. But any integral curve of this equation is an orthogonal trajectory of the given curves. Hence a family of curves admits of a family of orthogonal trajectories. They are defined by equation (31), when the differential equation of the curves is in the form (30). But when the family is defined by a finite equation, such as (27), the equation of the orthogonal trajectories is (32) 80 LINEAR ELEMENT OF A STJKFACE As an example, we consider the family of circles in the plane with centers on the x-axis whose equation is (i) x 2 + y 2 - 2 ux - a 2 , where u is the parameter of the family and a is a constant. In order to find the orthogonal trajectories of these curves, we take the lines x = const. , y = const. for parametric curves, in which case E = G = 1, F = 0, 3? and write the equation (i) in the form (27), thus x + i (y 2 - a 2 ) = 2 u. x Now equation (32) is 2 xy dx (x 2 y 2 + a 2 ) dy = 0, of which the integral is where v is the constant of integration. Hence the orthogonal trajectories are circles whose centers are on the y-axis. An ordinary differential equation of the second degree, such as (33) H (u, v) du 2 + 2 S(u, v) du dv + T(u, v) dv 2 = 0, is equivalent to two equations of the first degree, which are found by solving this equation as a quadratic in dv. Hence equation (33) defines two families of curves upon the surface. We seek the con dition that the curves of one family be the orthogonal trajectories of the other, or, in other words, the condition that (33) be the equa tion of an orthogonal system, as previously defined. If & x and Jc 2 denote the two values of - obtained from (33), we have du From (26) it follows that the condition that the two directions at a point corresponding to K I and K Z be perpendicular is E + FK + tc + GK = 0. If the above values are substituted in this equation, we have the condition sought; it is (34) MINIMAL CURVES 81 35. Minimal curves on a surface. An equation of the form (33) is obtained by equating to zero the first fundamental form of a surface. This gives Edv?+ ZFdudv + Gdv 2 = 0, and it defines the double family of imaginary curves of length zero which lie on the surface. In this case equation (34) reduces to JSG F 2 =0; hence the minimal lines on a surface form an orthogonal system only when the surface is an isotropic develop able ( 31). An important example of these lines is furnished by the system on the sphere. If we take a sphere of unit radius and center at the origin, its equation, x 2 + y*+ z 2 = l, can be written in either of the forms 1 z x -f iy where u and v denote the respective ratios, and evidently are conju gate imaginaries. If these four equations are solved for z, ?/, z, we find u -h v i(v u) uv 1 s** -, A i i -.--, \ f_ A , 9 ~uv+l uv +\ uv+1 From these expressions we find that the linear element, in terms of the parameters u and v^ is given by ,o 4:dudv (36) (1 + Hence the curves u = const, and v const, are the lines of length zero. Eliminating u from the first two and the last two of equations (35), we get x 4- (1 v*)y 2 iv = 0, (37) 1 i(i? + l)z + 2vy+i(l-v z ) = 0. Hence all the points of a curve v = const, lie on the line v z )Y 2iv = Q, 82 LINEAR ELEMENT OF A SURFACE where X, F, Z denote current coordinates. In consequence of (35), these equations can be written X-x, Y-y. Z-z, where # , y^ z are the coordinates of a particular point. In like manner the curves u = const, are the minimal lines X-x. Y-y. , = Z-z, EXAMPLES 1 . Show that the most general orthogonal system of circles in the plane is that of the example in 34. 2. Show that on the right conoid x = ucosv, y = usinv, z = au, the curves dw 2 (w 2 -f a 2 ) dv 2 = form an orthogonal system. 3. When the coefficients of the linear elements of two surfaces, ds 2 = Erfu* + 2 Fidudv + Gidv 2 , ds* = E 2 du? + 2 F 2 dudv + G 2 di; 2 , are not proportional, and points with the same curvilinear coordinates on each of the surfaces are said to correspond, there is a unique orthogonal system on one surface corresponding to an orthogonal system on the other; its equation is (Fi^a - FzEi)du* + (E a Gi - EiG z ) dudv +(GiF z - G 2 Fi)du 2 = 0. 4. If 61 and 2 are solutions of the equation a^ At/ _ u, dag/3 2 da 8? where X is any function of a and /J, the equations +*5 3 2* . define a surface referred to its minimal lines. 36. Variation of a function. Let S be a surface referred to any system of coordinates ?/, v, and let <j> (w, v) be a function of u and v. When the values of the coordinates of a point Tlif o f the surface are substituted in <, we obtain a number c ; and consequently the curve (38) VABIATION OF A FUNCTION 83 passes through M. In a displacement from M along this curve the value of (f> remains the same, but in any other direction it changes and the rate of change is given by d d$ k dc) du dv where k dv/du determines the direction. As thus written it is understood that the denominator of the right-hand member is positive. For the present we consider the absolute value of ~-t and write ds (39) ds du dv where e is 1 according as the sign of the numerator is positive or negative. The minimum value of A is zero and corresponds to the direction along the curve (38). In order to find the maximum value we equate to zero the derivative of A with respect to Jc. This gives From (32) it follows that this value of k determines the direction at right angles to the tangent to $ = c at the point. By substituting this value of k in (39) we get the maximum value of A. Hence: The differential quotient -^- of a function <f> (w, v) at a point on a ds surface varies in value with the direction from the point. It equals zero in the direction tangent to the curve <f>= c, and attains its greatest absolute value in the direction normal to this curve, this value being m^\-ZF-z-z-+G dv du dv < 4 > ; S A means of representing graphically the magnitude of the differ ential quotient A for any direction is given by the following theorem : If in the tangent plane to a surface at a point M the positive half tangents at M, corresponding to all values of k, positive and negative, 84 LINEAR ELEMENT OF A SURFACE be drawn, and on them the corresponding lengths A be laid off from M, the locus of the extremities of these lengths is a circle tangent to the curve <= const. The proof of this theorem is simplified if we effect a transfor mation of curvilinear coordinates. Thus we take for the new coor dinate lines the curves (f> const, and their orthogonal trajectories. We let the former be denoted by u v const, and the latter by v l = const., and indicate by subscript 1 functions in terms of these parameters. Now F l = 0, so that -t J. 7 where \ denotes the value of dvjdu^ which determines a given direction, and the maximum length is (J&\)~*. From (23) we have cos = . sin = where 6 Q is the angle which the given direction makes with the tangent to the curve v l = const. Hence if we regard the tangents at M to the curves v l const, and u l = const, as axes of coordinates in the tangent plane, the coordinates of the end of a segment of length A are The distance from this point to the mid-point of the maximum segment, measured along the tangent to v t = const., is readily found to be - =<> which proves the theorem. 37. Differential parameters of the first order. If we put (41) A^ = equation (40) can be written (3) -** where now the differential quotient corresponds to the direction normal to the curve <> = const. The left-hand member of this DIFFERENTIAL PARAMETERS 85 equation is evidently independent of the nature of the parameters u and v to which the surface is referred. Consequently the same is true of the right-hand member. Hence A^ is unchanged in value when there is any change of parameters whatever. The full significance of this result is as follows. Given a new set of parameters defined by M=/ I (M I , i^), v=/ 2 (w 1 , v^\ let ^(u^ vj denote the result of substituting these expressions for u and v in < (w, v), and write the linear element thus : ds 2 = E l du* + 2F l du l dv l + G l dv*. The invariance of A^ under this transformation is expressed by the identical equation , EG-F We leave it to the reader to verify this directly with the aid of equations (9). The invariant A^ is called the differential parame ter of the first order ; this name and the notation are due to Lame.* Consider for the moment the partial differential equation (42) A^ = and a solution </> = const. From the latter we get, by differentiation, d<l> , , 3$ , A - du -f dv = 0. du dv O J O J If we replace and in (42) by dv and du, which are evi dently proportional to them, we obtain Edu*+ 2 Fdudv + Gdv*= 0. Hence the integral curves of equation (42) are lines of length zero, and conversely if (/> = const, is a line of length zero, the function </> is a solution of equation (42). Another particular case is that in which A^ is a function of <, say (43) A,* = *<*). * Lemons sur les coordonnees curvilignes et leurs diverses applications, p. 5. Paris, 1859. 86 LINEAR ELEMENT OF A SURFACE From (41) it is seen that when we put equation (43) becomes (44) A 1 l9=l. As denned, 6 is a function of $; hence the family of curves 6 const, is the same as the family </> = const. Suppose we have such a family, and we take the curves 9 const, for the curves u = const, and their orthogonal trajectories for v = const., thus effecting a change of parameters. Since Aj%=l, it follows from (41) that ^ = 1, and consequently the linear element is (45) ds 2 =du? + Gdv 2 . Since now the linear element of a curve v const, is du, the length of the curve between its points of intersection with two curves u = u and u = u^ is u^ U Q . Moreover, this length is the same for the segment of every curve v const, between these two curves. For this reason the latter curves are said to be parallel. Con versely, in order that the curves u = const, of an orthogonal sys tem be parallel, it is necessary that the linear element of the curves v = const, be independent of v. Hence E must be a func tion of u alone, which, by a transformation of coordinates, can be made equal to unity. Hence we have the theorem : A necessary and sufficient condition that the curves of a family (/> = const, be parallel is that \(f> be a function of <f>. Let (f) = const, and ^ = const, be the equations of two curves upon a surface, through a point M, and let 6 denote the angle between the tangents at M. If we put E _ F + G dv dv \dv du du dv / du du (46) \(*,*)= -- EG -I" the expression (21) for cos 6 can be written (47) cos , DIFFERENTIAL PARAMETERS 87 Since cos 6 is an invariant for transformations of coordinates, it follows from this equation that A x (0, ^r) also is an invariant. It is called the mixed differential parameter of the first order. An imme diate consequence of (47) is that A,(*. * ) = is the condition of orthogonality of the curves (f> = const, and i/r = const. Now equation (22) can be written r \du dv dv c which by means of the function <e) (w, v), defined thus by Darboux,* can be written in the abbreviated form (49) sin 6 = Since all the functions in this identity except (<^, ^r) are known to be invariants, we have a proof that it also is an invariant. It is a mixed differential parameter of the first order. From (47) and (49) it follows that (50) A, 2 (<, f ) + 2 ((/>, t) - A^ A^ ; consequently the three invariants denned thus far are not inde pendent of one another. From (41) and (46) it follows that ri _ Tf< jfi ^ U = W A ^ ^=l^ AlV = ^ and from these we find (51) 2 (u, v) = A,H V - A i 2 (. ) = ^ Consequently @ 2 (w, t;) 2 (w, v) Hence i, ^, and G 1 are differential invariants of the first order. * Lemons, Vol. Ill, p. 197. 88 LINEAR ELEMENT OF A SURFACE Another result of these equations is the following. If the param eters of the surface are changed in accordance with the equations Ui = Ui(u, v), v^v^u, v), and the resulting linear element is written, ds* = E l du* + 2 F l du^dv^ + 6^ dv*, the value of E l is given by and 7^ and 6^ are found in like manner. In consequence of (51) these equations are equivalent to (13), which were found by direct calculation. 38. Differential parameters of the second order. Thus far we have considered differential invariants of the first order only. We introduce now one of the second order, discovered by Beltrami.* To this end we study the integral n = for an ordinary portion of the surface bounded by a closed curve C (cf. 33). For convenience we put G z_ F d E z_ F 3 * du dv dv du (53) M= - - - , N= - - , so that, in consequence of (46), we have This may be written If we apply Green s theorem to the first integral, this equation reduces to (54) n= C(j>(Mdv-Ndu)- ff$(^+ ( j^} dudv > *Ricerche di analisi applicata alia geometria, Giornale di matematiche, Vol. II (1864), p. 365. DIFFERENTIAL PARAMETERS 89 where the first integral is curvilinear and is taken about C in the customary manner. Evidently du and dv refer to a displacement along C. If we indicate by 8 variations in directions normal to C and directed toward the interior of the contour, then from (23) and (25) it follows that Edu + Fdv _-H8v F du + G dv __ ~dT ~ST ds 8s Hence M dv du Ss dv 8*/ 8s All of the terms in this equation, with the exception of ( - h -r ) > H\du dv J are independent of the choice of parameters. Hence the latter is an invariant. It is called the differential parameter of the second order and is denoted by A 2 i/r. In consequence of (53) we have ..... (56) In the foregoing discussion it has been assumed that only real quantities appear. But all these results can be obtained directly from algebraic considerations of quadratic differential forms * without any hypothesis regarding the character of the variables ; hence the differential parameters can be used for any kind of curvilinear coordinates. In addition to A 2 c/> there are other differential invariants of the second order, such as And AA Q, i/r), A, (A^, A^), (A are mixed invariants of the second order. In like manner we can find a group of invariants of the third order ; for instance, AAM>, AA(4>.M>)> A.A,*, A A*. * Cf. Bianchi, Lezioni di geometria differ enziale, Vol. I, chap. ii. Pisa, 1902. 90 LINEAR ELEMENT OF A SURFACE These invariants and others, which can be obtained by an evident extension of this method, involve functions c/>, A/T, . , E, F, G, and their derivatives. Conversely, we shall show * that every invariant of the form T ftw v r dE dG *W I ^ /== /(A ^ tr, -, -, 0, -^-, -, i/r, -^L, ), dw di> dw du where <, -^, . . . are independent functions, is expressible by means of the symbols A and . Already we have seen that E, F, and G can be expressed in terms of A x w, A x v, and A 1 (w, v). Moreover, from (48) it follows that when X is any function whatever. Hence all the terms in / can be expressed in terms of the symbols A and , applied to Since u and v do not appear explicitly in /, we can effect a change of parameters, replacing u and v by </> and ty respectively, and con sequently we express / in terms of (/>, ^, , and the differential invariants obtained by applying the operators A and <*) to these functions. In case $ is the only function appearing in /, we can take for i/r, in the change of parameters, any invariant of c/>, such as A^ or A 2 <, so long as it is not a function of (/>, E, F> or G. EXAMPLES 4 1 . When the linear element of a surface is in the form ds 2 = \(du^ + dv^), where X is a function of u and D, both u and v are solutions of the equation A 2 = 0, the differential parameter being formed with respect to the right-hand member. 2. Show that on the surface x = u cos u, y = u sin v, z = av -f- (u), the curves it = const, are parallel. 3. When the linear element is in the form ds 2 = cos^adu 2 + sin 2 a: eh? 2 , where a is a function of u and u, both u and v are solutions of the equation * Cf. Beltrami, I.e., p. 357. SYMMETRIC COORDINATES 91 4. If the curves = const., \p = const, form an orthogonal system on a surface, the projection on the x-axis of any displacement on the surface is given by dx d\b dx dd> dx = - = + 2= , * A0 where ds and da- are the elements of length of the curves = const., ^ = const. respectively. 5. If /and are any functions of u and u, then . a/ a0 i if a0 /3A A . , a/a0. , 0) = ^- --^ AIU + (^ - + - - A! (w, u) + ^ ^ Aii>, du du \cu cv cv du/ cv cv A 2 / = ^A 2 w + ^A 2 u + ^A lW + 2^- Ai(u, v) + ^AiU. CM CU SU 2 0ttCtJ SV 2 39. Symmetric co drdinates. We have seen that through every point of a surface there pass two minimal curves which lie entirely on the surface, and that these curves are defined by the differential equation Edv?+2 Fdudv + G dv 2 = 0. If the finite equations of these curves be written a (w, v) = const., fi (w, v) = const., it follows from (42) that (5T) A, ()=<), A 1 (/3) = 0. Since for any parameters /^\ w = when the curves a ~ const., ft = const., are taken as parametric, the corresponding coefficients E and G are zero, and consequently the linear element of the surface has the form (59) ds 2 = \ dad/3, where, in general, X is a function of a and {3. Conversely, as fol lows from (58), when the linear element has the form (59) equa tions (57) are satisfied and the parametric curves are minimal. Hence the only transformations of coordinates which preserve this form of the linear element are those which leave the minimal lines parametric, that is (60) or a = - 92 LINEAR ELEMENT OF A SURFACE where F and F l are arbitrary functions. Whenever the linear ele ment has the form (59), we say that the parameters are symmetric. The above results are given by the theorem : When a and ft are symmetric coordinates of a surface, any two arbitrary functions of a and ft respectively are symmetric coordi nates, and they are the only ones. The general linear element of a surface can be written as the product of two factors, namely (61) d**: If t and t 1 denote integrating factors of the respective terms of the right-hand member of this equation, a pair of symmetric coordinates is given by the quadratures (62) When these values are substituted in (61), and the result is com pared with (59), it is seen that X = tt l The first of equations (62) can be replaced by , da ^FiH da> = > t du Eliminating t from these equations, we have E^-F^ dv du . ccc, < 63 > - IT =l Tu If this equation be multiplied by : ~ Z the result can be reduced to r*-o dv cu . dec ISOTHERMIC PARAMETERS 93 From these equations it follows that or, by (56), (65) A 2 tf = 0. It is readily found that /3 also satisfies this condition. 40. Isothermic and isometric parameters. When the surface is real, and the coordinates also, the factors in (61) are conjugate imaginary. Hence the conjugate imaginary of t can be taken for t r In this case a and fi are conjugate imaginary also. In what follows we assume that this choice has been made, and write (66) a = <+ty, = </> iyfr. If these values be substituted in (59), we get (67) ds* = \(d<t> 2 +d^). At once we see that the curves c/> = const, and ^r = const, form an orthogonal system. Moreover, the elements of arc of these lines are V\d-*fr and ^\d(f> respectively. Consequently when the increments d<f> and d^ are taken equal, the four points (<, i/r), (<f) -f c?<, i/r), ($, i/r -f efo/r), (< -f tity, ^ -f eityr) are the vertices of a small square. Hence the curves (f> = const, and ^|r = const, divide the surface into a network of small squares. On this account these curves are called isometric curves, and <f> and ty isometric parameters. These lines are of importance in the theory of heat, and are termed isothermal or isothermic, which names are used in this connection as synonymous with isometric. Whenever the linear element can be put in the symmetric form, equations similar to (66) give at once a set of isometric parameters. And conversely, the knowledge of a set of isometric parameters leads at once to a set of symmetric parameters. But we have seen that when one system of symmetric parameters is known, all the others are given by equations of the form (60). Hence we have the theorem : Given any pair of real isometric parameters <, -v/r for a surface ; every other pair < x , ty 1 is given by equations of the form where F and F Q are any functions conjugate imaginary to one another. 94 LINEAR ELEMENT OF A SURFACE Consider, for instance, the case (68) * 1 +^ 1 = ^(0+i. From the Cauchy-Riemann differential equations (69) ?*i = *i, ?& = _?*i, d<l> c^ 3^ 8<f> it follows that (f) l and ^ l are functions of both < and T/T. Hence the curves </> 1 = const., ^ 1 = const, are different from the system <f> const., T/T = const. Similar results hold when + i is replaced by i in the argument of the right-hand member of (68). Hence There is a double infinity of isometric systems of lines upon a sur face; when one system is known all the others can be found directly. If the value (66) for a be substituted in the first of equations (57), the resulting equation is reducible to Since <f> and ^r are real, this equation is equivalent to (70) A^A.VT, A 1 (^,f) = o. From (58) it is seen that these equations are the condition that E G, F= 0, when <f> and i/r are the parameters. Hence equations (70) are the necessary and sufficient conditions that $ and i/r be isometric parameters. Again, when a in (65) is replaced by </>+ i^r, and all the func tions are real, we have f (71) A 2 *=0, Conversely, when we have a function (f> satisfying the first of these equations, the expression cu cv , on ov , dv is an exact differential. Call it d^r ; then jr^ E G F d u _ dv^_c^r du c)v H ~~du H ISOTHERMIC ORTHOGONAL SYSTEMS 95 If these equations be solved for we get du dv /r _ ox dv du dd> dv du d<f> (<o) = "> = H du H dv When we express the condition ( ) = ( ) > we find that dv\duj du\dvj A 2 -/r=0. Moreover, these two functions (f> and ^ satisfy (70), in consequence of (72) and (73), and therefore they are isometric parameters. Hence : A necessary and sufficient condition that <f> be the isometric param eter of one family of an isometric system on a surface is that A 2 c/> = ; the isometric parameter of the other family is given by a quadrature. Incidentally we remark that if u and v are a pair of isometric parameters, equations (72) and (73) reduce to (69). 41. Isothermic orthogonal systems. If the linear element of a surface is given in the form (67) and the parameters are changed in accordance with the equations the linear element becomes where the accents indicate differentiation. However, this trans formation of parameters has not changed the coordinate lines ; the coefficients are now no longer equal, but in the relation <> i-f where U and V denote functions of u and v respectively. Conversely, when this relation is satisfied the linear element may be written and by the transformation of coordinates, (75) 4> = C^/lfdu, ^ = C 96 LINEAR ELEMENT OF A SURFACE it is brought to the form (67), whatever be U and V\ and the coor dinate lines are unaltered. Hence : A necessary and sufficient condition that an orthogonal system of parametric lines on a surface form an isothermic system is that the coefficients of the corresponding linear element satisfy a relation of the form (74). We seek now the necessary and sufficient condition which a function o> (w, v) must satisfy in order that the curves o> = const. and their orthogonal trajectories form an isothermic system. Either o>, or a function of it, is the isothermic parameter of the curves o> = const. We denote this parameter by </>; then </>=/() . Since </> must be a solution of equations (71), we have, on substitution, (76) A 2 o> ./ (G>) + \a> ./"(a>) = 0, where the primes indicate differentiation with respect to &>. If this equation is written in the form we see that the ratio of the two differential parameters is a func tion of co. Conversely, if this ratio is a function of o>, the function /(a>), obtained by two quadratures from (77) / (*>) = *-/>, will satisfy equations (71). Hence: A necessary and sufficient condition that a family of curves a) = const, and their orthogonal trajectories form an isothermic sys tem is that the ratio of A 2 &> and AjO) be a function of &>. Suppose we have such a function w ; then the orthogonal tra jectories of the curves &> = const, can be found by quadrature ; for, the differential equation of these trajectories is (78 ^ \~ dv - du/ "- \~ dv - Su If equation (76) be written in the form I *f> dv du v i , . wu t/i/ n f (&)) H I r (ft)) = u, O I *^ \ / TT- * r\ I 4/ \ / -TT- <7V ISOTHEEMIC ORTHOGONAL SYSTEMS 97 it is seen that an integrating factor of equation (78) is f (a))/H, where f (co) is given by (77). Hence /() and the function <f, obtained by the quadrature __#&> ^cto s^fo -rJo<* are a pair of isometric parameters. From these equations and (77) it follows that and consequently, by means of (52), the linear element can be given the form (80) ds* = (da* + 8 /sS d " ctyA A^ x The linear element of the plane referred to rectangular axes is ds 2 = dx 2 4- dy 2 . Consequently x and y are isothermic parameters, and we have the theorem : The plane curves whose equations are obtained by equating to constants the real and imaginary parts of any function of x + iy or x - iy form an isothermal orthog onal system ; and every such system can be obtained in this way. c 2 For example, consider 4- ty = -- x iy where c is any constant. From this it follows that x 2 4- y z x 2 4- y 2 Hence the circles = const., $ = const, form an isothermal orthogonal system, and and ^ are isothermic parameters. The above system of circles is a particular case of the system considered in 34. We inquire whether the latter also form an isothermal system. If we put u = x 4- i (2/ 2 - 2 ), 1 2d) we find that AIO> = (w 2 4- 4 a 2 ), A^u = x 2 x 2 Hence the ratio of AIW and A2W is a function of w, and consequently the system of circles is isothermal. From (77) it follows that the isothermic parameter of the first family is = tan- 1 , and the parameter of the orthogonal family is 2 a 2 a 1 w x 2 4- a 2 \b tanh -1 > w = y 4 2 a 2 a y 98 LINEAR ELEMENT OF A SURFACE EXAMPLES 1. Show that the meridians and parallels on a sphere form an isothermal orthog onal system, and determine the isothermic parameters. 2. Show that a system of confocal ellipses and hyperbolas form an isothermal orthogonal system in the plane. 3. Show that the surface x _ I (a 2 - u) (a^v) y _ I (b* - u) (b 2 - v) z _ ! a ~ \ (a 2 - & 2 ) (a 2 - c 2 ) b ~ \ (6 2 - a 2 ) (6 2 - c 2 ) c ~~ \ ( C 2 _ U ) ( C 2 _ f(c 2 -a 2 )(c 2 -6 2 ) is an ellipsoid, and that the parametric curves form an isothermal orthogonal system. 4. Find the curves which bisect the angles between the parametric curves on the surface % _ u + v y _ u _ v _ uv a~ 2 b~"~2~ = 2 and show that they form an isothermal orthogonal system. 5. Determine (v) so that on the right conoid x u cos v, y = u sin v, z = <f> (v) the parametric curves form an isothermal orthogonal system, and show that the curves which bisect the angles between the parametric curves form a system of the same kind. 6. Express the results of Ex. 4, page 82, in terms of the parameters and ^ defined by (66). 42. Conformal representation. When a one-to-one correspond ence of any kind is established between the points of two sur faces, either surface may be said to be represented on the other. Thus, if we roll out a cylindrical surface upon a plane and say that the points of the surface correspond to the respective points of the plane into which they are developed, we have a representa tion of the surface upon the plane. Furthermore, as there is no stretching or folding of the surface in this development of it upon the plane, lengths of lines and the magnitude of angles are unal tered. It is evidently impossible to make such a representation of every surface upon a plane, and, in general, two surfaces of this kind do not admit of such a representation upon one another. However, it is possible, as we shall see, to represent one surface upon another in such a way that the angles between correspond ing lines on the surfaces are equal. In this case we say that one surface has conformal representation on the other. In order to obtain the condition to be satisfied for a conformal representation of two surfaces S and S r , we imagine that they are referred to a corresponding system of real lines in terms of the CONFOKMAL KEPKESENTATION 99 same parameters w, v, and that corresponding points have the same curvilinear coordinates. We write their linear elements in the respective forms ds 2 = Edu*+ 2 Fdudv + G dv 2 , ds 2 = du*+ 2 F dudv + G dv*. Since the angles co and to between the coordinate lines at corre sponding points must be equal, it is necessary that F F (81) y/EG If and Q denote the angles which a curve on S and the corre sponding curve on S r respectively make with the curves v = const. at points of the former curves, we have, from (23) and (25), . n H dv . . Q H du sin = - , sin (to - ) = = ds . a , H dv , , ai\ H du sm 6 = =, sin (to 1 - 6[) = -= s V <&? By hypothesis a> r =a) and 6[ Q , according as the angles have the same or opposite sense. Hence we have H - H - H du ~ ds ~ according to the sense of the angles. From these equations we find which, in combination with (81), may be written where t 2 denotes the factor of proportionality, a function of u and v in general. From (83) it follows at once that (84) ds *=t * And so when the proportion (83) is satisfied, the equations (81) and (82) follow. Hence we have the theorem : A necessary and sufficient condition that the representation of two surfaces referred to a corresponding system of lines be conformal is 100 LINEAR ELEMENT OF A SUKPACE that the first fundamental coefficients of the two surfaces be propor tional, the factor of proportionality being a function of the param eters ; the representation is direct or inverse according as the relative positions of the positive half tangents to the parametric curves on the two surfaces are the same or symmetric. Later we shall find means of obtaining conformal representations. From (84) it follows that small arcs measured from correspond ing points on S and S f along corresponding curves are in the same ratio, the factor of proportionality being in general a function of the position of the point. Conversely, when the ratio is the same for all curves at a point, there is a relation such as (84), with t a function of u and v at most. And since it holds for all directions, we must have the proportion (83). On this account we may say that two surfaces are represented conformally upon one another when in the neighborhood of each pair of homologous points corre sponding small lengths are proportional. 43. Isometric representation. Applicable surfaces. When in par ticular the factor t is equal to unity, corresponding small lengths are equal as well as angles. In this case the representation is said to be isometric, and the two surfaces are said to be applicable. The significance of the latter term is that the portion of one surface in the neighborhood of every point can be so bent as to be made to coincide with the corresponding portion of the other surface with out stretching or duplication. It is evident that such an applica tion of one surface upon another necessitates a continuous array of surfaces applicable to both S and $ r This process of transformation is called deformation, and S l is called a deform of S and vice versa. An example of this is afforded by the rolling of a cylinder on a plane. Although a conformal representation can be established between any two surfaces, it is not true, as we shall see later, that any two surfaces admit of an isometric representation upon one another. From time to time we shall meet with examples of applicable sur faces, and in a later chapter we shall discuss at length problems which arise concerning the applicability of surfaces. However, we consider here an example afforded by the tangent surface of a twisted curve. APPLICABLE SURFACES 101 We recall that if #, y, z are the coordinates of a point on the curve, expressed in terms of the arc, the equations of the surface are of the form f = x + ^ v = y + y t, =z + z% and the linear element of the surface is d <r* = /I + -\ ds z + 2 dsdt + dt\ where p denotes the radius of curvature of the curve. Since this expression does not involve the radius of torsion, it follows that the tangent surfaces to all curves which have the same intrinsic equation p =f(s) are applicable in such a way that points on the curves determined by the same value of s correspond. As there is a plane curve with this equation, the surface is appli cable to the plane in such a way that points of the surface corre spond to points of the plane on the convex side of the plane curve. The tangents to a curve are the characteristics of the osculating planes as the point of osculation moves along the curve, and con sequently they are the axes of rotation of the osculating plane as it moves enveloping the surface. Instead of rolling the plane over the tangent surface, we may roll the surface over the plane and bring all of its points into coincidence with the plane. It is in this sense that the surface is developable upon a plane, and for this reason it is called a developable surface (cf. 27). Later it will be shown that every surface applicable to the plane is the tangent surface of a curve ( 64). 44. Conformal representation of a surface upon itself. We return to the consideration of conformal representation, and remark that another consequence of equations (83) is that the minimal curves correspond upon S and S r . Conversely, when two surfaces are referred to a corresponding system of lines, if the minimal lines on the two surfaces correspond, equations (83) must hold. Hence : A necessary and sufficient condition that the representation of two surfaces upon one another be conformal is that the minimal lines correspond. If the minimal lines upon the two surfaces are known and taken as parametric, the linear elements are of the form (85) ds 2 = X dadfr ds 2 = \ da^dft v 102 LINEAR ELEMENT OF A SURFACE Hence a conformal representation is defined in the most general way by the equations W l = F(a), ft = *;(), or (87) ^ = F(ft), ft = *;(), where F and F 1 are arbitrary functions which must be conjugate imaginary when the surfaces are real. Instead of interpreting (85) as the linear elements of two sur faces referred to their minimal lines, we can look upon them as the linear element of the same surface in terms of two sets of parameters referring to the minimal lines. From this point of view equations (86) and (87) define the most general conformal representation of a surface upon itself. If we limit our considera tion to real surfaces and put, as before, a = $ + i^, = <-ty, a 1 =<^ 1 +i> 1 , ft=0 1 -*^ 1 , the functions fa i/r and fa, ^ are pairs of isothermic parameters. Now equations (86), (87) may be written (88) <#> 1+i > 1 = 7^^). Consequently we have the theorem : When a pair of isothermic parameters fa ty of a surface are known and the surface is referred to the lines <j> = const., ^r = const., the most general conformal representation of the surface upon itself is obtained by making a point (fa \fr) correspond to the point (fa, i^), into which it can be transformed in accordance with equation (88). As a corollary of this theorem, we have : When a pair of isothermic parameters is known for each of two surfaces, all the conformal representations of one surface upon the other can be found directly. Consider two pairs of isothermic parameters fa ty and fa, ^ for a surface S, and suppose their relation is (89) &+*+! = F(t + i+). If two curves C and C l are in correspondence in this representa tion, their parametric equations must be the same functional rela tion between the parameters, namely, *,) =0. CONFORMAL REPRESENTATION 103 Denote by 9 and l the angles which C and C^ make with the curves ^ = const, and ^ 1 = const, respectively. If we write the linear element of S in the two forms it follows from (23) that a deb . a cos = y = , sin = cos = - -- =, sin = From these expressions we derive the following = ^ d(f> i c?i/r so that in consequence of (89) we have (90) ..,-= :*^ where 7^ is the function conjugate to 7^, and the accents indicate differentiation with respect to the argument. If T and F x are another pair of corresponding curves, and their angles are denoted by 6 and V it follows from (90) that ,, OI> For, the right-hand member of (90) is merely a function of the position of the point and is independent of directions. Hence in any conformal representation defined by an equation of the form (89) the angles between corresponding curves have the same sense. When, now, the correspondence satisfies the equation the equation analogous to (90) is Hence l -0 l =0-0 i consequently the corresponding angles are equal in the inverse sense. 104 LINEAB ELEMENT OF A SURFACE 45. Conformal representation of the plane. For the plane the preceding theorem may be stated thus : The most general real conformal representation of the plane upon itself is obtained by making a point (x, y) correspond to the point (x^ y^), where x^iy^ is any function of x + iy or x iy. We recall the example of 41, namely 0) Xl + iyi = ;rrfc where c is a real constant. This equation is equivalent to and also to C 2 X Hence the parallels x = const, and y const., in the xy-plane, are represented in the z^-plane by circles which pass through the origin and have their centers on the respective axes. Conversely, these circles in the xy-plane correspond to the parallels in the Xi^/i-plane. If we put o;2 + y* = r 2 , x* + y* = rf, equations (ii) and (iii) may be written <*> ?-? f-S- Hence corresponding points are on the same line through the origin, and their distances from it are such that rr\ = c 2 . On this account equations (iv) are said to define an inversion with respect to the circle x 2 + y 2 = c 2 , or, since TI = c 2 /r, o transformation by reciprocal radii vector -es. From 44 it follows that corresponding angles are equal in the inverse sense. For the case c 2 (v) xi + iy\ = x + iy the equations analogous to (iv) are - = ?, V = -Vl. r n r ri* Hence the point PI (xi, y\) corresponding to P (x, y) lies on the line which is the reflection in the x-axis of the line OP, and at the distance OPi = c 2 /r. Evidently this transformation is the combination of an inversion and the transformation *i = *, y\ = - y- One finds that the transformations (i) and (v) have the following properties : Every straight line is transformed into a circle which passes through the origin ; and conversely. Every circle which does not pass through the origin is transformed into a circle. CONFOKMAL REPRESENTATION 105 We propose now the problem of finding the most general con- formal transformation of the plane into itself, which changes circles not passing through the origin into circles. In solving it we refer the plane to symmetric parameters #, fi, where a = x -f- iy, f$x iy. The equation of any circle which does not pass through the origin is of the form (91) ca(S+ aa + 5/3 + d = 0, where <z, 5, c, d are constants ; when the circle is real a and b must be conjugate imaginaries and c real. Equation (91) defines @ as a function of a. If we differentiate the equation three times with respect to #, and eliminate the constants from the resulting equations, we find (92) 3/3" 2 -2/3 /3 "=0, where the accent indicates differentiation with respect to a. Moreover, as equation (91) contains three independent constants, it is the general integral of (92). We know that the most general conformal representation of the plane upon itself is given by (93) a 1 = A(a), ft = (), or (94) ! = (), 13, = A (a). Our problem reduces, therefore, to the determination of functions A and B, such that the equation (95) 3 ft 2 - 2 ft ft" =0, where the accent indicates differentiation with respect to a v can be transformed by (93) or (94) into (92). We consider first equations (93), which we write Now ff^*!L*pta_ * 30 da da In like manner we find ft and ft". When their values are sub stituted in (95) we get, since A( and B are different from zero, 3 ft" 2 - 2 ffff" + - (3 B" 2 - 2 B B ") 4 + (3 A - 2 A[Al") ft 2 = 0. B A l 106 LINEAR ELEMENT OF A SURFACE Since equation (95) must be directly transformable into (92), it follows that (96) 3 " 2 - 2 />" " = 0, 3 A^~ 2A[A = 0. As these equations are of the form (92), their general integrals are similar to (91). Hence the most general forms of (93) for our problem are > Moreover, when, these values are substituted in an equation in a^ (S l of the form (91), the resulting equation in a and ft is of this form. Equation (91) may likewise be looked upon as defining a in terms of ft, so that a, as a function of ft, satisfies an equation of the form (92) ; similarly for a l as a function of ft r Hence if we had used (94), we should have been brought to results analogous to (97) ; and therefore the most general forms of (94) for our problem are (98) i=!4 T ft- 54 **- b s /3+b^ , + , Hence : When a plane is defined in symmetric parameters a, ft, the most general conformal representation of the plane upon itself, for which circles correspond to circles or straight lines, is given by (97) or (98).* EXAMPLES 1. Deduce the equations which define the most general conformal representation of a surface with the linear element cZs 2 = dv? + (a 2 u^dv 2 upon itself. 2. Show that the surfaces x u cos v, y = u sin u, z = au, x u cos v, y u sin v, z = a cosh -* - , are applicable. Find the curve in which a plane through the z-axis cuts the latter surface, and deduce the equations of the conformal representation of these surfaces on the plane. 3. When the representation is defined by (97), what are the coordinates of the center and radius of the circle in the <n-plane which corresponds to the circle of center (c, d) and radius r in the or-plane ? * The transformations (97) and (98) play an important role in the theory of functions. For a more detailed study of them the reader is referred to the treatises of Picard, Darboux, and Forsyth. SURFACES OF REVOLUTION 107 4. Show that in the conformal representation (97) there are, in general, two distinct points, each of which corresponds to itself ; also that if 7 and 5 are the values of a at these points, then K = ai d a 5 ai + ai + V(ai a 4 ) 2 + 4 a 2 a 3 5 . Find the condition that the origin be the only point which corresponds to itself, and show that if the quantities 01, ag, ^3, a are real, a circle in the a-plane through the origin corresponds to a circle in the a^plane through and touching the other circle ; also that a circle touching the x-axis at corresponds to itself. 6. The equation 2 ai = (a b) a -f - - ? where a and 6 are constants, defines a conformal representation of the plane upon itself, such that circles about the origin and straight lines through the latter in the a-plane correspond to confocal ellipses and hyperbolas in the ai-plane. 7. In the conformal representation i = logo: to lines parallel to the x- and y-axes in the ai-plane there correspond lines through the origin and circles about it in the a-plane, and to any orthogonal system of straight lines in the ai-plane an orthogonal system of logarithmic spirals in the a-plane. 46. Surfaces of revolution. By definition a surface of revolution is the surface generated by a plane curve when the plane of the curve is made to rotate about a line in the plane. The various positions of the curve are called the meridians of the surface, and the circles described by each point of the curve in the revolution are called the parallels. We take the axis of rotation for the 2-axis, and for o>axis and ?/-axis any two lines^perpendicular to one another, and to the z-axis, and meeting it in the same point. For any posi tion of the plane the equation of the curve may be written z = </>(r), Avhere r denotes the distance of a point of the curve from the 2-axis. We let v denote the angle which the plane, in any of its positions, makes with the #2-plane. Hence the equations of the surface are (99) x = r cos,v, y = rsinv, z=(f>(r). The linear element is (100) ds 2 = [1 + < 2 (r)] dr 2 + If we put a 01 ) the linear element is transformed into (102) 108 LINEAR ELEMENT OF A SURFACE where X is a function of u, which shows that the meridians and parallels form an isothermal system. As this change of parameters does not change the parametric lines, the equations x = u, y = v, define a conformal representation of the surface of revolution upon the plane in which the meridians and parallels correspond to the straight lines x = const, and y = const, respectively. By definition a loxodromic curve on a surface of revolution is a curve which cuts the meridians under constant angle. Evidently it is represented on the plane by a straight line. Hence loxodromic curves on a surface of revolution (99) are given by C- Vl + $* + bv + c = 0, where a, , c are constants. Incidentally we have the theorem : When the linear element of a surface is reducible to the form where \ is a function of u or v alone, the surface is applicable to a surface of revolution. For, suppose that X is a function of u alone. Put r = Vx and solve this equation for u as a function of r. If the resulting expression be substituted in (101), we find, bya quadrature, the function <f>(r) y for which equations (99) define the surface of revolution with the given linear element. r , When, in particular, the surface of revolution is the unit sphere, with center at the origin, we have r = sin w, z= Vl r 2 = cos w, where u is the angle which the radius vector of the point makes with the positive z-axis. Now = log tan | . Hence the equations of correspondence are , u x = log tan-, y = v. MERCATOR REPRESENTATION 109 This representation is called a Mercator chart of the sphere upon the plane. It is used in making maps of the earth for mariners. A path represented by a straight line on the chart cuts the meridians at constant angle. 47. Conformal representations of the sphere. We have found ( 35) that when the unit sphere, with center at the origin, is referred to minimal lines, its equations are a + /3 () a/3-l (103) "- where a and j3 are conjugate imaginary. Hence the parametric equation of any real circle on the sphere is of the form ca{3+aa + b/3+d=Q, where a and b are conjugate imaginary and c and d are real. From this it follows that the problem of finding any conformal representation of the sphere upon the plane with circles of the former in correspondence with circles or straight lines of the latter, is the same problem analytically as the determination of this kind of representation of the plane upon itself. Hence, from the results of 45, it follows that All conformal representations of the sphere (103) upon a plane, with circles of the former corresponding to circles or straight lines of the latter, are defined by a.a + a,, . bfi+b. * < 104) ***-;{?+ ^ y = ^A We wish to consider in particular the case in which the sphere is represented on the ^-plane in such a way that the great cir cle determined by this plane corresponds with itself point for point. From (103) we have that the equations of this circle are * The representation with the lower signs is the combination of the one with the upper sign and the transformation &i = /3, /Si= , which from (103) is seen to transform a figure bn the sphere into the figure symmetrical with respect to the zz-plane. 110 LINEAR ELEMENT OF A SURFACE When these values are substituted in (104) it is found that we must have , r i A a i=4 b l =t> V a z =<* 3 =0 Z =0 3 =i), so that the particular form of (104)* is equivalent to * 1 =|(+/9), y, = (-) From these equations and (103) we find that the equations of the straight lines joining corresponding points on the sphere and plane are reducible to X Y 1-Z For all values of a and ft these lines pass through the point (0, 0, 1). Hence a point of the plane corresponding to a given point P upon the sphere is the point of intersection with the plane of the line joining P with the pole (0, 0, 1). This form of representation is called the stereographic projection of the sphere upon the plane. It is evident that a line in the plane corresponds to a circle on the sphere ; this circle is determined by the plane of the pole and the given line. We will close this chapter with a few remarks about the con- formal representation of the sphere upon itself. From the fore going results we know that every such representation of the sphere (103) is given by equations of similar form in a^ ft v where the latter are given by (86) or (87), and that for conformal repre sentations with circles in correspondence a l and ^ have the values (97) or (98). We consider in particular the case a.a The expressions of the linear elements of the sphere are found to be reducible to 4 dad/3 4 da^ft, 4 dad ft 2 ~ * Here we have used the upper signs in (104). STEREOGRAPHIC PROJECTION 111 Hence, equations (105) define an isometric representation of the sphere upon itself. Since angles are preserved in the same sense by (105), this representation may be looked upon as determining a motion of configuration upon the sphere into new positions upon it. The stationary points in the general motion, if there are any, correspond to values of a and /3, which are roots of the respective equations If t l and 2 are the roots of the former, those of the latter are l/^ and 1/ 2 . Hence there are four points stationary in the motion; their curvilinear coordinates are 1 -L\ /. -*- X / j " \ I J. L n i ~ " From (103) it is seen that the first two are at infinity, and the last two determine points on the sphere, so that the motion is a rotation about these points. If the z-axis is taken for the axis of rotation, we have from (103) that the roots of (106) must be oo and ; hence # 2 = 3 = 0, so that (105) becomes If the rotation is real, these equations must be of the form = e where o> is the angle of rotation. EXAMPLES 1. Find the equations of the surface of revolution with the linear element ds 2 = dw 2 + (a 2 - w 2 )du 2 . 2. Find the loxodromic curves on the surface i i u X = MCOSU, y = usmv, z = a cosh- 1 -, and find the equations of the surface when referred to an orthogonal system of these curves. 3. Find the general equations of the conformal representation of the oblate spheroid upon the plane. 4. Show that for the surface generated by the revolution of the evolute of the catenary about the base of the latter the linear element is reducible to ds 2 = du" 2 + u dv 2 . 112 LINEAR ELEMENT OF A SURFACE 5. A great circle on the unit sphere cuts the meridian v = in latitude <x under angle a. Find the equation of its stereographic projection. 6. Determine the stereographic projection of the curve x = asinwcosw, y = acos 2 w, z asinw from the pole (0, a, 0). GENERAL EXAMPLES 1. When there is a one-to-one point correspondence between two surfaces, the cross-ratio of four tangents to one surface at a point is equal to the cross-ratio of the corresponding tangents to the other. 2. Given the paraboloid x = 2awcosu, y=2&Msinv, z = 2 w 2 (a cos 2 u + 6sin 2 u), where a and b are constants. Determine the equation of the curves on the surface, such that the tangent planes along a curve make a constant angle with the xy-plane. Show that the generators of the developable 2, enveloped by these planes, make a constant angle with the z-axis, and express the coordinates of the edge of regression in terms of v. 3. Find the orthogonal trajectories of the generators of the surface S in Ex. 2. Show that they are plane curves and that their projections on the xy-plane are involutes of the projection of the edge of regression. 4. Let C be a curve on a cone of revolution which cuts the generators under constant angle, and Ci the locus of the centers of curvature of C. Show that C\ lies upon a cone whose elements it cuts under constant angle. 5. When the polar developable of a curve is developed upon a plane, the curve degenerates into a point. 6. When the rectifying developable of a curve is developed upon a plane, the curve becomes a straight line. 7. Determine <f>(o) so that the right conoid, x = ucosv, y=usinv, z = (f>(v), shall be applicable to a surface of revolution. 8. Determine the equations of a conformal representation of the plane upon itself for which the parallels to the axes in the ai-plane correspond to lines through a point (a, b) and circles concentric about it in the a-plane. 9. The equation a\ = c sin a, where c is a constant, defines a conformal repre sentation of the plane upon itself such that the lines parallel to the axes in the a-plane correspond to confocal ellipses and hyperbolas in the ai-plane. 10. In the conformal representation of the plane upon itself, given by ai = a 2 , to lines parallel to the axes in the ori-plane there correspond equilateral hyperbolas in the a-plane, and to the pencil of rays through a point in the ori-plane and the cir cles concentric about it there corresponds a system of equilateral hyperbolas through the corresponding point in the or-plane and a family of confocal Cassini ovals. 11. When the sides of a triangle upon a surface of revolution are loxodromic curves, the sum of the three angles is equal to two right angles. 12. The only conformal perspective representation of a sphere upon a plane is given by (104). GENERAL EXAMPLES 113 13. Show that equations (105) and the equations obtained from (105) by the interchange of cc. and /3 define the most general isometric representation of the sphere upon itself. 14. Let each of two surfaces S, S\ be defined in terms of parameters w, u, and let points on each with the same values of the parameters correspond. If H H\, where the latter is the function for Si analogous to H for S, corresponding elements of area are equal and the representation is said to be equivalent.* If H ^ HI and the parameters of S are changed in accordance with the equations u <f> (w, v), v = $ (u, a), the condition that the equations u = M, v = v define an equivalent rep resentation of S and Si is H du dv cv du HI (0, \[<) 15. Under what conditions do the equations x aix + a z y + a 3 , y = b& + b 2 y + 63 define an equivalent representation of the plane upon itself ? 16. Show that the equations determine an equivalent representation of the surface of revolution (99) upon the plane. 17. Given a sphere and circumscribed circular cylinder. If the points at which a perpendicular to the axis of the latter meets the two surfaces correspond, the representation is equivalent. 18. Find an equivalent representation of the sphere upon the plane such that the parallel circles correspond to lines parallel to the y-axis and the meridians to ellipses for which the extremities of one of the principal axes are (a, 0), ( a, 0). * German writers call it " flachentreu." CHAPTER IV GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD OF A POINT 48. Fundamental coefficients of the second order. In this chapter we study the form of a surface in the neighborhood of a point M of it, and the character of the curves which lie upon the surface and pass through the point. We recall that the tangents at M to all these curves lie in a plane, the tangent plane to the surface at the point. The equation of the tangent plane at M(x, y, 2), namely (II, 11), may be written (1) (f- where we have put H _ _ du du dy^ dz ~dv dv H dz dx du du dz dx do dv H dx dy dti du dx dy dv dv We define the positive direction of the normal ( 25) to be that for which the functions X, I 7 , Z are the direction-cosines. From this definition it follows that the tangents to the curves v const, and u = const, at a point and the normal at the point have the same mutual orientation as the #-, ?/-, and 2-axes. From (2) follow the identities (3) F - fl ^ 7T u dv which express the fact that the normal is perpendicular to the tan gents to the coordinate curves. In consequence of these identities the expression for the distance p from a point M (u + du, v + dv) to the tangent plane at M is of the second order in du and dv. It may be written (4) p = ^X dx = 1 (D du 2 +2D dudv + &" dv 2 ) + e, lit COEFFICIENTS OF THE SECOND OKDEK 115 where e denotes the aggregate of terms of the third and higher orders in du arid dv, and the functions Z>, D\ D" are defined by (5) dudv If equations (3) be differentiated with respect to u and v respec tively, we get (6) dudv 1 = 0, dv dv du dX dx _ ~ dv dv And so equations (5) may be written (7) _ V Y ~^^ dudv~ Ztdu d ,,_y ^__y^^ ** -9* ^ dv dv dv du The quadratic differential form (8) <1> = D du* + 2 D dudv -f D"dv 2 is called the second fundamental form of the surface, and the func tions D, D , D" the fundamental coefficients of the second order. We leave it to the reader to show that these coefficients, like those of the first order, are invariant for any displacement of the surface in space. Later we shall have occasion to use two sets of formulas which will now be derived. From the equations of definition, toy ~ I cv) we get, by differentiation and simple reduction, the following : a^^_ia^ y^^ = ?^_l?^, dE v dx &x _ l dG ~2 du (9) ^ \du] ^ du dv (10) dx dv dudv 2 dv 116 GEOMETRY OF A SURFACE ABOUT A POINT Again, if the expressions (9) be substituted in the left-hand mem bers of the following equations, the reduced results may be written by means of (2) in the form indicated : dv " du \ du du, dv du \ dv dv, Similar identities can be found by permuting the letters x, y, z ; X, F, Z. From the fundamental relation we obtain, by differentiation with respect to u and v respectively, the identities These equations and (7) constitute a system of three equations linear in * and a system linear in - , , - Solving du du du dv dv dv ij. -y O -y for and for we find, by means of (11), du dv dX FD -GDdx FD-ED dx ^ ~du~ H* du H* dv ^dX _FD"-GD dx FZ> -RD"dx dv " H* du H* dv 7} V /} 7 The expressions for ,, -- are obtained by replacing x by y .. , du dv and z respectively. By means of these equations we shall prove that a real surface whose first and second fundamental coefficients are in proportion, thus D V V" (14) = = =\ where X denotes the factor of proportionality, is a sphere or a plane. We assume that the minimal lines are parametric. In consequence we have E = G = D = D" = 0, so that equations (13) become dX -\ \ du du dv dv RADIUS OF NORMAL CURVATURE 117 The function X must satisfy the condition dv \ du/ du \ dv which reduces to ---- = 0. Moreover, we have two other equations of dv du du dv condition, obtained from the above by replacing x by y and z respectively. Since the proportion to.ay . to = to .0y . to du du du dv dv dv is not possible for a real surface, we must have = = : that is, X is a con- du dv stant. When X is zero the functions X, F, Z given by (15) are constant, and consequently the surface is a plane. When X is any other constant, we get, by integration from (15), X \x + a, Y = \y 4- 6, Z = Xz + c, where a, 6, c are constants. From these equations we obtain (\x -f a) 2 4- (\y 4- 6) 2 4- (Xz 4- c) 2 1. Since this is the general equation of a sphere, it follows that the above condition is necessary as well as sufficient. v/ 49. Radius of normal curvature. Consider on a surface S any curve C through a point M. The direction of its tangent, MT, is determined by a value of dv/du. Let o> denote the angle which the positive direction of the normal to the surface makes with the positive direction of the principal normal to C at Jf, angles being measured toward the positive binormal. If we use the notation of the first chapter, and take the arc of C for its parameter, we have In terms of and the derivatives in the parenthesis have ,, . as as the forms fo = Zfa/duV aV ~~ du 2 \ds) du dv ds ds dv 2 \ds so that the above equation is equivalent to cos w D du 2 + 2 D dudv + D"dv 2 (16) Edu*+ ZFdudv 4- Gdv* As the right-hand member of this equation depends only upon the curvilinear coordinates of the point and the direction of MT, it is the same for all curves with this tangent at M. Since p is positive, the angle o> cannot be greater than a right angle for one curve tangent to MT, if it is less than a right angle for any other 118 GEOMETRY OF A SURFACE ABOUT A POINT curve tangent to MT; and vice versa. We consider in particular the curve in which the surface is cut by the plane determined by MT and the normal to the surface at M. We call it the normal section tangent to MT, and let p n denote its radius. Since the right-hand member of equation (16) is the same for C and the normal section tangent to it, we have (17) P Pn where e is +1 or 1, according as w is less or greater than a right angle; for p and p n are positive. Equation (17) gives the follow ing theorem of Meusnier: The center of curvature of any curve upon a surface is the pro jection upon its osculating plane of the center of curvature of the normal section tangent to the curve at the point. In order to avoid the ambiguous sign in (17), we introduce a new function R which is equal to p n when < o> < TT/%, and to p n when 7r/Z<a><7r, and call it the radius of normal curvature of the surface for the given direction MT. As thus defined, E is given by R Edu 2 + 2 Fdudv + Gdv* Now we may state Meusnier s theorem as follows : If a segment, equal to twice the radius of normal curvature for a given direction at a point on a surface, be laid off from the point on the normal to the surface, and a sphere be described with the segment for diameter, the circle in ivliich the sphere is met by the osculating plane of a curve with the given direction at the point is the circle of curvature of the curve. 50. Principal radii of normal curvature. If we put t = > equa tion (18) becomes I D+2D t+D"t When the proportion (14) is satisfied, R is the same for all values of t, being oc for the plane, and the constant 1/X for the sphere. For any other surface R varies continuously with t. And so we PRINCIPAL RADII OF CURVATURE 119 seek the values of t for which 11 is a maximum or minimum. To this end we differentiate the above expression with respect to t and pnt the result equal to zero. This gives (20) (J} +D"t)(E+2Ft+Gt 2 )-(F+Gt)(D + 2D t + D"t 2 ) = Q, or (21) (FD"-GD l )t 2 +(FD"-GD)t+(ED -FD) = (). Without any loss of generality we can assume that the parametric curves are such that E = 0, so that we have the identity (22) (ED"- GDf- 4 (FD" D r G) (ED -FD) 7T" 2 T 2 F H 2 = 4 (FD -FDf+\ ED"GD--(ED FD)\. E \_ E J When the surface is real, and the parameters also, the right-hand member of this equation is positive. Since the left-hand member is the discriminant of equation (21), the latter has two real and distinct roots.* When the test (III, 34) is applied to equation (21), it is found that the two directions at a point determined by the roots of (21) are perpendicular. Hence: At every ordinary point of a surface there is a direction for which the radius of normal curvature is a maximum and a direction for which it is a minimum, and they are at right angles to one another. These limiting values of R are called the principal radii of normal curvature at the point. They are equal to each other for the plane and the sphere, and these are the only real surfaces with this property. From (20) and (19) we have D +D"t_D + D t _ 1 F+Gt E + Ft~~R Hence the following relations hold between the principal radii and the corresponding values of t : f E + Ft-R(D + D t) = Q, \F+Gt-R(D + D"t) = 0. * In order that the two roots he equal, the discriminant must vanish. This is impos sible for real surfaces other than spheres and planes, as seen from (22). For an imaginary surface of this kind referred to its lines of length zero, we have from (21) that D or D" is zero, since F ^ 0. The vanishing of the discriminant is also the necessary and sufficient condition that the numerator and denominator in (19) have a common factor. 120 GEOMETRY OF A SURFACE ABOUT A POINT When t is eliminated from these equations, we get the equation (24) (DD" - D 1 2 ) A )2 - (ED" + GD-2 FD ) R + (EG - F 2 ) = 0, whose roots are the principal radii. If these roots be denoted by p l and /3 2 , we have (25) !_ ^^ DD"-D 2 PiP* H Although equations (14) hold at all points of a sphere and a plane, and for no other surface, it may happen that for certain par ticular points of a surface they are satisfied. At such points R, as given by (19), is the same for all directions, and the equa tion (21) vanishes identically. When points of this kind exist they are called umbilical points of the surface. EXAMPLES 1. When the equation of the surface is z =f(x, ?/), show that x,Y,z = ^J^, D, D , D" = dz dz 8 2 z d 2 z where p = > a > r = s dx dy dx* dxdy 2. Show that the normals to the right conoid along a generator form a hyperbolic paraboloid. 3. Show that the principal radii of normal curvature of a right conoid at a point differ in sign. 4. Find the expression for the radius of normal curvature of a surface of revolu tion at a point in the direction of the loxodromic curve through it, which makes the angle a with the meridians. 5. Show that the meridians and parallels on a surf ace of revolution, x = u cosu, y = u sin i>, z = (w), are the directions in which the radius of normal curvature is maximum and minimum ; that the principal radii are given by Pl (1 + /2 ) P2 M and that /> 2 is the segment of the normal between the point of the surface and the intersection of the normal with the z-axis. 6. Show that AIX = 1 - X 2 and AI (x, y) = - XY, where the differential param eters are formed with respect to the linear element of the surface. LINES OF CURVATUBE 121 51. Lines of curvature. Equations of Rodrigues. We have seen that the curves defined by equation (21), written (26) (ED - FD) du 2 + (ED" - GD) dudv + (FD" - GD f ) dv z =0, form an orthogonal system. As defined, the two curves of the sys tem through a point on the surface determine the directions at the point for which the radii of normal curvature have their maximum and minimum values. These curves are called the lines of curvature, and their tangents at a point the principal directions for the point. They possess another geometric property which we shall now find. The normals to a surface along a curve form a ruled surface. In order that the sur face be developable, the normals must be tangent to a curve ( 27), as in fig. 12. If the coordinates of a point M l on the normal at a point M be denoted by x r y^ z^ we have FIG. 12 where r denotes the length MM r If M^ be a point of the edge of regression, we must have dx+rdX+Xdr _dy-{-r dY+ Ydr _dz + r dZ + Zdr X Y Z Multiplying the numerators and the denominators of the respec tive members by X, F, Z, and combining, we find that the common ratio is dr. Hence the above equations reduce to or, when the parametric coordinates are used, ( 8x , dx , dX , , dX , du H d v + r I du H d v du dv \du dv (27) dv fa dv du dv du dv 122 GEOMETRY OF A SURFACE ABOUT A POINT If these equations be multiplied by > respectively and dx o- dz ^ u ^ u ^ u added, and by ^ respectively and added, we get dv dv dv Fdu + Gdv r(D du + D" dv) = 0. But these equations are the same as (23). Hence: The normals to a surface along a curve of it form a ruled surface which is a developable only when the curve is a line of curvature ; in this case the points of the edge of regression are the centers of normal curvature of the surface in the direction of the curve. The coordinates of the principal centers of curvature are (28) When the parametric curves are the lines of curvature, equa tion (26) is necessarily of the form (29) X dudv = 0, and consequently we must have ED FD = 0, FD" GD = 0. Since ED"GD = 0, these equations are equivalent to (30) ^=0, D =Q. Conversely, when these conditions are satisfied equation (26) reduces to the form (29). Hence: A necessary and sufficient condition that the lines of curvature be parametric is that F and D be zero. Let the lines of curvature be parametric, and let p^ and p 2 denote the principal radii of normal curvature of the surface in the direc tions of the lines of curvature v = const, and u = const, respectively. From (19) we find (31) ^ = f ~ = ^T and equations (13) become By dY dz _ dZ (32) du du du ri du du These equations are called the equations of Rodrigues. TOTAL AND MEAN CURVATURE 123 52. Total and mean curvature. Of fundamental importance in the discussion of the nature of a surface in the neighborhood of a point are the product and the sum of the principal curvatures at the point. They are called the total curvature * of the surface at the point and the mean curvature respectively. If they be denoted by K and K m1 we have, from (25), 1 (33) K Mb" ^ JL-i+i- Pi P* ^ When K is positive at a point J/, the two principal radii have the same sign, and consequently the two centers of principal curva ture lie on the same side of the tangent plane. As all the centers of curvature of other normal sections lie between these two, the portion of the surface in the neighborhood of M lies entirely on one side of the tangent plane. This can be seen also in another way. Since H 2 is positive, we must have DD f D 2 > 0. Hence the distance from a near-by point to the tangent plane at Jf, since it is proportional to the fundamental form <l> ( 48), does not change sign as dv/du is varied. When K is negative at M, the principal radii differ in sign, and consequently part of the surface lies on one side of the tangent plane and part on the other. In particular there are two directions, given by j>du*+2 D dudv + D" dv 2 = 0, for which the normal curvature is zero. In these directions the dis tances of the near-by points of the surface from the tangent plane, as given by (4), are quantities of the third order at least. Hence these lines are the tangents at M to the curve in which the tangent plane at M meets the surface. At the points for which K is zero, one of the principal radii is infinite. At these points < has the form (^/J) du -f ^W dvf and vanishes in the direction Vl)du + ^/D"dv = 0. But as dv/du passes through the value given by this equation, <1> does not change sign. Hence the surface lies on one side of the tangent plane and is tan gent to it along the above direction. * The total curvature is sometimes called the Gaussian curvature, after the celebrated geometer who suggested it as a suitable measure of the curvature at a point. Cf . Gauss, p. 15. 124 GEOMETEY OF A SUEFACE ABOUT A POINT An anchor ring, or tore, is a surface with points of all three kinds. Such a sur face may be generated by the rotation of a circle of radius a about an axis in the plane of the circle and at a distance b (> a) from the center of the circle. The points at the distance b from the axis lie in two circles, and the tangent plane to the tore at a point of either of the circles is tangent all along the circle. Hence the surface has zero curvature at all points of these circles. At every point whose dis tance from the axis is greater than b the surface lies on one side of the tangent plane, whereas, when the distance is less than 6, the tangent plane cuts the surface. There are surfaces for which K is positive at every point, as, for example, the ellipsoid and the elliptic paraboloid. Moreover, for the hyperboloid of one sheet and the hyperbolic paraboloid the curvature is negative at every point. Surfaces of the former type are called surfaces of positive curvature, of the latter type surfaces of negative curvature. Later ( 64) we shall prove that when K is zero at all points of a surface the latter is developable, and conversely. 53. Equation of Euler. Dupin indicatrix. When the lines of curvature are parametric, equation (18) can be written, in con sequence of (III, 23) and (31), in the form (34) cos 2 # sin 2 6> I Pi Pi where is the angle between the directions whose radii of normal curvature are R and p r Equation (34) is called the equation of Euler. When the total curvature K at a point is positive, p l and p 2 for the point have the same sign, and R has this sign for all directions. If the tangents to the lines of curvature at the point M be taken for coordinate axes, with respect to which % FlG - 13 and T? are coordinates, and segments of length VTIFi be laid off from M in the two directions correspond ing to R, the locus of the end points of these segments is the ellipse (fig. 13) whose equation is & N + ra This ellipse is called the Dupin indicatrix for the point. When, in particular, p l and p 2 are equal, the indicatrix is a circle. Hence the Dupin indicatrix at an umbilical point is a circle ( 50). For this reason such a point is sometimes called a circular point. = 1. DUPIN INDICATRIX 125 When K is negative p l and p 2 differ in sign, and consequently certain values of R are positive and the others are negative. In the directions for which R is positive we lay off the segments V.Z2, and in the other directions V R. The locus of the end points of these segments con sists of the conjugate hyperbolas (fig. 14) whose equations are Pl T]_ Pt We remark that R is infinite for the directions given by (35) tah 2 0=-^, FIG. 14 or, in other words, in the directions of the asymptotes to the hyperbolas. The above locus is the Dupin indicatrix for the point. Finally, when K = the equation of the indicatrix is of one of the forms e-2_ i that is, a pair of parallel straight lines. In view of the foregoing results, a point of a surface is called elliptic, hyperbolic, or parabolic, according as the total curvature at the point is positive, negative, or zero. 1 In consequence of (4) the expression for the distance p upon the tangent plane to a surface at a point M from a near-by point P of the surface is given by Edu 2 Gdv 2 n + = 2 , Pl to within terms of higher order. But ^/Edu and ^/Gdv are the distances, to within terms of higher order, of P from the normal planes to the surface at M in the directions of the lines of curva ture. Hence the plane parallel to the tangent plane and at a dis tance p from it cuts the surface in the curve Evidently this is a conic similar to the Dupin indicatrix at an elliptic or parabolic point, and to a part of the indicatrix at a hyperbolic point. 126 GEOMETRY OF A SURFACE ABOUT A POINT EXAMPLES 1. Show that the meridians and parallels of a surface of revolution are its lines of curvature, and determine the character of the developable surfaces formed by the normals to the surface along these lines. 2. Show that the parametric lines on the surface a , b , uv X = -(tt-M), y = -( U -v), z = -, are straight lines. Find the lines of curvature. 3. When a surface is denned by z = /(x, ?/), the expressions for the curvatures are and the equation of the lines of curvature is [(1 + p2) s - pqr] (to* + [(i + p-2) t-(l + g2) r -j dxdy + [ pqt _ (1 + ?2) s] dy z = . 4. The principal radii of the surface y cos x sin - = at a point (x, y, z) are 2 i o 2 _i_ n2 equal to Find the lines of curvature. 5. Derive the equations of the tore, defined in 52, and prove therefrom the results stated. 6. The sum of the normal curvatures in two orthogonal directions is constant. 7. The Euler equation can be written E = 2plp * Pi + P2 - (PI - pa) cos 2 6 54. Conjugate directions at a point. Conjugate systems. Two curves on a surface through a point M are said to have conjugate directions when their tangents at M coincide with conjugate diam eters of the Dupin indicatrix for the point. These tangents are also parallel to conjugate diameters of the conicr in which the sur face is cut by a plane parallel to the tangent plane to M and very near it. Let P denote a point of this conic and N the point in which its plane a cuts the normal at M. The tangent plane to the surface at P meets the plane a in the tangent line at P to the conic. Moreover, this tangent line is parallel to the diameter conju gate to NP. Hence as P approaches 3/this tangent line approaches the diameter of the Dupin indicatrix, which is conjugate to the diameter in the direction MP. Hence we have (cf. 27) : The characteristic of the tangent plane to a surface, as the point of contact moves along a curve, is the tangent to the surface in the direction conjugate to the curve. CONJUGATE DIRECTIONS 127 By means of this theorem we derive the analytical condition for conjugate directions. If the equation of the tangent plane is f, 77, f being current coordinates, the characteristic is denned by this equation, and where s is the arc of the curve along which the point of contact moves. If &c, 8y, Bz denote increments of #, ?/, z in the direction conjugate to the curve, we have, from the above equations, If increments of u and v in the conjugate direction be denoted by Bu and 8v, this equation may be written (36) D duBu + D f (du8v + dvSu) + D"dvv = 0. The directions conjugate to any curve of the family (37) <(%, v) = const. are given by ( 38) cv du dv du As this is a differential equation of the first order and first degree, it defines a one-parameter family of curves. These curves and the curves </> = const, are said to form a conjugate system. Moreover, any two families of curves are said to form a conjugate system when the tangents to a curve of each family at their point of inter section have conjugate directions. From (36) it follows that the curves conjugate to the curves v = const, are defined by D Su + D Sv = 0. Consequently, in order that they be the curves u = const., we must have D 1 equal to zero. As the converse also is true, we have : A necessary and sufficient condition that the parametric curves form a conjugate system is that D be zero. 128 GEO vlETRY OF A SURFACE ABOUT A POINT We have seen (51) that the lines of curvature are characterized by the property that, when they are parametric, the coefficients F and D are zero. Hence : The lines of curvature form a conjugate system and the only orthogonal conjugate system. If the lines of curvature are parametric, and the angles which a pair of conjugate directions make with the tangent to the curve v = const, are denoted by and 6 , we have a [G dv ar [GSv tan = ^J , tan 6 = x - , MjE < du N^ Su so that equation (36) may be put in the form (39) tan0tan0 = -?H, which is the well-known equation of conjugate directions of a conic. 55. Asymptotic lines. Characteristic lines. When is equal to 0, equation (39) reduces to (35). Hence the asymptotic directions are self-conjugate. If in equation (36) we put Sv/Su = dv/du, we obtain (40) D du 2 + 2 D dudv + D" dv 2 = 0, which determines, consequently, the asymptotic directions at each point of the surface. This equation defines a double family of curves upon the surface, two of which pass through each point and admit as tangents the asymptotic directions at the point. They are called the asymptotic lines of the surface. The asymptotic lines are imaginary on surfaces of positive curva ture, real on surfaces of negative curvature, and consist of a single real family on a surface of zero curvature. Recalling the results of 52, we say that the tangent plane to a surface at a point cuts the surface in asymptotic lines in the neighborhood of the point. As an immediate consequence, we have that the generators of a ruled surface form one family of asymptotic lines. Since an asymptotic line is self-conjugate, the characteristics of the tangent plane as the point of contact moves along an asymp totic line are the tangents to the latter. Hence the osculating plane of an asymptotic line at a point is the tangent plane to the ASYMPTOTIC LINES 129 surface at the point, and consequently the asymptotic line is the edge of regression of the developable circumscribing the surface along the asymptotic line. This follows also from equation (16). From (40) we have the theorem : A necessary and sufficient condition that the asymptotic lines upon a surface be parametric is that D=D"=Q. If these equations hold, and, furthermore, the parametric curves are orthogonal, it is seen from (33) that the mean curvature is zero, and conversely. Hence : A necessary and sufficient condition that the asymptotic lines form an orthogonal system is that the mean curvature of the surface be zero. A surface whose mean curvature is zero at every point is called a minimal surface. At each of its points the Dupin indicatrix con sists of two conjugate equilateral hyperbolas. By means of (39) we find that the angle between conjugate directions is given by P-/>1 If we consider only real lines, this angle can be zero only for sur faces of negative curvature, in which case the directions are asymp totic. It is natural, therefore, to seek the conjugate directions upon a surface of positive curvature for which the included angle is a minimum. To this end we differentiate the right-hand member of the above equation with respect to 6 and equate the result to zero. The result is reducible to (41) tan 6 = : Then from (39) we have From these equations it follows that # = 0, and Conversely, when = 6 equation (39) becomes (41). Hence: Upon a surface of positive curvature there is a unique conjugate system for which the angle between the directions at any point is the 130 GEOMETRY OF A SURFACE ABOUT A POINT minimum angle between conjugate directions at the point ; it is the only conjugate system whose directions are symmetric with respect to the directions of the lines of curvature. These lines are called the characteristic lines. It is of interest to note that equations (35) and (41) are similar, and that the real asymptotic directions upon a surface of negative curvature are symmetric with respect to the directions of the lines of curvature. As just seen, if 6 is the angle which one characteristic line makes with the line of curvature v = const, at a point, the other charac teristic line makes the angle 6. Hence the radii of normal curva ture for these directions are equal, and consequently a necessary and sufficient condition that the characteristic curves of a surface be parametric is (42) f = 7T iy= 56. Corresponding systems on two surfaces. By reasoning similar to that of 34 we establish the theorem : A necessary and sufficient condition that the curves defined by fidu 2 +2S dudv + T dv 2 = form a conjugate system upon a sur- f ace * s RD" + TD 2 SD = 0. From this we have at once : If the second quadratic forms of two surfaces S and S l are D du 2 + 2 D dudv + D" dv 2 and D l du 2 + 2 D[ dudv + D[ f dv\ and if a point on one surface is said to correspond to the point on the other with the same values of u and tf, the equation du 2 D? D" (43) dudv D[ D dv 2 DI D defines a system of curves which is conjugate for both surfaces. By the methods of 50 we prove that these curves are real when either or both of the surfaces S, S l is of positive curvature. If the curvature of S is negative and it is referred to its asymptotic lines, the above equation reduces to GEODESIC CURVATUKE 131 Hence the system is real when D l and D[ have the same sign, that is, when the curvature of S l is positive. Another consequence of the above theorem is : A necessary and sufficient condition that asymptotic lines on one of two surfaces $, S l correspond to a conjugate system on the other is (44) DD J + D"Di - 2 D D[ = 0. EXAMPLES 1. Find the curves on the general surface of revolution which are conjugate to the loxodromic curves which cut the meridians under the angle a. 2. Find the curves on the general right conoid, Ex. 1, p. 56, which are conju gate to the orthogonal trajectories of the generators. 