>c CORNELL UNIVERSITY, LIBRARIES iS/lathematics Library White Hail 3 1924 060 288 986 DATE DUE _jfLiU^ lOCt 5- I99R ^AP 1 9 Mrt^i wmK 1 ^ /«w CAY1.0RD PRINTEOIN U.S A. Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991. ^^ Cornell University Library The original of this bool< is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924060288986 University of QNaNNATi BULLETIN OF MATHEMATICS no LECTURES ON THE CALCULUS OF VARIATIONS (The Weierstrassian Theory) By HARRIS HANCOCK, Ph.D. (Berlin), Dr. Sc. (Paris), Professor of Mathematics 1904 Entered at the Post Oifice at Cincinnati, Ohio, as second-class matter. -Sir^N v. ■<^. 2: University of Cincinnati BULLETIN OF MATHEMATICS No. j LECTURES ON THE CALCULUS OF VARIATIONS (The Weierstrassian Theory) By HARRIS HANCOCK, Ph.D. (Berlin), Dr. Sc. (Paris), " Professor of Mathematics 1904 EiUi-Tuil at the Post Office at Cincinnati. (Jliii>. ;i'- second-das'.^ matter. Copyright, 1904. BY Harris Hancock. All Rights Reserved. PREFACE. Since the time of Newton and the BernouUis, problems have been solved by methods to which the general name of the Calculus of Variations has been applied. These methods were generalized and systematized by Euler, Lagrange, Legendre and their fol- lowers; but numerous difficulties arose. Some of these were re- moved by Jacobi and his contemporaries. Still many of the methods had to be extended and it was necessary to supply much that was deficient and to make clear what remained obscure. The progress of Analysis is indebted to the genius of Weierstrass for the perfection of this theory. While a student in the University of Berlin it was my privil- ege to hear the lectures of Professor H. A. Schwarz on the Cal- culus of Variations. In its presentation this eminent mathema- tician followed his great teacher, Weierstrass, who had established the theory on a firm foundation, free from objection, simple and at the same time more comprehensive than it had been hitherto. I also took the opportunity to study Weierstrass's lectures, of which there were copies in the Mathematischer Verein. Through the courtesy of Professor Ormond Stone abstracts of this theory were published in Volumes IX, X, XI and XII of the Annals of Mathematics^ and from time to time the Calculus of Variations has been included in my University lecture courses. I have delayed the publication of these lectures with the hope that Weierstrass's lectures would be published by the commission IV PREFACE. to which has been intrusted the editing of his complete works. This publication, however, seems remote, and commentaries on the Calculus of Variations are becoming so numerous that I have deemed it expedient to bring out my work at the present time. As one would naturally expect, I have followed Weierstrass's treatment of the subject; in many places, especially in the latter part of the book, my lectures are little more than a repetition of his. It is from the Weierstrassian standpoint that I have devel- oped mv own ideas and have presented those derived from other writers. Thus, instead of giving separate accounts of Legendre's and Jacobi's works introductory to the general treatment, I have produced their discoveries in the proper places in the text, and I believe that by this means confusion has been avoided which otherwise might be experienced by students who are reading the subject for the first time. I hope that this exposition of the fundamental principles may prove attractive. The reader will then naturally desire a more extensive knowledge regarding the literature and the various improvements that have been made by successive mathematicians. He will wish to follow the methods which they have employed^ and will seek further information re- garding the historical development. References are given on Pages 18 and 19 of the text from which the original sources are easily obtained. The necessary and suflBcient conditions as they arise for the existence of a maximum or a minimum are illustrated by six problems, which are worked out step by step in the theory. They have been chosen to represent the different phases of the subject, the exceptional cases which may occur, the discontinuous solu- tions, etc. For example, in the first problem it is found that a minimum may be offered by an irregular curve, whereas seemingly PREFACE. V the problem is satisfied by a regular curve, the catenary. Atten- tion is thereby called to the fact that although our integrals have a meaning only when taken over regular curves, we have to guard against discontinuous solutions, and consequently further condi- tions for the existence of a maximum or a minimum must be derived. The case of the discontinuous solution is considered in this problem as also when the limits of integration are two con- jugate points. Newton's problem is introduced to show that one of the necessary conditions is not satisfied and that there is no curve which fulfills the given requirements. By the formulation of such problems in Chapter I we come readily to the statement of the general problem of the Calculus of Variations. In the general discussion attention has been confined for the most part to the realm of two variables, and in this realm only the first derivatives of the variables have been admitted. Gener- alizations and extensions are suggested which, as a rule, may be executed with little difficulty. The second part of the work beginning with Chapter XIII treats of the theory of Relative Maxima and Minima, where the isoperimetrical problems are considered. Here also the existence of a field about the curve which is to maximize or minimize a given integral is emphasized and the necessary and sufficient con- ditions are derived and proved in a manner similar to that by which the analogous conditions are found in the first part of the work. My wish in these lectures has been to give a connected and simple treatment of what may be called the Weierstrassian Theory of the Calculus of Variations. Many instructive theo- VI PREFACE. rems of older writers have been omitted. I regret too that there has not been room to take up some of the investigations which have recently appeared. It is seldom that the first edition of a book is the final form in which an author wishes to leave his work. As I expect to make additions and alterations in my University lectures from time to time, I shall receive with pleas- ure any suggestions that ma}' be offered. In conclusion, I vrish to take the opportunity here of return- ing my sincere thanks to the Board of Directors of the University of Cincinnati for their liberality in the publication of this work. My thanks are also due to Mr. Harold P. Murray, Manager of the University Press, for his careful supervision of the printing. Harris Hancock. Auburn Hotei,, Ctncinnati, O. ApeilIS, 1904. CONTENTS. CHAPTER I. PHESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS. ARTS. PAGBS 1 The connection between the Calculus of Variations and the Theory of Max- ima and Minima. Problem J. The curve which generates a minimal surface-area when rotated about a given axis 1 2-4 The solution of this problem by the methods of Maxima and Minima. 1-4 5, 6 The difference between the Calculus of Variations and the Theory of Max- ima and Minima. ............ 5, 6 7 The coordinates x, y expressed as functions of a parameter t. Problem I formulated in terms of the parameter t. ..... . 7 8, 9 Problem II. The brachistochronc . ... ... 7-9 10 Problem III. The shortest line on a given surface. . . . . . . 9, 10 11 The advantage of formulating the problems in terms of the parameter /. 10 12 Problem IV. The surface of revolution of least resistance 11,12 13 The general problem stated. . . 12 14 The variation of the end-points. 12 15 Problem V. The isoperimetrical problem 13 16 Problem VI. The curve whose center of gravity lies lowest. ... 13 17 Statement of the general problem in Relative Maxima and Minima. . . 14 13 Generalizations that may be made 14 19 Variation of a curve. Analytical definition of maximum and minimum. Neighboring curves. .......... IS, 16 20 A different statement of the general problem. ... . 17 21 Inadmissibility of the presupposed existence of a maximum or a minimum. 17, 18 Problems. .... 18, 19 CHAPTER II. EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLICATIONS TO THE CATENARY. 22 Total variation in the case of Problem I, Chapter 1 20 23 A bundle of neighboring curves 21 24 The first variation. 22 ^. 25 The integral /= r/"[jr,>,^j rfar 23 x„ vii vili CONTENTS. AKTS. PAGES 26 The vanishing of the first variation. 24 27 Application to Problem I. . ... . 24, 25 28 The differential equation of this problem. ... ... 25 29 The integral /- \F(y,y')dx. .26 X 30 Solution of the differential equation of Art. 26. . 27 31 The notion of a region within which two neighboring curves do not inter- sect. ...... . . .... 28 32 The catenary. ..... 29 CHAPTER III. PROPERTIES OF THE CATENARY. 33 Preliminary remarks 34 The general equation of the catenary. ... . . 35 Geometrical construction of its tangent. . ..... 36 Geometrical construction of the catenary. .... 37 The catenary uniquely determined when a point on it and the direction of the tangent at this point are given 38 Limits within which the catenary must lie. 39-41 The number of catenaries that may be drawn through two fixed points. 42 The functions/,(>«) and/,(»») ... 43 The discussion of the function f^{m) 44 The discussion of the function /,(»«) 45 An approximate geometrical construction for the root of a transcendental equation. . . ... .... 46-48 Graphical representation of the functions /,(»i) and /,{»»). 49, SO The different cases that arise and the corresponding number of catenaries. 50-52 The position of the intersection of the tangents through the two fixed points for each case. ... 53, 54 The common tangents to two catenaries. . . . . 55 Catenaries having the same parameter which intersect in only one point. 56 Lfindelof s Theorem. .... 57 A second proof of the same theorem. . . ..... 58, 59 Discussion of the several cases for the possibility of a minimal surface of rotation. . . ........ 60, 61 Application to soap bubbles. CHAPTER IV. PROPERTIES OF THE FUNCTION F{^X, y, X , y). 62 The function F defined as a function of its arguments. .... 54 63-67 Necessary conditions and sufficient conditions . 55-57 68 The function F(x,y, x'yy) must be homogeneous of the first deg^ree in X- and >' . . 57, 58 30 30 31 32 33, 34 34, 35 35-37 37 38 39 40 40, 41 42, 43 43-45 46 47 47, 48 48-50 50, 51 52, 53 CONTENTS. IX ARTS. 69 Integrability of the function /''(^.j', ;r',y) X, 70 The integral | ^ jr, v, 1, — dx, when x and y are one-valued functions J I. dx I of each other. 71 Introduction of the variable t ox s .... 72 Analytical condition for the function F. . . 73 Introduction of the function F^ ... PAGBS 59 S9 59,60 60 60, 61 CHAPTER V. THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION. 74 General forms of the variations hitherto employed. ... 62 75 The functions i and i). Their continuity 62, 63 76 Neighboring curves. The first variation 64 77 The functions C, C, and G, ... 65 78 Proof of an important lemma. ... ... . 66-70 79 The vanishing of the first variation and the di£Ferential equation G=o. . 70, 71 80 The cwrz/a/io-f expressed in terms of y^ and .F,. 72 81 The components w^ and Wi in the directions of the normal and the tangent. 72, 73 82 Variations in the direction of the tangent and in the direction of the normal. 73, 74 83 Discontinuities in the path of integration. Irregular curves. . .74, 75 84, 85 Problem of Euler illustrating the preceding article 75, 76 86 Summary 76, 77 CHAPTER VI. THE FORM OF THE SOLUTION OF THE DIFFERENTIAL EQUATION G = 0. 87 Another form of the differential equation G=o 78-81 88 Another form of the function /", 81,82 89 Integration in power-series . . 82, 83 90 The solutions A'=<^(/, a, /3), 3'=^(/, <*> ^) of the differential equation C=o. 83, 84 91, 92 The case when F^=o at the initial-point of the curve 84, 85 93 The form of the differential equation C=o when i is the independent vari- able 85, 86 94 The solution of this equation 87 95 The curve can have no singular points within the interval in question. The coordinates of any point of the curve expressed as power-series in s. . 87, 88 CONTENTS. CHAPTER VII. REMOVAL OF CERTAIN LIMITATIONS THAT HAVE BEEN MADE. INTEGRATION OF THE DIFFERENTIAL EQUATION G=,0 FOR THE PROBLEMS OF CHAPTER I. ARTS. PACKS 96 Instead of a sing-le regular trace, the curve may consist of a finite number of such traces. . . . .89 97 The first derivatives of F with respect to x" and y' vary in a continuous manner for the curve G=o, even if there are sudden changes in the direction of this curve. . ... . 89-92 98 Explanation of the result given in the preceding article. . . 92 99 Summary. ..... . 92, 93 100 Solution of the differential equation G=o for Problem I, Chapter I. 93 101,102 The discontinuous solution. . 94,95 103 The equation G=o solved for Problem II, Article 9. . 95-99 104 The two fixed points must lie on the same loop of the cycloid. . . 99-101 105 Through two points may be drawn one and only one cycloidal-loop, which does not include a cusp. ... . 101, 102 106 Probletn III. Problem of the shortest line on a surface. 102-106 107 The same result derived in a difi'erent manner. . . . 106-108 108 Problem IV. Surface of rotation which offers the least resistance. . 108,109 109,110 Solution of the equation G=o for Problem IV of Chapter I. . 109-111 CHAPTER VIII. THE SECOND VARIATION; ITS SIGN DETERMINED BY THAT OF THE FITNCTION F\. 111 Nature and existence of the substitutions introduced. 112,113 112 The total variation. . . 114 113,114 The second variation of the function /^ ... 115-119 lis The second variation of the integral /. The sign of the second variation in the determination of maximum or minimum values. . . . 119,120 116 Discontinuities. ... . 120-122 117 The sign of the second variation made to depend upon that of F^. . 122-124 118 The admissibility of a transformation that has been made. The differen- tial equation J=o. . . ... . 124 125 119 A simple form, of the second variation. . . 125 120 A general property of a linear differential equation of the second order. 125, 126 121 The second variation and the function F^. The function F^ cannot change sign and must be different from o and cx> in order that there may be a maximum or a minimum. ... . , 126-128 CONTENTS. XI CHAPTER IX. CONJUGATE POINTS. AKTS. PAGES 122 The second variation of the differential equation J=o. . 129-131 123, 124 The solutions of the equations G=o and J-=o. The second variation de- rived from the first variation. ... .... 131-134 125 Variations of the constants in the solutions of G=o. . 134 126 The solutions «, and u, of the differential equation J^o. . . 135,136 127 These solutions are independent of each other. . . 136, 137 128 The function 0(/, ;;'). Conjuf;atc points. . 137,138 129 The relative position of conjugrate points on a curve 138-140 130 Graphical representation of the ratio — ' . . ... 140, 141 131 Summary. . ......... 141-143 132 Points of intersection of the curves G=o and SG=o. . . 143 133 The second variation when two conjugate points are the limits of integ-ra- tion, and when a pair of conjugate points are situated between these limits. .... ... .... 143-145 CHAPTER X. THE CRITERIA THAT HAVE BEEN DERIVED UNDER THE ASSUMPTION OF CERTAIN SPECIAL VARIATIONS ARE ALSO SUFFICIENT FOR THE ESTABLISHMENT OF THE FORMULA HITHERTO EMPLOYED. 134 The process employed is one of progressive exclusion 146 135 Summary of the three necessary conditions that have been derived. . 146,147 136,137 Special variations. The total variation. . . . . 147-149 138 A theorem in quadratic forms 150, 151 139 Establishment of the conditions that have been derived from the second variation. .... . 151-154 140-144 Application to the first four problems of Chapter I. .... 154-160 CHAPTER XI. THE NOTION OF A FIELD ABOUT THE CURVE WHICH OFFERS A MIN- IMUM OR A MAXIMUM VALUE OF THE INTEGRAL. THE GEO- METRICAL MEANING OF THE CONJUGATE POINTS. 145 Notion of a field 161,162 146 Neighboring curves which belong to the family of curves G=o. . 162 147 A general theorem in the reversion of series. . 163, 164 148 The coordinates of a neighboring curve expressed in power-series of k, where k is the trigonometric tangent between the initial directions of the neighboring curve and the original curve 165, 166 Xll CONTENTS. ARTS. PAGES 149 A curve which satisfies the equation G=o is determined as soon as its ini- tial point and the direction of the tangent at this point are known. . 166-168 150 Limits assigned to k. Extension of the notion of a field. . . . 168, 169 151 Intersection of two neighboring curves. Conjugate points. . . 169-171 152 A point cannot be its own conjugate. The derivative of 0(/> t') does not vanish at a point which causes the function itself to vanish. . 171-173 CHAPTER XII. A FOURTH AND FINAL CONDITION FOR THE EXISTENCE OF A MAXI- MUM OR A MINIMUM, AND A PROOF THAT THE CONDITIONS WHICH HAVE BEEN GIVEN ARE SUFFICIENT. 153 The notion of a field continued from the preceding Chapter. . . . 174,175 154 The function (S'C^.J',/*, ?,/>?)■ ... 175-177 155 The function (s must have the same sign for every point of the curve. 177, 178 156 The sufficiency of the above condition. . .... 178-180 157 Another form of the function ^. . 180-182 158 Still another form . 182-184 159,160 The signs of the function (5" and .F,. . .184,185 161 Another proof of the sufficiency of the condition as gfiven in Article 156. 186, 187 162 The function (5 cannot be zero along an entire curve in the given field. 187-189 163 The envelope of conjugate points. . . 189-191 164 The curve may be composed of a finite number of regular traces. 191, 192 165 Cases where the traces are not regular. ... . . 192-194 166 Generalizations in the Integral Calculus. ... ... 194, 195 167-172 Applications to the four problems already considered. . . . 195-203 173 When F{x, y, x',y') is a rational function of x' and y', there can exist neither a maximum nor a minimum value of the integral. . 203 174 General summary. . . 203,204 175 Extensions and generalizations: Instead of the determination of a struc- ture of the first kind in the domain of two variables, it may be required to determine a structure of the first kind in the domain of n quantities. 204-206 176 When equations of condition exist among the variables. .... 206, 207 177 When the second and higher derivatives appear. . . . 207,208 178 The Calculus of variations applied to the determination of structures of a higher kind. The minimal surfaces 208,209 CONTENTS. xill RELATIVE MAXIMA AND MINIMA. CHAPTER XIII. STATEMENT OF THE PROBLEM. DERIVATION OF THE NECESSARY CONDITIONS. ARTS. PAGBS 179 The g-eneral problem stated. ... 210,211 180 Existence of substitutions by which one integral remains unchang-ed while the other is caused to vary. An exceptional case. ... 211-213 181 Case of two variables. Converg'ence of the series that appear. . 213-215 182 The nature of the substitutions that have been introduced. . 215 183 Formation of certain quotients which depend only upon the nature of the curve . . 215,216 184 Generalization, in which several integrals are to retain fixed values. . 217-219 185 The quotient of two definite integrals being denoted by X, it is shown that X has the same constant value for the whole curve. . . . 219 186 The differential equation C(o)_X C(l) = o. ... 220 187 Extension of the theorem of Article 97. .... 220-222 188 Discontinuities, etc. . . 222 189 The second variation: the three conditions formulated in Article 135 are also necessary here. ... 222,223 CHAPTER XIV. THE ISOPERIMETRICAL PROBLEM. 190 Statement of the problem. ... .... . 224,225 191 A simpler form of the integral that appears. . .... 225, 226 192 The function F, for this problem 226 193 Integration of the differential equation that occurs 226,227 194 An immediate consequence is the theorem of Steiner: Those portions of curve that are free to vary, are the arcs of equal circles 228,229 195 If there exists a curve, which with a given periphery incloses the greatest surface-area, that curve is a circle 229, 230 196 The admissibility that this property belongs to the circle. 230 CHAPTER XV. RESTRICTED VARIATIONS. THE THEOREMS OF STEINER. 197 Variations along two different portions of curve. . ... 231 198 Variation where a point must remain upon a fixed curve 231, 232 199 Application to a particular case 232,233 XIV CONTENTS. ARTS. PAGES 200 Variation where a part of the curve coincides with a fixed curve. . 233,234 201 Generalizations involving' several variables and several integ^rals. • 234-236 202 The isoperimetrical problem when the circle (Art. 195) which incloses the given area cannot be inscribed within fixed boundaries. . 236 203 Statement of two problems due to Steiner. Criticism of his assertion that the Calculus of Variations was not sufficient for the proof of these problems. . • 236,237 204 Two problems due to Weierstrass which are more general than Steiner's and their proof by means of the Calculus of Variations. . 237 205 The behavior of the ©-function at fixed boundaries. . . 237-239 206 Further discussion regarding: this function. . . . • 239 207 The case that there is a sudden change in the direction of the boundary curve at the point where it is approached by the curve that is free to vary. ... ... 240,241 208 The case where the curve meets the boundary at a point and then leaves it. 241,242 209 The tangents to the two portions of curve make equal angles witli the tan- gent to the fixed curve . . • 242,243 210 The isoperimetrical problem reversed. . ... . 243,244 211 Consideration of the problem: Three points not lying in the same straight line are given in the plane. It is required to draw a line through them in a definite order which with a given length includes the greatest possi- ble surface-area. .... . . . . . 244, 245 212 Expression for portions of the curve that overlap. . . . 245-247 213 The solution of the differential equation may be straight lines or arcs of circles • . 247,248 214 The problem reduced to a problem in the Theory of Maxima and Minima. 248,249 215 The problem solved. . 249, 250 CHAPTER XVI. THE DETERMINATION OF THE CURVE OF GIVEN LENGTH AND GIVEN END-POINTS, WHOSE CENTER OF GRAVITY LIES LOWEST. 216 Statement of the problem 251,252 217 The necessary conditions 252 218 The number of catenaries having a prescribed length that may be drawn through two given points with respect to a fixed directrix. . . . 253,254 219 The constants uniquely determined. . ... 254,255 CHAPTER XVII. THE SUFFICIENT CONDITIONS. 220 The problem solved without using the second variation 256 221 The ©-function. .... . 256,257 CONTENTS. XV ARTS. PAGES 222 Consequences due to this function. ........ 258 223 A field about the curve which maximizes or minimizes the integral. . 258, 259 234 Further discussion of the nature of the enveloped space. . . 259 225 Properties of an enveloped space which lies within the first enveloped space. ... .... . ... 259,260 226 The sufficiency of the ,2S7 The ^-innciion ioT t\ie. curve whose center of gravity lies lowest. . . 291,292 CHAPTER I. PRESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS. 1. At the time when the DiflFerential Calculus, and in part also the Integral Calculus, were being formulated, certain prob- lems were proposed, which, although not belonging to the province of the Theory of Maxima and Minima, had a marked semblance to the problems of that theor)', and were often solvable by methods belonging to it. The following was one of the first problems pro- posed: Problem I. Two points P^ and P-^ with coordinates {^x^^, y^) and {xi,yi) respectively are given. Both points lie on the same side of the axis of X in the plane-xy. It is required to join Pt^and Pi by a curve which lies in the upper half of the xy-plane {axis of X inclusive) such thatwhen the plane is turned through one complete revolution about the axis of X, the zone generated by this curve may have the smallest possible surface-area. We may use this problem to illustrate the connection be- tween the Calculus of Variations and the Theory of Maxima and Minima; at the same time the difference between the two theories is evident. 2. If we try to solve the problem of the preceding article by the methods of the Theory of Maxima and Minima, we must pro- ceed as follows: Suppose that it is possible to draw a curve between P^ and P^ which satisfies the problem. Then every portion of this curve, however small, must have the property of generating a surface of smallest area. For, suppose a change is made in an arbitrary por- tion of the curve, however small, and let the remaining portion of the curve be unchanged. If by this change the surface-area gen- erated by this arbitrary portion of curve is less than it was before, CALCULUS OF VARIATIONS. then the curve containing the deformed portion of curve generates a smaller surface-area than the original curve. Also, if to one value of X there belong several values of y, then^ instead of the portion of curve belonging to the same abscissa, we might take the straight line vt^hich joins these two points. This line would generate a surface of smaller area than that generated by the curve that passes through the same two points. Hence the curve would gener- ate a surface which did not have a min- imum area. We may therefore consider the curve as divided into portions such that the projections of these portions on the axis of J^sTare all equal. The above hypotheses being granted, we suppose that the a. two points P' (x', y ) and P" (x", y") are taken on the curve, and we find another point P {x^ y) on the curve such that x—x'=x"—x—l^x. We suppose that P and P', P and P" are joined together by straight lines, and _^ later we suppose that these straight lines are taken so close together that there is a transition from the straight lines to the curve. The re- maining portions of curve on the left-hand side of P' and on the right-hand side of P" are supposed to remain unaltered. The portions of surface-area generated by the straight lines /''/^and PP" are O My'-^y) V ( [SxY^{y-yJ SindTiy+y") v' {tsx^ +(y"-yy. In order to have a minimum the sum of these two expressions when differentiated with regard to y must be zero ; i. e., n V ( b^xy- + iy-y' )^ -f 771/ ( tsxj + ^y'-yY -njy' ^y) (y-y') __ ^ {y +y") ( /'- y) _^ r^-j ^■{l^xy+{y-y'y v' (l^xy^{y"-yy The quantity r may be determined from this equation as a func- CALCULUS OF VARIATIONS. 3 tion of X, so that y^/{x\ say. We therefore have jv'=/ {x— l^x) and y" = /{x-{- l^x). Hence by Taylor's Theorem, y =/(x- l^x)=/{x)-/'{x) isx + K/" i.x) ( ISxY- , y" = /{x+ iSx) =/{x)+/'{x) rj^x + y^f" {x) ( l^xf^ ; and consequently, V ^y'^f\x) l^X-%/" {X) {l\xy+ , y'-y=/\x) t^x^y^f" (x) (/^xy+ Substituting these values in [^], we have, neglecting the factor ir. /sxr l+fixf-/' (x)/" (x) /^x+ + /^x^ 1+/' {xy+/'{x)/" (x)lSx +.... + [2/(^)-r {^)^^ -:- ■■■■K/' (x)isx-y/" (x)/^x^... .] l^xV \J^f'{xf-f' {x)f" (x) A;c + . . . . ^ [2/(x)+/'(x) /\x + .. ..][/-(;>;) A;.+ y/"(x)i^x+ . . . j ^ ^ A X V l + f'{xy^/' (x)/" (x) ^x+.... Expand this expression in ascending powers of A a;, divide through by A ;t; and then make iSx =o. We then have or l^rf (xy-f{x)f"ix) = o; Therefore in order to have a minimum value,/ (;»;) or jy must sat- isfy this differential equation; however, when jv satisfies this differ- ential eqation we do not always have a minimum, as will be shown later. In other words, the differential equation {B'] is a necessary consequence of the supposed existence of a minimal surface of rev- olution. As a condition, however, it is not sufficient to assure the existence of a curve giving such a surface. Differentiate the equation [B] with regard to x, and we have CALCULUS OF VARIATIONS. dy d}y ^ (Py dx ' dx^ dx^ or dx dx ix \dxy Integrating, we have d^y d^ id^y ^ dx" c where c^ is the constant of integration. Since y—e ^ and y^e are two solutions of this last differential equation, the general solution is .1- X y=Ci e'^ -^ Cjg ^ , where c, and c-^ are constants. This last equation is that of the catenary curve. 4. Thus, by the help of the Theory of Maxima and Minima, we have, it is true, come to a certain result; but, on the other hand, we have yet to ask whether this curve gives a true minimum; and owing to the manner in which we have arrived at these con- clusions, we have yet to see whether this curve only in a definite portion or throughout its whole extent possesses the property re- quired in the problem. That we are justified in insisting upon this last statement is seen from what follows later, where it will be shown that the curve found above satisfies the required conditions only between given limits. A simple consideration shows that the method we have fol- lowed above is not at all rigorous; since it presupposes, which of itself is not admissible, that the curve which satisfies the prob- lems is regular in its whole extent, for otherwise the portions of curve between the two points {^x— l^x,y') and {x,y) could not be replaced by straight lines joining these two points; also, the ex- pansion by Taylor's Theorem would not have been admissible. 5. The characteristic difference between problems relative to Maxima and Minima and the problems which have to do with CALCULUS OF VARIATIONS. 5 the Calculus of Variations consists in the fact that, in the first case, we have to deal with only a finite number of discrete points, while in the Calculus of Variations, the question is concerning- a continuous series of points. If we wish to substitute in the place of the curve first a po- lygonal line and afterwards apply to this line methods similar to those used above, then it turns out that, after we have found a line which satisfies all the conditions, it is necessary yet to prove that the required limiting transition from polygonal line to curve in reality results in a definite curve which satisfies the conditions of the problem. 6. Bvery limiting transition, as from polygon to curve, is made of itself, if we make use of the conception of integration, since an integral represents the limiting value of a sum of quanti- ties which, following a definite law, increase so as to become infi- nite in number, the quantities themselves becoming smaller in a corresponding manner. If we therefore define the surface-area of the cuxv^ y—f {x), which we have to find, by 5":^= 2 TtX y d s, or Xi 3T-JW -(£)"-' then this integral will have a definite value for every curve that is drawn between /'o and P^, and consequently the problem may be stated as follows: Problem I. y is to be so determined as a function of x that the above integral shall have the sm^allest possible value. The solution of this problem will be given later. The two methods given above have been chosen to make clear what there is in common in the Theory of Maxima and Minima and the Cal- culus of Variations, and also to show the difference between them. In the Differential Calculus a definite function is given, and a special value of the variable or variables (if there are more than one variable) is sought, for which the function takes the greatest CALCULUS OF VARIATIONS. or least possible value; in the Calculus of Variation a function is sought and an expression is given which depends upon this function in a certain known manner. A definite integral is con- sidered, in which the integrand depends upon the unknown func- tion in a known manner, and it is asked what form must the un- known function have in order that the definite integral may have a maximum or a minimum value. We treat only real values of the variables. 7. If A < ^3 and the point P2 corresponds to U and P^, to t^, then Pj with reference to P-i is known as a later point; and P-^ with reference to /j is known as an earlier point. As was shown in Art. 2, the ordinate y of the required curve O ~^ is a one-valued function of the abscissa x. It often happens that one cannot know a priori that one of the ordinates is a one-valued function of the other. Poincare* has shown that it is always possible to express the two variables x, y, when there is an analytic relation between them, as one-valued functions of a third variable t. The only property that is required of this variable is, when it traverses all valiies between two given limits, the corresponding point {^x,y^ traverses the curve from the initial point to the end point, and in such a way that for a greater value of / there belongs a later point of the curve. For example, suppose that z^xy where x and y are two in- dependent variables. Then in virtue of this equation there is no way of expressing the dependence of one of these variables upon the other without the introduction of transcendental functions. But if we write X- then or .y' z=e-' _log -g y t *Poiiicar6 (Bulletin de la Soci^t^ Math^matique de France, T XI. 1883.) See also my lectures on the Theory of Maxima and Minima, etc. Page 13. CALCULUS OP VARIATIONS. 