3. When the equations of a surface are of the form x=U l , y=Vi, z=U 2 +V 2 , where U\ and U 2 are functions of u alone, and V\ and F 2 of v alone, the para metric curves are plane and form a conjugate system. 4. Prove that the sum of normal radii at a point in conjugate directions is constant. 5. When a surface of revolution is referred to its meridians and parallels, the asymptotic lines can be found by quadratures. 6. Find the asymptotic lines on the surface acosw x = a(l + cos it) cot v, y = a(l + cosw), z= -- 7. Determine the asymptotic lines upon the surface z y sin a: and their orthog onal trajectories. Show that the x-axis belongs to one of the latter families. 8. Find the asymptotic lines on the surface 2 ?/ 3 - 2 xyz + z 2 = 0, and determine their projections on the xy-plane. 9. Prove that the product of the normal radii in conjugate directions is a maxi mum for characteristic lines and a minimum for lines of curvature. 10. When the parametric lines are any whatever, the equation of character istic lines is [D(GD - ED") - 2D (FD - ED )] tin* + 2 [D (GD + ED") - 2 FDD"] dudv + [2D (GD - FD") - D"(GD - ED")] dv* = 0. 57. Geodesic curvature. Geodesies. Consider a curve C upon a surface and the tangent plane to the surface at a point M of C. Project orthogonally upon this tangent plane the portion of the curve in the neighborhood of M, and let C 1 denote this projection. The curve C 1 is a normal section of the projecting cylinder, and C is a curve upon the latter, tangent to C" at M. Hence the theorem 132 GEOMETRY OF A SURFACE ABOUT A POINT of Meusnier can be applied to these two curves. If l/p g denotes the curvature of C and -^ the angle between the principal normal to C and the positive direction of the normal to the cylinder at Jf, we have (45) i = c ^. P, P In order to connect this result with others, it is necessary to define the positive direction of the normal to the cylinder. This normal lies in the tangent plane to the surface. We make the convention that the positive directions of the tangent to the curve, the normal to the cylinder, and the normal to the surface shall have the same mutual orientations as the positive or-, y-, and 2-axes. From this choice of direction it follows that if, as usual, the direc tion-cosines of the tangent to the curve be dx/ds, dy/ds, dz/ds, then those of the normal to the cylinder are (46 ) Y*-Z*/, Z~-*~> *f-4- ds ds ds ds ds ds The curvature of C f is called the geodesic curvature of (7, and p g the radius of geodesic curvature. And the center of curvature of C is called the center of geodesic curvature of C. From its definition the geodesic curvature is positive or nega tive according as the osculating plane of C lies on one side or the other of the normal plane to the surface through the tangent to C. From (45) it follows that the center of first curvature of C is the projection upon its osculating plane of the center of geodesic curvature. Moreover, the former is also the projection of the center of curvature of the normal section tangent to C (49). Hence the plane through a point M of (7, normal to the line joining the centers of normal and geodesic curvature at M, is the osculating plane of C for this point, and its intersection with the join is the center of first curvature. By definition ( 49) w denotes the angle which the positive direction of the normal to the surface makes with the positive direction of the principal normal to (7, angles being measured toward the binomial. Hence equation (45) can be written 1 sin w (47) - = GEODESIC CUEVATUKE 133 These various quantities are represented in fig. 15, for which the tangent to the curve is normal to the plane of the paper, and is directed toward the reader. The directed lines MP, MB, MK, MN represent respectively the positive directions of the principal normal and binomial of the curve and the normals to the projecting cylinder and to the surface. A curve whose principal normal at every point coincides with the normal to the surface upon which it lies, is called a geodesic. From (45) it follows that a geodesic may also be defined as a curve whose geodesic curvature is zero at every point. For example, the meridians of a surface of revolution are geodesies, as follows from the results in 46. A twisted curve is a geodesic on its rectifying developable, and when a straight line lies 011 a surface, it is a geodesic for the surface. Later we shall make an extensive study of geodesies, but now we desire to find an expression for the geodesic curvature in terms of the fundamental quantities of the surface and the equation of the curve. 58. Fundamental formulas. The direction-cosines of the prin cipal normal are ( 8) d 2 x ~ d 2 y d 2 z f\ _ Q Q H ds 2 ^ ds 2 r ds 2 . Consequently, by means of (46), equation (45) may be put in the form 1 ^\ / dz dy\ d x ( 48 ) 7 g = *\ ds~ ~ds)~ds 2 Expressed as functions of u and v, the quantities -j- -ji are ^ the form dx _ fa du dx dv ds du ds dv ds _ g d 2 x dudv d*x ^Y+ + ~ds 2 ~ du 2 \ds) dudv ds ds dv 2 \ds) du ds 2 dv ds 2 134 GEOMETRY OF A SURFACE ABOUT A POINT When these expressions are substituted in (48), and in the reduction we make use of (10) and (11), we obtain ds ds where L and M have the significance _ ,_ , F (Fv ~ + ds ds^\dv 29/Vb/ <*r ds^ -- . \\ ^^ + ~ (\\ F G- ~ du ~ 2 ~dv) \ds) du ds ds 2 dv \ds) ds* * From this it is seen that the geodesic curvature of a curve depends upon E, F, G, and is entirely independent of D, D , D". Suppose that the parametric lines form an orthogonal system, and that the radius of geodesic curvature of a curve v = const, be denoted by p gu . In this case F= 0, ds = Vfldu. Hence the above equation reduces to (50) r - In like manner we find that the geodesic curvature of a curve u = const, is given by _! As an immediate consequence c^ these equations we have the theorem : f When the parametric lines upon a surface form an orthogonal system, a necessary and sufficient condition that the curves v = const, or u = const, be geodesies is that E be a function of u alone or G of v alone respectively. It will now be shown that p gu is expressible as a function of differential parameters of v formed with respect to the linear ele ment (III, 4). From the definition of these parameters ( 37, 38) it follows that when ^=0 l d IE ~V\G GEODESIC CURVATURE 135 Hence, by substitution in (50), we obtain (52) JUPAL ./ 1 P gn [y&^v In like manner, we find (53) -i- = Thus we have shown that the geodesic curvature of a parametric line is a differential parameter of the curvilinear coordinate of the line. Since this curvature is a geometrical property of a line, it is necessarily independent of the choice of parameters, and thus is an invariant. This was evident a priori, but we have just shown that it is an invariant of the differential parameter type. From the definition of the positive direction of the normal to a surface ( 48), and the normal to the cylinder of projection, it fol lows that the latter for a curve v = const, is the direction in which v increases, whereas, for a curve u const., it is the direction in which u decreases. Hence, if the latter curves be defined by u = const., equations (52) and (53) have the same sign. If, now, we imagine the surface referred to another parametric system, for which the linear element is (54) ds* = Edu 2 + 2 Fdudv + G dv\ the curve whose geodesic curvature is given by (50) will be defined by an equation such as c/> (u, v) = const. And if the sign of $ be such that < is increasing in the direction of the normal of its pro jecting cylinder, its geodesic curvature will be given by (55) p. where the differential parameters are formed with respect to (54). If two surfaces are applicable, and points on each with the same curvilinear coordinates correspond, the geodesic curvature of the curve <= const, on each at corresponding points will be the same in consequence of (55). Hence : Upon two applicable surfaces the geodesic curvature of corresponding curves, at corresponding points, is the same. 136 GEOMETRY OF A SURFACE ABOUT A POINT When the second member of equation (55) is developed by (III, 46, 56), we have 1 1 ff(V*d d \ du dv\ d \ dv du cu H L 1 cu dv R dv du d 1 H F W\ I I d I du dv\ d^\ dv du, Hence we have the formula of Bonnet*: (56) i-^ du dv In particular, the geodesic curvature of the parametric curves, when the latter do not form an orthogonal system, is given by (57) ft. dv du The geodesic curvature of a curve of the family, defined by the differential equation Mdu + Ndv*sQ, has the value 1 _ 1 f d I FN GM \ p g " H \ du \^EN*- 2 FMN+ GM 2 / d / FM-EN ZFMN+ GM *Memoire sur la theorie generale des surfaces, Journal de VEcole Poll/technique, Cahier 32 (1848), p. 1. GEODESIC TOKSION 137 In illustration of the preceding results, we establish the theorem : When the curves of an orthogonal system have constant geodesic curvature, the system is isothermal. When the surface is referred to these lines, and the linear element is written ds z = Edu 2 + Gdv 2 , the condition that the geodesic curvature of these curves be constant is, by (50) and (51), 1 dVE l/i, - = KI, VEG fa where Ui and V\ are functions of u and v respectively. If these equations are differentiated with respect to D and u respectively, we get dudv dv cu cucv dv du ""S 75^ Subtracting, we obtain - log = 0. CUCV G Hence E/G is equal to the ratio of a function of u and a function of u, and the system is isothermic. In terms of isothermic parameters, equations (i) are of the form 10X = r r, J.3X \2 5U \2 dV ~ and the linear element is (II) tf It is evident that the meridians and parallels on a surface of revolution form such a system. The same is true likewise of an orthogonal system of small circles on a sphere. 59. Geodesic torsion. We have just seen that when a curve is denned by a finite equation or a differential equation, its geodesic curvature can be found directly. The same is true of the normal curvature of the surface in the direction of the curve by (18). Then from (16) and (47) follow the expressions for p and tw. In order to define the curve it remains for us to obtain an expression for the torsion. From the definition of o> it follows that (59) sin o) = X\ -f Y/JL + Zv, where X, /Lt, v are the direction-cosines of the binormal. If this equation is differentiated with respect to the arc of the curve, and the Frenet formulas (I, 50) are used in the reduction, we get ( 60 ) v ^ _i *- ds 138 GEOMETRY OF A SURFACE ABOUT A POINT From (I, 37, 41) Ave have dx d"*z\ ~d~s~dfj _ /dy d"z dz d*y\ _ /dz d*x _ dx d"*z ~~~ ^~ ~ ~~~~ j ds 2 ds ds 2 r d 2 x -V-K , and r- / . 7 . 6?S^ Moreover, from (13), we obtain the identity ds ds ds ds H\_ \ds/ ds ds Consequently equation (60) is equivalent to (61) eos where l/T has the value 1 (FD - ED } du 2 +(GD ED") dudv + ( GD - FD") dv* " ~T ~ H(Edu 2 + 2 Fdudv + G dv 2 ) When cos w is different from zero, that is, when the curve is not an asymptotic line, equation (61) becomes hf-F As the expression for T involves only the fundamental coeffi cients and dv/du, we have the following theorem of Bonnet: The function is the same for all curves which have the same T ds tangent at a common point. Among these curves there is one geodesic, and only one, for it will be shown later ( 85) that one geodesic and only one passes through a given point and has a given direction at the point. At every point of this geodesic w is equal to or 180, and conse quently T T. Hence the value of T for a given point and direc tion is that of the radius of torsion of the geodesic with this direction. The function T is therefore called the radius of geodesic torsion of GEODESIC TORSION 139 the curve. From (63) it is seen that T is the radius of torsion of any curve whose osculating plane makes a constant angle with the tangent plane.* When the numerator of the right-hand member of equation (62) is equated to zero, we have the differential equation of lines of curvature. Hence : A necessary and sufficient condition that the geodesic torsion of a curve be zero at a point is that the curve be tangent to a line of curvature at the point. The geodesic torsion of the parametric lines is given by 1 _ FD-ED l_ _ GD -FD" ~T~ EH ~T V ~ GH When these lines form an orthogonal system T u and T v differ only in sign. Consequently the geodesic torsion at the point of meeting of two curves cutting orthogonally is the same to within the sign. Thus far in the consideration of equation (61) we have excluded the case of asymptotic lines. In considering them now, we assume that they are parametric. The direction-cosines of the tangent and binomial to a curve v = const, in this case are JL^, =-L^, 7= ; ~ where e is +1 or 1. Consequently the direction-cosines of the principal normal have the values and similar expressions for m and n. When in the Frenet formulas d\ I dfju _m dv _n ds ~~ T ds T ds r we substitute the above values, and in the reduction make use of (11) and (13), we get (65) * Thus far exception must be made of asymptotic lines, but later this restriction will be removed. 140 GEOMETRY OF A SURFACE ABOUT A POINT In like manner, the torsion of the asymptotic lines u = const, is found to be V K. But from (64) we find that the geodesic torsiqn in the direction of the asymptotic lines is qp V JT. Hence equation (63) is true for the asymptotic lines as well as for all other curves on the surface. Incidentally we have established the following theorem of Enneper : The square of the torsion of a real asymptotic line at a point is equal to the abso lute value of the total curvature of the surface at the point; the radii of torsion of the asymptotic lines through a point differ only in sign. The following theorem of Joachimsthal is an immediate consequence of (63) : When two surfaces meet under a constant angle, the line of intersection is a line of curvature of both or neither; and conversely, when the curve of intersection of two surfaces is. a line of curvature of both they meet under constant angle. For, if &>!, o> 2 denote the values of w for the two surfaces, and Z\, T 2 the values of T, we have, by subtracting the two equations of the form (63), that TI = T 2 , which proves the first part of the theorem. Conversely, if \/T\ = l/T 2 = 0, we have (wi uz) = 0, and consequently the surfaces meet under constant angle. ds EXAMPLES 1. Show that the radius of geodesic curvature of a parallel on a surface of revolution is the same at all points of the parallel, and determine its geometrical significance. 2. Find the geodesic curvature of the parametric lines on the surface a , b / uv X = -(u + v), y = -(u-v), z = > 3. Given a family of loxodromic curves upon a surface of revolution which cut the meridians under the same angle a ; show that the geodesic curvature of all these curves is the same at their points of intersection with a parallel. 4. Straight lines on a surface are the only asymptotic lines which are geodesies. 5. Show that the geodesic torsion of a curve is given by 1 1/1 1\ . = -I ) sin 2 0, T *\fi pj where 6 denotes the angle which the direction of the curve at a point makes with the line of curvature v = const, through the point. 6. Every geodesic line of curvature is a plane curve. 7. Every plane geodesic line is a line of curvature. 8. When a surface is cut by a plane or a sphere under constant angle, it is a line of curvature on the surface, and conversely. 9. If the curves of one family of an isothermal orthogonal system have constant geodesic curvature, the curves of the other family have the same property. SPHERICAL REPRESENTATION 141 60. Spherical representation. In the discussion of certain prop erties of a surface S it is of advantage to make a representation of S upon the unit sphere * by drawing radii of the sphere parallel to the positive directions of the normals to S, and taking the extrem ities of the radii as spherical images of the corresponding points on S. As a point M moves along a curve on $, its image m describes a curve on the sphere. If we limit our consideration to a portion of the surface in which no two normals are parallel, the portions of the surface and sphere will be in a one-to-one correspondence. This map of the surface upon the sphere is called the spherical representation of the surface, or the Gf-aussian representation. It was first employed by Gauss in his treatment of the curvature of surfaces. f The coordinates of m are the direction-cosines of the normal to the surface, namely X, F, Z, so that if we put the square of the linear element of the spherical representation is In 48 we established the following equations : _f FD 1 GD dx . FD ED dx % (68) du~ H 2 du H 2 dv dX FD"GD ex FD ED" dx ] v H z du H 2 dv By means of these relations and similar ones in F and Z, the func tions (o, c^, & may be given the forms (69) = ~ [GIf - 2 H _ JL H F(DD" + D 2 ) + ED D"], - 2 or, in terms of the total and mean curvatures ( 52), (70) * The sphere of unit radius and center at the origin of coordinates. t L.c., p. 9. 142 GEOMETRY OF A SUEFACE ABOUT A POINT In consequence of these relations the linear element (67) may be given the form (71) do- z It m (Ddu-+ 2D dudv+D"dv 2 ) K(Edi? + 2 Fdudv and, by (18), (72) ^=( \ J* From (70) we have also (73) ff. where e is 1, according as K is positive or negative. Equations (69) are linear in E, F, G. Solving for the latter, we have E = (74) In seeking the differential equation of the lines of curvature from the definition that the normals to the surface along such a curve form a developable surface, we found ( 51) that for a dis placement in the direction of a line of curvature we have fa 7 & 7 l^ X 3 , 2X , \ . du + dv + r( du + - dv = 0, du dv \du dv / and similar equations in y and 2, where r denotes the radius of principal curvature for the direction. If these equations be multi- o -T7" QT/" Q *7 7} "jf plied respectively by - - > -- and added, and likewise by - > du du du dv and added, th dv dv ?., _ an( l added, the resulting equations may be written D du +D dv - r(fdu+di>) = 0, D du +D n dv -r(3du + gdv} = 0. Eliminating r, we have as the equation of the lines of curvature (75) - 2 -"dudv D -D"&dv* = 0. SPHERICAL REPRESENTATION 143 Again, the elimination of du and dv gives the equation of the principal radii in the form (76) (<o c^ 2 ) r*(<oD" + 3D 2 &D ) r + (DD" D ) = 0, so that (77) These results enable us to write equations (74) thus : (78) 61. Relations between a surface and its spherical representation. Since the radius of normal curvature R is a function of the direc tion except when the surface is a sphere, we obtain from (72) the following theorem : A necessary and sufficient condition that the spherical representation of a surface be conformal is that the surface be minimal or a sphere. As a consequence of this theorem we have that every orthog onal system on a minimal surface is represented on the sphere by an orthogonal system. From (70) it is seen that if a surface is not minimal, the parametric systems on both the surface and the sphere can be orthogonal only when If is zero, that is, when the lines of curvature are parametric. Hence we have : The lines of curvature of a surface are represented on the sphere by an orthogonal system ; this is a characteristic property of lines of curvature, unless the surface be minimal. This theorem follows also as a direct consequence of the theorem : A necessary and sufficient condition that the tangents to a curve upon a surface and to its image at corresponding points be parallel is that the curve be a line of curvature. In order to prove this theorem we assume that the curve is parametric, v = const. Then the condition of parallelism is &.?.- to ;&.* du du du du du du 144 GEOMETRY OF A SURFACE ABOUT A POINT From (68) it follows that in this case (FD ED 1 ) must be zero. But the latter is the condition that the curves v = const, be lines of curvature ( 51). Moreover, from (32) it follows that the positive half-tangents to a line of curvature and its spherical representa tion have the same or contrary sense according as the correspond ing radius of normal curvature is negative or positive. In consequence of (7) the equation (40) of the asymptotic directions may be written dxdX+ dydY+ dzdZ = 0. And so we have the theorem : The tangents to an asymptotic line and to its spherical representa tion at corresponding points are perpendicular to one another ; this property is characteristic of asymptotic lines. It is evident that the direction-cosines of the normal to the sphere are equal to X, Y, Z, to within sign at most. Let them be denoted by X><> I/, ^; then (79) x> = ( - ? #\du dv du dv When expressions similar to (68) are substituted for the quantities in the parentheses, the latter expression is reducible to KHX. Hence, in consequence of (73), we have (80) x> = ex, ^ = er, l = *z, where e = 1 according as the curvature of the surface is positive or negative. From the above it follows that according as a point of a surface is elliptic or hyperbolic the positive sides of the tangent planes at corresponding points of the surface and the sphere are the same or different. Suppose, for the moment, that the lines of curvature are parametric. From our convention about the positive direction of the normal to a surface, and the above results, it follows that both the tangents to the parametric curves through a point M have the same sense as the corresponding tangents to the sphere, or both have the opposite sense, when M is an elliptic point; but that one tangent has the same sense as the corresponding tangent to the sphere, and the other the opposite sense, when the point is GAUSSIAN CURVATURE 145 hyperbolic. Hence, when a point describes a closed curve on a surface its image describes a closed curve on the sphere in the same or opposite sense according as the surface has positive or negative curvature. We say that the areas inclosed by these curves have the same or opposite signs in these respective cases. Suppose now that we consider a small parallelogram on the sur face, whose vertices are the points (u, v), (u -f du, v), (u, v -f dv), and (u + du, v + dv). The vertices of the corresponding parallelo gram on the sphere have the same curvilinear coordinates, and the areas are Ifdudv and e/tdudv, where e 1 according as the sur face has positive or negative curvature in the neighborhood of the point (u, v). The limiting value of the ratio of the spherical and the surface areas as the vertices of the latter approach the point (u, v) is a measure of the curvature of the surface similar to that of a plane curve. In consequence of (73) this limiting value is the Gaussian curvature K. Since any closed area may be looked upon as made up of such small parallelograms, we have the following theorem of Gauss : The limit of the ratio of the area of a closed portion of a surface to the area of the spherical image of it, as the former converges to a point, is equal in value to the product of the principal radii at the point. Since the normals to a developable surface along a generator are parallel, there can be no closed area for which there are not two nor mals which are parallel. Hence spherical representation, as defined in 60, applies only to nondevelopable surfaces, but so far as the preceding theorem goes, it is not necessary to make this exception ; for the total curvature of a developable surface is zero ( 64), and the area of the spherical image of any closed area on such a surface is zero. The fact that the Gaussian curvature is zero at all points of a developable surface, whereas such a surface is surely curved, makes this measure not altogether satis factory, and so others have been suggested. Thus, Sophie Germain* advocated the mean curvature, and Casorati f has put forward the expression - [ 1 ) 2 \Pi fill But according to the first, the curvature of a minimal surface is zero, and according to the second, a minimal surface has the same curvature as a sphere. Hence the Gaussian curvature continues to be the one most frequently used, which may be due largely to an important property of it to be discussed later ( 64). * Crelle, Vol. VII (1831), p. 1. f Acta Mathematica, Vol. XIV (1890), p. 95. 146 GEOMETRY OF A SURFACE ABOUT A POINT 62. Helicoids. We apply the preceding results in a study of an important class of surfaces called the helicoids. A helicoid is generated by a curve, plane or twisted, which is rotated about a fixed line as axis, and at the same time translated in the direction of the axis with a velocity which is in constant ratio with the velocity of rotation. A section of the surface by a plane through the axis is called a meridian. All the meridians are equal plane curves, and the surface can be generated by a meridian moving with the same velocities as the given curve. The particular motion described is called helicoidal motion, and so we may say that any helicoid can be generated by a plane curve with helicoidal motion. In order to determine the equations of a helicoid in parametric form, we take the axis of rotation for the 2-axis, and let u denote the distance of a point of the surface from the axis, and v the angle made by the plane through the point and the axis with the #z-plane in the positive direction of rotation. If the equation of the gen erating curve in any position of its plane is z = <f> (w), the equations of the surface are (81) x = u cos v, y = u sin v, z = </> (u) + av, where a denotes the constant ratio of the velocities ; it is called the parameter of the helicoidal motion. When, in particular, a is zero, these equations define any surface of revolution. Moreover, when < (u) is a constant, the curves v = const, are straight lines perpendicular to the axis, and so the surface is a right conoid. It is called the right helicoid, By calculation we obtain from (81) (82) ^ = l + c/) /2 , F=a<t> , G=u 2 +a 2 , where the accent indicates differentiation with respect to u. From the method of generation it follows that the curves v = const, are meridians, and u = const, are helices on the helicoids, and circles on surfaces of revolution. From (82) it is seen that these curves form an orthogonal system only on surfaces of revolution and on the right helicoid. Moreover, from (57) it is found that the geo desic curvature of the meridians is zero only when a is zero or < is a constant. In the latter case the meridian is a straight line perpendicular to the axis or oblique, according as (/> is zero or not. HELICOIDS 147 Hence the meridians of surfaces of revolution and of the ruled helicoids are geodesies. The orthogonal trajectories of the helices upon a helicoid are determined by the equation (cf. Ill, 31) afidu + (u 2 + a 2 ) dv = 0. Hence, if we put v 1 = / >2 ^ 2 du + v, J It -J- Q the curves v x = const, are the orthogonal trajectories, and their equations in finite form are found by a quadrature. In terms of the parameters u and v l the linear element is (83) t?s 2 u As an immediate consequence of this result we have that the helices and their orthogonal trajectories on any helicoid form an isothermal system. From (83) and ( 46) we have the theorem of Bour: Every helicoid is applicable to some surface of revolution, and helices on the former correspond to the parallels on the latter. We derive also the following expressions : , r a sin v u<f> cos v, (a cos v + u$ ! sin v), u (o4) JL, y, z = - - V^ 2 (l + f 2 )+a 2 and (85) />,/> .!>"= From (84) it follows that a meridian is a normal section of a sur face of revolution at all its points, and consequently is a line of curvature (Ex. 7, p. 140). This is evident also from the equation of the lines of curvature of a helicoid, namely (86) a [1 + </> 2 + u<t> $ ] dii 2 + [(u 2 + a 2 ) M<" - (1 + $ *) u^ ] dudv -a[u 2 (t> - 2 +u 2 +a 2 ]dv 2 =Q. Moreover, the meridians are lines of curvature of those helicoids, for which </> satisfies the condition 148 GEOMETRY OF A SURFACE ABOUT A POINT By integration this gives <f) = Vtf" U 2 C log When the surface is the right helicoid the expressions for D and D" vanish. Hence the meridians and helices are the asymp totic lines. Moreover, these lines form an orthogonal system, so that the surface is a minimal surface ( 55). Since the tan- gent planes to a surface along an asymptotic line are its oscu lating planes, if the surface is a ruled minimal surface, the gener ators are the principal normals of all the curved asymptotic lines. But a circular helix is the only Bertrand curve whose principal normals are the principal normals of an infinity of curves ( 19). Hence we have the theorem of Catalan : The right helicoid is the only real minimal ruled surface. In fig. 16 are represented the asymptotic lines and lines of curvature of a right helicoid. For any other helicoid the equation of the asymptotic lines is (87) ufi du 2 - Zadudv + uty di? = 0. As the coefficients in (86) and (87) are functions of u alone, we have the theorem : When a helicoid is referred to its meridians and helices, the asymp totic lines and the lines of curvature can be found by quadratures. EXAMPLES 1. Show that the spherical representation of the lines of curvature of a surface of revolution is isothermal. 2. The osculating planes of a line of curvature and of its spherical representa tion at corresponding points are parallel. 3. The angles between the asymptotic directions at a point on a surface and between their spherical representation are equal or supplementary, according as the surface has positive or negative curvature at the point. FIG. 16 GENERAL EXAMPLES 149 4. Show that the helicoidal surface x = u cos v, y = u sin v, z = bv is minimal. 5. The total curvature of a helicoid is constant along a helix. 6. The orthogonal trajectories of the helices upon a helicoid are geodesies. 7. If the fundamental functions E, F, G of a surface are functions of a single parameter w, the surface is applicable to a surface of revolution. 8. Find the equations of the helicoid generated by a circle of constant radius whose plane passes through the axis and the lines of curvature on the surface ; also find the equations of the surface in terms of parameters referring to the meridians and their orthogonal trajectories. GENERAL EXAMPLES 1. If a pencil of planes be drawn through a tangent MT to a surface, and if lengths be laid off on the normals at M to the sections of the surface by these planes equal to the curvature of the sections, the locus of the end points is a straight line normal to the plane determined by MT and the normal to the surface at M. 2. If P is a point of a developable surface, P the point where the generator through P touches the edge of regression, t the length PoP, p and r the radii of curvature and torsion of the edge of regression, then the principal radii of the surface are given by -i -i = 0, - = - 3. For the surface of revolution of a parabola about its directrix, the principal radii are in constant ratio. 4. The equations x = a cos it, y = asinw, z = uv define a family of circular helices which pass through the point A (a, 0, 0) of the cylinder ; each helix has an involute whose points are at the distance c from A (cf. I, 106). Find the surface which is the locus of these involutes ; show that the tangents to the helices are normal to this surface ; find also the lines of curvature upon the latter. 5. The surfaces defined by the equations (cf. 25) l+p 2 + ? 2 = q*f(y), x + pz=<t>(p) have a system of lines of curvature in planes parallel to the xz-plane and to the y-axis respectively. 6. The equations y - ax = 0, a: 2 + y 2 + z 2 - 2 px - a 2 = 0, where a and ft are parameters, define all the circles through the points (0, 0, a), (0,0, a). Show that the circles determined by a relation ft=f(a) are the characteristics of a family of spheres, except when f(a) is a linear function ; also that the circles are lines of curvature on the envelope of these spheres. 7. If one of the lines of curvature of a developable surface lies upon a sphere, the other nonrectilinear lines of curvature lie on concentric spheres. 150 GEOMETRY OF A SURFACE ABOUT A POINT 8. If P is a point on a surface, P the center of normal curvature of the line bisecting the angle between the lines of curvature, and PI, P 2 the centers of normal curvature in two directions equally inclined to the first, then the four points P, PI, PO, PZ form a harmonic range. 9. If EI, R a , R-s, , R m denote the radii of normal curvature of m sections of a surface which make equal angles 2 tr/m with one another, and m > 2, then I /-I V--1 + J_\ = * /i + IV m \Ri R 2 RJ 2 Vx T P J 10. If the Dupin indicatrix at a point P of a surface is an ellipse, and through either one of the asymptotes of its focal hyperbola two planes be drawn perpen dicular to one another, their intersections with the tangent plane are conjugate directions on the surface. 1 1 . All curves tangent to an asymptotic line at a point M , and whose osculating planes are not tangent to the surface at 3f, have M for a point of inflection. 12. The normal curvature of an orthogonal trajectory of an asymptotic line is equal to the mean curvature of the surface at the point of intersection. 13. The surface of revolution whose equations are x = u cos w, y = u sin w, z = a log (u 4- vV 2 a 2 ) is generated by the rotation of a catenary about its axis ; it is called the catenoid. Show that it is the only minimal surface of revolution. 14. When the osculating plane of a line of curvature makes a constant angle with the tangent plane to the surface, the line of curvature is plane. 15. A plane line of curvature is represented on the unit sphere by a circle. 16. The cylinder whose right section is the curve defined by the intrinsic equa tion p = a - s 2 /6, where a and b are positive constants, has the characteristic prop erty that upon it lie curves of curvature ^l a + b , whose geodesic curvature is I/Void. ^ a26 17. When a surface is referred to an orthogonal system of lines, and the radii of geodesic curvature of the curves v = const, and u - const. are p^, p gv respectively, the geodesic curvature of the curve which makes an angle Q with the lines v = const, is given by 1 _ dd cos0 sin > Pg ~~ dS Pgu Pgv 18. When a surface is referred to an orthogonal system of lines, and p vi s denote the radius of geodesic curvature and the arc for one system of isogonal trajectories of the parametric lines, and pj, 8 the similar functions for the orthogonal trajectories of the former, then whatever be the direction of the first curves the quantity 4. to constant at a point. 19. If p and p denote the radii of first curvature of a line of curvature and its spherical representation, and also p, and p g the radii of geodesic curvature of these curves, then fi s d ff fa dtr 7 = 7 P~8~P? where ds and d<r are the linear elements of the curves. GENERAL EXAMPLES 151 20. When a surface is referred to its lines of curvature, and , #o denote the angles which a curve on the surface and its spherical representation make with the curves v = const., the radii of geodesic curvature of these curves, denoted by p g and pg respectively, are in the relation ds dcr ddo = d&o -- - Py Pg 21. When the curve x=f(u)cosu, y=f(u)sinu, z = - is subjected to a helicoidal motion of parameter a about the z-axis, the various positions of this curve are orthogonal trajectories of the helices, and also geodesies on the surface. 22. When a curve is subjected to a continuous rotation about an axis, and at the same time to a homothetic transformation with respect to a point of the axis, such that the tangent to the locus described by a point of the curve makes a con stant angle with the axis, the locus of the resulting curves is called a spiral surface. Show that if the z-axis be taken for the axis of rotation and the origin for the center of the transformation, the equations of the surface are of the form x = f(u) e hv cos (u + v) , y = f(u) e hv sin (u + v) , z = (u) e* v , where h is a constant. 23. A spiral surface can be generated in the following manner: Let C be a curve, I any line, and P a point on the latter ; if each point M on C describes an isogonal trajectory of the generators on the circular cone with vertex P and axis I in such a way that the perpendicular upon I, from the moving point M , revolves about I with constant velocity, the locus of these curves is a spiral surface (cf . Ex. 5, 33). 24. Show that the orthogonal trajectories of the curves u = const., in Ex. 22, can be found by quadratures, and that the linear element can be put in the form where A is a function of a alone. 25. Show that the lines of curvature, minimal lines, and asymptotic lines upon a spiral surface can be found by quadrature. CHAPTER V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63. Christoffel symbols. In this chapter we derive the necessary and sufficient equations of condition to be satisfied by six func tions, E, F, G ; D, D\ D", in order that they may be the fundamental quantities for a surface. For the sake of brevity we make use of two sets of symbols, suggested by Christoffel,* which represent certain functions of the coefficients of a quadratic differential form and their derivatives of the first order. If the differential form is a n du 2 -f- 2 a^du^du^ + c the first set of symbols is defined by R&l !/^ + ^_<HA [l J 2\du k du ( duj where each of the subscripts i, k, I has one of the values 1 and 2.f From this definition it follows that When these symbols are used in connection ,with the first fun damental quadratic form of a surface ds 2 = E du 2 + 2F dudv + G dv 2 , they are found to have the following significance : (1) [iriia* [""L^ia* L 1 J 2Su L 2 J Su 2 2dv I ~l I Q/* O ,* I O I L J- I cv & cu L ^ J ^ 2 J 2 di< 2~dv * Crelle, Vol. LXX, pp. 241-245. t This equation defines these symbols for a quadratic form of any number of vari ables wi, , u n . In this case i, k, I take the values 1, , n. 152 CHRISTOFFEL SYMBOLS 153 The second set of symbols is defined by the equation where A vl denotes the algebraic complement of a vl in the discrimi nant a n a 22 a 2 2 divided by the discriminant itself. With reference to the first fundamental quadratic form these symbols mean 4i--Jv ^ =: ^ A =-!r* and du dv du fill du dv du f) TT 2 I n I ," Ti-O ri2\ ^g^ du I 12 \_ llJ 2^ l2/~ du dv f" G 2 G ^v 3t* dv f221 dv du dv From these equations we derive the following identities : With the aid of these identities we derive from (III, 15, 16) the expressions > (MVKff}> Sm-K?))- f^l From the above definition of the symbols <! > WQ obtain the following important relation : 64. The equations of Gauss and of Codazzi. The first two of equations (IV, 10) and the equation 154 FUNDAMENTAL EQUATIONS form a consistent set of equations linear in , - , and the 2 du 2 du 2 du 2 determinant is equal to H. Solving for -, we get similar equations hold for y and z. Proceeding in like manner with the other equations (IV, 10) and (6) we get the following equations of G-auss : a 2 * _ri2\az fiaiat ^ awa li/a* la/a* For convenience of reference we recall from 48 the equations (8) dX_FD GDdx FDED dx ~du = H 2 a^ 4 n 2 dv dX __ FD"-GD dx FD -ED" dx 3v ~ H 2 du H 2 dv The conditions of integrability of the Gauss equations (7) are du\dudv dv \dudv du\dv 2 By means of (7) and (8) these equations are reducible to the forms <* x v o ^ 4. A ?E _i_ x 1 dv 1 2 du 2 v (9) a where a v a 2 , >, c 2 are determinate functions of E, F, G ; D, D , D" and their derivatives. Since equations similar to (9) hold for y and z, we must have (10) !=0, a 2 =0, &j=0, 6 2 =0, ^=0, <? 2 =0. EQUATIONS OF GAUSS AND OF CODAZZI 155 When the expressions for a r a 2 , b^ and 5 2 are calculated, it is found that the first four equations are equivalent to the following : 12 12\ jMlll |12jfl2j fill J221 d /in d ri2i , rni/121 , rnif22 fi2i fill ri2i 2 HiAaJ-ta-J 221 d M21 , /221 /121 , /221 Ml f!2\ a /22\ , fl2\fl2\ f22\/ll When the expressions for the Christoffel symbols are substituted in these equations the latter reduce to the single equation Z?D"-jP a = 1 f g r F dE__ 1 3G1 H 2 ~2N \du \_EH dv H du\ j^r 2 dF i _ dv IH du H dv EH du_ This equation was discovered by Gauss, and is called the G-auss equation of condition upon the fundamental functions. The left- hand member of the equation is the expression for the total curva ture of the surface. Hence we have the celebrated theorem of Gauss * : The expression for the total curvature of a surface is a function of the fundamental coefficients of the first order and of their deriva tives of the first and second orders. When the expressions for c^ and c 2 are calculated, we find that the last two of equations (10) are (13) v du " dv *L.c., p. 20. 156 FUNDAMENTAL EQUATIONS These are the Codazzi equations, so called because they are equiva lent to the equations found by Codazzi * ; however, it should be mentioned that Mainardi was brought to similar results some what earlier. f It is sometimes convenient to have these equations written in the form (13 ) D d D , f22\D f!2\D fin ~ " ~~ which reduce readily to (13) by means of (3). With the aid of equations (7) we find that the conditions of integrability of equations (8) and similar ones in Y and Z reduce to (13). From the preceding theorem and the definition of applicable surfaces ( 43) follows the theorem : Two applicable surfaces have the same total curvature at corre sponding points. As a consequence we have : Every surface applicable to a plane is the tangent surface of a twisted curve. For, when a surface is applicable to a plane its linear element is reducible to ds 2 = du 2 + dv\ and consequently its total curvature is zero at every point by (12). From (IV, 73) it follows that 2 Hence X, Y, Z are functions of a single parameter, and therefore the surface is the tangent surface of a twisted curve (cf. 27). Incidentally we have proved the theorem : When K is zero at all points of a surface the latter is developable, and conversely. * Sulle coordinate curvilinee d una superficie e dello spazio, Annali, Ser. 3, Vol. II (18(W) , p. 269. t Giornale dell Istituto Lombardo, Vol. IX, p. 395. FUNDAMENTAL THEOREM 157 65. Fundamental theorem. When the lines of curvature are parametric, the Gauss and Codazzi equations (12), (13) reduce to (14) DP" _ G dv 2_ ( D "\ _ ^ ~ E du The direction-cosines of the tangents to the parametric curves, v = const, and u = const., have the respective values (15) 2 By means of equations (7) and (8) we find (16) . du ( du D" V ax and similar equations obtained by replacing X^ X^ X by Y v T 2 , Y respectively, and by Z^ Z^, Z. From (15) we have (17) We proceed to the proof of the converse theorem : Given four functions, E, G, D, D", satisfying equations (14); there exists a surface for which E, 0, G ; Z>, 0, D" are the fundamental quantities of the first and second order respectively. = C^EX = CVflY = CVEZ 158 FUNDAMENTAL EQUATIONS In the first place we remark that all the conditions of integra- bility of the equations (16) are satisfied in consequence of (14). Hence these equations admit sets of particular solutions whose values for the initial values of u and v are arbitrary. From the form of equations (16) it follows (cf. 13) that, if two such sets of particular solutions be denoted by X^ X z , X and Y^ F 2 , F, then I XI + XI + X* = const., Y?+ F 2 + F 2 = const., X 1 F x + X 2 Y 2 + XY= const. From the theory of differential equations we know that there exist three particular sets of solutions X^ X z , X\ Fp F 2 , F; Z^ Z^ Z, which for the initial values of u and v have the values 1, 0, 0; 0, 1, 0; 0, 0, 1. In this case equations (18) become X 1 2 + X 2 2 + ^ 2 = 1, (19) F 2 +F 2 2 + F 2 = l, X i Y l +X 2 Y 2 + XY=Q, which are true for all values of u and v. In like manner we have (19 ) From (16) it follows that the expressions in the right-hand mem bers of (17) are exact differentials, and that the surface denned by these equations has, for its linear element and its second quadratic form, the expressions (20) Edu?+G dv\ D dy? + D"dv 2 respectively. Suppose, now, that we had a second system of three sets of solutions of equations (16) satisfying the conditions (19), (19 ). By a motion in space we could make these X s, F s, and Z s equal to the corresponding ones of the first system for the initial values of u and v. But then, because of the relations similar to (18), they would be equal for all values of u and v, as shown in 13. Hence, to within a motion in space, a surface is determined by two quad ratic forms (20). As in 13, it can be shown that the solution of equations (16) reduces to the integration of an equation of Riccati. FUNDAMENTAL THEOEEM 159 Later * we shall find that the direction-cosines of any two per pendicular lines in the tangent plane to a surface, and of the normal to the surface, satisfy a system of equations similar in form to (16). Moreover, these equations possess the property that sets of solu tions satisfy the conditions (18) when the parametric lines are any whatever. Hence the choice of lines of curvature as parametric lines simplifies the preceding equations, but the result is a general one. Consequently we have the following fundamental theorem: When the coefficients of two quadratic forms, Edu* + 2 Fdudv + G dv\ Ddu 2 +2 D dudv + D"dv\ satisfy the equations of Gauss and Codazzi, there exists a surface, unique to within its position in space, for which these forms are respectively the first and second fundamental quadratic forms ; and the determination of the surface requires the integration of a Riccati equation and quadratures. From (III, 3), (5) and (6), it follows that if E, F, G; D, D , D" are the funda mental functions for a surface of coordinates (x, y, z), the surface symmetric with respect to the origin, that is, the surface with the coordinates ( x, y, z), has the fundamental functions E, F, G; - D, - D , - D". Moreover, in consequence of the above theorem, two surfaces whose fundamental quantities bear such a rela tion can be moved in space so that they will be symmetric with respect to a point. Two surfaces of this kind will be treated as the same surface. EXAMPLES 1. When the lines of curvature of a surface form an isothermal system, the surface is said to be isothermic. Show that surfaces of revolution are isothermic. 2. Show that the hyperbolic paraboloid a b uv x = -(t* + t>), y = -(*-), * = - is isothermic. 3. When a surface is isothermic, and the linear element, expressed in terms of parameters referring to the lines of curvature, is ds 2 = \ 2 (du 2 + dv 2 ), the equations of Codazzi and Gauss are reducible to Pl i dp 2 a _ PZ i api 4. Find the form of equations (11), (13) when the surface is defined in terms of symmetric coordinates (cf . 39). * Cf . 69. Consult also Scheffers, Vol. II, pp. 310 et seq. ; Bianchi, Vol. I, pp. 122-124. 160 FUNDAMENTAL EQUATIONS 5. Show that K is equal to zero for the tangent surface of a twisted curve, taking the linear element of the latter in the form (105), 20. 6. Show that the total curvature of the surface of revolution of the tractrix about its axis is negative and constant. 7. Establish the following formulas, in which the differential parameters are formed with respect to the form Edu? + 2Fdudv + Gdv 2 : )=~- where the quantities have the same significance as in 65. 8. Deduce the identity A 2 x = ( 1 ) JT, and show therefrom that the curves in which a minimal surface is cut by a family of parallel planes and the orthogonal trajectories of these curves form an isothermal system. 66. Fundamental equations in another form. We have seen in 61 that if X, F, Z denote the direction-cosines of the normal to a surface, the direction-cosines of the normal to the spherical rep resentation of the surface are eX, eF, eZ, where e is 1 according as the curvature of the surface is positive or negative. If, then, the second fundamental quantities for the sphere be denoted by A ^ , 3", we have (21) =-< ^ = -e^, ,&" = -e^ so that for the sphere equations (7) become rnvax_ 12J dv 2 2X f J 2X f 12V + l2J (22) F -^x; where the Christoffel symbols \ T ?\ are formed with respect to the linear element of the spherical representation, namely The conditions of integrability of equations (22) are reducible by means of the latter to = 0, EQUATIONS OF CODAZZI 161 where A., A 2J B^ B 2 are the functions obtained from the quantities DD"D 2 a v a 2 , b lt b 2 respectively of 64 by replacing > E, F, G by 1, , &, ^respectively. Since the above equations must be sat isfied by Y and Z, the quantities A^ A 2 , B v B z must be zero. This gives the single equation of condition (24} J_rA/^^__L^V-^- -- --^- ^1 = 1 J 2 ft [du \ft dv ft du ) to W fa ft 2v &ft du)\ Moreover, the Codazzi equations (13 ) become, in consequence of (21), (26). which vanish identically. o \ //"/ o V // / I 1 //" ~ * ~* * f ~* * "" 3w W / v \/// \.\jft dx dx If equations (IV, 13) be solved for and we get du cv f dx _ " (26) . i_ dv ft* du By means of equations (22) the condition of integrability of these equations, namely g /^\ ^ and similar conditions in y and 2, reduce to (27) - OU v^ ^Hu dv Hence two quadratic forms (odu 2 -f 2 & dudv + dv 2 , D du 2 +2Z> dudv + D"dv 2 , whose coefficients satisfy the conditions (24), (27), may be taken as the linear element of the spherical representation of a surface and as the Si3cond quadratic form of the latter. When X, F, Z are 162 FUNDAMENTAL EQUATIONS known, the cartesian coordinates of the surface can be found by quadratures (26) ; however, the determination of the former requires the solution of a Riccati equation. If the equations D = _^fc*x t be differentiated with respect to u and v, the resulting equations may be reduced by means of (7) and (22) to the form : * 55 (ii cu ( 1 = [ I2 l a c i cD cu ^-= Jl * JX + cv cu cv (V D D" 2) -f D + D -f 11 2 12 2 12 2 22 ) 2 D". 67. Tangential coordinates. Mean evolute. A surface may be looked upon not only as the locus of a point whose position depends upon two parameters, but also as the envelope of its tangent planes. This family of planes depend* upon one or two parameters according as the surface is developable or not. We considered the former case in 27, and now take up the latter. If W denotes the algebraic distance from the origin to the tan gent plane to a surface S at the point M(x, y, z), then (29) W=xX+yY+zZ. If this equation is differentiated with respect to u and v, the resulting equations are reducible, in consequence of (IV, 3), to X dW *Cf. Bianchi, Vol. I, p. 157. TANGENTIAL COORDINATES 163 The three equations (29), (30) are linear in #, y, z, and in con sequence of (IV, 79, 80) their determinant is equal to e/ Hence we have W Y Z e dW cY dZ x = tt cu du du dW cY dZ dv dv dv and similar expressions for y and z. From (IV, 11) we deduce the identities dz dY e / ^dX , ~dX\ Y Z = TH ex 1 r <o ) 01 du du rf \ du dv / ( dl ) a - y. v , ~Y zr< -r r C rrV^ I C^\- j^G-A- By means of these equations the above expression for x is reducible to cv Hence we have (32) x = WX+k[(W,X), y = WY+k((W,Y), z = WZ+ &((W,Z], the differential parameters being formed with respect to (23). Conversely, if we have four functions X, F, Z, W of u and i>, such that the first three satisfy the identity (33) x 2 +r 2 +^ 2 = l, equations (32) define the surface for which X, F, Z are the direction- cosines of the tangent plane, and W is the distance of the latter from the origin. For, from (33), we have dv = 0, in consequence of which and formulas (22) we find from (32) that >TA dx Moreover, equation (29) also follows from (32). Hence a surface is completely defined by the functions X, F, Z, W, which are called the tangential coordinates of the surface.* * Cf . Weingarten, Festschrift der Technischen Hochschule zu Berlin (1884) ; Bianchi, Vol. I, pp. 172-174 ; Darboux, Vol. I, pp. 234-248. 164 FUNDAMENTAL EQUATIONS When equations (30) are differentiated, we obtain ffw _ dv* By means of (22), (29), and (30) these equations are reducible to (34) D = - du 2 tfW \_dudv When these expressions for D, A &" are substituted in the expression (IV, 77) for p^+ p 2 , the latter becomes By means of (25) this equation can be written in the form (35) /> 1 4-^ 2 = -(A;TF4-2^), where the differential parameter is formed with respect to the linear element (23) of the sphere. Moreover, if A^ 2 denotes the following expression, _ r22y^__ 1 1 J an 2 J 12 12 it follows from (34) that (3T) MEAN EVOLUTE 165 In passing we shall prove that A 22 is a differential parameter by showing that it is expressible in the form (38) Without loss of generality we take (39) Edu*+Gdv 2 as the quadratic form, with respect to which these differential parameters are formed. Then 1 1 /I dG 1 dE\ 1 dE dv \du u = - F By substitution we find _ Since the terms in the right-hand member are differential param eters, their values are independent of the choice of parameters u and v, in terms of which (39) is expressed. Hence equation (38) is an identity. The coordinates # , y , Z Q of the point on the normal to a sur face halfway between the centers of principal curvature have the expressions The surface enveloped by the plane through this point, which is parallel to the tangent plane to the given surface, is called the mean evolute of the latter. If W denotes the distance from the origin to this plane, we have (40) W,= ZXf 9 =W+^(p 1 + Pt ). By means of (35) this^may be written (41) TF O =-JA;TF. 166 FUNDAMENTAL EQUATIONS EXAMPLES 1. Derive the equations of the lines of curvature and the expressions for the principal radii in terms of W, when the parametric lines on the sphere are (i) meridians and parallels ; (ii) the imaginary generators. Show that in the latter case the curves corresponding to the generators lie sym metrically with respect to the lines of curvature. 2. Let Wi and W 2 denote the distances from the origin to the planes through the normal to a surface and the tangents to the lines of curvature v = const. , u = const, respectively, so that we have Wi = xX l + yYi + zZi, W 2 = xX 2 + yY z + zZ 2 . Show that Pi the differential parameters being formed with respect to Edu* + 2 Fdudv + 3. If 2 q = x 2 + y z + z 2 , then we have 4. Show that when the lines of curvature are parametric = = Pi cu cu P 2 v ~ dv 5. The determination of surfaces whose mean evolute is a point is the same problem as finding isothermal systems of lines on the sphere. 68. The moving trihedral. The fundamental, equations of con dition may be given another form, in which they are frequently used by French writers. In deriving them we refer the surface to a moving set of rectangular axes called the trihedral T. Its ver tex M is a point of the surface, the a^-plane is tangent to the surface at M, and the positive 2-axis coincides with the positive direction of the normal to the surface at M. The position of the x- and ?/-axes is determined by the angle U which the tangent to the curve v = const, through M makes with the z-axis, U being a given function of u and v. In Chapter I we considered another moving trihedral, consisting of the tangent, principal normal, and binormal of a twisted curve. Let us associate such a trihedral with the curve v const, through THE MOVING TRIHEDRAL 16T M and call it the trihedral t u . We have found ( 16) that the varia tions of the direction-cosines a , 6 , c of a line L, fixed in space, with reference to M , as its vertex moves along the curve which we call (7 M , are given by (42 ) <^= b -, f W- + ds u p u ds u \ Pu where p u , r u denote the radii of first and second curvature of C u1 and ds u its linear element ; evidently the latter may be replaced by V^ 1 du. The direction-cosines of L with respect to the trihedral T have the values r a _ a > cos jj _ ^ s j n ^ _ c > cos ^ s j n ^r (43) I b a sin U + (b r sin w u c cos o> M ) cos ?7, [<7 = 6 f cos o> w -f c sin o) tt , where w u is the angle which the positive direction of the z-axis makes with the positive direction of the principal normal to C u at Jf, the angle being measured toward the positive direction of the binor- mal of C u . From equations (42) and (43) we obtain the following : (44) da , db do , - = br cq, = cp ar, = aq op, du du cu where j9, q, r have the following significance : (45) p = = cosU 1 . coso) )-f sin U rr/ c?&> 1\ rr coso). sm7( 2 I cosU ds,. V If, in like manner, we consider the trihedral t v of the curve u = const, through M, denoted by C v , we obtain the equations da , db do , where p v q v r l can be obtained from (45) by replacing Vjg; Z7, * M , ^^ Pui T u ^7 ^^^ ^ 8 v-> i; Pv T V - A S ^ denotes the angle which the tangent to the curve C v at M makes with the a>axis, we have (46) V-U=G. 168 FUNDAMENTAL EQUATIONS If the vertex M moves along a curve other than a parametric line, that is, along a curve determined by a value of dv/du, the variations of a, , c are evidently given by da du da dv du ds dv ds ^ ^_ du ds dv ds do du do dv du ds dv ds in which the differential quotients have the above values. 69. Fundamental equations of condition. Suppose that we asso ciate with the trihedral T a second trihedral T Q whose vertex is fixed in space, about which it revolves in such a manner that its edges are always parallel to the corresponding edges of T, as the vertex of the latter moves over the surface in a given manner. The position of T is completely determined by the nine direction- cosines of its edges with three mutually perpendicular lines L v L 2 , L s through 0. Call these direction-cosines a v b { , c l ; a 2 , 5 2 , c 3 , <? 3 . These functions must satisfy the equations (47) da , - = 6r- ^w = br - dv~ TI If we equate the db = cp ar, du dc, , ^ = ?-ip, db _ dv dc , - = aq-l Pl . -irolnQO f\f . rkVvf.o i ma rl fvrvm cucv two of these equations, and in the reduction of the resulting equa tion make use of (47), we find Since this equation must be true when b and c have the values b , <? ; 5 , ; 6 3 , <? 8 , the expressions in parenthesis must be equal ** ^>O T /^9 to zero. Proceeding in the same manner with - obtain the following fundamental equations * : dp dpi and d*c dudv we (48) dq dq l dr dr l * These equations were first obtained by Combescure, Annales de VEcole Normale, Ser. 1, Vol. IV (1867), p. 108; cf. also Darboux, Vol. I, p. 48. ROTATIONS 169 These necessary conditions upon the six functions p, - - , r 1? in order that the nine functions a x , , c s may determine the position of the trihedral T , are also sufficient conditions. The proof of this is similar to that given in 65. Equations (47) have been obtained by Darboux * from a study of the motion of the trihedral T Q . He has called jt?, q, r l the rotations. We return to the consideration of the moving trihedrals T and t u . Let (x, y, z) and (# , ?/, z ) denote the coordinates of a point P with respect to T and t u respectively. Between these coordinates the following relations hold : (x = x f cos U (y sin o> M z cos W M ) sin 7, / = # sin CT + (# sin W u 2 cos &> M ) cos /, z y 1 cos <W M + 2 sin o> M . If in a displacement of P absolute increments with respect to the trihedral t u at M be indicated by S, and increments relative to these moving axes by c?, we have, from 16, ^L^-S^+i, = ^_ + - + -, ^. = dz L-y-. d u ds u p u ds u ds u p u T M C?S M ds u r u From (49), (50), and (45) we obtain the following! : $x dx r = 4- V E cos Ury + qz, du du -^ = + VfismUpz + ra, ^W ^M 2 dz = ox + PV. aw a% Equations similar to these follow also from the consideration of the trihedral t v . Hence, when the trihedral T moves over the surface with its vertex M describing a curve determined by a value of dv/du, the increments of the coordinates of a point P(x, y, z), in the directions of the axes of the trihedral, in the * L.c., Vol. I, chaps, i and v. t In deriving these equations we have made use of the fact that equations (49) define a transformation of coordinates, and consequently hold when the coordinates are replaced by the projections of an absolute displacement of P. 170 FUNDAMENTAL EQUATIONS absolute displacement of P, which may also be moving relative to these axes, have the values * where we have put The coordinates of M are (0, 0, 0), so that the increments of its displacements are (53) Sx = 1; du + ^dv, y = vidu + ri l dv, Sz = 0. If fa, y v 2J denote the coordinates of M with respect to the fixed axes formed by the lines L v 2 , L 3 previously defined, it follows that 2 , 2 and similar expressions for y l and z^ where a t , 6 1? ^; 2 , 6 2 a g , 6 8 , c 3 are the direction-cosines of the fixed axes with reference to the moving axes. Since the latter satisfy equations (47), the conditions that the two values of ^- obtained from (54) be equal, a 2 d z z cu and similarly for * and ---1 are J dudv (55) When we have ten functions f , f p ?/, 77^ p, p^ <?, q^ r, r x , satis fying these conditions and (48), the functions a 1? -, c s can be found by the solution of a Riccati equation, and x^ y# z l by quad ratures. Hence equations (48) and (55) are sufficient as well as necessary, and consequently are equivalent to the Gauss and Codazzi equations. * Cf. Darboux, Vol. II, p. 348. LINES OF CURVATUKE 171 70. Linear element. Lines of curvature. From (53) we see that the linear element of the surface is (56) ds 2 = (%du + ^ dvf +(ndu + r )i dv}\ Hence a necessary and sufficient condition that the parametric lines be orthogonal is (57) ff!+^i==0. For a sphere of radius c the coordinates of the center are (0, 0, - c), it being assumed that the positive normal is directed outwards. As this is a fixed point, it follows from equations (51) that whatever be the value of dv/du we must have du + hdv - (qdu + qidv)c = 0, ydu + 171 dv + (pdu + pidv)c = 0, and consequently i) 1 ^i /KQ\ zr = = =: C. q -P qi -Pi Conversely, when these equations are satisfied, the point (0, 0, c) is fixed in space, and therefore the surface is a sphere. Moreover, suppose that we have a propor tion such as (58), where the factor of proportionality is not necessarily constant. For the moment call it t. When the values from (58) are substituted in (55) and reduction is made in accordance with (48) we get dt dt 3t dt r? ^" 7?1 ^ = *to~* l to Hence t is constant unless ^ - f^ is zero, which, from (56) and 31, is seen to be possible only in case the surface is isotropic developable. By definition (51) a line of curvature is a curve along which the normals to the surface form a developable surface. When the vertex is displaced along one of these lines, a point (0, 0, p) must move in such a way that Bx and % are zero. Hence we must have f du -h ^dv + (qdu + q l dv)p = 0, r]du + i] 1 dv (pdu+p l dv)p = 0. Eliminating p and dv/du respectively, we obtain the equation of the lines of curvature, (59) (f du + ^dv) (p du + p v dv) + (17 du + rj^v) (q du + q^v) = 0, and the equation of the principal radii, (60) p z (pq, - q Pl ) + p (qrj, - q,rj + p^ - p ) + (fa - ^) = 0. From (59) it follows that a necessary and sufficient condition that the parametric lines be the lines of curvature is (61) fe> + i# = 0, f^i+ih^O. 172 FUNDAMENTAL EQUATIONS We may replace these equations by P = \rj, ? = -xf, ^^V?!, ?i = -\fn thus introducing two auxiliary functions X and \. When these values are substituted in the third of (55), we have If X and \ are equal, the above equations are of the form (58), which were seen to be characteristic of the sphere and the iso- tropic developable. Hence the second factor is zero, so that equa tions (61) may be replaced by (62) ffi+ ^=0, m+??i=<> or (63) ^=17 = 0, P = q 1 =Q- From (52) it follows that in the latter case the x- and ?/-axes are tangent to the curves v = const, and u = const. We shall consider this case later. From (60) and (52) we find that the expression for the total curvature of the surface is where co denotes the angle between the parametric curves. Hence the third of equations (48) may be written /g4\ V76rsin co H dr dr. PiP* PiP 71. Conjugate directions and asymptotic directions. Spherical representation. We have found ( 54) that the direction in the tangent plane conjugate to a given direction is the characteristic of this plane as it envelopes the surface in the given direction. Hence, from the point of view of the moving trihedral, the direc tion conjugate to a displacement, determined by a value of dv/du, is the line in the #?/-plane which passes through the origin, and which does not experience an absolute displacement in the direction of the 2;-axis. From the third of equations (51) it is seen that the equation of this line is (65) (p du H- p^v) y (q du + q^dv) x = 0. CONJUGATE DIRECTIONS 173 If the increments of u and v, corresponding to a displacement in the direction of this line, be indicated by d^ and d^v, the quan tities x and y are proportional to (f d^u + f ^v) and (r; d^ 4- rj^v). When x and ?/ in (65) are replaced by these values, the resulting equation may be reduced to (66) (prj - gf) dudjU + (pr ]l - qgj dudy + (p& - qg) d^udv In consequence of (55) the coefficients of dud^v and d^udv are equal, so that the equation is symmetrical with respect to the two sets of differentials, thus establishing the fact that the rela tion between a line and its conjugate is reciprocal. In order that the parametric lines be conjugate, equation (66) must be satisfied by du = and d^v = 0. Hence we must have (67) It should be noticed that equations (61) are a consequence of the first of (62) and (67). Hence we have the result that the lines of curvature form the only orthogonal conjugate system. From (66) it follows that the asymptotic directions are given by (68) (prj - gf ) du* + (prj l - q^ +p^ - q) dudv + (p^ - q^) dv 2 = 0. The spherical representation of a surface is traced out by the point m, whose coordinates are (0, 0, 1) with respect to the tri hedral T of fixed vertex. From (51) we find that the projections of a displacement of m, corresponding to a displacement along the surface, are (69) SX=qdu + q l dv, &Y= (pdu+p l dv), &= 0. Hence the linear element of the spherical representation is (70) da 2 = (qdu + q^v) 2 + (pdu+ p^dv)\ The line defined by (65) is evidently perpendicular to the direc tion of the displacement of m, as given by (69). Hence the tangent to the spherical representation of a curve upon a surface is perpen dicular to the direction conjugate to the curve at the corresponding point. Therefore the tangents to a line of curvature and its rep resentation are parallel, whereas an asymptotic direction and its representation are perpendicular ( 61). 174 FUNDAMENTAL EQUATIONS 72. Fundamental relations and formulas. From equations (53) and (69) we have, for the point M on the surface, = ^ = = du * (71) "" = ?, =i?i, = 0; and 5v ~V \7~ ^\ 7 cu du du (72) .. Consequently the following relations hold between the fundamental coefficients, the rotations, and the translations: F= f + ^, G = f (73) When, in particular, the parametric system on a surface is orthog onal, and the x- and y- axes of the trihedral are tangent to the curves v = const, and u = const, through the vertex, equations (52) are ( 74) f=V5, 17 = =0, and equations (55) reduce to (76) r L Moreover, equations (45) and the similar ones for p lt q^ r t become (76) P - ""^ T.. . The first two of equations (75) lead, by means of (76), to sin w 1 d^/~E shift\ 1 P U " VEG fo P V which follow also from 58. The third of equations (75) establishes the fact, previously remarked in 59, that the geodesic torsion in two orthogonal directions differs only in sign. FUNDAMENTAL RELATIONS 175 The variations of the direction-cosines X\, Y\, Z\ of the tangent to the curve u const, are represented by the motion of the point (1, 0, 0) of the trihedral T with fixed vertex. From (51) we have 5-5Ti 5^1 dZ\ du cu du (78) dv cv From these equations we see that as a point describes a curve v = const. , namely C M , the tangent to this curve undergoes an infinitesimal rotation consisting of two components, one in amount rdu about the normal to the surface and the other, qdu, about the line in the tangent plane perpendicular to the tangent to C u . Consequently, by their definition, the geodesic and normal curvature of C u are r/^/E and q/^/E respectively. Moreover, it is seen from (72) that as a point describes C u the normal to the surface undergoes a rotation consisting of the com ponents q du about the line in the tangent plane perpendicular to the tangent, and p du about the tangent. Hence, if C u were a geodesic, the torsion would be p/VE to within the sign at least. Thus by geometrical considerations we have obtained the fundamental relations (76). We suppose now that the parametric system is any whatever. From the definition of the differential parameters ( 37) it follows that E = G = Consequently if P, $, ^ denote functions similar to p, q, r, for a general curve . v) = const. which passes through M and whose tangent makes the angle with the moving z-axis, we have, from (45), (79) T, /^ 1\ , ^ cos ^1 cos < ( --- + sm <I> - , \ds r/ p P^VA^T , T idco 1\ ) = H. V A.cf) sin 4> cos < \ds rj cos P sn where by (III, 51) ff~ 2 = A any other family of curves Moreover, equations analogous to (44) are db __ cP aR dc ds A X 2 ((, T|T) and ty = const, defines da _ bE cQ aQ 176 FUNDAMENTAL EQUATIONS T . . da da du da dv If now in - = - H as cu as dv as we replace the expressions for and from (47), and similarly for db/ds and dc/ds, we obtain Pd8=H l \ / A l (f>(p du +p 1 dv), Qds=H^/~K^>(qdu + q^dv), Eds = 7^ VA~C/> (r du -f r^ dv). From these equations and (79) we derive the following funda mental formulas : (80) ( j ds = cos <&(p du + p l dv) + sin < (q du + q l dv), \ as T I ds = sin <I> (p du + p 1 dv) cos 4> (<? c?t* -f q^ dv), P sin oj d<& du dv l p ds ds ds By means of the last of equations (80) we shall express the geodesic curvature of a curve in terms of the functions E, F, G, of their derivatives, and of the angle 6 which the curve makes with the curve v = const. If we take the rr-axis of the trihedral tangent to the curve v = const., we obtain from the last of (80), in consequence of (45), 1 d0 ^/E du /V G dco\dv Po~~ ds Pg ds \Pffv dv/ds From (III, 15, 16) we obtain dv If 2 EG \ dv dv/ dv When this value and the expressions for p gu and p gv (IV, 57) are substituted in the above equation, we have the formula desired: - L __ __ + 2dvds 2H\du E dv ds EXAMPLES 1. A necessary and sufficient condition that the origin of the trihedral T be the only point in the moving zy-plane which generates a surface to which this plane is tangent, is that the surface be nondevelopable. PARALLEL SURFACES 177 2. Determine p so that the point of coordinates (p, 0, 0) with respect to T shall describe a surface to which the x-axis of T is normal ; examine the case when the lines of curvature are parametric and the x-axis is tangent to the curve v = const. 3. When the parametric curves are minimal lines for both the surface and the sphere, it is necessary that or 77 = i, ifji = ii, q = ip, q\ = ipi\ in this case the parametric curves on the surface form a conjugate system, and the surface is minimal (cf. 55). 4. When the asymptotic lines on a surface form an orthogonal system, we must have ^ + ^ = ^ ^ + ^ = Q> in which case the surface is minimal. 5. When the lines of curvature are parametric, and the x-axis of T is tangent to the curve v = const., equations (80) reduce to 1 dw /I 1\ . cosw cos 2 * sin 2 * -j- = ( ) sin * cos 4>, = H , T as \PI PZ/ p pi PZ sin w _ d< 1 / q dp\ du p\ dPz dv\ p ds Pz Pi \pi cv ds q du ds) 6. When the second equation in Ex. 5 is differentiated with respect to s, the resulting equation is reducible to cos u dp sinw/ dw 2\ _ 2 dp\ /du\ s 2 dPi/du\ 2 dv P 2 ds p \ ds T/ du \ds/ dv \ds/ ds dpo du /du\ 2 dp /dv\ 8 7. On a surface a given curve makes the angle * with the x-axis of a trihedral T; the point P of coordinates cos <t>, sin <J>, with reference to the parallel trihedral TO with fixed vertex, describes the spherical indicatrix of the tangent to the curve ; the direction-cosines of the tangent to this curve are sin * sin w, cos < sin w, cos w, where w has the significance indicated in 49, and the linear element is ds/p; derive therefrom by means of (51) the second and third of formulas (80). 8. The point #, whose coordinates with reference to T of Ex. 7 are sin * cos w, cos $ cos w, sin w, describes the spherical indicatrix of the binormal to the given curve on the surface, and its linear element is ds/r; derive therefrom the first of formulas (80). 73. Parallel surfaces. We inquire under what conditions the normals to a surface are normal to a second surface. In order that this be possible, there must exist a function t such that the point of coordinates (0, 0, Q, with reference to the trihedral 7", describes a surface to which the moving 2-axis is constantly normal. Hence 178 FUNDAMENTAL EQUATIONS we must have 8z = 0, and consequently, by equations (51), t must be a constant, which may have any value whatever. We have, therefore, the theorem : If segments of constant length be laid off upon the normals to a sur face, these segments being measured from the surface, the locus of their other end points is a surface with the same normals as the given surface. These two surfaces are said to be parallel. Evidently there is an infinity of surfaces parallel to a given surface, and all of them have the same spherical representation. Consider the surface for which t has the value a, and call it $. From (51) it follows that the projections on the axes of T of a dis placement on S have the values r f du + ^dv -f (q du + q^dv) a, (82) = 77 du -f- jj^dv (p du + Pidv) a. Comparing these results with (53), we see that the displacements on the two surfaces corresponding to the same value of dv/du are parallel only in case equation (59) is satisfied, that is, when the point describes a line of curvature on S. But from a characteristic property of lines of curvature ( 51) it follows that the lines of curva ture on the two surfaces correspond. Hence we have the theorem : The tangents to corresponding lines of curvature of two parallel surfaces at corresponding points are parallel. From (82) and (73) we have the following expressions for the first fundamental quantities of /S Y : y or, in consequence of (IV, 78), (84) / PI p i / PARALLEL SURFACES 179 The moving trihedral for S can be taken parallel to T for , and thus the rotations are the same for both trihedrals ; and from (82) it follows that the translations have the values = + 00, li=i+ fl ?i> ^ = i?-op, *?i=>?i-api- On substituting in the equations for analogous to (59), (60), (66), we obtain the fundamental equations for S in terms of the functions for S. Also from (73) we have the following expressions for the second fundamental coefficients for S: (85) D = D-a, D = D -a&, D" = D" - ag. Since the centers of principal curvature of a surface and its parallel at corresponding points are the same, it follows that (86) Pi = Pi+ a > P 2 = P2+ a Suppose that we have a surface whose total curvature is constant and equal to 1/c 2 . Evidently a sphere of radius c is of this kind, but later (Chapter VIII) it will be shown that there is a large group of surfaces with this property. We call them spherical surfaces. From (86) we have ^ _ ^ ( ^ _ a ) = ^ so that if we take a = c, we obtain I+l-i. Pi P* c Hence we have the theorem of Bonnet : * With every surface of constant total curvature 1/c 2 there are asso ciated two surfaces of mean curvature 1/ey they are parallel to the former and at the distances :p c from it. And conversely, With every surface whose mean curvature is constant and different from zero there are associated two parallel surfaces, one of which has constant total curvature and the other constant mean curvature. 74. Surfaces of center. As a point M moves over a surface S the corresponding centers of principal curvature M l and M z describe two surfaces S 1 and S 2 , which are called the surfaces of center of S. Let C l and (7 2 be the lines of curvature of S through M, and D l and 7> 2 the developable surfaces formed by the normals to S along C l *Nouvelles annales de mathematiques, Ser. 1, Vol. XII (1853), p. 433. 180 FUNDAMENTAL EQUATIONS and C 2 respectively. The edge of regression of D v denoted by I\, is a curve on S l (see fig. 17), and consequently S l is the locus of one set of evolutes of the curves C l on S. Similarly $ 2 is the locus of a set of evolutes of the curves C z on S. For this reason S 1 and $ 2 are said to constitute the evolute of S, and S is their involute. Evidently any surface parallel to S is also an involute of S l and S 2 . The line M^M^ as a generator of D v is tangent to I\ at M v and, as a generator of D 2 , it is tangent to F 2 at M z . Hence it is a common tangent of the surfaces S l and S z . From this it follows that the developable surface D 1 meets S { along T l and envelopes S z along a curve F 2 . Its generators are con sequently tangent to the curves conjugate to Fg ( 54). In particular, the generator -flfjJfg is tangent to F 2 , and therefore the directions of F 2 and T 2 at Jf 2 are conjugate. Similar results follow from the considera tion of Z> 2 . Hence : On the surfaces of center of a surface S the curves corresponding to the lines of cur vature of S form a conjugate system. FIG. 17 Since the developable D 1 envelopes *Sf 2 , the tangent plane to $ at M 2 is the tangent plane to D l at this point. But the tangent plane at M 2 is tangent to D l all along M 1 M Z ( 25), and consequently it is determined by M^M Z and the tangent to C[ at M. Hence the normal to S at M 2 is parallel to the tangent to C 2 at M. In like manner, the normal to S l at M l is parallel to the tangent to C l at M. Thus, through each normal to S we have two perpendicular planes, of which one is tangent to one surface of center and the other to the second surface. But each of these planes is at the same time tangent to one of the developables, and is the osculating plane of its edge of regression. Hence, at every point of one of these curves, the osculating plane is perpendicular to the tangent plane to the sheet of the evolute upon which it lies, and so we have the theorem : The edges of regression of the developable surfaces formed by the normals to a surface along the lines of curvature of one family are SURFACES* OF CENTER 181 geodesies on the surface of center which is the locus of these edges ; and the developable surf aces formed ly the normals along the lines of curvature in the other family envelope this surface of center along the curves conjugate to these geodesies. In the following sections we shall obtain, in an analytical manner, the results just deduced geometrically. 75. Fundamental quantities for surfaces of center. As the trihe dral T moves over the surface S the point (0, 0, p^ describes the surface of center S r Let the lines of curvature on S be parametric, and the z-axis of T be tangent, to the curve v const. Now \ / & L J. J. L f^ -I. A f\ ri rz so that the first two of equations (48) may be put in the form JI, ___,,=,_ -,_ __. (88) The projections on the moving axes of the absolute displace ment of J/J corresponding to a displacement of M on S are found from (51) to be (89) Bx l = 0, S^ = (rj l p^pj dv = V6r ( 1 )dv, Szj = dp r Hence the linear element of S l is / p V (90) ds*= dri.+ Q(I-^]dfi consequently the curves p^= const, on S l are the orthogonal tra jectories of the curves v = const., which are the edges of regression, I\, of the developables of the normals to S along the lines of curvature v = const. Let us consider the moving trihedral T^ for S l formed by the tangents to the curves v = const, and p l const, at M^ and the nor mal at this point. From (89) it follows that the first tangent has the same direction and sense as the normal to S, and that the sec ond tangent has the same direction as the tangent to u = const, on S, the sense being the same or different according as (1 p l /p 2 ) is 182 FUNDAMENTAL EQUATIONS positive or negative. And the normal to S l has the same direction as the tangent to v = const, on , and the contrary or same sense accordingly. If then we indicate with an accent quantities referring to the moving trihedral T v we have (a =c, l =bj c = ea, where e is 1 according as (1 pjp^ is positive or negative. From (89) it follows that (92) When the values (91) are substituted in equations for 2\ similar to equations (47), we find Since / is zero, it follows from (76) that the curves v = const, are geodesies, as found geometrically. The various fundamental equations for S t may now be obtained by substituting these values in the corresponding equations of the preceding sections. Thus, from (73) we have which follow likewise from (90); and also Hence the parametric curves on S l form a conjugate system (cf. 54). The equation of the lines of curvature may be written and the equation of the asymptotic directions is ^^-41^=0. p? du pl$u SURFACES OF CENTER 183 The expression for K^ the total curvature of S^ is (98) ^-L-Jj. ~du From (80) and (93) it follows that the geodesic curvature at M l of the curve on S l which makes the angle <& l with the curve v = const, through M^ is given by Hence the radius of geodesic curvature of a curve p l = const., that is, a curve for which <J> t is a right angle, has, in consequence of (87), the value p l p . In accordance with 57 the center of geo desic curvature is found by measuring off the distance p l /> 2 , in the negative direction, on the 2-axis of the trihedral T. Consequently M z is this center of curvature. Hence we have the following theo rem of Beltrami: The centers of geodesic curvature of the curves p^ = const, on S t and of p 2 = const, on S., are the corresponding points on $ 2 and S l respectively. For the sheet $ 2 of the evolute we find the following results : (90 ) d** = E\- du 2 + the equation of the lines of curvature is (96 ) r**^ the equation of the asymptotic lines is ^ 5* -BS" the expression for the total curvature is 8ft *"5FS-5 dv 184 FUNDAMENTAL EQUATIONS In consequence of these results we are led to the following theorems of Ribaucour,* the proof of which we leave to the reader : A necessary and sufficient condition that the lines of curvature upon Sj and S 2 correspond is that p l p 2 = c (a constant); then K^ K^ = 1/c 2 , and the asymptotic lines upon S 1 and $ 2 correspond. A necessary and sufficient condition that the asymptotic lines on S l and S 2 correspond is that there exist a functional relation between p^ and p 2 . 76. Surfaces complementary to a given surface. We have just seen that the normals to a surface are tangent to a family of geo desies on each surface of centers. Now we prove the converse : The tangents to a family of geodesies on a surface S l are normal to an infinity of parallel surfaces. Let the geodesies and their orthogonal trajectories be taken for the curves v const, and u = const, respectively, and the param eters chosen so that the linear element has the form We refer the surface to the trihedral formed by the tangents to the parametric curves and the normal, the z-axis being tangent to the curve v = const. Upon the latter we lay off from the point M l of the surface a length X, and let P denote the other extremity. As M 1 moves over the surface the projections of the corresponding displacements of P have the values (99) d\ + du, V X + X ~l dv, - X (y,du + q,dv). In order that the locus of P be normal to the lines J^P, we must have d\ + du = 0, and consequently X = u + , where c denotes the constant of integration whose value determines a particular one of the family of parallel surfaces. If the direction- cosines of M^P with reference to fixed axes be X v Y v Z^ the coordinates of the surface /S, for which c = 0, are given by where x^ y^ z l are the coordinates of M r * Comptes Rendus, Vol. LXXIV (1872), p. 1399. COMPLEMENTAKY SUKFACES 185 The surface S 1 is one of the surfaces of center of S. In order to find the other, $ 2 , we must determine X so that the locus of P is tangent at P to the zz-plane of the moving trihedral. The con dition for this is Hence S 2 is given by ^L V ^i y<i y\ i *!> ^2 ^i / aV^ gVg. tfM dw dM and the principal radii of S are expressed by ( 10 ) Pl = u, Pz =u- du Bianchi* calls S 2 the surface complementary to S l for the given geodesic system. Beltrami has suggested the following geometrical proof of the above theorem. Of the involutes of the geodesies v const, we consider the single infinity which meet S^ in one of the orthogonal trajectories u = U Q . We shall prove that the locus of these curves is a surface S, normal to the tangents to the geodesies. Consider the tangents to the geodesies at the points of meeting of the latter with a second orthogonal trajectory u = u r The segments of these tangents between the points of contact M and the points P of meeting with S are equal to one another, because they are equal to the length of the geodesies between the curves u U Q and u = u r Hence, as M moves along an orthogonal trajectory u = u l of the lines JfP, P describes a second orthogonal trajectory of the latter. Moreover, as M moves along a geodesic, P describes an involute which is necessarily orthogonal to MP. Since two directions on S are perpendicular to JfP, the latter is normal to S. EXAMPLES 1. Obtain the results of 73 concerning parallel surfaces without making use of the moving trihedral. 2. Show that the surfaces parallel to a surface of revolution are surfaces of revolution. *Vol. I, p. 293. 186 FUNDAMENTAL EQUATIONS 3. Determine the conjugate systems upon a surface such that the corresponding curves on a parallel surface form a conjugate system. 4. Determine the character of a surface S such that its asymptotic lines corre spond to conjugate lines upon a parallel surface, and find the latter surface. 5. Show that when the parametric curves are the lines of curvature of a surface, the characteristics of the 7/z-plane and zz-plane respectively of the moving trihe dral whose x-axis is tangent to the curve v = const, at the point are given by (r du + ri dv) y q (z pi) du = 0, (r du + r\ dv) x pi(z p%) dv = ; and show that these equations give the directions on the surfaces Si and S 2 which are conjugate to the direction determined by dv/du. 6. Show that for a canal surface ( 29) one surface of centers is the curve of centers of the spheres and the other is the polar developable of this curve. 7. The surfaces of center of a helicoid are helicoids of the same axis and parameter as the given surface. GENERAL EXAMPLES 1. If t is an integrating factor of ^Edu-\ -- - dv, and t the conjugate v^ imaginary function, then A 2 log V# is equal to the total curvature of the quadratic form E du 2 + 2 Fdudv + Gdv 2 , all the functions in the latter being real. 2. Show that the sphere is the only real surface such that its first and second fundamental quadratic forms can be the second and first forms respectively of another surface. 3. Show that there exists a surface referred to its lines of curvature with the linear element ds 2 = e au (du* + du 2 ), where a is a constant, and that the surface is developable. 4. When a minimal surface is referred to its minimal lines hence the lines of curvature and asymptotic lines can be found by quadratures. 5. Establish the following identities in which the differential parameters are formed with respect to the linear element : . 6. Prove that (cf. Ex. 2, p. 1G6) A 2 * = - 4k I- + -}- -(- + -}- x( l + -V VJE^PI f* -VGCV\PI f) \PI PZ/ GENEEAL EXAMPLES 187 7. Show that z 2 + ?/ 2 + z 2 = "FT 2 + Ai TF, the differential parameter being formed with respect to (23). 8. A necessary and sufficient condition that all the curves of an orthogonal system on a surface be geodesies is that the surface be developable. 9. If the geodesic curvature of the curves of an orthogonal system is constant (different from zero) all over the surface, the latter is a surface of constant negative curvature. 10. When the linear element of a surface is in the form ds 2 = du 2 + 2 cos u dudv + dto 2 , the parametric curves are said to form an equidistantial system. Show that in this case the coordinates of the surface are integrals of the system du dv du dv du dv dy dz dz dy cz dx dz dx ex dy _ dx dy_ cu dv du dv cu dv dv cu cu dv dv cu 11. If the curves v = const., u = const, form an equidistantial system, the tan gents to the curves v = const, are orthogonal to the lines joining the centers of geo desic curvature of the curves u = const, and of their orthogonal trajectories. 12. Of all the displacements of a trihedral T corresponding to a small displace ment of its vertex M over the surface there are two which reduce to rotations ; they occur when M describes either of the lines of curvature through the point, and the axes of rotation are situated in the planes perpendicular to the lines of curvature, each axis passing through one of the centers of principal curvature. 13. When a surface is referred to its lines of curvature, the curves defined by a 2 irl d M 3 + 3 g 2 du z dv + 3p? dudv 2 + P? dv s = du dv du dv possess the property that the normal sections in these directions at a point are straight lines, or are superosculated by their circles of curvature (cf . Ex. 9, p. 21 ; Ex. 6, p. 177). These curves are called the superosculating lines of the surface. 14. Show that the superosculating lines on a surface and on a parallel surface correspond. 15. Show that the Gauss equation (64) can be put in the following form due to Liouville : p gu du\ p gv ) du dv where p gu and p gv denote the radii of geodesic curvature of the curves v = const, and u = const, respectively. 16. When the parametric curves form an orthogonal system, the equation of Ex. 15 may be written _!\_J: ___ L p gv ) p% u P% V VE du \ 17. Determine the surfaces which are such that one of them and a parallel divide harmonically the segment between the centers of principal curvature. 188 FUNDAMENTAL EQUATIONS 18. Determine the surfaces which are such that one of them and a parallel admit of an equivalent representation (cf. Ex. 14, p. 113) with lines of curvature . corresponding. 19. Derive the following properties of the surface a 2 _ ft2 uv Va 2 - 6 2 v V& 2 w 2 _ Va 2 & 2 u Vu 2 a 2 _ ab u + u b M + U a w + v (i) the parametric lines are plane lines of curvature ; (ii) the principal radii of curvature are p\ = i>, p% = u ; (iii) the surface is algebraic of the fourth order ; (iv) the surfaces of center are focal conies. 20. Given a curve C upon a surface S and the ruled surface formed by the tan gents to S which are perpendicular to C at its points M ; the point of each generator M. N at which the tangent plane to the ruled surface is perpendicular to the tan gent plane at M to S is the center of geodesic curvature of C at M ; when the ruled surface is developable, this center of geodesic curvature is the point of contact of MN with the edge of regression. 21. If two surfaces have the same spherical representation of their lines of curvature, the locus of the point dividing the join of corresponding points in con stant ratio is a surface with the same representation. 22. The locus of the centers of geodesic curvature of a line of curvature is an evolute of the latter. 23. Show that when E, F, G ; D, IX, IX of a surface are functions of a single parameter, the surface is a helicoid, or a surface of revolution. CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. Asymptotic lines. We have said that the asymptotic lines on a surface are the double family of curves whose tangents at any point are determined in direction by the differential equation D du 2 + 2 D dudv + D"dv 2 = 0. These directions are imaginary and distinct at an elliptic point, real and distinct at a hyperbolic point, and real and coincident at a parabolic point. If we exclude the latter points from our discussion, the asymptotic lines may be taken for parametric curves. A neces sary and sufficient condition that they be parametric is (55) (1) D = D n =Q. Then from (IV, 25) we have _ _D^_ !_ where p as thus denned is called the radius of total curvature. The Codazzi equations (V, 13 ) may be written of which the condition of integrability is a ri2i d ri2i < 4 > ail i -hail 2 r In consequence of (V, 3) this is equivalent to In 64 we saw that K is a function of E, F, G and their deriva tives. Hence equations (3) are two conditions upon the coefficients of a quadratic form (6) E du 2 + 2 Fdudv + G dv 2 , 189 190 SYSTEMS OF CURVES that it may be the linear element of a surface referred to its asymp totic lines. When these conditions are satisfied the function D is given by (2) to within sign. Hence, if we make no distinction be tween a surface and its symmetric with respect to a point, from 65 follows the theorem : A necessary and sufficient condition that a quadratic form (6) be the linear element of a surface referred to its asymptotic lines is that its coefficients satisfy equations (3); when they are satisfied, the surface is unique. For example, suppose that the total curvature of the surface is the same at every point, thus j a 2 where a is a constant. In this case equations (3) are cv cu cv du which, since H 2 ^ 0, are equivalent to dv du Hence E is a function of u alone, and G a function of v alone. By a suitable choice of the parameters these two functions may be given the value a 2 , so that the linear element of the surface can be written (7) ds 2 = a 2 (du 2 + 2 cos o> dudv + dv 2 ), where w denotes the angle between the asymptotic lines. Thus far the Codazzi equa tions are satisfied and only the Gauss equation (V, 12) remains to be considered. When the above values are substituted, this becomes (8) sinw. dudv Hence to every solution of this equation there corresponds a surface of constant curvature whose linear element is given by (7). a 2 The equation of the lines of curvature is du 2 dv 2 = 0, so that if we put u -f v 2 M!, u v = 2 !, the quantities u\ and v\ are parameters of the lines of cur vature, and in terms of these the equation of the asymptotic lines is du} dv} = 0. Hence, when either the asymptotic lines or the lines of curvature are known upon a surface of constant curvature, the other system can be found by quadratures. When the asymptotic lines are parametric, the Gauss equations (V, 7) may be written ^ + ^ + 5^1 = 0, /OX i du du dv ( y ) dv 72 ai fru l dv~ ASYMPTOTIC LINES 191 where a, 5, a x , b 1 are determinate functions of u and v, and in consequence of (5) (10) da Jo du Conversely, if two such equations admit three real linearly inde pendent integrals f^u, v), f z (u, v), f 3 (u, v), the equations <* =/l(w V), # =/ 2 (M, V), ^ =/>(!*, 1 ) define a surface on which the parametric curves are the asymptotic lines. For, by the elimination of a, 6, a^ b l from the six equations obtained by replacing 6 in (9) by x, y, z we get d 2 x tfy d z z du* du* i? dx tji dz du du du dx dy_ dz dv dv dv . n ju uy v*> f\ 7 7 7 = " = 0, which are equivalent to (1), in consequence of (IV, 2, 5).* As an example, consider the equations dv 2 s dv 2 dx ty dz du du du dx dy dz dv dv dv of which the general integral is auv + bu + cv + d, where a, b, c, rf are constants. By choosing the fixed axes suitably, the most general form of the equations of the surface may be put in the form From these equations it is seen that all the asymptotic lines are straight lines, so that the surface is a quadric. Moreover, by the elimination of u and v from these equations we have an equation of the form z = ax- -\- 2hxy + by 2 + ex + dy. Hence the surface is a paraboloid. 78. Spherical representation of asymptotic lines. From (IV, 77) we have that the total curvature of a surface, referred to its asymp totic lines, may be expressed in the form (ii) A =-^ where ff-" = (o o^ 2 , the linear element of the spherical represen- tation being da 2 = (odu 2 + 2 &dudv + * Darbonx, Vol. I, p. 138. It should be noticed that the above result shows that the condition that equations (9) admit three independent integrals carries with it not only (10) but all other conditions of integrability. 192 SYSTEMS OF CURVES From this result and (2) it follows that * 5! # Hence the fundamental relations (IV, 74) reduce to /- o\ Jf n^/"" Jf _ r? "- and equations (V, 26) may be written " A ^ ^ X P / **dX e$X\ 3X__^__ _ /{ \ du cv Moreover, the Codazzi equations (V, 27) are reducible to Consider now the converse problem : To determine the condition to be satisfied by a parametric system of lines on the sphere in order that they may serve as the spherical representation of the asymptotic lines on a surface. First of all, equations (15) must satisfy the condition of integra- bility. Then p is obtainable by a quadrature. The corresponding values of x, y, z found from equations (14) and from similar ones are the coordinates of a surface upon which the asymptotic lines are parametric. For, it follows from (14) that du ** dv dv Furthermore, p is determined to within a constant factor ; conse quently the same is true of x,y,z\ therefore the surface is unique to within homothetic transformations. Hence we have the following theorem of Dini : A necessary and sufficient condition that a double family of curves upon the sphere be the spherical representation of the asymptotic lines upon a surface is that &, &, $ satisfy the equation V d ri2 the corresponding surfaces are homothetic transforms of one another, and their Cartesian coordinates are found by quadratures. * The choice P = D /ft gives the surface symmetric to the one corresponding to (12), as is seen from (14), and hence may be neglected. FORMULAS OF LELIEUVKE 193 When equations (1) obtain, the fundamental equations (V, 28) lead to the identities |in = r 11 v 2 ri2V |22 1 = r 22V_ r ri2i 121 ri2v ri2i ri2v (18) i rm l2/ = - rny r22j py \2J llJ- llJ The third and fourth of these equations are consequences also of (3) and (15). 79. Formulas of Lelieuvre. Tangential equations. In conse quence of (V, 31) equations (14) may be put in the form where e is 1 according as the curvature of the surface is positive or negative. Hence, if we put (20) ^ = V-/)X, v 2 = V^epY, v s = ^/ we have the following formulas due to Lelieuvre : * dx _ du du dx _ dv fa<2 fa* dv dv (21) dy _ ^w du du dv ~dv dv~ a^ _ du V l V 2 du du dz _ V l V * fai fa% dv dv The conditions of integrability of these equations arc du dv _ du dv _ dudv ~ v * v * Bulletin des Sciences Mathtmatiques, Vol. XII (1888), p. 126. 194 ^ SYSTEMS OF CURVES By means of (V, 22) and (15) we find from (20) that the common ratio of these equations i , ,. . , , ,. solutions of the equation ratio of these equations is = ^ -- <& Consequently i^, i/ 2 , v s are dudv \^/p dudv Conversely, we have the theorem : G-iven three particular integrals v^ i> 2 , v s of an equation of the form d 2 (22) -^- = X0, where \ is any function whatever of u and v ; the surface, whose co ordinates are given by the corresponding quadratures (21), has the parametric curves for asymptotic lines, and the total curvature of the surface is measured by /93\ K ~ For, from (21), it is readily seen that v^ i/ 2 , v z are proportional to the direction-cosines of the normal to the surface. And if these direction-cosines be given by (20), we are brought to (19), from which we see that the conditions (16) are satisfied. Take, for example, the simplest case - = 0, and three solutions j / \ i / / \ //; 1 9 Q\ V{ 0| (U) -(- Yi(V). (I = *j &, &) The coordinates of the surface are /r j and similar expressions for y and z. When, in particular, we take 0,- (u) = a,-w + &, $i (v) = a to + /S,-, the expressions for x, y, z are of the form auv + bu + cv + d, and consequently the surface is a paraboloid. From equations (V, 22, 34) it follows that when the asymptotic lines are parametric, the tangential coordinates X, Y, Z, W are solutions of the equations HVd0 18^-llJ du \ I ^2/j ^22") 30 CONJUGATE SYSTEMS 195 EXAMPLES 1. Upon a nondevelopable surface straight lines are the only plane asymptotic lines. 2. The asymptotic lines on a minimal surface form an orthogonal isothermal system, and their spherical images also form such a system. 3. Show that of all the surfaces with the linear element ds 2 = du* + (u 2 + a 2 ) du 2 , one has the parametric curves for asymptotic lines and another for lines of curva ture. Determine these two surfaces. 4. The normals to a ruled surface along a generator are parallel to a plane. Prove conversely, by means of the formulas of Lelieuvre, that if the normals to a surface along the asymptotic lines in one system are parallel to a plane, which differs with the curve, the surface is ruled. 5. If we take v^ = u, v z = D, j> 3 = 0(u), the formulas of Lelieuvre define the most general right conoid. 6. If the asymptotic lines in one system on a surface be represented on the sphere by great circles, the surface is ruled. 80. Conjugate systems of parametric lines. Inversions. It is our purpose now to consider the case where the parametric lines of a surface form a conjugate system. As thus defined, the character istics of the tangent plane, as it envelops the surface along a curve v = const., are the tangents to the curves u = const, at their points of intersection with the former curve ; and similarly for a plane enveloping along a curve u = const. The analytical condition that the parametric lines form a conju gate system is ( 54) (25) D =0. It follows immediately from equations (V, 7) that x, y, z are solu tions of an equation of the type (26) J^ + a^ + 6^0, cudv du dv where a and b are functions of u and v, or constants. By a method similar to that of 77 we prove the converse theorem : Iffi(u, v),/ 2 (M, v),/ 3 (w, v) be three linearly independent real solu tions of an equation of the type (26), the equations (27) * = />,*), y=f z (u,v), *=f t (u,v) define a surface upon which the parametric curves form a conjugate system.* 9 * Cf . Darboux, Vol. I, p. 122. 196 SYSTEMS OF CURVES We have seen that the lines of curvature form the only orthog onal conjugate system. Hence, in order that the parametric lines on the surface (27) be lines of curvature, we must have F^fa + tyty+tete^^ du dv du dv du dv But this is equivalent to the condition that x z +y z +z 2 also be a solution of equation (26), as is seen by substitution. Hence we have the theorem of Darboux * : If x, y, z, # 2 -{- ?/ 2 -f- z* are particular solutions of an equation of the form (26), the first three serve for the rectangular coordinates of a surface, upon which the parametric lines are the lines of curvature. Darboux f has applied this result to the proof of the following theorem : When a surface is transformed ly an inversion into a second sur face, the lines of curvature of the former become lines of curvature of the latter. By definition an inversion, or a transformation by reciprocal radii, is given by * where c denotes a constant. From these equations we find that (29) (if + f + z*) (x? + y + z, ) = c , and by solving for x, y, z, ( , f ~ " yt+*l *?+**-- , If, now, the substitution Q _ _ " ?+*+? be effected upon equation (26), the resulting equation in or will admit, in consequence of (29) and (30), the solutions x v y^ z v c 4 , and therefore is of the form (31) * Vol. I, p. 136. t Vol. I, p. 207. SURFACES OF TRANSLATION 197 Moreover, equation (26) admits unity for a particular solution, and consequently x* + yl + z? is a solution of (31), which proves the theorem. As an example, we consider a cone of revolution. Its lines of curvature are the elements of the cone and the circular sections. When a transformation by recip rocal radii, whose pole is any point, is applied to the cone, the transform S has two families of circles for its lines of curvature, in consequence of the above theorem and the fact that circles and straight lines, not through the pole, are transformed into circles. Moreover, the cone is the envelope of a family of spheres whose cen ters lie on its axis, and also of the one-parameter family of tangent planes ; the latter pass through the vertex. Since tangency is preserved in this transformation, the surface S is in two ways the envelope of a family of spheres : all the spheres of one family pass through a point, and the centers of the spheres of the other family lie in the plane determined by the axis of the cone and the pole. 81. Surfaces of translation. The simplest form of equation (26) is dudv in which case equations (27) are of the type (32) x = U 1 +r y = u^+V e = V>+r where U^ Z7 , U s are any functions whatever of u alone, and V^ F 2 , F 3 any functions of v alone. This surface may be generated by effecting upon the curve X l= U V Vl= U 21=^3 a translation in which each of its points describes a curve con gruent with the curve *, = F,, y,= r,, Z . 2 =F 3 . In like manner it may be generated by a translation of the second curve in which each of its points describes a curve congruent with the first curve. For this reason the surface is called a surface of translation. From this method of generation, as also from equa tions (32), it follows that the tangents to the curves of one family at their points of intersection with a curve of the second family are parallel to one another. Hence we have the theorem of Lie * : The developable enveloping a surface of translation along a gener ating curve is a cylinder. * Math. Annalen, Vol. XIV (1879), pp. 332-367. 198 SYSTEMS OF CURVES Lie has observed that the surface defined by (32) is the locus of the mid-points of the joins of points on the curves It may be that these two sets of equations define the same curve in terms of different parameters. In this case the surface is the locus of the mid-points of all chords of the curve. These results are only a particular case of the following theorem, whose proof is immediate : The locus of the point which divides in constant ratio the joins of points on two curves, .or all the chords of one curve, is a surface of translation ; in the latter case the curve is an asymptotic line of the surface. When the equations of a surface of translation are of the form x=U, y = V, 9=Ui+V the generators are plane curves whose planes are perpendicular. We leave it to the reader to show that in this case the asymptotic lines can be found by quadratures. 82. Isothermal-conjugate systems. When the asymptotic lines upon a surface are parametric, the second quadratic form may be written X dudv. When the surface is real, so also is this quadratic form. Therefore, according as the curvature of the surface is posi tive or negative, the parameters u and v are conjugate-imaginary or real. r We consider the former case and put when u^ and v l are real. In terms of these parameters the second quadratic form is \(du+dvj). Hence the curves M t = const., v l == const, form a conjugate system, for which (33) D = D", D =0. Bianchi * has called a system of this sort isothermal-conjugate. Evi dently such a system bears to the second quadratic form an ana lytical relation similar to that of an isothermal-orthogonal system * Vol. I, p. 107. ISOTHERMAL-CONJUGATE SYSTEMS 199 to the first quadratic form. In the latter case it was only necessary that EGF* be positive, and the analogous requirement, namely DD" I) 2 > 0, is satisfied by surfaces of positive curvature. Hence all the theorems for isothermal-orthogonal systems ( 40, 41) are translated into theorems concerning isothermal-conjugate systems by substituting Z>, IX, D" for E, F, G respectively in the formulas. In particular, we remark that if the curves u = const., v = const. on a surface form an isothermal-conjugate system, all other real isothermal-conjugate systems are given by u = const., v 1 = const., the quantities u l and v l being defined by u i+ i l \ ( t ) ( u i w) where < is any analytic function. When the curvature of the surface is negative and we put in the second quadratic form \dudv, it becomes \(du* dv*). In this case (34) D = -!>", D =0. Hence the curves w 1 const, and i\ = const, form a conjugate sys tem which may be called isothermal-conjugate. With each change of the parameters u and v of the asymptotic lines there is obtained a new isothermal-conjugate system. Hence if u and v are parame ters of an isothermal-conjugate system upon a surface of negative curvature, the parameters of all such systems are given by where < and ^r denote arbitrary functions. It is evident that if the parameters for a surface are such that (35) = -, Z/=0, D" V where U and V are functions of u and v respectively, then by a change of parameters which does not change the parametric curves we can reduce (35) to one of the forms (33) or (34). Hence equa tions (35) are a necessary and sufficient condition that the para metric curves form an isothermal-conjugate system. Referring to 200 SYSTEMS OF CUKVES 77, we see that the lines of curvature upon a surface of constant total curvature form an isothermal-conjugate system. When equation (35) is of the form (33) or (34), we say that the parameters u and v are isothermal-conjugate. 83. Spherical representation of conjugate systems. When the parametric curves are conjugate, equations (IV, 69) reduce to - GI? FDD" ., ED m ~-~W^ ~W -Ji- From these equations and (III, 15) it follows that the angle CD between the parametric curves on the sphere is given by , & F COS o> = - = qp - == if COS O), where the upper sign corresponds to the case of an elliptic point and the lower to a hyperbolic point. Hence we have the theorem: The angles between two conjugate directions at a point on a sur face, and between the corresponding directions on the sphere, are equal or supplementary, according as the point is hyperbolic or elliptic. When the parametric curves form a conjugate system, the Codazzi equations (V, 27) reduce to and equations (V, 26) become (dx D / X dX du dv Hence, when a system of curves upon the sphere is given, the problem of finding the surfaces with this representation of a conjugate system reduces to the solution of equations (36) and quadratures of the form (37), after X, Y, Z have been determined by the solution of a Riccati equation. By the elimination of D or D" from equations (36) we obtain a partial differential equation of the second order. CONJUGATE PARAMETRIC SYSTEMS 201 From the general equations (V, 28) we derive the following, when the parametric curves form a conjugate system : * (38) (fii\_8io S D rnv /22\ aiogD" /22V li/ = ~w ~\ir i2/ = ~fo~ ~i2/ /12\ D" fllV f!2\ D /22V ilJ = "T12/ l2/ = ~I7 ll/ 22 D" f!2 ll !2V 84. Tangential coordinates. Pro jective transformations. The prob lem of finding the surfaces with a given representation of a con jugate system is treated more readily from the point of view of tangential coordinates. For, from (V, 22) and (V, 34) it is seen that -3T, r, Z, and W are particular solutions of the equation <> Hence every solution of this equation linearly independent of A", F, Z determines a surface with the given representation of a conjugate system, and the calculation of the coordinates 2-, y, z does not involve quadratures ( 67). Conversely, it is readily seen that if the tangential coordinates satisfy an equation of the form d*e 30 00 h a -- h b -- f-c#=0. du dv the coordinate lines form a conjugate system on the surface. As an example, we determine the surfaces whose lines of curvature are repre sented on the sphere by a family of curves of ccinstant geodesic curvature and their orthogonal trajectories. If the former family be the curves v = const., and if the linear element on the sphere be written da- 2 Edu 2 -f Gdu 2 , we must have (IV, 60) where (u) is a function of v alone. By a change of the parameter v this may be made equal to unity. In this case equation (39) is reducible to du *Cf. Bianhi, Vol. I, p. 167. 202 SYSTEMS OF CUKVES The general integral of this equation is where v denotes a constant value of t>, and U and V are arbitrary functions of u and u respectively. Hence : The determination of all the surfaces whose lines of curvature are represented on the sphere by a family of curves of constant geodesic curvature and their orthogonal trajectories, requires two quadratures. In order that among all the surfaces with the same represen tation of a conjugate system there may be a surface for which the system is isothermal-conjugate, and the parameters be isothermal- conjugate, it is necessary that equations (36) be satisfied by iX ssiD, according as the total curvature is positive or negative. In this case equations (36) are 01og.D_/12\ fllV alocrT) 12V 22V u The condition of integrability is a rri2V my-i z 17121 /22V ] ^Llim2)rdl2mi)J When this is satisfied D may be found by quadratures, and then the coordinates, by (37). Hence we have the theorem: A necessary and sufficient condition that a family of curves upon the sphere represent an isothermal-conjugate system on a surface, and that u and v be isothermal-conjugate parameters, is that , <^, $ satisfy (40); then the surface is unique to within its homothetics, and its coordinates are given by quadratures. ,-< The following theorem concerning the in variance of conjugate directions and asymptotic lines is due to Darboux : When a surface is subjected to a protective transformation or a transformation by reciprocal polars, conjugate directions and asymp totic lines are preserved. We prove this theorem geometrically. Consider a curve C on a surface $and the developable D circumscribing the surface along C. When a projective transformation is effected upon S we obtain a surface S 19 corresponding point with point to S, and C goes into a curve CjUpon S^ and D in to a developable D l circumscribing S l along PROJECTIVE TRANSFORMATIONS 203 Cj ; moreover, the tangents to C and (7 X correspond, as do the gener ators of D and D r Since the generators are in each case tangent to the curves conjugate to C and C l respectively, the theorem is proved. In the case of a polar reciprocal transformation a plane corre sponds to a point and vice versa, in such a way that a plane and a point of it go into a point and a plane through it. Hence S goes into Sv C into D 1 , D into C v and the tangents to C and generators of D into the generators of J\ and tangents to <7 r Hence the theorem is proved. EXAMPLES 1. Show that the parametric curves on the surface - ^I + F I _ _ C7 AZ 2 - u * + Fs : U + V - u + V* "1TTF" where the I7 s are functions of u alone and the F s of v alone, form a conjugate system. 2. On the surface x U\V\, y = U Z V\, z F 2 , where U\, U 2 are functions of u alone and FI, F 2 of v alone, the parametric curves form a conjugate system and the asymptotic lines can be found by quadratures. 3. The generators of a surface of translation form an equidistantial system (cf. Ex. 10, p. 187). 4. Show that a paraboloid is a surface of translation in more than one way. 5 . The locus of the mid-points of the chords of a circular helix is a right helicoid. Q. Discuss the surface of translation which is the locus of points dividing in constant ratio the chords of a twisted cubic. 7. From (28) it follows that _ 2 , , a c 4 (dx 2 + dy 2 + dz 2 ) dx? + dy? + dz? = : (x* + y* + 2 2 ) 2 consequently the transformation by reciprocal radii is conformal. 8. Determine the condition to be satisfied by the function u so that a surface with the linear element 9/ <>,79, = a? (cos 2 w du 2 + shall have the total curvature I/a 2 . Show that if the parametric curves are the lines of curvature, they form an isothermal-conjugate system. 9. A necessary and sufficient condition that the linear element of a surface referred to a conjugate system can be written is that the parametric curves be the characteristic lines. Find the condition imposed upon the curves on the unit sphere in order that they may represent these lines. 10. Conjugate systems and asymptotic lines are transformed into curves of the same sort when a surface is transformed by the general protective transformation ABC X = D y = D * = D where A, 2>, C, D are linear functions of the new coordinates Xi, y\, z\. 204 GEODESICS 85. Equations of geodesic lines. We have defined a geodesic to be a curve whose geodesic curvature is zero at every point ; conse quently its osculating plane at any point is perpendicular to the tangent plane to the surface. From (IV, 49) it follows that every geodesic upon a surface is an integral curve of the differential equation (41) , ds ds/\ ds 2 ds*/ \ ds . /V + 7< T W -- Y Y-4- 4 i f \ ds ds)l\du 2 to/\ds) du ds ds 2 dv \ds \\- (\ 2 -i- fi -\( Y = ds/[2 du \ds) dv ds ds \dv 2 du)\ds ~ _/ \ ds If the fundamental identity +2+ \ds/ ds ds \ds/ which gives the relation between w, v, s along the curve, be differ entiated with respect to s, we have du d*u d*v\ dv d*u d*v L+ 2 4-24- -4--= dv du)\ds) ds \ dv du) ds \dsj dv If this equation and (41) be solved with respect to / 72 72 \ , T,tfu ~d*v\ ,, . and F- + G -^ ) we obtain \ ds 2 ds*/ F + F 4- - 1^ Y+ +( - - V V= ds 2 ds 2 2 du \ds) dv ds ds \dv 2 du)\ds/ cFv (W_lMy<faV dGdu dv IdG/dv \ 2 = Q dt \9* 2 dv )\ds) du ds ds 2 Bv \ds) If these equations be solved with respect to - and ^ we have, in consequence of (V, 2), (42) d*u riii/<fov 2 i 12 \ ^+{ 22 V Y=o 111 /du\* . rt fl21 (fw dv . f 22- EQUATIONS OF GEODESICS 205 Every pair of solutions of these equations of the form u =/j(), v =/ 2 (), determines a geodesic on the surface, and s is its arc. But a geodesic may be defined in terms of u and v alone, without the introduction of the parameter s. If v = <f>(u) defines such a curve, then dv du d * v /du\* .,d*u ct u Substituting these expressions in (42) and eliminating > we have, to within the factor (du/ds) 2 , (43) <// From (42) it follows that when du/ds is zero, Hence, when this condition is not satisfied, equation (43) defines the geodesies on a surface ; and when it is satisfied, equations (43) and u = const, define them. From the theory of differential equations it follows that there exists a unique integral of (43) which takes a given value for u M O , and whose first derivative takes a given value for u = U Q . Hence we have the fundamental theorem : , Through every point on a surface there passes a unique geodesic with a given direction. As an example, we consider the geodesies on a surface of revolution. We have found ( 46) that the linear element of such a surface referred to its meridians and parallels is of the form (45) ds 2 = (1 + 2 ) du 2 -f wW, where z = (u) is the equation of the meridian curve. If we put (46) and indicate the inverse of this equation by u ^(wi), we have (47) <Zs 2 = d^ + fdw 2 , and the meridians and parallels are still the parametric curves. For this case equa tions (42) are (48) ^i - w*y = - + - = - ( <W " W <W ds ds 206 GEODESICS The first integral of the second is .do <// 2 = c, ds where c is a constant. Eliminating ds from this equation and (47), and integrating, we have (49) c f - / , = w + 2 _ C 2 where Ci is a constant. The meridians v = const, correspond to the case c = 0. Hence we have the theorem : The geodesies upon a surface of revolution referred to its meridians and parallels can be found by quadratures. It should be remarked that equation (49) defines the geodesies upon any surface applicable to a surface of revolution. 86. Geodesic parallels. Geodesic parameters. From (43) it fol lows that a necessary and sufficient condition that the curves v = const, on a surface be geodesies is that If the parametric system be orthogonal, this condition makes it necessary that E be a function of u alone, say E = U 2 . By replacing I U du by u we do not change the parametric lines, and E becomes equal to unity. And the linear element has the form (51) d**=du*+Gdi?, where in general G is a function of both u and v. From this it follows that the length of the segment of a curve v = const, between the curves u = U Q and u = u^ is given by /! X! I ds u = I du = u^u^ V0 WO Since this length is independent of v, it follows that the segments of all the geodesies v = const, included between any two orthog onal trajectories are of equal length. In consequence of the funda mental theorem, we have that there is a unique family of geodesies which are the orthogonal trajectories of a given curve C. The above results enable us to state the following theorem of Gauss * : If geodesies be drawn orthogonal to a curve C, and equal lengths be measured upon them from C, the locus of their ends is an orthogonal trajectory of the geodesies. * L.C., p. 25. GEODESIC PARALLELS 207 This gives us a means of finding all the orthogonal trajectories of a family of geodesies, when one of them is known. And it sug gests the name geodesic parallels for these trajectories. Referring to 37, we see that these are the curves there called parallels, and so the theorem of 37 may be stated thus : A necessary and sufficient condition that the curves <f> = const, be geodesic parallels is that (52) A,* =/(*), where the differential parameter is formed with respect to the linear element of the surface, and f denotes any function. In order that <f> be the length of the geodesic curves measured from the curve </> = 0, it is necessary and sufficient that (53) A,* = l. Moreover, we have seen that when a function $ satisfies (52), a new function satisfying (53) can be found by quadrature. When this function is taken as u, the linear element has the form (51). In this case we shall call u and v geodesic parameters. 87. Geodesic polar coordinates. The following theorem, due to Gauss,* suggests an important system of geodesic parameters: If equal lengths be laid off from a point P on the geodesies through P, the locus of the end points is an orthogonal trajectory of the geodesies. In proving the theorem we take the geodesies for the curves v = const., and let u denote distances measured along these geo desies from P. The points of a curve u = const, are consequently at the same geodesic distance from P, and so we call them geodesic circles. It is our problem to show that this parametric system is orthogonal. From the choice of u we know that E = \, and hence from (50) it follows that F is independent of u. At P, that is for u = 0, the derivatives are zero. Consequently F and G are zero Zv dv dv for u = 0, and the former, being independent of w, is always zero. Hence the theorem is proved. We consider such a system and two points M Q (u, 0), J/^w, vj on the geodesic circle of radius u. The length of the arc M Q M 1 *L.c.,p. 24. 208 GEODESICS is given by / ^Gdv. As u approaches zero the ratio Jo approaches the angle between the tangents at P to the geodesies v = and v = v^ If 6 denotes this angle, we have a .. 6 = lim - = I dv. u =o v r = I /o In order that v be 6, it is necessary and sufficient that - = 1. L du Ju= These particular geodesic coordinates are similar to polar coordi nates in the plane, and for this reason are called geodesic polar coordinates. The above results may now be stated thus : The necessary and sufficient conditions that a system of geodesic coordinates be polar are (54 ) =0, -i. L J=o L Bu J M=0 It should be noticed, however, that it may be necessary to limit the part of the surface under consideration in order that there be a one-to-one correspondence between a point and a pair of coordinates. For, it may happen that two geodesies starting from P meet again, in which case the second point of meeting would be defined by two sets of coordinates.* For example, the helices are geodesies on a cylinder ( 12), and it is evident that any number of them can be made to pass through two points at a finite distance from one another by varying the angle under which they cut the elements of the cylinder. Hence, in using a system of geodesic polar coordinates with pole at P, we consider the portion of the surface inclosed by a geodesic circle of radius r, where r is such that no two geodesies through P meet within the circle, t When the linear element is in the form (51), the equation of Gauss (V, 12) reduces to (55) If K Q denotes the total curvature of the surface at the pole P, which by hypothesis is not a parabolic point, from (54) and (55) it follows that _ _ K o L <* ~ * * Notice that the pole is a singular point for such a system, because H* = for u = 0. tDarboux (Vol. II, p. 408) shows that such a function r exists; this is suggested also by 94. GEODESIC POLAK COOKDINATES 209 Therefore, for sufficiently small values of w, we have . O Hence the circumference and area of a geodesic circle of radius u have the values * /* 2 = I Jo 2 ITU - + where e t and e 2 denote terms of orders higher than the third and fourth respectively. EXAMPLES 1. Find the geodesies of an ellipsoid of revolution. 2. The equations x u, y = v define a representation of a surface with the linear element ds 2 = v (du 2 + dv 2 ) upon the xy-plane in such a way that geodesies on the former are represented by parabolas on the latter. 3. Find the total curvature of a surface with the linear element (a 2 - v 2 ) du? + 2 w> dudv + (a 2 - u 2 ) dv* US T XV* - -- - - 9 ( a 2 _ W 2 _ W 2)2 where R and a are constants and integrate the equation of geodesies for the surface. 4. A twisted curve is a geodesic on its rectifying developable. 5. The evolutes of a twisted curve are geodesies on its polar developable. 6. Along a geodesic on a surface of revolution the product of the radius of the parallel through a point and the sine of the angle of inclination of the geodesic with the meridian is constant. 7. Upon a surface of revolution a curve cannot be a geodesic and loxodromic at the same time unless the surface be cylindrical. 8. Upon a helicoid the orthogonal trajectories of the helices are geodesies and the other geodesies can be found by quadratures. 9. If a family of geodesies and their orthogonal trajectories on a surface form an isothermal system, the surface is applicable to a surface of revolution. 10. The radius of curvature of a geodesic on a cone of revolution at a point P varies as the cube of the distance of P from the vertex. 88. Area of a geodesic triangle. With the aid of geodesic polar coordinates Gauss proved the following important theorem f : The excess over 180 of the sum of the angles of a triangle formed by geodesies on a surface of positive curvature, or the deficit from 180 * Bertrand, Journal de Mathematiques, Ser. 1, Vol. XIII (1848), pp. 80-86. t L.c., p. 30. 210 GEODESICS of the sum of the angles of such a triangle on a surface of negative curvature, is measured by the area of the part of the sphere which represents that triangle. In the proof of this theorem Gauss made use of the equation of geodesic lines in the form where 6 denotes the angle which the tangent to a geodesic at a point makes with the curve v const, through the point. This equation is an immediate consequence of formula (V, 81). When the parametric system is polar geodesic, this becomes (57) M = - -*. Let ABC be a triangle whose sides are geodesies, and let a, /3, 7 denote the included angles. From (IV, 7 3) it follows that the inclosed area on the sphere is given by (58) d = f f // dudv = Ipcff dudv, where e is 1 according as the curvature is positive or negative, and the double integrals are taken over the respective areas. Let A be the pole of a polar geodesic system and AB the curve v = 0. From (55) and (58) we have rr ] Jo Jo , -- dvdu o o In consequence of (54) we have, upon integration with respect to u, which, by (57), is equivalent to & = e f dv + e f dd. Jo Jn-ft For, at B the geodesic BC makes the angle TT fi with the curve v = 0, and at C it makes the angle 7 with the curve v = a. Hence we have (7i = e(a + /3 + 7 - TT), which proves the theorem. AREA OF A GEODESIC TRIANGLE 211 Because of the form of the second part of (58) Ci may be said to measure the total curvature of the geodesic triangle, so that the above theorem may also be stated thus : The total curvature of a geodesic triangle is equal to the excess over 180, or deficit from 180, of the sum of the angles of the tri angle, according as the curvature is positive or negative. The extension of these theorems to the case of geodesic polygons is straightforward. In the preceding discussion it has been tacitly assumed that all the points of the triangle ABC can be uniquely denned by polar coordinates with pole at A. We shall show that this theorem is true, even if this assumption is not made. If the theorem is not true for ABC, it cannot be true for both of the triangles ABD and ACD obtained by joining A and the middle point of BC with a geodesic AD (fig. 18). For, by adding the results for the two triangles, we should have the theorem holding for ABC. Suppose that it is not true for A BD. Divide the latter into two triangles and apply the same reason ing. By continuing this process we should obtain a triangle as small as we please, inside of which a polar geodesic system would not uniquely determine each point. But a domain can be chosen about a point so that a unique geodesic passes through the given point and any other point of the domain.* Consequently the above theorem is perfectly general. By means of the above result we prove the theorem : Tivo geodesies on a surface of negative curvature cannot meet in two points and inclose a simply connected area. Suppose that two geodesies through a point A pass through a second point B, the two geodesies inclosing a simply connected portion of the surface (fig. 19). Take any geodesic cutting these two segments AB in points C and D. Since the four angles ACD, ADC, BCD, BDC are together equal to four right angles, the sum /D of the angles of the two triangles ADC arid BDC exceed four right angles by the sum of the angles at A and B. Therefore, in consequence of the above theorem of Gauss, the total curvature of the surface cannot be negative at all points of the area ADBC. On the contrary, it can be shown that for a surface of positive curvature geodesies through a point meet again in general. In * Darboux, Vol. II, p. 408 ; cf. 94. 212 GEODESICS fact, the exceptional points, if there are any, lie in a finite portion of the surface, which may consist of one or more simply connected parts.* For example, the geodesies on a sphere are great circles, and all of these through a point pass through the diametrically opposite point. Again, the helices are geodesies on a cylinder ( 12), and it is evident tnat any number of them can be made to pass through two points at a finite distance from one another by varying the angle under A hich they cut the elements of the cyl inder. Hence the domain oi a system of polar geodesic coordinates is restricted on a surface of oositive curvature. 89. Lines of shortest length. Geodesic curvature. We are now in a position to prove the theorem : If two points on a surface are such that only one geodesic passes through them, the segment of the geodesic measures the shortest dis tance on the surface between the two points. Take one of the points for the pole of a polar geodesic system and the geodesic for the curve v = 0. The coordinates of the second point are (u^ 0). The parametric equation of any other curve through the two points is of the form v = ^>(w), and the length of its arc is f Jo Since G > 0, the value of this in tegral is necessarily greater than Wj, and the theprem is proved. By means of equation (57) we derive another definition of geo desic curvature. Consider two points M and M upon a curve C, and the unique geodesies g, g tangent to C at these points (fig. 20). Let P denote the point of intersection of g and g\ and Sty the angle under which they cut. Liouville f has called Sty the angle of geodesic contingence, because of its analogy to the ordinary angle of contingence. Now we shall prove the theorem: The limit of the ratio Sty/Ss, as M approaches M, is the geodesic curvature of C at M. * For a proof of this the reader is referred to a memoir by H. v. Mangoldt, in Crelle, Vol. XCI (1881), pp. 23-53. t Journal de Mathtmatiques, Vol. XVI (1851), p. 132. GEODESIC ELLIPSES AKD HYPERBOLAS In the proof of this theorem we take for parametric curves the given curve (7, its geodesic parallels and their geodesic orthogonals, the parameter u being the distance measured along the latter from C. Since the geodesic g meets the curve v = v orthogonally, the angle under which it meets v = v may be denoted by ?r/2 4- SO. As M approaches Jf, W approaches dd given by y/>7), and the sum of the angles of the triangle M PQ approaches 18C. Hence S^/r approaches dQ, so that we have Ss ds v- which is the expression for the geodesic curvature of the curve C. 90. Geodesic ellipses and hyperbolas. An important system of parametric lines for a surface is formed by two families of geodesic parallels. Such a system may be obtained by constructing the geo desic parallels of two curves C^ and (7 2 , which are not themselves geodesic parallels of one another, or by taking the two families of geodesic circles with centers at any two points F^ and F 2 . Let u and v measure the geodesic distances from C^ and C 2 , or from F l and F 2 . They must be solutions of (53). Consequently, in terms of them, we must have ^ Q. EG-F*~~ EG-F*~ If, as usual, o> denotes the angle between these parametric lines, we have, from (III, 15, 16), v n * T? COSft) -U = U = > JF = . ? sin 2 ft) sin <o so that the linear element has the following form, due to Weingarten : /rrix , 9 du 2 + 2 cos ft) dudv -f- dv 2 (59) ds*=- . a - sin 2 &) Conversely, when the linear element is reducible to this form, u and v are solutions of (53), and consequently the parametric curves are geodesic parallels. In terms of the parameters u v and v^ denned by u = i^-h v l and v = u l i\, the linear element (59) has the form ( 60) df^^aL + JuL. . o o sm S 2 214 GEODESICS The geometrical significance of the curves of parameter i^ and v l is seen when the above equations are written The curves w 1 = const, and v l = const, are respectively the loci of points the sum and difference of whose geodesic distances from C 1 and Cg, or from F t and F z , are constant. In the latter case these curves are analogous to ellipses and hyperbolas in the plane, the points F l and F 2 corresponding to the foci. For this reason they are called geodesic ellipses and hyperbolas, which names are given likewise to the curves u^ = const., v l = const., when the distances are measured from two curves, C l and C 2 . From (60) follows at once the theorem of Weingarten : * A system of geodesic ellipses and hyperbolas is orthogonal. By means of (61) equation (60) can be transformed into (59), thus proving that when the linear element of a surface is in the form (60), the parametric curves are geodesic ellipses and hyperbolas. If 6 denotes the angle which the tangent to the curve v^= const, through a point makes with the curve v = const., it follows from (III, 23) that ft) ... , cos u = cos i sin = sin Hence we have the theorem : Given any two systems of geodesic parallels upon a surface ; the corresponding geodesic ellipses and hyperbolas bisect the angles included by the former. 91. Surfaces of Liouville. Dini f inquired whether there were any surfaces with an isothermal system of geodesic ellipses and hyperbolas. A necessary and sufficient condition that such a sur face exist is that the coefficients of (60) satisfy a condition of the form (41) ^ 8in2 = r/i CQ8 , | , where U^ and V r denote functions of u l and i\ respectively. In this case the linear element may be written V i *\ *Ueber die Oberfliichen fur welche einer der beiden Hauptkrummungshalbmesser eine Function des anderen ist, Crelle, Vol. LXII (1863), pp. 160-173. t Annali, Ser. 2, Vol. Ill (1869), pp. 269-293. SURFACES OF LIOUVILLE 215 By the change of parameters defined by 1 this linear element is transformed into (63) ds* = ( U 2 + r 8 ) (du* + dvt), where U 2 and V z are functions of u 2 and v 2 respectively, such that Conversely, if the linear element is in the form (63), it may be changed into (62) by the transformation of coordinates Surfaces whose linear element is reducible to the form (63) were first studied by Liouville, and on that account are called surfaces of Liouville.* To this class belong the surfaces of revolution and the quadrics ( 96, 97). We may state the above results in the form : When the linear element of a surface is in the Liouville form, the parametric curves are geodesic ellipses and hyperbolas ; these systems are the only isothermal orthogonal families of geodesic conies.^ 92. Integration of the equation of geodesic lines. Having thus discussed the various properties of geodesic lines, and having seen the advantage of knowing their equations in finite form, we return to the consideration of their differential equation and derive certain theorems concerning its integration. Suppose, in the first place, that we know a particular first inte gral of the general equation, that is, a family of geodesies defined by an equation of the form (64) From (IV, 58) it follows that M and N must satisfy the equation _2_ / ___ FN-GM \ d_ I FM-EN \ = du \^/EN 2 - 2 FMN + GM 2 / ^ \^EN 2 - 2 FMN + GM 2 / * Journal de Mathematiques, Vol. XI (1846), p. 345. t The reader is referred to Darboux, Vol. II, p. 208, for a discussion of the conditions under which a surface is of the Liouville type. 216 GEODESICS In consequence of this equation we know that there exists a func tion <f> denned by ._. dc#> EN-FM d<f> FN-GM (DO ) = ====================== ? ==: -=============================== du ^EN*-1FMN+GM* to V EN*-2FMN+GM* Moreover, we find that (66) A^=l. From (III, 31) and (65) it follows that the curves </> = const, are the orthogonal trajectories of the given geodesies, and from (66) it is seen that < measures distance along the geodesies from the curve $ = 0. Hence we have the theorem of Darboux * : When a one-parameter family of geodesies is defined by a differ ential equation of the first order, the finite equation of their orthogonal trajectories can be obtained by a quadrature, which gives the geodesic parameter at the same time. Therefore, when the general first integral of the equation of geodesies is known, all the geodesic parallels can be found by quadratures. We consider now the converse problem of finding the geodesies when the geodesic parallels are known. Suppose that we have a solution of equation (66) involving an arbitrary constant a, which is not additive. If this equation be differentiated with respect to a, we get (67) where the differential parameter is formed with respect to the linear element. But this is a necessary and sufficient condition ( 37) that the curves </> = const, and the curves (68) ^ = const.= a da form an orthogonal system. Hence the curves defined by (68) are geodesies. In general, this equation involves two arbitrary constants, a and a , which, as will now be shown, enter in such a way that this equation gives the general integral of the differ ential equation of geodesic lines. * Lemons, Vol. II, p. 430; cf. also Bianchi, Vol. I, p. 202. EQUATIONS OF GEODESIC LINES 217 Suppose that a appears in equation (68), and write the latter thus : (69) f (u, v, a) = a r , in which case equation (67) becomes (70) A l (*,^)=0. The direction of each of the curves (69) is given by -^- / If this 36 /dd> ratio be independent of a, so also by (70) is the ratio -^-1 Write the latter in the form If this equation and (66) be solved for and we obtain values cu dv independent of a, so that a would have been additive. Hence / involves a, and so also does iJL/l_, an d therefore a direction at cu / dv a point (M O , v ) determines the value of a; call it . If then a Q be such that ^. (, *,) = 4, the geodesic -v/r (%, v, a ) = ^ passes through the point (w , ?; ) and has the given direction at the point. Hence all the geodesies are defined by equation (68), and we have the theorem: Criven a solution of the equation A 1 <^ = 1, involving an arbitrary r\ II constant a, in such a way that involves a; the equation da da for all values of a is the finite equation of the geodesies, and the arc of the geodesies is measured by (/>.* By means of this result we establish the following theorem due to Jacobi : If a first integral of the differential equation of geodesic lines be known, the finite equation can be found by one quadrature. Such an integral is of the form dv - = ^(u, v, a), du Cf. Darboux, Vol. II, p. 429. 218 GEODESICS where a is an arbitrary constant. As this equation is of the form (64), the function c/>, defined by (# + = P is a solution of equation (66). As $ involves a in the manner specified in the preceding theorem, the finite equation of the d(f> , geodesies is = a. 93. Geodesies on surfaces of Liouville. The surfaces of Liouville ( 91) afford an excellent application of the theorem of Jacobi. We take the linear element in the form * (71) ds 2 = (U- V) (U?du 2 + V?dv 2 ), which evidently is no more general than (63). In this case equa tion (66) becomes When this equation is written in the form u*\du. one sees that it belongs to the class of partial differential equa tions admitting an integral which is the sum of functions of u and v alone, f In order to obtain this integral, we put each side equal to a constant a and integrate. This gives (72) </> = C l\ -\/Ua du f F! Va Vdv. Hence the equation of geodesies is (73) If 6 denotes the angle which a geodesic through a point makes with the line v = const, through the point, it follows from (III, 24) and (71) that y dv tan 6 = - * Cf. Darboux, Vol. Ill, p. 9. t Forsyth, Differential Equations (1888), p. 310. SURFACES OF LIOUVILLE 219 If the value of dv/du from equation (73) be substituted in this equation, we obtain the following first integral of the Gauss equation (56): (74) ?7sin 2 + Fcos 2 = a. This equation is due to Liouville. * EXAMPLES 1. On a surface of constant curvature the area of a geodesic triangle is pro portional to the difference between the sum of the angles of the triangle and two right angles. 2. Show that for a developable surface the first integral of equation (56) can be found by quadratures. 3. Given any curve C upon a surface and the developable surface which is the envelope of the tangent planes to the surface along C; show that the geodesic curvature of C is equal to the curvature of the plane curve into which C is trans formed when the developable is developed upon a plane. 4. When the plane is referred to a system of confocal ellipses and hyperbolas whose foci are at the distance 2 c apart, the linear element can be written 5. A necessary and sufficient condition that be a solution of Ai0 = 1 is that ds 2 d<p 2 be a perfect square. 6. If = did + 62 , where 6\ and 62 are functions of u and v, is a solution of Ai0 =: 1, the curves 0i = const, are lines of length zero, and the curves B\a -j- 6 2 = const, are their orthogonal trajectories. 7. When the linear element of a spiral surface is in the form ds 2 = e 2 " (du 2 -\- U" 2 do 2 ), the equation Ai0 = 1 admits the solution e ? Z7i, where U\ is a function of M, which satisfies an equation of the first order whose integration gives thus all the geodesies on the surface. 8. For a surface with the linear element where V and V\ are functions of v alone, the equation Ai0 = 1 admits the solution (f> u\fsi (v) -f ^2 ( v ), the determination of the functions \f>i and ^ 2 requiring the solu tion of a differential equation of the first order and quadratures. 9. If denotes a solution of Ai0 = 1 involving a nonadditive constant a, the linear element of the surface can be written ca where (0, $} indicates the mixed differential parameter (III, 48). *i.c.,p. 348. 220 GEODESICS 94. Lines of shortest length. Envelope of geodesies. We can go a step farther than the first theorem of 89 and show that whether one or more geodesies pass through two points M l and M 2 on a sur face, the shortest distance on the surface between these points, if it exists, is measured along one of these geodesies. Thus, let v =f(u) and v =f l (u) define two curves C and C l passing through the points M^ M# the parametric values of u at the points being u^ and u 2 . The arc of C between these points has the length (75) = 2Fv +Gv 2 du, where v denotes the derivative of v with respect to u. For con venience we write the above thus : (76) s= f *4>(ui v, v )du. Jiti Furthermore, we put f 1 (u)=f(u) + ea>(u), where w(u) is a function of u vanishing when u is equal to u l and M 2 , and e is a constant whose absolute value may be taken so small that the curve C l will lie in any prescribed neighborhood of C. Hence the length of the arc M 1 M 2 of C l is = fc/tt (u, v -f- e tw, v -f- e CD ) C?M. Thus j is a function of e, reducing for e = to s. Hence, in order that the curve C be the shortest of all the neai?-by curves which pass through M l and J/ 2 , it is necessary that the derivative of s l with respect to e be zero for e = 0. This gives On the assumption that admits a continuous first derivative in the interval (u^ u z ), and <f> continuous first and second deriva tives, the left-hand member of this equation may be integrated by parts with the result " 1 /& d d<l>\ , n wl-2- ^lauaaO: \v du v LINES OF SHORTEST LENGTH 221 for o> vanishes when u equals u^ and u 2 . As the function &> is arbi trary except for the above conditions upon it, this equation is equivalent to the following equation of Euler * : (77) du When this result is applied to the particular form of (f> in equa- tkm (75), we have d F+ Gv __ I / _ 71 I _ ?? dv cv dv _ ~ " which is readily reducible to equation (43). Hence the shortest distance between two points, if existent, is measured along a geodesic through the points. This geodesic is unique if the surface has negative total curvature at all points. For other surfaces more than one geo desic may pass through the points if the latter are sufficiently far apart. We shall now investigate the nature of this problem. Let v f(u, a) define the family of geo desies through a point J/ (w , v ), and let v = g (u) be the equation of their envel ope (o. We consider two of the geodesies C l and C 2 (fig. 21), and let MI(UV vj and M,,(u , v z ) denote their points of contact with the envelope. Suppose that the arc M M 2 is greater than Jf Jf r The distance from M Q to ^ measured along C l and <~is equal to D = f JttQ f J^ If 3/ 2 is considered fixed and M l variable, the position of the latter is determined by a. The variation of D with M 1 is given by j da J M=MI * Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, chap, ii, 21 (Lausanne, 1744) ; cf. Bolza, Lectures on the Calculus of Variations, p. 22 (Chicago, 1994). 222 GEODESICS B u t f or u = u., f = g and f = g \ consequently the last term is zero. f)f Integrating the first member hy parts, and noting that -f- is zero for u = u and u = u x ( 26), we have Since C^ is a geodesic, the expression in parenthesis is zero, and hence D does not vary with M r This shows that the envelope of the geodesies through a point bears to them the relation which the evolute of a curve does to a family of normals to the curve. Moreover, the curve ("is not a geodesic, for at each point of it there is tangent a geodesic. Hence there is an arc connecting M^ and M 2 which is shorter than the arc of &. In this way, by taking different points M l on & we obtain any number of arcs connecting M Q and M z which are shorter than the arc of C 2 , each consisting of an arc of a geodesic such as C l and the geodesic distance M V M^ It is then necessarily true that the shortest distance from M Q to a point M of C 2 beyond M 2 is not measured along <7 2 . However, when M lies within the arc M Q M 2 , a domain can be chosen about (7 2 so small that the arc M Q M of <7 2 is shorter than the arc M Q M of any other curve within the domain and passing through these points.* Another historical problem associated with this problem is the following : t Given an arc C joining two points A, B on a surface ; to find the curve of shortest length joining A and B, and inclosing with Co a given area. The area is given by CClfdudv. It is evident that two functions M and N can be found in an infinity of ways such that r < __ 8N dM ~ du dv By the application of Green s theorem we have lldudv =- // where the last integral is curvilinear and is taken around the contour of the area. Since C Q is fixed, our problem reduces to the determination of a curve C along which the integral C *Mdu + Ndv is constant, and whose arc AB, that is, the / A * For a more complete discussion of this problem the reader is referred to Darboux, Vol. Ill, pp. 86-112; Bolza, chap. v. tin fact, it was in the solution of this problem that Minding (Crelle, Vol. V (18.30), p. 297) discovered the function to which Bonnet (Journal de I Ecole Poly technique, Vol. XIX (1848), p. 44) gave the name geodesic curvature.. ENVELOPE OF GEODESICS 223 integral C V2? -f 2 Fv + Gv ^du, is a minimum. From the calculus of variations we know that, so far as the differential equations of the solution is concerned, this is the same problem as finding the curve C along which the integral f B VE + 2 Fv + Gv *du + c(M + Nv )du JA is a minimum, c being a constant. Euler s equation for this integral is + > ^ + ^.!? d / F 4- GV \ cto cu dy_ _ -f 6rv 2 / V E + 2 .FV -f Crt/ 2 Comparing this result with the formula of Bonnet (IV, 56), we see that C has con stant geodesic curvature 1/c, and c evidently depends upon the magnitude of the area between the curves. Hence we have the theorem of Minding :* In order that a curve C joining two points shall be the shortest which, together with a given curve through these points, incloses a portion of the surface with a given area, it is necessary that the geodesic curvature of C be constant. GENERAL EXAMPLES 1. When the parametric curves on the unit sphere satisfy the condition 12 ) a I 12 ) ( 12 1 J dv - 2 12 ) j 12 ) i n 2 r they represent the asymptotic lines on a surface whose total curvature is 2. When the equations of the sphere have the form (III, 35), the parametric curves are asymptotic and the equation (22) is (1 + wu) 2 = - 20, of which the CU vV general integral is ^ 2 ^(K) + ^(.)_ 1 + uv where (u) and \f/ (v) denote arbitrary functions. 3. The sections of a surface by all the planes through a fixed line L in space, and the curves of contact of the tangent cones to the surface whose vertices are on L, form a conjugate system. 4. Given a surface of translation x = u, y = v, z =f(u) + 0(0). Determine the functions/ and so that ( Pl + P2 )Z = const., where Z denotes the cosine of the angle which the normal makes with the z-axis, and determine the lines of curva ture on the surface. 5. Determine the relations between the exponents m< and n t - in the equations x = U mi V ni , y = U m *V"* t z = U m3 V Hs , so that on the surface so defined the parametric curves shall form a conjugate sys tem, and show that the asymptotic lines can be found by quadratures. *Z.c., p. 207. 224 GEODESICS 6. The envelope of the family of planes (Ui + Fi)z + (U z + V 2 )y + (U s + F 8 )z + (U* + F 4 ) = 0, where the U" s are functions of u alone and the F s of , is a surface upon which the parametric curves are plane, and form a conjugate system. 7. The condition that the parametric curves form a conjugate system on the envelope of the plane x cos u + y sin u -f z cot v =/(u, u), is that / be the sum of a function of u alone and of v alone ; in this case these curves are plane lines of curvature. 8. Find the geodesies on the surface of Ex. 7, p. 219, and determine the expres sions for the radii of curvature and torsion of a geodesic. 9. A representation of two surfaces upon one another is said to be conformal- conjugate when it is at the same time conformal, and every conjugate system on one surface corresponds to a conjugate system on the other. Show that the lines of curvature correspond and that the characteristic lines also correspond. 10. Given a surface of revolution z = ucosu, y = wsinw, zf(u), and the function defined by (i) where A and c are constants ; a conf orjual-conjugate representation of the surface upon a second surface x\ = MI cosi, y\ MI sin I?!, z\ = <f>(ui) is defined by - du V CUi, C log Ui = where F denotes the function of M found by solving (i) for < . 11. If two families of geodesies cut under constant angle, the surface is developable. 12. If a surface with the linear element ds* = (aM 2 - bv 2 - c) (du? + cto 2 ), where a, 6, c are constants, is represented on the xy-plane by u = x, v = y, the geodesies correspond to the Lissajous figures defined by where A, -Z?, C are constants. 13. When there is upon a surface more than one family of geodesies which, together with their orthogonal trajectories, form an isothermal system, the curva ture of the surface is constant. 14. If the principal normals of a curve meet a fixed straight line, the curve is a geodesic on a surface of revolution whose axis is this line. Examine the case where the principal normals meet the line under constant angle. GENERAL EXAMPLES 225 15. A representation of two surfaces upon one another is said to be a geodesic representation when to a geodesic on one surface there corresponds a geodesic on the other. Show that the representation is geodesic when points with the same parametric values correspond on surfaces with the linear elements where the IPs are functions of u alone, the F s of v alone, and h is a constant. 16. A surface with the linear element ds 2 = (w* - v 4 ) [0 /- where is any function whatever, admits of a geodesic representation upon itself. 17. A necessary and sufficient condition that an orthogonal system upon a sur face may be regarded as geodesic ellipses and hyperbolas in two ways, is that when the curves are parametric the linear element be of the Liouville form ; in this case these curves may be so regarded in an infinity of ways. 18. Of all the curves of equal length joining two points, the one which, together with a fixed curve through the points, incloses the area of greatest extent, has con stant geodesic curvature. 19. Let T be any curve upon a surface, and at two near-by points P, P draw the geodesies g, g perpendicular to T; let C be the curve through P conjugate to gr, P" the point where it meets g , and Q the intersection of the tangents to g and g at P and P" ; the limiting position of Q, as P f approaches P, is the center of geodesic curvature of T at P. 20. Show that if a surface S admits of geodesic representation upon a plane in such a way that four families of geodesies are represented by four families of par allel lines, each geodesic on the surface is represented by a straight line (cf . Ex. 3, p. 209). CHAPTER VII QUADRICS. RULED SURFACES. MINIMAL SURFACES 95. Confocal quadrics. Elliptic coordinates. Two quadrics are confocal when the foci, real or imaginary, of their principal sec tions coincide. Hence a family of confocal quadrics is defined by the equation a) -A+/-+-A-1, a 2 u b 2 u c 2 u where u is the parameter of the family and a, 6, c are constants, such that (2) a 2 > b 2 > c 2 . For each value of u, positive or negative, less than a 2 , equation (1) defines a quadric which is Ian ellipsoid when c 2 > u > oo, an hyperboloid of one sheet when b 2 > u > c*, an hyperboloid of two sheets when a 2 > u > b 2 . As u approaches c 2 the smallest axis of the ellipsoid approaches zero. Hence the surface u = c 2 is the portion of the zy-plane, counted twice, bounded by the ellipse (4) 2 a 2 c 2 b 2 c 2 Again, the surface u = b 2 is the portion of the ^-plane, counted twice, bounded by the hyperbola which contains the center of the curve. Equations (4) and (5) define the focal ellipse and focal hyperbola of the system. CONFOCAL QUADEICS 227 Through each point (x, y, z) in space there pass three quadries of the family; they are determined by the values of w, which are roots of the equation (6) < (u) = (a 2 - u) (b 2 - u) (c 2 - u) - x 2 (b 2 - u) (c 2 - u) - y 2 (a 2 -u) (c 2 - u) - z 2 (a 2 - u) (b 2 - u) = 0. Since </> (a 2 ) < 0, c/> (b 2 ) > 0, < (c 2 ) < 0, </> (- oo) > 0, the roots of equation (6), denoted by u l9 u , w 3 , are contained in the following intervals : (7) a 2 > u, > b 2 , b 2 >u,> c 2 , c 2 > u^> oo. From (3) it is seen that the surfaces corresponding to u v w 2 , u s are respectively hyperboloids of two and one sheets and an ellipsoid. Fig. 22 represents three confocal quadrics; the curves on the ellipsoid are lines of cur vature, and on the hyper- boloid of one sheet they are asymptotic lines. From the definition of u v w 2 , u s it follows that < (u) is equal to (u^u) (u 2 u) (u s u). When (/> in (6) is replaced by this expression and u is given successively the values a 2 , b 2 , c 2 , we obtain * FIG. 22 (8) or = = (*- )(>-?) These formulas express the Cartesian coordinates of a point in space in terms of the parameters of the three quadrics which pass through the point. These parameters are called the elliptic coordinates of the point. It is evident that to each set of these * Kirchhoff , Mechanik, p. 203. Leipsic, 1877. 228 QUABEICS coordinates there correspond eight points in space, one in each of the eight compartments bounded by the coordinate planes. If one of the parameters - t . in (8) be made constant, and the others u^ %, where i = j = Ar, be allowed to vary, these equations define in parametric form the surface, also defined by equation (1), in which u has this constant value u r The parametric curves 1^.= const., u k const, are the curves of intersection of the given quadric and the double system of quadrics corresponding to the parameters Uj and u k , If we put ,r\\ o 12 7* 2 . ng .. the equation of the surface becomes (10) - + ^ + -=1, a b c and the parametric equations (8) reduce to (11) la(a u)(a v) " N (a - b) (a - c) fb (b u)(b v) y ~~~ \ (b - a) (b - c) \c(c u) (c v) Z ~~ N (c a}(c b} Moreover, the quadrics which cut (10) in the parametric curves have the equations: (12) a u b u c u av bv cv = 1, = 1. In consequence of (3) and (9) we have that equations (10) or (11) define an ellipsoid when a>u>b>v>c>0, (13) - an hyperboloid of one sheet when a>u>b>Q>c>v, an hyperboloid of two sheets when a>0>b>u>c>v. FUNDAMENTAL QUANTITIES 229 96. Fundamental quantities for central quadrics. By direct cal culation we find from (11) *. U ( U -V) _A ^_V(V-U) (14) /() where for the sake of brevity we have put (15) f(0) = 4 (a - 6) (b - 0) (c - 6). We derive also the following : (16) and (17) (a-b)(a- = \ ^ a)(c b) lobe u v JL> \ abc u v uv f(u) N uv f(v) Since F and D are zero, the parametric curves are lines of curva ture. And since the change of parameters (9) did not change the parametric curves, we have the theorem : The quadrics of a confocal system cut one another along lines of curvature, and the three surfaces through a point cut one another orthogonally at the point. This result is illustrated by fig. 22. From (14) and (17) we have ., 1 lobe 1 _ \abc 1 _ abc Pl ~ NtfV p 2 ~ N^ 3 />^ 2 ~wV Hence the ellipsoid and hyperboloid of two sheets have positive curvature at all points, whereas the curvature is negative at all points of the hyperboloid of one sheet. If formulas (16) be written \abc x uv a abc z uv c > uv a \ uv b the distance W from the center to the tangent plane is (19) 230 QUADEICS Hence : The tangent planes to a central quadric along a curve, at points of which the total curvature of the surface is the same, are equally distant from the center. From (18) we see that the umbilical points correspond to the values of the parameters such that u = v. The conditions (13) show that this common value of u and v for an ellipsoid is b, and c for an hyperboloid of two sheets, whereas there are no real umbilical points for the hyperboloid of one sheet. When these values are substituted in (11), we have as the coordinates of these points on the ellipsoid \c(b-c) and on the hyperboloid of two sheets It should be noticed that these points lie on the focal hyperbola and focal ellipse respectively. 97. Fundamental quantities for the paraboloids. The equation of a paraboloid (22) 2z = ax 2 +by* may be replaced by (23) a:=V^, y=V^, z = -(au l +bv l ). Hence the paraboloids are surfaces of translation ( 81) whose generating curves are parabolas which lie in perpendicular planes. By direct calculation we find D = 0, // =-- ^VS^ + ftX + l 4 ^ so that the equation of the lines of curvature is a dv, b dv. . b FUNDAMENTAL QUANTITIES The general integral of this equation is (24) ^ 231 where c is an arbitrary constant. When u l and v 1 in (24) are given particular values, equation (24) determines two values of c, c l and 2 , in general distinct. If these latter values be substituted in (24) successively, we obtain in finite form the equations of the two lines of curvature through the point / \ Tf ,1 "U 1/11 A + au \ J A + aV (., tu. 11 c l and c z be replaced by ( - - ) and ( \ on I \ ov spectively, we have, in consequence of (23), the two equations re- (25) buy 2 + (1 + au) x 2 = u (1 + an} ab -f (1 -f- av) x 2 = v (1 + av) b-a ab When these equations are solved for x 2 and y 1 , we find that equa tion (22) can be replaced by a b b (26) 1 b ~ a /i , / = 2"^- (1+aW + ^ and the parametric curves are the lines of curvature. Now we have (27) a b a(a b)u E r (u v) * , F = 0, 2 b a G = ~7T5~ ( U ~ 4 6 2 u(I+au) a(a b)v au}(\ + av), Vab and (29) V [a (a b) u b] [a (a b) v b] (a-b)(u-v) 1 1 U 3 [ a (a-b)u-b][a(a-b)v-b] (a b)(u v) a(a b)u b][a(a b)v- b] v(l+av) 232 QLJADRICS From (27), (28), and (29) we obtain (30) W = ^Xx = - [a (a - b)u- b]*[a(a - b)v- and (31) From these results we find that the ratio W/z is constant along the curves for which the total curvature is constant. We suppose that b is positive and greater than a. From the first of (26) it follows that u and v at a real point differ in sign, or one is equal to zero. We consider the points at which both u and v are equal to zero. There are two such points, and their coordinates are (32) ,-0, = [ a (a b)u b]~ [ a (a-b)v- Evidently these points are real only on the elliptic paraboloid. From (31) it follows that p l and p z are then equal, and conse quently these are the umbilical points. Since at points other than these u and v must differ in sign, we may assume that u is always positive and v negative. Moreover, from (26) it is seen that u and v are unrestricted except in the case of the elliptic paraboloid, when v must be greater than I/a. 98. Lines of curvature and asymptotic lines on quadrics. From (14), (27), and 91 we have the theorem : The lines of curvature of a quadric surface form an isothermal system of the Liouville type. Bonnet * has shown that this property is characteristic of the quadrics. There are, however, many surfaces whose lines of curva ture form an isothermal system. They are called isothermic sur faces. The complete determination of all such surfaces has never been accomplished (cf. Ex. 3, 65). * Meraoire sur la theorie des surfaces applicables sur une surface donne e, Journal de V Ecole Poly technique, Vol. XXV (1867), pp. 121-132. ASYMPTOTIC LINES ON QUADRICS 233 From (17), (29), and 82 follows the theorem: The lines of curvature of a quadric surface form an isothermal- conjugate system, and consequently the asymptotic lines can be found by quadratures. We shall find the expressions for the coordinates in terms of the latter in another way. Equation (10) is equivalent to the pair of equations \vS V<y \ Vo/ \V# V< or the pair (34) where u and v are undetermined. For each value of u equations (33) define a line all of whose points lie on the surface. And to each point on the surface there corresponds a value of u determin ing a line through the point. Hence the surface is ruled, and it is nondevelopable, as seen from (18). Again, for each value of v equations (34) define a line whose points lie on the surface (10), and these lines are different from those of the other system. Hence the central quadrics are doubly ruled. These lines are necessarily the asymptotic lines. Consequently, if equations (33), (34) be solved for z, y, z, thus : x u + v y uv 1 z . v u V^i^r+i vP^TT vP ^TT we have the surface defined in terms of parameters referring to the asymptotic lines. In like manner equation (22) may be replaced by ^fax -f i^/by = 2 uz, ^/ax i ^Jby = - , or u V ax + i^Tby - > V ax i Vfo/ == 2 vz. v Solving these, we have I * / \ "1 \ * C% tJ c\ """" 2uv 234 QUADKICS As in the preceding case, we see that the surface is doubly ruled,* and the parameters in (36) refer to the asymptotic system of straight lines. Hence : The asymptotic lines on any quadric are straight lines. EXAMPLES 1. The focal conies of a family of confocal quadrics meet the latter in the umbilical points. 