7 Thus X and y are one- valued functions of the variable t. If then we introduce such a new variable t in the integral of Problem I, that integral becomes A 4 where we denote bv x' and y' the quantities -^ and -^ . at at We may now state Problem I as follows: The quantities x and y are to be determined as one-valued functions of a parameter t in such a way that the above inte- gral will have the smallest possible value. 8. That we may learn the essential properties of the Calcu- lus of Variations, we shall next formulate other simple problems; then, while we seek the general characteristics of these problems, we shall of our own accord come to a more exact statement of the problems which the Calculus of Variations has to solve. As a second problem may be given the very celebrated prob- lem of the Calculus of Variations, that of the brae histochr one* (curve of quickest descent), which may be stated as follows: Problem II. Two points A and B are situated in a ver- tical plane, the point B being situated lower than the point A ; a curve is to be drawn between these points in such a manner that a material point subject to the action of gravity and com- pelled to move upon this curve with a given initial velocity, shall go from the point A to the point B in the shortest possi- ble time. L/et the mass of the material point be 1, its initial velocity a^ the acceleration of gravity 2g, the time t, and the coordinates of *Woodhouse (A Treatise on Ipsoperimetrical Problems and the Calculus of Varia- tions, 1810) writes (p. 1): "The ordinary questions of maxima and minima were amongst the first that eng'aged the attention of mathematicians at the time of the invention of the Differential Calculus (1684), three years before the publication of the Principia. The first problem relative to a species of maxima and minima distinct from the ordinar3' was pro- posed by Newton in the Principia; it was that of the solid of least resistance. But the subject became not matter of discussion and controversy till John Bernoulli (Acta Erudit., 1696, p. 269) required the curve of quickest descent.'''' CALCULUS OF VARIATIONS. A and B respectively {_o,o) and {a, b). Let the direction of the positive F-axis be the direction of a falling body (due to gravity) and let the positive ^-axis be directed toward the side on which the point B lies. Then, according to the law of the Conservation of Energy, m-m-^^y--- or, whence b J 1/4 (dx\ T^ I ^ ^^y dy. y'^gy + a^ We have then as our problem: so determine x as a function of y that the above integral shall have the smallest possible value. As regards the signs of the roots that appear in the above integral, it is evident that these signs must be the same at the beginning of the motion and may be taken positive. For on me- chanical grounds it follows that the curve must at first descend; consequently at the beginning of the motion y increases with in- creasing t, and is therefore positive. Since 4^ + a^is always a — I + (t~)» ^'^^ *^^" never vanish, we may alwa^'s give to V \gy-^a!' the positive sign. Also at the beginning of the motion the quantity A\ + ( t~ ) must have the positive sign, since dt always represents a positive increment of time. However, in the further course of the motion, it may hap- pen that dy= o. Then the quantity .Jl + ( ;^ ) passes through infinity, so that dy and ^n. + { ;^ ) may simultaneously change their sign, while r'4^y-i-a^ continues with the positive sign. CALCULUS OF VARIATIONS. V 9. The assumption made in the statement of the problem that B must lie below A is not essential. For the material point has at ^ a certain velocity )8, which we may calculate from the initial velocity a and the height of A above B. When the point reaches B with this velocity it may rise again, and it will have the original velocity when it has reached the height A on the other side of B. The time which is necessary for the ascent is the same as that required in the descent, if we assume that the curve along which the ascent takes place is symmetrical with that of the descent. If, therefore, the point started from B, we could calculate from )8, which is now the initial velocity, the velocity a at the point A. We then have the curve in question, if we seek the curve along which the point with the initial velocity ^8 reaches A in the shortest time. In the case of the present problem we see from physical con- siderations that jv is a one-valued function of x. As this is not possible in all cases, it is expedient to represent the curve here also by two equations; that is, to consider x z.-aA y as one- valued func- tions of a third variable t* where t is subject to the only condi- tion, that when it goes through all values between two given lim- its, the corresponding point x, y traverses the curve from the beginning-point to the end-point and in such a way that to a greater value of / there corresponds a later point of the curve. The above integral becomes A J VAgy+a} to where we have written x' and y' for —— and -^ respectively. at at Our problem then is: Determine x and y as functions of a parameter t in such a way that the integral just written m^ay have the smallest possible value. 10. Problem III. Between two points on a regular sur- face fix, y, z)=o, a curve is to be drawn so that its length is a m,inimum. * This t is, of course, different from the time / of the preceding' article. 10 CALCULUS OF VARIATIONS. Consider the orthogonal coordinates x, y, 2 oi a. surface rep- resented as one-valued regular functions* of two parameters u and V. If we consider these as the rectangular coordinates of a point on the plane, then to every point of the surface there will corre- spond a definite point of the uv-plant, and these points in their collectivity fill out a definite portion of the plane, which may be looked upon as the image of the surface on the plane. To every curve on the surface corresponds a curve in this part of the uv- plane and reciprocally. Further, consider u and v as one-valued functions of a quan- tity t ; hence, to every value t there corresponds a point of the u v- plane, and therefore, also, in case this point lies in the definite por- tion of the u zz-plane, there is a corresponding definite point of the surface. Consequently if 4 and t^ are values of t which correspond to the two fixed points on the surface, then the length of any curve which lies between these two points is determined through L 4 '■I where H l^W j?z[V i^i' du ) I. du S \ du S Q _ "dx dx dy dy dz dz du dv du dv du dv' -- m- m- 111 2 We have then to determine u and v as functions of t, so that L is a minimtun. 11. In the case of the above problem it is necessary to apply the representation there given, whereas in Problem I and Prob- lem II the expression of x and y as one-valued functions of t may be regarded as expedient. In Problem III the variables u and v * See my lectures on the Theory of Maxima and Minima, etc. Pag-e 31. CALCULUS OF VARIATIONS. 11 ■must he regarded as functions of a third variable. We cannot re- gard V as a function of u, for we know nothing about the trace of the curve. If we wished to regard v only as a double-valued fun- ction of u, we would even then encounter many difficulties. Hence the requirements must be made that u and v be so determined as one-valued functions of t, that the integral in the preceding arti- cle be a minimum. 12- Problem IV. Find the form of the surface of rotation, which, having an axis lying in a fixed direction, offers the least resistance in moving through a liquid in the direction of the axis, it being supposed that the resistance of an element of sur- face is proportional to the square of the com^ponent of velocity in the direction of its normal. This problem is due to Newton.* It is assumed that the friction between the body and the fluid and that within the fluid itself may be neglected. Let the K-axis be the axis of rotation, ds an element of the generating curve, B the angle between the normal and the F-axis, so that -— — =cos Q. ds A zone of the surface is therefore given by 2'nxds^2TTX Vx''^ + y'^ dt. The component of velocity in the normal direction is z'cos^, and the resistance in the normal direction which the zone offers^ is iP- cos^ Q 2ttx\/x''^ + y^ dt. This quantity multiplied by cos 6 gives the resistance in the direction of the F-axis. We consequently have the required re- sistance of the body expressed by the integral R C ^^ = -. r dt- 2'rrv' J x'^'+y'^ * Newton, Principia, Book II, prop. 34. Thus Newton was the first to consider a prob- lem in the Calculus of Variations, and his problem involved a discontinuous solution. Solutions of it have been given by Euler and almost all other writers on the Calculus of Variations. We shaU see that one of the principal conditions for a minimum (the condi- tion of Weierstrass) is not satisfied, and that there can never be a maximum or a minimum. 12 CALCULUS OF VARIATIONS. Our problem then is to connect two points /^ and /^ by a curve so that the zone which it generates about the F-axis offers the least resistance. Neglecting the constant factor 2 tt z^, we have to determine x andy as one-valued functions of t so that the integral A ji = I dt to shall be a minimum. 13. That which is common to the four problems stated above consists in the determination of x and y as one-valued functions of a quantity t in such a way that an integral dependent upon them of the form A I=JF{,x,y,x\y')dt 4 will have the smallest possible value. Here 4 and A have fixed values so that the corresponding coordinates x, y of the initial and the final point of the curve are supposed to be known. F{x,y, x' y') represents a one-valued regular function of the four arguments x, y, x',y' of which x' andy' (since they represent the direction of the tangent to the curve) are to be regarded as unrestricted, while the region of the point x, y may be either the whole plane or only a continuous portion of it. 14. The condition that t^, A should have fixed values is not essential ; moreover both end-points may move, as in the case of the third problem, if we give it the following form : Tzuo curves are given on a surface; among all the possible curves between the points of the one curve and the points of the other, that curve is to be found which has the shortest length. We are accustomed to call this the geodesic distance of two curves. In order to solve this problem, we must first solve the special Problem III, since, if a curve has the property of being of minimum length such as is required above, it must also retain the same property, if we consider the end-points fixed. Hence from III the nature of the curve must be determined. The variation of the end- CALCULUS OF VARIATIONS. 13 points gives in addition certain special properties which the curve must possess. For example, the shortest distance betw^een two curves which lie in the same plane is clearly a straight line; through the varia- tion of the end-points it follows that this straight line must be perpendicular to both curves at the same time. 15. Essentially different from the four problems already given is the following: Problem V. It is required to draw a closed curve which with a given periphery inscribes the greatest possible area. Let X and y be one-valued functions of /, say x{^t^ and jv(/), such that for two definite values 4 and t^ of t the corresponding points X, y of the curve coincide, and that, if t goes from a smal- ler value 4 to a greater value t^, the point x, y completely tra- verses the curve in the positive direction. Then twice the area of the surface included by the curve is expressed by the integral A /<»'=J(;l;y-;^';r') dt, 4 and the periphery of the curve is given by the integral A T'^^=^ Vx!^j^y^dt. 4 Our problem then is: So determine x and y as one-valued functions of t that /"" shall have the greatest possible value, while at the same time /'*' has a given value. 16. Problem VI. What form is taken by an indefinitely thin, absolutely flexible, but inexpansible thread which is fixed at both ends, if the action of gravity alone acts upon it? This problem offers the characteristics of a minimum, for with stable equilibrium the center of gravity must be as low as possible. If the F-axis is taken vertical with the direction up- ward, and if 6" denotes the length of the curve, and f, 17 the coor- 14 CALCULUS OF VARIATIONS. dinates of the center of gravity, then 17 is determined from the equation k where t-^ S =^ Vl^y^ dt. 4 The problem may be stated thus: the variables x and y are to be determined as one-valued functions of a quantity t in such a way that the first of the above integrals has a minimum value, while the second retains a given fixed value. 17. Problems V and VI are usually classified under the name, Relative Maxima and Minima, a. term which requires no further explanation. In general they are included in the following prob- lem : Let F '*" i^x, y, x', y') and F ''' {x, y, x', y') be two functions of the same character as the function F {x, y, x', y') of Art. ij. It is required to determine x and y as one valued functions of a quantity t in such a way that the integral k 1^''''=^ F'''>'>{x,y,x',y')dt 4 has a m,axim.um or a m-inimum value, while at the sam-e time the integral k r^^^j F^^\x,y,x',y') dt to conserves a given value. 18. We shall give in the sequel what we believe to be a rig- orous treatment of the problems already formulated. The reader may propose for himself natural extensions of what is given; for example, instead of taking two variables, consider an integral hav- ing as integrand a function of n variables. Further, subject these variables to subsidiary conditions and also allow the second and CALCULUS OF VARIATIONS. 15 higher derivatives of the variables with respect to a quantity t to enter the discussion. Then double integrals which lead to the study of Minimal Surfaces may be treated by methods of variation (see Arts. 175 et seq.). 19. We may define the object of the Calculus of Variations in a still more general manner by the introduction of a fundamental conception, that of the variation of a curve. In former times the Calculus of Variations was considered one of the most difficult branches of analysis. It was wrongly thought that the difficulty was in the supposed lack of clearness in the fundamental concep- tions, especially in that of the variation of a curve. The difficul- ties that arise are mostly in other directions. In the Theory of Maxima and Minima we say that for a definite system of values of the variables the value of a function is a maximum or a minimum, if this value of the function for this system of values is greater or smaller than it is for all the neigh- boring systems of values. We say* of a function f{x) of one variable, it has, at a def- inite position x=^a, a maxitnum or a minifnum value, if this value for x=a is respectively greater or less than it is for all other values of x which are situated in the neighborhood of I x—a I < 8 a5 near as we wish to a. The analytical condition that /"(;»:) shall have for the position x=^a a maximutn, is expressed by /"(;»;) — /"(a Xo/) >for I x—a I ■'« where <^ is a given function of the variables x, y, z, which are connected by the equation y"(ar, y, z)=o,f being a known function. 11. Find the curve of minimum length between two fixed points in space, the ra- dius of curvature being a constant. 12. Find the form which a homogeneous body of given volume must take that its attraction upon a material point in a definite direction be as great as possible. 20 CALCULUS OF VARIATIONS. CHAPTER II. EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLI- CATIONS TO THE CATENARY. 22. Let us consider again the integral of Art. 6, JW'^-W^- ri] Suppose that there is a minimum surface-area that is generated by the rotation of a curve between the two fixed points Pg and Pi and let this curve be r^^/( a;). Let tj be the distance between this curve and any neighboring curve measured on the jy-ordinate, and sup- pose that 17 is a continuous function of X subject to the conditions: that for x=Xo, r) = o ; for x=Xi, -q—o ; and X for all other points | 17 | ■< p , where p may be as small as we choose. -^0 x^ The integral of any neighboring curve corresponding to [1] is 7 ^^ ?, i ". f^ *^v^ P y. f Ji, J j;^.„Vi-(^^±^y^- [2] Xt^ CALCULUS OF VARIATIONS. 21 Hence the total variation caused in [1] when, instead oiy=z/{x), we take a neighboring curve, is X^ X(, i^S has always a positive sign, since the surface in question is a minimum. 23. Instead of the one neighboring curve, we may consider a whole bundle of such curves, if for 17 we substitute e 17, where e is independent of x and has any value between — 1 and + 1. The ex- pression [3] becomes then Xq Xq and, developing A6" by Taylor's Theorem, ^S=.SS^^^'S+^^^S...... [5] There is no constant term in this last development, since when c is made zero in [4] the first and second integrals cancel each other. 85" is known as the Jirst variation , h^ S is called the second variation, etc. Instead of taking 17 a very small quantity, we may take e so small that e 77 is as small as we choose. With Lagrange (Misc. Taur., tom. II, p. 174), writing 17 = Sjy, it is seen that the total change in r is e tj =e 8 j)/ — Aj'. Remark. The sign of differentiation and the sign of varia- tion may be interchanged; for example, the 1st derivative of a variation is equal to the 1st variation of a derivative, as is seen by writing y)=ly, then ^==(8^)'= ^ (8r). [^1 22 CALCULUS OF VARIATIONS. Again rj — Sy; change y into jy + eij, and consequently y' into y + e Tj'. Hence -rj' is the first variation of y', so that V=8y = s(|): [«] and therefore from [i] and [n] It follows too that owing to the presupposed existence of 17', we must also assume the existence of the second differential coefficient of r. 24. Returning to [41, write y=-^ , rj' —-^ . Then expand - ax ax ing the expression under the sign of integration ( r-^erj)]' 1 J-(y-UeTj')2 —y \ 1 + y^ we have S . T , / / , ry I (y + ^^)(y + £VK ( , 2. , ) 1 i^(y + cV)Mc=o Hence, equating the coefficients of the 1st power of 6 in [4] and in [5] we have as- 27r which is a homogeneous function of the first degree in 17 and t] . The quantity t] cannot be indefinitely large, since then the devel- opment would not be necessarily convergent; but see Art. 116. In a similar manner we may find a definite integral for the second variation, in which the integrand is an integral homoge- neous function of the second degree in tj and 17'; similarly for the third variation, etc. CALCULUS OF VARIATIONS. 23 25. As a form of the integrals which were given in Problems I, II, III and IV of the preceding Chapter, consider the integral /=- j F(x,y,y) dx, where Fi^x, y, y ) is a known function of x, y and y\ and where the limits of this integral, x-^ and x^, are fixed. Hence, as above, X\ Xi A/= I F{x,y + €T), y ^er)') dx — \F{x,y,y') dx Xq Xq = I [F(x,y — €7),y'-^eri')—F{x,yjy')] dx. Xq This expression, when expanded by Taylor's Theorem, is X, We also have, as in Art. 23, 1.2 and by comparing the coefficients of e in these two expressions, it follows that X, «^=J(f " + §7'^')"- In the particular case given in Art. 22, F — yvl+y'^. Hence 5 — ^1 1 -L- y ^ and 5—7 — „ ' dy ■" dy 1 1 + /^ 24 CALCULUS OF VARIATIONS. and when these relations are substituted in (A) we have, as in Art. 24, ■^1 A 26. From the relation it is seen that when c is taken very small, ^ is as near as we wish to zero; and consequently when c is positive and indefinitely small, A/ is positive. On the other hand, when e is indefinitely small and negative. A/ is negative. Hence the total variation A/ of the integral will be either positive or negative according as e is positive or negative, so long as S/is different from zero; and consequently there can be neither a maximum nor a minimum value of the integral. We know, however, if / is a maximum A/ is always negative, and if / is a minimum A/ is always positive; and consequently in order to have a maximum or a minimum value of the integral, S/ must be zero. 27. Applying the above result to the example given in Art. 22 we have o=.[\ V\Zy-^^-l£^d^\dx. [6] J I -^ ' I'l+y^ dx) ^ -• Integrating by parts, Xq Xf. CALCULUS OF VARIATIONS. 25 and since, by hypothesis (see Art. 22), •»?=o at both of the fixed points Po and /'j, we have Hence [6] may be written X. [7] 28. We assert that in the expression above must always be zero between the limits x^ and x^. For, assuming that the contrary is the case; then, since t) is arbitrary, we may, with Heine,* write ,=(.-..)u-.))v-iTy'-X(7W")}' where ■*? becomes zero for the valued x=Xo and x^x^. Substitu- ting this value of i? in [7] , we have ^1 an expression which is positive within the whole interval x^,. . . .x^. The integrand in [8] , looked upon as a sum of infinitely small elements, has all its elements of the same sign and positive; so that the only possible way for the right-hand member of [8] to be zero is that d ^'l + y' dx •Heine, CreUe's Journal, bd. 54, p. 338. 26 CALCULUS OF VARIATIONS. We therefore have a diflFerential equation of the second order for the determination of the unknown quantity y. 29. This difEerential equation is a special case of the more general differential equation, which may be derived from the in- tegral 1=^^ F{^y,y')dx; Xo whence, as before (Arts. 25 and 27), X(, Xq As in Art. 27, we have dF dxldy'j ' or dy dx\Jd y e^=^r!4]. [9] But or dF(y,y)=^^dy + ^^dy, [10] Hence from [9] , or dF(y,y')-d\y'^^,\=o, and integrating, Fiy,y')-y'^^,^C, [11] where C is the constant of integration. CALCULUS OF VARIATIONS. 27 The relation [11] exists only when the integrand of the given integral does not contain explicitly the variable x; otherwise the relation [10] would not be true, and then we could not deduce [11]. 30. Applying this relation [11] to the special case above (Art. 28) where F{y,y'-)^yV\^y'\ we have y 1 1 + r'^ — — ^-^ = m , m being the constant of integration, a quantity which will be con- sidered more in detail later. The above expression may be written or y = m \ 1+y^. [I] From [I] it follows directly that fLm^=nt\^^: [II] and [II] , differentiated with respect to x, is Two solutions of this diflFerential equation are y^e^'"^ and y^e~'^''^ , so that the general solution is y=.c^ e^/" + Cj e -^/'" . [Ill] It appears that we have in this expression three arbitrary con- stants, m, Ci, and c^ ; but from [II] we have, after substituting for y and( — ) their values from [III], m^=4 Cj Cj. 28 CALCULUS OF VARIATIONS. Hence, writing in [III], Ci= Yitn e"^"''"^ and c^^ Yitn e*"''"", where x^ is a constant, we have y^ I^wCe'""^"''^"" -- g-l"-^-''/™] . [Ill'] The two constants Xq and m are determined from the two conditions that the curve is to pass through the two fixed points Po and P^. 31. From what was given in Art. 19 it would appear that two neighboring curves are distinct throughout at least certain portions of their extent. This implies the existence of a certain neighborhood about the curve C that is supposed to offer a mini- mum, within which this curve is not intersected by a neighboring curve. Suppose that the curve C^ is derived from the curve C by the substitution oi y^f.-r} for y (cf. Art. 22). Consider the family of curves (C^) obtained by varying e between —1 and +1. For sufficiently small values of € the curve C^ will lie within the neigh- borhood presupposed to exist, and a portion of our family of curves will lie within this neighborhood. This is a necessary conse- quence of the supposed existence of a minimal surface of revolu- tion. As a condition, however, it is not sufficient to assure the existence of a curve giving such a surface. The fact that the sur- faces generated by the curves C^ are all greater than that gener- ated by the curve C does not prevent the existence of a neighbor- ing curve constructed after a manner other than that by which the curves C^ are produced, which would generate a surface of revolution having less surface-area than that due to the revolution of C. It is useful to determine for just what curve C the above con- dition may be satisfied, and while this does not prove that the curve C gives a minimal surface of revolution, it will at least limit the range of curves among which we may hope to find a generator of a minimal surface. Further investigation of this more limited range of curves may locate the curve or curves giving a minimal surface^ if such exists, and in the other case may prove their non- existence. In the further investigation we shall derive the suf- ficient conditions to assure the existence of a maximum or a minimum. CALCULUS OP VARIATIONS. 29 32. The conclusions drawn from Art. 30 show that, if a curve exists which offers the required minimal surface, that curve must be a catenary. Since the catenary must pass through the two fixed points /o and P^ , we may determine the constants fn and Xq from the two relations (see formula [III'], Art. 30): We shall see in the next Chapter that three cases arise according as the solution of the above equations furnish us with two cate- naries, one catenary, or no catenary. In the first place, it may be shown that the catenary nearest the ^-axis can never furnish a minimal surface. The second case arises from the coincidence of the two catenaries just mentioned, and it will be seen that an infinite number of curves may in this case be drawn between the two points, each of which gives rise to the same rotation-area. These results are due to Todhunter (see references at the beginning of the next Chapter). 30 CALCULUS OF VARIATIONS. CHAPTER III. PROPERTIES OF THE CATENARY. 33. Owing to certain theorems that have been discovered by Lindelof and other writers, some of the very characteristics of a minimal surface of rotation, which are sought in the Calculus of Variations, may be obtained for the case of the revolution of the catenary without the use of that theory. We shall give these results here, as they offer a handy method of comparison when we come to the results that have been derived through the methods of the Calculus of Variations. In presenting the subject-matter of this Chapter, the lectures given by Prof. Schwarz at Berlin are followed rather closely. The results are derived by Todhunter in a somewhat different form in his Researches in the Calculus of Variations, p. 54; see also the prize essay of Goldschmidt, Monthly Notices of the Royal As- tronomical Society, Vol. 12, p. 84; Jellett, Calculus of Variations, 1850, p. 145; Moigno et Lindelof, Calcul des Variations, 1861^ p. 204; etc. 34. Take the equation of the catenary which "was given in the preceding Chapter, Art. 30^ in the form* It follows at once that m -^ =: ± V y'^—m^= >^w[e '"'"=''■'''''" _e-(x-x,')/m-|_ ax On the right-hand side of the equation stands a one-valued func- tion, but on the left-hand side, a two-valued function. It is there- fore necessary to define the left-hand side so that it will be a one- valued function corresponding to the right-hand side. "■Throughout this discussion the A'-axis is taken as the directrix. CALCULUS OF VARIATIONS. If we make x "^x^, then is 31 > (x— 3to')/m >e- -(x— Xii')/m and consequently V y^ — m^ is positive. But when x and {a, h). The equation of the catenary is now, since x^ = o, and consequently b = ^(e^^"' - e-^''^ ) = (m), say, [1] where we regard a as constant and m variable. We wish to determine whether this last equation gives a real value or real values for m. We see that (f>{fn) is infinite when m—O and also when ni='{m) is negative infinity when m is zero; is unity when m is infinite, and changes sign once and only once as m passes CALCULUS OF VARIAT10^"S. 35 from zero to infinity. The least value that (f>(fn) can have is for the value of m that satisfies '{fn)=o. If, then, the given value of d is greater than the least value of ^{m), there are two values of m vt'hich satisfy [1] ; if the given value of b be equal to the least value of {m), there is only one value of ffi/ and if the given value of b is less than the least value of {fn), there is no possible value of ?n. Moigno and Lindelof have shown that the value of — which satisfies w ,a/ni a _ 111 ,— a/nl m • g-a/m)^^ 1.19968. . . . ; and then from [1] it follows and therefore - = 1.50888. . . =tan (56° 28') is approximately that —=1.81017. approximately (see Todhunter, loc. cit.. Art. 60). Thus there are two catenaries satisfying the prescribed conditions, or one or «o«e^ according as -is greater than, equal to, or less than 1.50888. . . b a a If we write Jt = tan (56° 28'), it is seen that>'=^:»; and j^'= — ^ X are the two tangents to the catenary that may be drawn through the origin. As the ratio b/a is independent of ntj it also follows that all the catenaries of the form y=in/2 (e'"''"-'-e~''''"'), which may be derived by varying m, have the same two tangent lines through the origin, the points of contact being ;i; = ±1.19968. ... ^/ and r= 1.81017.... w. 39. Returning to the catenary i'=>^w[c''=-'"'''^'"4-e~'''-"°''^'"], we shall see that also here there are three cases which come under investigation according as: 36 CALCULUS OF VARIATIONS. P'. I. Two catenaries may be drawn through the fixed points; II. One catenary may be drawn through these points; III. No catenary may be drawn through the two points. We may assume that jVi^ jVo. ^i>.^o- ^or if Xi<^Xo, we would / only have to change the direc- tion of the ^-axis which we name positive and negative; or we might consider the case of Po and /*!', where /\' is the image of Pi\ that is, the point symmetrically situated to /\ on the other side of the ro-or- dmate. 40. From the equation of the catenary it follows that \ and Therefore y^=y2mle ^^'-^'""^ + g-(=c,-x,')/m j ^ jKo — ^W fh^ fg (xo— Xo')/m g— (xn— Xo')/m"l 2 .p v' y^—rn^ —±y2fn[e^''° ^"' )/m n — (xo — Xo')/ni ]; [I] and from this relation it is seen that V y^—tn^ has a positive or negative sign according as Xf^—Xo^o. Hence, also, {xa—x^)/m=±\og nat [(>'o+ v'jJ'o^— w^)/w]■ [a] 41. Under the assumption that y-t^yo, we must first show that such a figure as the one which follows cannot exist in the present discussion. We know that That Xi—Xa is necessarily positive is seen from the fact that the ordinate y^=in corresponds to the value x^, and is a minimum. (See Art. 34.) Suppose that x^ >^i. By hypoth- esis y^^yoi and further fn^yf,, and con- '' '' '• sequently m^y^. The form of the curve is then that given in the figure; and we have within the interval x^ to x^ a value of x, for which the ordinate y is greater than it is at the end-points, y CALCULUS OF VARIATIONS. 37 must therefore have within this interval a maximum value. But we have shown (Art. 34) that there is no maximum value* of y; hence, and there cannot be the minus sign as in equation [I] ; hence, (;ri-V)/w=+lognat [{y^^V y^^-m^) /m\. [d] 42. Eliminate x\ from [a] and [^] and noting that in [a\ there is the ± sign, we have two different functions of m, which may be written : A ( w) =log nat [( >'i + 1/ y^—rn^ )/m\ —log nat [(>'o+ V y^—m^ )/m\ — {x^—Xo)/m , and /j (w) =log nat [( y^ + V y^—m^ )/w] + log nat [( Jo + v/ y^— ni^ Ym] — ( x^— x^)/m, two functions of a transcendental nature, which we have now to consider. We must see whether y^(»?)=o,/^(7»)=c have roots with regard to in; that is^ whether it is possible to give to ^w posi- tive real values, so that the equations f^{ni)^o, f-i{rn)=o will be satisfied. If it is possible thus to determine tn^ we must then see whether the values x^ which may be derived from equations \a\ and [5] are one-valued. The first derivative oif-^{m) is mLi/jKo— ^ V y^—rnr ^ J On the right-hand side of this expression 1/m is positive, also (^x-^—Xf^)/m is positive, and is positive, '\i yC^y^. V 1- myyi V\- myy^ Hence /i(m) is positive in the interval o jo * In other words, >-, cannot be greater than y„ and at the same time jr/ greater than j:,. 38 CALCULUS OF VARIATIONS. Further, y^(o)=log nat2>', — log oat (^1=0) —log nat2>'o-rlog nat (m=o) — [{zi-Xo)/m]t^^,i = — CO. 43. It is further seen th.a.t/[(m) continuously increases with- in the interval o. . . .Vo, so that — 00 is the least value that /i{fn) can take. Again Xi — X(, /(>'o)=log nat Then if When -y^' y^ [11] I- /i(>'o)<<3./(w) has no root; II. f-^ i^y^) = o,/i{^m) has one root, m^=zy^ ; III. fiiyo) > o./i (w) has a root, m^<^y^. AijKo) 'n ) > <5^ /'i is within the catenary. This mav be show as follows : y=j4 yo[e' ,(x— x„)/y„ j^ /,— (x— Xo)/y ■•'] ; since when;)^=w^ ^=^0'; and, therefore, when jv=ji'o=w, ;i;=;»;o. We also have y^--y„^= % y^- [e (''-""'/vn _g-(x-x„)/y„-] 2_ Hence where the positive sign is to be taken, when x>Xo, and the nega- tive sign, when x'oJ • Comparing this equation with equation [II] above, and noticing the figure, it is seen that, when Xx—Xo=y>'olog nat [(jKi-j- ^'"^ yi—yo)/yo\^ then P^ is outside the cat- enary, ^1— ^oo, there is one and only one real root in the interval o. . . .y^, and we can draw through the points Pi and P(, a catenary, for which the abscissa of the lowest point is '«+ V y^--m^ym\ — {xi—x^ym. Therefore m \^v y^—mr V y^—rrr J When in changes from o to y^, the quantity V y}/ rri?'