2. Find the characteristic lines on the quadrics of positive curvature. 3. The normal section of an ellipsoid at a point in the direction of the curve along which the total curvature is constant is an ellipse with one of its vertices at the point. 4. Find the equation of the form = Md (cf . 79) when the corresponding cu dv surface is a hyperboloid of one sheet ; when a hyperbolic paraboloid. 5. Find the evolute of the hyperboloid of one sheet and derive the following properties : (a) the surface is algebraic of the twelfth order ; (b) the section by a principal plane of the hyperboloid consists of a conic and the evolute of a conic ; (c) these sections are edges on the surface ; (d) the curve of intersection of the two sheets of the surface is cut by each of the principal planes in four ordinary points, four double points, and four cusps, and consequently is of the twenty-fourth order. 6. Determine for the evolute of a hyperbolic paraboloid the properties analogous to those for the surface of Ex. 5. 7. Deduce the equations of the surfaces parallel to a central quadric ; determine their order and the character of the sections of the surface by the principal planes of the quadric ; find the normal curvature of the curves corresponding to the asymp totic lines on the quadric. 99. Geodesies on quadrics. Since the quadrics are isothermic surfaces of the Liouville type, the finite equation of the geodesies can be found by quadratures ( 93). From (VI, 74), (14) and (27), * Moreover, the quadrics are the only doubly ruled surfaces. For consider such a sur face, and denote by a, b, c three of the generators in one system. A plane a through a meets 6 and c in unique points B and C , and the line B(J meets a in a point A. The line ABC is a generator of the second system, and the only one of this system in the plane a. The other lines of this system meet a in the line a. On this account the plane a cuts the surface in two lines, a and ABC, that is, in a degenerate conic. Hence the surface is of the second degree. GEODESICS ON QUADEICS 235 it follows that the first integral of the differential equation of geodesies on any one of the quadrics is (37) u sin 2 + v cos 2 = a, where a is a constant of integration and 6 measures the angle which a geodesic, determined by a value of a, makes with the lines of curvature v = const. We recall that in equations (11) and (26) the parameter u is greater than v, except at the umbilical points, where they are equal. We shall discuss the general case first. Consider a particular point M (u\ v ). According as a is given the value u 1 or v f , equation (37) defines the geodesic tangent at M 1 to the line of curvature u = u or v v respectively. It is readily seen that the other values of a, determining other geo desies through M.\ lie in the interval between u and v . More over, to each value of a in this domain there correspond two geodesies through M r whose tangents are symmetrically placed with respect to the directions of the lines of curvature. From this result it follows also that the whole system of geodesies is defined by (37), when a is given the limiting values of u and v and all the intermediate values. We write equation (37) in the form (38) (u a) sin 2 + (va) cos 2 = 0, and consider the geodesies on a central quadric defined by this equation when a has a particular value a . Suppose, first, that a is in the domain of the values of u. Then at each point of these geodesies v< a and consequently from (38) u > a . We have seen that these geodesies are tangent to the line of curvature u = a - From (11) it follows that they lie within the zone of the surface bounded by the two branches of the curve u = a . When, now, a is in the domain of the values of v, u a 1 is positive, and con sequently from (38) v < a . Hence the geodesies tangent to the curve v a lie outside the zone bounded by the two branches of the line of curvature v a f . Similar results are true for the parabo loids, with the difference, as seen from (26), that the geodesies tangent to u a lie outside the region bounded by this curve, whereas the curves tangent to v = a lie inside the region bounded by v = a . 236 QUADKICS 100. Geodesies through the umbilical points. There remains for consideration the case where a takes the unique value which u and v have at the umbilical points. Let it be denoted by , so that the curves defined by (39) (u a Q ) sin 2 + (v a ) cos 2 = are the umbilical geodesies. We have, at once, the theorem : Through each point on a quadric with real umbilical points there pass two umbilical geodesies which are equally inclined to the lines of curvature through the point. Hence two diametrically opposite umbilical points of an ellipsoid are joined by an infinity of geodesies, and no two geodesies through the same umbilical point meet again except at the diametrically opposite point. These properties are possessed also by a family of great circles on a sphere through two opposite points. On the elliptic paraboloid and on each sheet of the hyperboloid of two sheets there are two families of umbilical geodesies, but no two of the same family meet except at the umbilical point common to all curves of the family. For the ellipsoid (11) = b and equations (VI, 72, 73) become a^_ _1 C \ u ~~ du 1 C \ v _dv ~db~ ~4J ^\(a u)(uc)u b 4 J N(a v)(v c) v Similar results hold for the hyperboloid of tw sheets and the elliptic paraboloid. Hence the distances of a point P from two umbilical points (not diametrically opposite) are of the form Hence we have : The lines of curvature on the quadrics with real umbilical points are geodesic ellipses and hyperbolas with the umbilical points for foci. 101. Ellipsoid referred to a polar geodesic system. A family of umbilical geodesies and their orthogonal trajectories constitute an excellent system for polar geodesic coordinates, because the domain is unrestricted (87) except in the case of the ellipsoid, UMBILICAL GEODESICS 237 and then only the diametrically opposite point must be excluded. We consider such a system on the ellipsoid, and let denote the pole of the system and , O", " the other umbilical points (fig. 23). If we put (40) i r I , i r r v <h = - I A du - l\ 2J \(a-u)(u-c) 2J N(a *)( _ 1 r I u du _ 1 C I v ~2 J \(a u)(u c) u b~2jv( a v)(v it is readily found that 00 1 dv C)V ( ( j_. )(&_) By means of (11) we may reduce the linear element to the form (41) ds 2 = dp + -= In order that the coordinates be polar geodesic, ^r must be replaced by another parameter measuring the angles between the geodesies. For the ellipsoid equation (39) is (42) (u-b)s FIG. 23 As previously seen, 6 is half of one of the angles between the two geodesies through a point M. As M approaches along the geodesic joining these two points, the geodesic O MO" ap proaches the section # = 0. Consequently the angle 2 approaches the angle MOO 1 , denoted by &>, or its supplementary angle. Hence we have from (42) (43) ib-V\ im = b,r=b \U b/ lim u=b, ? = We take &> in place of t/r and indicate the relation between them by ^fr =/(o>). From (41) we have 238 QUADRICS This expression satisfies the first of conditions (VI, 54). The second is (44) lim 1 -7=^ [<* - *) - ( - > 1 - 1 -j 2 7^-6)^-1.) [ ^ *<H If we make use of the formulas (III, 11) and (40), we find du n \(a u)(u c)u b dv \(a v)(v c) b v - = % -yj- -- i - = t \\ -- j d<p M ^ u vd(f) M v w v so th.it equation (44) reduces to lim m V(^M^) r |(c-o(u- g ) + i(a-^(.-^i =1 u=,r = b U-V \_\ U N V J By means of (43) we pass from this to Hence the linear element has the following form due to Roberts * : siir&> The second of equations (40) may now be put in the form 1 fjl HI 1 fjl HI dv 2J \l( ar . u )(p c) u b 2J \(a v)(v- b &) ~ log tan - -f (7, I (a - b) (b - c) \vhere C denotes the constant of integration. In order to evaluate this constant, we consider the geodesic through the point (0, ft, 0). At this point the parameters have the values u = a, v = <?, and the angle co has a definite value o>. Hence the above equation may be replaced by i r r ~^~ du _i r r ~^~ ^ (-*)<*-) , 2 . * Journal de Mathematiques, Vol. XIII (1848), pp. 1-11. PROPERTIES OF QUADRICS 239 In like manner, for the umbilical geodesies through one of the other points (not diametrically opposite) we have i r I u du [ i r \ v dv 2Ja \(a u)(u c) u b 2j c \(a v)(v c) v (a-b)(b-c) It follows at once from these formulas that if M is any point on a line of curvature u const, or v const., we have respectively tan - -- tan - = const., tan - cot - = const. 102. Properties of quadrics. From (18) it follows that for the central quadrics Euler s equation (IV, 34) takes the form By means of (19) and (37) this reduces to (47) I? R abc In like manner, we have for the paraboloids (48) I = _J! [& + (&_)]. Hence we have : Along a geodesic or line of curvature on a central quadric the product RW* is constant, and on a paraboloid the ratio fiW 3 /z s . Consider any point P on a central quadric and a direction through P. Let a, ft, 7 be the direction-cosines of the latter. The semi-diameter of the ellipsoid (10) parallel to this direction is given by (49) = a - + . By definition 240 QUADRICS and similarly for /3 and 7. When the values of #, /, 2, E, G from (11) and (14) are substituted, equation (49) reduces to 1 = cos 2 fl sin 2 fl p* u v By means of (19) and (37) this may be reduced to (50) ap 2 W 2 =abc. From this follows the theorem of Joachimsthal : Along a geodesic or a line of curvature on a central quadric the product of the semi-diameter of the quadric parallel to the tangent to the curve at a point P and the distance from the center to the tangent plane at P is constant. From (47) and (50) we obtain the equation for all points on the quadric. Since W is the same for all direc tions at a point, the maximum and minimum values of p and R correspond. Hence we have the theorem : In the central section of a quadric parallel to the tangent plane at a point P the principal axes are parallel to the directions of the lines of curvature at P.* EXAMPLES 1. On a hyperbolic paraboloid, of which the principal parabolas are equal, the locus of a point, the sum or difference of whose distances frotn the generators through the vertex of the paraboloid is constant, is a line of curvature. 2. Find the radii of curvature and torsion, at the extremity of the mean diam eter of an ellipsoid, of an umbilical geodesic through the pokit. 3. Find the surfaces normal to the tangents to a family of umbilical geodesies on an ellipsoid, and determine the complementary surface (cf. 76). 4. The geodesic distance of two diametrically opposite umbilical points on an ellipsoid is equal to one half the length of the principal section through the umbilical points. 5. Find the form of the linear element of the hyperboloid of two sheets or the elliptic paraboloid, when the parametric system is polar geodesic with an umbilical point for pole. 6. If MI and M 2 are two points of intersection of a geodesic through the umbilical point with a line of curvature v = const. , then tan - - cot - = const. * For a more complete discussion of the geodesies on quadrics, the reader is referred to a memoir by v. Braunmuhl, in Math. Annalen, Vol. XX (1882), pp. 556-686. EQUATIONS OF A RULED SURFACE 241 7. Given a line of curvature on an ellipsoid and the geodesies tangent to it; the points of intersection of pairs of these geodesies, meeting orthogonally, lie on a sphere. 8. Given the geodesies tangent to two lines of curvature ; the points of inter section of pairs of these geodesies, meeting orthogonally, lie on a sphere. 103. Equations of a ruled surface. A surface which can be gen erated by the motion of a straight line is called a ruled surface. Developables are ruled surfaces for which the lines, called the generators, are tangent to a curve. As a general thing, ruled sur faces do not possess this property, and in this case they are called skew surfaces. Now we make a direct study of ruled surfaces, par ticularly those of the skew type, limiting our discussion to the case where the generators are real.* A ruled surface is completely determined by a curve upon it and the direction of the generators at their points of meeting with the curve. We call the latter the directrix __ Z>, and the cone formed by drawing through a point lines parallel to the generators the FIG 24 director-cone. If the coordinates of a point M Q of D are # , y^ 2 , expressed in terms of the arc v measured from a point of it, and Z, m, n are the direction-cosines of the gen erator through Jf , the equations of the surface are (51) x = x lu, y = y Q +mu, z = z + nu, where u is the distance from M Q to a point M on the generator through M Q . If denotes the angle which the generator through M Q makes with the tangent at Jf to Z>, then (52) cos = x J, + y r m + z Q n, where the accent indicates differentiation with respect to v (fig. 24). From (51) we find for the linear element the expression (53) ds* =du 2 +2 cos dudv + (aV + 2 bu + 1) dv*, where we have put for the sake of brevity * We shall use the term ruled to specify the surfaces of the skew type, and developable for the others. 242 RULED SURFACES Since the generators are geodesies, their orthogonal trajectories can be found by quadratures ( 92). We arrive at this result directly by remarking that the equation of these trajectories is (HI, 26) du + cos 6 dv = 0, and that is a function of v alone. 104. Line of striction. Developable surfaces. We shall now con sider the quantities which determine the relative positions of the generators of a ruled surface. Let g and g be two generators determined by parametric values v and v + Sv, and let X, /*, v denote the direction-cosines of their common perpendicular. If the direction-cosines of g and g be denoted by I, m, n ; I + SZ, m + 8w, n + Bn respectively, we have ( l\ + nifJL + nv = 0, | (I + 81) \ + (m + &m) A* + (w + Sw) v = 0, and consequently (56) \:fji:v = (m$n n$m) : (n&l l&n) : (ISm mSl). From (54) it follows that (mn 1 - nm )* + (nl - ln )*+(lm - ml * 2 arid by Taylor s theorem, (57) l+ = l + rftr + gW+--. Hence equations (56) may be replaced by (58) where e t , e 2 , e 3 denote expressions of the first and higher orders in Sv. If Mfa y, z) and Jf (z+8s, y + % 2 + ^2) are the points of meeting of this common perpendicular with # and g respectively (fig. 24), the length MM , denoted by A, is given by or (60) A = \&x + /x% -f LINE OF STRICTION 243 From (51), after the manner of (57), we obtain Sjc = (x -+ ul ) Bv -f IBu -f- <7, where cr involves the second and higher powers of Bv. When this and similar values for By and Bz are substituted in (60), we have (61) ^=p + , where (62) I m .n I m 1 n and e involves first and higher powers of Bv. In consequence of (52) and (54) we have (63) /= ^. As Bv approaches zero, the point M approaches a limiting posi tion C, which is called the central point of the generator. Let a Denote the value of u for this point. In order to find its value we remark that it follows from the equations (55) and (59) that Sx Bl By Bm Bz Bn _ ~ Bv Bv Sv Bv Bv 8v If the above expressions for these quantities be substituted in this equation, we have in the limit, as Bv approaches zero, (64) a*u + b = 0. Consequently (65) The locus of the central points is called the line of striction. Its parametric equation is (64). Evidently b is a necessary and sufficient condition that the line of striction be the directrix. From (61) and (63) it is seen that the distance between near-by generators is of the second order when (66) a 2 sm 2 6> -6 2 =:0. Without loss of generality we may take the line of striction for directrix,, in which case we may have sin# =:0, that is, the 244 KULED SURFACES generators are tangent to the directrix. Another possibility is afforded by a 0. From (54) it is seen that the only real sur faces satisfying this condition are cylinders. Hence (cf. 4) : A necessary and sufficient condition that a ruled surface, other than a cylinder, be developable is that the distance between near-by genera tors be of the second or higher orders ; in this case the edge of regres sion is the line of striction. 105. Central plane. Parameter of distribution. The tangent plane to a ruled surface at a point M necessarily contains the generator through M. It has been found ( 25) that for a devel opable surface this plane is tangent at all points of the generator. We shall see that in the case of skew surfaces the tangent plane varies as M moves along the generator. We deter mine the character of this variation by finding the angle which the tangent plane at M makes with the tangent plane at the central point C of the gen erator through M. The tangent plane at C is called the central plane. Let g and g l be two generators, and MM their common per pendicular (fig. 25). Through the point M of g draw the plane normal to g ; it meets g^ in M v and the line through M parallel to ff l in M 2 . The limiting positions of the planes M^MM and M MM, as g^ approaches g, are the tangent planes at M and at C, the limiting position of M. The angle between these planes, de noted by ( , is equal to MMJtt^ and the angle between g and g v denoted by cr, is equal to MMM 2 . By construction MM 2 M l and MMM 2 are right angles. Hence ., MM. MM ton a- tan d> = - - = MM In the limit M is the central point (7, and so we have v ,, (u a)da- , (u a)a m tan<f>=lim tan<f> = - ^ = for we have pdv p da* = lim (SI 2 + 8m 2 -f &i 2 ) = a W. PARAMETER OF DISTRIBUTION 245 It is customary to write the above equation in the form (67) The function ft thus defined is called the parameter of distribution. It is the limit of the ratio of the shortest distance between two generators and their included angle. As it is independent of the parameter u, we have the theorem : The tangent of the angle between the tangent plane to a ruled surface at a point M and the central plane is proportional to the distance of M from the central point. From this it follows that as M moves along a generator from oo to + oo, (/> varies from Tr/2 to 7r/2. Hence the tangent planes at the infinitely distant points are perpendicular to the central plane. Since /3=0 is the condition that a surface be developable, the tangent plane is the same at all points of the generator. We shall now derive equation (67) analytically. From (51) we find that the direction-cosines of the normal to the surface are of the form (mz ny Q ) + (mn m n) u ( ) X=- 2 ^ ; the expressions for Y and Z are similar to the above. The direction-cosines JT , F , Z Q of the normal at the central point are obtained from these by replacing u by a. From this we have /\Q\ r\o x/\ ^T ~V~V _ 2 (mz[ ny ^f + 2 (mz[ ny Q } (mn m n} (u + a) -f a 2 ua (aV + 2 bu + sin 2 )* (a V + 2 ba + sin 2 )* which leads to 4/^2 ^2J _. a (u a) From this equation and (67) we have (70) ._ a -_ ( / ^~ 2 " 2 I m n When the surface is defined by its linear element, @ is thus deter mined only to within an algebraic sign. We shall find, however, that this is not the case when the surface is defined by equations of the form (51). 246 EJJLED SURFACES To this end we take a particular generator g for the z-axis. Then for g we have Let also the central plane be taken for the zz-plane and the central point for the origin. From (68) it follows that y Q = 0. Since the origin is the central point, b = and consequently I = 0. Hence the equation of the tangent plane at a point of g has the simple form (71) m u% X O T) = 0, f and rj being current coordinates. If the coordinate axes have the usual orientation, and the angle * is measured positively in the direction from the positive #-axis to the positive ^/-axis, from equation (71) we have m u (7 2) tan * = Comparing this with equation (67), we find for ft the value x Jm . In order to obtain the same value from (70) for these particular values, we must take the negative sign. Hence we have, in general, \ / / (73) = -i I m n 2 I m 1 n It is seen from (72) that, as a point moves along a generator in the direction of u increasing, the motion of the tangent plane is that of a right-handed or left-handed screw, according as ft is negative or positive. EXAMPLES 1 . Show that for the ruled surface denned by 2 J 2 _ i r ,- . . ^ . -, . * y ~2> = Cu<t>du where and \f/ are any functions of w, the directrix and the generators are minimal. Determine under what condition the curvature of the surface is constant. 2. Determine the condition that the directrix of a ruled surface be a geodesic. PARAMETER OF DISTRIBUTION 247 3. Prove, by means of (62), that the lines of curvature of a surface F(x, y, z) = are defined by ^ dy, dz dF d_F cF dx dy dz >*, a**, *?* dx dy dz 4. The right helicoid is the only ruled surface whose generators are the principal normals of their orthogonal trajectories. Find the parameter of distribution. 5 . Prove for the hyperboloid of revolution of one sheet that : (a) the minimum circle is the line of striction and a geodesic ; (6) the parameter of distribution is constant. 6. With every point P on a ruled surface there is associated another point P on the same generator, such that the tangent planes at these points are perpendicular. Prove that the product OP OP , where denotes the central point, has the same value for all points P on the same generator. 7. The normals to a ruled surface along a generator form a hyperbolic paraboloid. 8. The cross-ratio of four tangent planes to a ruled surface at points of a gen erator is equal to the cross-ratio of the points. 9. If two ruled surfaces are symmetric with respect to a plane, the values of the parameter of distribution for homologous generators differ only in sign. 106. Particular form of the linear element. A number of prop erties of ruled surfaces are readily obtained when the linear element is given a particular form, which we will now deduce. Let an orthogonal trajectory of the generators be taken for the directrix. In this case (74) *,=, f,-*. If we make the change of parameters, C v (75) u = u, v l = I a dv, Jo the linear element (53) is reducible to (76) ds* = du* + [(u - a) 2 + /3 2 ] dv*. The angle 6 which a curve v l =f(u) makes with the generators is given by (77) tan0 = V(w Also the expression for the total curvature is (78) JT = -- f* 248 KULED SURFACES Hence a real ruled surface has no elliptic points. All the points are hyperbolic except along the generators for which /3 = 0, and at the infinitely distant points on each generator. Consequently the linear element of a developable surface may be put in the form (79) ds* = du" + (u - a) 2 dv*. Also, in the region of the infinitely distant points of a ruled sur face the latter has the character of a developable surface. As another consequence of (78) we have that, for the points of a generator the curvature is greatest in absolute value at the cen tral point, and that at points equally distant from the latter it has the same value. When the linear element is in the form (76), the Gauss equation of geodesies (VI, 56) has the form V(M - a) 2 + @*d6 + (u-a) dv 1 = 0. An immediate consequence is the theorem of Bonnet : If a curve upon a ruled surface has two of the following properties, it has the third also, namely that it cut the generators under constant angle, that it be a geodesic and that it be the line of striction. A surface of this kind is formed by the family of straight lines which cut a twisted curve under constant angle and are perpen dicular to its principal normals. A particular case is the surface formed of the binomials of a curve. It is readily shown from (73) that the parameter of distribution of this surface is equal to the radius of torsion of the curve. ,- 107. Asymptotic lines. Orthogonal parametric systems. The gen erators are necessarily asymptotic lines on a ruled surface. We consider now the other family of these lines. From (51) and (68) we find r 1 (80) i> = 0, D =- I m n H " z"+n"u n z + n u m Hence the differential equation of the other family of asymp totic lines is of the form dv ASYMPTOTIC LINES 249 where , Jf, N are functions of v. As this is an equation of the Riccati t}^pe, we have, from 14, the theorem of Serret: The four points in which each generator of a ruled surface is cut by four curved asymptotic lines are in constant cross-ratio. From 14 it follows also that when one of these asymptotic lines is known the others can be found by quadratures. When the surface is referred to an orthogonal system and the linear element is in the form (76), written (81) ds 2 = du 2 + a 2 [(u - a) 2 + /3 2 ] dv 2 , the expressions (80) can be given a simpler form. From (73) and (81) we have From the equations Lx ol = 0, Sz6 2 = 1, 2Z 2 = 1, and (54) we obtain, by differentiation, Zzfceo = 0, ZM = 0, Zatf J = -b, -Ll l" = aa , ZK" = - a 2 , ZJ x6 = & -, where t is defined by zr 6 = t. If the expression for D" in (80) be multiplied by the determinant of the right- hand member of (73), and the result be divided by its equal, a 2 /3, we have, in consequence of the above identities, D" = - i [w 2 (to? - aa fc) + u (2 tb - aa - 66 ) + t - & ]. 2 If equations (74) be solved for a and 6 as functions of a and 0, and the resulting expressions be substituted in this equation, we have D" = -~{r[(u - a) 2 + ^] + ?(u - a) + /3a }, where the primes indicate differentiation with respect to Vi, given by (75), and r is defined by / From the above equations it follows that the mean curvature (cf . 52) is express ible in the form (82) J - + * = - r ^ u ~ a)2 + ^ + Pi Pz [(u - a-) 2 EXAMPLES 1. When the linear element of a ruled surface is in the form (76), the direction- cosines of the limiting position of the common perpendicular to two generators are ?<L?, V* + < t z + n /3 aft o/3 250 MINIMAL SURFACES 2. Prove that the developable surfaces are the only ruled surfaces with real generators whose total curvature is constant. 3. Show that the perpendicular upon the z-axis from any point of the cubic x _ u ^y M 2 ? z w a lies i n the osculating plane at the point, and lind the asymp totic lines on the ruled surface generated by this perpendicular. 4. Determine the function in the equations x = w, y = u n , z = 0(u), so that the osculating plane at any point M of this curve shall pass through the projection P of M on the y-axis. Find the asymptotic lines on the surface gener ated by the line MP. 5. Show that the equations x = M sin cos ^, y = wsinflsin^, z = u-fwcos0, where 6 and ^ are functions of t>, define the most general ruled surface with a rec tilinear directrix, and prove that the equation of asymptotic lines can be integrated by two quadratures. Discuss the case where is constant. 6. Concerning the curved asymptotic lines on a ruled surface the following are to be proved : (a) if one of them is an orthogonal trajectory of the generators, the determina tion of the rest reduces to quadratures ; (6) if two of them are orthogonal trajectories, they are curves of Bertrand ; (c) if all of them are orthogonal trajectories, the surface is a right helicoid. 7. Determine the condition that the line of striction be an asymptotic line, and show that in this case the other curved asymptotic lines can be found by quadratures. 8. Find a ruled surface of the fourth degree which is generated by a line pass ing through the two lines x = 0, y = ; z = 0, + y-fz = l. Show that these lines and the line x = 0, x + y + z = 1 are double lines. Find the line of striction. 9. The right helicoid is the only ruled surface each of whose lines of curvature cuts the generators under constant angle ; however, on any other ruled surface there are in general four lines of curvature which have this property. 108. Minimal surfaces. In 1760 Lagrange extended to double integrals the Euler theorems about simple integrals in the calculus of variations, and as an example he proposed the following problem * : Given a closed curve C and a connected surface S bounded by the curve; to determine 8 so that the inclosed area shall be a minimum. If the surface be denned by the equation z =f(x, y), the problem requires the determination of f(x, y) so that the inte gral (cf. Ex. 1, p. 77) * CEuvres de Lagrange, Vol. I, pp. 354-357. Paris, 1867. MINIMAL SURFACES 251 extended over the portion of the surface bounded by C shall be a minimum. As shown by Lagrange, the condition for this is (83) or, in other form, (84) (1 + q z )r - 2pqs + (1 + p*)t = 0. Lagrange left the solution of the problem in this form, and Meusnier,* sixteen years later, proved that this equation is equivalent to the vanishing of the mean curvature ( 52), thus showing that the surfaces furnishing the solution of Lagrange s problem are characterized by the geometrical property which now is usually taken as the definition of minimal surfaces; however, the name indicates the connection with the definition of Lagrange. f In what follows we purpose giving a discussion of minimal sur faces from the standpoint of their definition as the surfaces whose mean curvature is zero at all the points. At each point of such a surface the principal radii differ only in sign, and so every point is a hyperbolic point and its Dupin indicatrix is an equilateral hyperbola. Consequently minimal surfaces are characterized by the property that their asymptotic lines form an orthogonal sys tem. Moreover, the tangents to the two asymptotic lines at a point bisect the angles between the lines of curvature at the point, and vice versa. We recall the formulas giving the relations between the funda mental quantities of a surface and its spherical representation (IV, 70) : (85) (o = From these we have at once the theorem : The necessary and sufficient condition that the spherical represen tation of a surface be conformal is that the surface be minimal. * Memoire sur la courbure des surfaces, Memoires des Savants Strangers, Vol. X (1785), p. 477. t For a historical sketch of the development of the theory of minimal surfaces and a complete discussion of them the reader is referred to the Lemons of Darboux (Vol. I, pp. 267 et seq.). The questions in the calculus of variations involved in the study of mini mal surfaces are treated by Riemann, Gesammelte Werke, p. 287 (Leipzig, 1876) ; and by Schwarz, Gesammelte Abhandlungen, Vol. I, pp. 223, 270 (Berlin, 1890). 252 MINIMAL SURFACES Hence isothermal orthogonal systems on the surface are repre sented by similar systems on the sphere, and conversely. All the isothermal orthogonal systems on the sphere are known ( 35, 40). Suppose that one of these systems is parametric and that the linear element is * From the general condition for minimal surfaces (IV, 77), namely (86) <D" + 3D - 2 &D f = 0, it follows that in this case n 1 In consequence of this the Codazzi equations (V, 27) are reducible to (87) ?-^-0, f + ~? = - dv du du dv By eliminating D or D we find that both D and D 1 are integrals of the equation ~IQ ^Q a? + ^? = * Hence the most general form of D is (88) D = $ (u + iv) + ^(u- iv), where <f> and i/r are arbitrary functions. Then from (87) we have (89) D = ~D" = - i((j) -ty+c, where c is the constant of integration. To each pair of functions <, T/T there corresponds a minimal surface whose Cartesian coordi nates are given by the quadratures (V, 26), namely ( 90) = - du \\ du dv dv \ du dv and similar expressions in y and z. Evidently the surface is real only when <f> and i/r are conjugate functions. In obtaining the preceding results we have tacitly assumed that neither D nor D 1 is zero. We notice that either may be zero and then the other is a constant, which is zero only for the plane. These results may be stated thus: Every isothermal system on the sphere is the representation of the lines of curvature of a unique minimal surface and of the asymptotic lines of another minimal surface. LINES OF CURVATURE AND ASYMPTOTIC LINES 253 The converse also is true, namely: The spherical representations of the lines of curvature and of the asymptotic lines of a minimal surface are isothermal systems. For, if the lines of curvature are parametric, equation (86) may be replaced by D = p ^ D >, = _ p g where p is equal to either principal radius to within its algebraic sign. When these values and D = & = are substituted in the Codazzi equations (V, 27), we obtain so that /g=*U/V, which proves the first part of the theorem ( 41). When the asymptotic lines are parametric, we have Z>=D"=c^=0, and equations (V, 27) reduce to (>!")= cu from which it follows that <~/^= U/V. 109. Lines of curvature and asymptotic lines. Adjoint minimal surfaces. We return to the consideration of equations (87) and investigate first the minimal surface with its lines of curvature represented by an isothermal system. Without loss of generality,* we may take (91) D = -D" = 1, > =0. From (IV, 77) it follows that = -- 2 = ~X 2 , E=G = p, PiP 2 Pi where /> = |^| = |p 2 J. Hence we have the theorem : The parameters of the lines of curvature of a minimal surface may be so chosen that the linear elements of the surface and of its spher ical representation have the respective forms ds 2 = p (du 2 + dv 2 ), dd 2 = - (du 2 + dv 2 ), P where p is the absolute value of each principal radius. * Any other value of the constant leads to homothetic surfaces. 254 MINIMAL SURFACES In like manner we may take, for the solution of equations (87), (92) D = I>"=Q, D = l. Again we find .. -, J- = - = -\\ E=-G = p, PiP* Pi so that we have a result similar to the above : The parameters of the asymptotic lines of a minimal surface may be so chosen that the linear elements of the surface and of its spherical representation have the respective forms ds* = p (du* + di?), d<r* = - (du* + dv 2 ), where p is the absolute value of each principal radius. From the symmetric form of equations (87) it follows that if (88) and (89) represent one set of solutions, another set is given by These values are such that which is the condition that asymptotic lines on either surface cor respond to a conjugate system on the other ( 56). When this condition is satisfied by two minimal surfaces, and the tangent planes at corresponding points are parallel, the two surfaces are said to be the adjoints of one another. Hence a pair of functions <, ->/r determines a pair of adjoint minimal surfaces. When, in par ticular, the asymptotic lines on one surface a*e parametric, the functions have the values (92), and on the other the values (91). It follows, then, from (90), that between the Cartesian coordinates of a minimal surface and its adjoint the following relations hold: cjx\_ _dx foi = fa. cu dv dv du and similar expressions in the / s and z s, when the parametric curves are asymptotic on the locus of (#, #, z). 110. Minimal curves on a minimal surface. The lines of length zero upon a minimal surface are of fundamental importance. When they are taken for parametric curves, the equations of the surface take a simple form, which we shall now obtain. MINIMAL CURVES 255 Since the lines of length zero, or minimal lines, are parametric, we have (94) ^ = = 0. From (85) it follows that the parametric lines on the sphere also are minimal lines, that is, the imaginary rectilinear generators. And from (86) we find that 1) is zero. Conversely, when the latter is zero, and the parametric lines are minimal curves, it follows from (IV, 33) that K m is equal to zero. Hence : A necessary and sufficient condition that a surface be minimal is that the lines of length zero form a conjugate system.* In consequence of (94) and (VI, 26) the point equation of a minimal surface, referred to its minimal lines, is ducv Hence the finite equations of the surface are of the form where U^ T/ 2 , U s are functions of u alone, and F x , F 2 , F 3 are functions of v alone, satisfying the conditions (96) U? + V? + U? = 0, F{ 2 + Fi a + Fj 2 = 0. From (95) it is seen that minimal surfaces are surfaces of trans lation ( 81), and from (96) that the generators are minimal curves ( 22). In consequence of the second theorem of 81 we may state this result thus : A minimal surface is the locus of the mid-points of the joins of points on two minimal curves. In 22 we found that the Cartesian coordinates of any minimal curve are expressible in the form (97) f (1 - u*)F(u) du, i f (1 + u 2 ) F(u) du, 2 Cu F(u) du. *This follows also from the fact that an equilateral hyperbola is the only conic for which the directions with angular coefficients i are conjugate. 256 MINIMAL SURFACES Hence by the above theorem the following equations, due to Enneper *, define a minimal surface referred to its minimal lines : (98) z= I u F(u) du + I v&(v) dv, where F and 4> are any analytic functions whatever. Moreover, any minimal surface can be defined by equations of this form. For, the only apparent lack of generality is due to the fact that the algebraic signs of the expressions (98) are not determined by equations (96), and consequently the signs preceding the terms in the right-hand members of equations (98) could be positive or negative. But it can be shown that by a suitable change of the parameters and of the functions F and 3> all of these cases reduce to (98). Thus, for example, we consider the surface defined by the equations which result when the second terms of the right-hand members of (98) are replaced by In order that the surface thus defined can be brought into coin cidence, by a translation, with the surface (98), we must have Dividing these equations, member by member, we have from which it follows that Substituting this value in the last of the above equations, we find * Zeitschrift fur Mathematik und Physik, Vol. IX (1864), p. 107. MINIMAL CUEVES 257 and this value satisfies the other equations. Similar results fol low when another choice of signs is made. The reason for the particular choice made in (98) will be seen when we discuss the reality of the surfaces. Incidentally we have proved the theorem : When a minimal surface is defined by equations (98), the necessary and sufficient condition that the two generating curves be congruent is that (99) , ( )._1 From (98) we obtain so that the linear element is (100) ds 2 = (l + uv}*F(u)3>(v)dudv. We find for the expressions of the direction-cosines of the normal 1 H- uv 1 + uv 1 + uv and the linear element of the sphere is , 2 4 dudv d<r - - > Alt, (1 + ) Also we have (102) D = so that the equations of the lines of curvature and of the asymp totic lines are respectively (103) F(u) du 2 - 3> (v) dv* = 0, (104) F(u) du 2 + <$> (v) dv* = 0. These equations are of such a form that we have the theorem : When a minimal surface is referred to its minimal lines, the finite equations of the lines of curvature and asymptotic lines are given by quadratures, which are the same in both cases. In order that a surface be real its spherical representation must be real. Consequently u and v must be conjugate imaginaries, as 258 MINIMAL SURFACES is seen from (101) and 13, and the functions F and < must be conjugate imaginary. Hence if RO denotes the real part of a function 9, all real minimal surfaces are defined by x = fi f (1 - u 2 ) F(u) du, y = R ft (1 + u 2 ) F(u) du, z=R I 2uF(u)du, where F(u) is any function whatever of a complex variable u. In like manner the equations of the lines of curvature may be written in the form (105) 72 / ^/F(u)du = const., 11 \ iVF(u)du = const. 111. Double minimal surfaces. It is natural to inquire whether the same minimal surface can be denned in more than one way by equations of the form (98). We assume that this is possible, and indicate by u v v^ and F^(u^ ^V^) the corresponding parameters and functions. As the parameters u^ v l refer to the lines of length zero on the surface, each is a function of either u or v. In order to determine the forms of the latter we make use of the fact that the positive directions of the normal to the surface in the two forms of parametric representation may have the same or opposite senses. When they have the same sense, the expressions (101) and similar ones in u v and v l must be equal respectively. In this case (106) %!=!*, v^v. If the senses are opposite, the respective expressions are equal to within algebraic signs. From the resulting equations we find (107) u When we compare equations (98) with analogous equations in u l and v v we find that for the case (106) we must have and for the case (107) DOUBLE MINIMAL SUBFACES 259 Hence we have the theorem : A necessary and sufficient condition that two minimal surfaces, deter mined by the pairs of functions F, <& and F v <& v be congruent is that (108) ^w to the point (u, v) on one surface corresponds the point ( -- -- on u the other, and the normals at these points are parallel but of different sense. In general, the functions F and F l as given by (108) are not the same. If they are, so also are < and 4> r Ih this case the right-hand members of equations (98) are unaltered when u and v are replaced by l/v and 1/u respectively. Hence the Cartesian coordinates of the points (u, v) and ( -- > -- j differ at most by constants. And so the regions of the surface about these points either coincide or can be brought into coincidence by a translation. In the latter case the surface is periodic and consequently transcendental. Suppose that it is not periodic, and consider a point -ZjJ(w , V Q ). As u varies continuously from U Q to l/t> , v varies from v to l/w , and the point describes a closed curve on the surface by returning to P Q . But now the positive normal is on the other side of the sur face. Hence these surfaces have the property that a point can pass continuously from one side to the other without going through the surface. On this account they were called double minimal surfaces by Lie,* who was the first to study them. From the third theorem of 110 it follows that double minimal surfaces are characterized by the property that the minimal curves in both systems are congruent. The equations of such a surface may be written The surface is consequently the locus of the mid-points of the chords of the curve f =/,(), i =/,(), ?=/,(), which lies upon the surface and is the envelope of the parametric curves. * Math. Annalen, Vol. XIV (1878), pp. 345-350. 260 MINIMAL SURFACES EXAMPLES 1. The focal sheets of a minimal surface are applicable to one another and to the surface of revolution of the evolute of the catenary about the axis of the latter. 2. Show that there are no minimal surfaces with the minimal lines in one family straight. 3. If two minimal surfaces correspond with parallelism of tangent planes, the minimal curves on the two surfaces correspond. 4. If two minimal surfaces correspond with parallelism of tangent planes, and the joins of corresponding points be divided in the same ratio, the locus of the points of division is a minimal surface. 5. Show that the right helicoid is defined by F(u) = im/2 w 2 , where m is a real constant, and that it is a double surface. 2 6. The surface for which F(u) = - is called the surface of Scherk. Find its equation in the Monge form z = f(x, y). Show that it is doubly periodic and that it is a surface of translation with real generators which are in perpendicular planes. 7. By definition a meridian curve on a surface is one whose spherical representa tion is a great circle on the unit sphere. Show that the surface of Scherk possesses two families of plane meridian curves. 112. Algebraic minimal surfaces. Weierstrass * remarked that formulas (98) can be put in a form free of all quadratures. This is done by replacing F(u) and <J>(v) by f "(u) and # "(v), where the accents indicate differentiation, and then integrating by parts. This gives (109) x p. 2 + uf (u) -f(u) + ^- 4>"(v) + v$(v) "(u) iuf(u) + if(u) i - -^- <l>"(v) -h iv( = uf"(u) -) + v<f>"(v) - 4> (v). It is clear that the surface so denned is real when / and <f> are conjugate imaginary functions. In this case the above formulas may be written : = R[(l- u*)f"(u) + 2 uf (u) - (110) y = Ri [(1 + u*)f"(u) - 2 uf (u) + * Monatsberichte der Berliner Akademie (1866), p. 619. ALGEBRAIC MINIMAL SURFACES 261 However, it is not necessary, as Darboux * has pointed out, that f and (f> be conjugate imaginaries in order that the surface be real. For, equations (109) are unaltered if /and be replaced by f t (u) =f(u) + A (1 - u 2 ) + Bi (1 + u*) + 2 Cu, ^(v) = (f>(v) - A(l - v 2 ) + Bi ({.-{- v 2 )- 2 Cv, where A, B, C are any constants whatever. Evidently, if / and </> are conjugate imaginaries, the same is not true in general of /,_ and <f> l ; but the surface was real for the former and consequently is real for the latter also. It is readily found that / t and </> x are conjugate imaginary functions only in case J, J5, C are pure imaginaries. Formulas (109) are of particular value in the study of algebraic surfaces. Thus, it is evident that the surface is algebraic when/ and < are algebraic. Conversely, every algebraic minimal surface is determined by algebraic functions / and </>. In proving this we follow the method suggested by Weierstrass.f We establish first the following lemma : Gttven a function $*(? + *??) and let "^(f, 77) denote the real part of 4> ; if in a certain domain an algebraic relation exists between M*, , and 77, 4> is an algebraic function of j~ + irj. If the point f = 0, rj = does not lie within the domain under consideration, this can be effected by a change of variables without vitiating the argument. Assuming that this has been done, we develop the function <E> in a power series, thus : 4> = a + #o + K + ibj (| + irj) + (a 2 + ib 2 ) (f + irj) 2 + . . . , where the a s and 5 s are real constants. Evidently M* is given by 1 (a, - ^) (f - ^) + J (a, - i6 8 ) (f - ii?) 2 + - - - . Let J^(^, f , ?;) = denote a rational integral relation between "SP, f, and 77. When M* has been replaced by the above value, and the resulting expression is arranged in powers of | and 77, the coeffi cient of every term is identically zero. They will continue to be zero when f and 77 have been replaced by two complex quantities * Vol. I, p. 293. f Monatsberichte tier Berliner Akademie (1867), pp. 511-518. 262 MINIMAL SURFACES a and /3, provided that the development remains convergent. The condition for the latter is that the moduli of a and /3 be each one half the modulus of f + irj. This condition is satisfied if we take Now we have - -. - K- i5 ) + - < (f t irt, f + ti,] = 0, which proves the lemma. In applying this lemma to real minimal surfaces we note from (101) that X _u + v Y _u v, l-Z = ~Y~ l-Z = 2i consequently the left-hand members of these equations are equal to u l and v l respectively, where u = u^ + iv r When the surface is algebraic there exists an algebraic relation between the functions X Y , and each of the Cartesian coordinates.* Since, then, 7 7 .L j A. /j there is an algebraic relation between u^ v^ and each of the coordinates given by (110), it follows from the lemma that each of the three expressions <k( M ) = (1 - ?/)/ + 2 uf (u) - 2/(w), fa(u) = i (1 + u 2 )f"(u) - 2 iuf (u) + 2 if(u), 4> 9 (u) = 2uf"(u)2f(u) are algebraic functions of w, and so also isf(u) ; for, Hence we have demonstrated the theorem of Weierstrass : The necessary and sufficient condition that equation (110) define an algebraic surface is that f(u) be algebraic. * For. if the surface is defined by F(x. y, z) = 0, the direction-cosines of the normal X Y are functions of x, y, z. Eliminating two of the latter between _ > _ _> and F(x, y, z) = 0, we have a relation of the kind described. ASSOCIATE SURFACES 263 113. Associate surfaces. When the equations of a minimal surface S are written in the abbreviated form (95), the linear element is This is the linear element also of a surface defined by where a is any constant. There are an infinity of such surfaces, called associate minimal surfaces. It is readily found that the direc tion-cosines of the normal to any one have the values (101). Hence any two associate minimal surfaces defined by (111) have their tan gent planes at corresponding points parallel, and are applicable. Of particular interest is the surface S l for which a = ?r/2. Its equations are i C i)du - I (1 v 2 ) < (v) dv, (112) {y = - ^ / (1 + u 2 ) F(u) du - I & J * \ = i I uF(u) du i I v<& (v) dv. In order to show that S l is the adjoint ( 109) of S, we have only to prove that the asymptotic lines on either surface correspond to the lines of curvature on the other. For S l the equations of the lines of curvature and asymptotic lines are iF(u) du 2 - i (v) dv 2 = 0, respectively. Comparing these with (103) and (104), we see that the desired condition is satisfied. From (98) and (112) we obtain the identities \ dx dx l + dy dy l + dz dz^ = 0. The latter has the following interpretation : On two adjoint minimal surfaces at points corresponding with par allelism of tangent planes the tangents to corresponding curves are perpendicular. 264 MINIMAL SURFACES From (105) it follows that if we put u + iv = / ^/F(u) du, the curves u = const, and v = const, on the surface are its lines of curvature. Moreover, for an associate surface the lines of curva ture are given by ia ttf R [e 2 (u + iv)] = const., R [ie 2 (u -h iv)] = const. t or a - . a - . a _ a u cos v sin = const., u sin -f- v cos - = const. 22 22 From this result follows the theorem : The lines of curvature on a minimal surface associate to a surface S correspond to the curves on S which cut its lines of curvature under the constant angle a/ 2. Since equations (111) may be written x a = x cos a + x l sin #, (114) - ^^ycosa + ^sina, . z a z cos a -|- z l sin #, the plane determined by the origin of coordinates, a point P on a minimal surface and the corresponding point on its adjoint, con tains the point P a corresponding to P on every associate minimal surface. Moreover, the locus of these points P a is an ellipse with its center at the origin. Combining this result and the first one of this section, we have *- A minimal surface admits of a continuous deformation into a series of minimal surfaces, and each point of the surface describes an ellipse whose plane passes through a fixed point which is the center of the ellipse. 114. Formulas of Schwarz. Since the tangent planes to a minimal surface and its adjoint at corresponding points are parallel, we have From this and the second of (113) we obtain the proportion dx l __ dy l _ dz l Zdy Ydz = Xdz Z dx~ Y dx X dy FORMULAS OF SCHWAKZ 265 In consequence of the first of (113) the sums of the squares of the numerators and of the denominators are equal. And so the com mon ratio is -|-1 or 1. If the expressions for the various quanti ties be substituted from (98), (101), and (112), it is found that the value is 1. Hence we have (115) dx^Ydz Zdy, dy l = Zdx Xdz, dz 1 = Xdy Ydx. From these equations and the formulas (95), (112) we have Zdy - Ydz, (116) and (117) 1 = x + i i=y + * (xdz - Zdx, l = z-^-i \ Ydx Xdy, 1 = xi \ Zdy Ydz, i \ ^z i \Ydx X dy. These equations are known as the formulas of Schwarz* Their importance is due to their ready applicability to the solution of the problem : To determine a minimal surface passing through a given curve and admitting at each point of the curve a given tangent plane.\ In solving this problem we let C be a curve whose coordinates #, y, z are analytic functions of a parameter f, and let JT, Y, Z be analytic functions of t satisfying the conditions X 2 + F 2 + Z 2 = 1, Xdx + Ydy + Zdz = 0. * Crelle, Vol. LXXX (1875), p. 291. t This problem is a special case of the more general one solved by Cauchy : To deter mine an integral surface of a differential equation passing through a curve and admitting at each point of the curve a given tangent plane. For minimal surfaces the equation is (84). Cauchy showed that such a surface exists in general, and that it is unique unless the curve is a characteristic for the equation. His researches are inserted in Vols. XIV, XV of the Comptes Rendus. The reader may consult also Kowalewski, Theorie der partiellen Differentialgleichungen, Crelle, Vol. LXXX (1875), p. 1; and Goursat, Cours d Analyse Mathematique, Vol. II, pp. 563-567 (Paris, 1905). 266 MINIMAL SUKFACES If x uJ y u , z u denote the values of x, y, z when t is replaced by a complex variable u, and x v , y v , z v the values when t is replaced by v, the equations (118) l - f"(Ydx-Xdy) Jv define a minimal surface which passes through C and admits at each point for tangent plane the plane through the point with direction-cosines X, I 7 , Z. For, when u and v are replaced by , these equations define C. And the conditions (96) and are satisfied. Furthermore, the surface defined by (118) affords the unique solution, as is seen from (116) and (117). When, in particular, C and t are real, the equations of the real minimal surface, satisfying the conditions of the problem, may be put in the form ,- -i x = R\x + il (Zdy-Ydz}\, y = R \y + i C\Xdz - Zdx)] , z = R \z + i r\Ydx - Xdy\\ As an application of these formulas, we consider minimal surfaces containing a straight line. If we take the latter for the z-axis, and let denote the angle which the normal to the surface at a point of the line makes with the x-axis, we have x = y = 0, z=t, JT=cos0, Y=sin<t>, Z = 0. Hence the equations of the surface are x = - RiTsm <f>dt, y = B{J**C08^ctt, z = R(u). Here <#> is an analytic function of t, whose form determines the character of the surface. For two points corresponding to conjugate values of M, the z-coordinates are equal, and the x- and ^-coordinates differ in sign. Hence : Every straight line upon a minimal surface is an axis of symmetry. FORMULAS OF SCHWAKZ 267 EXAMPLES 1. The tangents to corresponding curves on two associate minimal surfaces meet under constant angle. 2. If corresponding directions on two applicable surfaces meet under constant angle, the latter are associate minimal surfaces. 3. Show that the catenoid and the right helicoid are adjoint surfaces and deter mine the function F(u) which defines the former. 4. Let C be a geodesic on a minimal surface S. Show that (a) the equations of the surface may be put in the form y = where f , 77, f are the coordinates of a point on C, and X, /*, v the direction-cosines of its binomial ; (6) if C denotes the curve on the adjoint S t corresponding to C, the radii of first and second curvature of C are the radii of second and first curvature of C ; (c) if C is a plane curve, the surface is symmetric with respect to its plane. 5. The surface for which F(u) = 1 is called the surface of Henneberg ; it is u 4 a double algebraic surface of the fifteenth order and fifth class. GENERAL EXAMPLES 1. The edge of regression of the developable surface circumscribed to two con- focal quadrics has for projections on the three principal planes the evolutes of the focal conies. 2. By definition a tetrahedral surface is one whose equations are of the form x = A (u - a) m (v - a), y = B(u- b) m (v - 6) n , z = C(u- c) m (v - c) n , where A, B, 0, w, n are any constants. Show that the parametric curves are con jugate, and that the asymptotic lines can be found by quadratures ; also that when m = n, the equation of the surface is III ^)> - c) + (|)"(c - a) + (0"<a - b) = (a - b) (b - c) (a - c). 3. Determine the tetrahedral surfaces, defined as in Ex. 2, upon which the parametric curves are the lines of curvature. 4. Find the surfaces normal to the tangents to a family of umbilical geodesies on an elliptic paraboloid, and find the complementary surface. 5. At every point of a geodesic circle with center at an umbilical point on the ellipsoid (10) abc = f W <i (a + c _ r ^ where r is the radius vector of the point (cf. 102). 6. The tangent plane to the director-cone of a ruled surface along a generator is parallel to the tangent plane to the surface at the infinitely distant point on the corresponding generator. 268 MINIMAL SURFACES 7. Upon the hyperboloid of one sheet, and likewise upon the hyperbolic parab oloid, the two lines of striction coincide. 8. The line of striction of a ruled surface is an orthogonal trajectory of the generators only in case the latter are the binormals of a curve or the surface is a right conoid. 9. Determine for a geodesic on a developable surface the relation existing between the curvature, torsion, and angle of inclination of the geodesic with the generators. 10. If h denotes the shortest distance and a the angle between two lines li and Z 2 , and the latter revolves about the former with a helicoidal motion of parameter a (cf . 62), the locus of 1 2 is a developable surface if a = h cot a. If a = h tan a, the surface is the locus of the binormals of a circular helix. 11. If the lines of curvature in one family upon a ruled surface are such that the segments of the generators between two curves of the family are of the same length, the parameter of distribution is constant and the line of striction is a line of curvature. 12. If two ruled surfaces meet one another in a generator, they are tangent to one another at two points of the generator or at every point ; in the latter case the central point for the common generator is the same, and the parameter of distribu tion has the same value. 13. If tangents be drawn to a ruled surface at points of the line of striction and in directions perpendicular to the generators, these tangents form the conju gate ruled surface. It has the same line of striction as the given surface. More over, a generator of the given surface, the normal to the surface at the central point C of this generator, and the generator of the conjugate surface through C are parallel to the tangent, principal normal, and binormal of a twisted curve. 14. Let C be a curve on a surface S, and S the ruled surface formed by the normals to S along C. Derive the following results : (a) the distance between near-by generators of S is of the first order unless C is a line of curvature ; (6) if r denotes the distance from the central point of a generator to the point of intersection with S, rS (dX) 2 Z dxd X ; (c) the tangent to C at a point M is conjugate to the tangent to the surface at M parallel to the line of shortest distance ; (d) the maximum and minimum values of r are the principal radii of -S, pi, and p 2 , and the above equation may be written r = pisin 2 </> -f p 2 cos 2 tf>, where <f> is the angle which the corresponding line of shortest distance makes with the tangent to the line of curvature corresponding to pz> 15. If C and 6" are two orthogonal curves on a surface, then at the point of intersection (cf. Ex. 14) 1111 rB 4 >.* ~tf + *| 16. If C and C are two conjugate curves on a surface, then at the point of intersection (cf. Ex. 14) j j i \ r R GEKEBAL EXAMPLES 269 17. If two surfaces are applicable, and the radii of first and second curvature of every geodesic on one surface are equal to the radii of second and first curvature of the corresponding geodesic on the other, the surfaces are minimal. 1 8. The surface for which F in (98) is constant, say 3, is called the minimal sur face of Enneper ; it possesses the following properties : (a) it is an algebraic surface of the ninth degree whose equation is unaltered when x, y, z are replaced by y, x, z respectively ; (6) it meets the plane z = in two orthogonal straight lines ; (c) if we put u = a i/3, the equations of the surface are x = 3 a + 3 ap? - a 3 , y = 3 ft + 3 a 2 ft -ft 3 , z = 3 a 2 - 3 /3 2 , and the curves a const. , ft = const, are the lines of curvature ; (d) the lines of curvature are rectifiable unicursal curves of the third order and they are plane curves, the equations of the planes being x + az -3a-2a 3 = 0, y - ftz - 3ft - 2 ^ = 0; (e) the lines of curvature are represented on the unit sphere by a double family of circles whose planes form two pencils with perpendicular axes which are tangent to the sphere at the same point ; (/) the asymptotic lines are twisted cubics ; (g) the sections of the surface by the planes x = and y = are cubics, which are double curves on the surface and the locus of the double points of the lines of curvature ; (h) the associate minimal surfaces are positions of the original surface rotated through the angle a/2, about the z-axis, where a has the same meaning as in 113 ; (i) the surface is the envelope of the plane normal, at the mid-point, to the join of any two points, one on each of the focal parabolas X = 4 cr, y = 0, z - 2 a 2 - 1 ; x - 0, y = 4 ft, z = 1-2 ft 2 - the planes normal to the two parabolas at the extremities of the join are the planes of the lines of curvature through the point of contact of the first plane. 19. Find the equations of Schwarz of a minimal surface when the given curve is an asymptotic line. 20. Let S and S be two surfaces, and let the points at which the normals are parallel correspond ; for convenience let S and S be referred to their common con jugate system. Show that if the correspondence is conformal, either S and S are homothetic ; or both are minimal surfaces ; or the parametric curves are the lines of curvature on both surfaces, and form an isothermal system. 21. Find the coordinates of the surface which corresponds to the ellipsoid after the manner of Ex. 20. Show that the surface is periodic, and investigate the points corresponding to the umbilical points on the ellipsoid. 22. When the equations of an ellipsoid are in the form (11), the curves u + v = const, lie on spheres whose centers coincide with the origin ; and at all points of such a curve the product pW is constant ( 102). CHAPTER VIII SURFACES OF CONSTANT TOTAL CURVATURE. W-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CURVATURE 115. Spherical surfaces of revolution. Surfaces whose total cur vature K is the same at all points are called surfaces of constant curvature. When this constant value is zero, the surface is devel opable ( 64). The nondevelopable surfaces of this kind are called spherical or pseudospherical, according as K is positive or negative. We consider these two kinds and begin our study of them with the determination of surfaces of revolution of constant curvature. When upon a surface of revolution the curves v = const, are the meridians and u = const, the parallels, the linear element is reducible to the form (1) d8*=du*+Gdif, where G is a function of u alone ( 46). In this case the expres sion for the total curvature (V, 12) is (2) K = For spherical surfaces we have 7f=l/a 2 , where a is a real constant. Substituting this value in equation (2) and integrating, we have (3) where b and c are constants of integration. From (1) it is seen that a change in b means simply a different choice of the parallel u = 0. If we take 6 0, the linear element is (4) ds 2 =du 2 + c 2 cos 2 -^ 2 . a From (III, 99, 100) it follows that the equations of the meridian curve are - . u C it c 2 . 9 u (5) r = cos-> z = / \1 jsm- a J \ a 2 a 270 SPHERICAL SURFACES OF REVOLUTION 271 and that v measures the angle between the meridian planes. There are three cases to be considered, according as c is equal to, greater than, or less than, a. CASE I. c = a. Now r = a cos-) a . u z a sin - > a FIG. 26 and consequently the surface is a sphere. CASE II. c > a. From the expression for z it follows that sin 2 - < 1 and con- a sequently r > 0. Hence the surface is made up of zones bounded by minimum parallels whose radii are equal to the ?/ minimum value of cos - and the greatest parallel of each zone is of radius c ; as in fig. 26, where the curves represent geodesies. CASE III. c < a. Now r varies from to c, the former correspond ing to the value u = mcnr/2, where m is any odd integer. At these points on the axis the meridians meet the latter under the angle v? sin" 1 -. Hence the surface is made up of a series of spindles a (fig. 27). For the cases II and III the expression for z can be integrated in terms of elliptic functions.* It is readily found that these two surfaces are applicable to the sphere with the meridians and parallels of each in correspondence. Thus, if we write the linear element of the sphere in the form ds 2 du 2 4- a 2 cos 2 - dv 2 , a it follows from (4) that the equations u = u. FIG. 27 determine the correspondence desired. It is evident that for values of b other than zero we should be brought to the same results. However, for the sake of future *Cf. Bianchi, Vol. I, p. 233. 272 SURFACES OF CONSTANT CURVATURE reference we write down the expressions for the linear element when b = 7r/2 and Tr/4 together with (4), thus : (6) (i) ds 2 =du* (ii) ds 2 =du*- (iii) ds 2 =du* cos u TT\ , --- }dv\ a 4/ Let S be a surface with the linear element (6, i), and consider the zone between the parallels u = const, and rt 1 = const. A point of the zone is determined by values of u and v such that The parametric values of the corresponding point on the sphere are such that 9 _ Hence when c < , the given zone on S does not cover the zone on the sphere between the parallels M O = const, and u^ = const. ; but when c > a it not only covers it, but there is an overlapping. 116. Pseudospherical surfaces of revolution. In order to find the pseudospherical surfaces of revolution we replace K in (2) by I/a 2 and integrate. This gives V5 = c. cosh - + <?_ sinh - > a a where c t and c 2 are constants of integration. We consider first the particular forms of the linear element arising when either of these constants is zero or both are equal. They may be written (i) ds 2 = (ii) ?/ a i oU = du + c sinh 2 - a (iii) ds^dtf+fe" dv*. Any case other than these may be obtained by taking for either of the values cosh - or sinh(- where b is a constant. PSEUDOSPHEKICAL SURFACES OF REVOLUTION 273 By a change of the parameter u the corresponding linear elements are reducible to (i) or (ii). Hence the forms (7) are the most general. The corresponding meridian curves are defined by (8) = c cosh - > 2 = C . , 2 U , sum - aw ; w C \ <? .u (ii) r = tfsinh- 2= I \ 1 -costf-du; v a J N a 2 a (iii) r = ce a . z = We consider these three cases in detail. CASE I. The maximum and minimum values of sinh 2 - are a 2 /e 2 a and 0. Hence the maximum and minimum values of r are Va 2 + c 2 and c. At points of a maximum parallel the tangents to the merid ians are perpendicular to the axis, and at points of a minimum parallel they are par allel to the axis. Hence the former is a cus pidal edge, and the latter a circle of gorge, so that the surface is made up of spool-like sections. It is represented by fig. 28, upon which the closed curves are geodesic circles and the other curves are geodesies. These pseudospherical surfaces are said to be of the hyperbolic type.* CASE II. In order that the surface be real c 2 cannot be greater than a 2 , a restriction not necessary in either of the other cases. If we put e = asino:,f the maximum and minimum values of cosh 2 are cosec 2 o; and 1, and the correspond- a ing values of r are a cos a and 0. The tangents to the meridians at points of the former circle are perpendicular to the axis, and at the points for which r is zero they meet the axis under the angle a. Hence the surface is made up of a series of parts similar in shape FIG. 28 * Cf . Bianchi, Vol. I, p. 223. f Cf. Bianchi, Vol. I, p. 220. 274 SURFACES OF CONSTANT CURVATURE to hour-glasses. Fig. 29 represents one half of such a part ; one of the curves is an asymptotic line and the others are parallel geodesies. The surface is called a pseudospherical surface of the elliptic type. CASE III. In the preceding cases the equations of the meridian curve can be expressed without the quadrature sign by means of elliptic functions.* In this case the same can be done by means of trigonometric functions. For, if we put sin d) = e a . a FIG. 29 equations (iii) of (8) become (9) r = asin<, 2 = a (log tan^-f cos(/>). We find that c/> is the angle which the tangent to a meridian at a point makes with the axis. Hence the axis is an asymptote to the curve. Since the length of the segment of a tangent between the point of contact and the intersection with the axis is r cosec c/> or a, the length of the segment is independent of the point of contact. Therefore the meridian curve is a tractrix. The surface of revolution of a tractrix about its asymptote is called the pseudo- sphere, or the pseudospherical surface of the parabolic type. The surface is shown in fig. 30, which also pictures a family of parallel geodesies and an asymptotic line. If the integral (3) be written in the form u = c, cos - -f- c sin - 1 a a the cases (i), (ii), (iii) of (6) are seen to correspond to the similar cases of (7). We shall find other marks of similarity between these cases, but now we desire to call at tention to differences. Each of the three forms (7) determines a particular kind of pseudospherical surface of revolution, and c is restricted in value FIG. 30 *Cf. Bianchi, Vol. I, pp. 226-228. APPLICABILITY 275 only for the second case. On the contrary each of the three forms (6) serves to define any of the three types of spherical surfaces of revolution according to the magnitude of c. From (IV, 51) we find that the geodesic curvature of the par allels on the surfaces with the linear elements (7) is measured by the expressions -. -, -, 1 . , * 1 ., M 1 - tann i - cotn - a a a a a Since no two of these expressions can be transformed into the other if u be replaced by u plus any constant, it follows that two pseudospherical surfaces of revolution of different types are not applicable to one another with meridians in correspondence. 117. Geodesic parametric systems. Applicability. Now we shall show that in corresponding cases of (6) and (7) the parametric geodesic systems are of the same kind, and then we shall prove that when such a geodesic system is chosen for any surface of constant curvature, not necessarily one of revolution, the linear element can be brought to the corresponding form of (6) or (T). In the first place we recall that when on any surface the curves v = const, are geodesies, and u = const, their orthogonal trajectories, the linear element is reducible to the form (1), where G is, in general, a function of both u and v ; and the geodesic curvature of the curves u const, is given by (IV, 51), namely - p ff When, in particular, the curvature of the surface is constant, is given by equation (2) in which K may by replaced by l/a 2 . Hence, for spherical surfaces, the general form of V& is (11) V& = </> (v) cos - + A/T (v) sin - , a a and for pseudospherical surfaces (12) VG = < (v) cosh - + i/r (v) sinh - , ci a where < and i/r are, at most, functions of v. We consider now the three cases of (6) and (7). 276 SURFACES OF CONSTANT CUKVATUKE CASE I. From the forms (i) of (6) and (7), and from (10), it follows that the curve u = is a geodesic and that its arc is measured by cv. Moreover, a necessary and sufficient condition that the curve u = on any surface with the linear element (1) satisfy these conditions is =o. Applying these conditions to (11) and (12), we are brought to the forms (i) of (6) and (7) respectively. CASE II. The forms (ii) of (6) and (7) satisfy the conditions = 0, which are necessary and sufficient that the parametric system be geodesic polar, in which cv measures angles (cf. VI, 54). When these conditions are applied to (11) and (12), we obtain (ii) of (6) and of (7) respectively. CASE III. For (iii) of (6) the curve u = has constant geodesic curvature I/a, and for (iii) of (7) all of the curves u = const, have the same geodesic curvature I/a. Conversely, we find from (11) and (12) that when this condition is satisfied on any sur face of constant curvature the linear element is reducible to one of the forms (iii). We gather these results together into the theorem : The linear element of any surface of constant curvature is reducible to the forms (i), (ii), (iii) of (6) or (7) according as the parametric geodesies are orthogonal to a geodesic, pass through a point, or are orthogonal to a curve of constant geodesic curvature. When the linear element of a surface of constant curvature is in one of the forms (i), (ii), (iii) of (6) and (7), it is said to be of the hyperbolic, elliptic, or parabolic type accordingly. The above theorem may be stated as follows : Any spherical surface of curvature l/a z is applicable to a sphere of radius a in such a way that to a family of great circles with the same diameter there correspond the geodesies orthogonal to a APPLICABILITY 277 given geodesic on the surface, or all the geodesios through any point of it, or those which are orthogonal to a curve of geodesic curvature I/ a. Any pseudo spherical surface of curvature I/a 2 is applicable to a pseudospherical surface of revolution of any of the three types ; according as the latter surface is of the hyperbolic, elliptic, or par abolic type, to its meridians correspond on the given surface geodesies which are orthogonal to a geodesic, or pass through a point, or are orthogonal to a curve of geodesic curvature I/a. In the case of spherical surfaces one system of geodesies can satisfy all three conditions ; for in the case of the sphere the great circles with the same diameter are orthogonal to the equator, pass through both poles, and are orthogonal to two small circles of radius a/V2, whose geodesic curvature is I/a. But on a pseudo- spherical surface a geodesic system can satisfy only one of these conditions. Otherwise it would be possible to apply two surfaces of revolution of different types in such a way that meridians and parallels correspond. From the foregoing theorems it follows that, in order to carry out the applicability of a surface of constant curvature upon any one of the surfaces of revolution, it is only necessary to find the geodesies on the given surface. The nature of this problem is set forth in the theorem : The determination of the geodesic lines on a surface of constant curvature requires the solution of a Riccati equation. In proving this theorem we consider first a spherical surface defined in terms of any parametric system. It is applicable to a sphere of the same curvature with center at the origin. The coordinates of this sphere, expressed as functions of the parameters u, v, can be found by the solution of a Riccati equa tion ( 65). To great circles on the sphere correspond geodesic lines on the spherical surface ; hence the finite equation of the geodesies is ax + by + cz = 0, where a, b, c are arbitrary constants. When the surface is pseudospherical we use an imaginary sphere of the same curvature, and the analysis is similar. 278 SURFACES OF CONSTANT CURVATURE 118. Transformation of Hazzidakis. Let a spherical surface of curvature I/a 2 be defined in terms of isothermal-conjugate parame ters. Then * D D" 1 and the Codazzi equations (V, 13 ) reduce to 1 dE dG ^dF T _ / - - II dv dv du From these equations follows the theorem : The lines of curvature of a spherical surface form an isothermal- conjugate system. For, a solution of these equations is E G = const., F0. When this constant is zero the surface is a sphere because of (13). Excluding this case, we replace the above by (15) E = a 2 cosh 2 a>, F = 0, G = a 2 sinh 2 a>. Now (16) D D n = a sinh a> cosh a>. When these values are substituted in the Gauss equation (V, 12), namely - _ + 2 # I a^ L^ ^ H ^ HE du \ du ULE jv R it is found that o> must satisfy the equation a 2 ft> a 2 o) A /18} H --- - + smh a) cosh o> = 0. du 2 dv 2 Conversely, for each solution of this equation the quantities (15) and (16) determine a spherical surface. If equations (14) be differentiated with respect to u and v respec tively, and the resulting equations be added, we have (19) ^ + 0^ "" du 2 dv 2 * The ambiguity of sign may be neglected, as a change of sign gives a surface sym metrical with respect to the origin. TRANSFORMATION OF HAZZIDAKIS 279 In consequence of (14) equation (17) is reducible to 4 H 4 ( \\du] \dv / J L#ti $ v d v du Equations (14) are unaltered if E and G be interchanged and the sign of F be changed. The same is true of (17) because of (19) and (20). Hence we have : If the linear element of a spherical surface referred to an isothermal- conjugate system of parameters be ds 2 = E du 2 +2F dudv + G dv 2 , there exists a second spherical surface of the same curvature referred to a similar parametric system with the linear element ds 2 = Gdu 2 2F dudv -f E dv 2 , and with the same second quadratic form as the given surface ; moreover, the lines of curvature correspond on the two surfaces. The latter fact is evident from the equation of the lines of curva ture (IV, 26), which reduces to Fdu 2 + (G - E) dudv - F dv 2 = 0. From (IV, 69) it is seen that the linear elements of the spherical representation of the respective surfaces are da 2 = -(Gdu 2 -2F dudv + E dv 2 ), CL da 2 = (E du 2 + ZFdudv -f- G dv 2 ). a/ In particular we have the theorem : Each solution co of equation (18) determines two spherical surfaces of curvature I/ a 2 ; the linear elements of the surfaces are ds 2 = a 2 (cosh 2 co du 2 + sinh 2 co dv 2 ), ds 2 = a 2 (sinh 2 co du 2 -f cosh 2 co dv-), and of their spherical representations , 9 -, v { da 2 = sinh 2 co du 2 -f- cosh 2 co dv\ j d<r*= cosh 2 co du 2 + sinh 2 co dv 2 ; moreover, their principal radii are respectively p l = a coth &), p 2 = a tanh co, p[ = a tanh co, p 2 a coth co. 280 SURFACES OF CONSTANT CURVATURE Bianchi * has given the name Hazzidakis transformation to the relation between these two surfaces. It is evident that the former theorem defines this transformation in a more general way. 119. Transformation of Bianchi. We consider now a pseudo- spherical surface of curvature I/a 2 , defined in terms of isothermal- conjugate parameters. We have D__ # _ _ 1 4 H~ H~~ a and the Codazzi equations reduce to ^+ -2^=0, ^+-2^=0. du du dv dv dv du These equations are satisfied by the values (22) E= a- cos 2 w, ^=0, G = a 2 sin 2 to, where &> is a function which, because of the Gauss equation (V, 12), must satisfy the equation o2 <"2 /c . n , &> (0 (23) ___ = 8in coB. Conversely, every solution of this equation determines a pseudo- spherical surface whose fundamental quantities are given by (22) and by (24) D = D" = a sin w cos to. Moreover, the linear element of the spherical representation is (25) do- 2 =sin 2 o>c^ 2 +cos 2 a>dv 2 . f There is not a transformation for pseudospherical surfaces sim ilar to the Hazzidakis transformation of spherical surfaces, but there are transformations of other kinds which are of great im portance. One of these is involved in the following theorem of Ribaucour : If in the tangent planes to a pseudo spherical surface of curvature I/ a 2 circles of radius a be described with centers at the points of contact, these circles are the orthogonal trajectories of an infinity of surfaces of curvature 1J a 2 . * Vol. II, p. 437. t This choice of sign is made so that the following formulas may have the custom ary form. TKANSFOKMATION OF BIANCHI 281 In proving this theorem we imagine the given surface S referred to its lines of curvature, and we associate with it the moving trihe dral whose axes are tangent to the parametric lines. From (22) and (V, 75, 76) it follows that rt n d(0 3(0 P P\ = cos w = sin , t =s 0, r = , r 1 = cv vU = a cos ft), t] l = a sin &>, f x = 77 = 0. In the tangent z^-plane we draw from the origin M a segment of length #, and let 6 denote its angle of inclination with the #-axis. The coordinates of the other extremity M 1 with respect to these axes are a cos 0, a sin 0, 0, and the projections upon these axes of a displacement of M l as M moves over S are, by (V, 51), a sin 6 dO -f cos oadui du -\ - dv } sin 6 L L \dv du I J a cos 6 d6 -f sin &) c?y +( rfw + dv } cos , L \cv du / J a [cos &> sin c?v sin &) cos du\. We seek now the conditions which must satisfy in order that the line MM l be tangent to the locus of M l denoted by S^ and that the tangent plane to S l at M 1 be perpendicular to the tangent plane to S at M. Under these conditions the direction-cosines of the tangent plane to S l with reference to the moving trihedral are (26) sin0, -COS0, 0, and since the tangent to the above displacement must be in this plane, we have (27) dO + ( - sin cos co] du+(+ cos sin a>\ dv = 0. \dv / \cu ) As this equation must hold for all displacements of Jtf", it is equivalent to dd da) ( = cos &) sin 0, (28) du dv Q/J /)M 0" fft) = sin &) cos 0. dv du These equations satisfy the condition of integrability in conse quence of (23). Moreover, is a solution of equation (23), as is seen by differentiating equations (28) with respect to u and v respectively and subtracting. 282 SURFACES OF CONSTANT CURVATURE By means of (28) the above expressions for the projections of a displacement of M^ can be put in the form a cos (cos ft) cos 6 du + sin &> sin 6 dv), a sin (cos o> cos 9 du + sin o> sin 6 dv), a (cos &) sin # c?t> sin &) cos $ du). From these it follows that the linear element of S l is ds? = a 2 (cos 2 9 du 2 + sin 2 cfrr 2 ). In order to prove that S l is a pseudospherical surface referred to its lines of curvature, it remains for us to show that the spherical representation of these curves forms an orthogonal system. We obtain this representation with the aid of a trihedral whose vertex is fixed, and which rotates so that its axes are always parallel to the corresponding axes of the trihedral for S. The point whose coordinates with reference to the new trihedral are given by (26) serves for the spherical representation of S r The projections upon these axes of a displacement of this point are reducible, by means of (28), to cog e ^ cos m sin du _ sin a cos dv ^ sin #(cos &) sin du sin <w cos dv), sin &) sin du cos ft) cos 9 dv, from which it follows that the linear element is Since is a solution of (23), the surface S l is pseudospherical, of curvature 1/ 2 , and the lines of curvature are parametric. To each solution 9 of equations (28) there corresponds a surface S r Darboux * has called this process of finding S 1 the transformation of Bianchi. As the complete integral of equations (28) involves an arbitrary constant, there are an infinity of surfaces >S\, as remarked by Ribaucour. Moreover, if we put (29) *-tan|. these equations are of the Riccati type in <. Hence, by 14, When one transform of Bianchi of a pseudospherical surface is known, the determination of the others requires only quadratures. * Vol. Ill, p. 422. TRANSFORMATION OF BIANCHI 283 From (III, 24) it follows that the differential equation of the curves to which the lines joining corresponding points on S and $! are tangent is (30) cos co smddu sin o> cosddv = 0. Hence, along such a curve, equation (27) reduces to 7/1 da) , d(o i f. d6-\ -- du H -- dv 0. dv du But from (VI, 56) it is seen that this is the Gauss equation of geodesies upon a surface whose first fundamental coefficients have the values (22). Hence : The curves on S to which the lines joining corresponding points on S and S l are tangent are geodesies. The orthogonal trajectories of the curves (30) are defined by (31) coswcostfdtt + sinw sinflcto = 0. In consequence of (28) the left-hand member of this equation is an exact differential. d = a (cos w cos 0du + sin w sin 6dv), the quantity e~& a is an integrating factor of the left-hand member of (30). Conse quently we may define a function rj thus : drj = ae~ / a (cos w sin 6 du sin w cos 6 dv) . In terms of and i\ the linear element of S is expressible in the parabolic form (7), (32) <Zs 2 = d 2 + e^A cfys. Equation (31) defines also the orthogonal trajectories of the curves on Si to which the lines MMi are tangent, and the equation of the latter curves is sin w cos 6 du cos w sin 6 dv = 0. The quantity e* /a is an integrating factor of this equation, and if we put accordingly d = ae /a (sin w cos 6 du cos a; sin dv) , the linear element of Si may be expressed in the parabolic form (33) ) dsf = dp + e-*/ a dp. As the expressions (32) and (33) are of the form of the linear element of a surface of revolution, the finite equations of the geodesies can be found by quadratures. Hence : When a Bianchi transformation is known for a surface, the finite equation of its geodesies can be found by quadratures. This follows also from the preceding theorem and the last one of 117. 