—\ contin- uously decreases, and consequently — -^^ becomes greater V y^/fi^—X and greater. Hence if the expression —m^f-^i^m) takes the value o, it takes it only once in the interval from o to jVo- That this ex- pression does take the value o within this interval is seen from the fact that, for in=o, —trP- f-l {^m)=L —{^x^— x^ , where x^—x^o, so that —in^f-l{m) has a negative value; but, for m^y^, — nP'f^i^y^)—^^, so that the expression must take the value zero between these two values of m. Let /A be this value of m which satisfies the equation, so that Vy^—v^' V y^—v''- which is an algebraical equation of the eight degree in /a, or an algebraical equation of the fourth degree in \i?. 40 CALCULUS OP VARIATIONS. 45. An approximate geometrical construction for the root ,/>, that lies between o and y^ In the figure it is seen that the tri- angles Pq Q^ Ao and Po Qo Q are similar, as are also the triangles Pi Q\ A I and P^ Q^ C\; hence, if m is the length of the line Qq C^= Qi Q, we have ■yr,m 1/ y^— m^ and GiA = y^m V y,' m'- By taking equal lengths So C^=^ Qi Cx on the two semi-circles and prolonging P^ C^ and /\ Cj until they intersect, we have as the locus of the intersections a certain curve. This curve must intersect the A'-axis in a point S, say. Noting that it follows that — ■^i — Xf), which, compared with the equation _ [A] above, shows that 46. Graphical representation of the functions f{ni) andfi^m). The lengths m are measured on the A'- axis. Equation [c] gives y^'( ;>/„)= CO ; that is, the tangent to the curve y—f{x) at the point y^ is parallel to the axis of y. Fur- / n. \ ^ / / if 1 / 1 / / 1 / 1 B i 1 , / CALCULUS OP VARIATIONS. 41 ther,/i((?)= — CO, so that the negative half of the axis of jy is asym- ptotic to the curve yz=f.^(^x). The branch of the curve is here algebraic, since ;^'=/i (;?;), for x=^o, is algebraically infinite. 47. Consider next the cvrve y^^f-J^m). It is seen that/i( jVo) =y^(>'o); and also y^'(;v'o)= — co , so that the tangent at this point* is also parallel to the axis of the y. Further, the negative half of the axis of the y is an asymptote to the curve; but the branch of the curve jv=y^(^w) is transcendental at the point ■m=o; because logarithms enter in the development of this function in the neigh- borhood of m=^o, as may be seen as follows: /^{m) = \og [{y^+ V y^—m')/m\^\og \iy<,^v yo'—ni'') / m\- l(x^~Xo)/m] = ~l{x^—Xo)/m]—2 log m+P(m), where P{m) denotes a power series in positive and integral as- cending powers of fn/ hence, the function behaves in the neigh- borhood of ■m=o as a logarithm. 48. We saw that 1 r _y^^ m'-Vv y^- m + y^rn rtv' V yo m'- ■ {x-,—x^)\. For the value ^w=/x the expression within the brackets is zero, and when tn=^o, this expression becomes —(xi—Xo), and is nega- tive. As seen above in the interval m=o to m=^yo, the expression yiffi + y^m — {x^ — x^) V yi — pt^ V yo — w^ becomes greater and greater, so that between the value m=o and ?w=/i., it is negative. Furthermore, /zim) is positive between m=o and w=/i., and negative between ^w=ju. and m^yo. Hence f-lrn) increases between fn=o and ni=i^, and decreases be- tween m=i^ and m=yo\ and conse- quently y^(fi) is a maximum. TL \/!L. \ / r I / ! / / ■f.f"^ / i/ *The distance >„ is, of course, measured on the A'^axis. 42 CALCULUS OF VARIATIONS. 49. We must consider the function f^{m ) when m is given different values and see how many catenaries may be laid between the points Po and P-^. We have: Case I. /2(>)o. We have here two catenaries. One root ofy^(/«)=o lies be- tween o and /x, and often another between /x and y^, as is seen from what follows: /2(-l-c) = — 00 and fz{y.)>o. Since y^(;w) continuously increases in the interval +o. . . .^ it can take the value o only once within this interval. In the interval fi. . . ■yo^/z^m) continuously decreases, so that if /^(;>'o)>-o^ there is no root oifi{m)^^o within this interval; but if_/^(j>'o)f o^ then there is one and onlj' one root within this inter- val, and in the latter case there are two catenaries. We must next consider the roots of /i\nt). Wheny^(j)'o)'o) = ^. A)'o). ^\ (But when/^(jyo)>'?. then there is only one root for_/^(w)=:o, (which lies between o. . . .M-; this root is denoted by fn^. 50. From the formula ( Art. 42 ) f or y^( m ) and y^( w ) we have : /2(m)=/,(m)+2 log [(>•„+! ;^'„2_w2)/w]. We consider the val ues of m within the interval o....Vo; for m=:0,{^o+ y' _yo^—m'^)/m= oo ; and for m^y^,, (>'o+ V ya—m^^/m = 1. Consequentl)% within this interval log [(>'o+ i ;>'n^— m^)/;»] is positive, and therefore alsoy^(w)>/"i(m); and since y^(wi) = c^ it follows that/i(mi)<^o)=y^(>'o); and since y^( jFo )> o^ we have /i(j>'o)>o. Moreover, within the interval o . . . . y^, f^{m) continuously increases, and/i( +c>)o. A) fly,)^o. Then, as already shown, y^(^w)=c> has two roots, one of which lies between o and /x, and the other between /t and y^ . Let these roots be m-^ and m-^ respectively. For the root Wi, we have y^i^\ . QoT,= Vy.'- -m-; Q.T, yifn^ Vy^ -m-: We assert that here the inter- 'T section of the tangents at P^ and /*! lies on the other side of the X-axis from the curve. In order to show this we need only prove that Q. T, ^Q. T,„^-Wi^ (;Ci — ^Ko) . Now, since f-liin) within the interval o ... .ft. is positive, and since Wj lies within this interval, it follows that f-lim^ is positive. Therefore —{m^Y-lim^ is negative, and consequently QoT^+QiTi — <2o Qx is negative. Remark. In this consideration the whole interpretation de- pends upon the fact that the root lies in the interval o . . . . /i, and the same discussion is applicable to Case B), where f-^y^^o, and where the root lies between o . . . ./i. CALCULUS OF VARIATIONS. 45 52. On the consideration of the root Wj . r. When/,(j,^„)ft,. The root lies within the interval [l. . . .y^ and here fzim) is negative within the interval; therefore —m^f-^im) is positive, and consequently V y^* — m-2, V y}—m} \X^—X^Y>0; therefore so that 7" is on the same side of the J^-axis as the curve. 2°. When/j ( y^) >o; then the root m^ is a root of the equation f.^(^ni)^o, so we have here to con- ^_ sider the sign of V y^' — mi V y^ — m{ within the interval o. . . .jVo- We have proved that within this interval fiint) is positive, and since y--^ y^^ {x,-x,)\ iince ^ Li/ y^ y^ — m} V y} — m} is positive, it follows that r yx'^x y^'rnr y{—m. is negative. Hence V y^— m.^ Xi — Xf)) - /°/^^ <(^.-^o). i/ yi^ — ^2- V y^ — mi Consequently ^0^^ y-'^-' >{x,-x,). V y^—m} Vy^ — m^ 46 CALCULUS OP VARIATIONS. Since — -^^ ^ is a positive quantity, it follows a fortiori \.\izX V y^ — mi yx 1^2 + ^"^^ >U.-^o). and the intersection lies on the same side of the ^-axis as the curve. 53. We have seen that two catenaries having the same direc- trix cannot intersect in more than two points /{, and P^. Denote as above the smaller parameter of these two curves by m^ and the larger by m-2,. Then it is seen that Q, the curve of smaller par- ameter, comes up from below and crosses Cj, the catenary of larger parameter, and, having crossed Cj, never finds its way out again. For, consider the tangent PT to the curve Q as the point P moves along this curve. This tangent must at first intersect C2, but at the vertex it is parallel to the A'-axis and evidently has no point in common with Cj. Hence, for some position between these two positions the tangent to G must also be tangent to €2- We see that there are two tangents common to Q and Q, and we shall next show that they intersect on the directrix. 54. Draw the common tangent A T^ and draw a tangent A 7\ to the curve C^. Then between these lines we may lay an infinite number of catenaries that have the same directrix. One of these catenaries must be C2, for it touches A T^ and is the only catenary that can be drawn through the point of tangency made by A To (Art. 37). Consequently A T■^ is the other common tangent to both curves. We see also that the points P^ and P^ are beyond the points of contact of C, with the two common tangents, while for Cj the points of contact of the tangents are beyond P^ and P-^. It is also seen that, as the two curves Q and Cj tend to coincide, the com- mon tangents to the distinct curve become tangents to the single curve at the points P^ and P^ (see Art. 51). If we call ^ the value of m corresponding to this latter curve^ we have m^p- ^l^ m^. CALCULUS OF VARIATIONS. 47 55. Suppose we have two catenaries which are not coincident and which have the same parameter m. Denote their equations by These catenaries intersect in only one point. For we have at once therefore or g(x— x„')/in I g— (x— Xo')/m__g(x— x„")/in , g— (x— x/')/m gx/m Pg— XoVm g— Xo"/m"| __g— x/m r^Xo'Vin gXo'/ra"| 1 V & """Z™ p Xo'/m ~| p— (x„'+x„"Vml _gXo'/m I gXo'Vra y-,Xo"/ra -,Xo'/m ,2x/m ^ ^ ' ~l TTl p — Xo /m p — Xo /ni Therefore g2x/in__ g(xo'+Xo")/ni and consequently (x„"-x„')/2m _L g— (x„"-x„72m"| which are the coordinates of one point. 56. Lindelof's Theorem ( 1860 ). If we suppose the catenary to revolve around the .AT-axis, as also the lines /{, T and P-^ T, then the surface-area generated by the revolution of the catenary is equal to the sum of the surface- areas generated by the revolution of the two lines P^ T and P^ T about the ^-axis. Suppose that with 7" as center of similarity (Aehnlichkeits- -^ punkt), the curve P^ Pi is sub- jected to a strain so that P^ goes into the point P^, and P^ into the point P-^, the distance Pq Pa being very small and equal, say, to a=/\ /*/. Then P^T: Po'T=l : 1-a. 48 CALCULUS OF VARIATIONS. To abbreviate, let Mq denote the surface generated by /*« T; M^ that generated by P^ T\ M^ denote the surface generated by P-^ T; M-l that generated by P^ T; 6" that by the catenary P^P^; S' that by the catenary P^ P/. From the nature of the strain, the tangents P^ T and P^ T are tangents to the new curve at the points P^ and P-l, so that we may consider P^P^ P^P^ as a variation of the curve P^Pi . It is seen that ^ : S' =1: (1-a)^ M, : ^/o'-l : (l-a)2; M^ : M^=l : (l-af \2 Now from the figure we have as the surface of rotation of /Jj/o'/'i-^i' where [^a^)] denotes a variation of the second order. Therefore Hence Sll-(l-ay] =M,[l-(l-ay] +M,[l-{l~ay] + [( a^)], and consequently 2aS=2aAf, + 2aM,+ l(a?}'\, or finally a result which is correct to a differential of the first order. In a similar manner so that S~S'=( Mo-M,' ) + ( M^-M;-) ; or s={m,-m:~)+s' ^^m^-m;) is an expression which is absolutely correct. 57. Another proof . We have seen that 7i^ - 7i^ -(-->-. w CALCULUS OP VARIATIONS. 49 and (see Fig. in Art. 45) ,2 PoS = — ^g ; P,S= y^ . [2] The surfaces of the two cones are, therefore, equal to y<^ -y^-^ ^„A yi -y^"" and V y}—\i^ V y^—\i^ The surface generated by the catenary is Xx J 'Zyj^ds. X. In the catenary ds^y/m dx (see Art. 35), so that X, X, \2y IT ds = \{2y^ ir dx)/in Xa Xn "1 = 2Trf m^/A [e ^''^"-^»'' ^" +2+e -z'^-'"''' /■" ] dx/m Xcs = 7r[±jv V y^—m^+mx^ = ■n-[_yi V y^—w? +y^V y^—m^+m{xx—Xo)], [^] where we have taken the + sign with jKo V y^—rn^ because x^—x^ is negative, hence e(^-*«')/>«_e-(»-=^»')/°' in \A\ is negative. But from [1] X\ — Xa — — = ~r V y^—i^ V y^—v: 50 CALCULUS OF VARIATIONS. Substituting in [^],we have, after making ???=/x, for the area gen- erated by the revolution of the catenary ttI >'l Ji h' y y^ — ^^ :yoV ^'o'^—i^^ + yoi^ v V— M^ ] which, as shown above, is the sum of the surface-areas of the two cones. 58. Let us consider* again the following figure, in which the strain is represented. In order to have a minimum surface of rev- olution, the curve which we rotate must satisfy the differential equation of the problem. If, then, we had a minimum, this would be brought about by the rotation of the catenary; for the catenary is the curve which satisfies the differential equation. But in our figure this curve can produce no minimal surface of revolution for two reasons: 1° because, drawing tangents (in Art. 59 it is proved that there exists an infinite number) which intersect on the X-axis, it is seen that the rotation of PoP\ is the same as that of the two lines P^ T and P^ T, as shown ^ above, so that there are an infi- nite number of lines that may be drawn between /•„ and P^ which give the same surface of revolution as the catenary be- tween these points; 2° because between /o a^nd P\ lines may be drawn which, when caused to revolve about the X-axis, would produce a smaller surface-area than that produced by the rev- olution of the catenar}'. For the surface-area generated by the revolution of P^P-l is the same as that generated by P^P^'Pl'Pl. But the straight lines P^P^' and PlP^' do not satisfy the differen- tial equation of the problem, since they are not catenaries. Hence the first variation along these lines is < o, so that between the points * See also Todhunter, Researches iu the Calculus of Variations, p. 29. CALCULUS OP VARIATIONS. 51 Po'. Pd' and P^, P-l' curves may be drawn whose surface of rotation is smaller than that generated by the straight lines /o'/'o" and pip:'. The Case II, given above and known as the transition case, i. e., where the point of intersection of the tangents pass from one side to the other side of the j'^-axis, affords also no minimal surface, since, as already seen, there are, by varying the quantity a (Art. 56), an infinite number of surfaces of revolution that have the same area. 59. In Case III we had two roots of m, which we called m^ and ^»2, where m^^m-^. We consider first the catenary with par- ameter m^. This parameter satisfies the inequality 2 /**7 2 -x/ ^f 2 A*7_ 2 V y^ — ■m^'- y y} — m/ The equation of the tangent to the curve is dy _ y' — y ax X — X where x' and y' are the running coordinates. The intersection of this line with the ^-axis is I _ y '_ y X — X — — — ;; — / , , or X — X -J / _, , dy / ax dy/dx X = X — m I (x-x„')/m _ g-(x-x/)/m J ■ J. e., Hence, when x=.xd, ;r'= — oo, and when ;i:= + co, ;t'== + oo. On the other hand, dx'/dx is always positive, so that x' always increases when x increases, and the tangent passes from — oo along the ^-axis to + oo, and never passes twice through the same point. It is thus seen that there are an infinite number of pairs of points on the catenary between the points Pt> and P^ such that the tan- gents at any of these pairs of points intersect on the ^-axis, and there can consequently be no minimum. Such pairs of points are known as conjugate points. When 'm=m2, the tangents intersect above the X-axis, and there is in reality a minimum, as will be seen later. 52 CALCULUS OP VARIATIONS. 60. Application. Suppose we have two rings of equal size attached to the same axis which passes perpendicularly through their centers. If the rims of these rings are connected by a free film of liquid (soap solution), what form does the film take? By a law in physics the film has a tendency to make its area as small as possible. Hence, only as a minimal surface will the film be in a state of equilibrium. Let O be midway between O' and O" . The film is sym- metric with respect to the 00" and OL axes and has the form of a surface of revolution about the 00" axis, this surface being a catenoid. The line OL is the axis of symmetry of the gen- erating catenary. Construct the tangents OP" and OP' from the origin to the cate- nary. Only when P' and P" are situated beyond the rims of the circles will the generat- ing arc of the catenary be free from conjugate points, and only then will we have a minimal surface and a position of stable equilibrium of the film. 61. We saw (Art. 38) that all catenaries having the same axis of symmetry and the same directrix may be laid between two lines inclined approximately at an angle tan ~' (3/2) to the directrix and which pass through the intersection of the directrix and the axis of symmetry. All catenaries under consideration then are en- sconced within the lines OP' and OP" and have these lines as tan- gents. The arcs of these catenaries between their points of con- CALCULUS OF VARIATIONS. 53 tact with O T' and O T" do not intersect one another. Through any point P^ inside the angle T' O T" will evidently pass one of these arcs, and the same arc (on account of the axis of symmetry OL of the catenary ) will contain the point P-^ symmetrical to P^ on the other side of OL. The arc P^Px contains no conjugate point (Chap. IX, Art. 128), and therefore generates a minimal surface of revolution. Further, this is the only arc of a catenary through the points P^, and /\ which generates a minimal surface. Suppose that we started out with our two rings in contact and shoved them along the axis at the same rate and in opposite directions from the point O. As long as P^ and P-^ are situated within the angle T'OT" (or what is the same thing, as long as PoOPx< T'O T") then the tangents at P^ and P^ meet on the upper side of the X-axis and there exists an arc of a catenary which gives a minimal surface of revolution and the film has a tendency to take a definite position and hold itself there. But as soon as the angle PoOP^ becomes equal to or greater than T'O T" this ten- dency ceases and the equilibrium of the film becomes unstable. As a matter of fact (see Art. 101), the only minimum which now ex- ists is that given by the surface of the two rings, the film having broken and gone into this form. 54 CALCULUS OF VARIATIONS. CHAPTER IV. PROPERTIES OF THE FUNCTION F{^X,y, x',y). ' 62. Consider the general integral of Art. 13: 4 where /^ is a given function of the four arguments, x, y, x\ y' , the quantities x' and y' being written for dx/ dt and dy/dt; further we must regard i^ as a one-valued regular function of these four arguments, one-valued not in the analytical sense, but only for real values of the arguments; x and y are defined for the whole plane or for a connected portion of it, while x' and y' are to be consid- ered as variables that are not limited, since they determine the direction of the tangent, and it is supposed that we may go in any direction from the point x, y. In our problem new assumptions are made regarding x and y, but not regarding x' and y'* We further assume that the functions x, y, x' and y' , are capable of being differentiated, and that the curve is regular throughout its whole extent, or is composed of regular portions. Consequently X and y considered as functions of t and written x{^ t), y{ t) are one- valued regular functions of / throughout its whole extent or throughout the regular portions ; in the latter case we shall limit ourselves to one regular portion. If we did not make this assump- tion, the curve could not be the subject of mathematical investiga- tion, since there is no method of treating irregular curves in their generality ; and, if we wish the rules of the differential and integral * A limitation has to be made, however, if for certain values of x', y' the function F becomes infintely large. Such cases must be excluded from the present discussion. CALCULUS OF VARIATIONS. 55 calculus to be sufficient, then we must first apply our investigation to such functions, to which the rules are applicable without any limitation ; that is, to functions having the above properties. 63. If we find a curve which is regular and which satisfies the conditions of the problem, then it still remains as a supple- ment to prove that it is the only curve which satisfies the condi- tions of the problem. For example, it is found that of all regular closed curves of given perimeter the circle is the one which encloses the greatest surface-area; a priori, however, it is not known that a regular curve satisfies the problem. We know that of all polygons with a given number of sides and having a given perimeter the regular polygon has the greatest surface-area, and we thus come to the conclusion that the circle, to which the polygon approaches when the number of sides is increased, will have the greatest surface- area of all the closed curves ; however, no one will recognize in this a rigorous proof, and in fact there still remains a peculiar artifice to prove this property of the circle. 64. The chief difficulty in all analysis consists in giving a strenuous proof that the necessary conditions, that have been found for the existence of a certain property, are also sufficient. In analytical researches we make conclusions in the following man- ner : If the analytical quantities exist, which are required through the problems that have been set, then they must have certain properties ; this gives the necessary conditions for the sought functions. It remains yet reciprocally to prove : If the conditions for an analytical object (curve, surface, etc. ) are ful- filled, then the analytical object satisfies the conditions of the problem. We therefore presuppose in our investigations, that the re- quired functions are regular in their whole extent, and we seek the necessary conditions for the function which are given from the problems. Finally we will free ourselves from the limitations as far as it is possible, and see whether also the functions which have been found correspond to the conditions of the problem. 65. The development of a mathematical idea is, as a rule, first suggested by a concrete instance. We assume, for example, the existence in nature of something which we call the area of a limited plane. This area we express by a mathematical formula. 56 CALCULUS OF VARIATIONS. We extend our formula and talk of the area of a curved surface. The mathematical formula exists. That to which it corresponds in nature may or may not have an objective existence. The word "area," however, is defined for us, and is limited by the mathe- matical formula. When the formula ceases to be intelligible, ceases to have a meaning and to give a value, then also does the idea "area" cease to exist for us. We must always presuppose those limitations to be involved in our symbols which permit of the formula having a meaning. 66. Only for regular curves do we compare our integrals; for such curves alone have they a meaning. Among this class of curves we seek one which gives a maximum or a minimum value of our integral. And when we put our theory into practice we as- sume the non-existence of quantites other than those which our theory has actuall)'^ compared. Here we run a risk. It may be that in some particular problem we have assigned a certain role; it may be also that, as far as our theory goes, we are correct in assuming the possibility of the existence of all the regular curves that are compared with one another and that their roles relative to one another has not been misstated. But it may be that there exists in nature the possibility of quantities other than those defined by our definite integrals along regular curves, and these quantities may have the same essential properties rela- tive to the problem in question as our various definite integrals. It may be also that to one of these quantities nature has assigned that very role which w^e have been seeking among our definite integrals. When we apply any mathematical theory to objective reality, we make assumptions in the way of continuity, differentiation, etc., regarding the possibilities which are permitted in nature. The question arises, do our hypotheses include all possibilities? 67. We may emphasize the fact that in the development of a general theory, as a rule its scope is not determined beforehand. The quantities and functions to which we must apply the opera- tions involved are named a priori, but formulas are developed on the supposition that the operations involved are feasible and have a meaning. The scope of the formulae is afterwards defined by the territory in which all the steps involved have some significa- CALCULUS OF VARIATIONS. 57 tion, or by the exclusion of any realm in which they would be in- capable of interpretation. 68. We will now prove some important properties of the func- tion F (Art. 62). In the problems which we have discussed the fol- lowing is to be observed : the value of the integral, which is to be a minimum, depends in all cases only upon the form of the curve which is to be determined, not upon the manner in which x, y are represented as functions of a quantity /. For example, if in the first problem we write the integral in the form X, JWi+ (§)''='-- Xff then t is exactly equal to x, and it is clear that the value of this integral is the same as it was for the previous form (Art. 7). If we write for t any function of another quantity t of such a nature that to the values 4 and /j of / the values Tq and t^ of t cor- respond, and that the curve with increasing t will be traversed in the Scune direction as in the first case with increasing t, then the integral must remain unaltered, if it is to be independent of the manner in which x, y are represented as functions of the quantity t; that is, we must have as the integral / of Art. 62 The simplest function of this kind that we can write for ^^is t=^kT, where k represents any arbitrary but positive quantity. Hence considering x, y as functions of r in the left-hand side of 1), we have h ^1 r„/ dx dy\.. f;^)', ^ ^ dx \ dy\,. J ^f'>'^ TV dt) ^^ =J ^y^'^'-k -dr'TcdrS ^^' 58 CALCULUS OF VAWATIOMS. hence J I dr dr^ J I k dT' k dj) Since this equation must be true for any arbitrary positive vakie of k, which however is not necessarily a constant, but may be any continuous positive function, it follows that the functions to be integrated must themselves be equal for every positive value of /',• and consequently f{x V "^'"^ '^^^-UfIx r i— i-^V '^ V'^' Tt- dr) ~^^y^^^' j^ ^i^' J. ar) ' or, writing k=-\ / k . 3 ) F(x, y, kx . ky' ) =k F{x,y, x ,y' ). That is, if the integral / is to depend only upon the form of the curve (or in other words, upon the analytical connection be- tween X and jv), then F{x, y, x , y'), with regard to x and y 7nust be a homogeneous /unction of the first degree. This con- dition is also sufficient to assure that the integral depends only upon the form of the curve; for consider x,y first expressed as func- tions of a quantity t and then as functions of a quantity t, and if the.se functions are of such a nature that the curve is traversed from the beginning-point to the end-point when t takes all values from ta to t-^, and r all values Tq to Tj, then we can write dt/dT = k, if t increases at the same time as t . Since ^ is a positive quantity, the correctness of the expression 2) follows from the existence of 3) and at the same time also the correctness of 1). It follows also that dF{x,y,kx;,ky') _ j ZF{^x,y,x',y') _ dF (x,y,x',y') d(kx') d(kx') ~' dx' -.--F^'-''{x,y,x! ,y'). say. In the same way the partial derivative of F with respect to its fourth argument is invariantive and may be denoted by F'-\x, y, X , y ). CALCULUS OF VARIATIONS. 59 69. The condition that F{x,y,x',y) must be a homoge- neous function of the first degree with regard to x' and r' is gen- erally expressed in another manner. In fact, it is nothing else than the condition of integrability of F. For if F{x,v, x ,y') dt is to be an exact differential, so that, say, Fi^x, y,x' , y )—d^, then the equation F ( x,y,x',y' ) = ( d^/dx) x' + ( d/dy )y' -l ( d<^/dx' ) x" ;- id'i>/dy' )y" must exist identically. Since no second differential quotient is present in F, it fol- lows that d(f>/dx' = o and d^/dy' = o, /'. e., 4> does not contain ex- plicitly x' or y'j and therefore F{x, y, x\ y' ) ^ {d<^/dx ) x' -;- ( d(^/dy )y'. But this is nothing more than that /^ is a homogeneous function of the first degree in x , y' . This is everywhere the case in the examples given in Chap. I. 70. If the curve is of such a nature that one may regard the one coordinate as a one-valued function of the other and in such a way that for every value of x between two limits x^ and x^, there corresponds only one definite value of y, and that x continuously increases when we traverse the curve from the beginning-point to the end-point, then we may choose for t the quantity x itself, and therefore write the integral in the form X, 4) / -- f Fix, y, 1, dy/dx) dx, Xf^ as it is usually written. 71. This representation is not always true, since the above conditions which are necessary are not always fulfilled; for ex- ample, in the fourth problem of Chapter I we must distribute the )-et unknown curve into several parts, and this is not always con- venient. On the other hand, a representation such as given above is always possible, if we introduce the quantity /, since one could in- troduce as the variable / the arc 5 of the curve measured from the 60 CALCULUS OF VARIATIONS. beginning-point. Besides in the form 4) it sometimes unavoid- ably happens that dy/dx and consequently F becomes infinite within the limits of integration; on the other hand it is generally possible so to choose t that this is not the case. For these reasons, in spite of the fact that many developments become more cumbrous, it is preferable to treat the integral / in the form A f --= j F {x, y, x',y') dt; 4 for on the other hand its great symmetry overbalances the fault just mentioned. 72. Analytical condition for F{x, y, x , y'). In the relation (Art. 68) F{x, y, kx\ ky^ = kF{x, y, x\y'), write k=l + hj then is F[x,y,(l + /t)x',(l+A)y']=(l + h)F(x,y,x',y'), or, where F=F(x,y,x',y'). Therefore equating the coefficients of h : which again is the condition of homogeneity. 73. DifEerentiate the above equation [1] first with regard to x' and then with regard to y', which is allowable, since i^ is a regular function, and x', y' vary in a continuous manner, and we have [2] ^ d-F , , d^'F a.) ^^-i-,x -;- ^'P ' . ^-F , ( CALCULUS OF VARIATIONS. Hence, from < x) and from /3) d^F d^F , , , , ,. therefore d^F , d^F d^F ,. , , ,. 61 and, if F^ denotes the factor of proportionality, we have: dx'^ ~ '^ ' dx'dv' ~ ~^'^^-^' ^' 3/2-^^^ ' L-iJ and consequentl}-^ d'F d^F d^F dx'^ _ dx'dy' _ dy'^ _ ^ y'^ - -x'y' " x'^ ^ •■ i^ is of the first dimension in ;i;' and r'/ ^=i — -,i ^ — > are of the di- o X o y • , , , d^F d'F d'F r., . ... mension o m x and y ; ^r— r:, . ,, , , -^^^^ are of the —1st dimension oy^ oxoy ax^ in x' and y' ; consequently F-^ is of the dimension — 3 in ;»;' and y' . This function F-^ plays an exceedingly important role in the whole theory. 62 CALCULUS OF VARIATIONS. CHAPTER V. THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION. 74. In Chapter II we considered examples of special variations. The method followed provided for the displacement of a curve in one direction only, in the direction parallel to the X-axis, and is consequently applicable only to the comparison of integrals along curves obtained from one another by such a deformation. We shall now give a more general form to the variations em- ployed and shall seek strenuous methods for the solution of the general problem of variations proposed in Chapter I. After de- riving the necessar}' conditions we shall then proceed to discuss the stifficient conditions. In order to develop the conditions for the appearance of a maximum or a minimum of the integral H 1) I -^. jJF{x,y,x\y')dif, it is necessary to study more closely the conception of the varia- tion of a curve and fix this conception analytically. By writing instead of each point x, y oia. curve (presupposed regular) another point x-^^, y-{-T}, we transform the first curve into another regular curve. This second curve is neighboring the first curve if we make sufficiently small the quantities ^ and tj which like x and y we consider as one-valued continuous functions of/. 75. The following is one of the methods of effecting this re- sult. Let ^ and 17 be continuous functions of t and also of a quan- CALCULUS OF VARIATIONS. 63 tity k. We further suppose that i and t) vanish when k=o for ever}^ value of /, for example ^=k iiit) ; 17 r-_=/^ &.(/), u and V being finite and continuous functions of /. The functions u and v and consequently also | and t] are sub- ject to further conditions. It is in general required to construct a curve between two given points which first may be regarded as fixed. Later the condition of their variability may be introduced. Consequently we have to consider only such values of t that f , t] and consequently u, v vanish on the limits. If for X, y we vi-rite x^-^, .I'-l-i?,, then for x\ y' we must write ■*^' + r.y + V- Further, the function F(x+^, y-\-y], x -rt . y' -r)' ) must be developed in powers of ^, tj, |' and 17'. For the conver- gence of this series, it is necessary that f, 77, ^' and 7?' have finite values. Now, if we write ^=k sin t/k" ; ri = k cos t/t\ then we have -J}=k^~" cos t/t'\ ^= -^''"" sin t/ k\ so that, whereas ^ and 17 have infinitely small values for infinitely small values of ^, the quantities — ^ , -^ vacillate for « = 1 be- dt at tween +1 and —1 and become infinite for «>-l. We shall conse- quently consider only such special variations in which u and v are functions of / alone, and which with their derivatives are finite and continuous between the limits 4 and t^. We thus restrict, in a great measure, the arbitrariness of the indefinitely small variations of the curve, and thus exclude a great many neighboring curves from the discussion. However, there exist among all the possible neighboring curves also such which satisfy the above conditions, and with these we shall first establish the necessary conditions and later show that the necessary conditions thus established are also sufficient for the establishment of the existence of a maximum or a minimum value of the integral. (See Arts. 134 et seq.) 64 CALCULUS OP VARIATIONS. 76. We have instead of one neighboring curve a whole bun- dle of such curves if we make the substitutions y y +ei? , x' x' + ^^\ y' \y'+^v', and let e, a quantity independent of the variables in F(x,y, x\y'), vary between +1 and —1. The total variation that is thereby introduced in the integral of the preceding article is k to which developed b}^ Maclaurin's Theorem is 4 Further (see Art. 25) 1.2 hence, equating coefficients of e, 4 But 4 i CALCULUS OF VARIATIONS. 65 SO that 2) becomes ^fdF > , dF 7- or 4 where r - ^ _ -^ / aF\ ///■ // 77. Owing to the hypotheses that x'^ y' ,-ri ~r vary in a con- at at tinuous manner with t [that is, within the portion of curve con- sidered no sudden change enters in the direction of the curve] , the first variation of the integral / may be transformed in a remark- able manner. We had r -^^ dSdF\ ' ~ dx dt\ dx' \ and also (Art. 72) Therefore /^(^.^.^'.y) = ^'|5+y|f,. dF yJ!^, y ^'F dx dx'dx dy'dx' d F and differentiating ^5—, with respect to /, we have ax d(dF\ _ d^F dx d^F dy d^F dx' d'F dy ~dAdx)~ dxdx' dt dydx' dt dx'^ dt dy'dx' dt' 66 CALCULUS OF VARIATIONS. Hence, ' '" '•' \dxdy' dydx'l \dx'' dt ~' dy'dx' dt J Writing ^ = y'^F, (Art. 73 ), and ^^, = -x'v'F,, and de- dx^ oy ax fining G by the equation d^F d^F p(,,'dx' ,dy'\ dxdy' dydx' 'V dt dt J' it is seen that G, = y'G. In a similar manner it may be shown that G. = -x'G. 78. Lemma. If 4>{t) and i|/(/) are two continuous func- tions of t between the limits 4 cind t^ and if the integral k J(t) ^lj(t) dt to is always zero, in whatever m.anner t/i(/) is chosen, then neces- sarily { t) must vanish for every value of t between 4 and t^. The following proof, due to Prof. Schwarz, is a geometrical interpretation of a method due to Heine.* Suppose it possible that the function (j>{t) has a finite value for a point t—t' situated between 4 and t^. Then owing to the continuity of (l>{t} we can find an interval t'—d. . . .t'-\-d within which 4'{t) also has a finite value. We write the integral in the form A t'-d t' + d f^{t) ^{t) dt=j'(t) t/»(/) dt+fif>(t) i|»(/) dt to 4 t'—d ^fcl>{t)^{,t)dt. t'+d 'Heine, Crelle, bd. 54, p. 338. CALCULUS OF VARIATIONS. 67 The second integral on the right hand side may be written t' -^d M J»/»( i) dt, t'-d where M is the mean value of ^( t) for a value of t within the in- interval t' —d. . . .t' ^d. We shall show that it is possible to determine a function i/»(/) which will render this integral positive and greater than the sum of the first and third integrals in the above expression, while at the same time »/»( 4) = »!»( /,) = o. Let us form the equation d'-S ;>^-i + ^.i>'=^. which represents the parabola y = 1 — — and the .Y-axis. Consider next the equation where e is a small quantity. By taking c sufficiently small, this curve can be made to approach as near as we wish the parabola and the ^-axis. 68 CALCULUS OF VARIATIONS. Solving the above equation for y, we have as the two roots (two branches) The branch 2 is symmetrical with respect to the K-axis, and for values of x, such that —dLxL^d, the ordinate of any point of the curve is greater than the corresponding ordinate of the parabola. For the parabola ;»'=1 + -^., the integral + d j ydx=A/3d. -d It follows then that for the curve we must have ^d J[ (i-}i)WKf-S'*"']*>it -d for \x\-=d, we have>'=e; and from the inequality it follows for \x\'> d, that>' is positive and <€. For the lower branch y is negative and the curve follows the parabola to infinity as shown in the figure. It is how- ever the upper branch which we use, since y is less than e, as soon as X passes the value d on either side of the origin. CALCULUS OP VARIATIONS. 69 Instead of the integral last written, take the integral which has the same value t'-d Writing we have h t'—d t'-^d /i ^^{t)x{t)dt^^4>{t)x{t)dt-^\{t)x{t)dt+\{t)x{t)dt 4 4 t'~d f + d t'~d t' + d A =M' \x{t)dt+ M \x{t)dt +M"\x{t)dt 4 t'-d t'^d +M"{<^-\{t,-t'-d), where M\ M and M" are mean values of <^(^) in the respective in- tervals and where [•< e ] denotes that the quantity that stands within the brackets is less than e. It is seen that by taking e sufficiently small that the sign of A the integral J ^ ( /) x ( /) ^^f is determined by that of M 4/3 fi?and 4 consequently this integral is different form zero. Instead of the function x(/), write where fn and n are positive integers. 70 CALCULUS OF VARIATIONS. We see that i/(( 4) = c>=t|>(/i), and as above, it follows that f(t. xjj( t )clt^O. 4 Hence, on the supposition that <^( / )^o for a point of the curve alonjj^ which we integrate, it follows that a function »//(/) can be found which causes the above integral to be different from zero. But as this integral was supposed to be zero for all functions i/»(/f ), it follows that we must have ^{t") = o for all values of t be- tween tf, and /, . 79. In the expression ■ 1.2 unless 8/ and e always retain the same sign, it is necessary that 8/ be zero in order that A/ be continuously negative or continu- ously positive ; i. e., in order that the integral / be a maximum or a minimum ( see Art. 26 ). Substitute for G-^ and G2 their values in terms of G from Art. 77, in the expression for 8/ of Art. 76, and we have If we suppose that the points P^, and P^ are fixed so that $=o=r) for them, then the boundary terms vanish. Further, since y^—x'ri is an arbitrary and continuous function of t, it follows from the above lemma that, in order for 8/ to be zero, G=^o for every point of the curve within the interval 4- • ■ • A- G can not have a finite value different from zero for isolated points on the curve, since this portion of curve must be continuous in order that the integral may have a meaning. The differential equation G=^o of the second order is a ne- cessary condition for a maximum or a minimum value of I, and CALCULUS OF VARIATIONS. 71 will afford the required curve if such curve exists. We note that it is independent of the manner of variation, as the quantities ^ and 1) do not appear in it. From the relation (see Art. 77) k 8/=J {G^^-^G.j])dt^o. to it follows that I Gi$dt-ir \ G-^i]dt^ o. to tg Among all possible variations there are those for which -q^o, and consequentlv A fG,^dt=^o. k As above we have then Gi--=o, and similarly Further, if we multiply y' G=G^ and —!^G=Gt, respectively by v' and — ,r', we have by addition {y"-^x"-)G^o. and as x and y cannot both vanish simultaneously, it follows that G=o. Hence^ the equations Gi=o, G2=o on the one hand, and G^o on the other, are necessary consequences of one another. The equations (7i=o=6^2 ^^^ often more convenient than G=^o; especially is this the case if the function F does not con- tain explicitly one of the two quantities x and y. If .-c, for instance, is wanting, then -^— = f , and from ^ dF d dF ox dt ox d F it follows that -^7 ■■=-■ constant. 72 CALCULUS OF VARIATIONS. 80. The curvature at any point of a curve is denoted by xy —y X p [x'^ + y'^y/^' and owing to the equation we have d^F d^J^ T^ ( I II II i\ 1 dydx' dxdy' an expression which depends upon ;t:,>'^;r'^jv' alone and not upon the higher derivatives. It is thus seen that through the equation G^o, a definite relation is expressed between the curvature of a curve at a point, the coordinates and the direction of the tangent of the curve at this point. 81. Let a point P on the curve be transformed by a variation into the point P\ and let the displacement PP' be denoted by v ; further, let the components in the x and y directions be ^ and 17, while w-s. and w ^ denote the components of this displacement in the direction of the normal and the tangent to the curve at the point P. Let X denote the angle between these directions and let the direction cosines of the normal be denoted by d X 'dt dy It v(^r -^ (f J " ~mw} X V ; /. e., by -p and ^ CALCULUS OF VARIATIONS. 73 Then from analytical geometry, Wr=^ cos X + 7, sin X=^' ^+->^'^ , zVjf=T} COS \—i sin X and therefore and w^T + tt^N=^^+V- These expressions substituted in the formula for 8/ (Art. 79) give 8/=— I WjiGds + to ,dF ,dF F Wr , oy ex A From this it is seen that only the component of the variation which is in the direction of the normal enters under the inte- gral sign. 82. By means of the formula below we will prove that the variation in the direction of the tangent brings forth only such terms for the first variation that are free from, the sign of in- tegration. As in Art. 76, write A or ^[^F . ^F ., ZF dF ,\ ,, 4 and, substituting in this expression £=2/— — , tj — v -f-, we have ds ds k Ci IdF dx dFdy\ , dF d I ^dx\ ^dF d 1^ dy \\ ,, 74 CALCULUS OF VARIATIONS. Noting that k ^^^(^^)=[if^^];:-f^^iif)* J 'dx dt it is seen that K ^^ Cj dxVdF dldF\\^dy\dF d{dF\\., 4 + \_dx' ds ^ dy' dsX^ J' = \G{y'^-x'r))dt+ But \jdx' ds ^ ay ^s J,; ,> , t dx , dy v[ idx ^i dy\j. — ■ so that everything under the sign of integration drops out, leaving s. [dF dx , dF dy~V^ Hence, if we make a sliding of the curve by the substitution X y and resolve this sliding into two components, of which the one is parallel to the direction of the tangent and the other is parallel to the direction of the normal, then the result of the sliding in the direction of the tangent is seen only in the terms which have reference to the limits, and all these terms are exact differentials under the sign of integration, while the eflEect due to a sliding in the direction of the normal is shown in the formula of the preced- ing article. 83. The expression for the first variation has been obtained CALCULUS OF VARIATIONS. 75 on the hypothesis that the elements of integration have for each point of the path of integration a one-valued meaning. In case the path involved, discontinuities, it could be resolved into a finite number of portions of regular curve, and along each portion S/ would have a meaning similar to that of the preceding article. It is assumed, then, that the required curve, which is to furnish a maximum or a minimum value of the integral, is regular in its whole trace, or, at least, that it consists of regular portions of curve. In the latter case we shall at first limit ourselves to the consideration of one such portion. Within this portion of curve not only x,y but also x! , y' will be one-valued functions of t. This assumption is already implicitly contained in the assumption of the possibility of the development of A/ by Taylor's Theorem; for otherwise the derivatives of F according to x' and y' for the curve which has to be developed could not be formed. That these assumptions have been made is due to the fact that otherwise the curve could not be the object of a mathemat- ical investigation, since there are no methods of representing ir- regular curves in their entire generality. If therefore one is con- tented with the rules of differentiation and integration, he must extend these considerations over only such functions to which the rules may be applied, that is, to the functions having the proper- ties above. There are many problems in geometry and mechanics for which the above hypotheses cannot be made. 84. The following problem proposed by Euler illustrates what has just been said : Required a curve connecting two fixed points such that the area between the curve, its evolute and the radii of curva- ture at its extremities may be a minimum. The analytical solution of this problem is the arc of a cycloid, if indeed there exists a minimum. We shall now show that such is not the case. For join the two fixed points A and ^ by a straight line which divide into n equal parts, and draw alternately above and below the line n semi-circles having the n parts of the line as diameters. 76 CALCULUS OF VARIATIONS. All the radii of curvature of each semi-circle, *. e., of each por- tion of curve which is to be a minimum, intersect on the line A B and it is evident that 2 \ln] •n AB^ must be a minimum. If we increase the number n sufficiently, we may make the above expression become arbitrarily small; and in the limit « = c» the curve will tend to become the straight line AB. From this it is evident that there is not present a minimum surface-area. 85. The same result would have been obtained, if instead of the straight line AB we had taken the arc of a cycloid through these points, and had then drawn a system of small cycloids having their cusps along the large cycloid. (See Todhunter, Re- searches in the Calculus of Variations, p. 252.) The reason that a minimum is not given through the large cycloid is due to the fact that such a minimum is ofiEered by an irregular curve, and that this irregular curve is not included in our analytical research.* It follows that our assumption made regarding the regularity of the curve is out of place and leads to something untrue. But in spite of the not improbable possibility that the curve which is to satisfy a given proposition is irregular, we must make the hypothesis that the curve is regular, since we come to analyt- ical difEerential equations only by limiting our investigations to such regular curves, and the most general theory of functions teaches that in turn through these diflEerential equations are de- fined the analytical functions which in their whole extent have ex- isting derivatives. 86. To avoid any misunderstanding, we repeat what we have already said in the previous Chapter: it is not asserted that there is anything in the nature of the problem whereby one may a priori conclude that the required curve must be regular. Hav- ing these hypotheses, we fix our ideas and draw deductions. After the solution of the problem has been effected, we have to make in ' See also Moigno et Liadelof , Calcul de Variations, p. 252. CALCULUS OF VARIATIONS. 77 addition a special proof that the derived curve has all the required properties, and that this curve is the only one which has them. The chief difficulty in all such problems, as we have shown above in the special problem of approximation (or of the passing to a limit), consists in showing that the regular curve that has been found, found indeed from the necessary conditions, also at the same time satisfies the sufficient conditions, and therefore satis- fies all the requirements of the problem. 78 CALCULUS OF VARIATIONS. CHAPTER VI. THE FORM OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATION G=0. 87. Before we proceed to the development of further condi- tions for the existence of a maximum or a minimum of the integral k r=^^F{x,y^x!^y')dt, [1] 4 we shall endeavor to investigate more closely the nature and form of the differential equation G=^o. We assume that a curve satisfying the differential equation > + ^'(-'^-^t) = ''' ra dx dy dy dx' for which F-^ is different from zero, is known. Let A be the initial point of the curve, and let A A' be the direction of the curve at A. We suppose that the differen- tial equation G=o takes iits simplest form, if we regard one of the coordi- nates as a one -valued function of the other. In the integral f above, the X dependence of the quantities x,y upon C* the quantity t is subject only to the condition that a point is to traverse the curve from the beginning- point to the end-point, when t, continuously increasing, goes from 4 to ti. CALCULUS OF VARIATIONS. 79 In an infinite number of ways we may introduce in the place of / another variable t, where It is only necessary that the function / be so formed, that with increasing t also t increases. In place of t we may reciprocally introduce again The form of the integral / and of the differential equation G=o does not change with these transformations. Under certain conditions we may choose for t the coordinate X iself as independent variable, this being the case when on trav- ersing the curve from the initial-point to the end-point, x contin- uously increases. In particular, we may take as the X-axis the tangent at the initial point A and take the direction of the curve as the positive direction of the X-axis. Since we consider only regular curves or curves composed of regular portions^ it follows, if the point P traverses the curve starting from the point A, that its distance from the normal at A continuously increases for a cer- tain portion of curve. Hence for this portion of curve, if we take the positive direction of the normal at A as the positive F-axis, there is only one value of ^ for every value of X. Consequently for a definite portion of curve we may always assume that, by a suitable choice of the system of coordinates, the second coordinate may be regarded as a one-valued func- tion of the first. We have therefore only to mcike a transformation of coor- A. dinates. Let the coordinates of the new origin of coordinates be a, 6, and further let x=a+gu+£-'v,y=5+h'U+h'v, [3] where u and v are the new coordinates. If X is the angle between the X-axis and the «-axis^ we have the well-known relations ^=cos X, A=sin X, ^'=— sin X^ h'=cos X. [4] 80 CALCULUS OP VARIATIONS. The integral / becomes then^ since u may be regarded as the independent variable, u o which we shall for brevity denote by u o If we further write -7— = v' then [5] becomes au u 1=^ /{u,v,v')du. [5-] o We have a differential equation to determine z' as a function of u, if we apply to this integral the same methods as were used in Arts. 74-80 of the last Chapter. Let the curve be subjected to a sliding in the direction only of the ordinate v, and therefore write v-\-v in the place of v, where z' is a very small quantity which vanishes at the end-point and the initial-point of the portion of curve under consideration. The in- tegral which has been subjected to variation is We next develop A/ according to powers of v and —7— • The aggregate of the terms of the first dimension is o CALCULUS OF VARIATIONS. 81 or, since 3/ ^^_^/- 3/\ - d df dv' du du \ "dv' I du dv' ' we have The quantity in the square brackets vanishes, because v=o on the limits. Further, we must have Since v is arbitrary, subjected only to the condition that it must vanish on the limits, it follows from the lemma of the preceding Chapter that _^d/ _d£ _^^ du dv' dv or dv'^ du ^ dvdv' ^ dudv' dv ' ^ ^ 88. If one of the coordinates can be regarded as a one-valued function of the other, the equation [2"] may take the place of the form [2] for G=o. cP'f We shall now show that the quantity -^ which enters in [2"] ov is identical with F-^, provided that in the function F {^x,y, x' , y'~) X and y may be regarded as functions of u alone. Since f{u,v,v')^F{a+gu+g'v, b+hu+h'v, g+g'v' .h+h' v') ^F{x,y,x',y'), it follows that 3/ _ 9^^' , 3^ h' W " dx'^^ dy' and consequently 3y _3!^^'. + 2-^^yA'+ ^h'\ W^~ W^^ + ^ ^x'dy' ^"^ ^ dy'^ 82 CALCULUS OF VARIATIONS. On the other hand by its definition F^ was determined by any of the relations : From this it follows that |Z= {g'^y^-2^'h'x'y+h''x''}F, = {g'y'-h;x'YF, ^{g'h-h'gYF, = {— sin^X— cos^X}^i^i; or finally, ^'-^ -F rsi Hence [2"] may be written ^ du dvdv' dudv' dv Since we have I _dv du' and /{u,v,v')^F{x,y,x',y'), where ^r^jV are determined in terms of u,v from [3], it follows that ^^ d^v ^ d^F dv ^ d^F ^F ^c [2*] dt^ dvdv' du dudv' dv 89. In the theory of differential equations it is known that every differential equation of the form [2*] may be integrated in the form of a power-series of the independent variable u. As a special case we have the following : Suppose 1) that at the initial point of the curve represented by the power-series which is to be formed, we have u = o,v=^ba, where bg is an arbitrary constant; CALCULUS OF VARIATIONS. 83 2) that the direction of the curve at the initial point is deter- mined by the arbitrary constant dv _ , du = 2'o. then V for sufficiently small values of u may be expressed in the power-series v=b^JrVo u+...., [9] where we have assumed that F^ is different from zero at the initial point {o, bo). The second and higher derivatives of v on the position (o, bo, Vo) may all be derived from the differential equation [2*]. Hence, in [9] we have z' as a power-series in u, whose coefficients contain^ besides the constants had in each problem, only the two arbitrary constants bo and Vo, which change from curve to curve. 90. If we substitute the expression for v given by equation [9] in the formulae [3] , we have x and y expressed in terms of u. In these expressions there appear the constants g, g , h, h' , which depend upon \ and also upon the coordinates a, b of the origin of the u, V system of coordinates^ and the two constants of integra- tion bo and Vo, defined in Art. 89. These latter constants vary from curve to curve. In these formulae, just as in Art. 89, we can ascribe only small values to the quantity u. We know, however^ as is seen in the theory of functions^ that if a curve is given only in a small portion^ its continuation is thereby completely determined. We therefore need to know the curve only for indefinitely small «'s in order to be able to follow its trace at pleasure. The coordinates x, y of the curve may be represented as func- tions of / and two arbitrary constants a and ^. Instead of «, we may introduce an arbitrary function of another quantity^ if only this quantity increases in a continuous manner, when the curve is traversed from the beginning-point to the end-point. As already mentioned, the two constants of integration vary from curve to curve. If we determine suitably these constants, we can compel the curve, which satisfies the differential equation G=o, to pass through two prescribed points. 84 CALCULUS OF VARIATIONS. In this manner we have a clear representation of the manner in which the analytical expressions giving x and y are derived ; X and y are found in general from the equation G—O in the form x^^<^t,a,^~), y=^{^t,o.,^\ [10] At the same time it is seen that up to a certain point, at least, x and y are one-valued and regular functions of ^ and of the two con- stants of integration a and yS, so that eventually we can also differ- entiate with respect to these two constants. 91. It seems desirable here in connection with what was given in Arts. 89, 90 to consider the exceptional case, viz., the one in which is equal to zero for the origin {u=o, v=b^ of the curve which satisfies the equation G^=^o. We shall see that this is only an exceptional case by showing the following : If we draw around the point («=(?^ f=^o) a small circle, then this circle may be so distributed into sectors that within each sec- tor ^=^ is not equal to zero. For we may regard the radius of a sufficiently small circle about the initial point («=<7^ v—6o) of the curve in question as the initial direction of this curve. If for v we write in [8] the power-series given in [9] , we have, by putting Fj^=o, an equation for the determination of v^,', that is, the quan- tity which fixes the initial direction. This equation has either no real roots, and then there will exist no curve starting from the point {u=Oj v—ba), or Fy vanishes for single z'o"s, and then the radii determine separate sectors. Within these sectors curves may be drawn starting from the point {u—o, v=b^) in every direction, for which F-^ is different from zero. Consequently one can always assign limits for v^ within which the corresponding curves, satis- fying the equation G=o and starting from the point {u=o, v=b^)^ have at the origin, at least, an F^ different from zero. CALCULUS OF VARIATIONS. 85 92. Finally we shall show that the curves starting from the same point («=o, v=b^) which satisfy the equation G^o lie com- pletely separated from one another at their initial point. If we draw a small circle around the point («=o, v^b^) , then on its periphery we can easily determine the point u, v in which it is cut by one of the curves in question. For let p be the radius of the small circle, so that u^^{v-h,y=p\ [11] Writing for v the power-series [9] , we have p^=(l+z;o'='V+C3«^+...., or p^l/l + Z^o'^W + («)2-f We may revert this series and have u expressed as a function of p, so that and therefore \ [12] v-h^ ''^ P+(p\"+... These series are convergent for all p's within a certain limit po , so that therefore u and Vj the coordinates of the point to be deter- mined upon the periphery of the circle with the radius p, are uniquely found for all values of p— r are independent of the manner ox ay in which x and ;>' are expressed as functions of /, and depend only upon the point in question of the curve and the direction of the tangent at this point. We have at once d^F _ d" ^ / dx dy\ 1 ■ dx' dy g ^;r g ^ \ ' ' ds ds I ds ds ds dt and since ^_ 1 d'F',^, f' • as as For, from the relation (fr+(g)-' it follows, through diflFerentiation, that ds d^ ^ ds d^ ' '--' If, for brevity, we write ^' f(x y^ ^A- ^' f(x y^ a dxd^ V ' VsVs/ a^a^ V ' ' ds ds) ds ds ^(--■i-i)' we may write the differential equation of the last article in the form: \' ^'ds"ds)\dsd^~ds dsy ~ V ' ^' ^' ds)- With the aid of [*] , we have ds' ^A ' ^' ds' dsl~ds ^ \^' •^' ds' ds) ' / V [»v] ds' A ' •^' ds' dsl~ ds I ' ■^' ds ds/-) It requires no further explanation to show how from these relations one can express the third and higher derivatives of x and y with respect to 5 in terms of x, y, -r and -^ . 95. If, then, /\ nowhere vanishes, and like /^ is a continuous 88 CALCULUS OF VARIATIONS. function of its arguments, and if H never becomes infinite (see Art. 149), it follows that — — r and —4 can never become infinitely large, and are also continuous functions of the arc. It follows that the curve has no singular point within the in- terval in question and that the curvature nowhere becomes infi- nitely large. This may be shown in the following manner : Let the points x^^, y^ of the curve correspond to the value Sa of 5^ then owing to the equations [«] of the preceding article, the curve in the neighborhood of this point may be represented by the equa- tions x=x^-^A {s—SoY -f , y=y^+B{s-s^y + ...., where the constants A and B do not vanish simultaneously. When the values of -^ and -4- derived from these equations are ds ds substituted in it follows that l=,i.\A^-^B^) {s-s.y^^ j.A.is-s.y^' + ...., and since this equation is true for all points in the neighborhood of Xa, y^, it is seen that A^=A^=....^o, and that further |Li=l, A^'+B^^l. Hence the coordinates of every point of the curve situated in the neighborhood of x^^, y^ may be represented through the regular functions x=x^+A{s—s<,)^ , y=y^^B{s-Sa)+.... , where A and B do not simultaneously vanish. Since this is true for every point x^, yo, it follows that the curve can have no singu- lar points. Hence also the quantities x' a.ndy' can not both vanish at the same time. CALCULUS OF VARIATIONS. 89 CHAPTER VII. REMOVAL OP CERTAIN LIMITATIONS THAT HAVE BEEN MADE. INTEGRATION OF THE DIFFERENTIAL EQUATION 6^=0 FOR THE PROBLEMS OF CHAPTER I. 96. In the derivation of the formulae of Chapter V, it was presupposed that the portion of curve under consideration changed its direction in a continuous manner throughout its whole trace ; that is, ;»:', y' varied in a continuous manner. We shall now assume only that the curve is composed of regular portions of curve ; so that, there- fore, the tangent need not vary continu- ously at every point of the curve. Then it may be shown as follows that each por- tion of curve must satisfy the differential equation G=o. For if the curve consists of two regular portions A C and CB, then among all possible variations of AB there exist those in which CB re- mains unchanged^ and only AC is subjected to variation. As above, we conclude that this portion of curve must satisfy the differential equation G—o. The same is true of CB. We may now do away with the restriction that the curve consists of one regular trace, and assume that it consists of a finite number of regular traces. 97. Suppose that the function F does not contain explicitly the variable x, and consequently -^=^o. Instead of the equation G=o, let us take Gi=o, or dF ddF ^ — — ——. = o. ox at ax 90 CALCULUS OP VARIATIONS. It follows that ^ — J = constant, ox the constant being independent of // a priori, however, we do not know that -^.^-r does not undergo a sudden change at points of ax , discontinuity of x' and y' . Consequently, the more important is the following theorem for the integration of the differential equa- tion G^o : Even if x', y', and thereby also the direction of the curve, suffer at certain points sudden changes, nevertheless, the quan- tities ^ — ,, ^ — 7 vary in a continuous manner throughout the ox ay whole curve for which G=o. If /' is a point of discontinuity in the curve, then on both sides of /' we take the points t and t in such a manner that within the portions t. . . . /' and t' . . . .t there is no other discontinuity in the direc- tion of the curve. Then a possible varia- tion of the curve is also the one by which /q. . . .T and t' .... 4 remain unaltered and */ \b only the portion t .... t' is varied. Here the points r and / are supposed to remain fixed, while /' is sub- jected to any kind of sliding. The variation of the integral I^ §F{x, y,x', y')dt, 4 then depends only upon the variations of the sum of the integrals f r' ^F{x,y, x', y ) dt +/i^ {x,y, x' , / ) dt. T /' CALCULUS OF VARIATIONS. 91 Since the first variation of this expression must vanish, we neces- sarily have (Art. 79) t' 0=^ JG{y' i-x' ■n)dt + JG (y ^-x' v) dt T /' Since G=o along the whole curve, it follows that [dF .dF -^^' , [dF > , dF y' The quantities ^ and tj are both zero at the fixed points t and t' ; and, if we denote the values that may belong to the quantity ^4" ' dx according as we approach the point t' from the points r or t' by m] - mi the above expression becomes where {f)' and (17)' are the values of f and tj at the point t' . Since the quantities (f)' and (17)' are quite arbitrary, it follows that their coefficients in the above expression must respectively vanish, so that that is, the quantities -^r-r and ^^—jvary in a continuous manner ox oy by the transition from one regular part of the curve to the other, even if x' and y' at this point suffer sudden changes. 92 CALCULUS OF VARIATIONS. This is a new necessary condition for the existence of a max- imum or a minimum of the integral I, which does not depend upon the nature of the differential equation G — o. 98. The question naturally arises : How is it possible that the functions -^^-j, -;^—f, which depend upon x' and y' , vary in ox ay a continuous manner, even when x' and y' experience discon- tinuities? To answer this question we may say that the compo- sition of these functions is of a peculiar nature, viz., the terms which contain x\ y' are multiplied by functions which vanish at the points considered. This is illustrated more clearly in the example treated in Art. 100. The theorem is of the greatest importance in the determination of the constant. In the special dp case of the preceding article, where 3— ^ = constant^ it is clear ox that this constant must have the same value for all points of the curve. The theorem may also be used in many cases to prove that the direction of the curve nowhere changes in a discontinuous manner, and consequently does not consist of several regular por- tions but of one single regular trace. This is also illustrated in the examples which follow (Arts. 100 et seq.). 99. We may give here a summary of what has been obtained through the vanishing of the first variation as necessary conditions for the existence of a maximum or a minimum of the integral I: 1) The curve offering the maximum or minimum must satisfy the differential equation ^_ d'^F d'F ^ j,l ,dy' ,dx'\ ^ = d^'-dy^'^^\''dt-^^r'^ or, what is the same thing, the two equations aF dldF\_^ ^—9^ d ldF\ ^'=dx dt\dx']~^' ^^=^~di\dyi' 2) The two derivatives of the function F with respect to x! and y' must vary in a continuous m,anner even at the points where the direction of the curve does not vary continuously. CALCULUS OF VARIATIONS. 93 In order to establish the criteria by means of which it may be ascertained whether the curve determined through the equation G = o offers a maximum or a minimum, we must investigate the terms of the second dimension in A / of Chapter V. First, how- ever, to make clear what has already been written, we may apply our deductions to some of the problems already proposed. SOLUTION OF THE DIFFERENTIAL EQUATION G=^0 FOR THE PROBLEMS OF CHAPTER I. 100. Let us consider Problem I of Art. 7. The integral which we have to minimize is yVx'^ + y'^dt. [1] Hence F = yVx'^+y\ [2] and consequently dF _ yx' . dF _ yy dx' y'x'^ + y'^' Sy Vx'^' + y'^' [3] From this it is seen that ^^-y and ^^—^ are proportional to the direc- ax ay tion cosines of the tangent to the curve at any point xit), y{t); and, since -^ and -^r-r must vary everywhere in a continuous man- ax ay ner, it follows also that the direction of the curve varies every- where in a continuous manner except for the case where y — o. But the quantity Vx''^-{- y''^ varies in a discontinuous manner if x' and y' are discontinuous ; at the same time, however, y is equal to zero, as is more clearly seen in the figure below. Since F does not contain x explicitly, we may use the equation G, = o, or 1^ = -^=:=^ [4] ox. y'x^ -I- y'^ where /8 is the constant of integration. Hence 94 CALCULUS OF VARIATIONS. The solution of this equation is the catenary : -.), } [5] [6] where a is a second arbitrary constant. 