284 SURFACES OF CONSTANT CURVATURE 120. Transformation of Backlund. The transformation of Bianchi is only a particular case of a transformation discovered by Backlund,* by means of which from one pseudospherical surface S another S^ of the same curvature, can be found. Moreover, on these two sur faces the lines of curvature correspond, the join of corresponding points is tangent at these points to the surfaces and is of constant length, and the tangent planes at corresponding points meet under constant angle. We refer S to the same moving trihedral as in the preceding case, and let X and 6 denote the length of MM l and the angle which the latter makes with the o>axis. The coordinates of 3/ x are X cos 0, X sin 0, 0, and the projections of a displacement of M l are (34) X sin d0 -f a cos wdu \ sin 6 ( du-\ dv } , \0t> (?M / \cosOdO -f a sin&xi*; + X cos#( du -\ dv ), \dv du I X (cos ft) sin 6 dv sin o> cos 6 du) . If cr denotes the constant angle between the tangent planes tP S and S l at M and Jf t respectively, since these planes are to inter sect in MMv the direction-cosines of the normal to S l are sin & sin 0, sin a cos 0, cos a. Hence must satisfy the condition X sin a- dB a sin or (cos G> sin 6 du sin &> cos dv} , 7 -f X sin <r du H --- dv \dv cu 4- X cos cr (sin &) cos 6du cos &) sin 0dv) = 0. Since this condition must be satisfied for every displacement, it is equivalent to X sin a ( -- [-) = # sin a- cos &) sin 6 X cos a sin &) cos 0, \dw fltf/ /Q /I Q \ X sin <r ( -- h ) = a sin <r sin w cos + X cos a cos &) sin 6. v cu *Om ytor med konstant negativ krokning, Lunds Universitets Arsskrift, Vol. XIX (1883). An English translation of this memoir has been made by Miss Emily Coddington of New York, and privately printed. TRANSFORMATION OF BACKLUKD 285 If these equations be differentiated with respect to v and u respect ively, and the resulting equations be subtracted, we have a 2 sin 2 cr-X 2 =0, from which it follows that X is a constant. Without loss of gen erality we take X = a sin cr. If this value be substituted in the above equations, we have (35) f . (d6 dco\ . a a . sin cr ( = sin cos co cos & cos 6 sin o>, \du dv/ ( 1 ) = cos sin <w + cos er sin cos co, \dv du/ smcr and these equations satisfy the condition of integrability. If they be differentiated with respect to u and v respectively, and the resulting equations be subtracted, it is found that is a solution of (23). In consequence of (35) the expressions (34) reduce to a cos (cos &) cos -f cos a sin &> sin 0) du + a sin (sin &> cos cos a cos &> sin 0) dv, a cos (cos co sin cos cr sin &) cos 0) C?M + <* sin #(sin &) sin 6 + cos cr cos &) cos a sin cr(cosft) sin0o?v sin CD cosOdu), and the linear element of ^ is d** = a 2 (cos 2 <9 dw 2 -f- sin 2 In a manner similar to that of 119 it can be shown that the spherical representation of the parametric curves is orthogonal, and consequently these curves are the lines of curvature on S^ Equations (35) are reducible to the Riccati form by the change of variable (29). Moreover, the general solution of these equations involves two constants, namely cr and the constant of integration. Hence we have the theorem : By the integration of a Riccati equation a double infinity of pseudo- spherical surfaces can be obtained from a given surface of this kind. We refer to this as the transformation of Backlund, and indicate it by B v , thus putting in evidence the constant cr. 286 SURFACES OF CONSTANT CURVATURE 121. Theorem of permutability. Let S l be a transform of S by means of the functions (0 X , o^). Since conversely S is a transform of Sj, and the equations for the latter similar to (35) are reducible to the Riccati type, all the transforms of S l can be found by quad ratures. But even these quadratures can be dispensed with because of the following theorem of permutability due to Bianehi*: If S 1 and S 2 are transforms of S by means of the respective pairs of functions (0 1? a^ and (0 2 , <r 2 ), a function <f> can be found without quadratures which is such that by means of the pairs ((/>, <r 2 ) and (<, o-j) the surfaces S l and S 2 respectively are transformable into a pseudospherical surface S f . By hypothesis </> is a solution of the equations sin o-J -- + * )= sin $ cos l cos cr 2 cos (/> sin 0^ \H */ /p I ^/l \ sin <T 9 ( -^- 4- -^ ) = cos 6 sin 0. + cos cr 9 sin 6 cos ^, 2 and also of the equations (37) pi p sin <7, - 4- - - = sn d> cos 9 cos a. cos 6 sn l * = cos <> sn + coso-Sn) cos The projections of the line If^Tf on the tangents to the lines of curvature of S l and on its normal, where M l and M 1 are correspond ing points on $ l and S , are (38) a sin <r 2 cos </>, a sin <r 2 sin (#>, ; 0. The direction-cosines of the tangents to the lines of curvature of S l with respect to the line JOf 1? the line MQ^ perpendicular to the latter and in the tangent plane at J/, and the normal to S are cos to, cos <r 1 sin o>, sin <T I sin &), sin a), cos cr 1 cos w, sin o-j cos w. From these and (38) it follows that the coordinates of M with respect to MM^ MQ^ and the normal to S are a [sin a- l + sin cr 2 cos (j> a))], a [sin <r 2 cos o-j sin (<f> &))], [sin o- 1 sin <r 2 sin (</> w)]. * Vol. II, p. 418. THEOREM OF PEEMUTABILITY 287 Hence the coordinates of M 1 with respect to the axes of the moving trihedral for S are x = cos 0j sin o- l 4- cos l sin & 2 cos (< &>) (39) sin 0, sin <r 2 cos o^ sin (< <w), = sin l sin cr 1 + sin 6 l sin cr 2 cos ((/> + cos 0j sin <7 2 cos o^ sin ($ <w), = sin cr sm & sin (9 a If $ 2 be transformed by means of a l and the same function c, the coordinates x", y", z" of the resulting surface can be obtained from (39) by interchanging the subscripts 1 and 2. Evidently z and z f are equal. A necessary and sufficient condition that x\ y be equal to #", y" respectively is cos 1 (d x") + sin B^(y y") 0, cos 2 (x r x") + sin z (y r y") = 0. If the above values be substituted in these equations, we obtain [sin a l cos (# 2 0^ sin o- 2 ] cos ((/> co) sin a l cos cr 2 sin (# 2 0^ sin (^> &>) = sin a l sin cr 2 cos (Q n #J, [sin <7 2 cos (^ 2 0j) siu crj cos (</> o>) -f- sin <7 2 cos cr 1 sin(^ 2 ^)sin(^) w) = sin cr z sin a^ cos(# 9 6^). Solving these equations with respect to sin (</> ) and cos (</> &>), we get sin o-j sin <7 2 cos (^ 2 X ) + cos o-j cos o- 2 1 ,. sin o-, sin o-- + (cos <r. coscr 9 l)cos(^ 9 ^.) cos (<f> (0) = - - ^ /i ^ - - - - - " - r sin cr 1 sin <r 2 cos (0 2 c/j) 4- cos ^ cos cr 2 1 These two expressions satisfy the condition that the sum of their squares be unity, and the function (j> satisfies equations (36) and (37). Hence our hypotheses are consistent and the theorem of permutability is demonstrated. We may replace the above equations by 288 SURFACES OF CONSTANT CURVATURE The preceding result may be expressed in the following form : When the transforms of a given pseudospherical surface are known, all the transformations of the former can be effected by algebraic processes and differentiation. Thus, suppose that the complete integral of equations (35) is (41) =/(w, v, <r, c), and that a particular integral is ^i=/( v > *v c i)i corresponding to particular values of the constants, and let ^ denote the transform of S by means of ^ and <7 r All the trans formations of S l are determined by the functions <f> and <r, where Exceptional cases arise when cr has the value <r r For all values of c other than c l formula (42) gives $ = to + WTT, where m is an odd integer. When this is substituted in equations (36) they re duce to (35). In this case S coincides with S. We consider now the remaining case where c has the value c 1? whereupon the right-hand member of (42) is indeterminate. In order to handle this case we consider c in (41) to be a function of o-, reducing to c l for <r = a-^ If we apply the ordinary methods to the function tan L Ism ~ * which becomes indeterminate for a = o- v differentiating numerator and denominator with respect to <r, we have or /6-w\ . /a/ . , tan ^ = sin - 4- c . / = sin ^ V where c is an arbitrary constant. It is necessary to verify that this value of ^ satisfies the equations (36), which is easily done.* * Cf . Bianchi, Vol. II, p. 418. TRANSFORMATION OF LIE 289 122. Transformation of Lie. Another transformation of pseudo- spherical surfaces which, however, is analytical in character was discovered by Lie.* It is immediate when the surface is referred to its asymptotic lines, or to any isothermal-conjugate system of lines. Since the parameters in terms of which the surface is defined in 119 are isothermal-conjugate, the parameters of the asymptotic lines may be given by In terms of these curvilinear coordinates the linear elements of the surface and its spherical representation have the forms ds 2 = a 2 (da 2 + 2 cos 2 &) dad/3 + d/3 2 ), da- 2 = da 2 2 cos 2 ft) dad/3 -f- and equation (23) takes the form dad/3 sin &) cos ft). From the form of this equation it is evident that if &> = <(#, ft) be a solution, so also is co l = (f>(am, ft/m) 9 where m is any constant. Hence from one pseudospherical surface we can obtain an infinity of others by the transformation of Lie. It should be remarked, however, that only the fundamental quantities of the new surfaces are thus given, and that the determination of the coordinates re quires the solution of a Riccati equation which may be different from that for the given surface. Lie has called attention to the fact that every Biicklund trans formation is a combination of transformations of Lie and Bianchi.f In order to prove this we effect the change of parameters (43) upon equations (35) and obtain (44) d , n . 1 + cos a . Q . (0 + co) = - sin (6 &)), da v sin tr d , a x 1 cos er . Q x (6 (w) = - : - sin (9 + <). d3 V sin <r *Archivfor Mathematik og Naturvidenskab, Vol. IV (1879), p. 150. t Cf. Bianchi, Vol. II, p. 434; Darboux, Vol. Ill, p. 432. 290 . SUEFACES OF CONSTANT CUEVATUEE In particular, for a transformation of Bianchi we have (B 4- o)) = sin (B o>), (6 &>) = sin (6 + o>). ccc dp Suppose that we have a pair of functions 6 and &> satisfying these equations, and that we effect upon them the Lie transforma tion for which m has the value (1 + cos cr)/sin a-. This gives 1 4- cos a 1 cos cr _ a , sin or sin cr /1 + \ s sin cr sm cr As these functions satisfy (44), they determine a transformation B a . But O l may be obtained from o^ by effecting upon the latter an inverse Lie transformation, denoted by Z" 1 , upon this result a Bianchi transformation, B n/2 , and then a direct Lie transformation, Z a . Hence we may write symbolically which may be expressed thus : A Backlund transformation B ff is the transform of a Bianchi transformation ly means of a Lie transformation L a * EXAMPLES 1. The asymptotic lines on a pseudospherical surface are curves of constant torsion. 2. Every surface whose asymptotic lines are of the same length as their spherical images is a pseudospherical surface of curvature 1. 3. Show that on the pseudosphere, defined by (9), the curves = 0, where 6 is a constant, are geodesies, and find the radius of curvature of these curves. 4. When the linear element of a pseudospherical surface is in the parabolic form (iii) of (7), the surface defined by dx dy dz x = x a y y a z z a du cu du is pseudospherical (cf . 76) ; it is a Bianchi transform of the given surface. * Spherical surfaces admit of transformations similar to those of Lie and Backlund. The latter are imaginary, but such combinations of them can be made that the resulting surface is real. For a complete discussion of these the reader is referred to chap. v. of the Lezioni of Bianchi. TF-SURFACES 291 5. The helicoids X = U COS V. y = u sin u, z= f */ Idu + hv, J \ a k~u z u 2 where a, A, fc are constants, are spherical surfaces. 6. The helicoid whose meridian curve is the tractrix is called the surface of Dini. Find its equations when sin <r denotes the helicoidal parameter and cos <r the con stant length of the segment of the tangent between the curve and its axis. Show that the surface is pseudospherical. 7. The curves tangent to the joins of corresponding points on a pseudospherical surface and on a Backlund transform are geodesies only when <r = ir/2. 8. Let S be a pseudospherical surface and Si a Bianchi transform by means of a function d ( 119). Show that X{ cosw(cos0X 1 -f sin0JT 2 ) sinwJT, X % = sin w (cos<? Xi -f sin0JT 2 ) + coswJT, X = where .Xi, X 2 , X are direction-cosines, with respect to the x-axis, of the tangents to the lines of curvature on S and of the normal to S, and JT{, X %, X are the similar functions for Si. 123. W-surfaces. Fundamental quantities. Minimal surfaces and surfaces of constant curvature possess, in common with a great many other surfaces, the property that each of the prin cipal radii is a function of the other. Surfaces of this kind were first studied in detail by Weingarten, * and, in consequence, are called Weingarten surfaces, or simply W-surfaces. Since the prin cipal radii of surfaces of revolution and of the general helicoids are functions of a single parameter ( 46, 62), these are TF-surfaces. We shall find other surfaces of this kind, but now we consider the properties which are common to TF-surfaces. When a surface S is referred to its lines of curvature, the Codazzi equations may be given the form (45) glogV^ = 1 dp^ dv P 2 Pi dv d If a relation exists between p l and /> 2 , as the integration of equations (45) is reducible to quadratures, thus : r dpi r </p 2 =Ue J *-*, V ^ = Ve J Pl ~ P2 , Crelle, Vol. LXII (1863), pp. 160-173. 292 JF-SUKFACES where U and V are functions of u and v respectively. Without changing the parametric lines the parameters can be so chosen that the above expressions reduce to r dp, f_dpj / A T \ re I a o r^ I Thus and ^ are expressible as functions of p l or /3 2 , and conse quently they are functions of one another. This relation becomes more clear when we introduce an additional parameter K defined by / <*pi (48) * = * *-* By the elimination of p 2 from this equation and (46) we have a relation of the form j\( \ When this value is substituted in (48) we obtain where the accent indicates differentiation with respect to K. From (47) it follows that -, -. V^=, ^=T, K </> When these values are substituted in the Gauss equation for the sphere (V, 24), the latter becomes 1/* M , jL/* aY _1. = du \ </>" du) dv \tc* dv) K<f> This equation places a restriction upon the forms of K and <(), but it is the only restriction, for the Codazzi equations (45) are satisfied. Hence we have the theorem of Weingarten : * When one has an orthogonal system on the unit sphere for which the linear element is reducible to the form there exists a W-surface whose lines of curvature are represented by this system and whose principal radii are expressed by (50) ft =*(*), P 2 = *(*)- * (*) Z.c., p. 163. FUNDAMENTAL QUANTITIES 293 If the coordinates of the sphere, namely X, Y, Z, are known functions of u and v, the determination of the JF-surface with this representation reduces to quadratures. For, from the formulas of Kodrigues (IV, 32) we have r dX , dX , x = Pi du + p 2 dv, J cu dv C dY , dY , y = / p i du + /? 2 ^r~ dv, r cz 7 a^ , = ~ / ft ~^~ ^ + P* v" dv - J du dv The right-hand members of these equations are exact differentials, since the Codazzi equations (45) have been satisfied. If A", F, Z are not known, their determination requires the solution of a Riccati equation. The relation between the radii of the form (46) is obtained by eliminating K from equations (50). We find readily that the fundamental quantities for the sur face have the values (51) And from (48), (50), and (51) we obtain t < pi _ r Pi (52) Ve = p^ "- ft , vG = p,e J f - p >. Consider the quadratic form (53) [(EJJ -FD) du 1 + (El)"- GD) dudv + (FD"- GD ) dv*], H . which when equated to zero defines the lines of curvature. When these lines are parametric, this quadratic form is reducible by means of (IV, 74) to But in consequence of (47) this is further reducible for JF-surfaces to dudv. Since the curvature of this latter form is zero, the curvature of (53) also is zero, and consequently ( 135) the form (53) is redu cible by quadratures to dudv. Hence we have the theorem of Lie : The lines of curvature of a W-surface can be found by quadratures. 294 JF-SURFACES 124. Evolute of a W-surface. The evolute of a JF-surface pos sesses several properties which are characteristic. Referring to the results of 75, we see that by means of (52) the linear elements of the sheets of the evolute of a JF-surface are reducible to the form or, in terms of K, (55) From these results and the remarks of 46 we obtain at once the following theorem of Weingarten : Each surface of center of a W-surface is applicable to a surface of revolution whose meridian curve is determined by the relation between the radii of the given surface. We have also the converse theorem, likewise due to Weingarten : If a surface S l be applicable to a surface of revolution, the tan gents to the geodesies on S^ corresponding to the meridians of the surface of revolution are normal to a family of parallel W-surfaces; if S l be deformed in any manner whatever, the relation between the radii of these W-surfaces is unaltered. In proving this theorem we apply the results of 76. If the linear element of S l be r t i i*** ? the principal radii of S are given by /tM . V (56) p^u, ft--^7- Since both are functions of a single parameter, a relation exists between them which depends upon U alone, and consequently is unaltered in the deformation of S r From (V, 99) the projections upon the moving trihedral for 8^ of a displacement of a point on the complementary surface 2 are U (___), 0, (qdu-, ai ~ /ir EVOLUTE OF A JF-SUBFACE 295 In consequence of formulas (V, 48, 75) the expression U(q du + q t dv) is an exact differential, which will be denoted by dw. Hence the linear element of 2 is (57) dl = l from which it follows that S z also is applicable to a surface of revolution.* The last theorem of 75 may be stated thus : A necessary and sufficient condition that the asymptotic lines on the surfaces of center S^ S 2 of a surface S correspond is that S be a W-surface ; in this case to every conjugate system on S l or S 2 there corresponds a conjugate system on the other. From (V, 98, 98 ) it follows that when S is a TF-surface, and only in this case, we have (58) . . . ^-E^b- Hence at corresponding points the curvature is of the same kind. An exceptional form of equation (46) is afforded by the case where one or both of the principal radii is constant. For the plane both radii are infinite ; for a circular cylinder one is infinite and the other has a finite constant value. The sphere is the only surface with both radii finite and constant. For, if p r and p 2 are different constants, from (45) it follows that and ^ are functions of u and v respec tively, which is true only of developable surfaces. When one of the radii is infinite, the surface is developable. There remains the case where one has a finite constant value ; then S is a canal surface ( 29). In considering the last case we take then, from (48), we have and the linear element of the sphere is do* = ~ + dv\ K Conversely, when the linear element of the sphere is reducible to this form, the curves on the sphere represent the lines of curvature on an infinity of parallel canal surfaces. * Cf. Darboux, Vol. Ill, p. 329. 296 TF-SURFACES 125. Surfaces of constant mean curvature. For surfaces of con stant total curvature the relation (46) may be written where c denotes a constant. When this value is substituted in (48) we have, by integration, (59) P so that the linear element of the sphere is (60) Conversely, when we have an orthogonal system on the sphere for which the linear element is reducible to the form (60), it serves for the representation of the lines of curvature of a surface of constant curvature, and of an infinity of parallel surfaces. When c is positive, two of these parallel surfaces have constant mean curvature, as follows from the theorem of Bonnet (73). In fact, the radii of these surfaces tifcT (61) p l =^/ K *+cy/~c J p 9 = -=L== V~ C . v K ~r~ c If we put (62) c = a 2 , ic = a csch &>, and replace u by au, the linear element (60) becomes da-- = sinlr co du 2 + cosh 2 o> dv 2 . r In like manner, if we replace u by iau, v by iv, and take (63) c = a 2 , K = ai sech CD, the linear element of the sphere is da- 2 = cosh 2 w du 2 + sinh 2 &) dv*. For the values (62) we have, from (61), and the linear elements of the corresponding surfaces are (65) <f* a =ffV aw (dw a SURFACES OF CONSTANT MEAN CURVATURE 297 Moreover, for the values (63) the radii have the values cosh co sinh co but the linear elements are the same (65). In each case the mean curvature is l/. We state these results in the following form: The lines of curvature upon a surface of constant mean curvature form an isothermic system, the parameters of which can be chosen so that the linear element has one of the forms (65), where co is a solution of the equation (67) l ^ 4- ^ 4- sinh co cosh co = 0. du 2 dv 2 Conversely, each solution of this equation determines two pair* of applicable surfaces of constant mean curvature l/a, whose lines of curvature correspond, and for which the radii p^ p 2 of one surface are equal to the radii of p 2 , p^ of the applicable surface. It can be shown that if co = $(u, v) is a solution of equation (67), so also is (68) co 1 = cf)(u cos cr v sin cr, u sin <r + v cos cr), where cr is any constant whatever. Hence there exists for spherical surfaces a transformation analogous to the Lie transformation of pseudospherical surfaces. This transformation can be given a geo metrical interpretation if it is considered in connection with the sur faces of constant mean curvature parallel to the spherical surfaces. Let S l denote the surface with the linear element (69) ds 2 = aV w > (du 2 + dv 2 ). If we put (70) u v =u cos cr v sin cr, v 1 = u sin cr + v cos cr, the solution (68) becomes co l = cf)(u l , v^), and (69) reduces to Hence if we make a point (u, v) on S with the linear element (65), in which the positive sign is taken, correspond to the point (u v vj on 8^ the surfaces are applicable, and to the lines of curvature u = const., v = const, on S correspond on S l the curves u cos cr v sin a = const., u sin cr -f- v cos cr = const. 298 ^-SURFACES But the latter cut the lines of curvature u = const., v = const, on S l under the angle a-. Moreover, the corresponding principal radii of S and S l are equal at corresponding points. Hence we have tha following theorem of Bonnet : * A surface of constant mean curvature admits an infinity of appli cable surfaces of the same kind with preservation of the principal radii at corresponding points, and the lines of curvature on one surface correspond to lines on the other which cut the lires of curvature under constant angle. Weingarten has considered the IF-surfaces whose lines of curvature are represented on the sphere by geodesic ellipses and hyperbolas. In this case the linear element of the sphere is reducible to the form ( 90) do* = sm *2 C S *2 Comparing this with (49), we have .to ., ft) /c = sin-> < =cos- . from which it follows that to -f- sin ft) 4 Hence &) -f sin ft) ft) sin and the relation between the radii is found, by the elimination of w, to be (72) 2(^-^)=sin2(^+/) 2 ).t * Memoire sur la theorie des surfaces applicables sur une surface donnce, Journal de VEcole Poly technique, Cahier 42 (1867) , pp. 72 et seq. In this memoir Bonnet solves com pletely the problem of finding applicable surfaces with corresponding principal radii equal. When a surface possesses an infinity of applicable surfaces of this kind, its lines of curv ature form an isothermal system. tDarboux (Vol. Ill, p. 373) proves that these surfaces may be generated as follows: Let C and Ci be two curves of constant torsion, differing only in sign. The locus of the mid-points M of the join of any points P and PI of these curves is a surface of translation. If a line be drawn through M parallel to the intersection of the osculating planes of C and Ci at P and Pi, this line is normal to a IP-surface of the above type for all positions of M. RULED JF-SURFACES 299 126. Ruled W-surfaces. We conclude the present study of Tr-surfaces with the solution of the problem : To determine the W-surfaces which are ruled. This problem was proposed and solved simultaneously by Beltrami* and Dini.f We follow the method of the latter. In 106, 107 we found that when the linear element of a ruled surface is in the form ds 2 = du 2 + [(u - a) 2 + /3 2 ] dv\ the expressions for the total and mean curvatures are /3 2 ~ = where r is a function of v at most, and /=(tt~a) a -h^. In order that a relation exist between the principal radii it is necessary and sufficient that the equation 1* a* *jr.-l*:-o du dv dv du be satisfied identically. If the above values be substituted, the resulting equation reduces to 2u a d rr 2 +/3 ! u-a a l \ As this is an identical equation, it is true when u = a, in which case it reduces to /3 =0. Hence /3 is a constant and the above equation becomes r (u of + r ft 2 + /3a" = 0. Since this equation must be true independently of the value of w, both r and a" are zero. Therefore we have (73) a=cv + d, P = e, r = k, where <?, d, e, k are constants. The linear element is ds 2 = du 2 + [(t* - cv - d) 2 + e 2 ] dv 2 . * Annali, Vol. VII (1865), pp. 13&-150. t Annali, Vol. VII (1865), pp. 205-210. 300 SURFACES WITH PLANE LINES OF CURVATURE In order to interpret this result we calculate the expression for the tangent of the angle which the generators v = const, make with the line of striction u cv d = 0. From (III, 24) we have tan d = - ; c consequently the angle is constant. Conversely, if 6 and the param eter of distribution j3 be constant, a has the form (73). Hence we have the theorem : A necessary and sufficient condition that a ruled surface be a W-surface is that the parameter of distribution be constant and that the generators be inclined at a constant angle to the line of stric tion, which consequently is a geodesic. EXAMPLES 1. Show that the helicoids are ^surfaces. 2. Find the form of equation (49), when the surface is minimal, and show that each conformal representation of the sphere upon the plane determines a minimal surface. 3. Show that the tangents to the curves v = const, on a spherical surface with the linear element (i) of (6) are normal to a TT-surface for which P-2 - PI = COt - 4. The helicoids are the only >F-surfaces which are such that the curves Pi = const, meet the lines of curvature under constant angle (cf. Ex. 23, p. 188). 5. The asymptotic lines on the surfaces of center of a surface for which Pl + Pz const, correspond to the minimal lines on the spherical representation of the surface ; and, when />i p 2 = const., to a rectangular system on the sphere. 127. Spherical representation of surfaces with plane lines of curvature in both systems. Surfaces whose lines of curvature in one or both systems are plane curves have been an object of study by many geometers. Since the tangents to a line of curvature and to its spherical representation at corresponding points are parallel, a plane line of curvature is represented on the sphere by a plane curve, that is, a circle ; and conversely, a line of curvature is plane when its spherical representation is a circle. SPHERICAL REPRESENTATION 301 We consider first the determination of surfaces with plane lines of curvature in both systems from the point of view of their spherical representation.* To this end we must find orthog onal systems of circles on the sphere. If two circles cut one another orthogonally, the plane of each must pass through the pole of the plane of the other. Hence the planes of the circles of one system pass through a point in the plane of each circle of the second system, and consequently the planes of each family form a pencil, the two axes being polar reciprocal with respect to the sphere.f We consider separately the two cases : I, when one axis is tan gent to the sphere, and therefore the other is tangent at the same point and perpendicular to it ; II, when neither is tangent. CASE I. We take the center of the unit sphere for origin 0, the x- and ?/-axes parallel to the axes of the pencils, and let the coor dinates of the point of contact be (0, 0, 1). The equations of the- pencils of planes may be put in the form (74) x + u(z 1)=0, y + v(z 1) = 0, where u and v are the parameters of the respective families. If these equations be solved simultaneously with the equation of the sphere, and, as usual, X, I 7 , Z denote coordinates of the latter, we have v ^ v r7 _u?- - ~ Now the linear element of the sphere is (T6) ^= JtXl?- CASE II. As in the preceding case, we take for the z-axis the common perpendicular to the axes of the pencils, and for the x- and ?/-axes we take lines through parallel to the axes of the pencils. The coordinates of the points of meeting of the latter with the z-axis are of the form (0, 0, a), (0, 0, I/a). The equa tions of the two pencils of planes could be written in forms * Bianchi, Vol. II, p. 256; Darboux, Vol. I, p. 128, and Vol. IV, p. 180. t Bonnet, Journal de I Ecole Poly technique, Vol. XX (1853), pp. 136, 137. 302 SURFACES WITH PLANE LINES OF CURVATURE similar to (74), but the expressions for X, Y, Z will be found to be of a more suitable form if the equations of the families of planes be written tanw atanhv Proceeding as in Case I, we find Vl a 2 sin u (77) Y=- Z = cosh v + a cos u 1 a 2 sinh v cosh v -f a cos u cos u -\- a cosh v cosh v -f- a cos w and the linear element is (78) (cosh v + a cos w) From the preceding discussion we have tacitly excluded the sys tem of meridians and parallels. As before, the planes of the two families of circles form pencils, but now the axis of one pencil passes through the center of the sphere and the other is at infinity. Hence this case corresponds to the value zero for a in Case II. In fact, if we put a = in (77), the resulting equations define a sphere referred to a system of meridians and parallels, namely _ Q sinw sinhv cosw ( I V ) JL - - - t JL -- - - > Z/ - - - cosh v cosh v cosh v Since the planes of the lines of curvature on a surface are parallel to the planes of their spherical images, the curves v const, on a surface with the representation (79) lie in parallel planes, and the planes of the curves u = const, envelop a cylinder. These surfaces are called the molding surfaces.* We shall consider them later. 128. Surfaces with plane lines of curvature in both systems. By a suitable choice of coordinate axes and parameters the expressions for the direction-cosines of the normal to a surface with plane lines of curvature in both systems can be given one * These surfaces were first studied by Monge, Application de L Analyse a la Geomt- trie, 17. Paris, 1849. IN BOTH SYSTEMS 303 of the forms (75) or (77). For the complete determination of all surfaces of this kind it remains then for us to find the expres sion for the other tangential coordinate W, that is, the distance from the origin to the tangent plane. The linear element of the sphere in both cases is of the form 7 2 du 2 + dv 2 d(T = - - - -> where \ is such that (80) -^- = 0. cudv From (VI, 39) we see that the equation satisfied by W is gfl g log X d6 _ _ Q cucv dv du du dv In consequence of (80), if we change the unknown function in accordance with B l =\0 > the equation in 6 l is of the form (80). Hence the most general value *for W is where U and V are arbitrary functions of u and v respectively. Hence any surface with plane lines of curvature in both systems is the envelope of a family of planes whose equation is of the form (81) 2 ux + 2 vy + (u*+ v 2 -l)z = 2 (U+V), or (82) Vl a" sin ux Vl a 2 sinh vy + (cos u + a cosh v) z = (U+ F)Vl-a 2 . The expressions for the Cartesian coordinates of these surfaces can be found without quadrature by the methods of 67. Thus, for the surface envelope of (81) we have to solve for x, y, z equa tion (81) and its derivatives with respect to u and v. The latter are (83) x + uz = Z7 , y + vz = V\ where the accents indicate differentiation. We shall not carry out this solution, but remark that as each of these equations contains a single parameter they define the planes of the lines of curvature. 304 SURFACES WITH PLANE LINES OF CURVATURE From the form of (83) it is seen that these planes in each sys tem envelop a cylinder, and that the axes of these two cylin ders are perpendicular. This fact was remarked by Darboux, who also observed that equation (81) defines the radical plane of the two spheres These are the equations of two one-parameter families of spheres, whose centers lie on the focal parabolas -U, 2/1=0, and whose radii are determined by the arbitrary functions U and V. The characteristics of each famity are defined by its equation and the corresponding equation of the pair (83). Consequently the orig inal surface is the locus of the point of intersection of the planes of these characteristics and the radical planes of the spheres. Similar results follow for the equation (82), which defines the radical planes of two families of spheres whose centers are on the focal ellipse and hyperbola (86) a; 2 =0, 2/ 2 = When in particular a = 0, these curves of center are a circle and its axis. r i From the foregoing results it follows that these surfaces may be generated by the following geometrical method due to Darboux : * Every surface with plane lines of curvature in two systems can be obtained from two singly infinite families of spheres whose centers lie on focal conies and whose radii vary according to an arbitrary law. The surface is the envelope of the radical plane of two spheres S and 2, belonging to two different families. If one associate with S and 2 two infinitely near spheres S f and 2 , the radical center of these four spheres describes the surface ; and the radical planes of S and S and of 2 and 2 are the planes of the lines of curvature. * Vol. i, p. 132. SURFACES OF MONGE 305 129. Surfaces with plane lines of curvature in one system. Surfaces of Monge. When the lines of curvature in one system are plane, the curves on the sphere are a family of circles and their orthogonal trajectories ; and conversely. Every system of this kind may be obtained from a system of circles and their orthogonal trajectories in a plane by a stereographic projection. The determination of such a system in the plane reduces to the integration of a Riccati equation (Ex. 11, p. 50). Since the circles are curves of constant geodesic curvature we have, in consequence of the first theorem of 84, the theorem : The determination of all the surfaces with plane lines of curva ture in one system requires the solution of a Riccati equation and quadratures. We shall discuss at length several kinds of surfaces with plane lines of curvature in one system, and begin with the case where these curves are geodesies. They are consequently normal sections of the surface. Their planes envelop a developable surface, called the director-developable, and the lines of curvature in the other sys tem are the orthogonal trajectories of these planes. Conversely, the locus of any simple infinity of the orthogonal trajectories of a one-parameter system of planes is a surface of the kind sought. For, the planes cut the surface orthogonally, and consequently they are lines of curvature and geodesies ( 59). Since these planes are the osculating planes of the edge of regression of the developable, the orthogonal trajectories can be found by quadratures ( 17). Suppose that we have such a surface, and that C denotes one of the orthogonal trajectories of the family of plane lines of curvature. Let the coordinates of C be expressed in terms of the arc of the curve from a point of it, which will be denoted by v . As the plane of each plane line of curvature F is normal to C at its point of meeting with the latter, the coordinates of a point P of F with reference to the moving trihedral of C are 0, 77, f. Since P describes an orthogonal trajectory of the planes, we must have (I, 82) dv 306 SURFACES WITH PLANE LINES OF CURVATURE where r denotes the radius of torsion of C. If we change the parameter of C in accordance with the equation the above equations become The general integral of these equations is (88) ?; = U^ cos v l U 2 sin v^ f = U^ sin v l -f ?7 2 cos v lt where C^ and ?7 2 are functions of the parameter u of points of F. When v = we have v l = 0, and so the curve F in the plane through the point v = of C has the equations 77 = U^ ?= Z7 2 . Hence the character of the functions U^ and U 2 is determined by the form of the curve ; and conversely, the functions U } and U 2 determine the character of the curve. By definition (87) the function v t measures the angle swept out in the plane normal to C by the binormal of the latter, as this plane moves from v = to any other point. Hence equations (88) define the same curve, in this moving plane, for each value of v^ but it is defined with respect to axes which have rotated through the angle v r Hence we have the theorem : Any surface whose lines of curvature in one system are geodesies can be generated by a plane curve whose plane rolls, without slipping, over a developable surface. These surfaces are called the surfaces of Monge, by whom they were first studied. He proposed the problem of finding a surface with one sheet of the e volute a developable. It is evident that the above surfaces satisfy this condition. Moreover, they furnish the only solution. For, the tangents to a developable along an ele ment lie in the plane tangent along this element, and if these tangents are normals to a surface, the latter is cut normally by this plane, and consequently the curve of intersection is a line of curvature. In particular, a molding surface ( 127) is a surface of Monge with a cylindrical director-developable. Since every curve in the moving plane of the lines of curva ture generates a surface of Monge, a straight line in this plane MOLDING SURFACES 307 generates a developable surface of Monge. For, all the normals to the surface along a generator lie in a plane ( 25). Hence: A necessary and sufficient condition that a curve F in a plane normal to a curve C at a point Q generate a surface of Monge as the plane moves, remaining normal to the curve, is that the . line joining a point of T to Q generate a developable. 130. Molding surfaces. When the orthogonal trajectory C is a plane curve, the planes of the curves F are perpendicular to the plane of C, and consequently the director-developable is a cylinder whose right section is the plane evolute of C. The surface is a molding surface ( 127), and all the lines of curvature of the sec ond system are plane curves, involutes of the right section of the cylinder. Hence a molding surface may be generated by a plane curve whose plane rolls without slipping over a cylinder. We shall apply the preceding formulas to this particular case. Since 1/r is equal to zero, it follows from (88) that ?; and are functions of u alone. If u be taken as a measure of the arc of the curve F, we have, in all generality, ?; = U, f = I Vl U 2 du, where the function U determines the form of F. If we take the plane of the curve C for 2 = 0, and X Q , y Q denote the coordinates of a point of C, the equations of the surface may be written x = x + U cos v, y = 2/o + u sin v i 2 = / Vl U 2 du, where v denotes the angle which the principal normal to C makes with the a>axis. Since ^ x ^ ( , ^ = sin v = cos v, if V denote the radius of curvature of C, then ds Q = V dv, and the equations of the surface can be put in the following form, given by Darboux * : ( r v -f- I Fsin v dv, J (89) = U sin v I V cos v dv, * Vol. I, p. 105. 308 SURFACES WITH PLANE LINES OF CURVATURE The equations of the right section of the cylinder are x = X Q + V cos v = I V cos v dv, y = y Q -f- V sin v = I V sin v dv. In passing, we remark that surfaces of revolution are molding sur faces, whose director-cylinder is a line ; this corresponds to the case V 0. EXAMPLES 1. When the spherical representation of the lines of curvature of a surface is isothermal and the curves in one family on the sphere are circles, the curves in the other family also are circles. 2. If the lines of curvature in one system on a minimal surface are plane, those in the other system also are plane. 3. Show that the surface x au _|_ sin u cosh v, y = v + a cos u sinh v, z V 1 a 2 cos u cosh v, is minimal and that its lines of curvature are plane. Find the spherical representa tion of these curves and determine the form of the curves. 4. Show that the surface of Ex. 3 and the Enneper surface (Ex. 18, p. 209) are the only minimal surfaces with plane lines of curvature. 5. When the lines of curvature in one system lie in parallel planes, the surface is of the molding type. 6. A necessary and sufficient condition that the lines of curvature in one system on a surface be represented on the unit sphere by great circles is that it be a sur face of Monge. 7. Derive the expressions for the point coordinates of a molding surface by the method of 67. 131. Surfaces of Joachimsthal. Another interesting class of surfaces with plane lines of curvature in one system are those for which all the planes pass through a straight line. Let one of these lines of curvature be denoted by F, and one of the other system by C. The developable enveloping the surface along the latter has for its elements the tangents to the curves F at their points of intersection with 0. Since these elements lie in the planes of the curves F, the developable is a cone with its vertex on the line Z>, through which all these planes pass. This cone is tangent to the surface along (7, and its elements are orthogonal to the latter. Con sequently C is the intersection of the surface and a sphere with SURFACES OF JOACHIMSTHAL 309 center at the vertex of the cone which cuts the surface orthogo nally. Hence we have the following result, due to Joachimsthal * : When the lines of curvature in one system lie in planes passing through a line D, the lines of curvature in the second system lie on spheres whose centers are on D and which cut the surface orthogonally. Such surfaces are called surfaces of Joachimsthal. Each of the curves of the first system is an orthogonal trajectory of the circles in which the spheres are cut by its plane. Therefore, in order to derive the equations of such a surface, we consider first the orthog onal trajectories of a family of circles whose centers are on a line. If the latter be taken for the ?;-axis, the circles are defined by f = r sin 0, 77 = r cos 6 + u, where r denotes the radius, 6 the angle which the latter makes with the ?;-axis, and u the distance of the center from the origin. Now r is a function of u, and 6 is independent of u. In order that these same equations may define an orthogonal trajectory of the circles, 6 must be such a function of u that cos 0^- sin 0^ = 0, du cu or r f^_ sin = . du By integration we have (90) tan| = F</ r , where V denotes the constant of integration. Since each section of a surface of Joachimsthal by a plane through its axis is an orthogonal trajectory of a family of circles whose centers are on this axis, the equations of the most general surface of this kind are of the form x = r sin 6 cos v, y = r sin 6 sin v, z u -f r cos #, where v denotes the angle which the plane through a point and the axis makes with the plane y 0, and 6 is given by (90), in which now V is a function of v. * Crelle, Vol. LIV (1857), pp. 181-192. 310 SURFACES WITH PLANE LINES OF CURVATURE When V is constant is a function of u alone, and the surface is one of revolution. For other forms of Vihe geometrical genera tion of the surfaces is given by the theorem : Given the orthogonal trajectories of a family of circles whose cen ters lie on a right line D ; if they be rotated about D through dif ferent angles, according to a given law, the locus of the curves is a surface of Joachimsthal. 132. Surfaces with circular lines of curvature. We consider next surfaces whose lines of curvature in one system are circles. Let o- denote the constant angle between the plane of the circle C and the tangent planes to the surface along C (cf. 59), p the radius of normal curvature in the direction of C, and r the radius of the latter. Now equation (IV, 17) may be written (91) r = p sin a. As an immediate consequence we have the theorem : A necessary and sufficient condition that a plane line of curvature be a circle is that the normal curvature of the surface in its direction be the same at all of its points. Since the normals to the surface along C are inclined to its plane under constant angle, they form a right circular cone whose vertex is on the axis of C. Moreover, the cone cuts the surface at right angles, and consequently the sphere of radius p and center at the vertex of the cone is tangent to the surface along C. Hence the surface is the envelope of a family of spheres pf variable or con stant radius, whose centers lie on a curve. Conversely, we have seen in 29 that the characteristics of the family of spheres where x, y, z are the coordinates of a curve expressed in terms of its arc, and 11 is a function of the same parameter, are circles of radius (92) r whose axes are tangent to the curve of centers and whose centers have the coordinates (93) x l = x - aRR , y CIRCULAR LIKES OF CURVATURE 311 where a, ft, 7 are the direction-cosines of the axis, and the accent indicates differentiation. The normals to the envelope along a characteristic form a cone, and consequently these circles are lines of curvature upon it. Hence : A necessary and sufficient condition that the lines of curvature in one family be circles is that the surface be the envelope of a single infinity of spheres, the locus of whose centers is a curve, the radii being determined by an arbitrary law. From equations (91), (92) it follows that R cos a. Hence the circles are geodesies only when R is constant, that is, for canal surfaces ( 29). In this case, as is seen from (92), all the circles are equal. The circles are likewise of equal radius a when where s is the arc of the curve of centers and c is a constant of integration. Now equations (93) become ^ = x (s + c) a. y l =^y (s + c}IB, z l =z (s + c)y, which are the equations also of an involute of the curve of centers ( 21). This result may be stated thus* : If a string be unwound from a curve in such a way that its moving extremity M generates an involute of the curve, and if at M a circle be constructed whose center is M and whose plane is normal to the string, then as the string is unwound this circle generates a surface with a family of equal circles for lines of curvature. The locus of the centers of the spheres enveloped by a surface is evidently one sheet of the evolute of the surface, and the radius of the sphere is the radius of normal curvature in the direction of the circle. Consequently this radius is a function of the parameter of the spheres. Conversely, from 75, we have that when 2 is a curve H 2 = 0, and consequently Cf. Bianchi, Vol. II, p. 272. 312 SURFACES WITH PLANE LINES OF CURVATURE Excluding the case of the sphere, we have that p. 2 is a function of u alone. From the formulas of Rodrigues (IV, 32), 2x _ dX dy _ dY dz _ a d^~~~ Pz ~^ 1 fo~~ p2 ~fo Tv~~ pz ^v we have, by integration, Hence the points of the surface lie on the spheres (x - U, ) a + (y - tg 2 + (z - P,) = ft , and the spheres are tangent to the surface. Since the normals to a surface along a circular line of curvature form a cone of revolution, the second sheet of the e volute is the envelope of a family of such cones. The characteristics of such a family are conies. Hence we have the theorem : A necessary and sufficient condition that one sheet of the evolute of a surface be a curve is that the surface be the envelope of a single infinity of spheres ; the second focal sheet is the locus of a family of conies. 133. Cyclides of Dupin. From the preceding theorem it results that if also the second sheet of the evolute of a surface be a curve, it is a conic, and then the first sheet also is a conic. Moreover, these conies are so placed that the cone formed by joining any point on one conic to all the points of the other is a cone of revolution. A pair of focal conies is characterized by this property. And so we have the theorem : A necessary and sufficient condition that the lines of curvature in both families be circles is that the sheets of the evolute be a pair of focal conies.* These surfaces are called the cy elides of Dupin. They are the envelopes of two one-parameter families of spheres, and all such envelopes are cyclides of Dupin. A sphere of one family touches each sphere of the other family. Consequently the spheres of which the cyclide is the envelope are tangent to three spheres. We shall prove the converse theorem of Dupin f : The envelope of a family of spheres tangent to three fixed spheres is a cyclide. * Cf. Ex. 19, p. 188. t Applications de geomttrie et de mechanique, pp. 200-210. Paris, 1822. CYCLIDES OF DUPIN 313 The plane determined by the centers of the three spheres cuts the latter in three circles. If any point on the circumference (7, orthogonal to these circles, be taken for the pole of a transforma tion by reciprocal radii (cf. 80), C is transformed into a straight line L. Since angles are preserved in this transformation, the three fixed spheres are changed into three spheres whose centers are on L. Evidently the envelope of a family of spheres tangent to these three spheres is a tore with L as axis. Hence the given envelope is trans formed into a tore. However, the latter surface is the envelope of a second family of spheres whose centers lie on L. Therefore, if the above transformation be reversed, we have a second family of spheres tangent to the envelope, and so the latter is a cyclide of Dupin. We shall now find the equations of these surfaces. Let (x^ y^ zj and (# 2 , y 2 , z 2 ) denote the coordinates of the points on the focal conies which are the curves of centers of the spheres, and jR 1? E 2 the radii of the spheres. The condition of tangency is (94) ( Xi - x We consider first the case where the evolute curves are the focal parabolas defined by (85). Now equation (94) reduces to Since 2i l and R z are functions of u and v respectively, this equation is equivalent to where a is an arbitrary constant whose variation gives parallel surfaces. By the method of 132 we find that the coordinates (f, 77, f) of the centers of the circular lines of curvature u const, and the radius p are 9-0, 314 SURFACES WITH PLANE LINES OF CURVATURE Hence if P be a point on the circle and 6 denote the angle which the radius to P makes with the positive direction of the normal to the parabola (85), the coordinates of P are x = f H ^ cos 0, ^ = p sin 0, 2 = ? -- cos 9. v 1 + u 2 Vl + if This surface is algebraic and of the third order. If the evolute curves are the focal ellipse and hyperbola (86), we have (96) R l = - (a cos u + *), A 2 = - (cosh v /c), -! - where /c is an arbitrary constant whose variation gives parallel surfaces. This cyclide of Dupin is of the fourth degree. When in particular the constanta is zero, the surface is the ordinary tore, or anchor ring.* 134. Surfaces with spherical lines of curvature in one system. Surfaces with circular lines of curvature in one system belong evi dently to the general class of surfaces with spherical lines of curva ture in one system. We consider now surfaces of the latter kind. Let S be such a surface referred to its lines of curvature, and in particular let the lines v = const, be spherical. The coordinates of the centers of the spheres as well as their radii are functions of v alone. They will be denoted by (V^ F 2 , F 3 ) and It. By Joachims- thal s theorem ( 59) each sphere cuts the surface under the same angle at all its points. Hence for the family of spheres the expres sion for the angle is a function of v alone ; AVC call it V. Since the direction-cosines of the tangent to a curve u = const, are dX 3Y 1 dZ when the linear element of the spherical representation is written do- 2 = (odu 2 -}- dv\ the coordinates of S are of the form /07\ (97) , R sin VdX , = VA --- =- +XR cos F, Tr _ T _ y = F 2 + + YR cos F, 7 sin F dZ * For other geomotrical constructions of the cyclides of Dupin the reader is referred to the article in the Encyklopadie der Math. Wissenschaflen, Vol. Ill, 3, p. 290. SPHERICAL LINES OF CURVATURE 315 By hypothesis A, l r , Z are the direction-cosines of the normal to S ; consequently we must have YA-- = O, VA-- = O. ^ du ^ dv If the values of the derivatives obtained from (97) be reduced by means of (V, 22), and the results substituted in the above equa tions, the first vanishes identically and the second reduces to (98) XV[ + YV t + ZV Z + (R cos V) R sin FvV = 0, where the primes indicate differentiation with respect to v. Con versely, when this condition is satisfied, equations (97) define a surface on which the curves v = const, are spherical. Hence : A necessary and sufficient condition that the curves v = const, of an orthogonal system on the unit sphere represent spherical lines of cur vature upon a surface is that five functions of v, namely V r F 2 , F 3 , R, V, can I e found which satisfy the corresponding equation (98). We note that F 1? F 2 , F 3 , and R cos V are determined by (98) only to within additive constants. A change of these constants for the first three gives a translation of the surface. If R cos V be increased by a constant, we have a new surface parallel to the other one. Hence * : If the lines of curvature in one system upon a surface be spherical, the same is true of the corresponding system on each parallel surface. Since equation (98) is homogeneous in the quantities F/, F^, F 3 , (R cos F) , R sin F, the latter are determined only to within a factor which may be a function of v. This function may be chosen so that all the spheres pass through a point. From these results we have the theorem of Dobriner f : With each surface with spherical lines of curvature in one system there is associated an infinity of nonparallel surfaces of the same kind with the same spherical representation of these lines of curvature. Among these surfaces there is at least one for which all the spheres pass through a point. At corresponding points of the loci of the cen ters of spheres of two surfaces of the family the tangents are parallel. * Cf. Bianchi, Vol. II, p. 303. f Crelle, Vol. XCIV (1883), pp. 118, 125. 316 SURFACES WITH PLANE LINES OF CURVATURE If the values of x, y, z from (97) be substituted in the formulas of Rodrigues (IV, 32), dx dx dx dx < 99) "* a^-^ and similarly for y and z, we obtain by means of (V, 22), =R cos Conversely, when for a surface referred to its lines of curvature the principal radius p l is of the form -.-*." where (^ and $ 2 are any functions whatever of v, the curves v = const, are spherical. For, by (V, 22), dv V~ ov du Consequently, from the first of (99), in which p l is given the above value, we obtain by integration where V l is a function of v alone. Similar results follow for y and z. As these expressions are of the form (97), we have the theorem : A necessary and sufficient condition that the lines of curvature v = const, be spherical is that p l be of the form (100). EXAMPLES 1. If the lines of curvature in one system are plane and one is a circle, all are circles. 2. When the lines of curvature in one family on a surface are circles, their spherical images are circles whose spherical centers constitute the spherical indi- catrix of the tangents to the curve of centers of the spheres which are enveloped by the given surface. Show also that each one-parameter system of circles on the unit sphere represents the circular lines of curvature on an infinity of surfaces, for one of which the circles are equal. EXAMPLES 31T 3. If the lines of curvature of a surface are parametric, and the curves u = const. are spherical, we have j j cot F Pgu B sin F Pi where p gu , />i, E denote the radii of geodesic curvature and normal curvature in the direction v const, and of the sphere respectively, and F denotes the angle under which the sphere cuts the surface. 4. When a line of curvature is spherical, the developable circumscribing the surface along this line of curvature also circumscribes a sphere ; and conversely, if such a developable circumscribes a sphere, the line of curvature lies on a sphere concentric with the latter (cf. Ex. 7, p. 149). 5. Let S be a pseudospherical surface with the spherical representation (25) of its lines of curvature. Show that a necessary and sufficient condition that the curves v = const, be plane is a / 1 a&\ _ du \sin w dv/ show also that in this case w is given by V -U COS 0) = , where V and V are functions of u and v respectively, which satisfy the conditions U * = U" 4 + (a - 2) C72 + 6, F 2 = F* + aF 2 -f (a -f b - 1), a and b being constants, and the accent indicating differentiation, unless U or "V is zero. 6. When the lines of curvature v const, upon a pseudospherical surface are plane, the linear element is reducible to the form _ a 2 tanh 2 (u + v) dw 2 a 2 sech 2 (u 4- v} dv 2 ~ C -A cosh 2 u -f B sinh 2 u G + A cosh 2 v + B sinh 2 v - 1 where A, B, C are constants. Find the expressions for the principal radii. 7. When the lines of curvature v = const, on a spherical surface are plane, the linear element is reducible to _ a 2 cot 2 (u + v) dw 2 a 2 esc 2 (u + v} d i? 2 ~ .A sin 2 w -f B - I A sin 2 u - 7? where J. and J5 are constants. The surfaces of Exs. 5 and 6 are called the surfaces of Enneper of constant curvature. GENERAL EXAMPLES 1. The lines of curvature and the asymptotic lines on a surface of constant curvature can be found by quadratures. 2. When the linear element of a pseudospherical surface is in the form (iii) of (7), M the equations x = cw, y = ae~a determine a conformal representation of the surface upon the plane, which is such that any geodesic on the surface is represented on the plane by a circle with its center on the ic-axis, or by a line perpendicular to this axis. 318 TF-SURFACES 3. When the linear elements of a developable surface, a spherical surface, and a pseudospherical surface are in the respective forms ds~ = du 2 + u-dv 2 , ds 2 = a?(du 2 + sin 2 wdu 2 ), ds 2 = 2 (dw 2 + sinh^udw 2 ), the finite equations of the geodesies are respectively Au cos v -f Bu sin v -f C = 0, A tan u cos v -f- B tan u sin v + C = 0, A tanh u cos v + .B tanh u sin v + C = 0, where A, Z>, C are constants ; if the coefficients of A and B are in any case equated to x and y, the resulting equations define a correspondence between the surface and the plane such that geodesies on the former correspond to straight lines on the latter. Find the expression for each linear element in terms of x and y as parameters. 4. Each surf ace of center of a pseudospherical surf ace is applicable to the catenoid. 5. The asymptotic lines on the surfaces of center of a surface of constant mean curvature correspond to the minimal lines on the latter. 6. Surfaces of constant mean curvature are characterized by the property that if u = const. , v = const, are the minimal curves, then D is a function of u alone and D" of v alone. 7. Equation (23) admits the solution w = 0, in which case the surface degen erates into a curve. Show that the general integral of the corresponding equations M + r cos <r (35) is tan 0/2 = Ce Bin<r ; take for S the line x = 0, y - 0, z = an and derive the equations of the transforms of -S; shc^w that the latter are surfaces of Dini (Ex. C, 122), or a pseudosphere. 8. Show that the Backlund transforms of the surfaces of Dini and of the pseudo- sphere can be found without integration, and that if the pseudosphere be trans formed by the transformation of Bianchi, the resulting surface may be defined by 2 a cosh u 2 a cosh u . x = (sinu ucosu), y (cosv + vsm v), V V (2 sinh u cosh u\ U cosh 2 w + v 2 / Show that the lines of curvature v const, lie in planes through the 2-axis. 9. The tangents to a family of geodesies of the elliptic or hyperbolic type on a pseudospherical surface are normal to a W-surface ; the relations between the radii are respectively . P\ + c ., Pi 4- c P l PO = a tanh , p l p 2 = a coth , a a where a and c are constants (cf. 7(3). 10. Show that the linear elements of the second surfaces of center of the >F-surfaces of Ex. are reducible to the respective forms ds.? = tanh 4 - du" 2 + sech 2 - dv 2 , ds.? - coth 4 U du 2 + csch 2 ~ du 2 , a a a a and that consequently these surfaces are applicable to surfaces of revolution whose meridians are defined by a (log tan Hog tan + + COS j , COS j , V* W + a 2 Vi - a 2 , 2 where K denotes a constant. GENERAL EXAMPLES 319 11. Determine the particular form of the linear element (49), and the nature of the curves upon the surface to which the asymptotic lines on the sheets of the evolute correspond, when p. 11 (a) = const; (6) -- = const. Pz Pi Pz 12. When a JF-surface is of the type (72), the surfaces of center are applicable to one another and to an imaginary paraboloid of revolution. 13. When a IF-surface is of the type (72) and the linear element of the sphere has the form (VI, GO), the curves u + v const, and u y const, on the spherical representation are geodesic parallels whose orthogonal trajectories correspond to the asymptotic lines on the surfaces of center ; hence on each sheet there is a family of geodesies such that the tangents at their points of meeting with an asymptotic line are parallel to a plane, which varies in general with the asymptotic line. 14. Show that the equations -+ f Vs m-dv. y = aUsin- C a J a a J where a denotes an arbitrary constant, define a family of applicable molding surfaces. 15. When the lines of curvature in one system on a surface are plane, and the lines of the second system lie on spheres which cut the surface orthogonally, the latter is a surface of Joachimsthal. 16. The spherical lines of curvature on a surface of Joachimsthal have constant geodesic curvature, the radius of geodesic curvature being the radius of the sphere on which a curve lies. 17. When the lines of curvature in one system on a surface lie on concentric spheres, it is a surface of Monge, whose director-developable is a cone with its vertex at the center of the spheres ; and conversely. 18. The sheets of the evolute of a surface of Monge are the director-developable and a second surface of Monge, which has the same director-developable and whose generating curve is the evolute of the generating curve of the given surface. 19. If the lines of curvature in one system on a surface are plane, and two in the second system are plane, then all in the latter system are plane. 20. A surface with plane lines of curvature in both systems, in one of which they are circles, is (a) A surface of Joachimsthal. (5) The locus of the orthogonal trajectories of a family of spheres, with centers on a straight line, which pass through a circle on one of the spheres. (c) The envelope of a family of spheres whose centers lie on a plane curve C, and whose radii are proportional to the distances of these centers from a straight line fixed in the plane of C. 21. If an arbitrary curve C be drawn in a plane, and the plane be made to move in such a way that a fixed line of it envelop an arbitrary space curve T, and at the same time the plane be always normal to the principal normal to T, the curve C describes a surface of Monge. 320 TF-SUKFACES 22. If all the Bianchi transforms of a pseudospherical surface S are surfaces of Enneper (cf. Ex. 5, 134), S is a surface of revolution. 23. When u has the value in Ex. 5, 134, the surfaces with the spherical representation (25), and with the linear element ds* = (HI cos w + Ydu 2 + U? sin 2 u du 2 , where U\ is an arbitrary function of M, are surfaces of Joachimsthal. 24. If the lines of curvature in both systems be plane for a surface S with the same spherical representation of its lines of curvature as for a pseudospherical surface, S is a molding surface. 25. If S is a pseudospherical surface with the spherical representation (25) of its lines of curvature, and the curves v = const, are plane, the function 6, given by c 2 w aw aw . aw . sin 6 - + cos h sin w = 0, a 2 aw dv dv determines a transformation of Bianchi of S into a surface Si for which the lines of curvature v = const, are plane. 26. A necessary and sufficient condition that the lines of curvature v= const, on a pseudospherical surface with the representation (25) of its lines of curvature be spherical is that -\r a., cot w = y 1 + J , sin w dv where V and V\ are functions of v alone. Show that when w is a solution of (23) and of i aw a / i aw\ sin 2 w aw aw \sinwatv a / i aw\ a 2 / i aw\ au\sin 2 wau/ aM 2 \sinwau/ the curves v = const, are plane or spherical, and that in the latter case V and V\ can be found directly. 27. Show that when w is a solution of (23) and of dv aucv 2 a 2 awau du \au/ and ( ) ^. 0, the lines of curvature u = const, are spherical on the pseudo- du \cos w dv/ spherical surface with the spherical representation (25) ; and that when w is such a function, upon the surfaces with the linear element or /aw\ 2 \dv/ / r)w\ 2 I (sin w + F ) du 2 + cos w -f V + V \ dv/ where F is a function of t> alone, the curves t> = const, are spherical; in the former case the spheres cut the surface orthogonally. CHAPTER IX DEFORMATION OF SURFACES 135. Problem of Minding. Surfaces of constant curvature. Ac cording to 43 two surfaces are applicable when a one-to-one correspondence can be established between them which is of such a nature that in the neighborhood of corresponding points corresponding figures are congruent or symmetric. It was seen that two surfaces with the same linear element are applicable, the parametric curves on the two surfaces being in correspon dence. But the fact that the linear elements of two surfaces are unlike is not a sufficient condition that they are not applicable ; in evidence of this we have merely to recall the effect of a change of parameters, to say nothing of a change of parametric lines. Hence we are brought to the following problem, first proposed by Minding : * To find a necessary and sufficient condition that two surfaces be applicable. From the second theorem of 64 it follows that a necessary condition is that the total curvature of the two surfaces at corre sponding points be the same. We shall show that this condition is sufficient for surfaces of constant curvature. In 64 we found that when K is zero at all points of a surface, the surface is applicable to the plane. If the plane be referred to the system of straight lines parallel to the rectangular axes, its linear element is 797272 ds 2 =dx 2 + dy*. Hence the analytical problem of the application of a developable surface upon the plane reduces to the determination of orthogonal systems of geodesies such that when these curves are parametric the linear element takes the above form. * Crelle, Vol. XIX (1839), pp. 371-387. 321 322 DEFORMATION OF SURFACES Referring to the results of 39, we see that in this case the factor tt v must equal unity. Consequently we must find a function 6 such that the left-hand members of the equations du + -.dv = d(x + iy}, du + \ are exact differentials, in which case these equations give x and y by quadratures. Hence we must have du\ -^E 99 which are equivalent to du~Hu ZEHdu 2 H dv dd _ J_ d_G_ _ F d_E_ dv ~ ^H ~du 2 EH dv From (V, 12) it is seen that these equations are consistent when K= 0. In this case 6, and consequently x and y, can be found by quadratures. The additive constants of integration are of such a character that if ar , y are a particular set of solutions, the most general are x = x cos a y Q sin a + a, y = X Q sin a + y Q cos a + />, where a, #, 5 are arbitrary constants. In the above manner we can effect the isometric representation of any developable surface upon the plane, and consequently upon itself or any other developable. These results may be stated thus : A developable surface is applicable to itself, or to any other develop able, in a triple infinity of ways, and the complete determination of the applicability requires quadratures only. Incidentally we have the two theorems: The geodesies upon a developable surface can be foundby quadratures. If the total curvature of a quadratic form be zero, the quadratic form is reducible by quadratures to dad&. SUEFACES OF CONSTANT CURVATURE 323 Suppose now that the total curvature of two surfaces S, S l is I/a 2 , where a is a real constant. Let P and 7? be points on S and /S^ respectively, C and C l geodesies through these respective points, and take P and I[ for the poles and C and C l for the curves v = of a polar geodesic system on these surfaces. The linear elements are accordingly (VIII, 6) d** = du 2 + sin 2 - dv\ ds 2 = du* + sin 2 ^ dv 2 . Hence the equations u u v = v determine an isometric representation of one surface upon the other, in which P and C correspond to P and C l respectively. According as the upper or lower sign in the second equation is used, corresponding figures are equal or symmetric. Similar results obtain for pseudospherical surfaces. Hence we have: Any two surfaces of constant curvature, different from zero, are in two ways applicable so that a given point and geodesic through it on one surface correspond to a given point and geodesic through it on the other. In particular, a surface of constant curvature can be applied to itself so that a given point shall go into any other point and a geodesic through the former into one through the latter. Combin ing these results with the last theorem of 117, we have: A nondevelopable surface of constant curvature can be applied to itself, or to any surface of the same curvature, in a triple infinity of ways, and the complete realization of the applicability requires the solution of a Iliccati equation. 136. Solution of the problem of Minding. We proceed to the determination of a necessary and sufficient condition that two sur faces S, 8 of variable curvature be applicable. Let their linear elements be ds 2 = E du 2 + 2 Fdudv + G dv 2 , ds 2 = E du 2 + 2F du dv + G dv 2 . By definition S and S are applicable if there exist two independ ent equations (1) (/> (U, V) = $(U J , V 1 ), ^ (U, V) = ^<(U , V ), establishing a one-to-one correspondence between the surfaces of such a nature that by means of (1) either of the above quadratic forms can be transformed into the other. 324 DEF011MATION OF SURFACES It is evident that if the two surfaces are applicable, the differen tial parameters formed with respect to the two linear elements are equal. Hence a necessary condition is (2) A^ = A;</> , A^, f)=A!(<#> , f), A 1 f=A;t , where the primes indicate functions pertaining to S . These con ditions are likewise sufficient that the transformation (1) change either of the above quadratic forms into the other. For, if the curves (/> = const., ^r const. ; </> = const., ^ = const, be taken for the parametric curves on S and S f respectively, the respective linear elements may be written (cf. 37) , Hence when equations (1) and (2) hold, the surfaces are applicable. The next step is the determination of equations of the form (1). Since the curvature of two applicable surfaces at corresponding points is the same, one such equation is afforded by the necessary condition (3) K(u,v) = K (u ,v ). The first of equations (2) is (4) A^A;* . Both members of this equation cannot vanish identically. For, in this case the curves K const, and K = const, would be minimal ( 37), and consequently imaginary. If these two equations are independent of one another, that is, they establish a correspondence, and the condition that it be iso metric is, as seen from (2), If, however, (5) \K we may take for the second of (1) (6) unless (7) A 2J fiT PROBLEM OF MINDING 325 If this condition be not satisfied, the conditions that (3), (6) define an isometric correspondence are A 1 A a JST = Finally, we consider the case where both (5) and (7) hold. Since the ratio of \K and A 2 K is a function of JC, the curves K = const. and their orthogonal trajectories t = const, form an isothermal sys tem of lines on S ( 41). Moreover, the function t can be found by quadratures, and the linear element is reducible to - (8) ds 2 = - (dK 2 + e J < A > dt 2 ). J( K ) When in particular A 2 ^T= 0, the linear element is In like manner the linear element of S is reducible to or, in the particular case A^ = 0, to In either case the equations K = K i t = t +a, where a is an arbitrary constant, define the applicability of the surfaces. We have thus treated all possible cases and found that it can be determined without quadrature whether two surfaces are appli cable. Moreover, in the first two cases the equations defining the correspondence follow directly, but in the last case the determina tion requires a quadrature. The last case differs also in this respect : the application can be effected in an infinity of ways, whereas in the first two cases it is unique. * If the surface be referred to the curves <r = const, and their orthogonal trajectories, where a C= , equation (6) may be replaced by A 2 tr = A^ , and it can be shown J ~vf(K) that AI(<T, &<<?) = Ai(<r , A%<r ) is a consequence of the other conditions. Cf. Darboux, Vol. Ill, p. 227. 326 DEFORMATION OF SURFACES Furthermore, we notice from (8) that in the third case the sur face S is applicable to a surface of revolution, the parallels of the latter corresponding to the curves K= const, of the former. Con versely, the linear element of every surface applicable to a surface of revolution can be put in the form (8). For, a necessary and sufficient condition that a surface be applicable to a surface of revolution is that its linear element be reducible to where U is a function of u alone ( 46). Now U" Cdu A lW = l, X= --, From the second it follows that u F(K), and consequently F^K). When these values are substituted in the above equations, we have, in consequence of Ex. 5, p. 91, (9) A 1 /C=/(A"), A,JiT =*<*). Hence we have the theorem : Equations (9) constitute a necessary and sufficient condition that a surface be applicable to a surface of revolution. The equations K=K, t = t +a define an isometric representation of a surface with the linear ele ment (8) upon itself. Therefore we have : Every surface applicable to a surface of revolution admits of a continuous deformation into itself in such a way that each curve K = const, slides over itself. Conversely, every surface applicable to itself in an infinity of ways is applicable to a surface of revolution. For, if the curvature is constant, the surface is applicable to a surface of revolution ( 135), and the only case in which two surfaces of variable curva ture are applicable in an infinity of ways is that for which condi tions (5) and (7) are satisfied. DEFORMATION OF MINIMAL SURFACES 327 137. Deformation of minimal surfaces. These results suggest a means of determining the minimal surfaces* applicable to a surface of revolution. In the first place we inquire under what conditions two minimal surfaces are applicable. The latter problem reduces to the determination of two pairs of parameters, w, v and u^ v v and two pairs of functions, F(u), <f>(v)* and Ffa^, ^ > 1 (f 1 ), which satisfy the condition (10) (1 + uvfF(u)(v) dudv = (1 + u^)* f\(uj 4^) du^dv r From the nature of this equation it follows that the equations which serve to establish the correspondence between the two surfaces are either of the form (11) ^=0(M), V 1= ^(V), or (12) ,= *(), , = *() If either set of values for u l and v l be substituted in (10), and if after removing the common factor dudv we take the logarithmic derivative with respect to u and v, we obtain (1 + u^Y (1 + uvf As this may be written ,.. o du l dv l dudv (i + uft)* ~~ (I +^^y 2 the spherical images of corresponding parts on the two surfaces are equal or symmetric according as (11) or (12) obtains ( 47). The latter case reduces to the former when the sense of the normal to either surface is changed. When this has been done, corresponding spherical images are equal and can be made to coincide by a rota tion of the unit sphere about a diameter. Hence one surface can be so displaced in space that corresponding normals become parallel, in which case the two surfaces have the same representation, that is, Wj = u, v x = v. Now equation (10) is which is equivalent to F l (u)=cF(u), 3 * no. 328 DEFORMATION OF SURFACES where c denotes a constant. If the surfaces are real, c must be of the form e i<x . Hence, in consequence of 113, we have the theorem : A minimal surface admits of a continuous deformation into an infinity of minimal surfaces, which are either associate to it or can be made such by a suitable displacement. We pass to the determination of a minimal surface which admits of a continuous deformation into itself, and consequently is appli cable to a surface of revolution. In consequence of the interpre tation of equation (13) it follows that if a minimal surface be deformed continuously into itself, a point p on the sphere tends to move in the direction of the small circle through p, whose axis is the momentary axis of rotation, and consequently each of these small circles moves over itself. From 47 it follows that if the axis of rotation be taken for the 2-axis, these small circles are the curves uv = const. In the deformation each point of the surface moves along the curve K= const, through it. Hence K is a func tion of uv. From (VII, 100, 102) we have A= ___^l_ ; consequently F(u)Q(v) must be a function of uv, and hence uF (u) __ v<& (v) F(u) 4>(v) The common value of these two terms is a constant. If it be denoted by K, we have K , where c and c 1 are constants. Hence from (VII, 98) we have : Any minimal surface applicable to a surface of revolution can be defined by equations of the form * c f (1 _ u *) u du + | <?j f (1 - v 1 ) v*dv, -c Cfl + u^u du ^Ci 1(1 2 J { 2V /r <+i u ^du + c^Ji iv, ivhere c, c., and K are arbitrary constants. (14) y DEFOKMATION OF MINIMAL SUEFACES 329 Since the curves K const, are represented on the sphere by the small circles whose axis is the z-axis, in each finite deformation of the surface into itself, as well as in a very small one, the unit sphere undergoes a rotation about this axis. In 47 it was seen that such a rotation is equivalent to replacing u, v by ue ia , ve~ ia , where a denotes the angle of rotation. Hence the continuous deformation of a surface (14) is defined by the equations resulting from the substitution in (14) of ue ia , ve~ i<z for u, v respectively. An important property of the surfaces (14) is discovered when we submit such a surface to a rotation of angle a about the z-axis. Let S denote the surface in its new position, and write its equations in the form f (1 - tf and similarly for y and z. Between the parameters u, v and u, v the following relations hold: u = ue ta , v = ve~ tar , and we have also x = x cos a y sin a, ~y = x sin a -f y cos a:, z = z. Combining these equations with (14), we find F (u) = cu K e~ ia(K + 2) , 5> (v) = c { v K e ia(K + 2) . Hence, for the correspondence defined by u = ?/, v = v, the surface S is an associate of S, unless K -f- 2 = 0, in which case it is the same surface. We consider the latter case, and remark that its equations are (cf. 110) If u, v be replaced by ue ia , ve~ ia , and the resulting expressions be denoted by x v y^ z x , we have (15) x l = xcosa ysina 1 y^ = x sin a + y cos a, z l = z + c lR(iac}. Hence, in a continuous deformation, the surface slides over itself with a helicoidal motion. Consequently it is a helicoid. Moreover, it is the only minimal helicoid. For, every helicoid is applicable 330 DEFORMATION OF SURFACES to a surface of revolution, and each minimal surface applicable to a surface of revolution with the z-axis for the axis of revolution of the sphere is defined by (14). But only when ic = 2 will the substitution of we 1 ", ve~ ia give a set of equations such as (15). Hence we have : The helicoidal minimal surfaces are defined by the Weierstrass formulas when F(u)=c/u 2 . And we may state the other results thus : If any nonhelicoidal minimal surface, which is applicable to a surface of revolution, be rotated through any angle about the axis of the unit sphere whose small circles represent the curves K const. on the surface, and a correspondence with parallelism of tangent planes be established between the surfaces, they are associate ; con sequently the associates of such a minimal surface are supcrposable. EXAMPLES 1 . Find under what conditions the surfaces, whose equations are z\ F(r) + av, can be brought into a one-to-one correspondence, so that the total curvature at corresponding points is the same. Determine under what condition the surfaces are applicable. 