101. A discontinuous solution. If we take the arc 5 as independent variable instead of the variable t, the differential equation of the curve is dx o Suppose that ^=o, which value it must retain within the whole interval t^. . . .t^. Further, since y^o at the point /i. it dx follows that ---=cos^=Y-axis. Further investi- gation is necessary to determine when this curve offers in reality a minimum. 103. In the second problem (Art. 9)^ we have for the time of falling the integral J V^gy+a* to That this expression may, in reality, express the time of fall- ing (the time and, therefore, also the increment dt being essentially a positive quantity), the two roots that appear und er the i nte- gral sign must always have the same sign. Since VAgy + a^ can 96 CALCULUS OF VARIATIONS. always be chosen positive, it follows that Vx^+y'^ must be posi- tive within the interval 4 • • • • A- It might happen, however, if we express x and _y in terms of /, that x' and y might both vanish for a value of t within the inter- val 4 • • • • 4- In this case the curve has at the point x, jy, which belongs to this value of /, a singular point, at which the velocity of the moving point is zero. Suppose that this is the case for /=/, and that the corre- sponding point is Xq, jVo) so that we have x=Xo+a(t—t' )'"+...., where m^2, and at least one of the two quantities a and b is dif- ferent from zero. Then is and Vx'^+y^ ^m V a^ + b" {t-t'Y-^ + Here we may suppose V a^ -^ b^ positive. If now m is odd, then for small values of t—t', the expression on the right is positive, and hence V x''^ + jv'^ always has a positive sign. If on the contrary m is even, equal to 2, say, t hen the curve has at the point x^, jKo a, cusp, since here V x''^ + y^ has a positive or a negative value according as f^ t' or t>' ^ "- -^ From this we may conclude, in a similar manner as in the first example, that ^5— r. -^^-r are proportional to the direction cosines ox ay of the tangent of the curve at the point x, y. Since now -^^-y , ^5 — -, ox o y vary in a continuous manner along the whole curve, and since, fur- ther, V \gy Ar ^ has a definite value which is different from zero, it follows also that the direction of the required curve varies in a continuous manner, or the curve must consist of one single trace. Also here F is independent of x, and consequently we employ the differential equation Gi^o, from which we have ZF 1 x' C, [4] dx' VAgy + a? Vx'^ + y'^ where C is an arbitrary constant. If C is equal to zero, then in the whole extent o f the curve C must equal zero ; and consequently, since VA-gy+ a^ is neither o nor 00, — ^ = cos a must always equal zero; that is, the Vx'^ + y'^ ■ curve must be a vertical line. Neglecting this self-evident case, C must have a definite value which is always the same for the whole curve and different from zero. From [4] , it follows that dx" = C'iig^' + ol') (dx" + df), or, if we absorb \g in the arbitrary constant and write ^ = a, and 4^C^=c^, we have , , ^ \:—Z:l, dx'=c'{y+a){dx'+dy^); whence ,r, c{y+a)dy [-5-I 98 CALCULUS OF VARIATIONS. In order to perform this last integration, write ff "^y [6] V{_y^a) \X-c\y^a)-\ therefore dx = {y^a)di>. [5"] In the expression for d^, write 2c\y+a) = \-t [7] Then is 2[l_c^(^+a)] = l + ^, [8] and 2c^dy=^-dl [9] Therefore and hence [10] [11] Here the constant of integration may be omitted, since ^ itself is fully arbitrary. Hence, ^+a = — (1-cosf), and, from [5"], > [12] ^+ ^0= 2^ (^— sin <^); equations, which represent a cycloid. The constants of integration x^j, c are determined from the condition that the curve is to go through the two points A and B. Now develop x andy in powers of : then injv the lowest power is ^, and in ;«; it is <^; so that the curve has in reality a cusp for =o, and this is repeated for =2'n-, Att, A and B must lie between two consecutive cusps (Art. 104). The curve may be constructed, if we draw a horizontal line through the point —x^, —a, and construct on the under side of this line a circle with radius l/(2c^), which touches the horizontal line at the point —Xo, —a. I^et this circle roll in the positive CALCULUS OF VARIATIONS. 99 X-direction on the horizontal line, then the original point of contact describes a cycloid which goes through A and B and which satis- fies the differential equation. 104. That the points A and B cannot lie upon different loops of a cycloid may be seen as follows : For simplicity, let the initial velocity a be zero and shift the origin of coordinates so as to get rid of the constants. The equation of the cycloid is then x=r{4* — sin \) where we have written r in the place of 1/(2 c^). The cycloidal arc is seen from the accompanying figure. Take j{ two points lying upon dif- ferent loops very near and symmetrically situated with respect to an apex, and let us compare the time it would take to travel from one of these points to the other by the way of the apex with the time taken over a straight line joining them. The parameters of the two points may be expressed by d, ds=Vdx'+ dy =2r sin(<^/2) d, 100 CALCULUS OF VARIATIONS, and consequently ^ g ^ g s The component of velocity across the horizontal line from <^o to <^i is fz;^ 1 or, since ^ = sin ^ and v^=^2gy, this component is [_ ds J as 2 equal to [V2gr Vl-cos sin| p = 2y'^ sin^^. The length of the line to be traversed from <^o to <^i is Xi—Xo=r[i—o—sm <^i + sin «^oJ=2r[»|»o— sin i/»o]. Hence, the time required is ^ ^ 2 r (xlfQ—sm ^a) ' 2i/7^sinX«/'o/2) ' and consequently 7\ 2 r (\lio—sin xftp) ^ «/>o — sin i/ — sin 4>), y = r\l — cos (^), which are the coordinates of a new cycloid. The latter cycloid is similar to the first, since the transforma- tion moves the ordinate A'C and the chord A'B' parallel to them- selves. Their transformed lengths are respectively — A'C'=AC and ^A'B'^AB, r r giving us a cycloid with the requisite lengths for the ordinate ^C and the chord AB. Further, there is but one cycloid which answers the required conditions. For, if we already had A'B'=AB and A'C'=:AC, the only value of r' which could then make — A'B' = AB is r' = r. r Hence through the two points A and B there can be constructed one and only one cycloid-loop with respect to the X-axis* 106. Problem III. Problem of the shortest line on a surface. This problem cannot in general be solved, since the variables in the differential equation cannot be separated and the integration cannot be performed. Only in a few instances has one succeeded in carrying out the integration and thus represented the curve which satisfies the differential equation. This, for example, has been done in the case of the plane, the sphere and all the other surfaces of the second degree. As a simple example, we will take the problem of the shortest line between two points on the surface of a sphere. The radius of the sphere is put equal to 1, and the equation of the sphere is given in the form * This proof is due to Prof. Schwarz. CALCULUS OF VARIATIONS. 103 Now writing : a;=cos «, \ y=smu cosz',> [1] ^=sinu sin 2;,) then «= constant and z'= constant are the equations of the parallel circles and of the meridians respectively. The element of arc is ds = Vdu^ + sin^w diP', [2] and consequently the integral which is to be made a minimum is k L=jVu'^+ v'^ sin^ u dt ; [3] to so that here we have and F= Vu :'^ + Z''^sin%, dF u' du' Vu'^JrV'^^ATL^U dF v'sxv^u [4] [5] dv' Vu"^-^v''^&\vi^u Since F does not contain the quantity v, we will use the equation Gi=o, and have: dF _ v'sin^u _ dv Vu'^+v'^sin^u where c is an arbitrary constant, which has the same value along the whole curve. If for the initial point A of the curve u^o, and consequently, therefore^ not the north pole of the sphere, then c will be every- where equal to zero, only if v'=o. We must therefore have v constant. It follows as a solution of the problem that A and B must lie on the same meridian. 104 CALCULUS OF VARIATIONS. If this is not the case, then always c^o. It is easy to see that c < 1 ; we may therefore write sin c instead of c, and have v' sin^ u • r^T = sin c, [oj or If we write then is Vu'^ + sin^uv'^ dv= ^"^ ^cdu 1-7-j ^vo.u i/sin^?^— sin^c cos «<=:cos c cos Zf, [8] r sin c dzu 1 — cos^ c cos^ w since 1 may be replaced by sin^ zy-|-cos^ w, "we have dw -, tan w sm c — d ■ r sin c dw cos^ w sin c sin^ w + cos^ w sin^ c sin^ c + tan^ w ^ tan^ w sin^c Therefore v-^ =. tan-(^^^), V sm c / where /3 represents an arbitrary constant. It follows that tan(z;-^) = :tan_Z6; ^^-j sm c Eliminating Zf by means of [8] , we have tan u cos (z^— /8)— tan c. [10] This is the equation of the curve which we are seeking, expressed in the spherical coordinates «, v. In order to study their meaning more closely, we may express u, V separately through the arc s, where s is measured from the intersection of the zero meridian with the shortest line. Through [7] the expression [2] goes into sin u du ds: i^sin^w— sin^c CALCULUS OF VARIATIONS. 105 and this, owing to the substitution [8], becomes ds=^ dw, and, therefore, if ^ is a new constant, s—b=w. [11] Hence, from equations [8] and [9] we have the following equations : cos «==cosc cos (5— ^), \ cot (z;— j8)=:sin c cot {s—b).) But these are relations which exist among the sides and the angles of a right-angled spherical triangle. If we consider the meridian drawn from the north pole^ which cuts at right angles the curve we are seeking, then this meridian forms with the curve, and any other meridian, a triangle^ to which the above relations may be applied. Therefore, the curve which satisfies the differential equation must itself be the arc of a great circle. The constants of integra- tion c, b, /S are determined from the conditions that the curve is to pass through the two points A and B. The geometrical interpretation is: that c is the length of the geodetic normal from the point u=o to the shortest line ; s—b. 106 CALCULUS OF VARIATIONS. the arc from the foot of this normal to any point of the curve, that is, the difference of length between the end-points of this arc ; and v—fi, the angle opposite this arc. If we therefore assume that the zero meridian passes through A, then b is the length of arc of the shortest line from A to the normal, and 13 the geographical longitude of the foot of this normal. 107. We may derive the same results by considering the dif- ferential equation G=o. Since we have substituted in ^1 sin^ « This value, -(^ ,'^ + v'^sm^uy/''' causes this expression to become — [2 cos u u'^ z/'+sin^ u cos u v'^] +sin «*(z'' u"—u' v")=o, or In this equation write 1) zi'=sin« du dh) cos « + sin «^— -- = o. dtr dv and we have or 2) CALCULUS OP VARIATIONS. 107 cot u(zu + zi/) + ^^ z=0, du 5 4- cot u du—0. Integrating the last equation, it follows that and consequently^ 3) w^sin2«=Cni + «^'). Suppose that A is the north pole of the sphere, u the angular distance measured from A along the arc of a great circle, and v the angle which the plane of this great circle makes with the plane of a great circle through the point B. Hence for all curves of the family G—O that pass through A, we must have C=o, since sin u=o for u—o. It follows also that zu=o, and consequently, dv sin u — — = 0, du or 2^= constant. Hence, as above, A and B must lie on the arc of a great circle. Next, if A is not taken as the pole, then always C^o, and is less than unity. It follows then at once from equations 1) and 3) that ds^=du^ + sin^ u dip-^du^ [1 + w^] , or sin u du ds-- l/sin^ «— sin^ C (where we have written sin^ C for C^), and sin C du dv = sin u i/sin^ «— sin^ C 108 CALCULUS OF VARIATIONS. Writing cos «=cos C cos t, these two equations when integrated become, as in the last article, cos «^=cos C cos {s—6), cot (z*— /8)=sin Ccot(s— 5). 108. Problem IV. Surface of rotation which offers the least resistance. To solve this problem we saw (Art 12) that the integral to must be a minimum. We have here F x'^+y^' and we see that i^ is a rational function of the arguments x' and y. For such functions Weierstrass has shown that there can never be a maximum or a minimum value of the integral. But leaving the general problem for a later discussion (Art. 173), we shall confine our attention to the problem before us. We may determine the function Fi from the relation d^F , ,j^ dx'dy It is seen that „ _ 2xx'{Zy^-x'^') '" {x'^^y^r ' We may take x positive, and also confine our attention to a portion of curve along which x increases with t, so that ;*;' is also positive. Consequently F^ has the same sign as "iy^—x'^, or of 3 sin^ X — cos^ X, where X is the angle that the tangent to the curve at the point in question makes with the ^-axis. Fi is therefore positive, if | tan X|>- — = , V3 and is negative, if | tan X | <; — = , for the portion of curve considered. CALCULUS OP VARIATIONS. 109 We shall see later (Art. 117) that F^ must have a positive sign in order that the integral be a minimum. Hence, for the present problem, | tan X | must be greater than — = for the portion 1/3 of curve considered ; and as this must be true for all points of the curve at which x' has a positive sign, the tangent at any of these points cannot make an angle greater than 30° with the X^-axis (see Todhunter, Researches in the Calculus of Variations^ p. 168). 109. We shall next consider the difEerential equation G=o of the problem. Since F does not contain explicitly the variable y, we may best employ the equation ,^ ^ dF d dF We have at once dy or 7 = constant, 2xx^^y' {x'^^y'^y = c. Now, if there is any portion of the surface offering resistance, which lies indefinitely near the axis of rotation, then the constant must be zero, since x—o makes C=o. If C=o, we have x'^ y'=^o, and consequently x'=o or y'=o. From this we derive ;i;=const. or ^=:Const. In the first case, the surface would be a cylinder of indefinite length, with the K-axis as the axis of rotation, and with an indefi- nitely small radius (since by hypothesis a portion of the surface lies indefinitely near the K-axis); in the second case, the resisting- surface would be a disc of indefinitely large diameter. These solutions being without significance may be neglected, and we 110 CALCULUS OF VARIATIONS. may^ therefore, suppose that the surface offering resistance has no points in the neighborhood of the F-axis. This disproves the notion once held that the body was egg-shaped. 110. We consider next the differential equation where C is different from zero. We may take x positive, and as the constant C must always retain the same sign (Art. 97), it follows that the product x'y' cannot change sign. Instead of retaining the variable t, let us write t = -y, and dx _ dy The differential equation is then —2x1^ («'+!)' = C. That we may write —y'va. the place of /, is seen from the fact that x! .y' cannot change sign, and consequently either x is continuously increasing with increasing y, or is continuously decreasing when y is increased. Hence, corresponding to a given value of y there is one value of x. We have then and or dx _ dx_ dy_ _ dy_ ^(■\_'2 -2 -x -^\ du dy du du 2 consequently ^ 2 I log« + u-^ + i/iu~*\ + Ci. CALCULUS OP VARIATIONS. Ill The equations C x = —— {u + 2u-^+u-^), y c (log «+«^-2+^ «-^) +Ci, determine a family of curves, one of which is the arc, which gen- erates the surface of revolution that gives a minimum value, if such a minimum exists. For such a curve we have all the real points if we^give to u all real values from o to +00. Among these values is i/3, and as we saw above, it is necessary that — ^ = — > — = continuously, ax u 1/3 or — < — = continuously. In other words, if the acute angle which the tangent at any point of the arc makes with the ^-axis is less than 30", it must continue less than 30° for the other points of the arc, and if it is greater than 30" for any point of the arc^ it must remain greater than 30° for all points of the arc. Hence, if P is the point at which the inclination of the tan- gent with the ^-axis is 30°, we shall have on one side of P that portion of curve for which the inclination is less than 30°, and on the other side the portion of curve for which the inclination is greater than 30°. The arc in question must belong entirely to one of the two portions. 112 CALCULUS OP VARIATIONS. CHAPTER VIII. THE SECOND VARIATION ; ITS SIGN DETERMINED BY THAT OF THE FUNCTION F^. 111. The substitution x-\-i.^, y-\-irf for x,y causes any point of the original curve to move along a straight line, which makes an angle with the ^-axis whose tangent is -^. This deformation of the curve is insufiEcient, if we require that the point move along a curve other than a straight line. To avoid this inadequacy we make the more general substi- tution (by which the regular curve remains regular) : X y where, like i, rj in our previous development (Art. 75), the quan- tities fi, 17,, ^2' '?2. ■ • • • are functions of t, finite, continuous^ one- valued and capable of being differentiated (as far as necessary) between the limits 4. . . . /j. These series are supposed to be con- vergent for values of c such that |c|?' in the integration of the integral would have been assumed. These assumptions need not neces- sarily be fulfilled, since the variation of the curve is an arbitrary one, and it is quite possible that such variations may be intro- duced, where ^', rj' become discontinuous, as often as we please. We may, however, drop these assumptions without changing the final results, if only the first named conditions are satisfied. Since the quantities Z, M, N depend only upon x, y, x', y', x", y", and since these quantities are continuous, it follows that the introduc- tion of the integral X^dtin the form given above is always 122 CALCULUS OP VARIATIONS. admissible. For if i', ■>?' were not continuous for the whole trace of the curve, which has been subjected to variation, we could sup- pose that this curve has been divided into parts, within which the above derivatives varied in a continuous manner, and the integral would then become a sum of integrals of the form where /o, /o^^, .... are the coordinates of the points of division of corresponding values of /. But since |, rj vary in a continuous manner, we have through the summation of these quantities exactly the same expression \\Le+2M^r,+Nv'yi as before. The quantities $', 17' are also found under the sign of integration in the right-hand side of 8); but owing to the concep- tion of a definite integral, we may still write it in this form^ even when these quantities vary in a discontinuous manner ; however, in performing the integration, we must divide the integral corre- sponding to the positions at which the discontinuities enter into partial integrals. Therefore, we see that the possible discontinuity of $', T)' remains without influence upon the result, if only x, y, ;»' y\ x", y", i, t) are continuous. Consequently any assumptions regarding the continuity of f ', ij' are superfluous ; however, in an arbitrarily small portion of the curve which is subjected to varia- tion, the quantities i' and 17' must not become discontinuous an infinite number of times^ since such variation of the curve has been^ necessarily, once for all excluded. 117. Following the older mathematicians, Legendre, Jacobi, etc., we may give the second variation a form in which all terms appearing under the sign of integration will have the same sign (plus or minus). To accomplish this, we add an exact differential -^ (w^z*) at CALCULUS OF VARIATIONS. 123 under the integral sign in 8), and subtract it from R, the integral thus becoming to The expression under the sign of integration is an integral homogeneous quadratic form in w and —^. We choose the quan- dt tity V so that this expression becomes a perfect square ; that is, 9) e.-^. (/-. + !) = ., and consequently, 10) 8=/=j'/-.(^ + .^)'^.+ [^^..>]\ /< We shall see that it is possible to determine a function v, which is finite^ one-valued and continuous within the interval t^. . . .t^, and which satisfies the equation 9). The integral 10) becomes accordingly, if the end-points remain fixed, 10.) 8>/=jV.(^+.|.J* A ■0 Hence the second variation has the same sign as /\, and it is clear that /or the existence of a maximum F^ m-ust be negative, and for a minimum^ this function must be positive within the in- terval 4. . • • A. ^«^ i'n case there is a m^axim-um or a m,inim.um, F^ cannot change sign within this interval. This condition is due to Jacobi. Legendre had previously concluded that we have a maximum when a certain expression cor- responding to /\ was negative, and a minimum when it was posi- 124 CALCULUS OF VARIATIONS. tive. It is questionable whether the diflferential equation for v is always integrable. Following Jacobi we shall show that such is the case. 118. Before we go farther, we have yet to prove that the transformation, which we have introduced, is allowable. In spite of the simplicity of the equation 9) we cannot make conclusions regarding the continuity of the function v, which is necessary for the above transformation. • It is therefore essential to show that the equation 9) may be reduced to a system of two linear differ- ential equations, which may be reverted into a linear differential equation of the second order, since for this equation we have definite criteria of determining whether a function which satisfies it remains finite and continuous or not. Write U where u^ and u are continuous functions of t, and w^^o within the interval 4. . . . /j. Equation 9) becomes then or Since one of the functions u, u^ may be arbitrarily chosen, we take u so that 11) F,^^-u, = o, then, since u^o, we have 12) ^ + ^'- = o. CALCULUS OF VARIATIONS. 125 From 11) and 12) it follows that or where i^i and F^ are to be considered as given functions of t. We shall denote this difFerential equation by / = o. After u has been determined from this Equation, «i may be determined from 11), and from -^ = z; we have z' as a definite function of t. u 119. The expression which has been derived for v seems to contain two arbitrary constants, while the equation 9) has only one. The two constants in the first case, however, may be replaced by one, since the general solution of 13) is and hence from 11) an expression which depends only upon the ratio of the two constants. It follows from the above transformation that 14 ) SV == (f, (^ - i5^ ^Ydt ; ^ J ^\dt u dt) ' to but this transformation has a meaning only when it is possible to find a function u within the interval 4- • • • A) which is different from zero, and which satisfies the differential equation J=o. 120. If we have a linear differential equation of the second order 126 CALCULUS OP VARIATIONS. and if _j'i and y^ are a fundamental system of integrals of this equation, then we have the well known relation due to Abel (see Forsyth's Differential Equations, p. 99) or y^^ yz dy,^ dy^ dx ' dx = Ce -/■p(^) ( If A =0, then we would have y^= cy^, and the system is no longer a fundamental system of integrals. This determinant can become zero only at such positions for which P(^x) becomes infinitely large; or a change of sign for this determinant can enter only at such positions where P{x^ becomes infinite. In the differential equationy = o we have Z*— — (log F^), and if 2^1, «j form a fundamental system of integrals of this differen- tial equation, then du2 dui _ C ~dt ~ '^^It ~ T^' A = Ui -j^ — «2 ■ It follows that Fi cannot become infinite or zero within the interval under consideration or upon the boundaries of this inter- val. Hence, it is again seen that Fi cannot change sign within the interval 4- • • • A- If Fi and F2 are continuous within the interval 4 • • • • Ai "w^e have, through differentiating the equation J—O, all higher deriva- tives of u expressed in terms of u and — — . Hence, if values of u dt du and — — - are given for a definite value of t, say /', we have a dt power-series P {t—t') for u (see Art. 79), which satisfies the equation J=^ o. 121. Suppose that F-^^ has a definite, positive or negative value for a definite value t' oi t situated within the interval 4 ti, then on account of its continuity it will also be positive CALCULUS OF VARIATIONS. 127 or negative for a certain neighborhood of /', say t'—r^. . . .Z'+tj. We may vary the curve in such a manner that within the interval t'—T^. . . . Z'+Tj it takes any form^ while without this region it re- mains unchanged. Consequently the total variation, and therefore also the second variation of /, depends only upon the variation within the region just mentioned, and in accordance with the remarks made above^ since we may find a function u of the variable t, which is contin- uous within the given region, which satisfies the differential equa- tion J—o, and which is of such a nature that u and —r- have given at values for /= f, it follows that the transformation which was in- troduced is admissible, and we have A ^''-/^■{w-wlf^'- 4 This quantity is evidently positive when F^ is positive^ and negative when Fi is negative, so long as idw_duw\^^ (Art. 132). {dt at u) We have then for the total variation where {t V^ ^'. vli denotes an expression of the third dimension in the quantities included within the brackets. For small values of e it is seen that A/ has the same sign as the first term on the right-hand side of the above equation. We have, therefore, the following theorem : The total variation Mo/ the integral I is positive when Fi is positive, and negative when F^ is negative throughout the whole interval 4. ... 4- 128 CALCULUS OP VARIATIONS. If F^ could change sign for any position within the interval 4 .... 4. then there would be variations of the curve for which A / is positive^ and others for which A / is negative. Hence, for the existence of a maximum or a minimum of / we have the following necessary condition : In order that there exist a maximum or a minimum of the integral I taken over the curve G^^o within the interval to. . . .ti, it is necessary that F^ have always the same sign within this interval; in the case of a maxim,um F^ must be continuously negative, and in the case of a minimum this function must be continuously positive. In this connection it is interesting to note a paper by Prof. W. F. Osgood in the Transactions of the American Mathematical Society, Vol. II, p. 273, entitled: "(9« a fundamental property of a minimum in the Calculus of Variations and the proof of a theorem of Weierstrass' s.'" This paper, which is of great importance, may be much simplified. CALCULUS OF VARIATIONS. V2fi CHAPTER IX. CONJUGATE POINTS. 122. The condition given in the preceding Chapter is not sufficient to establish the existence of a maximum or a minimum. Under the assumption that F^ is neither zero nor infinite within the interval 4- • • • A. suppose that two functions <^i(/) and <^j(/) can be found which satisfy the differential equation 13) of the last Chapter, so that, consequently, is the general solution oij—o. Then, even if within the limits of integration it can be shown that u is not infinite, it may still hap- pen that, however the constants c^ and Cj be chosen, the function u vanishes, so that the transformation of the z;-equation into the ^-equation is not admissible ; consequently nothing can be deter- mined regarding the appearance of a maximum or a minimum. We are thus led again to the necessity of studying more closely the function u defined by the equation J^o, in order that we may determine under what conditions this function does not vanish within the interval /« A- It is seen that the equation /= o is satisfied, if for u we write Ui= —Fi u' [see Art. 118, equation 11 )] , and consequently V = —^ = — Fi — u u is a solution of the equation in v. 130 CALCULUS OF VARIATIONS. The integral 10) of the last Chapter may be then written A J \lV U } L M J4 t I From this we see that if — = — . or if zf = Cu, then the second ■w u variation is free from the sign of integration ; in other words, the second variation is free from the integral sign, if we make any deformation (normal [Art. 113, equation 5)] to the curve) such that the displacement is proportional to the value of any integral of the differential equation J — o. Again, if we deform any one of the family of curves G ^^o into a neighboring curve belonging to the family, we have an ex- pression which is also free from the integral sign. For (see Arts. 79 and 81), if we write p = Vx'^ + y'^=~, we have at and consequently, ^F=p^.SG+GHp..)+[^^ 8(f 1^ + ,^)];'. Hence, ii 8G= o, we have here also ^Hi^^-M: It may be shown as follows that the curve SG = o is one of the family of curves G = o. The curves belonging to the family of curves G = are given (Art. 90) by X = 4,{t,a,^), ^ = ,/,(/, a, ^), where a and fi are arbitrary constants. We have a neighboring curve of the family when for a, /3 we write a + e a', /8 4- c /8'. Then the function G becomes G + ^G=G + ^ZG + e{ )+ CALCULUS OF VARIATIONS. 131 Hence, when e is taken very small, it follows that is a solution of SG—o, since it is a solution of G-\-AG=o and of G=o. Now we may always choose normal displacements — which will take us from one of the curves G—o to a neighboring curve 8G=o. From this it appears that there is a relation between the differential equations 8 G=o and J ^^^^o. 123. In this connection a discovery made by Jacobi (Crelle's Journal, bd. 17, p. 68) is of great use. He showed that with the integration of the differential equation G=o, also that of the dif- ferential equation y= o is performed. We are then able to derive the general expression for «, and may determine completely whether and when u^o. We shall next derive the general solu- tion of the equation J=o, it being presupposed that the differen- tial equation G=o admits of a general solution. We derived the first variation in the form hi=^ ^Gwdt+ 1 y\ to We may form the second variation by causing in this expression G alone to vary, and then w alone, and by adding the results. It follows that h-'r=^\hGw+Ghw) dt+\ 1^'. (0 4 Since the differential equation G—o is supposed satisfied, we have t. r -,, hU=^ hGwdt+\ V. {a) 4 132 CALCULUS OF VARIATIONS. We had (Art. 76) r _9^ d{dF\ '~dx dAdx'i' r _ 9^ d{ ZF\ ^~ -dy dAdy'r and also Gi = y' G, G^=— x' G. When in the expression for G-^, the substitutions X x^^t y y + e-r] are made, we have and since 6^, + A(;, = (y + €V)(G^+AG^); it follows that and similarly t^G =€Sc; v'SG + G-q', -x'hG~G^'. • • J 124. When G is eliminated from the last two expressions, we have ^G,^' + hG,'n'={y'r-x'y^')hG. {U) On the other hand, it is seen that dy^ dxdy dxdx' dxdy' d ^ d^F t , ^^Ft> dM J ti Tiyf I d ( IT t dw\ CALCULUS OP VARIATIONS. 133 and if we take into consideration 3), 4)^ 6) and 7) of the last Chapter, we may write the above result in the form : In an analogous manner, we have When these values are substituted in («V), we have Hence from (a) we have By the previous method we found the second variation to be [see formula 8) of the last Chapter] These two expressions should agree as to a constant term. The difference of the integrals is 4 ^ -=/-#i-.^)-'^'-j'--(tr^'^ but since j#i^.^)---»^-s^-J^-(sr*' 134 CALCULUS OF VARIATIONS. it is seen that The formula (6) is BG^^,^-±(^/^). When we compare this with 12") of the preceding Chapter, the differential equation for u, viz.: 7- d { T^ du \ '-^^""-d-A^^wh it is seen that as soon as we find a quantity w for which 8G=o, we have a corresponding integral of the diflEerential equation for u. 125. The total variation of G is ^G=G (x+€^^+ -^^2+ ■ ■ ■ ■^J' + ^Vi+-^Vi+ . ^"+^i^" + -^i."+...,y'+er,,"+^v." +....) - G {x,y, x',y', x",y")^ehG + |-S^G^ + . . . ., where hG, as found in the preceding article, has the value Suppose that the equation G = ois integrable, and let be general expressions which satisfy it, where a, /3 are arbitrary constants of integration. The difEerential equation G — o will be satisfied, if we suppose that a and /3, having arbitrarily fixed values, are increased by two arbitrarily small quantities c8a and e 8 /3 ; that is, the functions CALCULUS OP VARIATIONS. 135 are also solutions of G — o. 126. Now choose the variation of the curve (Art. Ill) in such a way that and, whatever be the values of Sa and 8y8, we determine fi,^2,%,T72, etc., by the relations: > (in) da op I For all values of a and ^ the difEerential equation G = o \% satisfied; hence, the values of fi, tj^, etc., just written, when sub- stituted in AG^ above must make the right-hand side of that equa- tion vanish identically, and consequently also 86^. Hence, the corresponding normal displacement w^y'^^—id'i]^ transforms one of the system of curves 6^==o to another one of the same system. Since Sa and S^S are entirely arbitrary, the coeflEcients of Sa and S/3 must each vanish in the expansion of A (7 above. Owing to (**V) w=;v'^i— ^'•»?i becomes -=(^'a-!-^'l!)^"+(^'i-*'i)^''- Writing this value of w in the equation 86^= o, we have 136 CAI/CUI/US OF VARIATIONS. By equating the coefficients of 8a and Bp respectively to zero, we have the two equations: where, for brevity, we have written 2) It is seen at once that ^i(^) and O^ii) are the solutions of the differential equation If ^\ §-i\F^^\-F.u = o. Hence it is seen that the general solution of the differential equation for u is had from the integrals of the differential equa- tion G^=o, through simple differentiation. 127. We have next to prove that the two solutions ^i (/) and ^2 (^) are independent of each other. In order to make this proof as simple as possible, let x be written for the arbitrary quantity /. Then the expressions x=^{t,a, p), y = \^(^t, a, p,), etc., become x=x,y = y^{x,a,fi\ ax If 6i and O-,, are linearly dependent upon each other, we must have 02 = constant 0^, CALCULUS OF VARIATIONS, 137 from which it follows, at once, that $1 02 — 02 01 = o, where the accents denote differentiation with respect to x ; or, On the other hand, ;>/==«/» (x, a, fi) is the complete solution of the differential equation, which arises out of 0^= —x' G = o, when :*; is written for t; that is, of M'^-w^"- dx but here a and j8 are two arbitrary independent constants, and consequently »/» and i/»' ==: —^ are independent of each other with respect to a and fi, so that the determinant */»! ^2 — "h ^i is different from zero. Consequently 6^ and fi^ are independent of each other, since the contrary assumption stands in contradiction to the result just established. Hence, the general solution of the differential equation /=o, is of the form where Cj and c^ are arbitrary constants. 128. Following the methods of Weierstrass we have just proved the assertion of Jacobi ; since, as soon as we have the com- plete integral of G=o, it is easy to express the complete solution of the differential equation /=o. The constants Cj and C2 may be so determined that « vanishes on a definite position t', which may lie somewhere on the curve before we get to t^. This may be effected by writing c,=-0,{n c,=0m The solution of the equation J=o becomes 3) u=dit') 0it) - 02{n eit) = @u n 138 CALCULUS OF VARIATIONS. It may turn out that @ (/, ^') vanishes for no other value of / ; but it may also happen that there are other positions than t' at which @(/, ^') becomes zero. If t" is the first zero position of ( ^, /') which follows t\ then t" is called the conjugate point to /'. Since /' has been arbitrarily chosen, we may associate with every point of the curve a second point, its conjugate. This being premised, we come to the following theorem, also due to Jacobi : If within the interval 4. . . . /i there are no two points which are conjugate to each other in the above sense, then it is possi- ble so to determine u that it satisfies the differential equation J^=o, and nowhere vanishes within the interval 4- • • ■ ^i- 129. Let the point / = /' be a zero position of the function u = %{^t,t'\ and let t" be a conjugate point to t' , then 0(/, ^') will not again vanish within the interval t' . . . . t". Take in the neighborhood of the point t' a point /' + t, where - T>o, then the point which is f t" conjugate to t' + r can lie only on the other side of t". This may be shown as follows : li u= ®{t, t') is a solution of the equation then is u= @(t,t' + T) a solution of the same equation ; that is, of j-~ d^u , dF, du r~. - since u differs from u only through another choice of the arbi- trary constants Cj and Cj. If r is chosen sujB&ciently small, then (if'+r, /') is different from zero^ and consequently also (/', Z' + t) :^o. CALCULUS OF VARIATIONS. 139 Eliminate /'j from the two equations above, and we have Now write 5) ^^ du - du and the above equation becomes V Fy which, when integrated, is 7\ „. fi?« - du C The constant C in this expression cannot vanish, for, in that case, u = const, u, or ©(/, /') = const. 0(/;, Z' + t). Since, however, ©(/, /') vanishes for t=t', it results from the above that 0(/', /'4-t) = o, which is contrary to the hypothesis, and consequently C cannot vanish. It is further assumed that F^ does not change its sign or become zero within the interval 4- • • • A- If -^i vanishes without a transition from the positive to the negative or vice versa within the stretch /o- • • • A» then in general no further deductions can be drawn, and a special investigation has to be made for each par- ticular case. In the first case, however, v has a finite value, and the equa- tion 7), when divided through by «^ becomes du - du J u 11 u fl> — dt dt__ « _ C IP- " dt ~ F^u^' 'J' U=:^CU ^ ^2 140 CALCULUS OF VARIATIONS, an expression, which, when integrated, is dt t'+r Since the function u does not vanish between t' and t", it follows from the last expression that u cannot vanish between the limits /' + T and t". Accordingly, if there is a point conjugate to t' + r, it cannot lie before t" . If, therefore, we choose a point t'" before t" and as close to it as we wish, then i^ will certainly not vanish within the interval t' A^r . . . . t'". If t' is a point situated immediately before 4, and if we deter- mine the point t" conjugate to /', and choose a point t^ before t" and as near to it as we wish, , 7, — then from the preceding it is " ^ clear that no points conju- gate to each other lie within the interval 4- ■ • -^n the boundaries excluded. We may then, as shown above, find a function u, which satisfies the differential equation y=(? and which vanishes neither on the limits nor within the interval 4- ■ ■ ■ ^i- The transformation of Art. 117 is therefore admissible, and the sign of 8Y depends only upon the sign of F^. 130. We may investigate a little more closely the relation of Art. 120, where du^ dui C In the interval under consideration, boundaries included, we assume that Fi does not become zero or infinite, and consequently retains the same sign. Further, the constant C has always the same value and is different from zero, since u^ and «2 are linearly independent. It follows at once that — ^ cannot be zero at the same time dt that «i is zero; for then C would be zero contrary to our hypothesis. Owing to the form dt\u-2, / 1*2 Fi ' CALCULUS OF VARIATIONS. 141 it is clear that -7-.\ — ) has the same sign as ~ . We may take this sign positive, since otherwise owing to the expression du^ dux _ C 1*-,^r^ = we dt "^'dt F^ would have ;j^ ( — ) Positive. We may assume then that the U\ indices have been placed upon the «'s, so that — ^ is always on the increase with increasing /. u The ratio — ^ will become infinite for the zero values of U-, U'i (see Art. 120). Since this quotient is always increasing with in- creasing values of t, the trace of the corresponding curve must pass through + go, and return again (if it does return) from — oo. Values of t, for which this quotient has the same value, may be called congruent. It is evident, as shown in the accompanying figure, that such values are equi-distant from two values of t, say 4 and t^, which make «2 = o. The abscissae are values of t, and the ordinates are the corresponding values of the ratio 131. To summarize : We have supposed the cases excluded in which F^ is zero along the curve under consideration. If this function were zero at an isolated point of the curve, it would be a limiting case of what we have considered. If it were zero along a 142 CALCULUS OP VARIATIONS. stretch of this curve, we should have to consider variations of the third order, and would have, in general, neither a maximum nor a minimum value unless this variation also vanished, leaving us to investigate variations of the fourth order. We exclude these cases from the present treatment, and suppose also that F-^ and F^ are everywhere finite along our curve (otherwise the expression for the second variation, viz. — ^ {F^w"" + F^w") dt, would have no meaning). We also derived in Art. 124 the variation of G in the form and when this is compared with the differential equation 12-) o = F,u-f^ (^^ w) ^^^^ ^^^- ^^^^' it is seen that if an integral u of the differential equation 12") vanishes for any value of /, the corresponding integral w of the equation hG=o vanishes for the same value of t. In Art. 126 we had w =y^, - x'-n, = hadlt) + 8)8 ^j(/), where the displacement ^i, tjj takes us from a point of the curve G=^o to a point of the curve ZG= o. Consequently the normal displacement zt^u can be zero only at a point where the curves G = and hG = o intersect. At such a point we must have 8a^i(0 + 8/8^2(/)=0. When one of the family of curves G = o has been selected, the two associated constants a and /8 are fixed. These are the constants that occur in ^i(^) and Bj^f). If , further, the curve passes through a fixed point P^, the variable t is determined, and consequently the functions ^i(/) and O-i^t') are definitely determined, so that the CALCULUS OP VARIATIONS. 143 ratio Sa : 8^ is definitely known from the above relation. There may be a second point at which the curves G=o and BG^o inter- sect. This point is the point conjugate to Po (see Art. 128). 132. The geometrical significance of these conjugate points is more fully considered in Chapter XI. Writing the second vari- ation in the form A we see that the possibility of — — -^ = o is when «= Czu. Now w is zero at both of the end-points of the curve, since at these points there is no variation, but u is equal to zero at P^ only when /\ is conjugate to P^. Hence, unless the two curves G = o and hG = o intersect again at P^, u is not equal to zero at P^, and con- sequently \W 11/ In this case, if F^ has a positive sign throughout the interval to. . . .ti, there is a possibility of a minimum value of the inte- gral I, and there is a possibility of a maximum value when Fi has a negative sign throughout this interval. 133. Next, let P^ be conjugate to P^, so that at both of the limits of integration we have u=o=w. We may then take u=w at all other points of the curve, so that consequently A 8V =\F,w^i^^-^ Jdt = o. 4 We cannot then say anything regarding a maximum or a minimum until we have investigated the variations of a higher order.* * It is sometimes possible to establish the existence or the non-existence of a maximum or a minimum by other methods ; for example, the non-existence of a minimum is seen in Case II of Art. 58. In a very instructive paper (Trans, of the Am. Math. Soc, Vol. II, p. 166) Prof. Osgood has shown that there is a minimum in the case of the g-eodesics on an ellipsoid of revolution (due to the fact that the curve must lie on the ellipsoid). Prof. Osgood says (p. 166) that Kneser's Theorem "to the effect that there is not a minimum" is in general true. It seems that each separate case must be examined for itself, and in general nothing can be said regarding a maximum or a minimum. 144 CALCULUS OF VARIATIONS. Next, suppose that a pair of conjugate points are situated between P^ and Pj, and let these points be P' and P". We may then make a displacement of the curve so that w — kw from /{, to P', W — U + kw from P' to P" and w = kw from P" to /\, where >& is an indeterminate constant. The quantity w is sub- jected only to the condition that it must be zero at P^ and /*!, and u must be a solution of the difEerential equation J= o, and is zero at the conjugate points P' and P". The second variation takes the form t' W^k'fiF, w'^ + F2 w") dt t" + f {(F, u'^ + F2u') + 2k (F^ u' w' + F^uw) t' ^k^iF^w'^+F^w'^S dt + k^{Fi w"" + ^2 vJ^) dt. t" In the preceding article we saw (cf. also Art. 117) that t" ^{F^u'^ + F^u^)dt=o, t' and we may therefore write 8^/ in the form 8V =2kfiF^ u' w' -\-F^uw) dt^ k^ M, where i^ is a finite quantity. CALCULUS OF VARIATIOKS. 145 The integral t" J (7^1 u' w' -{- F2U w) dt t' may be written /" t' and since, in virtue of the formula 12") of Art. 118, the expres- sion under this latter integral sign is zero, it follows that m ^2k\ F^u' w\" + k'-M. Further, by hypothesis, F^ retains the same sign within the interval t' . . . . t", and does not become zero within or at these limits, the function u' is different from zero at the limits (Arts. 130 and 152), and of opposite sign at these limits, since u, always retaining the same sign, leaves the value zero at one limit and approaches it at the other limit. Consequently \^Fi «/'] is finite and of opposite signs at the two points P' and P", and it remains only that w be chosen finite and with the same sign, so that /^l «' zf be different from zero. Hence by the proper choice of k we may effect displacements for which 8V is positive, and also those for which it is negative. Hence when our interval includes {not, however, both as extremities) a pair of conjugate points, we have definitely estab- lished that the curve in question can give rise to neither a maxi- mum nor a m^inimum. The above semi-geometrical proof is due to a note given by Prof. Schwarz at Berlin (1898-99); see also Lefon V of a course of Lectures given by Prof.Picard at Paris (1899-1900) on ''Equa- tions aux dirivies partielles." 146 CALCULUS OF VARIATIONS. CHAPTER X. THE CRITERIA THAT HAVE BEEN DERIVED UNDER THE ASSUMP- TION OF CERTAIN SPECIAL VARIATIONS ARE ALSO SUFFICIENT FOR THE ESTABLISHMENT OF THE FORMULA HITHERTO EMPLOYED. 134. The methods which we have followed would indicate that the whole process of the Calculus of Variations is a process of progressive exclusion. We first exclude curves for which G is different from zero and limit ourselves to curves which satisfy the differential equation G =^o. From these latter curves we exclude all those along which F-^ does not retain the same sign. If, for any curves not yet excluded, Fi—O at isolated points, we have sim- ply a limiting case among those to which our conclusions apply. li Fi = o for a stretch of curve not excluded by the above condi- tion, we have to subject the curve to additional consideration in which the third and higher variations must be investigated. We further exclude all curves, in which conjugate points are found sit- uated between the limits of integration, as being impossible gen- erators of a maximum or a minimum. The cases in which no such pairs of points are to be found, or where such points are the limits of integration, require further investigation. This leads us to a fourth condition, a condition due to Weierstrass, which is dis- cussed in Chapter XII. In this process of exclusion let us next see whether the variations admitted are sufficient for the general treatment under consideration. 135. As necessary conditions for the appearance of a maxi- mum or a minimum, the following theorems have been established : CALCULUS OF VARIATIONS. 147 1) x,y as functions of t must be determined in such a ■manner that they satisfy the differential equation G=o. 2) Along the curve that has been so determ^ined the func- tion Fy cannot be positive for a maximum nor can it be negative for a minimum; m^oreover, the case that F^ = o at isolated points or along a certain stretch, cannot in its generality be treated, but the problems that thus arise m,ust be subjected to special investigation. 3) The integration may extend at most from a point to its conjugate point, but not beyond this point. The last two conditions, which were derived from the con- sideration of the second variation, require certain limitations. On the one hand, a proof has to be established that the sign of A/ is in reality the same as the sign of SV, if we choose for |, 17, etc., the most general variations of all those special variations, for which the developments hitherto made were true ; it then remains to investigate whether and how far the criteria which have been established remain true for the case where the curve varies quite arbitrarily. 136. We return to the proof of the theorem proposed in the preceding article. We have, in the case of the investigations hitherto made, always assumed that ^, tj, f ', 17' were sufficiently small quantities, since only under this assumption can we develop the right-hand side of A 4 in powers of these quantities. This means not only that the curve which has been subjected to variation must lie indefinitely near the original curve, but also that the direction of the two curves can differ only a little from each other at corresponding points. We retain the same assumptions, and limit ourselves always to special variations. 148 CALCULUS OF VARIATIONS. We shall first prove that for all these variations the sign of A/ and that of S^/ agree, so that for these variations the criteria already found are also sufficient. However, we no longer assume that the variations are expressible in the form e^, etj, where e denotes a sufficiently small quantity. Since the curvature of the original curve does not become infinitely large at any point (see Art. 95), and since further the original curve and the curve w^hich has been subjected to variation deviate only a little from each other at corresponding points both in their position and the direction of their tangents, it follows that with each point of the original curve is associated the point of the curve that has been varied, in which the latter curve is cut by the normal drawn through the point on the first curve. The equation of the normal at the point x, y is {X-x^x!^ (y-y)y = o; and from the remarks just made, the point x -\- ^, y -\- r] is to lie on this normal^ so that ^x' + r}y=^o. If we consider this equation in connection with the defini- tion of w : w — iy — 7) x', it follows that the variations may be represented in the form ^^ -\- y^ x^ + y^ In these expressions —j^ ^ is an indefinitely small quantity, since x' and y cannot both vanish at the same time (Art. 95), and it varies in a continuous manner with A L?', whose coefl&cients also contain these quantities and in such a way that they become indefinitely small with these quantities. Next, writing in A/ the values of ^, -q given in 1) and the fol- lowing values of $', -q also derived from 1): ^ - x'^^ y'^ dt ^ dt V;c'^ + yr ^^ '- _^__dvj ., d I x! \ "f-- x'^ + y'^ dt dt \x'' + yV' 150 CALCULUS OP VARIATIONS, we have 4) ^I^^{F.(^f)\F.J^dt 4 + 4 where /, g, h denote functions which still contain w and — |^, and in such a way that they become indefinitely small at the same time as these quantities. 138. After a known theorem* in quadratic forms, \'i may always, through linear substitutions not involving imaginaries, be brought to the form in such a way that at the same time the relation is true, and where ^ and y^ are roots of the quadratic equation in x: Since the coefficients in this equation become simultaneously small with w and -^, t at of this equation. with w and -j-, the same must also be true of /i and /j, the roots *Such substitutions are caUed by Cay ley orthogonal (Crelle, bd. 32, p. 119); see also Euler, Nov. Comm. Petrop., IS, p. 275; 20, p. 217; Cauchy, Exerc. de Math., 4, p. 140; Jacobi, Crelle, bd. 12, p. 7; bd. 30, p. 46; Baltzer, Theoiie und Anwendungen der Determi- nanten, 1881, p. 187; Rodrigves, Liouv. Journ., t. S, p. 405; Hesse, Crelle, bd. 57, p. 175. CALCULUS OF VARIATIONS. 151 If / is the mean value between f^ and /z, which also becomes div ~dt indefinitely small with w and ^~, we may bring the expression to the form and consequently we have for A/ the expression 4 4 or finally to and thus we have for A/ the same form as we had before for 8V (Art. 115). 139. We assume now that the necessary conditions for the existence of a maximum or a minimum are satisfied; that therefore along the whole curve G^o, the function F^ is different from zero or infinity, and always retains the same sign ; that a function u may be determined which satisfies the equation and nowhere vanishes within the interval 4 .... /j or upon the boundaries of this interval. If we therefore understand by ^ a positive quantity, and write l= — k + l-\-h, 152 CALCULUS OF VARIATIONS. then the expression for A / above becomes 4 + 4 J(^+^){(^)"+^}*- If k is given a fixed value, then we may choose ^, i] so small that the absolute value of the quantity / that depends upon them is less than k. The quantity k-\- 1 is therefore positive, and con- sequently also the second integral of the above expression. We have yet to show that the first integral is also positive, if F^ > o. After a known theorem in differential equations it is always possible, as soon as the equation dt (^^w)-^^^=^ is integrated through a continuous function u of t, which within and on the boundaries of the interval 4 .... /i nowhere vanishes, also to integrate the differential equation through a continuous function of t, which, if k does not exceed certain limits, deviates indefinitely little throughout its whole trace from u, and may therefore be represented in the form « = « + (/, ^), where (^t, k) becomes indefinitely small at the same time as k for all values of t that come into consideration. The function u will therefore vanish nowhere within the in- terval 4- • • • A- In this manner a certain limit has also been estab- lished for k, which it cannot exceed ; but if the condition is also CALCULUS OF VARIATIONS. 153 added that k must be so small that F^—k has the same sign as i^i, then ^, T) may always be chosen so small that \l\<^k. The first integral may then be transformed in a manner sim- ilar to that in which the integral 8) of Art. 115 was transformed into 14) of Art. 119, and we thus have rr t-\ i _. di dt u ) A/=C(F,-k)i^-^^^^di 4 A +j('^«{(^r+"'}* 4 which shows that A/ for all indefinitely small variations of the curve^ which have been brought about under the given assump- tions, is positive if /\ is positive. If F^ is negative, the same determinations regarding k remain ; only k must be chosen nega- tive and I / 1 -< — k. Both integrals on the right of the above equation are then negative, and consequently A/ is itself negative. We have therefore proved the assertion made above : //" in the interval t^. . . .t^ the necessary conditions which were derived from the consideration of the second variation of the integral for the existence of a maximum or a m-inimum, are satisfied, then the sign of the total variation will be the same as the sign of the second variation for all variations of the curve which have been so chosen, that not only the distances between corresponding points on the original curve and the curve sub- jected to variation are arbitrarily small, but also the directions of both curves at corresponding points deviate from, each other by an arbitrarily small quantity. It has thus been shown that the three conditions given in Art. 135 are necessary for the existence of a maximum or a mini- mum. A further examination will give a fourth condition (Weier- strass's condition, see Chapter XII) whose fulfillment is also suf- ficient. This condition, if fulfilled, is then decisive, after we have first assured ourselves that the other three conditions are satisfied. 154 CALCULUS OF VARIATIONS. APPLICATION OF THE ESTABLISHED CRITERIA TO THE PROB- LEMS I, II, III AND IV, WHICH WERE PROPOSED IN CHAPTER I AND FURTHER DISCUSSED IN CHAPTER VII. 140. Problem I. The problem of the minimal surface of rotation. As the solution of the equation G=o, we found (Art. 100) the two simultaneous equations of the catenary : 1) j X = o. + ^t = <^{t, a, /?), \ y =^/2(e' + e-') = xl>{t, a, fi). We have, therefore (Art. 125), ff(/) = )8, UO = h UO = t, ^'it)=^/2(e'-e-'), U^)=o, U0=y2ie'+e"); 2) -{ and consequently, e,(t)=ri,'(t)U0-ni)W)=^^/2(e^-e-')=y, d,{t)=.^p'(t)U^)-'(t)rij,(t) = ty-:y. If, now, Xo, jyo, V. 7o are the values of x,y, x\y' which correspond to the value 4, then is 3) ®{t,t,) = e,{t,)eif)-dit,)eit^ or, since t = X—a fi = x'^x: [cf.2)], we have 4) e(/, 4) = l/^iy:{xy'-yx') ~y'(x,y,'-yX)l In order to find the point conjugate to t^ we have to write in this expression for x, y, x', y' their values in terms of t and then solve the equation €>(/, /,) = o. CALCULUS OF VARIATIONS. 155 To avoid this somewhat complicated calculation, however, we may make use of a geometrical interpretation (Art. 58). The equation of the tangent to the catenary at the point x^, y^, is yo{X— x^) - V( Y-y^) = o. Therefore, the tangent cuts the .AT-axis in the point determined through the equation >'o'^o=^o>'o' — JJ'oV- The tangent at any point of the catenary cuts the ^-axis at a point determined by the equation y X— xy —yx . If, now, the point x, y is to be conjugate to x^, y^, then its coordi- nates must satisfy 4), which becomes y,'y'(X-X,)^o. Hence, since ^-q' andjv' do not vanish (Art. 101), we have that is, the conjugate points of the catenary have the property that the tangents drawn through them cut each other on the X-axis. We thus have an easy geometrical method of determining the point conjugate to any point on the catenary. Further we have F.= ^^ i^x'^+yy and since y is always positive, and x',y' cannot simultaneously vanish, it follows that F^ is always positive and different from zero and infinity. Hence, the portion of a catenary that is situated between two conjugate points, when rotated about the ^-axis, generates a surface of smallest area (cf. Art. 167). At the same time in this problem it is seen how small a role the condition regarding F, has played in the strenuous proof rel- ative to the existence of a minimum. 156 CALCULUS OF VARIATIONS, 141. Problem II. Problem of the brachistochrone. In this problem the expression for F-^ is found to be 1 1 1) F,= {Vx'^^y'^f V \gy-Vo^ We assumed from certain a priori reasons that between the points A and B of the curve there could be present no cusp (see also Art. 104); that is^ no point for which x' and y' are both equal to zero simultaneously. For such an arc of the curve ^i is then always positive and different from zero and infinity, since the quantities under the square root sign are always finite and differ- ent from zero (see also Art. 95). We obtained (Art. 103) the solution of the equation G^ = o in the form 2) (;t = a + y8 (/ - sin O = <^(^f, a, ^), ijK + a = ^(l-cos /) = »/, (i-, a, /8), where here t is written in the place of <^, and a in the place of — Xf,, and /8 instead of l/(2c^); a is a given quantity which is determined through the initial velocity. We consequently have (/>' (/)=/3 (1-cos /), <^i (0=1, «^2 (0 = ^-sin /; i//'(if)=;8sin t, i//i (2f)=0, 1^3 (/)=1— cos /; 3) \ 6'i (if)=;Ssin t, ^2(/)=/8sin t{t-^\n /)-/8(l-cos tf =2/3 sin(if/2) [icos(t/2)-2 sin(//2)]; and therefore @ (t, 4) =4/3^ sin A sini- |cos-| (/cos^- 2 sin|-J — COSy I 4 COS 4 s4)}. CAtCtfttfS OF VARIATIONS. 157 With the positions which we have assumed for A and B both 4 and t are different from o and 2 tt, and consequently the equa- tion for the determination of the point conjugate to 4 has the form cos-; 2 k(/cos^-2sini-)-cos|-(4cos|--2sin|-) = ^, or 4) ^_2tan^=4-2tanA which is a transcendental equation for the determination of /. We easily see that there is no other real root within the in- terval o. . . .2-^ except /=4, since the derivative of t—2 tan(^!/2), namely, 1 ^ , . . is negative, so that i — 2 tan ( t/2^ continu- COS ( T / ^ J ously decreases, if t deviates from 4, and can never again take the value 4—2 tan (4/2). Consequently there is no point conjugate to the point 4 on the arc of the cycloid upon which 4 lies, and therefore every arc of the cycloid situated between two cusps of this curve has the property that a material point which slides along it from a point A reaches another point B of the curve in the shortest time (Art. 168). In this problem we see that the condition F^^o was sufficient to establish the existence of a minimum. The case where the initial velocity is zero^ and the point A is situated at one of the cusps will be discussed later (Art. 169). 142. Problem III. Problem of the shortest line on the surface of a sphere. In this problem we find that 1) ^.= sin^ u (i/«'2 + z;'2sin^«*)^ This expression cannot become infinitely large, since u and v' cannot simultaneously vanish. However, the function F-, will vanish if sin «^ = o ; that is, 158 CALCULUS OF VARIATIONS, when u—0 or tt. Consequently, in this case, we must so choose the system of coordinates that u nowhere along the trace of the curve becomes equal to zero or to tt. If this has been done, then Fi for the whole stretch from ^4 to ^5 is positive, and does not become zero or infinitely large. The equation G^o furnishes the arc of a great circle, whose equations are (see Art. 106): cos u=cos, c cos (s— (5), cot (z;— y8)— sin c cot {s—b) ; 2) \ or, «=arc cos ]cos c cos {s—b)\=^ {s, a, ^), z; = ^-|-arc cot ]sin c cot (s — 5) [ =1/* (s, a^ )8). Accordingly, we have cos c sin (5—^) '(s)= t/1— cos^ c cos^ (5—^) Ms)- sins cos {s—5) i/l— cos^ c cos^ (s—d) sm c ^,(s)= 1— cos^ c cos^ (5—^) ' co s c sin (5 — 6) cos (5 — 6) , / \ , 1 — COS^ C COS^ (5 — 5) and consequently ^1(5) = cos (5 — 5) ^. / ^_ — cos c sin ( 5— 5) 1/1— cos^ccos^(5— 5) i/l— cos^c cos^ {s—b) Hence, since for the point A we have 5 = 5o = o, it follows that cos c sin 5 3) (5, 5o) = - i/ 1 — cos^c cos^5 1/1 — COS^C C0S^(5— 5) Therefore, in order to find the point conjugate to the point 5o= 0, we have to solve the equation 0(5, 5o) = o with respect to 5. CALCULUS OP VARIATIONS. 159 Since the denominator of 3) cannot become infinite, the con- jugate point is to be determined from the equation sins=(t,a, ft), y = xfs{t,a,P), {t=t,....t,), and one of the neighboring curves, which also satisfies the differ- ential equation G = o,hy the equations 2) ^=, (4) a' + , (4) y8' + (r', a', /3'),= 0, 4) < (f (4) r' + V/i (4) a' + xf,, (4) ^' + (r', a', fi'\=o, where {t' , a, ^'\ denotes the terms of the second and higher powers of t', a and y8'. CALCULUS OF VARIATIONS. 163 147. We may solve the equations 4) with respect to t', a' and /8' as follows. Suppose we have two equations a x-\-by-\-c ^ + (x, y, 2\=o, a' x-\-b' y+c' 2!-\-{x,y,z)i=o, where one of the three determinants a b'—a' b, a d —a' c, b c'—b' c is different from zero. It follows,* then, that we may express all values X, y, z which satisfy the two equations, and in which x, y, z do not exceed certain limits through three power-series of a single quantity. We may choose for this quantity 5 = Ci^r+Cjjv + Ca z, where Ci, c-2, and C3 need satisfy the only condition : ! (4) = -^1 (4), <^'(4)'/'2(4)-f (4)<^2(4) = -^2(4), )'/'2(4)-'/'i(4)«^3(4)=-^3(4), of which ^1(4) and ^2(4) cannot both simultaneously vanish (Art. 127). We may accordingly write Ci = o, c-f! + C3^' = ^1, •See my lectures on the Theory of Maxima and Minima, etc., p. 102 and p. 21. = c^diito) — Cj^iC 4) = 1. 164 CALCULUS OF VARIATIONS, and further impose upon the constants Cj and Cj the condition ^'(4). <^i(4), ^2(4) 6) ^/''(4), '/'i(4), '/'X^) If we consider only the linear terms in equations 4), we have f (4) r' + <^X4) <^' + <^2(4) )8' = o, f(4)r' + t/»i(4)a +»/'3(4)/8'=:(?, Cjtt' + C3/8' = ^1. From these equations we have as first approximations for t', a! and /8' the values / t' = —^1^3(4). 7) I a' =+>?;, ^,(4), ( /3'=-.{:,^,(4); and therefore, finally, 7") }a'=+k, ei 4) + k,^ Pi k„ 4 ), ( ^'=-k,d,{Q + k,^Plk„ 4), where /\(^i, 4). ^2(^1. 4) and PsC-^;!, 4) are power-series in k^ and 4- If we write these expressions in equations 2), or, what is the same thing, in 8) j ;r = «/.(/; + t', a + a', ^ + ;8'), where, now, / may take values less than 4. then we have 9) < _ CALCULUS OP VARIATIONS. 165 In these equations the symbol (t,^;^) is used to represent quanti- ties which for every value of t become indefinitely small at the same time as ^i. When ^1=0, the curve represented by equations 9) becomes the original curve, and we see that ^^ can be taken so small that the two curves at corresponding points, that is, at points that belong to the same value of /, may come as near to each other as we wish. We shall show in the following Article that by this process we have derived all the curves that satisfy the diflFerential equation G^o, which go through the point A and are neighboring the first curve. 148. Instead of the quantity ^i we may substitute a power- series in t', a, fi' which is subjected only to the condition that if Ci, Cj, C3 are the coefl&cients of the linear terms in t', a, ^', the de- terminant <^'(4), U), UO ^'(^o), ^lU), U^o) Ci, C2, C3 =1=0. This condition is satisfied by the power-series which expresses the trigonometric tangent of the angle which the initial directions of the two curves at the point A^t^ include with each other. For, denoting this tangent by k, we have ^ _ dxp dxo __ Xq yo—yo Xp l^^A^ Xo x^ + j>/o>o' dx^ dxo f (4), nto) f (4). f (4) r' + a' + /8' + (r',a',^')z It is assumed that the curve is regular at the point A, so that the quantities <^'(4) and V''(4) are not simultaneously zero, and consequently <^'(/o)^+ ^'i^oT is different from zero. 166 CALCULUS OF VARIATIONS. Hence, the determinant of the equations 4) and 10) is "U) + riio) f(4), f(4), ^"(4)- ^'(4) <^'(4), 'A'(4) "( 4). and add them both to the third row, which then becomes + f(4), Wo) «^2'(4), ^M) f(4), '/''(4) or, what is the same thing, Hence, the above determinant is 1 + <^2(4)> '/'2(4) na n4) 10) fX4)+'l''X4) [^/(4)^2(4)-^;(4)^i(4)] C an expression which ( loc. cit. ) is different from zero. We may accordingly write k in the place of k-^, and find in the same way as above : !x= x + k/iit, k) _ 149. In Art. 89 the form of the solution of the differential equation G = o was given. It follows that a curve which satis- fies the equation G = o is completely determined as soon as its initial point and the direction of the tangent at this point are known. CALCULUS OF VARIATIONS. 167 Let a, b be the coordinates of A and X (see Fig. of Art. 87) the angle which the initial direction makes with the ^-axis ; fur- ther, take instead of the coordinates x, y 2. new system of coordi- nates /, V with a new origin at A in such a way that !t^=.