2. If the tangent planes to two applicable surfaces at corresponding points are parallel, the surfaces are associate minimal surfaces. 3. Show that the equations x = e a u t y = e- a v, z ae a u z -f b<y~ a v 2 , where a is a real parameter, and a and 6 are constants, define a family of parab oloids which have the same total curvature at points with the same curvilinear coordinates. Are these surfaces applicable to one another ? 4. Find the geodesies on a surface with the linear element d u z _ 4 y dudv + 4 u dv 2 ds 2 =. -- 4(w-t> 2 ) Show that the surface is applicable to a surface of revolution, and determine the form of a meridian of the latter. 5 . Determine the values of the constants a and 6 in ds* = du 2 + [(u + au) 2 + 6 2 ] du 2 , so that a surface with this linear element shall be applicable to (a) the right helicoid. (&) the ellipsoid of revolution. SECOND GENERAL PROBLEM 331 6. A necessary and sufficient condition that a surface be applicable to a surface of revolution is that each curve of a family of geodesic parallels have constant geodesic curvature. 7. Show that the helicoidal minimal surfaces are applicable to the catenoid and to the right helicoid. 138. Second general problem of deformation. We have seen that it can always be determined whether or not two given surfaces are applicable to one another. The solution of this problem was an important contribution to the theory of deformation. An equally important problem, but a more difficult one, is the following : To determine all the surfaces applicable to a given one. This problem was proposed by the French Academy in 1859, and has been studied by the most distinguished geometers ever since. Although it has not been solved in the general case, its profound study has led to many interesting results, some of which we shall derive. If the linear element of the given surface be c?s 2 = Edu? + 2 F dudv -f- G dv*, every surface applicable to it is determined by this form and by a second, namely Ddu*+ 2 D dudv + D"dv 2 , whose coefficients satisfy the Gauss and Codazzi equations ( 64). Conversely, every set of solutions D, D , D" of these equations defines a surface applicable to the given one, and the determination of the Cartesian coordinates of the corresponding surface requires the solution of a Riccati equa tion. But neither the Codazzi equations, nor a Riccati equation, can be integrated in the general case with our present knowledge of differential equations. Later we shall make use of this method in the study of particular cases, but for the present we proceed to the exposition of another means of attacking the general problem. When the values of D, D , D n obtained from the Gauss equations (V, 7) are substituted in the equation H 2 K=J)D"D 2 , the result ing equation is reducible, inconsequence of the identity A^ =1 A 2 (cf. Ex. 6, p. 120), to (16) dtf " \du dv~\ 1 J ^ I 2 J dv 332 DEFORMATION OF SURFACES This equation, which is satisfied also by y and 2, involves only E, F, G and their derivatives, and consequently its integration will give the complete solution of the problem. It is linear in \tfxtfx / c 2 x Yl &x tfx tfx , , , , , ^, - ^ri^l^r-sr) oTi 1 T-T- Tl and therefore is of the form \_du cv \dudv / J dir ^<7v 0tr studied by Ampere. Hence we have the theorem: The determination of all surfaces applicable to a given one requires the integration of a partial differential equation of the second order of the Ampere type. In consequence of (16) and (V, 36) we have that the coordinates of a surface with the linear element (17) ds 2 = Edu 2 + 2 Fdudv + G dv 2 are integrals of (18) A 22 = (l A^JST, the differential parameters being 1 formed with respect to (17). We shall find that when one of these coordinates is known the other two can be found by quadratures. Our general problem may be stated thus : G-iven three functions E, F, G of u and v ; to find all functions x, y, z of u and v which satisfy the equation dx 2 + dy 1 + dz 2 = E du* + 2 Fdudv + G dv\ where du and dv may be chosen arbitrarily. Darboux * observed that as the equation may be written (19) dx 2 + dy* = Edu 2 +2Fdudv + Gdv 2 - dz\ whose left-hand member is the linear element of the plane, or of a developable surface, the total curvature of the quadratic form (20 ) \E- p?Y"U + 2\F- * **\ L WJ L dudv\ dudv dv\ must be zero ( 64). In order to find the condition for this, we assume that z is known, and take for parametric lines the curves z = const, and their *L.c., Vol. Ill, p. 253. SECOND GENERAL PROBLEM 333 orthogonal trajectories for v = const. With this choice of parame ters the right-hand member of (19) reduces to (E I)dz 2 + Gdv*. The condition that the curvature of this form be zero is tr where K denotes the curvature of the surface. But this is the condition also that z be a solution of (18) when the differential parameters are formed with respect to Edz 2 -}-Gdv 2 . However, the members of equation (18) are differential parameters ; consequently z is a solution of this equation whatever be the parametric curves. By reversing the above steps we prove the theorem : When z is any integral of the equation (18), the quadratic form (20) has zero curvature. When such a solution is known we can find by quadratures (cf. 135) two functions x, y such that the quadratic form (20) is equal to dx* + dy z , provided that that is, Ajg < 1. Hence we have the theorem : If z be a solution of A 22 (1 A X 0) K such that A^ < 1, it is one of the rectangular coordinates of a surface with the given linear ele ment, and the other two coordinates can be obtained by quadratures. 139. Deformations which change a curve on the surface into a given curve in space. We consider the problem : Can a surface be deformed in such a manner that a given curve C upon it comes into coincidence with a given curve F in Let the surface be referred to a family of curves orthogonal to C and to their orthogonal trajectories, C being the curve v = 0, and its arc being the parameter u, so that E \ for v = 0. The same conditions hold for F on the deform. 334 DEFORMATION OF SURFACES Since the geodesic curvature of C is unaltered in the deformation ( 58), it follows from the equation (IV, 47) for the new surface, namely (21) p = p g amw, that the deformation is impossible, if the curvature of F at any point is less than the geodesic curvature of C at the corresponding point. Since both p and p g are known, equation (21) determines to, and consequently the direction of the normal to the new surface along F is fixed. This being the case, the direction of the tangents to the curves u = const, on the new surface at points of F can be . 1 dx 1 c>f 1 cz found, and so we have the values of - r-i = tor v = U, V<? to \IG co vV; w as well as > for v = 0, the latter being the direction-cosines cu cu du of the tangent to F. If these expressions be differentiated with , tfx c~y d 2 z c 2 x tfy c 2 z respect to u, we obtain the values ot ^ 77, ; T r- > 77- 3u" cu" cu" cudv ducv cucv for v=0. Since F=Q and E = 1 for v = 0, the Gauss equations (V, 7) for v = are tfx 1 SEdx = r JJJL+ du 2 2G to to J. I/ Jjy t/*f J- *y*-* f**- " 2 "a^ a^ 2G du to dv* 2 cu du 2 6r 00 dv All the terms of the first two equations have been determined except D and D 1 ; hence the latter are given by these equations. Since the total curvature A" is unaltered by the deformation, it is known at all points of F; consequently // is given by H*K = />/>" /> 2 , unless D is zero, in which case F is an asymptotic line d 2 x and p =p (J . When /)" is found we can obtain the value of 5 from the last of equations (22). From the method of derivation of equa tion (16) it follows that the above process is equivalent to finding the value of ^ from equation (16), which is possible unless D = 0. to Excluding this exceptional case, we remark that if equations (22) PARTICULAR DEFORMATIONS 335 be differentiated with respect to u, we obtain the values of all the derivatives of x of the third order for v = except - The latter may be obtained from the equation which results from the differ entiation of equation (16) with respect to v. By continuing this process we obtain the values for v of the derivatives of x of all orders, and likewise of y and z. If we indicate by subscript null the values of functions, when u = U Q , v = 0, the expansions fdx\ idx = z + n \dufy \dvfy 2\#ir/o \dudvh and similar expansions for y and 2, are convergent in general, as Cauchy has shown,* and x, y, z thus defined are the solutions of equation (16) which for v = satisfy the given conditions. Hence : A surface S can be deformed in such a manner that a curve C upon it comes into coincidence with a given curve F, provided that the curvature of F at each point is greater than the geodesic curvature of C at the corresponding point. There remains the exceptional case p = p a . If the desired def ormation is possible, F is an asymptotic line on the deform, and consequently, by Enneper s theorem ( 59), its radius of torsion must satisfy the condition r 2 = 1/JC. Hence when C is given, F is determined, if it is to be an asymptotic line. If F satisfies these conditions, the value of D" for v = is arbi trary, as we have seen. But when it has been chosen, the further determination of the values of the derivatives of #, y, z of higher order for v = is unique, it being the same as that pursued in the general case. Hence equation (16) admits as solution a family of these surfaces, depending upon an arbitrary function. For all .of these surfaces the directions of the tangent planes at each point of F are the same. Hence we have the theorem : Criven a curve C upon a surface 8 ; there exists in space a unique curve F with which C can be brought into coincidence by a deforma tion of 8 in an infinity of ways ; moreover, all the new surfaces are tangent to one another along F. * Cf . Goursat, Lemons surT integration des Equations aux derivees partielles du second ordre, chap. ii. Paris, 1896. 336 DEFOKMATION OF SURFACES If C is an asymptotic line on S, it may be taken for F; hence : A surface may be subjected to a continuous deformation during which a given asymptotic line is unaltered in form and continues to be an asymptotic line on each deform. This result suggests the problem : Can a surface be subjected to a continuous deformation in which a curve other than an asymptotic line is unaltered? By hypothesis the curvature is not changed and the geodesic curvature is necessarily invariant; hence from (21) we have that sin o> must have the same value for all the surfaces. If o> is the same for all surfaces, the tangent plane is the same, and consequently the expansions (23) are the same. Hence all the surfaces coincide in this case. However, there are always two values of W for which sin o> has the same value, unless w is a right angle. Hence it is possible to have two applicable surfaces passing through a curve whose points are self-correspondent, but not an infinity of such surfaces. Therefore : An asymptotic line is the only curve on a surface which can remain unaltered in a continuous deformation. 140. Lines of curvature in correspondence. We inquire whether a surface S can be deformed in such a manner that a given curve C upon it may become a line of curvature on the new surface. Suppose it is possible, and let F denote this line of curvature. The radii of curvature and torsion of F must satisfy (21) and 1/T dto/ds (cf. 59), where p g is the same ; for F as for C. If we choose for w any function whatever, the functions p and r are thus determined, and F is unique. Since o> fixes the direction of the tangent plane to the new surface along F, there is only one deform of S of the kind desired for each choice of w (cf. 139). Hence : A surface can be deformed in an infinity of ways so that a given curve upon it becomes a line of curvature on the deform. This result suggests the following problem of Bonnet*: To determine the surfaces which can be deformed with preservation of their lines of curvature. * Memoire sur la theorie des surfaces applicables sur une surface donnee, Journal de I Ecole Poly technique, Cahier 42 (1867), p. 58. LINES OF CUBVATUBE IN COBBESPONDENCE 337 We follow the method of Bonnet in making use of the funda mental equations in the form (V, 48, 55). We assume that the lines of curvature are parametric. In this case these equations reduce to ($>i _ dc t-~~ Sr ^ (24) dv From these equations it follows that if $ and S are two applicable surfaces referred to corresponding lines of curvature, the functions r and r l have the same value for both surfaces, and consequently the same is true of the product qp r Hence our problem reduces to the determination of two sets of functions p v q ; p[, q , satisfying the above equations. In consequence of the identity (25) p(q =P& we have from the first two of (24) *- <S?-S- of which the integrals are p i 2 = p?+f(v), q 2 = <f+ <#>(M), where f(v) and <f>(u) are functions of v and u respectively. The parameters w, v may be chosen so that these functions become constants #, /3, and consequently (27) ;>I 2 = K+ ? 2 =<Z 2 +- If these equations be multiplied together, the resulting equation is reducible by means of (25) to either of the forms (28) pi ft + (fa + a(3 = 0, p(*P + z *a-a& = b. From the first we see that a and fi cannot both be positive if S is real, and from the second that they cannot both be negative. We assume that a is negative and j3 positive, and without loss of generality write (29) rf-rf-li ^-tf+l- The first of (28) reduces to pi q* = l. In conformity with this we introduce a function o>, thus p l = cosh &>, q = sinh o>. Then equations (29) may be replaced by p[ = sinh ft), ^ = cosh o>. 338 DEFORMATION OF SURFACES Moreover, the fundamental equations (24) reduce to _ dv du V ] du a 2 o2 o> , c o> . , , : H -- = smh o> cosh o>. dt* a 0v- Comparing these results with 118, we see that the spherical representation of lines of curvature of the surfaces S and S f respec tively is the same as of the lines of curvature of a spherical surface and of its Hazzidakis transform. Conversely, we have that every surface of this kind admits of an applicable surface with lines of curvature in correspondence. The preceding investigation rested on the hypothesis that neither the first nor second of equations (24) vanishes identically. Suppose that the second vanishes ; then q is a function of u alone, say $(u). Since the product p^q differs from the total curvature only by a factor (cf. 70), p l cannot be zero ; therefore r = and q ^(u). Equation (25) is now of the form p l <f>(u)=p[<t> l (u). If p[ be elimi nated from this equation and the first of (27), it is found that p l also is a function of u alone. Hence the curves v = const, on the sphere are great circles with a common diameter, and therefore S is a molding surface ( 130). The parameter u may be chosen so that we may take q = 1 and p^=U\ then from (27) and (25) we find />( = Vf/ 2 -h #, q = U/^/U* + a, where a is an arbitrary constant. Hence we have the theorem : A necessary and sufficient condition that a surface admit of an applicable surface with lines of curvature in correspondence is that the surface have the same spherical representation of its lines of cur vature as a spherical surface 2, or be a molding surface ; in the first case there is one applicable surface, and the spherical representation of its lines of curvature is the same as of the Hazzidakis transform of 2 ; in the second case there is an infinity of applicable surfaces.* 141. Conjugate systems in correspondence. When two surfaces are applicable to one another, there is a system of corresponding lines which is conjugate for both surfaces (cf. 56). The results of 140 show that for a given conjugate system on a surface S * Cf . EX. 14, p. 319. CONJUGATE SYSTEMS IN COEEESPONDENCE 339 there is not in general a surface S l applicable to S with the corre sponding system conjugate. We inquire under what conditions a given conjugate system of S possesses this property. Let S be referred to the given conjugate system. If the corre sponding system on an applicable surface S l is conjugate, we have D = D[= 0, Dip? = DD" ; for the total curvature of the two surfaces is the same. We replace this equation by the two D! - tanh 6 D, D[ = coth 6 . D", thus defining a function 6. The Codazzi equations for S are Since these equations must be satisfied by D l and D", we have 30 22 D , c6 fll Z lz The condition of integrability of (30) is reducible to 2 D m As the two roots of this equation differ only in sign, and thus lead to symmetric surfaces, we need consider only one. If it be substi tuted in (30), we obtain two conditions upon E, F, G ; 7), D", which are necessary in order that S admit of an applicable surface of the kind sought. Hence in general there is no solution of the problem. However, if the two expressions in the brackets of (31) vanish identically, the conditions of integrability of equations (30) are completely satisfied, and S admits of an infinity of applicable sur faces upon which the coordinate curves form a conjugate system. Consequently we have the theorem : If a conjugate system on a surface S corresponds to a conjugate system on more than one surface applicable to S, it corresponds to a conjugate system on an infinity of surfaces applicable to S. 340 DEFORMATION OF SURFACES We shall give this result another interpretation by considering the spherical representation of S. From (VI, 38) we have 1-22-1 .D f!2V firi-D". /12V ii/^ = la/ la/3" ~li/ {rtV > are formed with respect to the linear ele ment of the spherical representation of S. If we substitute these values in (30), we get d6 f!2\ d6 /12V ... = s f tanh 0, = \ \ coth 0, a^ 1 2 J 0* 1 1 J and the condition that these equations have an integral involving a parameter becomes a ri2V_ a ri2V py ri2V Sii/~Sla./ "-iJisr The first of these equations is the condition that the curves upon the sphere represent the asymptotic lines upon a certain sur face 2 (cf. 78). Moreover, if K denotes the total curvature of S, and we put K= l//> 2 we have (34) du 2 Now equations (33) are equivalent to (34), and & log p r,<2 which reduces to = 0. As the general integral of this equation cucv is p = cf)(u) 4- ^(i ), where (/> and ^ are arbitrary functions of u and v respectively, we have the following theorem due to Bianchi* : A necessary and sufficient condition that a surface S admit a con tinuous deformation in which a conjugate system remains conjugate is that the spherical representation of this system be that of the asymp totic lines of a surface whose total curvature, expressed in terms of parameters referring to these lines, is of the form * Annali, Ser. 2, Vol. XVIII (1890), p. 320; also Lezioni, Vol. II, p. 83. CONJUGATE SYSTEMS IN COKKESPONDENCE 341 The pseudospherical surfaces afford an example of surfaces with K of this form. In this case </> and ^r are constants, so that equa- ("12") f 12"| tions (34) reduce to 1 \ ~\ a | ** ^ which, in consequence of {11"| f22 > | f = 1 ^, f=0. But these are the condi tions that the parametric curves on S be geodesies. A surface with a conjugate system of geodesies is called a surface of l /r oss. We state these results thus : A surface of l^oss admits of a continuous deformation in which the geodesic conjugate system is preserved ; consequently all the new sur faces are of the same kind. EXAMPLES 1. Show that every integral of the equation Ai0 = 1 is an integral of the funda mental equation (18). 2. On a right helicoid the helices are asymptotic lines. Find the surfaces appli cable to the helicoid in such a way that one of the helices is unaltered in form and continues to be an asymptotic line. 3. A surface applicable to a surface of revolution with the lines of curvature on the two surfaces in correspondence is a surface of revolution. 4. Show that the equations X=KTCOS-, y = train-, z = /Vl /c 2 r 2 c?M, K K J define a family of applicable surfaces of revolution with lines of curvature in corre spondence. Discuss the effect of a variation of the parameter K. 5. Let S denote a surface parallel to a spherical surface S. Find the surface applicable to S with preservation of the lines of curvature. 6. It Si and S 2 be applicable surfaces referred to the common conjugate sys tem, their coordinates &i, y\,z\\ 2 , ?/ 2 , 2 are solutions of the same point equation (cf. VI, 26), and the function xf -f y? + zf (x| + y.| + z|) also is a solution. 7. Show that the locus of a point which divides in constant ratio the join of corresponding points on the surfaces Si and <S 2 of Ex. 6 is a surface upon which the parametric lines form a conjugate system. Under what condition is this surface applicable to Si and /S 2 ? 8. The tetrahedral surface x = A(a + u)*(a + )*, y = B(b + u)*(b + v)*, z = C(c + w)*(c + v) f , admits of an infinity of deforms The curves u = v upon these surfaces are congruent, and consequently each is an asymptotic line on the surface through it. 342 DEFORMATION OF SURFACES 9. If the equations of a surface are of the form x = U 1 V ll y = UiV!, z=V* t the equations sin 0, where h denotes a constant, define a family of applicable surfaces upon which the parametric lines form a conjugate system. 10. Show that the equations of the quadrics can be put in the form of Ex. 9, and apply the results to this case. 142. Asymptotic lines in correspondence. Deformation of a ruled surface. We have seen ( 139) that a surface can be subjected to a continuous deformation in which an asymptotic line remains asymptotic. We ask whether two surfaces are applicable with the asymptotic lines in one system corresponding to asymptotic lines of the other. We assume that there are two such surfaces, S, S lt and we take the corresponding asymptotic lines for the curves v = const, and their orthogonal trajectories for u = const. In con sequence of this choice and the fact that the total curvature of the two surfaces is the same, we have (36) J9 = D 1 =0, JF=0, D = D[. The Codazzi equations (V, 13 ) for S reduce to < Q Because of (36) the Codazzi equation for S 1 analogous to the first of (37) will differ from the latter only in the last term. Hence we must have either D" = Z>", or Ef(u}. In the former case the sur faces S and S l are congruent. Hence we are brought to the second, which is the condition that the curves v = const, be geodesies. As the latter are asymptotic lines also, they are straight, and conse quently 8 must be a ruled surface. By changing the parameter w, we have J5? = l, and equations (37) reduce to EULED SURFACES 343 By a suitable choice of the parameter v the first of these equations may be replaced by JX=1/V5, and the second becomes ra/i = I ~(- J 9*\& where </> is an arbitrary function. These results establish the fol lowing theorem of Bonnet : A necessary and sufficient condition that a surface admit an applicable surface with the asymptotic lines in one system on each surface corresponding is that the surface be ruled; moreover, a ruled surface admits of a continuous deformation in which the generators remain straight. To this may be added the theorem : If two surfaces are applicable and the asymptotic lines in both systems on each surface are in correspondence, the surfaces are con gruent, or symmetric. This is readily proved when the asymptotic lines are taken as parametric. We shall establish the second part of the above theorem in another manner. For this purpose we take the equations of the ruled surface in the form ( 103) (38) x = x Q +lu, y = y Q +mu, z = z +nu, where X Q , y^, z are the coordinates of the directrix C expressed as functions of its arc v, and I, m, n are the direction-cosines of the generators, also functions of v. They satisfy the conditions (39) aJ +jtf-K 1 -!. * 2 +w a +n a = l, where the accents indicate differentiation with respect to v. Furthermore, the linear element is (40) ds*= du 2 + 2 cos dudv + (aV+ 2 - 2 n b = l x Hence if we have a ruled surface with the linear element (40), the problem of finding a ruled surface applicable to it, with the gener ators of the two surfaces corresponding, reduces to the determi nation of six functions of v, namely X Q , y , z ; I, m, n, satisfying 344 DEFORMATION OF SURFACES the five conditions (39), (41). From this it follows that there is an arbitrary function of v involved in the problem, and consequently there is an infinity of ruled surfaces with the linear element (40). There are two general ways in which the choice of this arbi trary function may be made, either as determining the form of the director-cone of the required surface, or by a property of the directrix. We consider these two cases. 143. Method of Minding. The first case was studied by Mind ing.* He took /, m, n in the form (42) / = cos < cos i/r, m = cos < sin i/r, n = sin </>, which evidently satisfy the second of (39). The first of (41) reduces to (43) <J> 2 +^ 2 cos 2 =a a . If we solve equations (39) and (41) for x ^ y Q , z , the resulting expressions are reducible by means of (VII, 63) to (44) Q = I cos 6 Q + ^ [I b (mn f - m n) VV sin 2 - 6 2 ], and analogous expressions for y[ and z[. Hence, if < be an arbi trary function of v, and ^ be given by (46) +=(^f^ J COS (/> the functions # , # , z , obtained from (44) by quadratures, together with /, m, n from (42), determine a ruled surface with the linear element (40). Each choice of (/> gives a different director-cone, which is deter mined by the curve in which the cone cuts the unit sphere, whose center is at the vertex of the cone. Such a curve is defined by a relation /(c/>, -<fr) = 0, so that instead of choosing $ arbitrarily we may take / as arbitrary; for, by combining equations (43) and /(<, ifr) 0, we obtain the expressions for < and ^r as functions of v. Hence : A ruled surface may be deformed in such a way that the director- cone takes an arbitrary form. * Crelle, Vol. XVIII (1838); pp. 297-302. RULED SURFACES 345 When the given ruled surface is nondevelopable, the radicand in (44) is different from zero, and consequently there are two dif ferent sets of functions X QJ y Q1 Z Q . Hence there are two applicable ruled surfaces with the same director-cone. If the parameters of distribution of these two surfaces be calculated by (VII, 73), they are found to differ only in sign. Hence we have the theorem of Beltrami : * A ruled surface admits of an applicable ruled surface such that corresponding generators are parallel, and the parameters of distri bution differ only in sign. 144. Particular deformations of ruled surfaces. By means of the preceding results we prove the theorem : A ruled surface may be deformed in an infinity of ways so that a given curve becomes plane. Let the given curve be taken for the directrix of the original surface. Assuming that a deform of the kind desired exists, we take its plane for the zy-plane. From (44) we have a 2 n cos + bn f (lm f I m) a 2 sin 2 6 2 = 0, which, in consequence of (42) arid (43), reduces to b cosc/>.( + a 2 sine/) cos# cosc^Va 2 <// 2 Va 2 sin 2 b 2 = 0. The integral of this equation involves an arbitrary constant, and thus the theorem is proved. The preceding example belongs to the class of problems whose general statement is as follows: To deform a ruled surface into a ruled surface in such a way that the deform of a given curve C on the original surface shall possess a certain property on the resulting surface. We consider this general problem. Let the deform of C be the directrix of the required surface, and let , /3 , 7 ; J , TW O , n ; X , /* , v denote the direction-cosines of its tangent, principal normal, and binormal. If <r denotes the angle between the osculating plane to the curve and the tangent plane to the surface, we have (46) I = # cos # -f sin (/ cos <r + X sin <r), * Annali, Vol. VII (1865), p. 115. 346 DEFORMATION OF SURFACES and similar expressions for m and n. When these values are sub stituted in the first two of equations (41), the resulting equations are reducible, by means of the Frenet formulas (I, 50), to coscr /-, , b (47) P ~cos0 n ,, . sin cr sin . n . sincr sin# ~] 2 (cos cr sin ) H T/ aM COS cr sin ft"] 2 I2 + (sin cr sin Q ) \ab. These are two equations of condition on cr, /o, T, as functions of v. Each set of solutions determines a solution of the problem ; for, the directrix is determined by expressions for p and r, and equa tions (46) give the direction-cosines of the generators. We leave it to the reader to prove the above theorem by this means, and we proceed to the proof of the theorem: A ruled surface may be deformed in such a manner that a given curve C becomes an asymptotic line on the new ruled surface. On the deform we must have a = or a = TT, so that from (47) p the sign being fixed by the fact that p is necessarily positive. The second of (47) reduces to sin 2 6> If the curve with these intrinsic equations be constructed, and in the osculating plane at each point the line be drawn which makes the angle with the tangent, the locus of these lines is a ruled surface satisfying the given conditions. When the curve C is an orthogonal trajectory of the generators, the same is true of its deform. Hence : A ruled surface may be deformed in such a way that all the gener ators become the principal normals of the deform of any one of their orthogonal trajectories. Having thus considered the deformation of ruled surfaces in which the generators remain straight, we inquire whether two RULED SURFACES 347 ruled surfaces are applicable with the generators of each corre sponding to curves on the other. Assume that it is possible, and let v = const, be the generators of S and u = const, the curves on S corresponding to the generators of S r From (V, 13) it follows that the conditions for this are respectively where K= \/p\ But equations (48) are the necessary and sufficient conditions that there be a surface 2 applicable to S and Sv upon which the asymptotic lines are parametric (cf. VI, 3). But the curves v = const, and u = const, are geodesies on S and 8^ and consequently on 2. Therefore 2 is doubly ruled. Hence : If two ruled surfaces S and S l are applicable to one another, the generators correspond unless the surfaces are applicable to a quadric with the generators of S and S l corresponding to the two different systems of generators of the quadric. EXAMPLES 1. A ruled surface can be deformed into another ruled surface in such a way that a geodesic becomes a straight line. 2. A ruled surface formed by the binomials of a curve C can be deformed into a right conoid ; the latter is the right helicoid when the torsion of C is constant. Prove the converse also. 3. On the hyperboloid of revolution, defined by xwu.v y u . v v z u - = cos - + sin - , - = sin -- cos - , - = . c A c c c A c c d A where A 2 = c 2 + d 2 , the circle of gorge is a geodesic, which is met by the generators under the anle cos- 4. Show that the ruled surface which results from the deformation of the hyperboloid of Ex. 3, in which the circle of gorge becomes straight, is given by ud u ud . v uc x = cos - , y sin - , z = -- \- v. Ad Ad A 5. Show that the ruled surface to which the hyperboloid of Ex. 3 is applicable with parallelism of corresponding generators is the helicoid x u v c 2 d 2 . v y u . v c 2 d 2 v z u 2 c - = cos - H -- sin -i - = sin - --- cos - , - = I -- - v, c A c C 2 + d 2 c c A c c 2 + d 2 c d A A 2 and that the circle of gorge of the former corresponds to a helix upon the latter. 6. When the directrix is a geodesic, equations (47) reduce to Bin * .*; + 6 = 0, 348 DEFORMATION OF SURFACES 7. When an hyperboloid of revolution of one sheet is deformed into another ruled surface, the circle of gorge becomes a Bertraml curve and the generators are parallel to the corresponding bmormals of the conjugate Bertrand curve. 8. A ruled surface can be deformed in such a way that a given curve is made to lie upon a sphere of arbitrary radius. 9. When a ruled surface admits a continuous deformation into itself the total curvature of the surface is constant along the line of striction, the generators meet the latter under constant angle, and the parameter of distribution is constant (cf. 126). 10. Two applicable ruled surfaces whose corresponding generators are parallel cannot be obtained from one another by a continuous deformation. GENERAL EXAMPLES 1. Determine the systems of coordinate lines in the plane such that the linear element of the plane is ^ U 2 _j_ ^2 = where U and V are functions of u and u respectively. 2. Solve for the sphere the problem similar to Ex. 1. 3. Determine the functions 0(w) and ^ (u) so that the helicoids, defined by x = a\/U 2 - 6 2 cos-, y = shall be applicable to the surface whose equations are where U is any function of u. 4. Apply the method of Ex. 3 to find helicoids applicable to the pseudosphere ; to the catenoid. 5. The equations x = a V2 u 2 cos - , y = a V 2 it 2 sin -- , z = - (u 1) a a 2 define a paraboloid of revolution. Show that surfaces applicable to it are defined by id r /* /* X - / 3 02 ~/203 + J (fzdfz -fsdfz) - J (02^03 ~ 03 dfa) , -? 2 = -I /201 /102 where a is a real constant, and the / s and s are functions of a parameter a and /3 respectively such that 6. Investigate the special case of Ex. 5 for which a and /3 are conjugate imaginary functions, and 2 + a-2a* .2-a-2a* fl = - 7=- - /2 = l , - - / = * 2V2a 2V2a and the s are functions conjugate imaginary to the/ s. GENERAL EXAMPLES 349 7. Show that the surface of translation x a(cosw + cosv), y a(sinw + sinv), z = c(u + v) is applicable to a surface of revolution. 8. Show that the minimal surfaces applicable to a spiral surface (Ex. 22, p. 151) are determined by the functions F(u) = cu m + in , 4>(u) = cio m ~ in , and that the asso ciate surfaces are similar to the given one. 9. If the coefficients E, F, 6? of the linear element of a surface are homogeneous functions of u and v of order 2, the surface is applicable to a surface of revolution. 10. If z, y, z are the coordinates of a surface S referred to a conjugate system, the equations ctf__ dx W__pdy_ ^i_p^:. ^L-Q^L ^-Q^y. = Q~ aw ~ aw aw ~ au aw~ aw au au au ~ aw cv ~ au are integrable if P and Q satisfy the conditions where the Christoffel symbols are formed with respect to the linear element of S. Show that on the surface S , whose coordinates are x , y , z , the parametric curves form a conjugate system, and that the normals to S and S at corresponding points are parallel. 11. Show that for the surface x - f\fi(u)du + 0i(w), y = f A/ 2 (w)dit + 2 (u), z = j \f 3 (u)du + 3 (u), where \ is any function of u and u, and/!, / 2 , / 3 ; 0i, 2 , 3 *are functions of u and u respectively, the parametric curves form a conjugate system. Apply the results of Ex. 10 to this surface, and discuss the case for which X is independent of v. 12. If S and Si are two applicable surfaces, and S{ denotes the surface corre sponding to Si in the same manner as S to S in Ex. 10 and by means of the same functions P and Q, then S and S{ are applicable surfaces. 13. If x, ?/, z and i, 2/1, z\ are the coordinates of a pair of applicable surfaces S and Si, a second pair of applicable surfaces S and S{ is denned by x = x + h(z + zi) - k(y + T/J), x[ = x l - h(z + zi) + k(y + 2/1), y = y + k(x + xi)-g(z + *i), yi = z/i - k (x + xi) + g(z + zi), z = z + g(y + yi) - h(x + x x ), zi = z t - g (y + z/i) + h(x + KI), where #, ^-, and fc are constants. Show that the line segments joining correspond ing points of S and S are equal and parallel to those for Si and S{ ; that the lines joining corresponding points on S and Si meet the similar lines for S and S{ ; and that the common conjugate system on S and Si corresponds to the common conju gate system on S and Si. 14. Apply the results of Ex. 13 to the surfaces of translation x = w 2 - v 2 -|- 2 av, y = 2 w 2 -I- v 2 - 2 au - 2* V& 2 + 3 w 2 dw, z = 2 6u, 2 u 2 - 2 au - 2 Z! = 2 fa 2 - 3u 2 dv. Show that when g = h = 0, k = - 1, the surface S is an elliptic paraboloid. 350 DEFORMATION OF SURFACES 15. Show that the equations "a 2 " " y ~ J where the accent indicates differentiation with respect to the argument, define a family of applicable surfaces of translation. Apply the results of Ex. 12 to this case. 16. Show that when S and Si in Exs. 12 and 13 are surfaces of translation, and their generating curves correspond, the same is true of S / and S{. 17. If lines be drawn through points of a Bertrand curve parallel to the binor- mals of the conjugate curve, their locus is applicable to a surface of revolution. 18. If a real ruled surface is applicable to a surface of revolution, it is applicable to the right helicoid or to a hyperboloid of revolution of one sheet (cf. Ex. 9, 144). 19. A ruled surface can be deformed in an infinity of ways so that a curve not orthogonal to the generators shall be a line of curvature on the new ruled surface, unless the given curve is a geodesic ; in the latter case the deformation is unique and the line of curvature is plane. 20. Let P be any point of a twisted curve C, and MI, M 2 points on the principal normal to C such that / / /f = - PM 2 = a sin ( { H where a, 6 are constants and p is the radius of curvature of C. The loci of the lines through M\ and 3f 2 parallel to the tangent to C at P are applicable ruled surfaces. 21. On the surface whose equations are x = M, y =f(u)<f> (v) + i//(v), z = /(u)[0(i>) - 00 (u)] + t(o)- fl^ (u), the parametric curves form a conjugate system, the curves u = const, lie in planes parallel to the yz-plane, and the curves v const, in planes parallel to the x-axis ; hence the tangents to the curves u = const, at their points of intersection with a curve v = const, are parallel. 22. Investigate the character of the surfaces of Ex. 21 in the following cases : (a), (v) = Vv2 + l ; (b), (u) = const. ; (c), t(v) = Q; (d), f(u) = au + b. 23. If the equations of Ex. 21 be written the most general applicable surfaces of the same kind with parametric curves cor responding are defined by where AC is a parameter, and the functions 4>i, $2, ^i, ^2 satisfy the conditions $2 + <I>| = 02 + 0| - K, 4>i* + <J> 2 2 = 0{2 + 02, ~ $1(0212 Show also that the determination of 4>i and 3> 2 requires only a quadrature. CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN 145. Reduced form of the linear element. Weingarten has re marked that when we reduce the determination of all surfaces appli cable to a given one to the solution of the equation (IX, 18), namely (1) J = (\-\6)K, we make no use of our knowledge of the given surface, and in reality are trying to solve the problem of finding all the surfaces with an assigned linear element. In his celebrated memoir, Sur la deformation des surfaces,* which was awarded the grand prize of the French Academy in 1894, Weingarten showed that by taking account of the given surface the above equation can be replaced by another which can be solved in several important cases. This chapter is devoted to the exposition of this method. We begin by determining a particular moving trihedral for the given surface. It follows from (VII, 64) that the necessary and sufficient con dition that the directrix of a ruled surface be the line of striction is (2) 6 = a# +#X+*X=0. The functions // m/ n are proportional to the direction-cosines of the curve in which the director-cone of the surface meets the unit sphere with center at the vertex of the cone. We call this curve the spherical indicatrix of the surface. From (2) and the identity ll +mm + nn = it is seen that the tangent to the spherical indicatrix is perpen dicular to the tangent plane to the surface at the corresponding point of the line of striction. This fact is going to enable us to determine under what conditions a ruled surface 2, tangent to a curved surface S along a curve C, admits the latter for its line of striction. *Acta Mathematica, Vol. XX (1896), pp. 159-200. 351 352 DEFORMATION OF SURFACES We suppose that the parameters w, v are any whatever, and that the surface is referred to a moving trihedral. We consider the ruled surface formed by the z-axis of the trihedral as the origin of the latter describes the curve C. The point (1, 0, 0) of a second trihedral parallel to this one, but with origin fixed, describes the spherical indicatrix of 2. From equations (V, 51) we find that the components of a displacement of this point are 0, r du 4- r^v, (qdu + q^dv). In order that the displacement be perpendicular to the tangent plane to 2 at the corresponding point of (7, that is, perpendicular to the zy-plane of the moving trihedral, we must have (3) rdu + r l dv = Q. Hence if a trihedral T be associated with a surface S in any man ner, as the vertex of T describes an integral curve of equation (3), the 2>axis of T generates a ruled surface whose line of striction is this curve. When the parametric lines on S are given, and also the angle U which the a>axis of T makes with the tangent to the curve v const., the functions r and r l are completely determined, as follows from (V, 52, 55). They are Hll cU Hence if U be given the value /// C121 It") *+*<> where <f> (u) denotes an arbitrary function of M, the function r t is zero, and as the vertex of the trihedral describes a curve u = const., the z-axis describes a ruled surface whose line of striction is this curve. Suppose now that the trihedral is such that r x = 0. From (V, 48, 64) it follows that (6) consequently (1) r= C where ty is an arbitrary function of u. PARTICULAR TRIHEDRAL 353 Let the right-hand member of (7) be denoted by f(u, v), and change the parameters of the surface in accordance with the equations u l =u, v^ = f(u, v). From 32 and equation (7) it follows that i- dv Since K is unaltered by the transformation, in terms of the new coordinates H V K is equal to unity, and hence from (6) we have r = v r Therefore the coordinate curves and the moving trihedral of a surface can be chosen in such a way tnat (8) ^=0, r = v, HK=l. In this case we say that the linear element of the surface is in its reduced form. It should be remarked that for surfaces of negative curvature the parameters are imaginary. 146. General formulas. If X^ Y^ Z^ ; A;, F 2 , Zj> X, r, Z denote the direction-cosines of the axes of the moving trihedral with respect to fixed axes, we have, from (V, 47), (9) du il _ Xa - dv " qi dv - dv The rotations p,p^ q<> $i satisfy equations (V, 48) in the reduced form dv du dv du The coordinates x, y, z of S with reference to these fixed axes are given by c / (11) where y = 2 = f(^i + ^2) ^ + (f 1^1 + ^ 2 ) ^N and (13) dv du " 354 DEFORMATION OF SURFACES Weingarten s method consists in replacing the coefficients of f* 77, f t , rj l in the last of equations (13) by differential parameters of u formed with respect to the linear element of the spherical representation of the z-axis of the moving trihedral. By means of (9) this linear element is reducible to (14) da 2 = dX 2 + dY 2 + dZ 2 = (v 2 + q 2 ) du 2 + 2 qq^dudv + q 2 dv 2 . The differential parameters of u, formed with respect to this form, have the values * I _4(y2 + 2 2 ) (15) ^ v *ti A q q p A (u* A u} = Aoit = v^q l v q l vq l Because of the identity (V, 38) we have also (16) A 2 ^ = -^-- If the last of equations (13) be divided by q lt and the values of i Pi/2i obtained from (15) and (16) be substituted, we have 22 t 3 2 i 4 2 v In consequence of the first of equations (15), written (18) v = -L=, VAjt* the coefficients of f, TJ, f v , rj l in (17) are expressible in terms of differential parameters of u formed with respect to (14), as was to be proved. An exceptional case is that in which q^ 0. Under this condition the spherical representation of the z-axis reduces to a curve, as is seen from (14). * Previously we have indicated by a prime differential parameters formed with respect to the linear element of the spherical representation. For the sake of simplicity we dis regard this practice in this chapter. THEOREM OF WE1NGAKTEN 355 By means of (9) we find that (19) A, (A,, tO = ^. M^, ) = ^f. \(Z ) = f and consequently equations (11) may be written (20) x =[^ + W\ (X *>)] du + [f ^ + v,v\ (X u)] dv, and similarly for y and 2. 147. The theorem of Weingarten. Equation (17) is the equation which Weingarten has suggested as a substitute for equation (1). We notice that f, ?;, f x , T/ I are known functions of u and v when the surface S is given. By means of (18) equation (17) can be given a form which involves only u and differential parameters of u formed with respect to (14). On account of the invariant character of these differential parameters this linear element may be expressed in terms of any parameters, say u and v . We shall show that each solution of equation (17) determines a surface applicable to S. We formulate the theorem of Wein garten as follows : Let S be a surface whose linear element in the reduced form is (21 ) ds* = (? 2 + T? 2 ) du* + 2 (^ + wj dudv + (tf + then %, rj, fj, ?; 1 are functions of u and v such that Furthermore, let X v Y v Z^ le the coordinates of a point on the unit sphere, expressed in terms of any two parameters u 1 and v , the linear element of the sphere being (23) da 1 2 = & du * + 2 & du dv + >dv \ Any integral u l of the equation (24) Ju, L A M u - Ju, -l= - u, A,w t )= 0, 356 DEFORMATION OF SURFACES the differential parameters being formed with respect to (23), renders the following expression and similar ones in y and z total differentials: (25) where f Ae surface whose coordinates are the functions x, y, z thus defined has the linear element (21). Before proving this theorem we remark that the parameters u and v may be chosen either as known functions of u and v, or in such a way that the linear element (14) shall have a particular form. In the former case X^ Y v Z^ are known as functions of u f and v , and in the second their determination requires the solution of a Riccati equation. However, in what follows we assume that X v Y v Z l are known. Suppose now that u r and v f are any parameters whatever, and that we have a solution u^ of equation (24), where the differential parameters are formed with respect to (23). Let v^ denote the quantity (A^)"*. Both u^ and v l are functions of u and v , and consequently the latter are expressible as functions of the former. We express X^ Y^ Z { as functions of u l and v l and determine the corresponding linear element of the unit sphere, which we write (26) dffl = (; dul + 2 ^ dujvt + ^ c(v*. In terms of u l and v l we have , i From these expressions it follows that if we put we have ( 27 ) METHOD OF WEINGARTEN 357 Hence if we put x = Y& - Z,Y Y = z& - x&, z =x 1 Y a - r^, the functions A^, Y v - . , Z satisfy a set of equations similar to equa tions (V, 47). In consequence of (27) the corresponding rotations have the values dX dX .dx r 1= 0. It is readily shown that these functions satisfy equations similar to (10). Since the functions f, 77, , ^ are of the same form in (25) as in (21), equations similar to the first two of equations (13) are neces sarily satisfied. Hence the only other equation to be satisfied, in order that the expressions (25) be exact differentials, is But it can be shown that the coefficients of (26) are expressible in the form g _ v *\ ^ <% __ ^ _ -2 so that by means of differential parameters of u^ formed with respect to (26) the equation (28) can be given the form (17). Hence all the conditions are satisfied, and the theorem of Wein- garten has been established. 148. Other forms of the theorem of Weingarten. It is readily found that equations (22) are satisfied by the expressions (29) dv du dv dv where (/> is any function of u and v. Since now (30) ,7 + ^=0, equation (17) reduces to 358 DEFOKMATION OF SURFACES This equation will be simplified still more by the introduction of two new parameters which are suggested by the following considerations. As previously defined, the functions X^ Y v Z^ are the direction- cosines of lines tangent to the given surface S in such a way that the ruled surface formed by these tangents at points of a curve u = const, has this curve for its line of striction. Moreover, from the theorem of Weingarten it follows that the functions X^ Y v Z l have the same significance for the surface applicable to S which corresponds to a particular solution of equation (17). But X v Y v Z l may be taken also as the direction-cosines of the normals to a large group of surfaces, as shown in 67. In partic ular, we consider the surface S which is the envelope of the plane $ + Zj, = u. Each solution of equation (17) determines such a surface. If x, y, z denote the coordinates of the point of contact of this plane with S, we have from (V, 32) (32) x = uX l +\(u,X 1 ), which, in consequence of (19), may be written (32 ) i = wX-f-X. v Hence the point of contact of S lies in the plane through the origin parallel to the tangent plane to S at the corresponding point. If the square of the distance of the point of contact from the origin be denoted by 2 ^, and the distance from the origin to the tangent plane by p,* we have (33) 2g = s a +y a +i a =w a +^. p = u . From (V, 35, 37) it follows that the principal radii of 2 are given by (34) * The reader will observe that the functions p and q thus defirfed are different from the rotations designated by the same letters. As this notation is generally employed in the treatment of the theorem of Weingarten, it has seemed best to retain it, even at the risk of a confusion of notation. METHOD OF WEINGARTEN 359 where the differential parameters are formed with respect to (14). From these equations we have (35) We shall now effect a change of parameters, using p and q defined by (33) as the new ones. By direct calculation we obtain y^r *^r I ^*r . ^*r v *r o - - ^ - _ o _ -i ^ ^ . - -.;? o (36) du dp dq dv v* dq dif dp* + P 2 T + ~ a/?a^ ^ 2 a^ _i z. /^ Q^,2-r cudv vdpcq ^-1^ 1 dv 2 ~ v 6 dq 2 tf By means of the equations (33) and (36) the fundamental equa tion (31) can be reduced to (37) This is the form in whicli the fundamental equation was first con sidered by Weingarten.* The method of 146, 147 was a subse quent development. In terms of the parameters p and q the formulas (29) become dpdq dq (38) If these values and the expression for \(u, X^) given by (32) be substituted in (20), it is reducible to dp* dpdq/ " \ "tip tiq cq * Comptes Rendus, Vol. CXII (1891), p. 607. 360 DEFORMATION OF SURFACES Hence the equations for S may be written (39) and consequently the linear element of S is of the form Since these various expressions and equations differ only in form from those which figure in the theorem of Weingarten, the latter is just as true for these new equations. We remark also that the right- hand member of (40) depends only upon the form of c. Hence we have the theorem of Weingarten in the form : When (j) in equation (37) is a definite function of p and q, this equation defines a large group of surfaces with the same spherical representation, the functions p l and p 2 denoting the principal radii, and p and 2q the distance from the origin to the tangent plane and the square of the distance to the point of contact. Each surface 2 satisfying this condition gives by quadratures (39) a surface with the linear element (40). Conversely, each surface with this linear element stands in such relation to some surface satisfying the corresponding equation (37). As a corollary to the preceding results, we have the theorem : The linear element of any surface S is reducible to the form (41) di~ = du* + 2 ^ dudv + 2 ^ dv\ du dv where ^r is a function of u and v. For, we have seen that the linear element of any surface is reducible to the form (40). If, then, we change the parameters by means of the equations we have (43) ds* = du*+2p dudv + 2q dv\ METHOD OF WEINGARTEN 361 From (42) it follows that a 2 , 3 2 <t> , tf$ , , d 2 4> , <fu = ^ dp + -i- <fy, dt> = i- dp + ^ rfg, dp 1 ejpdg dp 3 a/ and consequently A (44) where A = dp 2 dq 2 \dp dq From (44) it is seen that ~- =-^j and consequently the inverse dv cu of equations (42) are of the form fAfii W W (45) = -X, q = ^L. du * dv Hence equation (43) is of the form (41), as was to be proved. Moreover, equations (44) reduce to (46) = A -= ---- A 2 dpdq In terms of these parameters w, v equations (39) reduce to (47) dx = X^du + ^c?v, (#?/ = I^c^tt + ^^v, dz = Z^du -f ^c?v. Hence the coordinates of 2 are given by AQ _ ^C _ 3v ^2 (48) X = T^ y = -- J z = ^" f dv dv dv and the direction-cosines of the normal to 2 are (49) X.A r l = ^, Z 1= =^, aw aw ^ that is, the normals to 2 are parallel to the corresponding tangents to the curves v = const, on S. Hence we have the following theorem : When the linear element of a surface is in the form (41), the sur face 2 whose coordinates are given by (48) has the same spherical 362 DEFORMATION OF SURFACES representation of its normals as the tangents to the curves v = const. on S. If p and 2q denote the distance from the origin to the tangent plane to S and the square of the distance to the point of contact, they have the values (45). Moreover, if the change of parameters defined by these equations be expressed in the inverse form /cn , d<f> d(f> (50) M = s v= * dp dq the principal radii of 2 satisfy the condition and the coordinates of S are given by quadratures of the form (52) dx Moreover, ever// surface with the same representation as 2, and whose functions p v /? 2 , p, q satisfy (51) for the same (f>, determines by equa tions of the form (52) a surface applicable to S.* 149. Surfaces applicable to a surface of revolution. When the linear element of a surface applicable to a surface of revolution is written (53) d?=du* + p*(u l )dv*, and the z-axis of the moving trihedral is tangent to the curve v const., the function r is equal to zero, as follows from (4). In order to obtain the conditions (8), we effect the transformation of variables u = v^ v = u^ so that the linear element becomes (54) ds* = p 2 du*+dv 2 . Now r = p f , 7^=0, and consequently in order to have the linear element in the reduced form we must take (55) u = u, v=p (v). * For a direct proof of this theorem the reader is referred to a memoir by Goursat, Sur un theoreme de M. Weingarten, et sur la the orie des surfaces applicables, Toulouse Annales, Vol. V (1891) ; also Darboux, Vol. IV, p. 316, and Bianchi, Vol. II, p. 198. SURFACES OF REVOLUTION 363 From these results and (32 ) we find that the coordinates of the surface 2 are given by p f du^ p dv l p cu v p dv l i== _lJl + !!l.^, p cu^ p d Vl and the direction-cosines of the normals to 2 are Y 1 fo v 1 fy ~ 1 8* .A., = -- > JL 1 -- 5 Zs* = -- p dv l p cv l p dv 1 Also, we have (56) P =^xX^v v 2 ? =^ 2 =^ + ~ Hence we have the theorem : To a curve which i* the deform of a meridian of a surface of revo lution there corresponds on the surface 2 a curve such that the tangent planes to 2 at points of the curve are at a constant distance from the origin, and to a deform of a parallel there corresponds a curve such that the projection of the radius vector upon the tangent plane at a point is constant. For the present case 77 = f 1 = ; consequently we have, from (38), Sf This equation is satisfied by (57) </>(>, #)=/(2 q jt? 2 ), where / is any function whatever. In terms of this function we have, from (38), where the accents indicate differentiation with respect to the argument, 2qp 2 . By means of (55) the linear element (54) can be transformed into the function a)(v) being defined by 364 DEFORMATION OF SURFACES Since 77 = f l = 0, we have and we know that r = v. Now equations (58) become and these are consistent because of the relation 2^ p~ = \/v 2 , which results from (56). Hence we have the theorem : When < (p, q) is a function of 2 q p\ the corresponding surface S is applicable to a surface of revolution, the tangents to the deforms of the parallels being parallel to the corresponding normals to 2. If we give <j> the form (57) and put ^ = 2f, the linear element of Sis (59) ds 2 = (^q p 2 ) d^ 2 + ^fr 2 dp 2 , as follows from (40) or (58). 150. Minimal lines on the sphere parametric. In 147 we re marked that the parametric curves on the sphere may be any what ever. An interesting case is that in which they are the imaginary generatrices. In 35 we saw that the parameters of these lines, say a and /3, can be so chosen that a/3-1 (60) X,= Consequently (61) - + a/3 da 2 =dX? + a/3 rp 4 dad /3 From (32) we find that the coordinates of 2, the envelope of the plane Xx + Yy+Zz p = are (62) z = From these we obtain (63) 2 MINIMAL LINES ON THE SPHEKE 365 By means of (34) the expressions for p l + p 2 and p^ in terms of p and its derivatives with respect to a and yS can be readily found, and thus the fundamental equation (37) put in a new form. However, it is not with the general case that we shall now concern ourselves, but with a particular form of the function <j>(p, q). This function has been considered by Weingarten * ; it is (64) In this case so that equation (37) reduces to (65) />i + ft = -(2;> + which, in consequence of (34), may be written (66) (P) dad j3 (l + a/3) 2 When the values from (62) are substituted in (52), we obtain (67) where (68) z = *& - Cu 1 dZ l + J \ occ From (42) and (64) we have u qp 2 a> f ( p), v=p. Hence the linear element (43) of *S is, in this case, (69) ds 2 =du 2 +2v dudv + 2 [u + v 2 + w (v)] dv 2 . *Acta Mathematica, Vol. XX (1896), p. 195. 366 DEFORMATION OF SURFACES However, from (68) it is seen that v 2 (70) Ul =u + -, so that (69) may be written (71) ds 2 = dul + 2 [ MI + (t>)] *> a . Gathering together these results, we have the theorem : 2%e determination of all the surfaces ivith the linear element (71) reduces to the integration of the equation The integral of this equation for o(_p) arbitrary is not known. However, the integral is known in certain cases. We consider several of these. 151. Surfaces of Goursat. Surfaces applicable to certain parab oloids. When we take (73) v (p)=im(l-m)p\ m being any constant, equation (72) becomes m(l-m)p dad ft (l + aj3f The general integral of this equation can be found by the method of Laplace,* in finite form or in terms of definite integrals, accord ing as m is integral or not. The linear element of the surface S is (75) ds 1 = du? + [2 u^ H- m (1 - m) ^] dv\ And the surfaces 2 are such that (76) p 1 + p t +2p = m(m-I)p, that is, the sum of the principal radii is proportional to the dis tance of the tangent plane from a fixed point. These surfaces were first studied by Goursat, f and are called, consequently, the surfaces of G- our sat. *Darboux, Vol. II, p. 66. t American Journal, Vol. X (1888), p. 187. SURFACES OF GOURSAT 367 Darboux has remarked* that equation (71) is similar to the linear element of ruled surfaces (VII, 53). In fact, if the equations of a ruled surface are written in the form (77) x = x +lu l , y where # , ; /, w, n are functions of v alone, which now is not necessarily the arc of the directrix, the linear element of the surface will have the form (71), provided that (78) 2J 2 = 1, 2a^ = 0, 2X 2 = 2 w (^), 2a# = l, 2/ 2 =0. In consequence of the equations 2ft =0, it follows that a ruled surface of this kind admits an isotropic plane director. If this plane be x + iy = 0, that is, if we have where V is a function of v. By means of these values and equa tions (78), we can put (77) in the form *dv (79) = %Vu^+% Cv v dv - C~ dv, /y yt dv - We shall find that among these surfaces there is an imaginary paraboloid to which are applicable certain surfaces to which Wein- garten called attention. To this end we consider the function _2p^ (80) (_p)=: ^ficp 2 tee v " where K denotes a constant. Now equation (66) becomes _2 A Vic _ 1 * Vol. IV, p. 333. 368 DEFOKMATION OF SURFACES In consequence of the identity the preceding equation is equivalent to log(l + a/3)V^ = - dad/3 If we put JL this equation takes the Liouville form -20 > dad IB of which the general integral is 1 A S where A and B are functions of a and /3 respectively, and the accents indicate differentiation with respect to these. Hence the general integral of (81) is c % = 2^ + AS) VAB (l + a/3) and the linear element of S is (82) ds 2 = du^2\u l - V^K - 2 If now, in addition to (80), we take V the equations (79) take such a form that (83) (x+iy)x = icz. Hence the surfaces with the linear element (82) are applicable to the imaginary paraboloid (83). The generator x + iy = Q of this paraboloid in the plane at infinity is tangent to the imaginary circle at the point (x:y:z = l:i:Q), which is a different point from that in which the plane at infinity touches the surface, that is, the point of intersection of the two generators. DEFOBMATION OF PARABOLOIDS 369 Another interesting case is afforded when m in (73) has the value 2. Then u> (v) = v 2 , and equation (71) becomes (84) ds 2 = duf + 2 ( Ml - If we take V=v/^/2^c, we obtain from equations (79) from which we find, by the elimination of u l and v, (85) (z + i 2 = K (x it/) . The generator x + iy = in the plane at infinity on the paraboloid (85) is tangent to the imaginary circle at the point (x: y: z 1 : i: 0), just as in the case of the paraboloid (83), but the paraboloid (85) is tangent to the plane at infinity at the same point. GENERAL EXAMPLES 1 . A moving trihedral can be associated with a surface in an infinity of ways so that as the vertex of the trihedral describes a curve u = const, the z-axis generates a ruled surface whose line of striction is this curve. 2. The tangents to the curves v = const, on a surface at the points where these curves are met by an integral curve of the equation form a ruled surface for which the latter curve is the line of striction. 3. If the ruled surface formed by an infinity of tangents to a surface S has the locus of the points of contact for its line of striction, this relation is unaltered by deformations of S. 4 . Show that if D, D , D" are the second fundamental coefficients of a sur face with the linear element (53), the equation of the lines of curvature of the associated surface S is reducible to Ddii! + D dvi V dui + D"d ; dui pp dv\ p 5 . Show that the surface S associated by the method of Weingarten with a sur face S applicable to a surface of revolution corresponds with parallelism of tangent planes to the surface S complementary to S with respect to the deforms of the meridians ; and that the lines of curvature on S and S correspond. 370 DEFORMATION OF SURFACES 6. Show that when has the form (57), the equation (51) is reducible to hence the determination of all the surfaces applicable to surfaces of revolution is equivalent to the determination of those surfaces S which are such that if MI and M 2 are the centers of principal curvature of 2 at a point I/, and N is the projection of the origin on the normal at M, the product NMi NM 2 is a function of ON. 7. Given any surface S applicable to a surface of revolution. Draw through a fixed point O segments parallel to the tangents to the deforms of the meridians and of lengths proportional to the radii of the corresponding parallels, and through the extremities of these segments draw lines parallel to the normals to S. Show that these lines form a normal congruence whose orthogonal surfaces 2 have the same spherical representation of their lines of curvature as S and are integral sur faces of the equation of Ex. 6. 8. Let fi be a surface applicable to a surface of revolution and S the surface complementary to S with respect to the deforms of the meridians ; let also S and S be surfaces associated with S and S respectively after the manner of Ex. 7. Show that corresponding normals to S and S are perpendicular to one another, and that the common perpendicular to these normals passes through the origin and is divided by it into two segments which are functions of one another. 9. Show that a surface determined by the equation 2 q + K + (PI + pz)p + PIPZ = 0, where K is a constant, possesses the property that the sphere described on the seg ment of each normal between the centers of principal curvature with this segment for diameter cuts the sphere with center at the origin and of radius V K in great circles, orthogonally, or passes through the origin, according as K is positive, nega tive, or zero. These surfaces are called the surfaces of Bianchi. 10. Show that for the surfaces of Bianchi the function 0(p, q) is of the form . = V2 q - p 2 + /c, and that the linear element of the associated surface S applicable to a surface of revolution is 1 Show also that according as /c = 0, > 0, or < the linear element of S is reducible to the respective forms ds 2 = dw 2 + e 2M dv 2 , ds 2 = tanh 4 u du 2 + sech 2 w du 2 , ds 2 = coth 4 u du 2 -f csch 2 u dv 2 . On account of this result and Ex. 10, p. 318, the surfaces of Bianchi are said to be of the parabolic, elliptic, or hyperbolic type, according as K = 0, > 0, or < 0. 11. Let S be a pseudospherical surface with its linear element in the form (VIII, 32), and Si the Bianchi transform whose linear element is (VIII, 33). Find the coordinates x, y, z of the surface S associated with Si by the method of Wein- garten, and show that by means of Ex. 8, p. 291, the expression for x is reducible to | x = ae a (cos 6X1 + sin 0JT 2 ) 4- fX, where X\, JT 2 , X are the direction-cosines with respect to the sc-axis of the tangents to the lines of curvature of S and of the normal to the latter. GENERAL EXAMPLES 371 12. Show that the surfaces S and S of Ex. 11 have the same spherical represen tation of their lines of curvature, that S is a surface of Bianchi of the parabolic type, and that consequently there is an infinity of these surfaces of the parabolic type which have the same spherical representation of their lines of curvature as a given pseudospherical surface S. 13. Show that if Si and S 2 are two surfaces of Bianchi of the parabolic type which have the same spherical representation of their lines of curvature, the locus of a point which divides in constant ratio the line joining corresponding points of Si and S 2 is a surface of Bianchi with the same representation of its lines of cur vature, and that it is of the elliptic or hyperbolic type according as the point divides the segment internally or externally. 14. When S is a pseudospherical surface with its linear element in the form (VIII, 32), the coordinates x~i, yi, z\ of the surface S determined by the method of Weingarten are reducible to A J- Xi = (ae a cos 6 + y sin 6) X\ -f (ae a sin 77 cos 0} JF 2 , and analogous expressions for yi and z 1? where JTi, FI, Z x ; JT 2 , F 2 , Z 2 are the direction-cosines of the tangents to the lines of curvature of S. Show also that S has the same spherical representation of its lines of curvature as the surface Si with the linear element (VIII, 33). 15. Derive from the equations xXi + yYi + zZ t =p, X2 + 2 + 2 = 2 g, by means of (44), (48), and (49), the equations where x, y, 2 are the coordinates of S. 16. Show that the equations for S similar to (IV, 27) are reducible to dudv cv* \cu* cucv and similar expressions in y and z. Derive therefrom (cf. Ex. 15) the equations D du + D"dv + r(Ddu + lYdv) - 0, du dpdq dp* \dq 2 cpdq where D, D , D" are the second fundamental coefficients of 8. 17. Show that the lines of curvature on S correspond to a conjugate system on S (cf. Ex. 16). 1 8. Show that for the surface S we have dx dX\ dx dXi . dXi 2p ~ plp< * e q dq~ dp dq 19. Let S be the surface defined by (67) and Si the surface whose coordinates are Xi = x u\X\, yi = y WiFi, z\ z u\L\. Show that Si is an involute of /S, that the curves p = const, are geodesies on S and lines of curvature on Si, and that the radii of principal curvature of Si are 372 DEFORMATION OF SURFACES 20. Show that when m in (73) is or 1, the function p is the sum of two arbi trary functions of a and /3 respectively, that the linear element of S is ds*= dw 1 2 + 2i*idw 2 , that S is an evolute of a minimal surface (cf. Ex. 19), and that the mean evolute of S is a point. 21. Show that when m in (73) is 2, the general integral of equation (74) is where /i and / 2 are arbitrary functions of a and respectively. Show also that the surface 2 is minimal (cf. 151). 22. Show that the mean evolute of a surface of Goursat is a surface of Goursat homothetic to the given one. 23. Show that when u>(p) = ^op 2 , then p = a log(l + a/3) +/i(nr) +/ 2 (/3), where /i and / 2 are arbitrary functions, that the linear element of S is ds 2 = du? + 2 (ui + aw) du 2 , and that the mean evolute of 2 is a sphere. 24. Show that the surfaces S of Ex. 23 are applicable to the surfaces of revolu tion S whose equations are v . v a a where a is an arbitrary constant. Show also that when a = ia, is a paraboloid. 25. Show that when c /~17 = I vu* a 2 J the surfaces S are spherical or pseudospherical according as m is positive or nega tive ; also that the surfaces <S are applicable to the surface 1)2 M 2 x + iy = v, x-iy = TT~~ mu z = w 2 2m which is a paraboloid tangent to the plane at infinity at a point of the circle at infinity. CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152. General problem. The preceding chapters deal with pairs of isometric surfaces which are such that in order that one may be applied to the other a finite deformation is necessary. In the present chapter we shall be concerned with the infinitesimal deformations which constitute the intermediate steps in such a finite deformation. Let x^y,z\ x , y\ z respectively be the coordinates of a surface S and a surface S\ the latter being obtained from the former by a very small deformation. If we put (1) x ^x + ex^, y f =y + cyv z ^z + ez^ where e denotes a small constant and x v y^ z l are determined func tions of u and v, these functions are proportional to the direction- cosines of the line through corresponding points of S and S f . From these equations we have dx * + dy * + dz n = da? + dy 2 + dz 2 + 2e(dx dx l + dy dy l + dz dzj If the functions satisfy the condition (2) dx dx l -f dy dy^ + dz dz l = 0, corresponding small lengths on S and S r are equal to within terms of the second order in e. When e is taken so small that e 2 may be neglected, the surface S defined by (1) is said to arise from S by an infinitesimal deformation of the latter. In such a deformation each point of S undergoes a displacement along the line through it whose direction-cosines are proportional to x v y v z r These lines are called the generatrices of the deformation. It is evident that the problem of infinitesimal deformation is equivalent to the solution of equation (2). Since x v y^ z l are 373 374 INFINITESIMAL DEFORMATION functions of u and v, they may be taken for the coordinates of a surface r Equation (2) expresses the fact that the tangent to any curve on S is perpendicular to the tangent to the correspond ing curve on S l at the homologous point. We say that in this case S and S l correspond with orthogonality of corresponding linear ele ments. And so we have: The problem of the infinitesimal deformation of a surface S is equivalent to the determination of the surfaces corresponding to it with orthogonality of linear elements. 153. Characteristic function. We proceed to the determination of these surfaces S v and to this end replace equation (2) by the equivalent system Weingarten * replaced the last of these equations by the two ^ ex fix. ^-v dx dx l (4) Sssw-** X^w=~* 7/ thus denning a function $, which Bianchi has called the character istic function ; as usual H = V EG F*. If the first of equations (3) be differentiated with respect to v, and the second with respect to M, we have -- dudv ** fa fr u $ v fa du dv * dv du dv With the aid of these identities, of the formulas (V, 3), and of the Gauss equations (V, 7), the equations obtained by the differentia tion of equations (4) with respect to u and v respectively are reducible to . H v H * Crelle, Vol. C (1887), pp. 296-310. CHARACTERISTIC FUNCTION 375 Excluding the case where S is a developable surface, we solve these <r-\ dx. TT-\ dx. equations for >,^ 2*X-zr anc * obtain *-l du ^ dv (5) cu u dv KH dv where K denotes the total curvature of S. If we solve equations (3), (4), and (5) for the derivatives of x^ y^ z l with respect to u and v, we obtain (6) v cu CU dv KH dv dv KH and similar expressions in y^ and z r Hence, when the characteristic function is known, the surface S l can be obtained by quadratures. Our problem reduces therefore to the determination of <. If equations (5) be differentiated with respect to v and u respec tively, and the resulting equations be subtracted from one another, we have V 1 V i +4 to du 2j fa. fa. When the derivatives of Jf, F, Z in the right-hand member are replaced by the expressions (V, 8), the above equation reduces to dv u d KH du du dv KH H Bianchi calls this the characteristic equation. In consequence of (IV, 73, 74) equation (7) is reducible to ft 376 INFINITESIMAL DEFORMATION where <", c^, $ are the coefficients of the linear element of the spherical representation of /S, namely (9) da* = &du* + 2 ^ dudv and /K By means of (V, 27) equation (8) is reducible to where the Christoffel symbols are formed with respect to (9). Since X, Y, Z are solutions of equations (V, 22), they are solu tions of (10), and consequently also of equation (7). Therefore the latter equation may be written dv dX du KH du v _ But this is the condition of integrability of equations (6). Hence we have the theorem : Each solution of the characteristic equation determines a surface S^ and consequently an infinitesimal deformation of S. 154. Asymptotic lines parametric. When the asymptotic lines on 8 are parametric, equation (10) is reducible, in consequence of (VI, 15), to where If we put dudv 2 dv du 2 du dv <f, V- ep = 6, ASYMPTOTIC LINES PARAMETIC 377 e being +1 or 1 according as the curvature of S is positive or negative, equation (11) becomes (12) ** Since X, Y, Z are solutions of (11), the functions v l = X V e/j, v 2 = F V ep, are solutions of (12). Now equations (6) may be put in the form e (13) du du dx, dv e cB cv The reader should compare these equations with the Lelieuvre formulas ( 79), which give the expressions for the derivatives of the coordinates of S in terms of i/ 1? i/ 2 , i> 3 . From these results it follows that any three solutions of an equation of the form ffQ = MQ, where M is- any function of u and w, determine a surface S upon which the parametric curves are the asymptotic lines, and every other solution linearly independent of these three gives by quad ratures an infinitesimal deformation of S. EXAMPLES 1. A necessary and sufficient condition that two surfaces satisfying the condi tion (2) be applicable is that they be minimal surfaces adjoint to one another. 2. If x, y, z and x l9 T/I, zi satisfy the condition (2), so also do , 17, f and &, ^j, ft, the latter being given by = aix + biy + ciz + di, xi = a^ + a 2 Tn + a s ft + ei, 77 = 2 + &22/ + C 2 Z + d 2 , 2/1 = &ll + &2^?l + &3ft + C 2 , f = a 3 x -f 6 8 y + c 3 z + d 8 , Zi = Ci^ + c 2 >?i + c 3 ft + c 8 , where a 1? a 2 , , ei, e 2 , e 3 are constants. 3. A necessary condition that the locus of the point (xi, ?/i, z\) be a curve is that S be a developable surface. In this case any orthogonal trajectory of the tangent planes to S satisfies the condition. 4. Investigate the cases = and = c, where c is a constant different from zero. 5. If Si and S{ correspond to S with orthogonality of linear elements, so also does the locus of a point dividing in constant ratio the line joining corresponding points on Si and S{. 378 INFINITESIMAL DEFORMATION 155. Associate surfaces. The expressions in the parentheses of equation (10) differ only in sign from the second fundamental co efficients, D , Z>J, Z> ", of the surface /7 enveloped by the plane (14) Hence equation (10) may be written (15) D"D Q + DD J - 2 D D[ = 0. This is the condition that to the asymptotic lines upon either of the surfaces S, S there corresponds a conjugate system on the other ( 56). Bianchi applies the term associate to two sur faces whose tangent planes at corresponding points are parallel, and for which the asymptotic lines on either correspond to a conjugate system on the other. Since the converse of the pre ceding results are readily shown to be true, we have the theorem of Bianchi f : When two surfaces are associate the expression for the distance from a fixed point in space to the tangent plane to one is the char acteristic function for an infinitesimal deformation of the other. Hence the problems of infinitesimal deformation and of the determination of surfaces associate to a given one are equivalent. We consider the latter problem. Since the tangent planes to S and S Q at corresponding points are parallel, we have dz n dx dx dx n dx dx -2 = X -- fji , -5 = 0- -- r du du dv dv du du and similar equations in y Q and 2 , where X, ft, <r, r are functions of u and v to be determined. J If these equations be multiplied by -- and added, and ~y ~Y 2V- $ U &M d U likewise by and added, we obtain dv dv dv \D< - * Cf. 67. t Lezioni, Vol. II, p. 9. J The negative signs before p. and r are taken so that subsequent results may have a suitable form. ASSOCIATE SURFACES 379 where Z> , Z> , D " are the second fundamental quantities for $. When these values are substituted in (15), we find (17) X-r=0. Consequently the above equations reduce to du du dv dv du dv If we make use of the Gauss equations (V, 7), the condition of integrability of equations (18) is reducible to du dv where A and B are determinate functions. Since similar equations hold in y and 2, both A and B must be identically zero. Calculating the expressions for these functions, we have the following equations to be satisfied by X, /A, and <r : JL d\ . f22i . rm . rii (19) da_d\ J22\ 2 fl21 fill du dv v 1 J I 1 J I 1 J To these equations we must add (20) 2 \D - pl>" <rZ> = 0, obtained from the last of (16). The determination of the asso ciate surfaces of a given surface referred to any parametric system requires the integration of this system of equations. Moreover, every set of solutions leads to an associate surface. We shall now consider several cases in which the parametric curves are of a particular kind. 156. Particular parametric curves. Suppose that S is a sur face upon which the parametric curves form a conjugate system. We inquire under what conditions there exists an associate sur face upon which also the corresponding curves form a conjugate system. 380 INFINITESIMAL DEFORMATION On this hypothesis we have, from (16), /* = a- = 6, so that equations (19) reduce to <> which are consistent only when that is, when the point equation of S, namely j^ fi21<tf ri2|<tf dudv \l J cu \ZJfo has equal invariants (cf. 165). Conversely, when condition (22) is satisfied, the function X given by the quadratures (21) makes the equations compatible, and thus the coordinates of an associate surface are obtained by quadratures. Hence we have the theorem of Cosserat*: The infinitesimal deformation of a surface S is the same problem as the determination of the conjugate systems with equal point invari ants on S. Since the relation between S and S is reciprocal and the parametric curves are conjugate for both surfaces, these curves on also have equal point invariants. If S be referred to its asymptotic lines, the corresponding lines on S Q form a conjugate system. In this case, as is seen from (16), X is zero and equations (18) reduce to . dx n dx dx n dx . /24 > ) 2 = u a- ; du ^dv dv 3u moreover, equations (19) become n Toulouse Annales, Vol. VII (1893), N. 60. KULED SUEFACES 381 The solution of this system is the same problem as the integra tion of a partial differential equation of the second order, as is seen by the elimination of either unknown. When a solution of the former is obtained, the corresponding value of the other unknown is given directly by one of equations (25). We make an application of these results to a ruled surface, which we suppose to be referred to its asymptotic lines. If the curves v const, are the generators, they are geodesies, and conse quently (VI, 50) p 1 12 Now /* can be found by a quadrature. When this value is sub stituted in the second of equations (25), we have a linear equa tion in <r, and consequently <r also can be obtained by quadratures. Hence we have the theorem : When the curved asymptotic lines on a ruled surface are known, its associate surfaces can be found by quadratures. If S were referred to its asymptotic lines, we should have equations similar to (24). These equations may be interpreted as follows: The tangent to an asymptotic line on one of two associate surfaces is parallel to the direction conjugate to the corresponding curve on the other surface. EXAMPLES 1. If two associate surfaces are applicable to one another, they are minimal surfaces. 2. Every surface of translation admits an associate surface of translation such that the generatrices of the two surfaces constitute the common conjugate system. 3. The surfaces associate to a sphere are minimal. 4. When the equations of the right helicoid are x u cos v, y u sin u, z cro, the characteristic function of any infinitesimal deformation is = ( U+ V) (u 2 + 2 )~ , where U and V are arbitrary functions of u and v respectively. Find the surfaces Si and So, and show that the latter are molding surfaces. 5. If S and S are associate surfaces of a surface S, the locus of a point dividing in constant ratio the joins of corresponding points of So and S6 is an associate of S. 382 INFINITESIMAL DEFOKMATION 157. Relations between three surfaces S, S 1? S . Having thus discussed the various ways in which the problem of infinitesimal deformation may be attacked, we proceed to the consideration of other properties which are possessed by a set of three surfaces $, Stf S Q . We recall the differential equation dxdx l + dydy v + dzdz l = 0, and remark that it may be replaced by the three (26) dXi=z Q dy y Q dz, dy^ x Q dz z Q dx, dz^y^dx x^dy, if the functions # , ?