(yX — a)cosX.— (jv — 5) sin X, z/= ( ;»:— a) sin X + ( jV— <5) cos X. ; or {x=^a-\-t cosX-|-z^ sinX, y=b—t sin 'k-^-v cos X. Now if we choose t as the independent variable, then is ;i;=cosX^ — smX, -^7 ^ tti ^^'^'^^ dt dt dt^ I ■ . . dv , dy' dh) ^^ , . •V= — sinXH — -- cosx -Tr=^3^ ^^^^! ■^ dt dt dr and consequently , dy' ,dx' d'^v X — — v = . dt ^ dt dt^ The differential equation G=^o, i. e., becomes then ^2», / . , . dv 14) o = ----^FAa +t co^x+ z/ smX, b—t&m\ + v cosx, cosx + -^ sinX, -sinX + ^cosx) + ^(/, v,^] (Art. 94). Following the method of integration given in Chapter VI, we solve the above equation in such a way that when t=o, both v=o and —^o, the z;-axis being the direction of the tangent at the dt point A. 168 CALCULUS OP VARIATIONS. Hence, if F-^ {a, d, cosX, — sin x) has a finite value different from zero^ and ii H it, v, —— | does not become infinitely large at the point A, as we have assumed was the case, since F^ together with its derivatives, of which H consists, is a regular function of its arguments, it follows that there is only one power-series of t that satisfies the differential equation, and which with its first deriva- tive vanishes for / = o. This power-series has the form v=P P(t). Writing this value of v in the equations 13), they become !x=^a+t cos \ + AiP+ , y = d—t&m\ + Bi t^+ , where the constants Ai, Bi, . . . . are definitely determined. Thus the equations 15) completely determine the curve which satisfies the differential equation G=o, where a, b are the coordi- nates of its initial point and \ the angle which its initial direction makes with the X-axis. From this it follows at once that through equations 11) we have all the neighboring curves of the original curve which pass through the same initial point and satisfy the differential equation 150. We may therefore give k an upper limit in such a way that all curves belonging to a value of k below this limit and sat- isfying the differential equation G=^o lie completely in the surface which envelops the original curve. This makes it possible to bring about a one-valued relation between both curves in such a way that, corresponding to every point of the original curve, we may determine the point of the neighboring curve at which this curve is cut by the normal at a point on the first curve. Let X, y be the coordinates of a point P on the original curve, and x + i,y + r) the coordinates of the corresponding point on the neighboring curve. CALCULUS OF VARIATIONS. 169 If P' is the point corresponding to P, its coordinates are ^ + f=<^(/ + -r, a+a', 13+^'), y + yj=xlj(t + T,a + a', ^ + /8'); and besides, since {X — x)x'+ ( V — y)y= o is the equation of the normal, and X=^x + i, K= r + tj is a point on it, we have x'i + /v = o. Hence, ^, rj and r are to be determined from the equations / f = f (/) r + <^i(0 a'+ (/..(O /8'+ . . . ., 16) \v = ^\t)r + Ut)^'^Ui)^'+----, The last of these equations combined with the first and second gives 17) o=['\i) + rlj'Xt)-]T + ^/{t) + it,n when for a', /8' we have written from 7" ) their power-series in ^. Since the portion of curve A B has no singularity, and conse- quently ^'\t)-\-^\t^ nowhere vanishes, we may from equation 17) express t and therefore also f and -17 as power-series in k. If, then, we limit ourselves to curves with which k remains within a certain limit, we may always determine the point where such a curve is cut by a normal of the original curve. 151. We ask if it is possible for the second curve to intersect the first curve. For this to be the case the length PP' must be zero ; that is, ^, 17 must for some value of / be equal to zero. Hence we have so to choose the quantities /, t, t', a', /8' that the equations 4) and 16), when in 16) ^ and 17 are put equal to zero, are satisfied. The terms of like dimension in 4) and 16) are homogeneous functions of t', a', /3' and of r, a' )8' respectively; these equations may be written : 170 CALCULUS OF VARIATIONS. where v, t>, q, v-^, p^, q^ represent functions of r', a', /8'; and v^, pz-, qz, Z's. A' 92 are functions of t, a', /8', which with these functions and therefore also with k, become infinitely small. The first two of these equations express that the two neigh- boring curves pass through the initial point A, and the last two that they are to go through another point. In order that these four equations exist simultaneously, their determinant must vanish. This determinant, when in it we make k=^o, is : f(4), 0, Wo), Uio) ^'{t,\ 0, U^ol UQ 0, f(0. un uo 0, ^v\ un, un and this is nothing other than the function — @(t, 4)- Hence the determinant of the above system of equations may be brought to the form — ©(/, 4) — ^(/f, 4. k); and, as this determinant is to van- ish, we must have 18) @{f,to) + k{t,t„k)=0. If, now, C is a point of the original curve for which f=f and which is not conjugate to A, then ®(t', 4) is different from zero, and we may therefore fix a limit for k so that for all values of k under this limit the expression ®{f, 4) + Jc(t', 4, k) is different from zero ; that is, none of the curves which lie very near the original curve can cut this curve at the point f or in the neighbor- hood of it, since we can always find a limit k of such a nature that for every value of t within the interval f—k t' + k the expres- sion is different from zero. And, reciprocally, every curve that lies CALCULUS OP VARIATIONS. l7l very near the original curve will cut this curve in the neighborhood of C, as soon as there is a point C in the interval AB which is conjugate to A. For one can then alwaj^s find for ^ a value sufficiently small that, with very small values of h, the sign of e{f—A, to) + k{t' — h, 4, k) is the same as the sign of @(t'—h, t^), and the sign of %(t' + k, to) + k(f + h, 4, k) is the same as that of &{f + h, 4)- But when the function @{t, 4) passes through the value zero it changes its sign, as is seen in the following Article. Hence, it follows, as @(t', 4) is to be zero, that the expression ®(^. 4) + ^(^. 4. ^) must vanish once within the interval f — h. . . f + h; or, in other words : //, in the interval AB of the original curve, there is a point t^t' conjugate to the initial point, then all the curves which lie very close to the first curve, which sat- isfy the differential equation G=o and which have the same initial point A, will cut again the first curve in the neighbor- hood of the point t' . Consequently the conjugate point is noth- ing other than the limiting position which the points of inter- section of a neighboring curve with the original curve approach, if we make smaller and smaller the angle which the initial directions of the two curves make with each other. If there is no such limiting position within the interval AB, then there is no conjugate point within this interval. 152. It remains yet to show that the point A cannot itself be this limiting position; that is, of all the neighboring curves there cannot be one which cuts the original curve as close as we wish to A. Analytically this case may be expressed in the following manner : If at the point t the original curve is cut by a neighbor- ing curve, we have the equation o, o, «/''(4) + z'i. «/'i(4) + A. V'2(4) + ?i '(t)-\-V2, O, lU)+p2^ <^2(^) + ?2 a determinant which becomes Q(/, 4)=^. when k=o. If for t, t', a, /8' expressed as power-series in k, their values be substituted in the determinant, it becomes an equation in t and k. Further, since 172 CALCULUS OP VARIATIONS. '{t), (i)i{t), .... ^2{^) 3.re power-series in t which are regular functions in the neighborhood of 4, the determinant may be de- veloped in a power-series in /— 4and ^, which converges for suf- ficiently small values of /— 4and k. If in the neighborhood of the original curve there exist curves which cut this curve as near as we wish to A, then, after sufficiently small limits have been given to t—^o and ^, it is possible to find values for these quantities within the given limits for which the equation is satisfied. If we write 2"= 4, the quantities v, p, q are respectively equal to v-i, p2i ?2 and the quantities v-^, p-^, q-^ to v^,, p^, q^. When this is the case, the determinant has the form o , a , b , c 0, (2i, ^1, c a, o, b, c a,. 0, b. c which is identically zero. Therefore the power-series in t— 4 and k will vanish for /=4. whatever be the value of k-y and conse- quently this series is divisible by t—t^. The determinant, then, when divided by t—t^ is for the value 19) We saw in Arts. 128 and 129 that and that If t' is a conjugate point to to, so that @(^', 4)-^i(4) eit')-dit,) e,(t')=o, CALCULUS OF VARIATIONS. 173 it follows that where A is a constant different from zero. We further have, since d the relation jf e{<, t,)=eit,) e:(t)-e,(i,)e,xi), [#.«('■'•>],.. = ^/Tfr which is different from zero. It is thus seen that the derivative of @(/, 4) does not vanish on the positions at which the function itself vanishes. At the same time it is shown that the equation 19) is not sat- isfied, so long as ^ and / — 4 remain within finite limits; and con- sequentl)"^ a neighboring curve cannot intersect the original curve a second time indefinitely near the initial point through which both curves pass. As there is a great range of choice regarding the variable /, and as the constants a and /3 may be chosen in many ways, it is possible to give many forms to the function ©. To be strictly rigorous, it would yet remain to prove that the solution of the equation 0(/, 4) leads always to the same conjugate point, what- ever be the form of 0; the geometrical significance of these points, however, make such a proof superfluous. 174 CALCULUS OP VARIATIONS. CHAPTER XII. A FOURTH AND FINAL CONDITION FOR THE EXISTENCE OF A MAXIMUM OR A MINIMUM, AND A PROOF THAT THE CONDITIONS WHICH HAVE BEEN GIVEN ARE SUFFICIENT. 153. In the preceding Chapter we considered the family of curves that have the same initial point A and satisfy the differen- tial equation G = o. These deviate very little from one another in their initial direction. We saw that the curves again intersect only in the neighborhood of points that are the conjugates of A, the conjugate point along any curve being the limiting position of the point of intersection of this curve and a neighboring curve when the angle between their initial directions becomes infinitesi- mally small. All points that lie on these curves before the points that are conjugate to A form a connected portion of surface ; that is, if /'i is a point belonging to this collectivity of points, a bound- ary may be described about P^ so that all points within this boundary also belong to the collectivity of points. For, let X = (t, a, ft), jv = »/»(/, a, /3) be the equations of a given curve which satisfies G^^o, and let the coordinates of a point on this curve be X, = <^( A, a, /3), y^ = t/»( /i, a, P). Further, let x^ -\- ^, y^ -\- ri be the coordinates of another point P^ that lies in the neighborhood of P^, so that ^, 17 are quantities arbitrarily small. CALCULUS OF VARIATIONS. 175 We may then (Art. 151) draw a curve between A and P^ which satisfies the differential equation G—o, if we can determine four quantities r, t', a', /3' as power-series in f , 17 in such a way that the following equations are true : O = f (4) t'+ M4) a'+ <^2(4) )8'+ (r', a', ^')„ O = f (4) t'+ ,A,(4) a'+ riflQ P'+ (r', a', )8')„ ^= f (A) r + <^,(A) a'+ U^,) ^'+ (t, a', y8')„ V=^'{A) r + U^,) a'+ ^a(A) ^' + (r, a\ fi'\. Since the determinant of these equations (Art. 151) is — ©(/"j, 4) and is different from zero, the point ^j not being conjugate to 4, it follows that the quantities t, t', a, yS' may be developed in power- series in $, T) which are convergent for small values of these quan- tities. Consequently a curve may be drawn through A and P2 which satisfies the differential equation G ^= o, and this curve will be neighboring the first curve and will deviate as little as we wish in direction from its initial direction, if ^, tj, and consequently also T, t', a', fi\ are sufficiently small. If we form the determinant for the curve AP2, which, when put equal to zero, is the equation for the determination of the point conjugate to A, it is seen that this determinant also may be developed as a power-series in f, 17, which becomes —®{ti, 4) when ^=rj=o. The function €)(/], t^) is different from zero when suf- ficiently small values are ascribed to $, t). Consequently within the interval AP2, there is present no point which is conjugate to A. We may therefore envelop the interval situated between two conjugate points of the original curve by a narrow surface area, which is of such a nattire that a curve, and only one, may be drawn from the point A to any point within it, which satisfies the differential equation G=o, is neighboring the first curve and deviates in its initial direction only a little from it. 154. Let a portion of curve Po^i. satisfying the differential equation G=o, be given, which is of such a nature that for no 176 CALCULUS OF VARIATIONS. point on it F-^^o or oo, and stippose that the point conjugate to /{, does not lie before P^. Between P^^ and /\ take an arbitrary point P2 and draw through P2 a regular curve.* On this curve we choose a point P^ so close to P2 that a curve may be drawn through /'o and P^ which satisfies the difEerential equation G = o, and which lies entirely within the strip of surface defined above. Let us consider the change in the integral when we take it over Pf^P^^P^P^, instead of over P^P-^. We may denote an integral taken over a curve that satisfies the differential equation by /, and one over an arbitrary curve by /, and we may denote the direction of integration by added indices. We have therefore to compute the expression A/=/o3 + 732— /02, or P P P P P P -'o-*3 ■'o-'a '■■3*1 an expression which (Art. 79) p, p. p, p, where ^, 77 are measured in the direction from P^ to P^. At the point P^ and along the curve P^ P^ in the direction P^ P^ we have ef=-^; dl-\-{dlf, ^l=-y;dt + {dty, dt denoting that this differential is taken with respect to the curve P^Pz- If we consider the arguments in F expressed as functions of 7 along the curve P3/2, it follows that ^Fdt = F(yX.,,y2,x{,y;^dl^(^dlf. p p ^ The shaded curves do not satisfy the differential equation G = o. CALCULUS OP VARIATIONS. 177 Hence at the point P^, which is an arbitrary point of the curve /o/*i, we have t.I^\^F{x^,y^,x;.yi^-\^; ~ F{x,, y,, x^,y^) +3^2' ^ F{x^, y^, X,', y,') ^dt +{d t)\ (a) The function F is homogeneous of the first order (Art. 68) with respect to its third and fourth arguments, so that (see Art. 72) F{x^, y^, X,,' y,') = x,' /^^ {x„ y,, x,', y,') + y,' F^^ (x„ y„ x,',y,'). We define by ^{x, y, x', y', x' y') the expression 1) ^ix,y,x',y',x',y')=x'{F^\x,y, x',y')-F^'Xx,y,x',y')\ +y'\F^\x, y, x', y')-F»\x, y, x', y')\. Hence at the point P2 it follows that A/ = ^(x, y, x', y', x', y') d J +{d tf, when in the function ^(A-A),?o+>^(?i-?o)](A-A) k—o +/'"[A+'^(A-A).?o+'^(?i-?o)](?i-?o))^>^- Hence, if we write i 9k =g + k{g-q) = {l—k)q + kq, CALCULUS OF VARIATIONS. 181 it is seen that F^\x, y, A q)-F^\x, y, p, g)= J^/rui) (^, y^ p^^ g^) (^p_p^^ k=o + F^'\x, y, p^, g^)(^-q)) dk, F^\x,y,prg)-F^\x,y,p, ^) ^ J(ir«i)(^, ^^^^, g^)G-P) k=o + F'-^\x,y,p^, g^){q-q)) dk. Note that (see Art. 73) ^'^"=?.' ^1, i^'^''= -A ?. ^1. F'-^^p^ F„ and further that i^'^^^i^'^'. By substituting these values in the above expressions, and in turn the resulting quantities in the expression for <£, we have Six,y,p, g, A g)=-/> [F^'\x,y,p, g)-F^\x,yp, g)] +'g [F^\x, y, p, 'g)-F'\x, y, p, ^)] k=:0 The expression in the square brackets is {g^P-P^g)\.gJ^P-p)-pJ.9-9)^=i'^-^)i9p-p'g)\ and consequently k^i 3) S{x, y, A ?. A q)={9P-P gy ^^(^^ y^ A' ?k) (l--^^) ^^• 182 CALCULUS OF VARIATIONS. This expression for =T— T {mod. 2 7r) ^{x, y, p, q, p, q)=pVF'\x, y, p, q)-F^\x, y, /, ^)] 4- ~qVF^\x, y, p, q)-F^\x, y, p, ^)] CALCULUS OF VARIATIONS. 183 = COS T [F^% X, y, cos r, sin r) —F^\x, y, cos t, sin t)] + sin ; [i^<^'(;i;, jv, cos ;, sin — i^«)(-^, jv, cos T, sin t)] X=a) =cost j ^x-'^'^'L^. :>'. cos(t + x), sin(T + \)] X=o X=', cos(t +x), sin(T + x)]. X=o But, if F'^' denotes the derivative of F with respect to its third argument, etc., d-^ F'>' [x, y, cos (t + x), sin (t + x)] = [— ^, ~q)= j [— cosr sin(r + x) + sin^ cos(t + x)] F^dx X=o X=/, cos(t + x), sin(T + x)]rfx. s- 184 CALCULUS OF VARIATIONS. If we write ct) — X=X. , the integral just written is sin X' Fi Ix, y, cos (t — X'), sin (i-— X')] d)J \'=o =Fi[x, y, cos ( T— X/), sin ( t— X2')] i d cos X', where X2' is intermediary between and — x) (a+2/8 cos 3x) ^x, x = o where we have written t + X =X or t =0 ; i. e., we have taken the ;^-axis as the initial direction, from which w is measured. Noting that sin(o>4-2x) + sin (w— 4x)=2sin (to — x) cos 3x, 186 CALCULUS OF VARIATIONS. it is seen that S{x,y,p, q, p, ip)=(l— C0S6j) [a + y8 (cos w + cos^w)]. The greatest and least values that cos a>+ cos^ w can have are 2 and ~%, the corresponding values of w being o and % tt. Hence, if we we make a=l and /8=1, the function .■, Pt,', Q3) is the differential quotient of the function j (5) at the point 3. If, then, along the curve Oi31 the function d" is nowhere positive, the function \{^s) continuously diminishes when the point 3 slides from Oj toward the point 1. Let the point Oj, which was taken very near the point 0, co- incide with this point ; then we can say : If the function ^ is nowhere positive and is not zero at every point of the arbitrary curve 031^ the integral taken over the original curve is always greater than the integral extended over the curve 031 ; and if the function 'it)r'+^^k' + (r',k'\, = ^'(t)r' + ^^k' + {r',k'\. The determinant of the linear terms of the equations just written gives, when put equal to zero, the equation for the deter- mination of the point conjugate to the initial point, i. e., CALCULUS OF VARIATIONS. 189 The smallest root of this equation, which is greater than the value to of t, gives the value of /, which belongs to the conjugate point. If this value is t^, then the coordinates of the point are x = j>{t^, k), y^xf,{ti, k). If we consider t^ as a function of ky defined through the equation (^), and if we give to ^ a series of values, the two equations just written represent the curve that is constituted of the points con- jugate to 0. The direction-cosines of the tangent to this curve are propor- ■ • • d X d'v tional to the quantities —r- . —rr • But we also have dk dk dx ^ d(f>(i„ k) dti 8<^(A, k) dk d t^ dk d k ' dy ^ d^{t^, k^ dt, a^KA. k) dk 3 ^1 dk d k ' Multiply the first of these equations by ^^^f^' ^^ = i/»'(A), and subtract from it the second after it has been multiplied by ^&i3 = ^'{t^y We have then, with the aid of M), Since <^'(A)) ^'(A) are proportional to the direction-cosines of the tangent at a point t^ of the curve through 4 and /j, which sat- isfies the differential equation G=o, it follows from the above equation that the tangents to both curves at the point /j coincide. Hence, the locus of the conjugate points to is the envelope of the curves through 0, which satisfy the differential equation G=o. 163. Let x=f{u) and r=^(«) be an arbitrary curve 031, which passes through the point 0, and is situated entirely within the region bounded by the envelope. Further, suppose that 031 does not coincide throughout its whole extent with any of the curves passing through 0, which satisfy the differential equation 190 CALCULUS OF VARIATIONS. G^=o. Suppose, however, that 031 is touched by the curves that pass through and satisfy the differential equation G—o. At the point of contact we must have and di du dt du The values of / and u, which belong to the point of contact, are determined as functions of k through the first two equations. These equations, being true for sufficiently small values of k, may be differentiated with respect to k, and we thus have : 9<^ dt d(j> _ df du d t dk dk du dk ' d^ dt dxj) _ dg du dt dk dk du dk ' If we multiply the first of these equations by -^ and the second by f— and add^ we have with the aid oi (B) du d^ dg_d^ df ^^ dk du dk du If between this equation and the equation (j9) we eliminate the quantities —f- and -^, we have du du di dk dt dk an equation, which served for the determination of the point con- jugate to the initial point. Consequently the point of contact of the curve, that passes through and satisfies the differen- tial equation G=o, with the arbitrary curve must be the point conjugate to 0. But this is possible only if the curve x= f{^u^, y=^ g{u) co- incides with the envelope ; while according to our supposition the curve 031 is to lie entirely within the region that is bounded by CALCULUS OF VARIATIONS. 191 the envelope. It follows that there can be within the region no curve 031 such that each of the curves which satisfies the differ- ential equation G^=o, and which joins the point with a point of 031, touches 031 at the same time. Hence, the quantity qp — Pq can be everywhere zero only when the arbitrary curve between and 1 coincides throughout its whole extent with one of the curves that passes through and satisfies the differential equation G=^o. But since, within the strip of surface inclosing the field as we have defined it, there can be only one curve draw^n through and 1 which satisfies the differen- tial equation G^o, it follows that the arbitrary curve 031 can coincide only w^ith the original curve 01, and then it is not a varia- tion of that curve. It therefore follows that the function (F can- not vanish for all the points of the curve that has been sub- jected to variation. 164. It is not necessary that the curve 031 be a single trace of a regular curve in its whole extent. If we assume that 031 is composed of an arbitrary number of regular portions of curve, the integral may be regarded as the sum of the integrals over the sin- gle portions, and the conclusions made above are also applicable. It may happen that one of the portions of curve coincides throughout its whole extent with a portion of one of the curves that goes through and satisfies the differential equation G — O. If this is the case for 23, for ex- ample, so that (5 is equal to zero along 23, then we may replace this portion of curve b}' an arbi- trary portion of curve 2'3, which lies very near 23. Then the the- orem proved above is true for the curve 02'31, viz., that -^ 03 < A2' + -' 2'3 > according as the function 'o=^'(4)('ri-4)+(Ti— 4)[ti-4], where [t^ — t^_i] denotes a quantity which becomes indefinitely small at the same time with t, — t,_i . For the first of the integrals in the expression (C) we write: X'=X,' + Xo"{ t~t,) + {t-to) [t- 4] , y'=-yo+yoV-^o)+{i-0[i-fo]; for the second integral we write x= X,+ X,'{i-T,) + (i-T^)[t—T,'], y^ yi+ yr{i-'ri)+{i-r,)[t-T,], and similarly for the other integrals. These expressions we write in the sum of integrals (C), and, developing them in power-series, we have through integration (ti— 4)/'(^o.>'o, ^o',>'o')+('-2-^i)^(^i.;>'i. V, j>'i')+ • • • • + (^l — rn)^(Xn,yn, X^ , yj) plus a similar number of terms, which become indefinitely small of the second order with respect to the quantities t,— t,_i. We may therefore write the integral in the form i^ { 2<^ "'- "^-^^ ^ ^^^-^' ^'-^' '''^-^' y'^-''^\ T=1.X n+1 194 CALCULUS OF VARIATIONS. where we must understand by to the value 4. and by 4+i the value t^. Since r^ — Ty_i are positive quantities, and the functions F in regard to x^, yl . . . . are homogeneous of the first degree, we may write the above limit in the form T=I,2 n+1 or, since the above expression is 7*=Q0 166. The integral in the above form has a more general meaning than the one hitherto employed, with which, however, it coincides in every particular where that one has a meaning. We may assume, with respect to any arbitrary variation, a series of points Xq, jKo; ^1, jKi; . . . .x„, jf„; ^n+i' jVn+i of such a nature that the distance between, say, two successive points does not exceed a certain quantity S. We then form the sum If we make 8 smaller and smaller by increasing the number of points, it may happen that this sum approaches a definite limit. We call this limit the value of the integral taken over the curve. It may also happen that the limit does not approach a definite value; for example, it may vacillate between two values. We then say the integral taken over this curve has no meaning. If we think of the series of points that are taken upon the curve, joined together successively by a broken line, the integral taken over this broken line will approach the same limit as will the integral taken over the curve, if the integral has a meaning. CALCULUS OP VARIATIONS. 195 If, therefore, a curve 01 is given, which satisfies all the con- ditions that have hitherto been made for a maximum or a mini- mum, and if this curve varies in an arbitrary manner, then if the integral taken over the curve, which has been subjected to varia- tion, has a meaning as defined above, we ma)' draw a broken line, the integral over which deviates as little as we wish from the integral taken over the curve that has been caused to vary and to which the theorem of Art. 161 is applicable. Consequently we may say, in the case of a maximum, the integral taken over the curve subjected to variation cannot be greater than the inte- gral taken over the original curve, and in the case of a m,ini- mum,, it cannot be less than the integral taken over the origi- nal curve. Since we may make the region as narrow as we wish within which all the variations are to lie, we ma)'^ assume that upon the curve which has been varied a point 3 lies so near to 01 (but not upon it) that two curves 03, 31 can be drawn between the points and 3 and between 3 and 1, which also satisfy all the conditions of the problem. For the sake of brevity, let us assume that we have to do with a maximum. Then, as we have just seen, the integrals over 03 and 31 cannot at all events be smaller than the integrals over the corresponding parts of the curve which has been varied ; but, after the preceding theorems, the integral taken over 01 is greater than the sum of the integrals taken over 03 and 31, and conse- quently also greater than the integral over the curve that has been varied. A maximum is therefore in reality present. 167. We may now investigate the behavior of the function T ^ =-'011 r>io. If, on the other hand, G=o along the whole curve AA^B, then this curve must consist of several cycloidal arcs ; since, if it were only one, the curves AA^B and AB would be identical. These arcs must have different tangents at the point where they come together ; for, since this point cannot lie on the X-axis, a consecu- tive point having the same direction must lie on the same cycloidal arc. If corners were present, however, they could be so rounded off that there would be a shorter path between the two points, and consequently, the velocity being the same, the time of falling would be shorter. CALCULUS OF VARIATIONS. 199 Hence the arc of a cycloid also minimizes the time of falling between A and B in the case where ^ is a singular point ; that is, when the material point starts from A with an initial velocity that is zero. The conclusions just made are also applicable, if j5 is a singu- lar point ; for it makes no difference whether the material point ascends from B to A or falls from A to B, if we allow the mate- rial point to go back with the same initial velocity with which it arrived at B. On the way back it will reach A with its original velocity. Its velocity will be the same in both cases at all points of the curve, but directed toward opposite directions. The inte- gral taken over the curve has the same value in both cases ; and consequently the curve which caused the integral to have a mini- mum value will also, in the second case, minimize the integral. 170. The problem of the geodesic line on a sphere offers here nothing of special interest. It is found that the function ', ^p, kq) = k (5 (x, j^^ x\ y\ and therefore <^(^. J' —A -q)=—^{x,y,p,q). It is thus seen that we have only to reverse the direction of the displacement to effect a change of sign in the function (s. 174. We have now completely solved the four problems that were proposed in Chapter I, and at the same time one of the prin- cipal parts of the Calculus of Variations has been finished. After stating succinctly the four criteria that have been established, we shall take up the second part, which has as its object the theoret- ical and practical solution of problems, a general type of which were the Problems V and VI of Chapter I. These criteria may be summarized as follows (cf. Art. 125): There exists a fninimum or a maximum value of the integral k I = JF{x,y,x\y')dt^ to where F is a one-valued, regular function of its four argu- ments and homogeneous of the first degree in x' and y' , if 1) the differential equation G=o is satisfied for every point of the curve; 204 CALCULUS OF VARIATIONS. 2) Fi is positive or negative throughout the whole interval, f . • • . A / 3) there are no conjugate points of the curve within the interval 4. . . . /j (^limits included^; 4) the function © is positive or negative throughout the whole interval t^,. . . .ti. In this discussion we have excluded the cases where 1) the extremities of the curve are conjugate points; 2) Fi=^o for some point of the curve; 3) Fi=^o for som-e stretch of the curve; 4) (S'=o for some point or stretch of the curve. A general treatment of the first three cases would require the extension of the theory to variations of a higher order. Otherwise particular devices must be employed in every example in which one of the above exceptional cases is found. 175. Before we begin the consideration of Relative Maxima and Minim,a, we may, at least, indicate the natural extensions and generalizations of the theory which has already been presented : Instead of the determination of a structure of the first kind* in the domain of two quantities, it may be required to determine a structure of the first hind in the domain of n quantities. If a structure of the first kind is determined in the domain of the n quantities x^, x-^,. . ., x^,, then n—loi these quantities may be expressed as functions of the remaining one, say, Xi. Writing u- Cf{x X X ^^^ ^^3 clxA , J \ dxi dxi dxj * See my Lectures on the Theory of Maxima and Minima of Functions of Several Variables, pp. IS and 86. CALCULUS OF VARIATIONS. 205 it is seen that u is so connected with the n—1 functions that dxx V ^' ^' ■ ■ ■ ■ ' °' dxi ' dxi '■■■■' cixiJ ' The difference of the values of u at the initial-point and at the end-point of the structure is expressed by a definite integral. This integral takes the form, when we consider the x's ex- pressed as functions of t, say, Xi==^Xi{i), x-^^^Xii^t),. . ., Xa—Xa{t), A I = /^(^i. ^. . ^n, V, x-l, , x^') dt. The function F must be a one-valued, regular function of its argu- ments in the whole or a limited portion of the fixed domain. The value of the integral / is independent of the manner in which the variables x-^, Xt_, . . . . , x^ have been expressed as functions of /. It therefore follows after the analogon of Art. 68 that the function F is subjected to the further restriction : Icr \Xxi X21 . . . . , x„, Xi , X2 f ■ • • • , x^ ) ^f yX-i^t X^t . . . . , ^n, ICX-^ , KX2 ,•••-, i^X.^ ), where ^ is a positive constant. The indicated generalization of the problem given in Art. 13 may accordingly be expressed as follows : The n quantities Xi,X2,....,x^ are to be determined as func- tions of a quantity t in such a manner that for the analytical structure that is defined through the equations Xx=Xi(t),X2=X2(t), , x„=xXO^ the value of the integral k 206 CALCULUS OF VARIATIONS. is a maximum or a m.inim,um,; in other words, if one causes the above analytical structure to vary indefinitely little, the change in the integral thereby produced must in the case of a maximum be constantly negative, and in the case of a minim^um it must be constantly positive. Further, the function F is to be considered a one-valued, regular function of its arguments, and indeed, with respect to x^, x^, . . . ., xj, a homogeneous function of the first degree. 176. The treatment of the above problem is found to be the complete analogon of the problem given in Art. 13. A greater complication arises when there are present equations of condition among the variables x^, X2, . . . ., x^. An example of this kind we had in Problem III of Chapter I. This problem may be expressed thus : Among all the curves in space which belong to the surface f{x,y, 2)^0, determine that one for which the integral t, to is a minimum,. The general problem may be formulated as follows : Among the structures of the first kind in the domain of the quantities Xi, X2,...., x„,for which the m equations f{xi, X2, xJ = o iH-=l, 2, m; m From this it is seen that the quotient ", is independent of the ^10 arbitrary functions |, tj, since it does not vary if we write for f, 17 as functions of t other functions ^1, tji. Consequently it follows that the value of the above quotient depends only upon the nature of the curve x=x{t), y=y{t). CALCULUS OF VARIATIONS. 