/ , z are such a form that the conditions of integrability of equations (26) are satisfied. These conditions are du dv du dv dx dy, du dv du dv dx_d_z_ du dv dv du o = fe?5, dv du dv du d_x_dz, dv du dy dx ( du dv dv du dv du If these equations be multiplied by - - respectively and added, and likewise by i ? and by JT, F, ^, we obtain, by (IV, 2), " 0y " (27) (28) From the first two of these equations it follows that the locus of the point with coordinates X QJ T/ O , z corresponds to S with paral lelism of tangent planes. In order to interpret the last of these equations we recall from 61 that -*-^ a% ^4 dv 7 A" r Z X Y Z . ^a; % dz a^ dy dz ^it ^?7 du fin du ~dv = fa du dv du dv du a d(Y, Z} ft d(u, v) Y= a d(Z, X) ft d(u, v) a d(X, Y) /if d (u, v) KELATIONS BETWEEN S, S 19 AND S 383 where a is 1 according as the curvature of the surface is positive or negative. If we substitute these values in the left-hand mem bers of the following equations, and add and subtract --- dx dX dX dU dv and --- from these equations respectively, the resulting dv du dv expressions are reducible to the form of the right-hand members du/ \ du dv dv \ du dv By means of these and similar identities, equation (28) can be transformed into ,_ ^. __ D ,^ * Y o Since this equation is equivalent to (15) because of (27), the quantities a; , # , Z Q in (26) are the coordinates of S . Hence when a surface S l is known, the coordinates of the correspond ing surface S are readily found. This result enables us to find another property of and S r If X^ Y^ Z t denote the direction-cosines of the normal to S^ they are given by 1 I d(u, v) l HI d(u, v) l H^ d(u, v) where 7/ t = ^^ F*, E^ F^ G l being the coefficients of the linear element of r If the values of the derivatives of x v y v , z x , as given by (26), be substituted in these expressions, we have, in consequence of (14), (30) X^-^ Y As an immediate consequence we have the theorem : A normal to S t is parallel to the radius vector of S at the corre sponding point. 384 INFINITESIMAL DEFORMATION By means of (30) we find readily the expressions for the second fundamental coefficients J9 t , D[, D J of $ L . If we notice that and substitute the values from (6) and (30) in du du 1 ^ du dv ^4 dv du we obtain (31) From these expressions follow (32) Combining this result with (15), we have : The asymptotic lines upon any one of a group of three surfaces S, S v S Q correspond to a conjugate system on the other two; or, in other words: The system of lines which is conjugate for any two of three surfaces *S f , S v S corresponds to the asymptotic lines on the other. If the curvature of S be negative, its asymptotic lines are real, and consequently the common conjugate system on S 1 and S is real. If these lines be parametric, the second of equations (32) reduces to As an odd number of the four quantities in this equation must be negative, either S Q or S 1 has positive curvature and the other negative. Similar results follow if we begin with the assumption that S l or S Q has negative curvature. If the curvature of S be positive, the conjugate system common to it and S l is real (cf. 56) ; consequently the asymptotic lines RELATIONS BETWEEN S, S AND S 385 on S Q are real, and the curvature of the latter is negative. But we saw that when the curvature of S Q is negative, and of S positive, that of S 1 also is negative. Hence : Given a set of three surfaces S, S^ S Q ; one and only one of them has positive curvature. Suppose that S is referred to the conjugate system corresponding to asymptotic lines on S Q . The point equation of S is We shall prove that this is the point equation of S^ also. If we differentiate the equation with respect to v, and make use of the fact that y and z are solu tions of (33), we have, in consequence of (26), dudv ~~ \dv du~~l)vduj I 1 / #M t 2 J~dv But the expression in parenthesis is zero in consequence of equa tions similar to (24), and hence x l is a solution of (33). Since the parametric curves on S are its asymptotic lines, the spherical representation of and consequently of S must satisfy the condition ^ f!2V d fl2V Hence we have the theorem of Cosserat: The problem of infinitesimal deformation of a surface is the same as the determination of the conjugate systems with equal tangential invariants upon the surface. 158. Surfaces resulting from an infinitesimal deformation. We pass to the consideration of the surface S arising from an infini tesimal deformation of 8. Its coordinates are given by where is a small constant whose powers higher than the first are neglected. Since the fundamental quantities of the first order for $ , namely J" , F f , G , are equal to the corresponding ones for , by 386 INFINITESIMAL DEFORMATION means of (26) the expressions for the direction-cosines X 1 , F , Z of the normal to S are reducible to and similar expressions for Y and Z . The derivatives of X with respect to u and v are reducible by means of (29) to dX^_d_X / dY_ dZ\ ea / D ,^_ D 3X dX dX I OJL vZ\ d I _ .1 uX i vJL i fly 77 I I - ( T) 7) dv dv \ dv dv / /i- \ du dv where a is 1 according as the curvature of $ is positive or negative. When these results are combined with (26) and (34), we obtain ^^_V V (D -D du du ^4 du du /if ^ du \ du dv e +- v **l \9* 9* du cu) *\du du du du The last expression is identically zero, as one sees by writing it fix 1 flX 9 out in full. From this and similar expressions for V- a- aTl , , v/ A ^ ay X , and V , the values for the second fundamental dv du ** dv dv coefficients of S can be given in the form = - 2) T T- =D + jf ** u cu (30) We know that ff is equal to UK according as the curvature of S is positive or negative (cf. 60). Also, by 157, one and only one of three surfaces S, S v S has positive curvature. Recalling that a in the above formulas is 1 according as the curvature of S is positive or negative, we can, in consequence of (31), write equations (36) in the form Z> .*" = .ZX J>, where the upper sign holds when S l has positive curvature. ISOTHERMIC SUBFACES 387 From these equations it is seen that & and D r can be zero sim ultaneously only when D[ is zero. Hence we have: The unique conjugate system which remains conjugate in an infini tesimal deformation of a surface is the one corresponding to a conju gate system on S^, or, what is the same thing, to the asymptotic lines on S Q . In particular, in order that the curves of this conjugate system be the lines of curvature, it is necessary and sufficient that the spherical representation be orthogonal, and consequently that be a minimal surface (cf. 55). From this it follows that the spherical representation of the lines of curvature of S is isothermal. Con versely, if a surface is of this kind, there is a unique minimal sur face with the same representation of its asymptotic lines, and this surface can be found by quadratures. Hence the required infinites imal deformation of the given surface can be effected by quadra tures (26), and so we have the theorem of Weingarten * : A necessary and sufficient condition that a surface admit an infini tesimal deformation which preserves its lines of curvature is that the spherical representation of the latter be isothermal; when such a surface is expressed in terms of parameters referring to its lines of curvature, the deformation can be effected by quadratures. 159. Isothermic surfaces. By means of the results of 158 we obtain an important theorem concerning surfaces whose lines of curvature form an isothermal system. They are called isothermic surfaces (cf. Exs. 1, 3, p. 159). From equations (23) it follows that if the common conjugate system on two associate surfaces is orthogonal for one it is the same for the other. In this case equation (22) reduces to of which the general integral is E U G=r where U and V are functions of u and v respectively. Hence the lines of curvature on S form an isothermal system (cf. 41). * Sitzungsberichte der Konig. Akademie zu Berlin, 1886. 388 INFINITESIMAL DEFORMATION If the parameters be isothermic and the linear element written ds 2 =r(du 2 +dv 2 ), it follows from (21) that (37) X = i, and equations (23) become du ~~ r du dv ~ r dv From these results we derive the following theorem of Bour * and Christoffel : If the linear element of an isothermic surface referred to its lines of curvature be ds * _ r / du * _|_ dl ?\ a second isothermic surface can be found by quadratures. It is asso ciate to the given one, and its linear element is 1 ds? (du + dv ). r From equations (16) and (17) it follows that the equation of the common conjugate system (IV, 43) on two associate surfaces $, S is reducible to (38) fi du 2 + 2 X dudv + a dv 2 = 0. The preceding results tell us that a necessary and sufficient condi tion that S be an isothermic surface is that there be a set of solu tions of equations (19) such that (38) is the equation of the lines of curvature on S. Hence there must be a function p such that p p (ED* FD), 2 X == p (ED 1 GD), <r =?<p (FD" GD f ) satisfy equations (19).f Upon substitution we are brought to two equations of the form S-s- = a;, - = p\ du dv where a and ft are determinate functions of u and v. In order that S be isothermic, these functions must satisfy the condition dv ~~ du When it is satisfied, p and consequently p, X, a are given by quad ratures. * Journal de I Ecole Poly technique, Cahier 39 (18G2), p. 118. / f Cf. Bianchi, Vol. II, p. 30. ISOTHEKMIC SURFACES 389 Consider furthermore the form (39) H(p du* + 2 X dudv + o- dv~). From (37) it is seen that when the lines of curvature are para metric, this expression reduces to 2 dudv. Hence its curvature is zero (cf. V, 12), and consequently the curvature of (39) is zero. From 135 it follows that this form is reducible to du 1 dv l by quad ratures. Hence we have the theorem of Weingarten : The lines of curvature upon an isothermic surface can be found by quadratures. We conclude this discussion of isothermic surfaces with the proof of a theorem of Ribaucour. He introduced the term limit surfaces of a group of applicable sur faces to designate the members of the group whose mean curvature is a maximum or minimum. According to Ribaucour, The limit surfaces of a group of applicable surfaces are isothermic. In proving it we consider a member S of the group referred to its lines of cur vature. Its mean curvature is given by D/E + D"/G. In consequence of equations (36) the mean curvature of a near-by surface is, to within terms of higher order, A necessary and sufficient condition that the mean curvature of S be a maximum or minimum is consequently /j) j>"\ Excluding the case of the sphere for which the expression in parenthesis is zero, we have that DO is zero. Hence the common conjugate system of S and <S is com posed of lines of curvature on the former, and therefore S is isothermic. GENERAL EXAMPLES 1. If x, y, and xi, y\, z\ are the coordinates of two surfaces corresponding with orthogonality of linear elements, the coordinates of a pair of applicable surfaces are given by fc = x + tei, m = y + ty\, n = + <zi, 2 = x txi, -r} 2 y tyi, f 2 = z tei, where t is any constant. 2. If two surfaces are applicable, the locus of the mid-point of the line joining corresponding points admits of an infinitesimal deformation in which this line is the generatrix. 3. Whatever be the surface S, the characteristic equation (7) admits the solu tion = aX + bY + cZ, where a, 6, c are constants. Show that S is the point (a, 6, c) and that equations (26) become xi = cy - bz + d, yi = az ex -\- e, z\ = bx ay + /, where d, e, /are constants; that consequently Si is a plane, and that the infinitesi mal deformation is in reality an infinitesimal displacement. 390 INFINITESIMAL DEFORMATION 4. Determine the form of the results of Exs. 1, 2, where has the value of Ex. 3. 5. Show that the first fundamental coefficients EI, FI, GI of a surface Si are of the form , E 1 = E<f>* - = -F0 2 , _.- dv 6. Let S denote the locus of the point which bisects the segment of the normal to a surface S between the centers of principal curvature of the latter. In order that the lines on 2 corresponding to the lines of curvature on S shall form a conju gate system, it is necessary and sufficient that S correspond to a minimal surface with orthogonality of linear elements, and that the latter surface and S correspond with parallelism of tangent planes. 7. Show that when the spherical representation of the asymptotic lines of a sur face S satisfies the condition \\y a (92 cu ( 2 } cv ( 1 equations (25) admit two pairs of solutions which are such that /x = <r and /j. = <r. On the two associate surfaces S , SQ thus found by quadratures the parametric systems are isothermal-conjugate, and S Q and S Q are associates of one another. 8. Show that the equation of Ex. 7 is a necessary and sufficient condition that two surfaces associate to S be associate to one another. 9. Show that when the sphere is referred to its minimal lines, the condition of Ex. 7 is satisfied, and investigate this case. 10. On any surface associate to a pseudospherical surface the curves correspond ing to the asymptotic lines of the latter are geodesies. A surface with a conjugate system of geodesies is called a surface of Voss (cf. 170). 11. Determine whether minimal surfaces and the surfaces associate to pseudo- spherical surfaces are the only surfaces of Voss. 12. When the equations of a central quadric are in the form (VII, 35), the asso ciate surfaces are given by 2/o = 2 V6 Fj Uu du + f Vv dv\ , z = i where U and V are arbitrary functions of u and v respectively ; hence the associates are surfaces of translation. 13. When the equations of a paraboloid are in the form x=Va(u + 1>), y=Vb(u-v), z = 2uv, the associate surfaces are surfaces of translation whose generators are plane curves ; their equations are x = Va(U + V), y =Vb(V-U), z = 2fuU du where U and V are arbitrary functions of u and v respectively. GENERAL EXAMPLES 391 14. Show that a quadric admits of an infinitesimal deformation which preserves its lines of curvature, and determine the corresponding associate surface. 15. Since the relation between S arid Si is reciprocal, there is a surface <S 3 associate to Si which bears to S a relation similar to that of SQ to Si. Show that the asymptotic lines on S and S 3 correspond, and that these surfaces are polar reciprocal with respect to the imaginary sphere z 2 + ?/ 2 + z 2 + 1 = 0. 16. Since the relation between S and So is reciprocal, there is a surface S% cor responding to S with orthogonality of linear elements which bears to S a relation similar to that of Si to So. Show that the asymptotic lines on Si and S z correspond, that the coordinates of the latter are such that xi-xz = yzo - zy , yi-y z = zx - zz , z\ z* = xy - yx , and that the line joining corresponding points on Si and S 2 is tangent to both surfaces. 17. Show that if S 5 denotes the surface corresponding to S 3 with orthogonality of linear elements which is determined by Si, associate to SB, the surfaces S and S& are related to one another in a manner similar to Si and Sz of Ex. 16. 18. Show that the surface S 4 , which is the associate to Sz determined by So, is the polar reciprocal of <S with respect to the imaginary sphere x 2 + y 2 + z 2 + 1 = 0. 19. If we continue the process introduced in the foregoing examples, we obtain two sequences of surfaces S, Si, $3, Sj, S 7 , Sg, Sn, - , S, So, S 2 , 84, Se, $8, Sio, Show that Sn and S 10 are the same surface, likewise S i2 and S 9 , and that conse quently there is a closed system of twelve surfaces ; they are called the twelve sur faces of Darboux. 20. A necessary and sufficient condition that a surface referred to its minimal lines be isothermic is that j) jj D" F where U and V are functions of u and v respectively. 21. A necessary and sufficient condition that the lines of curvature on an iso thermic surface be represented on the sphere by an isothermal system is that Pi_U P* ~ F where U and V are functions of u and v respectively, the latter being parameters referring to the lines of curvature. Show that the parameters of the asymptotic lines on such a surface can be so chosen that E = G. 22. Show that an isothermic surface is transformed by an inversion into an isothermic surface. 23. If Si and S 2 are the sheets of the envelope of a family of spheres of two parameters, which are not orthogonal to a fixed sphere, and the points of contact of any sphere are said to correspond, in order that the correspondence be conformal, it is necessary that the lines of curvature on Si and S 2 correspond and that these surfaces be isothermic (cf. Ex. 15, Chap. XIII). CHAPTER XII RECTILINEAR CONGRUENCES 160. Definition of a congruence. Spherical representation. A two- parameter system of straight lines in space is called a rectilinear congruence. The normals to a surface constitute such a system ; likewise the generatrices of an infinitesimal deformation of a sur face (cf. 152). Later we shall find that in general the lines of a congruence are not normal to a surface. Hence congruences of normals form a special class ; they are called normal congruences. They were the first studied, particularly in investigations of the effects of reflection and refraction upon rays of light. The first purely mathematical treatment of general rectilinear congruences was given by Kummer in his memoir, Allgemeine Theorie der gradlinigen Strahlensysteme.* We begin our treatment of the subject with the derivation of certain of Rummer s results by methods similar to his own. From the definition of a congruence it follows that its lines meet a given plane in such a way that through a point of the plane one line, or at most a finite number, pass. Similar results hold if a surface be taken instead of a plane ; this surface is called the surface of reference. And so we . r rnay define a con gruence analytically by means of the coordinates of the latter surface in terms of two parameters u, v, and by the direction- cosines of the lines in terms of these parameters. Thus, a con gruence is defined by a set of equations such as *f\( u i v } > y = fz( u -> v )-> z jz( u -> v ) > where the functions / and < are analytic in the domain of u and v under consideration, and the functions (/> are such that * Crelle, Vol. LVII (1860), pp. 189-230. 302 NORMAL CONGRUENCES 393 We make a representation of the congruence upon the unit sphere by drawing radii parallel to the lines of the congruence, and call it the spherical representation of the congruence. When We put the linear element of the spherical representation is (3) da 2 = <f(^ 2 + 2 &dudv If we put ^A dx dX ,, ex dX dx dX dx = we have the second quadratic form (5) ] dxdX= e du 2 + (/ +/ ) rfwdv + g dv\ which is fundamental in the theory of congruences. 161. Normal congruences. Ruled surfaces of a congruence. If there be a surface S normal to the congruence, the coordinates of S are given by (6) x =x + tX, y =y + tY, z =z + tZ, where t measures the distance from the surface of reference to S r . Since is normal to the congruence, we must have (7) which is equivalent to du du dv 3v If these equations be differentiated with respect to v and u respec tively, and the resulting equations be subtracted, we obtain (9) /=/ . Conversely, when this condition is satisfied, the function t given by the quadratures (8) satisfies equation (7). Since t involves an additive constant, equations (6) define a family of parallel surfaces normal to the congruence. Hence : A necessary and sufficient condition for a normal congruence is that f and f be equal. The lines of the congruence which pass through a curve on the surface of reference S form a ruled surface. Such a curve, and 394 RECTILINEAR CONGRUENCES consequently a ruled surface of the congruence, is determined by a relation between u and v. Hence a differential equation of the form (10) Mdu+Ndv = defines a family of ruled surfaces of the congruence. We consider a line l(u, v) of the congruence and the ruled surface 2 of this family upon which I is a generator ; we say that 2 passes through I. We apply to 2 the results of 103, 104. If dsQ denotes the linear element of the curve C in which 2 cuts the surface of reference, it follows from (VII, 54), (3), and (5) that the quantities a 2 and b for 2 have the values \* da 2 ^ dX dx ~ From (VII, 58) we have that the direction-cosi ies X, //., v of the common perpendicular to I and to the line I of parameters u 4- du, v + dv, where dv/du is given by (10), have the values (12) - \ da da which, by means of (V, 31), are reducible to dX ^dX\ , , / ^dX *>dX\-, - & )du + { <?- - 3 dv dv du / \ dv du/ (13) \= &* . da and similar expressions for /A and v. From (12) it follows that .dX , dY t dZ n X ^~ + ^^~" hz/ ^~ =0 - da da da Since dX/da, dY/da, dZ/da are the direction-cosines of the tangent to the spherical representation of the generators of 2, we have the theorem : Given a ruled surface 2 of a congruence ; let C be the curve on the unit sphere which represents 2, and M the point of C correspond ing to a generator L of S ; the limiting position of the common per pendicular to L and a near-by generator of 2 is perpendicular to the tangent to C at M. PRINCIPAL SURFACES 395 162. Limit points. Principal surfaces. By means of (VII, 62) and (12) we find that the expression for the shortest distance 8 between I and V is, to within terms of higher order, dx dy dz dsf da- X dX Y dY dS z dZ When the values (13) for X, /*, v are substituted in the right-hand member of this equation, the result is reducible to (14) /{do- <odu e du + gdv du + g dv If jY denotes the point where this line of shortest distance meets , the locus of jVis the line of striction of 2. Hence the distance of N from the surface $, measured along Z, is given by (VII, 65) ; if it be denoted by r, we have, from (11), n . _ edu*+(f+f) dudv + g dv 2 V 10 / r jC-J.2 i For the present* we exclude the case where the coefficients of the two quadratic forms are proportional. Hence r varies with the value of dv/du, that is, with the ruled surface 2 through I. If we limit our consideration to real surfaces 2, the denominator is always positive, and consequently the quantity r has a finite maxi mum and minimum. In order to find the surfaces 2 for which r has these limiting values, we replace dv/du by , and obtain (16) If we equate to zero the derivative of the right-hand member with respect to , we get a quadratic in t. Since $&*> 0, we may apply to this equation reasoning similar to that used in connection with equation (IV, 21), * Cf. Ex. 1, 171. 396 RECTILINEAR CONGRUENCES and thus prove that it has two real roots. The corresponding values of r follow from (16) when these values of t are substituted in the latter. Because of (17) the resulting equation may be written r where t indicates a root of (17) and r the corresponding value of r. When we write the preceding equations in the form \ r + e] + [$r + \ -o, and eliminate t, we obtain the following quadratic in r: If r^ and r 2 denote the roots of this equation, we have (19) The points on I corresponding to these values of r are called its limit points. They are the boundaries of the segment of I upon which lie the feet of each perpendicular common to it and to a near-by line of the congruence. The ruled surfaces of the con-, gruences which pass through I and are determined by equation (17) are called the principal surfaces for the line. There are two of them, and their tangent planes at the limit points are determined by I and by the perpendiculars of shortest distance at the limit points. They are called the principal planes. In order to find other properties of the principal surfaces, we imagine that the parametric curves upon the sphere represent these surfaces. If equation (17) be written du iu + l (20) = 0, PRINCIPAL SURFACES 397 it is seen that a necessary and sufficient condition that the ruled surfaces v = const., u = const, be the principal surfaces, is From thes"e it follows that since the coefficients of the two funda mental quadratic forms are not proportional, we must have (21) ^=0, /+/ =0. From the first of these equations and the preceding theorem follows the result: The principal surfaces of a congruence are represented on the sphere by an orthogonal system, and the two principal planes for each line are perpendicular to one another. For this particular parametric system equation (13) reduces to ^9X, ^dX. <o du & dv (22) so that the direction-cosines X x , JJL^ v l of the perpendicular whose foot is the limit point on I corresponding to v = const, have the values 1 ay l dz Hence the angle GO between the lines with these direction-cosines and those with (22) is given by . 7 du cos &> = The values of r 1 and r 2 are now e a ri=--r r ^~S ^ so that with the aid of (23) equation (15) can be put in the form (24) r = r l cos 2 &> -f r 2 sin 2 co. This is Hamilton s equation. We remark that it is independent of the choice of parameters. 398 KECTILINEAR CONGEUENCES 163. Developable surfaces of a congruence. Focal surfaces. In order that a ruled surface be developable, it is necessary and suffi cient that the perpendicular distance between very near generators be of the second or higher order. From (14) it follows that the ruled surfaces of a congruence satisfying the condition (25) e du -}-fdv, f du + g dv are developable. Unlike equation (20), the values of dv/du satis fying this equation are not necessarily real. We have then the theorem : Of all the ruled surfaces of a congruence through a line of it two are developable, but they are not necessarily real. The normals to a real surface afford an example of a congruence with real developables ; for, the normals along a line of curvature form a developable surface ( 51). Since /and/ are equal in this case, equations (20) and (25) are equivalent. And, conversely, they are equivalent only in this case. Hence : When a congruence is normal, and only then, the principal surfaces are developable. When a ruled surface is developable its generators are tangent to a curve at the points where the lines of shortest distance meet them. Hence each line of a congruence is tangent to two curves in space, real or imaginary according to the character of the roots of equation (25). The points of contact are called the focal points for the line. By means of (25) we find that the values of r for these points are given by e du -\-fdv _ f du+g dv If these equations be written in the form (p 4- e)du + (&p +f)dv = 0, (<&!> +/ ) du 4- (gp +g}dv = 0, and if du, dv be eliminated, we have (26) DEVELOPABLE SURFACES 399 If p l and p 2 denote the roots of this equation, it follows that (27) A= ^-// From (19) and (27) it is seen that (28) These results may be interpreted as follows : The mid-points of the two segments bounded respectively by the limit points and by the focal points coincide. This point is called the middle point of the line and its locus the middle surface of the congruence. The distance between the focal points is never greater than that between the limit points. They coincide when the congruence is normal. Equation (24) may be written in the forms cos 2 &) = - - 1 sin 2 &) = -- r r r r i 2 r i r i Hence if a) 1 and &) 2 denote the values of &) corresponding to the developable surfaces, we have A* T ^, A* From these and the first of (28) it follows that cos 2 ft) 1 = sin 2 &) 2 , sin 2 ft) 1 = cos 2 ft) 2 , so that (29) cos2&) 1 +cos2ft) 2 =0, and consequently (30) w 1 +o) 2 =|ww, jor (31) ft) 1 ~ft) 2 = 400 RECTILINEAR CONGRUENCES where n denotes any integer. If the latter equation be true, the developable surfaces are represented on the sphere by an orthog onal system, as follows from the theorem at the close of 161. But by 34 the condition that equation (25) define an orthogonal sys tem on the sphere is/=/ , that is, the congruence must be normal. Since in this case the principal surfaces are the developables, equa tion (30) as well as (31) is satisfied. Hence equation (30) is the general solution of (29). The planes through I which make the angles o^, &> 2 with the principal plane w = are called the focal planes for the line ; they are the tangent planes to the two developable surfaces through the line. Incidentally we have proved the theorem: A necessary and sufficient condition that the two focal planes for each line of a congruence be perpendicular is that the congruence be normal. And from equation (30) it follows that The focal planes are symmetrically placed with respect to the prin cipal planes in such a way that the angles formed by the two 2iairs of planes have the same bisecting planes. If 6 denote the angle between the focal planes, then and ^ (32) sin 6 cos 2 a) l cos 2 o) 1 cos 2 o> 2 = - l The loci of the focal points of a congruence are called its focal surfaces. Each line of the congruence touches both surfaces, being tangent to the edges of regression of the two developables through it. By reasoning similar to that employed in the discussion of surfaces of center ( 74) we prove the theorem : A congruence may be regarded as two families of developable sur faces. Each focal surface is touched by the developables of one family along their edges of regression and enveloped by those of the other family along the curves conjugate to these edges. The preceding theorem shows that of the two focal planes through a line I one is tangent to the focal surface S L and the other is the ASSOCIATE NORMAL CONGRUENCES 401 osculating plane of the edge of regression on /S\ to which I is tan gent ; similar results hold for S z . When the congruence is nor mal these planes are perpendicular, and consequently these edges of regression are geodesies on S l and S z . Since the converse is true ( 76), we have: A necessary and sufficient condition that the tangents to a family of curves on a surface form a normal congruence is that the curves be geodesies. EXAMPLES 1. If JT, Y", Z are the direction-cosines of the normal to a minimal surface at the point (cc, T/, z), the line whose direction-cosines are F, X, Z and which passes through the point (x, y, 0) generates a normal congruence. 2. Prove that the tangent planes to two confocal quadrics at the points of con tact of a common tangent are perpendicular, and consequently that the common tangents to two confocal quadrics form a normal congruence. 3. Find the congruence of common tangents to the paraboloids x 2 + y 2 = 2az, x 2 + y* = - 2 az, and determine the focal surfaces. 4. If two ruled surfaces through a line L are represented on the sphere by orthogonal lines, their lines of striction meet L at points equally distant from the middle point. 5. In order that the focal planes for each line of a congruence meet under the same angle, it is necessary and sufficient that the osculating planes of the edges of regression of the developables meet the tangent planes to the focal surfaces under constant angle. 6. A necessary and sufficient condition that a surface of reference of a congru ence be its middle surface is g - (/ + / )<^+ e& = 0. 164. Associate normal congruences. If we put dx dx equations (8) may be replaced by (34) t = c I 7 du + y^dv, where c is a constant. Now equation (9) is equivalent to 402 RECTILINEAR CONGRUENCES In consequence of this condition equation (34) may be written (36) t = c-$(u^ where u l is a function of u and v thus denned. If the orthogonal trajectories of the curves u^ const, be taken as parametric curves v l = const., it follows from (36) and from equations in u l and v l analogous to (33) and (34) that From this result follows the theorem : The lines of a normal congruence cut orthogonally the curves on the surface of reference at whose points t is constant. If denotes the angle which a line of the congruence makes with the normal to the surface of reference at the point of inter section, we have (37) sin *= where the linear element of the surface is If S be taken for the surface of reference of a second congruence whose direction-cosines X v Y v Z l satisfy the conditions where 4>i( u i) i g anv function whatever of u^ this congruence is normal and has the value Since 1 is any function, there is a family of these normal congru ences which we call the associates of the given congruence and of one another. Through any point of the surface of reference there passes a line of each congruence, and all of these lines lie in the plane normal to the curve u l const, through the point. Hence : The two lines of two associate congruences through the same point of the surface of reference lie in a plane normal to the surface. DERIVED CONGRUENCES 403 Combining with equation (37) a similar one for an associate con gruence, we have (38) E* = &) =/(W) sin^ #K) <*> Hence we have the theorem : The ratio of the sines of the angles which the lines of two associate congruences make with the normal to their surface of reference is con stant along the curves at whose points t is constant. When in particular f(u^ in (38) is a constant, the former theorem and equation (38) constitute the laws of reflection and refraction of rays of light, according as the constant is equal to or different from minus one. And so we have the theorem of Malus and Dupin : If a bundle of rays of light forming a normal congruence be reflected or refracted any number of times by the surfaces of successive homo geneous media, the rays continue to constitute a normal congruence. By means of (37) equation (36) can be put in the form t = c l \l E sin 6 du r From this result follows the theorem of Beltrami * : If a surface of reference of a normal congruence be deformed in such a way that the directions of the lines of the congruence with respect to the surface be unaltered, the congruence continues to be normal. 165. Derived congruences. It is evident that the tangents to the curves of any one-parameter family upon a surface S constitute a congruence. If these curves be taken for the parametric lines v = const., and their conjugates for u = const., the developables in one family have the curves v = const, for edges of regression, and the developables of the other family envelop S along the curves u const. We may take S for the surface of reference. If S l be the other focal surface, the lines of the congruence are tangent to the curves u = const, on S r The tangents to the curves v = const, on S 1 form a second congruence of which S l is one focal surface, and the second surface $ 2 is uniquely determined. Moreover, the * Giornale di matematiche, Vol. II (1864), p. 281. 404 RECTILINEAR CONGRUENCES lines of the second congruence are tangent to the curves u = const. on S z . In turn we may construct a third congruence of tangents to the curves v const, on S z . This process may be continued indefinitely unless one of these focal surfaces reduces to a curve, or is infinitely distant. In like manner we get a congruence by drawing tangents to the curves u = const, on S, which is one focal surface, and the other, S_ v is completely determined. The tangents to the curves u const, on S_ l form still another, and so on. In this way we obtain a suite of surfaces which is terminated only when a surface reduces to a curve, or its points are infinitely distant. Upon each of these surfaces the parametric curves form a conjugate system. The congruences thus obtained have been called derived congruences by Darboux.* It is clear that the problem of finding all the derived congruences of a given one reduces to the integration of the equation of its devel- opables (25); for, when the developables are known we have the conjugate system on its focal surfaces. In order to derive the analytical expressions for these results, we recall ( 80) that the coordinates x, y, z of S are solutions of an equation of the form (39) du dv du cv where a and b are determinate functions of u and v. If the coordi nates of S l be denoted by x^ y^ 2 t , they are given by dx By , , dz *-x+\-* fc-jr + x,-. v + Xi S . where \^J~E measures the distance between the focal points. But as the lines of the congruence are tangent to the curves u = const. on S v we must have dx. dx dy. dy dz. dz (40) 1 = M 1 -^ I = u 1 * i = u , dv Pl du dv Pl du dv l du * Vol. II, pp. 16-22. DERIVED CONGRUENCES 405 where /-^ is a determinate function of u and v. When the above value for x l is substituted in the first of these equations, the result is reducible, by means of (39), to L^LI _ flX n\te + (1 __ fog to = 0. dv V du l/ dv Since the same equation is true for y and z, the quantities in paren theses must be zero, that is, 1 a 1 a Hence the surface S l is defined by , 1 ex \dy , 1 dz and equations (40) become /42\ i __ __ ^y Vav b b]du dv \dv b b/du dv \dv b bjdu Proceeding in a similar manner, we find that $_i is defined by the equations /4 ox - 1 a* - 1 to . . 1 * V*; - and that and similar expressions in y_^ and 2_i. From (41) and (43) it is seen that the surface S l or S.i is at infinity, according as b or a is zero. When a and 5 are both zero, S is a surface of translation ( 81). Hence the tangents to the generators of a surface of translation form two congruences for each of which the other focal surface is at infinity. In order that S^ be a curve, x^ y^ z l must be functions of u alone. From (42) it follows that the condition for this is d 1 _a ~dv 5 ~6* In like manner the condition that $_i be a curve is l = i du a a 406 RECTILINEAR CONGRUENCES The functions h and &, denned by h __da , if * du dv are called the invariants of the differential equation (39). Hence the above results may be stated : A necessary and sufficient condition that the focal surface S l or _i be a curve is that the invariant k or h respectively of the point equation of S be zero. 166. Fundamental equations of condition. We have seen ( 160) that with every congruence there are associated two quadratic dif ferential forms. Now we shall investigate under what conditions two quadratic forms determine a congruence. We assume that we have two such forms and that there is a corresponding congruence. The tangents to the parametric curves on the surface of reference at a point are determined by the angles which they make with the tangents to the parametric curves of the spherical representation of the congruence at the corresponding point, and with the normal to the unit sphere. Hence we have the relations ,__._+ I Zti 7)ti (44) and similar equations in y and 2, where #, /3, 7; a r (S v r y l are functions of u and v. If we multiply these equations by > respec- dX dY dZ du dU 3U tively, and add; also by * and by A", Y, Z\ we obtain dv dv dv from which we derive e& j c/ j iy c-c/ a = ^ 2 * p=- * 7 =^.A > / \ f \ (O& ~~ cy (n & ~~~ c/ 01^ (45) Ctf M* fl CV^ *>2 1 foo/ CA (Q^/ c/ 1 yi^^rr FUNDAMENTAL EQUATIONS 407 In order that equations (44) be consistent, we must have du \dv 2v \du which, in consequence of equations (V, 22), is reducible to the form ]t?X+S d -X du dv where J?, S, T are determinate functions. Since this equation must be satisfied by Y and Z also, we must have R 0, S= 0, T= 0. When the values of a, y3, a^ fi v from (45), are substituted in these equations, we have (47) Conversely, when we have a quadratic form whose curvature is +1, it may be taken as the linear element of the spherical rep resentation of a congruence, which is determined by any set of functions e, f, / , #, 7, 7^ satisfying equations (4T). For, when these equations are satisfied, so also is (46), and consequently the coordinates of the surface of reference are given by the quadratures (44). Incidentally we remark that when the congruence is normal, and the surface of reference is one of the orthogonal surfaces, the last of equations (47) is satisfied identically, and the first two reduce to the Codazzi equations (V, 27). We apply these results to the determination of the congruences with an assigned spherical representation of their principal surfaces, and those with a given representation of their developables. 167. Spherical representation of principal surfaces and of devel opables. A necessary and sufficient condition that the principal surfaces of a congruence cut the surface of reference in the para metric lines is given by (21). 408 RECTILINEAR CONGRUENCES If we require that the surface of reference be the middle surface of the congruence, and if r denote half the distance between the limit points, we have, from (15), (48) e r<o, g = r& When these values are substituted in (47), the first two become 12, f? a/ / \ (49) 1 d , 7 = ^ (r < and the last is reducible to * ,50) 2 av i al g^ g? - i aiog^gTjg iog^,. dwy dv cu du dv dudv d \ \$ d i f YL a F W d { f l\4-2f + sLNtfsfe;J + 4N?s\^Jr^ * Moreover, equations (44) become ao: ax / ax , aa: / ax ax where 7 and y l are given by (49) ; and similar equations in y and z. Our problem reduces, therefore, to the determination of pairs of functions r and / which satisfy (50). Evidently either of these functions may be chosen arbitrarily and the other is found by the solution of a partial differential equation of the second order. Hence any orthogonal system on the unit sphere serves for the representation of the principal surfaces of a family of congruences, whose equations involve three arbitrary functions. In order that the parametric curves on the sphere represent the developables of a congruence, it is necessary and sufficient that as is seen from (25). If the surface of reference be the middle sur face, and p denotes half the distance between the focal points, it follows from (15) that e p " c $ * Cf. Bianchi, Vol. I, p. 314. DEVELOPABLES PARAMETRIC 409 Combining these equations with the above, we have (52) e=-p& f = -f = p & ff = p & When these values are substituted in the first two of equations (47) and the resulting equations are solved for 7 and 7^ we find and the last of equations (47) reduces to d ri2V a n 2 = Each solution of this equation determines a congruence with the given representation of its developables,* and the middle surface is given by the quadratures (54) and similar expressions in y and 2. When the values (52) are substituted in (18) the latter becomes Consequently equation (32) reduces to a 2 P sin ^ = -^ = Referring to equation (III, 16), we have: The angle between the focal planes of a congruence is equal to the angle between the lines on the sphere representing the corresponding developables. This result is obtained readily from geometrical considerations. 168. Fundamental quantities for the focal surfaces. We shall make use of these results in deriving the expressions for the funda mental quantities of the focal surfaces S l and $ 2 , which are defined by * This result is due to Guichard, Annales de I Ecole Normale, Ser. 3, Vol. VI (1889), pp. 342-344. 410 RECTILINEAR CONGRUENCES From these and (54) we get The coefficients of the linear elements of ^ and 2 , as derived from these formulas, are (56) and (57) . The direction-cosines of the normals to ^ and S 2 denoted by X^ Y v Z^ JT 2 , r 2 , ^ 2 respectively are found from the above equa tions and (V, 31) to have the values Si) = ._ 3V ^ V \ d(u, v) v . ^ _ 1 ZJ- 2/^, i\ /~^ ^/^s /./V^ /}?> ^ and similar expressions for Y t and Z { . If these equations be differ entiated, and the resulting equations be reduced by means of (V, 22), they can be put in the form K \\ fi2Vax I a*t_^/22Vwr "VL^^lJ"^ x \ dv-^lif dv ^^ "a^" == Til2/ FOCAL SURFACES From these expressions and (55) we obtain (58) and 411 ^ du du ~ V^ \du ^2 D[ = Y ^1 Mi = _ y ?i Mi = o, ^ dv du ^ du cv A"= ~2 ^r "^7 = * From the foregoing formulas we derive the following expressions for the total curvature of S l and of S z : (60) { 22V 1 J EXAMPLES 1. If upon a surface of reference S of a normal congruence the curves orthog onal to the lines of the congruence are defined by 0(u, u) = const., and 6 denotes the angle between a line of the congruence and the normal to the surface at the point of meeting, then sin 2 = AiF(0) where the differential parameter is formed with respect to the linear element of S. Show that 6 is constant along a line = const. only when the latter is a geodesic parallel. 2. When in the point equation of a surface, namely c 2 , c0 , , 30 n - + a + 6 = 0, du cv du cv a or 6 is zero, the coordinates of the surface can be found by quadratures. 3. Find the derived congruences of the tangents to the parametric curves on a tetrahedral surface (Ex. 2, p. 267), and determine under what conditions the sur face Si or 5-i is a curve. 4. Find the equation of the type (39) which admits as solutions the quantities *i, yi, zi given by (41). 5. When a congruence consists of the tangents to the lines of curvature in one system on a surface, the focal distances are equal to the radii of geodesic curvature of the lines of curvature in the other system. 412 RECTILINEAR CONGRUENCES 6. Let S be a surface referred to its lines of curvature, let i and s 2 denote the arcs of the curves v = const, and u = const, respectively, ri and r 2 their radii of first curvature, and RI and JR 2 their radii of geodesic curvature ; for the second focal sheet Si of the congruence of tangents to the curves v = const, the linear element is reducible to 2 hence the curves Si = const, are geodesies. 7. Show that 2 t of Ex. 6 is developable when n =/(si), and determine the most general form of r\ so that 2i shall be developable. 8. Determine the condition which p must satisfy in order that the asymptotic lines on either focal surface of a congruence shall correspond to a conjugate system on the other, and show that in this case where denotes the angle between the focal planes. 9. In order that the focal surfaces degenerate into curves, it is necessary and sufficient that the spherical representation satisfy the conditions { 12 \ ={ 12 \ = ( 12 du \ 1 ) cv \ 2 } ~ \ 1 10. Show that the surfaces orthogonal to a normal congruence of the type of Ex. 9 are cyclides of Dupin. 11. A necessary and sufficient condition that the second sheet of the congruence of tangents to a family of curves on a surface S be developable is that the curves be plane. 169. Isotropic congruences. An isotropic congruence is one whose focal surfaces are developables with minimal edges of regression. In 31 we saw that H = is a necessary and sufficient condition that a surface be of this kind. Referring to (56) and (57), we see that we must have From (54) it is seen that if p were zero the middle surface would be a point, and from (55) that if the expressions in parentheses were zero the surfaces S l and $ 2 would be curves. Consequently (61) < = g = 0. Conversely, if this condition be satisfied, S l and S. 2 are isotropic developables. Hence an isotropic congruence is one whose devel opables are represented on the sphere by minimal lines. ISOTROPIC CONGRUENCES 413 In consequence of (61) we have, from (52), and since f+f also is zero, it follows that (62) dxdX+ dydY+ dzdZ= 0. Therefore r is zero, so that all the lines of striction lie on the middle surface. Since (61) is a consequence of (62), we have the following theorem of Ribaucour,* which is sometimes taken for the definition of isotropic congruences : All the lines of striction of an isotropic congruence lie on the mid dle surface ; and, conversely, when all the lines of striction lie on the middle surface, the congruence is isotropic ; moreover, the middle sur face corresponds to the spherical representation with orthogonality of linear elements. Ribaucour has established also the following theorem : f TJie middle envelope of an isotropic congruence is a minimal surface. Since the minimal lines on the sphere are parametric, in order to prove this theorem it is only necessary to show that on the middle envelope, that is, the envelope of the middle planes, the corresponding lines form a conjugate system. If W denotes the distance of the middle plane from the origin, the condition necessary and sufficient that the parametric lines be conjugate is that W satisfy the equation (63) r + <^0 = 0. By definition and with the aid of (V, 22) we find ft du dv cu dv o2 Since equation (53) reduces to - + /><^=0, the function W> satisfies (63). * Etude des Elasso ides ou Surfaces a Courbure Moyenne Nulle, Memoires Couronnts par r Academic de Belgique, Vol. XLIV (1881), p. 63. t L.c., p. 31. 414 RECTILINEAR CONGRUENCES 170. Congruences of Guichard. Guichard* proposed and solved the problem : To determine the congruences whose focal surfaces are met by the developables in the lines of curvature. With Bianchi we call them congruences of Gruichard. We remark that a necessary and sufficient condition that a con gruence be of this kind is that F l and F 2 of 168 be zero. From (56) and (57) it is seen that this is equivalent to Comparing this result with 78, we have the theorem: A necessary and sufficient condition that the developables of a con gruence meet the focal surfaces in their lines of curvature is that the congruence be represented on the sphere by curves representing also the asymptotic lines on a pseudospherical surface. In this case the parameters can be so chosen thatf <F=^=1, c? = COSQ), where co is a solution of = sin ft). dudv In this case equation (53) is (65) - = p cos <0. In particular, this equation is satisfied by X, F, Z (V, 22). If we replace p by X in (54), we have consequently, for the congruence determined by this value of />, the middle surface is a plane. From (55) it follows that the lines of the congruence are tangent to the lines of curvature v = const, on *Sy Consequently they are *L.c., p. 346. f This is the only real solution of (64). CONGRUENCES OF GUICHARD 415 parallel to the normals to one of the sheets of the evolute of S l (cf. 74) ; call it 2 X . Hence the conjugate system on 2 t corre sponding to the lines of curvature on S^ is represented on the sphere by the same lines as the developables of the congruence. Referring to (VI, 38), we see that condition (64) is equivalent to where the Christoffel symbols are formed with respect to the linear element of 2 r But these are the conditions that the parametric curves on 2 X be geodesies (cf. 85). Surfaces with a conjugate system of geodesies were studied by Voss, * and on this account are called surfaces of Voss. Since the converse of the above results is true, we have the following theorem of Guichard : A necessary and sufficient condition that the tangents to the lines of curvature in one family of a surface form a congruence of Guichard is that one sheet of the evolute of the surface be a sur face of Voss, and that the tangents constituting the congruence be those which are parallel to the normals to the latter. If W l denotes the distance from the origin to the tangent plane to the surface of Voss 2 X , then W l is a solution of equation (65) (cf. 84). Hence W^ + Kp is a solution of this equation, provided K be a constant. But since the tangent plane to 2 X passes through the corresponding point of S v the above result shows that a plane normal to the lines of the congruence, and which divides in con stant ratio the segment between the focal points, envelopes a sur face of Voss. In particular, we have the corollary : The middle envelope of a congruence ofGruichardis a surface of Voss. 171. Pseudospherical congruences. The lines joining correspond ing points on a pseudospherical surface S and on one of its Backhand transforms S 1 (cf. 120) constitute an interesting congruence. We recall that the distance between corresponding points is constant, and that the tangent planes to the two surfaces at these points meet under constant angle. From (32) it follows that the distance between the limit points also is constant. *Miinchener JSerichte, Vol. XVIII (1888), pp. 95-102. 416 KECTILINEAB CONGRUENCES Conversely, when the angle between the focal planes of a con gruence is constant, and consequently also the angle 6 between the parametric lines on the sphere representing the developables, we have, from (V, 4), 111112 i Furthermore, if the distance between the focal points is constant, we have p = a, and by (60) g j n 2 Q K ^ = K * = "4^" Hence the two focal surfaces have the same constant curvature. Congruences of this kind were first studied by Bianchi.* He called them pseudospherical congruences. In order that the two focal surfaces of the congruence be Back- lund transforms of one another, it is necessary that their lines of curvature correspond. It is readily found that for both surfaces the equation of these lines is reducible by means of (66) to { 12V f 12V H2V 2 n^v 2 ! is\2j dw ~[^ + f \is + n 2 ) J Moreover, the differential equation of the asymptotic lines on each surface is dv? ^/di) 2 0. Hence we have the theorems: On the focal surfaces of a pseudospherical congruence the lines of curvature correspond, and likewise the asymptotic lines. The focal surfaces of a pseudospherical congruence are Backlund transforms of one another. EXAMPLES 1. When the parameters of a congruence are any whatsoever, and likewise the surface of reference, a condition necessary and sufficient that a congruence be isotropic is e f + f g ~~ 2^ = ^ 2. A necessary and sufficient condition that a congruence be isotropic is that the locus of two points on each line at an equal constant distance from the middle surface shall describe applicable surfaces. 3. Show that equation (65) admits and as solutions. Prove that in each 3u dv case one of the focal surfaces is a sphere. *Annali, Ser. 2, Vol. XV (1887), pp. 161-172; also Lezioni, Vol. I, pp. 323, 324. JF-CONGRUENCES 41T 4. Determine all the congruences of Guichard for which one of the focal surfaces is a sphere. 5. When a surface is referred to its lines of curvature, a necessary and suffi cient condition that the tangents to the curves v = const, shall form a congruence of Guichard is a / 1 3u\^ 6. Determine the surfaces which are such that the tangents to the lines of curvature in each system form a congruence of Guichard. 172. TF-congruences. We have just seen that the asymptotic lines on the focal surfaces of a pseudospherical congruence correspond ; the same is true in the case of the congruences of normals to a JF-surface (cf. 124). For this reason all congruences possessing this property are called W-congruences. We shall derive other prop erties of these congruences. The condition that asymptotic lines correspond, namely takes the following form in consequence of (58) and (59): 22V Hence from (60) it follows that a necessary and sufficient condition for a JF-congruence is In order to obtain an idea of the analytical problem involved in the determination of TF-congruences, we suppose that we have two surfaces , S referred to their asymptotic lines, and inquire under what conditions the lines joining corresponding points on the surfaces are tangent to them. We assume that the coordinates of the surfaces are defined* by means of the Lelieuvre formulas (cf. 79), thus: (68) dx du dx_ du du du du du dx dx dv dv ~dv dv *Cf. Guichard, Comptes Rendus, Vol. CX (1890), pp. 126-127. 418 RECTILINEAR CONGRUENCES and similar equations in y, z, y, and z. The functions v v v v i> 2 , v s respectively are solutions of equations of the form (69) dudv and they are such that (70) v? + l + vl = a, vl + v$ + vl = a, wliere a and a are defined by (71) J5T = - ,, K = -~. a- a 2 Since v^ v^ v s and v r v z , v s are proportional to the direction- cosines of the normals to S and S, the condition that the lines joining corresponding points be tangent to the surfaces S and S is v^x -z)+ v z (y -y}+ v^(z - z) = 0. Hence x x y v z z where w denotes a factor of proportionality. In order to find its value, we notice that from these equations follow the relations (2 />) 2 = ^(x - x)*= 7?i 2 2(^ 3 - iy> 2 ) a = w where ^ denotes the angle between the focal planes. If this value of 2p and the values of K and K from (71) be substituted in (67), it is found that m* = 1. We take w=l, thus fixing the signs of i/j, i> 2 , i> 3 , and the above equations become (72) x x = V& vfa y y = v^t *&, z - * = W- v 2 v r If the first of these equations be differentiated with respect to w, the result is reducible by (68) to JF-CONGRUENCES 419 Proceeding in like manner with the others, and also differentiating with respect to v, we are brought to /- 7 /- (73) . = 1,2,8) where Z and & are factors of proportionality to be determined. If the first of these equations be differentiated with respect to v, and in the reduction we make use of the second and of (69), we find In like manner, if the second of the above equations be differen tiated with respect to u, we obtain Since these equations are true for i=l, 2, 3, the quantities in parentheses must be zero. , This gives cl 3k du dv x =-^ + In accordance with the last we put and the others become a , i = IQOT - * Hence equations (69) may be written Bdudv dudv l dufo\0. i \ if from which it follows that l is a solution of the first of equa tions (69) and l/0 l of the second. Moreover, equations (73) may now be written in the form 0, v< du dv 420 RECTILINEAR CONGRUENCES Hence if Q l be a known solution of the first of equations (69), we obtain by quadratures three functions v f , which lead by the quadra tures (68) to a surface S. The latter is referred to its asymptotic lines and the joins of corresponding points on S and $ are tangent to the latter. And so we have : If a surface S be referred to its asymptotic lines, and the equations of the surface be in the Lelieuvre form, each solution of the corre sponding equation ffQ = \0 dudv determines a surface S, found by quadratures, such that S and S are the focal surfaces of a W-congruence. Comparing (74) with (XI, 13), we see that if we put ^1=^1. yi= i i=^8 the locus of the point (x^ y^ zj corresponds to S with orthogo nality of linear elements. Hence v v v 2 , v s are proportional to the direction-cosines of the generatrices of an infinitesimal deformation of , so that we have : Each focal surface of a W-congruence admits of an infinitesimal deformation whose generatrices are parallel to the normals to the other focal surface. Since the steps in the preceding argument are reversible, we have the theorem : The tangents to a surface which are perpendicular to the genera trices of an infinitesimal deformation of the latter constitute a W- congruence of the most general kind ; and the normals to the other surface are parallel to the generatrices of the deformation. 173. Congruences of Ribaucour. In his study of surfaces corre sponding with orthogonality of linear elements Ribaucour consid ered the congruence formed by the lines through points on one surface parallel to the normals to a surface corresponding with the former in this manner. Bianchi * calls such a congruence a con gruence of Ribaucour, and the second surface the director surface. In order to ascertain the properties of such a congruence, we recall the results of 153. Let S l be taken for the surface of *Vol. II, p. 17. CONGRUENCES OF RIBAUCOUR 421 reference, and draw lines parallel to the normals to S. If the latter be referred to its asymptotic lines, it follows from (XI, 6) that du dv Since these values satisfy the conditions civ du HK = y dx l dX 9 ~^ the ruled surfaces u = const., v = const, are the developables. And since also p^+ p 2 is equal to zero, S l is the middle surface of the congruence. But the parametric curves on 8 t form a conjugate system when the asymptotic lines on S are parametric. Hence we have the theorem : The developable surfaces of a congruence of Ribaucour cut the middle surface in a conjugate system. Guichard * proved that this property is characteristic of congru ences of Ribaucour. In order to obtain this result, we differentiate the first of equations (54) with respect to v, and in the reduction make use of the fact that X and p satisfy equations (V, 22 ) and (53) respectively. This gives /isyy*. is log p fi2\as. dv \\ } / du \ iu From this and similar equations in y v and z v it follows that a necessary and sufficient condition that the parametric curves form a conjugate system is ^ f!2V d T12V du\ 1 S to\ 2 J When this condition is satisfied by a system of curves on the sphere, they represent the asymptotic lines on a unique surface S, whose coordinates are given by the quadratures (VI, 14) *Annales L Ecole Nonnale, Ser. 3, Vol. VI (1889), pp. 344, 345. 422 RECTILINEAR CONGRUENCES and similar expressions for y and z. Combining these equations with (54), we find that dx _ _ ^\ 3x l dx ^ dx l dx ~ ^\ dx v dx du = Z du dv ** dv du = ^~dv~dv = Hence S and S i correspond with orthogonality of linear elements, and the normals to the former are parallel to the lines of the con gruence. Hence : A necessary and sufficient condition that the developables of a congruence cut the middle surface in a conjugate system is that their representation be that also of the asymptotic lines of a surface, in which case the latter and the middle surface correspond with orthogonality of linear elements. EXAMPLES 1. When the coordinates of the unit sphere are in the form (III, 35), the para metric curves are asymptotic lines. Find the IF-congruences for which the sphere is one of the focal sheets. 2. Let vi =fi(u) + 0i (w), where /; and </> t - are functions of u and u respectively, and i = 1, 2, 3, be three solutions of the first of equations (09), in which case X = 0, and let 61 in (74) be unity. Show that for the corresponding ^-congruence the mid dle surface is a surface of translation with the generatrices u = const., v = const., that the functions / t - and 0,- are proportional to the direction-cosines of the binor- inals to these generatrices, and that the intersections of the osculating planes of these generatrices are the lines of the congruence. 3. Show that isotropic congruences and congruences of Guichard are congru ences of Ribaucour. 4. A necessary and sufficient condition that a congruence of Ribaucour be nor mal is that the spherical representation of its developables be isothermic. 5. The normals to quadrics and to the cyclidesof Dupin constitute congruences of Ribaucour. 6. When the middle surface of a congruence is plane, the congruence is of the Ribaucour type. 7. Show that the congruence of Ribaucour, whose director surface is a skew helicoid, is a normal congruence, and that the normal surfaces are molding surfaces. 8. Show that a necessary and sufficient condition that a congruence of Ribaucour be normal is that the director surface be minimal. GENERAL EXAMPLES 1 . Through each line of a congruence there pass two ruled surfaces of the con gruence whose lines of striction lie on the middle surface ; their equation is edu* + (f + f )dudv they are called the mean ruled surfaces of the congruence. GENERAL EXAMPLES 423 2. Show that the mean ruled surfaces of a congruence are represented on the sphere by an orthogonal system of real lines, and that their central planes ( 105) bisect the angles between the focal planes. Let u = const. , v = const, be the mean ruled surfaces and develop a theory analogous to that in 167. 3. If the two focal surfaces of a congruence intersect, the intersection is the envelope of the edges of regression of the two families of developable surfaces of the congruence. 4. If a congruence consists of the lines joining points on two twisted curves, the focal planes for a line of the congruence are determined by the line and the tangent to each curve at the point where the curve is met by the line. 5. In order that the lines which join the centers of geodesic curvature of the curves of an orthogonal system on a surface shall form a normal congruence, it is necessary and sufficient that the corresponding radii of geodesic curvature be func tions of one another, or that the curves in one family have constant geodesic curvature. 6. Let S be a surface whose lines of curvature in one system are circles; let C denote the vertex of the cone circumscribing S along a circle, and L the corre sponding generator of the envelope of the planes of the circles ; a necessary and sufficient condition that the lines through the points C and the corresponding lines L form a normal congruence is that the distance from C to the points of the correspond ing circle shall be the same for every circle ; if this distance be denoted by a, the radius of the sphere is given by _ p /2 / jn \ a 2\ where the accent indicates differentiation with respect to the arc of the curve of centers of the spheres. 7. Let -S be a surface referred to its lines of curvature, Ci and C 2 the centers of principal normal curvature at a point, GI and G 2 the centers of geodesic curva ture of the lines of curvature at this point; a necessary and sufficient condition that the line joining C 2 and G\ form a normal congruence is that p 2 be a function of Pg u , or that one of these radii be a constant. 8. Let S be a surface of the kind defined in Ex. 6; the cone formed by the normals to the surface at points of a circle A is tangent to the second sheet of the evolute of -S in a conic T (cf. 132). Show that the lines through points of T and the vertex C of the cone which circumscribes 8 along A generate a normal con gruence, and that C lies in the plane of F. 9. Given an isothermal orthogonal system on the sphere for which the linear element is Z _ * + cto 2 ) ; on each tangent to a curve v = const, lay off the segment of length X measured from the point of contact, and through the extremity of the segment draw a line parallel to the radius of the sphere at the point of contact . Show that this congruence is iso tropic. 10. When a congruence is iso tropic and its direction-cosines are of the form (III, 35), equation (53) reduces to 8uBv (1-f-ww) Show that the general integral is p = 2 O0(v) - vf(u)](l + uv) where / and are arbitrary functions of u and v respectively. Find the equations of the middle surface. 424 RECTILINEAR CONGRUENCES 11. Show that the intersections of the planes (1 - M 2 )x -f i (1 + w 2 ) y + 2 uz + 4/(w) = 0, (1- v 2 )z - i(l + v z )y + 2vz +40(u)= constitute an isotropic congruence, for which these are the focal planes ; that the locus of the mid-points of the lines joining points on the edges of regression of the developables enveloped by these planes is the minimal surface which is the middle envelope of the congruence, by rinding the coordinates of the point in which the tangent plane to this surface meets the intersection of the above planes. 12. Show that the middle surface of an isotropic congruence is the most general surface which corresponds to a sphere with orthogonality of linear elements, and that the corresponding associate surface in the infinitesimal deformation of the sphere is the minimal surface adjoint to the middle envelope. 13. Find the surface associate to the middle surface of an isotropic congruence when the surface corresponding to the latter with orthogonality of linear elements is a sphere, and show that it is the polar reciprocal, "with respect to the imaginary sphere x 2 -f y 2 -f z 2 + 1 = 0, of the minimal surface adjoint to the middle envelope of the congruence. 14. The lines of intersection of the osculating planes of the generatrices of a surface of translation constitute a IT-congruence of which the given surface is the middle surface ; if the generatrices be curves of constant torsion, equal but of opposite sign, the congruence is normal to a TF-surface of the type (VIII, 72). 15. If the points of a surface S be projected orthogonally upon any plane A, and if, after the latter has been rotated about any line normal to it through a right angle, lines be drawn through points of A parallel to the corresponding nor mals to -S, these lines form a congruence of Ribaucour. 16. A necessary and sufficient condition that the tangents to the curves v const. on a surface, whose point equation is (VI, 26), shall form a congruence of Ribaucour is aa_S6 du dv dudv 17. Show that the tangents to each system of parametric* curves on a surface form congruences of Ribaucour when the point equation is where Ui and V\ are functions of u and v respectively, and the accents indicate differentiation. 18. Show that if the parametric curves on a surface S form a conjugate system, and the tangents to the curves of each family form a congruence of Ribaucour, the same is true of the surfaces Si and S_i, which together with S constitute the focal surfaces of the two congruences. 19. Show that the parameter of distribution p of the ruled surface of a con gruence, determined by a value of dv/du, is given by 1 P = -f e du + /du, f du -f g dv GENERAL EXAMPLES 425 20. Show that the mean ruled surfaces (cf. Ex. 1) of a congruence are char acterized by the property that for these surfaces the parameter of distribution has the maximum and minimum values. 21. If S and SQ are two associate surfaces, and through each point of one a line be drawn parallel to the corresponding radius vector of the other, the developables of the congruence thus formed correspond to the common conjugate system of S and SQ, 22. In order that two surfaces S and SQ corresponding with parallelism of tangent planes be associate surfaces, it is necessary and sufficient that for the congruence formed by the joins of corresponding points M and MQ of these sur faces the developables cut S and SQ in their common conjugate system, and that the focal points M and MQ form a harmonic range. 23. In order that a surface S be iso thermic, it is necessary and sufficient that there exist a congruence of Ribaucour of which S is the middle surface, such that the developables cut S in its lines of curvature. CHAPTER XIII CYCLIC SYSTEMS 174. General equations of cyclic systems. The term congruence is not restricted to two-parameter systems of straight lines, but is applied to two-parameter systems of any kind of curves. Darboux * has made a study of these general congruences and Ribaucourf has considered congruences of plane curves. Of particular interest is the case where these curves are circles. Ribaucour has given the name cyclic systems to congruences of circles which admit of a one- parameter family of orthogonal surfaces. This chapter is devoted to a study of cyclic systems. We begin with the general case where the planes of the circles envelop a nondevelopable surface S. We associate with the latter a moving trihedral ( 68), and for the present assume that the parametric curves on the surface are any whatever. As the circles lie in the tangent planes to S, the coordinates of a point on one of them with respect to the corresponding trihedral are of the form (1) a + Rcos0, b+Rsm0, 0, where a, b are the coordinates of the center, R the radius, and the angle which the latter to a given point makes with the moving In 69 we found the following expressions for the projections of a displacement of a point with respect to the moving axes : (dx+%du + ^dv + (qdu + q^v) z (rdu + r t dv) y, (") \ dy + 77 du 4- rj^dv + (rdu + r v dv] x (pdu + p^dv) z, \dz +(p du +p 1 dv) y (qdu+ q v dv) x, * Vol. II, pp. 1-10; also Eisenhart, Congruences of Curves, Transactions of the Amer. Math. Soc., Vol. IV (1903), pp. 470-488. t Memoire sur la theorie generale des surfaces courbes, Journal des Mathtmatiques, Ser. 4, Vol. VII (1891), 117 et. seq. 426 GENERAL EQUATIONS 427 where the translations f , f 1? T;, ^ and the rotations p,q,r-> p^ q v , r^ satisfy the conditions dp_di_ __ d|__Mi = (3) When the values (1) are substituted in (2) the latter are reducible to dr dr. ^Wl PH Ph-M^fl- J du + J^v -|- cos QdR (dd + rdu+ r^dv) R sin ^, # ^w + ^jrfu + sin 6dR + (dd+rdu + r^dv] R cos 6, (p du -f ^^v) (b + R sin 0) (y du + q^dv) (a+R cos where we have put, for the sake of brevity, (5) ^ du The conditions that (\ = (^ du \dv] dv \du/ du \dv are reducible, by means of (3), to \du (6) _ dv du The direction-cosines of the tangent to the given circle at the point (1) are (7) sin0, cos0, 0. Hence the condition that the locus of the point, as u and v vary, be orthogonal to the circle is that the sum of the expressions (4) multiplied respectively by the quantities (7) be zero. This gives 428 CYCLIC SYSTEMS In order that the system of circles be normal to a family of sur faces this equation must admit of a solution involving a parameter. Since it is of the form (9) the condition that such an integral exist is that the equation be satisfied identically. * For equation (8) this condition is reducible to In order that this equation be satisfied identically, the expressions in the brackets must be zero. If they are not zero, it is possible that the two solutions of this equation will satisfy (8), and thus determine two surfaces orthogonal to the congruence of circles. Hence we have the theorem of Ribaucour: If the circles of a congruence are normal to more than two surfaces, they form a cyclic system. The equations of condition that the system be cyclic are consequently dR . dR . The total curvature of S is given by (cf. 70) * Murray, Differential Equations, p. 137. New York, 1897; also Forsyth, Differential Equations, p. 257. London, 1888. THEOREMS OF RIBAUCOUR 429 From this and (5) it is seen that equations (12) involve only functions relating to the linear element of S and to the circle. Hence we have the theorem of Ribaucour: If the envelope of the planes of the circles of a cyclic system be deformed in any manner without disturbing the size or position of the circles relative to the point of contact, the congruence of circles continues to form a cyclic system. Furthermore, if we put Q t = tan > z* equation (8) assumes the Riccati form, dt + (af +a 2 t + a 3 ) du + (b/ + bjt + b 3 ) dv = 0, where the # s and 5 s are functions of u and v. Recalling a funda mental property of such equations ( 14), we have : Any four orthogonal surfaces of a cyclic system meet the circles in four points whose cross-ratio is constant. Since by hypothesis S is nondevelopable, equations (12) may be replaced by (13) du ^ dv AB, - - trf) JBT = 0. By (5) the first two of these equations are reducible to (14) du a The condition of integrability of these equations is (15) ^{ + g,-^f 1 -g, l -r( f ^-J 1 )-r 1 (,-^. dv cv du cu Instead of considering this equation, we introduce a function <j> by the equation (16) 24>=,K 2 -a 2 -& 2 , 430 CYCLIC SYSTEMS and determine the condition which </> must satisfy. We take for a and b the expressions obtained by solving (14); that is (17) Now the equation (15) vanishes identically, and the only other condition to be satisfied is the last of (13); this, by the substi tution of these values of a, 6, R, becomes a partial differential equation in < of the form (18) -r ^_r _(__r_] -\-J^ + L + M ~ + N= 0, du dv \dudv / du ducv du where 7, X, JHf, JV denote functions of <>, f , r t , and their deriva tives of the first order. Conversely, each solution of this equation gives a cyclic system whose circles lie in the tangent planes to S. EXAMPLES 1. Let S be a surface of revolution defined by (III, 99), and let Tbe the trihedral whose x-axis is tangent to the curve v = const. Determine the condition which the function \f/ (u) must satisfy in order that the quantities a, b in (1) may have the values a = _*w_. 6 = l, and determine also the expression for R. 2. A necessary and sufficient condition that all the circles of a cyclic system whose planes envelop a nondevelopable surface shall have the same radius, is that the planes of the circles touch their envelope S at the centers of the circles, and that S be pseudospherical. 3. Let S be a surface referred to an orthogonal system of lines, and let T be the trihedral whose z-axis is tangent to the curve v = const. With reference to the trihedral the equations of a curve in the tangent plane are of the form x = p cos 0, y = p sin 0, z = 0, where in general p is a function of 0, w, and v. Show that the condition that there be a surface orthogonal to these curves is that there exist a relation between 0, u, and v which satisfies the equation U sin -f prji cos 30 When this condition is satisfied by a function which involves an arbitrary con stant, there is an infinity of normal surfaces. In this case the curves are said to form a normal congruence. 4. When the surface enveloped by the planes of the curves of a normal con gruence of plane curves is deformed in such a way that the curves remain invari ably fixed to the surface, the congruence continues to be normal. CYCLIC CONGRUENCES 431 175. Cyclic congruences. The axes of the circles of a cyclic sys tem constitute a rectilinear congruence which Bianchi * has called a cyclic congruence. In order to derive the properties of this con gruence and further results concerning cyclic systems, we assume that the parametric curves on S correspond to the developables of the congruence. The coordinates of the focal points of a line of the congruence with reference to the corresponding trihedral are of the form a, ft, p^ a, ft, /> 2 . On the hypothesis that the former are the coordinates of the focal point for the developable v = const, through the line, we have, from (2), - + ? +?/i rft = 0, - + T)pp l +ra = Q. cu cu Proceeding in like manner with the other point, we obtain a pair of similar equations. All of these equations may be written in the abbreviated form (19) A + qPl =0, Ji- PPl =Q, ^+ ?lft =0, J?,-^p 2 =0, in consequence of (5). When these values are substituted in the last of equations (13), it is found that (20) Sf=-P 1 p f Hence the lines joining a point on the circle to the focal points are perpendicular. If we put thus indicating by 2 p the distance between the focal points, and by 8 the distance between the center of the circle and the mid-point of the line of the congruence, we find that We replace this equation by the two (21) 8 p cos <r, R p sin cr, thus defining a function a-. Now we have /5 1 =/E)(coso-+l), /? 2 =/o(cos<r 1), so that equations (19) may be written A = o. (22) ^ * i = _ q ip (cos o- 1), B l = prf (cos o- 1). *Vol. II, p. 161. 432 CYCLIC SYSTEMS By means of (5) equation (15) can be put in the form When the values (22) are substituted in this equation, it becomes Since by (3) the expression in the first parenthesis is zero, the same is true of the second, and so we have But these are the conditions (V, 67) that the parametric curves on S form a conjugate system. Hence we have the theorem of Ribaucour : On the envelope of the planes of the circles of a cyclic system the curves corresponding to the developables of the associated cyclic con gruence form a conjugate system. 176. Spherical representation of cyclic congruences. When the expressions (22) are substituted in (6), we obtain do da. dp dp, Since pq l p l q =t= unless Sis developable, the preceding equations may be replaced by [ /3 (coso--l)]=2 /0 { 1 2 2 }VH-^), d _ /12V [/3(cos a -hi)] = 2ps \ + ( a( l\P\) i cv L 1 J where the Christoff el symbols are formed with respect to (24) (pdu+p l dv) 2 -\-(qdu-{-q l dvY, the linear element of the spherical representation of S. SPHERICAL REPRESENTATION 433 When in like manner we substitute in the first two of equations J? 2 = p 2 sin 2 cr = p 2 (1 cos 2 o-), (13), taking we obtain . dp p cos o- d , cos a-) --- - cos a =pb qa, du 1 4- cos a du n , \ p (1 + cos o) -J- 7 P cos 1 cos o- From these equations and (23) we find * r cos a p o (25) d cos dv a = 2 (cos a 1 J12V )\~i i \~ 1. J and (cos<r 1)^ = 2 p cos ri2V oj r 4~ ^<^ pO) (26) (cosr+l)|; = 2 p cos J12V "\1 J "^^"^ The condition of integrability of equations (25) is reducible to If the expression for cos a- obtained from this equation be substi tuted in (25), we find two conditions upon the curves on the sphere in order that they may represent the developables of a cyclic con gruence. A particular case is that in which (27) is identically sat isfied, when the two conditions are (28) ri O"\ ^ n O"^ n o^ / f 1 o"\ / 121 1/121 _ 2 f 121 f 121 ll J 0v I 2 J " ll J 12 / It is now our purpose to show that if any system of curves on the sphere satisfies either set of conditions, all the congruences whose developables are thus represented on the sphere are cyclic. We assume that the sphere is referred to such a system and that we have a solution p of 434 CYCLIC SYSTEMS By the method of 167, or that hereinafter explained, we find the middle surface of the congruence. Then we take the point on each line at the distance p cos a- from the mid-point as the center of the circle of radius p sin a and for which the line is the axis. These cir cles form a cyclic system, as we shall show. In the first place we determine the middle surface with reference to a trihedral of fixed vertex, whose 2-axis coincides with the radius of the sphere parallel to the line of the congruence and whose x- and #-axes are any whatever. If # , # , z denote the coordinates of the mid-point of a line with reference to the corresponding tri hedral, the coordinates of the focal points are From (2) it is seen that if these points correspond to the develop- ables v = const, and u = const, respectively, we must have Since pq l p l q =t= 0, the conditions of integrability of these equa tions can be put in the form (30) It is readily found that the condition of integrability of these equa tions is reducible to (29). It will be to our advantage to have also the coordinates of the point of contact of the plane of the circle with its envelope S. If x, y, Z Q p cos a denote these coordinates with reference to the above trihedral, it follows from (2) that (z - p cos v) + py-qx = 0, o (z - p cos a) + p^y - q,x = 0. CYCLIC CONGRUENCES 435 If these equations be subtracted from the respective ones of (30), the results are reducible, by means of (25), to (cos a -1) - + 2 p cos a- 1 g j + p(y Q - y) - q(x - x) = 0, (cos cr + 1) + 2 p cos a- V^- y)- ftfo- x) = 0, which are the same as (26). For, the quantities x x, y Q y are the coordinates of the center of the circle with reference to the tri hedral parallel to the preceding one and with the corresponding point on S for vertex. If, then, we have a solution cr of (25) and p of (29), the corre sponding values of a and b given by (26) satisfy (22), since the latter are the conditions that the parametric curves on the sphere represent the developables of the congruence. However, we have seen that when the values (22) are substituted in (12), we obtain equations reducible to (25) and (26). Hence the circles constructed as indicated above form a cyclic system. Since equations (25) admit only one solution (27) unless the con dition (28) is satisfied, we have the theorem: With each cyclic congruence there is associated a unique cyclic sys tem unless it is at the same time a congruence of Ribaucour, in which case there is an infinity of associated cyclic systems. Recalling the results of 141, we have the theorem of Bianchi * : When the total curvature of a surface referred to its asymptotic lines is of the form - ~ [* it is the surface generatrix of a congruence of Ribaucour which is cyclic in an infinity of ways, and these are the only cyclic congru ences with an infinity of associated cyclic systems. In this case the general solution of equations (25) is (31) cos <,=>-* + *, <#> + ^ where a is an arbitrary constant. * Vol. II, p. 165. 436 CYCLIC SYSTEMS 177. Surfaces orthogonal to a cyclic system. In this section we consider the surfaces S l orthogonal to the circles of a cyclic sys tem. Since the direction-cosines of the normals to the surfaces with reference to the moving trihedral in 174 are sin 6, cos 0, 0, the spherical representation of these surfaces is given by the point whose coordinates are these with respect to a trihedral of fixed vertex parallel to the above trihedral. From (2) we find that the expressions for the projections of a displacement of this point are cos 0(d0 + rdu + ^ dv), sin 6 (dd + r du + r 1 dv), (p du + p^dv) cos 6 -f (q du + q v dv) sin 0. Moreover, by means of (8), (21), (22), we obtain the identity (32) sin a- (dd + r du + r^dv) = (1 4- cos a) ( p cos 6 -f q sin 6) du + (1 cos a) ( p l cos 6 + <?! sin 0) dv. Hence the linear element of the spherical representation of ^ is (33) da*= T - (p cosO + q sin 0fdu* -L COS O~ 1\ r 1 + COSO- Since the parametric curves on the sphere form an orthogonal system, the parametric curves on the surface are the lines of curvature, if they form an orthogonal system. In order to show that this condition is satisfied, we first reduce v the expressions (4) for the projections of a displacement of a point on S v by means of (21), (22), (25), (26), and (32), to (34) Cdu Ddv \ cos v sin <r ( ) ,1 cos a 1 + cos 07 , Cdu Ddv \ sin sin <T i cos <r 1 + cos cr, Cdu + Ddv, where we have put = pi(b + R sin 0)q 1 (a+E cos 0). NORMAL CYCLIC CONGRUENCES 437 Hence the linear element of S 1 is (36) ds* = 2 . du + 2 D V , 1 COS (T 1 + COS cr from which it is seen that the parametric curves on S l form an orthogonal system, and consequently are the lines of curvature. Furthermore, it is seen from (34) that the tangents to the curves v = const., u = const, make with the plane of the circle the respec tive angles .-. i/l COScrX ./ l-fCOSCT\ (37) tan" 1 : ), tan" 1 ! \ sin cr / \ sin a / But it follows from (21) that the lines joining a point on the cir cumference of a circle to the focal points of its axis make the angles (37) with the radius to the point. Hence we have : The lines of curvature on a surface orthogonal to a cyclic system correspond to the developables of the congruence of axes of the circles, and the tangents to the two lines of curvature through a point of the surface meet the corresponding axis in its focal points. 178. Normal cyclic congruences. Since the developables of a cyclic congruence correspond to a conjugate system on the enve lope S of the planes of the circles, this system consists of the lines of curvature when the congruence is normal, and only in this case (cf. 83). If, under these conditions, we take two of the edges of the trihedral tangent to the lines of curvature, we have and equations (25) become d -, 03, d , . cr d By a suitable choice of parameters we have " tt so that if we put &> = cr/2, the linear element of the sphere is (39) d(r*= sinW% 2 + cosWv 2 . Comparing this result with ( 119), we have the theorem: The normals to a surface 2 with the same spherical representation of its lines of curvature as a pseudospherical surface constitute the only kind of normal cyclic congruences. 438 CYCLIC SYSTEMS Since the surface 2 and the envelope of the planes of the circles have the same representation of their lines of curvature, the tangents to the latter at corresponding points on the two surfaces are parallel. Hence with reference to a trihedral for 2 parallel to the trihedral for S the coordinates of a point on the circle are R cos 0, R sin 6, p, where ft remains to be determined and 6 is given by (32), which can be put in the form ,*(\^ cd d(D . n dO da) . a (40) H = cos co sm 0, 1 = sin &> cos 6. du dv dv du If we express, by means of (2), the condition that all displace ments of this point be orthogonal to the line whose direction- cosines are sin 0, cos 0, 0, the resulting equation is reducible, by means of (40), to sin 9 (R cos o> fi sin o> f ) du cos (R sin &> -f- //, cos &> 77^ dv 0. Hence the quantities in parentheses are zero, from which we obtain (41) R = i; cos &) -h T] I sin w, ^ = f sin o> + rj l cos o>. When, in particular, 2 is a pseudospherical surface of curvature -I/a 2 , we have (VIII, 22) f = a cos &), rj l = a sin &>, so that R= a and /x = 0. Hence the circles are of constant radius and the envelope of their planes is the locus of their centers (cf. Ex. 2, 174). Conversely, when the latter < condition is sat isfied, it follows from (13) that R is constant. Moreover, in this case p^ and p 2 , as defined in 175, are the principal radii of the surface, which by (20) is pseudospherical. When these values are substituted in (36) and (33), it is found that the linear ele ment of each orthogonal surface is ds* = a? (cos 2 6 du* -f sin 2 6 dv z ), and of its spherical representation (42) d<r*=sm*0du*+ cos 2 <W. Hence these orthogonal surfaces are the transforms of 2 by means of the Bianchi transformation ( 119). PLANES OF CIRCLES TANGENT TO A CUKVE 439 The expression (42) is the linear element of the spherical rep resentation of the surfaces orthogonal to the circles associated with any surface 2, whether it be pseudospherical or not, whose spherical representation is given by (39). Since these orthogonal surfaces have this representation of their lines of curvature, they are of the same kind as 2. We have thus for all surfaces with the same rep resentation of their lines of curvature as pseudospherical surfaces, a transformation into similar surfaces of which the Bianchi trans formation is a particular case ; we call it a generalized Bianchi transformation.* 179. Cyclic systems for which the envelope of the planes of the circles is a curve. We consider now the particular cases which have been excluded from the preceding discussion, and begin with that for which the envelope S of the planes of the circles is a curve C. We take the moving trihedral such that its zy-plane, as before, is that of the circle, and take the z-axis tangent to C. If s denotes the arc of the latter, we have ds = f; du + ^dv, ?? = rj 1 = 0, and by (3) (43) r ^-r^ = Q, rfx-ftf = 0. From (14), (15), and (16) it follows that a and </> are functions of s, so that these equations may be replaced by (44) .K 2 =a 2 +& 2 If the parametric curves on the sphere represent the developables of the congruence, the conditions (19) must hold. But from (5), (15), and (43) we obtain ^. _^ _ Q If the values from (19) be substituted in this equation, we have, from (43), ^-^=0. Hence the focal surfaces coincide. If we put P = Pi=P* in (19) and substitute in the last of (12), we obtain (^+^ 2 )(^ 1 -^)==0. *Cf. American Journal, Vol. XXVI (1905), pp. 127-132. 440 CYCLIC SYSTEMS The vanishing of pq l p l q is the condition that there be a single infinity of planes, which case we exclude for the present. Hence p = iR ; that is, the developables of the cyclic congruence are imaginary. Instead of retaining as parametric curves those representing the developables, we make the following choice. We take the arc of C for the parameter u ; consequently f =1, ^=0. Since 77 = ^ x = also, we have, from (3), .- - 2MB-* hence we may choose the parameter v so that p = 0, ^=1. From (3) it follows, furthermore, that dq dr i= T > B-V = -* of which the general integral is q=U l cos v + U 2 sin v, r = U l sin v -f U z cos v, where U^ and U 2 are arbitrary functions of u. From (5) we have A = j>"(u)+\-rl, ^=0, A = 7T #y so that the third of equations (12) is reducible by (44) to (*"+!)- cv d U sin v U cos v Hence if we take for a any function of u denoted by <f> r (u), equation (45) gives 6, and R follows directly from (44). 180. Cyclic systems for which the planes of the circles pass through a point. If the planes of the circles of a cyclic system pass through a point 0, we take it for the origin and for the vertex of a moving trihedral whose z-axis is parallel to the axis of the circle under consideration. In this case equations (14) may be replaced by (46) A> 2 = tf 2 + 6 2 -*, where c denotes a constant. But this is the condition that all the circles are orthogonal to a sphere with center at 0, or cut it in PLANES OF CIKCLES THROUGH A POINT 441 diametrically opposite points, or pass through 0, according as c is positive, negative, or zero. Hence we have the theorem : If the planes of the circles of a cyclic system pass through a point, the circles are orthogonal to a sphere with its center at the point, or meet the sphere in opposite points, or pass through the center. From geometrical considerations we see that the converse of this theorem is true. When c in (46) is zero all the circles pass through 0. Then by (21) we have (47) a = p sin <r cos 6, b = p sin a sin 6, and equations (26) become (cos or 1) ^ = 2 cos a- \ f + sin a- (p sin 6 q cos , (cos <r +1) - S-L. . 2cos<rj f -\-smo-(p l sm0 These equations are obtained likewise when we substitute the values (47) in equations (22) and reduce by means of (25) and (32). Because of (22) the function p given by (26) is a solution of (29), arid therefore p given by (48) is a solution. But the solution 6 of (32) involves a parameter. Hence we have the theorem of Bianchi * : Among all the cyclic congruences with the same spherical repre sentation of their developables there are an infinity for which the circles of the associated cyclic system pass through a point. If we take the line through and the center of the circle for the z-axis of the trihedral, equation (11) must admit of the solu tion = TT, and consequently must be of the form In order that this equation admit of a solution other than TT, both L and M must be zero and the system cyclic. We combine this result with the preceding theorem to obtain the following: A two-parameter family of circles through a point and orthogonal to any surface constitute a cyclic system, and the most general spher ical representation of the developables of a cyclic congruence is afforded by the representation of the axes of such a system of circles.^ * Vol. II, p. 169. t Bianchi, Vol. II, p. 170. 442 CYCLIC SYSTEMS We consider finally the case where the planes of the circles depend upon a single parameter. If we take for moving axes the tangent, principal normal, and binormal of the edge of regression of these planes and its arc for the parameter w, we have and comparing (V, 50) with (2), we see that p -- <7 = 0, r = - T p where p and r are the radii of first and second curvature of the edge of regression. Now da , b , 8a _#, r> * b A = -- hi -- A= B = -- h- 2?! = du p dv du p dv The equations (12) reduce to two. One of the functions a, b may be chosen arbitrarily ; then the other and R can be obtained by the solution of partial differential equations of the first order. EXAMPLES 1. Show that a congruence of Ribaucour whose surface generator is the right helicoid is cyclic, and determine the cyclic systems. 2. A congruence of Guichard is a cyclic congruence, and the envelope of the planes of the circles of each associated cyclic system is a surface of Voss. 3. The surface generator of a cyclic congruence of Ribaucour is an associate surface of the planes of the circles of each associated cyclic system. 4. If S is a surface whose lines of curvature have the same spherical representa tion as a pseudospherical surface, and Si is a transform of S resulting from a gen eralized Bianchi transformation ( 178), the tangents to the lines of curvature of 81 pass through the centers of principal curvature of S. 5. When the focal segment of each line of a cyclic congruence is divided in constant ratio by the center of the circle, the envelope of the planes of the circles is a surface of Voss. 6. The circles of the cyclic system whose axes are normal to the surface S, defined in Ex. 11, p. 370, pass through a point, and the surfaces orthogonal to the circles are surfaces of Bianchi of the parabolic type. 7. If the spheres with the focal segments of the lines of a congruence for diameters pass through a point, the congruence is cyclic, and the circles pass through the point. 8. Show that the converse of Ex. 7 is true. GENERAL EXAMPLES 443 GENERAL EXAMPLES 1. Determine the normal congruences of Ribaucour which are cyclic. 2. If the envelope of the planes of the circles of a cyclic system is a surface of Voss whose conjugate geodesic system corresponds to the developables of the asso ciated cyclic congruence, any family of planes cutting the focal segments in con stant ratio and perpendicular to them envelop a surface of Voss. 3. A necessary and sufficient condition that a congruence be cyclic is that the developables have the same spherical representation as the conjugate lines of a sur face which remain conjugate in a deformation of the surface. If the developables of the congruence are real, the deforms of the surface are imaginary. 4. The planes of the cyclic systems associated with a cyclic congruence of Ribaucour touch their respective envelopes in such a way that the points of con tact of all the planes corresponding to the same line of the congruence lie on a straight line. 5. If the spheres described on the focal segments of a congruence as diameters cut a fixed sphere orthogonally or in great circles, the congruence is cyclic and the circles cut the fixed sphere orthogonally or in diametrically opposite points. 6. If one draws the circles which are normal to a surface S and which cut a fixed sphere S Q in diametrically opposite points or orthogonally, the spheres described on the focal segments of the congruence of axes as diameters cut S Q in great circles or orthogonally. 7. Determine the cyclic systems of equal circles whose planes envelop a devel opable surface. 8. Let Si be the surface defined in Ex. 14, p. 371, and let S be the sphere with center at the origin and radius r. Draw the circles which are normal to Si and which cut S orthogonally or in diametrically opposite points. Show that the cyclic congruence of the axes of these circles is a normal congruence, and that the coordinates of the normal surfaces are of the form [1 ( -- -) T j a?e a - (r? 2 + K) e cos 6 -f 77 sin 8 \Xi + fJL ja 2 e~ - (T; 2 + K )(P I sind - rj cose] X 2 + tX, L2 ( ) where K is equal to r 2 or + r 2 , according as the circles cut S orthogonally or in diametrically opposite points, and where t is given by [1 ( - ^) "1 \a 2 e a - (i? 2 -I- K) e a [ cos 6 + -r\ sin 6 sin du 2a( [1 ( -- -) ~\ _ j a?e a (j 2 4- K) e" j sin 6 t\ cos cos u dv. 9. Show that the surfaces of Ex. 8 are surfaces of Bianchi which have the same spherical representation of their lines of curvature as the pseudospherical surface S referred to in Ex. 14, p. 371. 10. Show that the surfaces orthogonal to the cyclic system of Ex. 8 are surfaces of Bianchi of the parabolic type. 444 CYCLIC SYSTEMS 11. Let S be a surface referred to an orthogonal system, and let T be the trihe dral whose x-axis is tangent to the curve u = const. The equations . x = p(l + cos0), y = 0, z = />sin0 define a circle normal to S. Show that the necessary and sufficient conditions that the circles so defined form a cyclic system are cu 12. A necessary and sufficient condition that a cyclic system remain cyclic when an orthogonal surface S is deformed is that S be applicable to a surface of revolution and that where c is a constant and the linear element of S is ds 2 = du 2 + 2 (w) dv* (cf . Ex. 11). 13. Determine under what conditions the lines of intersection of the planes of the circles of a cyclic system and the tangent planes to an orthogonal surface form a normal congruence. 14. Let Si and S 2 be two surfaces orthogonal to a cyclic system, and let MI and M 2 be the points of intersection of one of the circles with Si and S 2 . Show that the normals to Si and S 2 at the points MI and M 2 meet in a point M equidistant from these points, and show that Si and S 2 constitute the sheets of the envelope of a two-parameter family of spheres such that the lines of curvature on Si and S 2 correspond. 15. Let -S be the surface of centers of a two-parameter family of spheres of variable radius JR, and let Si and S 2 denote the two sheets of the envelope of these spheres. Show that the points of contact MI and M 2 of a sphere with these sheets are symmetric with respect to the tangent plane to S at the corresponding point M. Let S be referred to a moving trihedral whose plane y = is the plane MiMM 2 , and let the parametric curves be tangent to the x- and y-axes respectively. Show that if ff denotes the angle which the radius MMi makes with the x-axis of the trihedral, the lines of curvature on Si are given by sin <r (sin <rp r cos <r) du 2 + in[qi ) dv 2 \ dv/ H fllfl -) I sin <r (cos <rri + p sin <r) \dudv = 0. 16. Find the condition that the lines of curvature on S! and S 2 of Ex. 15 corre spond, and show that in this case these curves correspond to a conjugate system on S. 17. Show that the circles orthogonal to two surfaces form a cyclic system, pro vided that the lines of curvature on the two surfaces correspond. 18. Let <S be a pseudospherical surface with the linear element (VIII, 22), the lines of curvature being parametric, and let A be a surface with the same spher ical representation of its lines of curvature as S ; furthermore, let AI denote the envelope of the plane which makes the constant angle a with the tangent plane at a point M of A and meets this plane in a line I/, which forms with the tangent to the curve u = const, at M an angle defined by equations (VIII, 35). If MI GENERAL EXAMPLES 445 denotes the point of contact of this plane, we drop from MI a perpendicular on L, meeting the latter in N. Show that if X and p denote the lengths MN and NMi, they are given by X = ( V2? cos o> + V6? sin o>) sin <r, ^ = ( Vj sin w + Vt? cos w) sin o-, where E and (? are the first fundamental coefficients of A. 19. Show that when the surface A in Ex. 18 is the pseudospherical surface S, then AI is the Backhand transform Si of S by means of the functions (0, <r), and that when A is other than S the lines of curvature on the four surfaces S, .4, Si, AI correspond, and the last two have the same spherical representation. 20. Show that as is given all values satisfying equations (VIII, 35) for a given 0-, the locus of the point Jfi, defined in Ex. 18, is a circle whose axis is normal to the surface A at M. 21. Show that when A in Ex. 18 is a surface of Bianchi of the parabolic type (Ex. 11, p. 370) the surfaces AI are of the same kind, whatever be a-. CHAPTER XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181. Triple system of surfaces associated with a cyclic system. Let S 1 be one of the surfaces orthogonal to a cyclic system, and let its lines of curvature be parametric. The locus 2 t of the circles which meet S l in the line of curvature v = const, through a point M is a surface which cuts S l orthogonally. Hence, by Joachimsthal s theorem ( 59), the line of intersection is a line of curvature for 2 r In like manner, the locus 2 2 of the circles which meet S : in the line of curvature u = const, through M cuts S^ orthogonally, and the curve of intersection is a line of curva ture on S 2 also. Since the developables of the associated cyclic congruence correspond to the lines of curvature on all of the orthogonal surfaces, each of the latter is met by 2 X and 2 2 in a line of curvature of both surfaces. At each point of the circle through M the tangent to the circle is perpendicular to the line of curvature v = const, on 2 t through the point and to u = const, on 2 a . Hence the circle is a line of curvature for both 2 X and 2 2 , and these surfaces cut one another orthogonally along the circle. Since there is a surface 2 X for each curve v = const, on S l and a surface 2 2 for each u = const., the circles of a cyclic system and the orthogonal surfaces may be looked upon as a system of three families of surfaces such that through each point in space there passes a surface of each family. Moreover, each of these three sur faces meets the other two orthogonally, and each curve of intersec tion is a line of curvature on both surfaces. We have seen ( 96) that the confocal quadrics form such a system of surfaces, and another example is afforded by a family of parallel surfaces and the developables of the congruence of normals to these surfaces. When three families of surfaces are so constituted that through each point of space there passes a surface of each family and each of the three surfaces meets the other two orthogonally, they are 446 GENERAL EQUATIONS 447 said to form a triply orthogonal system. In the preceding examples the curve of intersection of any two surfaces is a line of curvature for both. Dupin showed that this is a property of all triply orthog onal systems. We shall prove this theorem in the next section. 182. General equations. Theorem of Dupin. The simplest exam ple of an orthogonal system is afforded by the planes parallel to the coordinate planes. The equations of the system are 3 = 1*!, y = M a , z = i* 8 , where u# u^ u 3 are parameters. Evidently the values of these parameters corresponding to the planes through a point are the rectangular coordinates of the point. In like manner, the surfaces of each family of any triply orthogonal system may be determined by a parameter, and the values of the three parameters for the three surfaces through a point constitute the curvilinear coordi nates of the point. Between the latter and the rectangular coor dinates there obtain equations of the form (1) x =/ 1 (w 1 , i* 8 , i*,), y =/ 2 K, i* a , i*,), z =/,(!*!, i* a , i* 8 ), where the functions / are analytic in the domain considered. An example of this is afforded by formulas (VII, 8), which define space referred to a system of confocal quadrics. In order that the system be orthogonal it is necessary and suffi cient that these functions satisfy the three conditions v dx dx _ v dx dx y dx dx _ ^aST *to t to.- Zto.dut Any one of the surfaces u t = const, is defined by (1) when u t is given this constant value. By the linear element of space at a point we mean the linear ele ment at the point of any curve through it. This is which, in consequence of (2), may be written in the parametric form (3) ds 2 = H l du* + H* du* + HI dui, As thus defined, the functions H# H 2 , H 3 are real and we shall assume that they are positive. 448 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES From (3) we have at once the linear element of any of the sur faces of the system. For instance, the linear element of a surface U= COnst. is , 2 rrl J. Now we shall find that the second quadratic forms of these surfaces are expressible in terms of the functions H and their derivatives. If X^ F., Z i denote the direction-cosines of the normals to the surfaces u i = const, we have (5) , du. We choose the axes such that (6) = 4-1. In consequence of (5) the second fundamental coefficients of a sur face u { const, are defined by dx d*x _, 1 ^ dx d 2 x_ _ 1 y dx d*x ^u t H i ** du t du? _ 1 ~ , = 1 y dx 4 where t, /c, I take the values 1, 2, 3 in cyclic order, and the sign 2 refers to the summation of terms in #, ?/, 2, as formerly. In order to evaluate these expressions we differentiate equations (2) with respect to u^ u v u 2 respectively. This gives dx dx . _JL__ = o, f dx ^ du. 0. If each of these equations be subtracted from one half of the sum of the three, we have = o, dx d*x ^ du z du consequently D-= 0. du 3 du l = 0, V = 0; THEOREM OF DUPIN. EQUATIONS OF LAME 449 If the first and third of (2) be differentiated with respect to u z and u s respectively, and the second and third of (4) with respect to u v we have dx d*z dx tfx a# 2 2 ~~ 2 du^ dx tfx y dx tfx 3H S ~ 3 J5T t Hence we have Proceeding in like manner, we find the expressions for the other Z> s, which we write as follows : where i, K, I take the values 1, 2, 3 in cyclic order. From the sec ond of these equations and the fact that the parametric system on each surface is orthogonal, follows the theorem of Dupin : The surfaces of a triply orthogonal system meet one another in lines of curvature of each. 183. Equations of Lame". By means of these results we find the conditions to be satisfied by H v H^ H z , in order that (3) may be the linear element of space referred to a triply orthogonal system of surfaces. For each surface the Codazzi and Gauss equations must be satisfied. When the above values are substituted in these equations, we find the following six equations which it is necessary and sufficient that the functions H satisfy : PH { 1 dH^H, 1 gJgjgjr^ ~ /ox H where t, /^, I take the values 1, 2, 3 in cyclic order. These are the equations of Lame", being named for the geometer who first deduced them.* * Lemons sur les coordonntes curvilignes et leurs diverses applications, pp. 73-79. Paris, 1859. 450 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES For each of the surfaces there is a system of equations of the form (V, 16). When the values from (7) are substituted in these equations we have < t du, Recalling the results of 65, we have that each set of solutions of equations (8), (9) determine a triply orthogonal system, unique to within a motion in space. In order to obtain the coordinates of space referred to this system, we must find nine functions JQ, r;., Zf which satisfy (10) and 1, 2^=0. (=*=*) Then the coordinates of space are given by quadratures of the form x = I H^XI du l + H^X Z du 2 + H Z X Z du s . If p. K denotes the principal radius of a surface u f = const, in the direction of the curve of parameter U K , we have, from (7), rm 1- Pi. Let p l denote the radius of first curvature of a curve of param eter u r In accordance with 49 we let w 1 and w[ ?r/2 denote the angles which the tangents to the curves of parameter u 3 and u 2 respectively through the given point make, in the positive sense, with the positive direction of the principal normal of the curve of parameter u r Hence, by (IV, 16), we have /i o\ Pi Pn Pi Pzi From these equations and similar ones for curves of parameter u and u s , we deduce the relations (13) 1 = 1 + 1, tan5 ( = 6t, Pi Pl> Pfi P ONE FAMILY OP SURFACES OF REVOLUTION 451 where 2, /c, I take the values 1, 2, 3 in cyclic order. Moreover, since the parametric curves are lines of curvature, it follows from (59) that the torsion of a curve of parameter u i is (14) l-l^i. r { Hi du, 184. Triple systems containing one family of surfaces of revolution. Given a family of plane curves and their orthogonal trajectories ; if the plane be revolved about a line of the plane as an axis, the two families of surfaces of revolution thus generated, and the planes through the axis, form a triply orthogonal system. We inquire whether there are any other triple systems containing a family of surfaces of revolution. Suppose that the surfaces u s = const, of a triple system are sur faces of revolution, and that the curves u 2 = const, upon them are the meridians. Since the latter are geodesies, we must have From (8) it follows that either dH. n dff s n i = 0, or 8 = 0. du s du 2 In the first case it follows from (11) that l//o 31 = 0. Consequently, the surfaces of revolution w 3 = const, are developables, that is, either circular cylinders or circular cones. Furthermore, from (15) and (11), we have l//o 21 =0, so that the surfaces u 2 = const, also are developables, and in addition we have, from (13), that l//^ = 0, that is, the curves of parameter u^ are straight lines and consequently the surfaces u^= const, are parallel. The latter are planes when the surfaces u s = const, are cylinders, and surfaces with circular lines of curvature when u s = const, are circular cones. Conversely, from the theorem of Darboux ( 187) and from 132, it follows that any system of circular cylinders with parallel generators, or any family of circular cones whose axes are tangent to the locus of the vertex, leads to a triple system of the kind sought. We consider now the second case, namely 452 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES From (11) we find that l//o 21 =0, and l//o 28 = 0; consequently the surfaces w a = const, are planes. Since these are the planes of the meridians, it follows that the axes of the surfaces coincide, and consequently the case cited at the beginning of this section is the only one for nondevelopable surfaces. 185. Triple systems of Bianchi and of Weingarten. In 119 it was found that all the Bianchi transforms of a given pseudo- spherical surface are pseudospherical surfaces of the same total curvature, and that they are the orthogonal surfaces of a cyclic system of circles of constant radius. Hence the totality of these circles and surfaces constitutes a triply orthogonal system, such that the surfaces in one family are pseudospherical. As systems of this sort were first considered by Ribaucour (cf. 119), they are called the triple si/stems of Ribaucour. We proceed to the consideration of all triple systems such that the surfaces of one family are pseudospherical. These systems were first studied by Bianchi, * and consequently Darboux f has called them the systems of Bianchi. From 119 it follows that the parameters of the lines of curva ture of a pseudospherical surface of curvature l/a a can be so chosen that the linear element takes the form (1 6) d a = cos a o) du* + sin a o> dv\ where o> is a solution of the equation d a o> d a o> _ sin o> cos a) In this case the principal radii are given by 1 tan &) 1 cot a) (18) Pl a p, a In general the total curvature of the pseudospherical surfaces of a system of Bianchi varies with the surfaces. If the surfaces w 3 = const, are the pseudospherical surfaces, we may write the curvature in the form 1/f^, where ?/ 8 is a function of ?/ alone. Annali, 8er. 2, Vol. XIII (188,",), pp. 177-234; Vol. XIV (1880), pp. 115-130; Lezioni, Vol. II, chap, xxvii. t 7>vow wr les ni/stemes orthogonaux et les coonlonntes curvilignes, pp. 308-323. Paris, 18U8. (19) TRIFLE SYSTEMS OF BIANCHI 453 In accordance with (11) and (18) we put _1 1 dff, tan<w P* , 1 g// cot <a s) tf ?) ff If these values of - - and = be substituted in equations (8) for* r/r, (K,, equal to 1 and 2 respectively, we obtain 1 BJf, 3to 1 dlf a do) l = tan o> , = = cot a) From these equations we have, by integration, (20) ff l = < 13 cos co, H z = </>., 3 sin CD, where $ 13 and $ 23 are functions independent of w a and u v respectively. \Vo shall show that both of them are independent of u 3 . When the values of H l and !! from (20) are substituted in (19), we have respectively (21) /. to 3 log <. jff = f r C ot cw ( tan to I / da) d log A \ // = f r tan o> [ cot o> --- \- - z ! \ ,5w, ^3 / From these equations it follows that Hence, unless l;l and ^>., ;l are independent of w 3 , tan o> is equal to the ratio of a function of u v and w 3 and of a function of w 2 and ?< 3 . We consider the latter case and study for the moment a partic ular surface i/ 3 = c. By the change of parameters (^..(MP cjau^ the linear element of the surface reduces to (16), and (22) becomes tan co = where f and V are functions of u and v respectively. When this is substituted in (17), we obtain a , I-- /C\^- i *S \ * -w-r-ft . w^Ox V ,,, iv n , v (u"T 454 TKIPLY ORTHOGONAL SYSTEMS OF SURFACES If this equation be differentiated successively with respect to u and v, we find /U"\ f 1 /V"\ r 1 \u) ~uu~ + \r) ~vv = unless V or V is equal to zero. From this it follows that where K denotes a constant. Integrating, we have U"=2icU*+aU, V"=-2tcV s + l a and y3 being constants, and another integration gives U *=KU*+a(r*+<y, F 2 =-*F 4 + /3F 2 +S. When these expressions are substituted in (23), we find This condition can be satisfied only when the curvature is zero. Hence U 1 or V must be zero, that is, &) must be a function of u or v alone. In this case the surface is a surface of revolution. In accord ance with 184 a triple system of Bianchi arises from an infinity of pseudospherical surfaces of revolution with the same axis. When exception is made of this case, the functions (f> 13 and </> 23 in (20) are independent of u s . Hence the parameters of the sys tems may be chosen so that we have /2^x H IT H f7 When these values are substituted in the six equations (8), (9), they reduce to the four equations ~ 2 &) 2 ft) sin to cos &) = U, (25) duf 0M* III cot ft) - -f- tan to = 0, 0/1 g 2 ft) \ 1 d /sinftA 1 g< 2 ft) =() cu v \cos ft) du 1 duj U 3 du 3 \ U 3 / sin G) du 2 du 2 du 3 _d_ / 1 d 2 &) \ 1 d /cos ft)\ 1 do d^co _ Q du 2 \sin &) du 2 duj U 3 du 3 \ U 3 / cos ft) cu v du l du z TRIPLE SYSTEMS OP WEINGAKTEN 455 Darboux has inquired into the generality of the solution of this system of equations, and he has found that the general solution involves five arbitrary functions of a single variable. We shall not give a proof of this fact, but refer the reader to the investi gation of Darboux.* We turn to the consideration of the particular case where the total curvature of all the pseudospherical surfaces is the same, which may be taken to be 1 without any loss of generality. As triple systems of this sort were first discussed by Weingarten, we follow Bianchi in calling them systems of Weingarten. Of this kind are the triple systems of Ribaucour. For this case we have U 3 = 1, so that the linear element of space is (26) ds 2 = cos 2 &) dul + sin 2 a) du* Since the second of equations (25) may be written in either of the forms c a / \ ff w \ \ cw c z a) )= , n ft) du 2 du s / cos ft) du 2 du^ du s du 2 \cos ft) du^ du 3 / sin ft) du l du 2 du s if We pUt / -j 2 \2 / -j o2 \2 /o ycos ft) du 1 du^l V sin &) du z cu z j \^ 3 it follows from the last two of (25) and from (27) that Hence <J> is a function of u s alone. But by changing the param eter u z , an operation which will not affect the form of (26), we can give <& a constant value, say c. Consequently we have (28 ) -- -+ ---= c. \cos ft) Bu l du 3 J \sm co du 2 du 3 / \^ 3 / Bianchi has shown f that equation (28) and the first of (25) are equivalent to the system (25), when Z7 8 = l. Consequently the problem of the determination of triple systems of Weingarten is the problem of finding common solutions of these two equations. *L.c., pp. 313, 314; Bianchi, Vol. II, pp. 531, 532. t Vol. II, p. 550. 456 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES EXAMPLES 1. Show that the equations x = r cos u cos v, y = r cos u sin u, z =: r sin w define space referred to a triply orthogonal system. 2. A necessary and sufficient condition that the surfaces u s = const, of a triply orthogonal system be parallel is that H s be a function of u 3 alone. What are the other surfaces u\ = const., u% = const.? 3. Two near-by surfaces u s = const, intercept equal segments on those orthog onal trajectories of the surfaces w 3 const, which pass through a curve H s = const. on the former; on this account the curves #3 = const, on the surfaces u 3 = const. are called curves of equidistance. 4. Let the surfaces w 3 = const, of a triple system be different positions of the same pseudosphere, obtained by translating the surface in the direction of its axis. Determine the character of the other surfaces of the system. 5. Derive the following results for a triple system of Weingarten : V/8w\2 C+ U where the differential parameter is formed with respect to the linear element of a surface u 3 = const., and p g is the radius of geodesic curvature of a curve = const w 3 on this surface. Show that the curves of equidistance on the surfaces u s = const. are geodesic parallels of constant geodesic curvature. 6. Show that when c in (28) is equal to zero, the first curvature l/p 8 of the curves of parameter u 3 is constant and equal to unity; that equations similar to (12) become 2 _ 8<a a 2 u> dot = sin o> cos w 3 , = cos u sin u> 3 ; that if we put - w 3 , the last two of equations (25), where U & = 1, may be written cd gw dO aw -- 1 -- = sm 6 cos w, ^ H -- = cos sin u ; &d / 1 8*6 \ 2 / 1 8*6 \ 2 /de\* -- ^ = Sin 6 COS 0, / --- J -f / --- ] = / - ) . 8u cos6 and that When c = in (28) the system is said to be of constant curvature. 7. A necessary and sufficient condition that the curves of parameter u s of a system of Weingarten be circles is that w 3 be independent of u 3 . In this case (cf. Ex. 6) the surfaces u s = const, are the Bianchi transforms of the pseudo- spherical surface with the linear element ds* = cos*0du* THEOREM OF RIBAUCOUR 457 186. Theorem of Ribaucour. The following theorem is due to Ribaucour * : Griven a family of surfaces of a triply orthogonal system and their orthogonal trajectories; the osculating circles to the latter at their points of meeting with any surface of the family form a cyclic system. In proving this theorem we first derive the conditions to be satis fied by a system of circles orthogonal to a surface S so that they may form a cyclic system. Let the lines of curvature on S be parametric and refer the surface to the moving trihedral whose x- and ?/-axes are tangent to the curves v = const., u = const. We have (V, 63) (29) ^=n=p = qi =0. If < denotes the angle which the plane of the circle through a point makes with the corresponding zz-plane, the angle which the radius to a point P of the circle makes with its projection in the z^-plane, and R the radius of the circle, the coordinates of P with reference to the moving axes are x = R(\ -f cos 0) cos $, y =^(1+ cos 0) sine/), z=lism0. Moreover, the direction-cosines of the tangent to the circle at P are sin 6 cos <, sin 6 sin $, cos 6. If we express the condition that every displacement of P must be at right angles to this line, we have, from (29) and (V, 51), dB - [sin B( - + lL^i)+ q cos 0(1 + cos 0)1 du [_ \R du R / J f . J\ dR 77, sin<f>\ -."I 7 - sin 6 - + A_ _ _ p sm 0(i + cos B)\dv = 0. [_ \jri cv K / J The condition that this equation admit an integral is reducible to cosjAI [sin 4, cose/) E /J L R Hence, as remarked before ( 174), if there are three surfaces orthog onal to a system of circles, the system is cyclic. * Comptes Rendus, Vol. LXX (1870), pp. 330-333. 458 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES The condition that it be cyclic is (30) d_ /i ?1 sin (f>\ d_ cu \ R / dv\ R sin <f> cos (/ , d /sin (f) \ d icos<f) \_ /> Since the principal radii of S are given by (31) i = -|. i = J. the second of equations (30) reduces to the first when S is a sphere or a plane. Hence we have incidentally the theorem : A two-parameter system of circles orthogonal to a sphere and to any other surface constitute a cyclic system. We return to the proof of the theorem of Ribaucour and apply the foregoing results to the system of osculating circles of the curves of parameter u 3 of an orthogonal system at their points of intersection with a surface u= const. o From equations similar to (12) we have, by (11), cos <j) 1 d//., sin <f> 1 dH z and the equations analogous to (31) are 1 q 1 211^ 1 _ p l _ 1 3H Z p n ^ H^H Z du 3 />, 2 // 2 7/ 2 // 3 du s When these values are substituted in equations (30) the first vanishes identically, likewise the second, in consequence of equa tions (8). Hence the theorem of Ribaucour is proved.* 187. Theorems of Darboux. The question naturally arises whether any family of surfaces whatever forms part of a triply orthogonal system. This question will be answered with the aid of the following theorem of Darboux, f which we establish by his methods : A necessary and sufficient condition that two families of surfaces orthogonal to one another admit of a third family orthogonal to both is that the first two meet one another in lines of curvature. * For a geometrical proof the reader is referred to Darboux, I.e., p. 77. t L.c., pp. 6-8. THEOREM OF DARBOUX 459 Let the two families of surfaces be defined by (32) a(x, y, z) = a, @(x, y, z) = b, where a and b are the parameters. The condition of orthogonality is dx ~dx dy^y ~dz~dz~ In order that a third family of surfaces exist orthogonal to the surfaces of the other families, there must be a function 7(2, y, z) satisfying the equations . , _ dx dx dydy dz dz~ _ ~dx dx ~d^ ~dz~dz~ If dx, dy, dz denote the projections on the axes of a displace ment of a point on one of the surfaces 7 = const., we must have dx dy dz da da da dx dy dz = 0. Idx ~dy ~dz This equation is of the form (XIII, 9). The condition (XIII, 10) that it admit of an integral involving a parameter is da dx dz* dz dxdz dx dz* dz dxdz ~^ydxdy~~dx^y* + dydxdy { ^x ty*\~ where S indicates the sum of the three terms obtained by permut ing x, y, z in this expression. If we add to this equation the identity d (a, /3) \da ^ 01* -^-\ v f~f u f-f ^-\ i/ LV i f* 5 ^ ~i \ = ~~ the resulting equation may be written in the form (34) da dp J d/3\ ,/ p da T~ T~ a ^~ ~ ^l& dx dx \ dx da dj3 da d/3 dz dz S( ill )_ gr/3 _ I r\ I I *1 r\ . = 0, 460 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES where, for the sake of brevity, we have introduced the symbol 8(0, 4>), defined by If equation (33) be differentiated with respect to x, the result may be written Consequently equation (34) is reducible to da df3 dx dx r\ O /O ccc cp (35) da dz dy dz =o, which is therefore the condition upon a and /S in order that the desired function 7 exist. A displacement along a curve orthogonal to the surfaces a = const, is given by ^ = ^_^. da da da dx dy dz Such a curve lies upon a surface /3 = const, and since, by (35), it satisfies the condition = 0, it is a line of curvature on the surface (cf. Ex. 3, p. 247). Hence the curves of intersection of the surfaces a = const., fi = const., being the orthogonal trajectories of the above curves, are lines of curvature on the surfaces ft = const. And by Joachimsthal s theo rem ( 59) they are lines of curvature on the surfaces a = const, also. Having thus established the theorem of Darboux, we are in a position to answer the question at the beginning of this section. dx dy dz c/3 dx a/8 9y 8/8 dz 0/3 *5 if dy a/3 rf fe TRANSFORMATION OF COMBESCURE 461 Given a family of surfaces a const. ; the lines of curvature in one family form a congruence of curves which must admit a family of orthogonal surfaces, if the surfaces a = const, are to form part of an orthogonal system. If this condition is satisfied, then, accord ing to the theorem of Darboux, there is a third family of surfaces which together with the other two form an orthogonal system. If X v Y v Z 1 denote the direction-cosines of the tangents to the lines of curvature in one family on the surfaces a const., the ana lytical condition that there be a family of surfaces orthogonal to these curves is that the equation admit an integral involving a parameter. The condition for this is In order to find X^ Y^ Z 1 we remark that since they are the direc tion-cosines of the tangents to a line of curvature we must have and similar equations in Y, Z, where the function X is a factor of proportionality to be determined arid Jf, Y, Z are the direction- cosines of the normal to the surface a = const. Hence, if the surfaces are defined by a = const., the functions X v Y^ Z^ are expressible in terms of the first and second derivatives of a, and so equation (36) is of the third order in these derivatives. There fore we have the theorem of Darboux*: The determination of all triply orthogonal systems requires the integration of a partial differential equation of the third order. Darboux has given the name family of Lame to a family of surfaces which forms part of a triply orthogonal system. 188. Transformation of Combescure. We close our study of triply orthogonal surfaces with an exposition of the transformation of Combescure,^ by means of which from a given orthogonal system others can be obtained such that the normals to the surfaces of one system are parallel to the normals to the corresponding sur faces of the other system at corresponding points. * L.c., p. 12. f Annales de I Ecole Normale Superieure, Vol. IV (1867), pp. 102-122. 462 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES We make use of a set of functions /3 iK , introduced by Dar- boux * in his development of a similar transformation in space of n dimensions. By definition In terms of these functions equations (8), (9) are expressible in the form < 37 > $-* t + and formulas (10) become (38) A.jr.-arr,, Equations (37), (38) are the necessary and sufficient conditions that the expression ^ ^ + ^ ^ + ^ ^ be an exact differential. From their form it is seen that if we have another set of functions H[, 77 2 , // 8 satisfying the six conditions < 39 > *--- where the functions /3 tK have the same values as for the given system, the expression XJI[ d Ul -f JT 2 // 2 du 2 + Xfi du a , and similar ones in F, Z, are exact differentials, and so by quadra tures we obtain an orthogonal system possessing the desired property. In order to ascertain the analytical character of this problem, we eliminate H[ and H^ from equations (39) and obtain the three equations n _ .._ , -^ du, cu^ The general integral of a system of equations of this kind involves three arbitrary functions each of a single parameter u t . When one *L.c., p. 161. GENERAL EXAMPLES 463 has an integral, the corresponding values of H^ H^ are given directly by (39). Hence we have the theorem : With every triply orthogonal system there is associated an infinity of others, depending upon three arbitrary functions, such that the normals to the surfaces of any two systems at corresponding points are parallel.* EXAMPLES 1. In every system of Weingarten for which c in (28) is zero, the system of cir cles osculating the curves of parameter u s at points of a surface w 3 = const, form a system of Ribaucour ( 185). 2. If the orthogonal trajectories of a family of Lame" are twisted curves of the same constant first curvature, the surfaces of the family are pseudospherical surfaces of equal curvature. 3. Every triply orthogonal system which is derived from a cyclic system by a transformation of Combescure possesses one family of plane orthogonal trajectories. 4. If the orthogonal trajectories of a family of Lame* are plane curves, the cyclic system of circles osculating these trajectories at the points of any surface of the family may be obtained from the given system by a transformation of Combescure. 5. Determine the triply orthogonal systems which result from the application of the transformation of Combescure to a system of Ribaucour ( 185). GENERAL EXAMPLES 1. If an inversion by reciprocal radii ( 80) be effected upon a triply orthogonal system, the resulting system will be of the same kind. 2. Determine the character of the surfaces of the system obtained by an inversion from the system of Ex. 1, 185, and show that all the curves of intersection are circles. 3. Establish the existence of a triply orthogonal system of spheres. 4. A necessary and sufficient condition that the asymptotic lines correspond on the surfaces u% = const, of a triply orthogonal system is that there exist a relation of the form 03 = ? where 0j, 2 <t>s are functions independent of w 3 . 5. When the condition of Ex. 4 is satisfied, those orthogonal trajectories of the surfaces u s = const, which pass through points of an asymptotic line on a sur face u s = const, constitute a surface S which meets the surfaces u s = const, in asymptotic lines of the latter and geodesies on <S. 6. Show that the asymptotic lines correspond on the pseudospherical surfaces of a triple system of Bianchi. 7. Show that there exist triply orthogonal systems for which the surfaces in one family, say u$ const., are spherical, and that the parameters can be chosen so that HI = cosh 8, Hz = sinh 6, H 3 = U S . CUz Find the equations of Lame" for this case. 8. Every one-parameter family of spheres or planes is a family of Lame . *Cf. Bianchi, Vol. II, p. 494. 464 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 9. In order to obtain the most general triply orthogonal system for which the surfaces in one family are planes, one need construct an orthogonal system of curves in a plane and allow the latter to roll over a developable surface, in which case the curves generate the other surfaces. When the developable is given, the determination of the system reduces to quadratures. 10. Show that the most general triply orthogonal system for which one family of Lame" consists of spheres passing through a point can be found by quadratures. 11. Show that a family of parallel surfaces is a family of Lame . 12. Show that the triply orthogonal systems for which the curves of parameter 3 are circles passing through a point can be found without quadrature. 13. By means of Ex. 6, 185, show that for a system of Weingarten of constant curvature the principal normals to the curves of parameter w 3 at the points of meet ing with a surface u 3 = const, form a normal pseudospherical congruence, and that the surfaces complementary to the surfaces w 3 = const, and their orthogonal tra jectories constitute a system of Weingarten of constant curvature. 14. By means of Ex. 13 show that for a triple system arising from a system of Weingarten of constant curvature by a transformation of Combescure the osculat ing planes of the curves w 3 = const., at points of a surface u s = const., envelop a surface S of the same kind as this surface M 3 = const. ; and these surfaces S and their orthogonal trajectories constitute a system of the same kind as the one result ing from the Combescure transformation of the given system of Weingarten. 15. Show that a necessary condition that the curves of parameter u\ of a triple system of Bianchi be plane is that w satisfy also the conditions d(*t . Set = 023 sm w, = 0i 3 sin w, di/2 dui where 23 and 0i 8 are independent of HI and w 2 respectively (cf. Ex. 5, p. 317). Show that if 0is and 02 3 satisfy the conditions where a and b are constants and U& is an arbitrary function of w 8 , the function w, given by g0 23 a0i 8 COS W = determines a triply orthogonal system of Bianchi of the kind sought. 16. When Z7 3 = 1 and w is independent of u 2 , the first and fourth of equations (25) may be replaced by g w = sin w. dui Show that for a value of satisfying this condition and the other equations (25) the expressions / rfadu s \ r^cosu, HI = cos w ( I - + 0i ) - / : - du 3 -f 0i, \J sin w / J sin w H z = sin w ( C^J^ _(- 0A- r0 3 cZM8 + 2 , \J sin w / J ( ffoduz \ du HB ~ ( I - + 0i \ J sin w / 5u 3 GENERAL .EXAMPLES 465 where 0i, 2 , 3 are functions of MI, u 2j w 3 respectively, and the accent indicates differentiation, define a triply orthogonal system for which the surfaces M 3 = const, are molding surfaces. 17. Under what conditions do the functions d 2 sin w where Z7 2 and Z7s are functions of M 2 and M 3 respectively, determine a triply orthog onal system arising from a triple system of Bianchi by a transformation of Combes- cure ? Show that in this case the surfaces w 2 = const, are spheres of radius Z7 2 , and that the curves of parameter M 2 in the system of Bianchi are plane or spherical. 18. Prove that the equations y B(UI b) m i(u 2 b) m *(u s 6) m s, Z = C(Ui C) wl i(lt 2 C) w 2(w 3 c) "3, where A, B, C, a, 6, c, m t - are constants, define space referred to a triple system of surfaces, such that each surface is cut by the surfaces of the other two families in a conjugate system. 19. Given a surface 8 and a sphere S; the circles orthogonal to both constitute a cyclic system ; hence the locus of a point upon these circles which is in constant cross-ratio with the points of intersection with S and S is a surface Si orthogonal to the circles ; Si may be looked upon as derived from S by a contact transformation which preserves lines of curvature ; such a transformation preserves planes and spheres. 20. When S of Ex. 19 is a cyclide of Dupin, so are the surfaces Si, and also the surface which is the locus of the circles which meet S in any line of curvature ; hence all of these surfaces form a triple system of cyclides of Dupin. 21. Given three functions Z7, defined by Ui = imuf + 2 mm + p^ (i = 1, 2, 3) where m t -, Wj, pi are constants satisfying the conditions Sm t - = 0, Sn t - = 0, Sp t - = ; and given also the function N= tti(M 2 - u 3 )VUi -f a 2 (M 8 - MI) VU^ + <* 3 (MI - + pZniiUi + 7 (PiMaWa + PZ^UI where ,-, 0, 7 are constants ; determine under what condition the functions TT W 2 - U S U S - Ui MI -Ma &l - ;= - 7= ^33= - p= N^lfi N^U 2 N-VU 3 determine a triply orthogonal system. Show that all of the surfaces are isothermic, and that they are cyclides of Dupin. 22. Determine whether there exist triply orthogonal systems of minimal surfaces. INDEX The numbers refer to pages. References to an author and his contributions are made in the form of the first Bianchi paragraph, whereas when a proper name is part of a title the reference is given the form as in the second Bianchi paragraph. Acceleration, 15, 60 Angle between curves, 74, 200 Angle of geodesic contingence, 212 Applicable surfaces, definition, 100 ; to the plane, 101, 156 ; invariance of geodesic curvature, 135 ; invariance of total curvature, 156 ; solution of the problem of determining whether two given surfaces are applicable, 321-326 ; pairs of, derived from a given pair, 349. See Deformation of surfaces Area, element of, 75, 145 Area of a portion of a surface, 145, 250 ; minimum, 222 Associate surfaces, definition, 378 ; de termination, 378-381; of a ruled surface, 381; of the sphere, 381; ap plicable, 381; of the right helicoid, 381 ; of an isothermic surface, 388 ; of pseudospherical surfaces, 390 ; of .quadrics, 390, 391; characteristic property, 425 Asymptotic directions, definition, \2S *Asymptotic lines, definition, 128 ; para metric, 129, 189-194 ; orthogonal, 129 ; straight, 140, 234 ; spherical represen tation, 144, 191-193; preserved by protective transformation, 202 ; pre served in a deformation, 342-347 Backhand, transformation of, 284-290 Beltrami (differential parameters), 88, 90; (geodesic curvature), 183; (ruled W-surfaces), 299 ; (applicable ruled surfaces), 345 ; (normal congruences), 403 Bertrand curves, definition, 39 ; proper ties, 39-41 ; parametric equations, 51 ; on a ruled surface, 250 ; deformation, 348 Bianchi (theorem of permutability), 286-288 ; (surfaces with circular lines of curvature), 311; (surfaces with spherical lines of curvature), 315 ; (associate surfaces), 378 ; (cyclic con gruences of Ribaucour), 435 ; (cyclic systems), 441 Bianchi, transformation of, 280-283, 290, 318, 320, 370, 456 ; surfaces of, 370, 371, 442, 443, 445 ; generalized transformation of, 439 ; triply orthog onal systems of, 452-454, 464, 465 Binormal to a curve, definition, 12 ; spherical indicatrix, 50 Bmormals which are the principal nor mals to another curve, 51 Bonnet (formula of geodesic curvature), 136 ; (surfaces of constant curvature), 179; (lines of curvature of Liouville type), 232 ; (ruled surf aces), 248 ; (sur faces of constant mean curvature), 298 IJour (helicoids), 147; (associate isother mic surfaces), 388 Canal surfaces, definition, 68 ; surfaces of center, 186 Catenoid, definition, 150 ; adjoint sur face of, 267 ; surfaces applicable to, 318 Cauchy, problem of, 265, 335 Central point, 243 Central plane, 244 Cesaro (moving trihedral), 8S Characteristic equation, 375 Characteristic function,, 374, 377 Characteristic lines, 13Q, 131 ; paramet ric, 203 Characteristics, of a fanUly of surfaces, 59-61 ; of the tangent pfones to a sur--/ face, 126 Christoffel (associate isothermic sur faces), 388 Christoffel symbols, definition, 152, 153 ; relations between, for a surface and its spherical representation, 162, 193, 201 Circle, of curvature, 14 ; osculating, 14 * References to asymptotic lines, geodesies, lines of curvature, etc., on particular kinds of surfaces are listed under the latter. 467 468 INDEX Circles, orthogonal system of, in the plane, 80, 97 ; on the sphere, 301 Circular lines of curvature, 149, 310, 316, 423, 446 Circular point on a surface, 124 Codazzi, equations of, 155-157, 161, 168, 170, 189, 200 Combescure transformation, of curves, 50 ; of triple systems, 401-465 Complementary surface, 184, 185, 283, 290, 370, 464 Conforinal representation, of two sur faces, 98-100, 391; of a surface and its spherical representation, 143 ; of a surface upon itself, 101-103 ; of a plane upon itself, 104, 112 ; of a sphere upon the plane, 109 ; of a sphere upon itself, 110, 111; of a pseudospherical surface upon the plane, 317 Conformal-con jugate representation of two surfaces, 224 Congruence of curves, 426 ; normal, 430 Congruence of straight lines (rectilinear), definition, 392 ; normal, 393, 398, 401, 402, 403, 412, 422, 423, 437; associate normal, 401-403, 411; ruled surfaces, 393, 398, 401 ; limit points, 396 ; prin cipal surfaces, 396-398, 408 ; principal planes, 396, 397; developable*, 398, 409, 414, 421, 432, 437; focal points, 398, 399, 425; middle point, 399; middle surface, 399, 401, 408, 413, 421-424 ; middle envelope, 413, 415 ; focal planes, 400, 401, 409, 416 ; focal surfaces, 400, 406, 409-411, 412, 414, 416, 420 ; derived, 403-405, 411, 412 ; isotropic, 412, 413, 416; of Guichard, 414,415,417,422,442 ; pseudospherical, 184, 415, 416, 464 ; W-, 417-420, 422, 424 ; of Ribaucour, 420-422, 424, 425, 435, 442, 443 ; mean ruled surfaces, 422, 423, 425 ; cyclic, 431-445 ; spher ical representation of cyclic, 432-433 ; cyclic of Ribaucour, 435, 442, 443 ; developables of cyclic, 437, 441 ; normal cyclic, 437 Conjugate directions, 126, 173 ; normal radii in, 131 Conjugate system, definition, 127, 223 ; parametric, 195, 203, 223, 224 ; spher ical representation of, 200 ; of plane curves, 224 ; preserved by projective transformation, 202 ; preserved in a deformation, 338-342, 348, 349 Conjugate systems in correspondence,- 130 Conoid, right, 56, 58, 59, 68, 82, 98, 112, 120, 195, 347 Coordinates, curvilinear, on a surface, 55 ; curvilinear, in space, 447 ; sym metric, 91-93 ; tangential, 163, 194, 201; elliptic, 227 Correspondence with orthogonality of linear elements, 374-377, 390 Corresponding conjugate systems, 130 Cosserat (infinitesimal deformation), 380, 385 Cross-ratio, of four solutions of a Riccati equation, 26 ; of points of intersection of four-curved asymptotic lines on a ruled surface, 249 ; of the points in which four surfaces orthogonal to a cyclic system meet the circles, 429 Cubic, twisted, 4, 8, 11, 12, 15, 269 Curvature, first, of a curve, 9; radius of, 9; center of, 14; circle of, 14; constant, 22, 38, 51 Curvature, Gaussian, 123 ; geodesic (see Geodesic) Curvature, mean, of a surface, 123, 126, 145 ; surfaces of constant (see Sur face) Curvature, normal, of a surface, radius of, 118, 120, 130, 131, 150; principal radii of, 119, 120, 291, 450 ; center of, 118, 150; principal centers of, 122 Curvature, second, of a curve, 16 ; con stant, 50. See Torsion Curvature, total, of a surface, 123, 126, 145, 155, 156, 160, 172, 186, 194, 208, 211 ; radius of, 189 ; surfaces of con stant (see Surface) Curve, definition, 2; of constant first curvature, 22, 38, 51; of constant torsion, 50 ; form of a, 18 Cyclic congruences. See Congruences Cyclic system, 426-445 ; definition, 426 ; of equal circles, 430, 443 ; surfaces orthogonal to, 436, 437, 444, 457; planes envelop a curve, 439, 440 ; planes through a point, 440, 441 ; planes depend on one parameter, 442 ; triple system associated with a, 446 ; associated with a, triple system, 457; 458 Cyclides of Dupin, 188, 312-314, 412, 422, 465 D, Z7, 7)", definition, 115 ; for the mov ing trihedral, 174 A Jb i &"> definition, 386 Darboux (moving trihedral), 168, 169, 170 ; (asymptotic lines parametric), 191 ; (conjugate lines parametric), 195 ; (lines of curvature preserved by an inversion), 196 ; (asymptotic lines and conjugate systems preserved by projective transformation), 202 ; (geo desic parallels), 216, 217 ; (genera tion of new surfaces of Weingarten), 298 ; (generation of surfaces with plane lines of curvature in both sys tems), 304 ; (general problem of INDEX 469 deformation), 332 ; (surfaces appli cable to paraboloids), 367 ; (triply orthogonal systems), 458-461 Darboux, twelve surfaces of, 391; de rived congruences of, 404, 405 Deformation of surfaces (see Applicable surfaces) ; of surfaces of revolution (see Surfaces of revolution) ; of mini mal surfaces, 264, 269, 327-330 ; of surfaces of constant curvature, 321- 323 ; general problem, 331-333 ; which changes a curve on the surface into a given curve in space, 333-336 ; which preserves asymptotic lines, 336, 342, 343 ; which preserves lines of curva ture, 336-338, 341 ; which preserves conjugate systems, 338-342, 349, 350, 443 ; of ruled surfaces, 343-348, 350, 367; method of Weingarten, 353-369 ; of paraboloids, 348, 368, 369 ; of the envelope of the planes of a cyclic system, 429, 430 Developable surface, definition, 61 ; equation, 64 ; particular kinds, 69 ; rectifying, 62, 64, 112, 209 ; polar, 64, 65, 112, 209 ; applicable to the plane, 101, 156, 219, 321, 322 ; formed by nor mals to a surface at points of a line of curvature, 122 ; principal radii, 149 ; total curvature, 156, 250 ; geodesies on a, 224, 268, 318, 322 ; fundamental property, 244; of a congruence (see Congruence) Dextrorsum, 19 Differential parameters, of the first order, 84-88, 90, 91, 120, 160, 166, 186 ; of the second order, 88-91, 160, 165, 166, 186 Diui (spherical representation of asymp totic lines), 192; (surf aces of Liouville), 214 ; (ruled TF-surfaces), 299 Dini, surface of, 291, 318 Director-cone of a ruled surface, 141 Director-developable of a surface of Monge, 305 Directrix of a ruled surface, 241 Dobriner (surfaces with spherical lines of curvature), 315 Dupin (triply orthogonal systems), 449 Dupin, indicatrix of, 124-126, 129, 150 ; cyclide of (see Cyclide) ; theorem of Malus and, 403 jE, F, G, definition, 70 ; for the moving trihedral, 174 &> &> ^ definition, 141 ; for the moving trihedral, 174 e,/,/, flr, definition, 393 Edge of regression, 43, 60, 69 Element, of are.a, 75, 145 ; linear (see Linear element) Ellipsoid, equations, 228 ; normal sec tion, 234 ; polar geodesic system, 236-238; umbilical geodesies, 236, 267; surface corresponding with par allelism of tangent plane, 269. See Quadrics Elliptic coordinates, 227 Elliptic point of a surface, 125, 200 Elliptic type, of pseudospherical sur faces, 274 ; of surfaces of Bianchi, 370, 371 Enneper (torsion of asymptotic lines), 140 ; (equations of a minimal surface) , 256 Enneper, minimal surface of, 269; sur faces of constant curvature of , 317, 320 Envelope, definition, 59, 60 ; of a one- parameter family of planes, 61-63, 64, 69, 442 ; of a one-parameter fam ily of spheres, 66-69 ; of a two-param eter family of planes, 162, 224, 426, 439; of geodesies, 221; of a two- parameter family of spheres, 391, 444 * Equations, parametric, 1, 2, 52, 53; of a curve, 1, 2, 3, 21; of a surface, 52, 53, 54 Equidistance, curves of, 456 Equidistantial system, 187, 203 Equivalent representation of two sur faces, 113, 188 Euler, equation of, 124, 221 Evolute, of a curve, 43, 45-47 ; of a surface, 180, 415 (see Surface of center) ; of the quadrics, 234 ; mean, of a surface, 165, 166, 372 F. SeeE & See // . Seee Family, one-parameter, of surfaces, 59, 446, 447, 451, 452, 457-461; of planes, 61-64, 69, 442, 463 ; of spheres, 66-69, 309, 319, 463 ; of curves, 78-80 ; of geo desies, 216, 221 Family, two-parameter, of planes, 162, 224, 426, 439 ; of spheres, 391, 444 Family of Lame", 461, 463, 464 Focal conic, 226, 234, 313, 314 Focal planes, 400, 401, 409, 416 Focal points, 398, 399, 425 Focal surface, of a congruence, 400 ; reduces to a curve, 406, 412 ; funda mental quantities, 409-411 ; develop able, 412; met by developables in lines of curvature, 414 ; of a pseudo- spherical congruence, 416; infinitesi mal deformation of, 420 ; intersect, 423 * For references such as Equations of Codazzi, see Codazzi. 4TO INDEX Form of a curve, 18 Frenet-Serret formulas, 17 Fundamental equations of a congruence, 406, 407 Fundamental quadratic form, of a sur face, first, 7 1 ; of a surface, second, 115; of a congruence, 393 Fundamental quantities, of the first order, 71; of the second order, 115 Fundamental theorem, of the theory of curves, 24 ; of the theory of surfaces, 159 G. See E & 8ee g. See e Gauss (parametric form of equations), 60 ; (spherical representation), 141 ; (total curvature of a surface), 155 ; (geodesic parallels), 200; (geodesic cir cles) 207 ; (area of geodesic triangle) 209 Gauss, equations of, 154, 155, 187 Generators, of a developable surface, 41 ; of a surface of translation, 198 ; of a ruled surface, 241 Geodesic circles, 207 Geodesic contingence, angle of, 212 Geodesic curvature, 132, 134, 135, 13(5, 140, 213, 223 ; radius of, 132, 150, 151, 174, 170, 209, 411 ; center of, 132, 225, 423 ; invariance of, 135 ; curves of constant, 137, 140, 187, 223, 319 Geodesic ellipses and hyperbolas, 213- 215, 225 Geodesic parallels, 207 Geodesic parameters, 207 Geodesic polar coordinates, 207-209, 230, 276 Geodesic representation, 225, 317 Geodesic torsion, 137-140, 174, 176 ; radius of, 138, 174, 176 Geodesic triangle, 209, 210 * Geodesies, definition, 133 ; plane, 140 ; equations of, 204, 205, 215-219; on surfaces of negative curvature, 211 ; on surfaces of Liouville, 218, 219 Goursat, surfaces of, 306, 372 Guichard (spherical representation of the developables of a congruence), 409; (congruences of Ribaucour), 421 Guichard, congruences of, 414, 415, 417, 422, 442 I/, definition, 71 //, definition, 142 Hamilton, equation of, 397 Hazzidakis, transformation of, 278, 279, 338 Helicoid, general, 146-148 ; parameter of, 140 ; meridian of, 140 ; geodesies, 149, 151, 209 ; surfaces of center of, 186 ; pseudospherical, 291 ; is a IP-sur face, 300 ; minimal, 329, 331 ; appli cable to a hyperboloid, 347 Helicoid, right, 146, 148, 203, 247, 250, 260, 267, 330, 347, 381, 422 Helix, circular, 2, 41, 45, 203 ; cylindri cal, 20, 21, 29, 30, 47, 64 Henneberg, surface of, 267 Hyperbolic point, 125, 200 Hyperbolic type, of pseudospherical sur face, 273 ; of surface of Bianchi,371, 379 Hyperboloid, equations, 228 ; fundamen tal quantities, 228-230; evolute of, 234 ; of revolution, 247, 348 ; lines of striction, 268 ; deformation of, 347, 348. See Quadrics Indicatrix, of Dupin (seel)upin); spheri cal (.see Spherical) Infinitesimal deformation of a surface, 373, 385-387 ; generatrices, 373, 420 ; of a right helicoid, 381 ; of ruled sur faces, 381 ; in which lines of curva ture are preserved, 387, 391; of the focal surfaces of a TF-congruence, 420 Intrinsic equations of a curve, 23, 29, 30, 30 Invariants, differential, 85-90 ; of a dif ferential equation, 380, 385, 406 Inversion, definition, 190 ; preserves lines of curvature, 190 ; preserves an isotherm ic system of lines of curva ture, 391 ; preserves a triply orthog onal system, 403. See Transformation by reciprocal radii Involute, of a curve, 43-45, 311 ; of a surface, 180, 184, 300 Isometric parameters. See Isothermic parameters ,- Isometric representation, 100, 113 Isothermal-conjugate systems of curves, 198-200; spherical representation, 202 ; formed of lines of curvature, 147, 203, 233, 278 ; on associate surfaces, 300 Isothermal -orthogonal system. See Iso thermic orthogonal system Isothermic orthogonal systems, 93-98, 209, 252, 254 ; formed of lines of curva ture (see Isothermic surface) Isothermic parameters, 93-97, 102 Isothermic surface, 108, 159, 232, 253, 269, 297, 387-389, 391, 425, 465 Isotropic congruence, 412, 413, 416, 422- 424 Isotropic developable, 72, 171, 412, 424 Isotropic plane, 49 * See footnote, p. 467. INDEX 471 Jacob! (geodesic lines), 217 Joachimsthal (geodesies and lines of curvature on central quadrics), 240 Joachimsthal, theorem of, 140 ; surfaces of, 308, 309, 319 Kummer (rectilinear congruences), 392 Lagrange (minimal surfaces), 251 Lame (differential parameters), 85 Lame", equations of, 449 ; family of, 401, 463, 464 Lelieuvre, formulas of, 193, 195, 417, 419, 420, 422 Lie (surfaces of translation), 197, 198 ; (double minimal surfaces), 259 ; (lines of curvature of JF-surfaces), 293 Lie, transformation of, 289, 297 Limit point, 396, 399 Limit surface, 389 Line, singular, 71 *Line of curvature, definition, 121, 122, 128; equation of, 121, 171, 247; par ametric, 122, 151, 186 ; normal cur vature of, 121, 131 ; geodesic torsion of, 139 ; geodesic, 140 ; two surfaces inter secting in, 140 ; spherical representa tion of, 143, 148, 150; osculating plane, 148; plane, 149, 150, 201, 305-314, 319, 320, 463 ; plane in both systems, 269, 300-304, 319, 320 ; spherical, 149, 314-317, 319, 320, 465 ; circular, 149, 310-314, 316, 446; on an isothermic surface, 389, Line of striction, 243, 244, 248, 268, 348, 351, 352, 369, 401, 422 Linear element, of a curve, 4, 5 ; of a surface, 42, 71, 171; of the spherical representation, 141, 173, 393; reduced form, 353 ; of space, 447 Lines of length zero. See Minimal lines Lines of shortest length, 212, 220 Liouville (form of Gauss equation), 187 ; (angle of geodesic contingence), 212 Liouville, surfaces of, 214, 215, 218, 232 Loxodromic curve, 78, 108, 112, 120, 131, 140, 209 Mainardi, equations of, 156 Malus and Dupin, theorem of, 403 v. Mangoldt (geodesies on surfaces of positive curvature), 212 Mean curvature, 123, 126, 145 Mean evolute, 165, 166, 372 Mean ruled surfaces of a congruence, 422, 423, 425 Mercator chart, 109 Meridian, of a surface of revolution, 107; of a helicoid, 146 Meridian curve on a surface, 260 Meusnier, theorem of, 118 Middle envelope of a congruence, 413, 415 Middle point of a line of a congruence, 399 Middle surface of a congruence, 399, 401, 408, 413, 421-424 Minding (geodesic curvature), 222, 223 Minding, problem of, 321, 323, 326 ; method of, 344 Minimal curves, 6, 47, 49, 255, 257 ; on a surface, 81, 82, 85, 91, 254-265, 318, 391 ; on a sphere, 81, 257, 364-366 390 Minimal straight lines, 48, 49, 260 Minimal surface, definition, 129, 251; asymptotic lines, 129, 186, 195, 254, 257, 269 ; spherical representation, 143, 251-254; ruled, 148; helicoidal, 149, 330, 331 ; of revolution, 160 ; parallel plane sections of, 160 ; mini mal lines, 177, 186, 254-265; lines of curvature, 186, 253, 257, 264, 269 ; double, 258-260 ; algebraic, 260-262 ; evolute, 260, 372 ; adjoint, 254, 263, 267, 377; associate, 263, 267, 269, 330, 381; of Scherk, 260; of Henneberg, 267; of Enneper, 269; deformation of, 264, 327-329, 349, 381 ; determi nation of, 265, 266 ; geodesies, 267 Molding surface, definition, 302 ; equa tions of, 307, 308 ; lines of curvature, 307, 308, 320 ; applicable, 319, 338 ; associate to right helicoid, 381 ; nor mal to a congruence of Ribaucour, 422 Molding surfaces, a family of Lame" of, 465 Monge (equations of a surface), 64 ; (molding surfaces), 302 Monge, surfaces of, 305-308, 319 Moving trihedral for a curve, 30-33 ; applications of, 33-36, 39, 40, 64-68 Moving trihedral for a surface, 166-170 ; rotationsof, 169; applications of, 171- 183, 281-288, 336-338, 352-364, 426-442 Normal, principal, definition, 12; par allel to a plane, 16, 21 Normal congruence of lines (see Con gruence) ; of curves (see Congruence) Normal curvature of a surface. See Curvature Normal plane to a curve, 8, 15, 65 Normal section of a surface, 118, 234 Normal to a curve, 12 Normal to a surface, 57, 114, 117, 120, 121, 141, 195 * See footnote, p. 467. 472 INDEX Normals, principal, which are principal normals of another curve, 41 ; which are binormals of another curve, 51 Order of contact, 8, 21 Orthogonal system of curves, 75, 77, 80-82, 91, 119, 129, 177, 187; par ametric, 75, 93, 122, 134 ; geodesies, 187 ; isothermic (see Iso thermic) Orthogonal trajectories, of a one-param eter family of planes, 35, 451 ; of a family of curves, 50, 79, 95, 112, 147, 149, 150 ; of a family of geodesies, 216 ; of a family of surfaces, 446, 451, 452, 456, 457, 460, 463, 464 Osculating circle, 14, 21, 65 Osculating plane, definition, 10 ; equa tion of, 11 ; stationary, 18 ; meets the curve, 19 ; passes through a fixed point, 22 ; orthogonal trajectories of, 35 ; of edge of regression, 57 ; of an asymp totic line, 128 ; of a geodesic, 133 Osculating planes of two curves parallel, 50 Osculating sphere, 37, 38, 47, 51, 65 Parabolic point on a surface, 125 Parabolic type, of pseudospherical sur faces, 274 ; of surfaces of Bianchi, 370, 371, 442, 443, 445 Paraboloid, a right conoid, 56 ; tangent plane, 112 ; asymptotic lines, 191, 233; a surface of translation, 203 ; equa tions, 230, 330 ; fundamental quanti ties, 231 ; lines of curvature, 232, 240 ; evolute of, 234 ; of normals to a ruled surface, 247 ; line of striction, 268 ; deformation of, 348, 349, 367-369, 372 ; congruence of tangents, 401. See Quadrics Parallel, geodesic, 86, 207 ; on a surface of revolution, 107 Parallel curves, 44 Parallel surface, definition, 177 ; lines of curvature, 178 ; fundamental quan tities, 178; of surface of constant cur vature, 179 ; of surface of revolution, 185 Parallel surfaces, a family of Lame" of, 446 Parameter, definition, 1 ; of distribution, 245, 247, 268, 348, 424, 425 Parametric curves, 54, 55 Parametric equations. See Equations Plane curve, condition for, 2, 16 ; curv ature, 15 ; equations, 28, 49 ; intrinsic equations, 36 Plane curves forming a conjugate sys tem,. 224 Plane lines of curvature. See Lines of curvature Point of a surface, singular, 71 ; elliptic, 125, 200 ; hyperbolic, 125, 200 ; para bolic, 125; focal (see Focal) ; middle (see Middle); limit (see Limit) Polar developable, 64, 65, 112, 209 Polar line of a curve, 15, 38, 46 Principal directions at a point, 121 Principal normal to a curve. See Normal Principal planes of a congruence, 396, 397 Principal radii of normal curvature, lit), 120, 291, 450 Principal surfaces of a congruence, 390- 398, 408 Projective transformation, preserves os culating planes, 49 ; preserves asymp totic lines and conjugate systems, 202 Pseudosphere, 274, 290 Pseudospherical congruence, 415, 416, 464 ; normal, 184 Pseudospherical surface, definition, 270 ; asymptotic lines, 190, 290, 414 ; lines of curvature, 190, 203, 280, 320 ; geo desies, 275-277, 283, 317, 318 ; defor mation, 277, 323 ; transformations of, 280-290, 318, 320, 370, 45(5 ; of Dini, 291, 318; of Enneper, 317, 820; evo lute, 318 ; involute, 318 ; surfaces with the same spherical representation of their lines of curvature as, 320, 371, 437, 439, 443, 444. See Surface of constant total curvature Pseudospherical surface of revolution, of hyperbolic type, 273 ; of elliptic type, 274 ; of parabolic type, 274 Pseudospherical surfaces, a family of Lam6 of, 452-456, 464 Quadratic form. See Fundamental Quadrics, confocal, 226,401 ; fundamen tal quantities, 229 ; lines of curvature, 233, 239, 240 ; asymptotic lines, 233 ; geodesies, 234-236, 239, 240 ; associate surfaces, 390, 391 ; normals to, 422. See Ellipsoid, Hyperboloid, Paraboloid Representation, conformal (see Con- formal); isometric, 100, 113; equiv alent, 113, 188; Gaussian, 141; conformal-con jugate, 224 ; geodesic, 225, 317 ; spherical (see Spherical) Revolution, surfaces of. See Surface Ribaucour (asymptotic lines on surfaces of center), 184 ; (cyclic systems of equal circles), 280; (limit surfaces), 389 ; (middle envelope of an isotropic congruence), 413 ; (cyclic systems), 426, 428, 432 ; (deformation of the envelope of the planes of a cyclic system), 429, 430 ; (cyclic systems associated with a triply orthogonal system), 457 INDEX 473 Ribaucour, congruence of, 420-422, 424, 425, 435, 442, 443 ; triple systems of, 452, 455, 463 Riccati equation, 25, 26, 50, 248, 429 Rodrigues, equations of, 122 * Ruled surface, definition, 241 ; of tan gents to a surface, 188 ; generators, 241; directrix, 241; linear element, 241, 247 ; director-cone, 241 ; line of striction, 243, 244, 248, 268, 348, 351, 352, 309, 401, 422 ; central point, 243 ; central plane, 244 ; parameter of dis tribution, 245, 247, 208, 348, 424, 425 ; doubly, 234; normals to, 195, 247; tangent plane, 246, 247, 268 ; total curvature, 247 ; asymptotic lines, 248- 250 ; mean curvature, 249 ; lines of curvature, 250, 268 ; conjugate, 268 ; deformation, 343-348, 350, 367; spher ical indicatrix of, 351; infinitesimal deformation, 381 ; of a congruence, 393-395, 398, 401, 422, 423. See Right conoid, Hyperboloid, Paraboloid Scheffers (equations of a curve), 28 Scherk, surface of, 260 Schwarz, formulas of, 264-267, 269 Singular line of a surface, 71 Singular point of a surface, 71 Sinistrorsum, 19 Sphere, equations, 62, 77, 81 ; minimal lines, 81 ; conformal representation, 109-111 ; equivalent representation, 113 ; fundamental quantities, 116, 171; principal radii, 120; asymptotic lines, 223, 422 Spheres, family of. See Family Spherical curve, 36, 38, 47, 50, 149, 314-316, 317, 319, 320, 465 Spherical indicatrix, of the tangents to a curve, 9, 13, 50, 177 ; of the binormals to a curve, 50, 177 ; of a ruled surface, 351 Spherical representation of a congruence, definition, 393 ; principal surfaces, 397, 408 ; developables, 409, 412-414, 422, 432-435, 437, 441 Spherical representation of a surface, definition, 141 ; fundamental quan tities, 141-143, 160-165, 173; lines of curvature, 143, 148, 150, 151, 188, 201, 253, 279, 280, 292, 296, 301, 302, 308, 314, 315, 320, 371, 387, 437, 442- 445 ; asymptotic lines, 144, 148, 191- 195, 254, 340, 390, 414 ; area of closed portion, 145 ; conjugate system, 200- 202, 257, 385 Spherical representation of an axis of a moving trihedral, 354 Spherical surface, definition, 270; par allels to, 179 ; of revolution, 270-272 ; geodesies, 275-279, 318 ; deformation, 276, 323 ; lines of curvature, 278 ; transformation, 278-280, 297 ; invo lute, 300 ; of Enneper, 317 ; surface with the same spherical representation of its lines of curvature as, 338. See Surface of constant total curvature Spherical surfaces, a family of Lame" of, 463 Spiral surface, definition, 151 ; gener ation, 151 ; lines of curvature, 151 ; minimal lines, 151 ; asymptotic lines, 151 ; geodesies, 219 ; deformation, 349 Stereographic projection, 110, 112 Superosculating circle, 21 Superosculating lines on a surface. 187 t Surface, definition, 53 Surface, limit, 389 \ Surface of center, definition, 179 ; met by developables in a conjugate sys tem, 180, 181 ; fundamental quantities, 181, 182 ; total curvature, 183 ; asymp totic lines, 183, 184 ; lines of curva ture, 183, 184 ; a curve, 186, 188, 308- 314 ; developable, 186, 305-308 Surface of constant mean curvature, definition, 179 ; parallels to, 179 ; lines of curvature, 296-298 ; transforma tion, 297 ; deformation, 298 ; minimal curves, 318 Surface of constant total curvature, definition, 179 ; area of geodesic tri angle, 219 ; geodesies, 224 ; lines of curvature, 317; asymptotic lines, 317; spherical representation, 372. See Pseudospherical surface and Spheri cal surface Surface of reference, 392 Surface of revolution, definition, 107 ; fundamental quantities, 107, 147 ; loxodromic curve (see Loxodromic) ; deformation, 108, 112, 147, 149, 260, 276, 277, 283, 326-331, 341, 349-350, 362-364, 369, 370, 372, 444; partic ular, 111, 160, 320 ; equivalent repre sentation, 113 ; lines of curvature, 126 ; asymptotic lines, 131 ; parallel sur faces, 185 ; geodesies, 20.5, 209, 224 Surface of translation, definition, 197, 198; equations, 197; asymptotic lines, 198; generators, 198, 203; deformation, 349, 350 ; associate surface, 381, 390; * Tim reference is to nondevelopable ruled surfaces. For developable ruled surfaces, see Developables. t For references such as Surface of Bianchi, see Bianchi. I Surfaces of center of certain surfaces are referred to under these surfaces. 474 INDEX congruence of tangents, 406 ; middle surface of a >F-congruence, 422, 424 Surface with plane lines of curvature. See Lines of curvature Surface with spherical lines of curvature. See Lines of curvature Surface with the same spherical repre sentation of its lines of curvature as a pseudospherical surface. See Pseudo- spherical surface Surface with the same spherical repre sentation of its lines of curvature as a spherical surface. See Spherical sur face Surfaces of revolution, a family of Lame" of, 451 Tangent plane to a surface, definition, 50, 114; equation, 57; developable sur face, 67; distance to, 114 ; meets the surface, 123 ; characteristic of, 126 ; is the osculating plane of asymptotic line, 128 Tangent surface of a curve, 41-44, 57; applicable to the plane, 101, 150 Tangent to a curve, 6, 7, 41), 50, 51) ; spherical indicatrix of, 9, 13, 50, 177 Tangent to a surface, 112 Tangential coordinates, 103, 104, 201 Tetrahedral surface, definition, 207 ; asymptotic lines, 207 ; deformation, 341 Tetrahedral surfaces, triple system of, 465 Tore, 124 Torsion, geodesic, 137-140, 174, 170 Torsion of a curve, definition, 10 ; radius of, 16, 17, 21; of a plane curve, 10 ; sign of, 19; constant, 60; of asymp totic line, 140 Tractrix, equations, 35 ; surface of revo lution of, 274, 290 ; helicoid whose meridian is a, 291 * Transformation, of curvilinear coordi nates, 53-55, 73, 74 ; of rectangular coordinates, 72 ; by reciprocal radii, 104, 196, 203 (see Inversion) ; project- ive (see Projective) Triply orthogonal system of surfaces, definition, 447; associated with a cyc lic system, 440 ; fundamental quan tities, 447-451 ; with one family of surfaces of revolution, 451, 452 ; of Kibaucour, 452, 403 ; of Bianchi, 452- 454, 404, 465 ; of Weingarten, 455, 456, 403, 404 ; transformation of, 462, 463 ; with one family of molding sur faces, 405 ; of cyclides of Dupin, 405 ; of isothermic surfaces, 405 Umbilical point of a surface, definition, 120 ; of quadrics, 230, 232, 234, 230- 238, 240, 207 Variation of a function, 82, 83 Voss, surface of, 341, 390, 415, 442, 443 W-congruence, 417-420, 422, 424 >F-surface, definition, 291 ; fundamental quantities, 291-293 ; particular, 291, 300, 318, 319; spherical representation, 292; lines of curvature, 293; evolute, 294, 295, 318, 319; of Weingarten, 298, 424; ruled, 299, 319 Weierstrass (equations of a minimal sur face), 200 ; (algebraic minimal sur faces), 201 Weingarten (tangential coordinates). 103 ; (geodesic ellipses and hyperbo las), 214; (>F-surfaces), 291, 292, 294 ; (infinitesimal deformation), 374, 387 ; (lines of curvature on an isothermic surface), 389 Weingarten, surface of, 298, 424 ; method of, 353-372 ; triple system of, 455, 450, 403, 464 * For references such as Transformation of BJickluud, see Bitcklund. 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