217 184. We might generalize the problem treated above by re- quiring the curve x^x{t), y=y{t) which minimizes or maxi- mizes the integral A r^^ = ^F'^\x,y,x',y')dt, 4 while at the sam,e tim,e the following integrals have a pre- scribed value: A r^'^^F'\x,y,x\y')dt, A I^^' = ^F'\x,y,x\y')dt, A m^^F^>^\x,y,x\y'^dt, to the /unctions /^*", /"'*', . . . . , /'('') being of the same nature as the function F defined in Chapter I. We must now consider the deformation of the curve caused by the variations We have, then, if we write w^=y'^^— x'% (* — 1, 2, . . . , /i), and sup- pose that the f s and tj's vanish for t = t^ and /= Ai A A M'''^ = ejG^'^wdt + e,jG''^w,dt+.... A 4- c/i Jg^"^ Wndt + (€,ei,..., c^ )2, 4 218 CALCULUS OF VARIATIONS. 4 4 4 M^ = o^ejG^^'zudf + e,jG'^zUidt+ .... 4 4 4 A A 4 4 A + e^ jG'-'')7V^,dl + {€, e^,. . .,e^ j^. 4 By means of the last ^ equations, if the determinant A jG^'^Widf 4 (i. j = 1.2, ^ is different from zero, we may, for sufficiently small values of Ci, €2,. . . ., e^, express these quantities as convergent power-series in c* These power-series when substituted in A/"" cause it to have the form A/'" = eZ»+(e)2, where A jG^'^w^dl 4 n (i. j = 0, 1, juand w© = w). *Cf. Lectures on the Theory of Maxima and Minima of Functions of Several Vari- ables, p. 21. CALCULUS OF VARIATIONS. 219 In order that the integral /'^' have a maximum or a minimum value, it is therefore necessary that This determinant, when expanded, may be written in the form 4 A where X; is the first minor of | G' w dt in the determinant D. 4 Hence, as before (cf. Art. 79, where we had G=o), we have here \ 6^°' + Xi 6^'^+ . . . . +^M 6^M=o. 185. Similarly, if in Art. 183 we denote the quotient — ^ by X and then give to W^^^ and W^^ their values, we have 4 JCG^'O-XG'^') w dt=o. 4 From this it follows that G''-\G'^ =^0. We may prove a very important theorem regarding the con- stant X, viz -.—it has one and the same value for the whole curve; i. e., we always have the same value of >^, whatever part of the curve x=x (t), y=y (t) we may vary. Consider the values of t laid off on a straight line, and suppose that the constant X has a definite value for, say, the interval 4 4 which also corresponds to a certain portion of curve. This value (see Art. 183) is inde- pendent of the manner in which the portion of curve 4 • • • • 4 has ^ *3 been varied. Next consider an interval t' t" which includes the interval 4 4; then, there belongs to all the possible variations of the interval t f , also 220 CALCULUS OF VARIATIONS. that variation by which t' 4 and t^ /" remain unchanged and only t^, t^, varies. As X has a definite value for this inter- val and is independent of the manner in which the curve has been varied, it must have the same value for t' . . . . /'. 186. The differential equation G""— XG^'''=o is the same as the one we would have if we require that the integral to have a maximum or a minimum value, where F is written for the function /r(o)_x^(i). Through this differential equation (See Art. 90) ;r and y are expressible in terms of / and X and two constants of integration a and /8 in the form ;r=^ (A a, A X), The curve represented by these equations is a solution of the prob- lem, when indeed a solution is possible. 187. We prove next a very important theorem which often gives a criterion whether a sudden change in direction can take place or not within a stretch where the variation is unrestricted (cf. Art. 97). Suppose that on a position ^=/, where the varia- tion is unrestricted, a sudden change in direction is experienced. On either side of / take two points t^ and t-^ so near to f that within the intervals i^. . . .f and / . . . . /j a similar discontinuity in change of direction is not had. Among the possible variations there is one such that the whole curve remains unchanged except the interval /j . . . . /j, which is, of course, varied in such a way that the integral /*'' retains its value. The variation of the integral /"" depends then only upon the variation of the sum of integrals f h //-'»> {^x,y, x', y ) dt+JF'" (x, y, x', y') dt. /i t' CALCULUS OF VARIATIONS. 221 We cause a variation in the stretch A .... /j by writing V^^V + ^iVi, where we assume that U) ^. ^1. '?. Vi are all zero for i—ti and /=/2, f . fi> ''Ji are zero for t— /„ _'>j#:o for t— ti. We may then always determine e^.as a power-series in « so that If by <^ we denote an expression of the form <^*' — X<^"*, we have (Art. 79) If the curve x=x(t),y=y{t) minimizes or maximizes the inte- gral /'•", it is necessary that the coefficient of e on the right-hand side of the above expression be zero. Since G=o for unrestricted vari- ation, it follows from the assumption {A) that If in the assumptions (^) we assume for t=ti that r)—o and $^o, we have an analogous equation for x'. It therefore follows (cf. Art. 97) that r 3(/^""-x/''^') ']-_ r a(/^""-x/^'^') "| + L dx' Jt'~L Sx' Jf' r a(/r(o)_x/''i!) 1- _ r a (/-'»' -x/^'") "|+ L a;»/' Jf L dy J/ 222 CALCULUS OF VARIATIONS. We have then the theorem : Along those positions which are free to vary of the curve which satisfies the differential equation G=o, the quantities ^ — j and -^ — -. vary everywhere in a contin- o X oy uous manner, even on such positions of the curve where a sud- den change in its direction takes place. 188. It is obvious that these discontinuities may all be avoided, if we assume that ^, tj, fj, f]^ vanish at such points. This we may suppose has been done. We may also impose many other restrictions upon the curve ; for example, that it is to go through certain fixed points, or that it is to contain certain given portions of curve, or that it is to pass through a certain limited region. In all these cases there are points on the curve which cannot vary in a free manner. But whatever condition may be imposed upon the curve, the following theorem is true. All points which are free to vary {and there always exist such points^ must satisfy the differential equation G^ '"' — \G "' =o, and for all such points the constant \ has the same value. 189. The second variation. We assume that the variations at the limits and at all points of the curve where there is a dis- continuity in the direction, vanish. We also suppose that the variations ^, 17 have been so chosen that A/'''=o. We then have (cf. Art. 115): A A 2 and consequently J [/■■.«(^J+^?^'] ' = | J(^y-;v^')^/, and its perimeter by A 2) n'=^Vx'^^ y'^dt. 4 The problem proposed consists in expressing x and y as functions of t in such a manner that the first integral shall have the greatest possible value, while at the same time the second integral retains a given value. It makes no difference vi^here the origin of coordinates has been chosen ; for by a transformation of the origin the second integral remains unchanged while the first integral is changed only by a constant. This does not alter the maximum property of the integral. One may also add other conditions ; for example : That the curve go through a certain number of fixed points in a given order, or that it is to include certain portions of curve in a given order, etc. The curve will then contain portions along which the variation is not free. 191. The function F is here F^t (xy'-~ x'y)-\ V x'^ + y'\ Instead of this function we may substitute another, since d{xy)_^ dt -"^y+y^ and consequently. - {xy'-x'y)= - —{xy)-yx'. 226 CALCULUS OF VARIATIONS. Now, if we integrate between the limits to. . . .t^, the first term of the right-hand side of the above equation vanishes, since the end- point and the initial-point of the curve coincide. It follows, then, that 2 j {x y - y x') d t^- j y x' d t. We may consequently give the function F the value 3) F^ -x'y- \ Vx'^+y'\ From this we have dF \x' dF \y' dx' -^ Vx'^^-y'^' 9/ Vx'^^y'^' dF dF But since (Art. 187) g — i and g — i vary in a continuous manner along the portions of curve that vary freely, since also \ has the same constant value for the whole curve (Art. 185), and since the quantities that are multiplied by x are nothing other than the direction-cosines of the tangent to the curve, it follows that the curve at every point, where the variation is free, changes its direc- tion in a continuous manner. 192. The function F^ has the value 4) ^1 = (^Vx'^^y'^y It is evident that F^ does not change sign, and since a maximum is to enter and consequently F^ is to be continuously negative, it follows that A must be a positive constant. 193. In order to find the curve itself, we have to integrate the differential equation G^""— xC' = o. This equation is equiva- lent (Art. 79) to the two equations d_dF^_d^^^ ddF d F ^^ dtdx' dx ' dt d y' dy CALCULUS OF VARIATIONS. 227 Since F does not contain x explicitly, the first of these equations gives <\ :^ — , = const., or y -\ =b, ' dx' ^ ^ Vx'^+y^ dF where 6 is an arbitrary constant. Since ^ — j varies in a continuous ox manner for a portion of curve where there is free variation, it fol- lows that the constant 5 retains the same value throughout such a portion of curve. The curve may, however, consist of separate portions which are free to vary, and for these the constant d may have different values. If we take as the independent variable the arcs of curve meas- ured from the origin, we have from 5), 6) ^=_i(^_^), as k and consequently, since ( — — ) + ( —7- ) =1, it follows that and ds^ X^ ^^ ^ A ds It is seen at once, if we integrate the last equation, that 7) ^ = hx-a), as A where a is an arbitrary constant ; and consequently the equation of the curve is 8) {x-af +{y-dy=\\ From the nature of the curve it is evident that A is a positive constant. 228 CALCULUS OF VARIATIONS. 194. An immediate consequence is the theorem of Steiner, that those portions of the cure, which are free to vary, must be the arcs of equal circles. These circles may have different cen- ters, since a and b are not determined. Each such arc of the circle may, however, lie on different sides of the chord joining two end- points ; we have, therefore, to ascertain which of the two arcs is the one required. The solutions of the differential equation are x—a- --\ cos^--^° = x cos /, y — b^\ sin" = X sin t, as is seen from equations 6) and 7), when differentiated. Since A is positive, s increases with t and since with increasing / the curve is traversed in the positive direction, we must take that arc for which this is also true. Let C be the center of the circle, A^ the initial-point, and A-^ the end-point of the arc. That arc will be the right one which lies on the positive side of C A-^, that is, on the side of the increasing /'s. For if A is the angle which the radius CALCULUS OF VARIATIONS. 229 CAi makes with the A'-axis, and if x^, y^ are the coordinates of the point A^, then we have cos ti= -{xi—a), A sin /i^-(yi—d'), A. and further the angle, which the tangent A^ B^ drawn to the arc at the point A^ includes with the ^-axis is t-^-\--- Consequently we have cos(A+|) = -sin A==-^(>'.-^) = (^^X. sin(/, + |)=^cos A=^(^.-«) = (g)^, formulae, which have the right signs. This would not be true if we took the other arc and also the tangent which is drawn in the other direction. Hence that arc is always to be taken^ which, looking out from the center, is traversed in the positive direction. 195. If no conditions are imposed upon the curve and it is required to find among all isoperimetrical lines that one which offers the greatest surface-area, then the question is not of an ab- solute maximum, since the curve may be shoved anywhere in the plane without an alteration in its shape. The problem may be stated more accurately by saying that the integral which repre- sents the surface-area is not to admit of a positive increment, when all possible variations are introduced. The problem thus formulated leads to exactly the same necessary conditions as be- fore, namely that the first variation is to vanish, and consequently we have the same differential equation to solve. We have also the same condition for \. Since the second variation can never be positive, and consequently F-^ can not change its sign, we conclude as above that x is positive. Since the whole curve is free to vary and since -^ — -. and n, — -. are continuous functions for the whole trace, ox oy the constants a and b are the same for the whole curve; however, they remain undetermined. We have, consequently, the following result: 230 CALCULUS OF VARIATIONS. If there exists a closed curve which with a given periphery includes the greatest surface-area, this curve is a circle. 196. However, it has not as yet been proved that this prop- erty belongs to the circle. The treatment of the second variation is not sufficient, since only such variations have been employed vi^here the distance betvi^een two corresponding points, and also the difEerence in direction at these points do not exceed certain limits. The further proof has to be made that every other curve forms the boundary of a smaller surface-area. The proof that the circle has this maximum property, ( a proof which is omitted in all previous solutions of the problem), has been considered so dif- ficult that its solution has been denied to be in the province of the Calculus of Variations. We shall, however, in the next Chapter show that in the theorems already treated a means of overcoming this difficulty is ofFered. It will be seen that without the use of the second variation the desired result is reached in all cases where the function ^i does not change sign, not only at any point of the curve but also for any direction at any point. CALCULUS OF VARIATIONS. 231 CHAPTBR XV. RESTRICTED VARIATIONS. THE THEOREMS OF STEINER. 197. We shall consider in this Chapter some special cases of y». restricted variations. Sup- pose first that the path of integration is taken over two traces PoPi, and P^P^,. We have for the first vari- ation of the integral (Art. 79) -J' 4 Since the variation along the traces (Co) and (Ci) is free, it fol- lows that G^^o for them, and consequently In order then for the first variation to be zero, it is necessary that La^ -dj' J^+Lay "ay y°' 198. If /irst the conditions of the problem leave P2 free to vary in any direction, we must have, since ^ and 17 are arbitrary, dF-_ dF + . aZ~_ If.^ dx' ~dx' dy ~dy' ' or, the curve consists of a single trace 232 CALCULUS OF VARIATIONS. Secondly, if the conditions of the problem require P^, to re- main upon a fixed curve ( C), then since the displacement is in the direction of the tangent to this curve, the expres- sion \dx' U ^ ^^^o may be replaced by CdF dF . T where \ is the angle betvsreen the fixed curve and the J^-axis. 199. We may apply the above results to the function Fix,y,x',y) = /(x,y) Vx''+y\ In the first case, w^here the point P2 can move at pleasure, we have dF+_dF- dx' ~dx' ' dF + _dF- dy dy ' or r/Ujv) x'-rj /{x,y) x' lr |_ y^x'^+y^j L Vx'^+y^j ' r /(x,y)y Yr/(x,y)y'Y so that, unless /"(;!;, y) vanishes at Pj, we must have cos T"*" = cos T" sin T"*" = sin t" and therefore where t + and t ~ are the angles that the tangents to the variable curve at the point Pj make with the ^-axis. From this it follows that Po Pi and P2 Pi are not different traces but constitute a single curve with one tangent at the point P^.. In the second case, where P2 is constrained to lie upon the fixed curve (C) (see Fig. in the preceding article), we have /(^.7)[ cos T cos X + sin T sin \ ]:- CALCULUS OF VARIATIONS. 233 From this it follows, unless /{x,^)=o at the point P^, that cos (r— x)~=cos (t— x)+, or (t-x)-=±(t— \)+ [modTr]. It is seen that the tangents to the two traces Pq ^2 and P^ P^ at the point P2 have either one and the same tangent at P-i and are parts of one and the same curve, so that this case is the same as if P-^ were not constrained, or they make with the tangent to the fixed curve equal angles T-^P^^M-^ and T-iP^M-i. A limiting case is where r=A., when again P^ P^ and P^ /\ form a continuous curve touching the fixed curve at the point P2. The function -J'^"-' 4 A A CALCULUS OF VARIATIONS. 235 to t, A 4 'o As in our previous discussion (Art. 184), it follows that m^-w^hS'"'"-'']- Since further 4^'^-^''']:-^'-' k J ] Ao 6^'"+A, C'"+ . . . . + x^ (;('') } wdt=o, 4 236 CALCULUS OF VARIATIONS, we have \_ ox ox ax J + If we write and denote by t', the angle which the tangent to the fixed curve at the point P' makes with the ^-axis, the above expression becomes fdF , dF . ,y ^— ^cos T + ,5—, sin T =0. \_dx oy J + If the point P' were not restricted, then ^ and 17 w^ould be arbi- trary, and we would have here which results compare with those of Art. 199. 202. We saw in the previous Chapter, if there existed a closed curve which with a given length bounded a maximum sur- face-area, that this curve was a circle. We supposed that it was possible for the circle to be situated entirely within the boundary of a given region. Suppose that this is not the case. The curve must then at least touch the given boundaries in two points or have a portion of the boundary in common. For we saw that the curve consisted of arcs of equal radii, and if these arcs did not touch the boundaries, there would necessarily be discontinuous changes in the direction of the variable curve. At such places, however, the surface -area could be increased without changing the perimeter. 203. Regarding the nature of the curve when it touches the boundaries, Steiner has given the two following theorems : 1 ) If the curve coincides with a portion of the boundary , then the free portions of this curve are arcs of circles of equal radii, which are tangent to the boundary at the points of con- tact. CALCULUS OF VARIATIONS. 237 2) If the curve touches the boundary of the region in a point, then both parts of the curve are arcs of circles of equal radii, and the tangents to these two arcs at the point of contact with the boundary make with the tangent to the boundary at this point, equal angles. Steiner proved these thorems in a synthetic manner, and remarked that a synthetic-geometrical treatment seemed necessary, because the principles of the Calculus of Variations were not suf- ficient. Such remarks were, in a measure, justifiable, since up to that time only curves had been considered which satisfied the dif- ferential equation throughout their whole extent, and, therefore, no analytical means were known for the treatment of curves which in part coincided with given curves. However, there was no rea- son for saying that a method for the treatment of such problems was not within the province of the Calculus of Variations. 204. We shall show that the principles of the Calculus of Variations are sufficient to establish Steiner's theorems by proving two theorems due to Weierstrass, which are more general than the theorems of Steiner, and which have reference to the behavior of a curve at the points where it touches the boundary. The two theorems of Steiner are special cases of these theorems. Suppose that the curve which satisfies the differential equa- tion approaches the boundary at the point 1 and coincides with it up to the point 2. On the part of the curve which is traversed before we come to the boundary at 1, we take a point so near to 1 that between and 1 there is no sudden change in the direction of the curve. The portion of curve 012 shall be so varied that we come to the boundary along another path from to a point 3 before 1 or from to a point 4 after 1 and then traverse the boundary to 2. 205. As we have already seen (Art. 161) the variation there- by produced in the integrals /'"' and /"* may be expressed as fol- lows: Let />i, ^1 be the direction-cosines of the curve 01 at the 238 CALCULUS OF VARIATIONS. point 1; />!, g^ the direction-cosines of the boundary at this point ; Xi, yi the coordinates of the point 1, and o- the element of length of the boundary. Then we have, if the boundary is approached be- fore the point 1 [see formula 5) Art. 161] k 1 A and if the boundary is approached after the point 1 [see formula 6) Art. 161] A .A/'« = -6'!''(;r,^,A,^.,A,9,V+ ^G^'"«^^^+('^.f.>?.^.^X• Hence f or case 1) : A/;-' = A/" - xA/!»^ = (<^ «» -\<^ '»')o- + ( 0-, ^, -r,, ^, ^ ) , \ at at 1 2 for case 2: A/'<"=A/««-xA/» = _(k, ?k) (1—^) ^^ is different from o, (which must be determined in each separate case), it follows that ?iA— A?i=^> and therefore, A=-A.?i=-?i- We wrote (Art. 157) A=(l->t)/^+^A and consequentl)% if we take the lower sign, so that A =— An ^1 = —gu then it may happen that F^ becomes infinitely large within the limits of integration, because for the value i:=}i both pi^ and $'k are zero (see Art. 157). In general, we have A=A. ^1=^1 (cf. Art. 199). A special investigation must be made in the other case for every particular problem. We, therefore, have the theorem : // the curve which satisfies the differential eguation ap- proaches the boundary at a point and then coincides with a portion of the boundary , the direction at the point of contact can suffer no discontinuous change. The same result is derived in an analogous manner for the point where the curve leaves the boundary after having coincided with a portion of it. 240 CALCULUS OP VARIATIONS. 207. We have tacitly assumed that there is no sudden change in the direction of the boundary at the point 1. But if this is the case and if ^2> ^j arethe direction-cosines with which one approaches the point 1, and^j, q-^ those with which one leaves the point 1, then we have for A/'*" the expression: in the first case: A/«» = d^ d'n\ 242 CALCULUS OF VARIATIONS. If we assume that Pq\—pvq is different from zero, and con- sequently that the tangents to the two portions of curve 1 and 1 2 at the point 1 do not coincide, then we may write yz~yx--=Q^-\-qxK The geometrical meaning of S and \ is seen, if we consider that in virtue of the two above relations^ the length 1 3 is the geometrical sum of the two lengths />S, ^S and/j 8i, q^ Sj and that consequently S and Sj are the coordinates of the line 1 3 with re- spect to an oblique system of coordinates whose positive axes have the directions^, q and/>j, ^i, and are consequently represented by the tangents of the two portions of curve at the point 1. If we write these values for X:^ — x-^, y^ — y-^^ in the above expressions, we have + (e^.t^,^U(FS- ' dt dt/2 <^x8, + (^, ^d^dn\ ^'dt'dt)2 209. The straight line whose equation is o, and for the other half, (S^S— ' in the place of i^=F'<»-A./^''' which occurred before; still on this account the nature of the difEerential equation is not changed, since there is only a change in the con- stants. It is, however, a priori clear that the solution of the two problems must be the same; for, if it were possible to keep the surface-area constant and shorten the perimeter, it is evident that with the original perimeter we could have inclosed a greater sur- face-area. Hence, the curve, which has been derived from the differential equation of the first problem, satisfies also the inverse problem. We consequently have as the solution of the second problem the theorem: The curve j wherever there is free varia- tion, consists of arcs of circles which have equal radii. 211. Problem. Three points 1, 7,, 3 not lying in the same straight line are given in the plane and it is required to draw a line through them in a definite order, which includes a given surface-area and at the sam-e time has the shortest possible length. We know that a circle W, say, fulfils these requirements, if the given area is the same as that included by a circle, which is determined by the three points 1, 2, 3. But if the surface-area is greater or smaller than W, then the arcs of circles must be drawn outward or inward. If, however^ the area is very small, we cannot draw arcs of circles so as to in- close this area without crossing one another, and we do not ad- mit into consideration the areas that are described in the op- posite directions. The problem may be solved as follows : The curve, al- though not being limited by further conditions, need not vary everywhere in a free man- ner, and, consequently, it is not necessarily constituted out of arcs of circles. For if we assume that the curve is not to cross itself, then of itself it may offer barriers which obstruct free variation. CALCULUS OF VARIATIONS. 245 If, for example, the curve 12 3 partially overlaps so that the portion 1 3 coincides up to the point 2 with the portion 1, then among all possible variations, there are present those where 1 remain unchanged and only 1 3 varies ; and since the curve is not to cross itself, the variation of the portion 1 2 can take place only on the side of 1 on which the point 3 lies, and, consequentl}^ the freedom of the variation of the curve is essentially limited. In itself the requirement that the curve is not to cut itself is not necessary, as the integrals that appear have a meaning also for this case. If there are overlapping portions of curve, then we may allow such variations to enter that points coincident before the variation may also coincide after the variation, without the second integral changing its value. We shall investigate the kind of differential equation that is thereby produced for these portions of curve. 212. The following investigation is also applicable to the case where the second integral is not present. We have simply to make \=o. We introduce the variations It has been shown that the first variation of iT"" is identical with 8j(ir(o)_x/^(«)^if, provided that Ci can be expressed as a power-series in e in such a way that the total variation of the second integral vanishes. This 8/'*" can be brought to the form In the former treatment ^ and tj were entirely arbitrary, except that at certain points and along certain portions of curve they van- ished. Wherever they were arbitrary it was necessary that G=o. 246 CALCULUS OF VARIATIONS. In the case before us we have in addition those portions of curve which overlap the curve several times without crossing it. The differential equation, which these portions of curve satisfy, may be obtained as follows: We have, since dt is a positive increment of t (Art. 68), F{^x,y, x',y)df=F(x,y, x'dt,y'di) = F{^x,y, dx, dy). Let 1 2 be a portion of curve that is traversed several times. The integral over this portion of curve, after it has been traversed once from the point 1 to the point 2, may be written in the form ^ F{x,y, dx, dy)=j F(x,y, dx, dy). The portion of the integral taken over the curve in the opposite direction is 1 jF(x,y,dx,dy). 2 If this portion of curve is traversed /u. times in the first direction and V times in the second, and if all the variations except those that relate to this portion of curve be put equal to zero, then the variation of the whole integral is equal to the variation of the sum of integrals: 2 1 fif F{x, y, dx, dy ) + v f F(x, y, dx, dy). 1 2 But since 1 2 fF{x,y, dx,dy)=fF(x,y,~dx,—dy), 2 1 the above sum is equal to 2 2 fif F{x,y, dx, dy) +vf F{x,y,—dx, — dy); 1 1 or, if we put IJiF(x,y, dx, dy) + v F{x, y,—dx,—dy)^F{x, y, dx, dy). CALCULUS OF VARIATIONS. 247 the sum is 2 J F{x,y, dx, dy). 1 The portion of curve 1 2 is traversed only once for this integral, and consequently the variations are quite free. The interval 1 2 must therefore satisfy the differential equation which is derived for the function F{x,y, x',y') in the same manner as in the former investigations, where F{xyy,x',y') was the function considered. 213. If, for example, the problem is to determine the curve which with a given surface-area has the shortest perimeter , then F(^x.,y, dx, dy)=\/dx^-[-dy^~\ydx, and for yi.=v, F{x,y, dx, dy)=fi[Vdx^+dy^—\ydx+ Vdx^ + dy^+Xydx] =2fJiVdx^+dy^. Consequently the differential equation leads to a straight line. But if /A ^ J/, we have F=(fi + v) Vdx^+dy^—\(ii—v)ydx. The corresponding differential equation is of the form where Xj=X ^~ ; it, therefore, leads to the arc of a circle which fl+V has a different radius than the one belonging to the portions of curve where the variation is free. This case, however, does not in reality appear unless there are certain modifications ; for, if we traverse such an arc of circle twice in opposite directions, the portion of surface-area thereby obtained is zero. We may, however, shorten the perimeter by taking instead of the arc of a circle the chord which joins its end- points, this being the first solution above. If, further, the same arc of circle was traversed several times, then in case there are 248 CALCULUS OF VARIATIONS. not special modifications, we may neglect the first two times or the first 2n times that the arc is traversed (owing to which the perimeter is shortened) without changing the surface-area, Taking also into consideration the case where ft — j'=l, when a straight line enters, we have to see which of these portions of curve (straight line or arc) can be used to form the required curve and how they are to be grouped. We have then to seek all pos- sible kinds of combinations and make proof of their admissibility. We consider any configuration and cause it to vary. Since the nature of the curve is known and only the end-points of the in- dividual portions are undetermined, we have to subject these to variations. The previous theorems are fully sufficient for carry- ing this out. We, therefore, have a means of determining whether such a configuration of the individual portions is, or is not possible. Since the individual portions satisfy their differential equa- tions, the first variations of the corresponding integrals will de- pend only upon the variation of the end-points; and, if we apply this to all the portions of the curve, we will have a linear function of all the variations of the coordinates of the individual end-points. These end-points may be subjected to further restrictions; for example, they may be compelled to lie upon given curves, etc. By the application of previously developed theorems, we have certain equations for the determination of the possible position of the end -points of the individual portions and w^e may thus see whether a definite configuration is, or is not possible. 214. At all events, for the grouping which has been thus de- termined the first variation of the integral vanishes, but this does not of itself denote that a maximum or a minimum has appeared. This determination is a problem in the usual Theory of Maxima and Minima. Since, as soon as the individual portions of curve have been found, we can also determine the integrals for them whose values depend only upon the constants \ that have been introduced and the coordinates of the end-points. We have thus an ordinary function of a finite number of variables, and the question is whether CALCULUS OF VARIATIONS. 249 this function really satisfies the conditions of a maximum or a min- imum. This subject is treated in the Theory of Maxima and Min- ima, involving several variables. Thus we may at least determine whether or not a certain formation of the curve satisfies the problem. For example, a curve is required to pass in a definite order through the points 1, 2 and 3 and which having the smallest possible perimeter is to inscribe a given surface-area. The curve in question consists of three por- tions which pass through 1 and 2, 2 and 3, 3 and 1. These por- tions are the arcs of circles with equal radii, if the given surface - area is sufficiently large. This radius is to be determined from the given value of the surface. The integral /"*' is a function of the constants that appear, and it may be shown that this integral is in reality a minimum when the constants have been correctly determined. But if the surface-area is not sufficiently large, then the por- tions of curve must partially overlap one another, and the portions along which this happens are straight lines. The curve cannot end in points which are perfectly free to vary; for if this were the case, we could so vary the point that the surface-area remained the same while its length became shorter. These points must lie along straight lines which pass through the three given points. It is thus found that the curve consists in reality of three arcs of circles which are described with equal radii and which mutu- ally touch one another and go oflF into straight lines that pass through the given points, as shown in the figure. 215. It is seen that the solution of the problem is independent of the position of the points 1, 2, 3 relative to one another; for we can slide the points 1, 2, 3 backward and forward upon the straight lines without causing the curve to lose the property of having the minimum length. It is essential only in what manner the points 250 CALCULUS OF VARIATIONS. are chosen where the straight lines come together with the arcs of the circles. These points corresponding to the points 1, 2, 3 may be denoted by 1', 2', 3'. If the portion 2' 1' 1 be considered as a fixed boundary and the end-point of 3' 1' varies along it, it fol- lows from a theorem already given ( Art. 206 ), that 3'1' must so touch the boundary, that the curve 3' 1' 1 does not change its direction abruptly. Hence every two arcs of circles must touch at the points where they come together. Since the radii of the arcs of circles are equal, it follows that the three centers of the arcs of circles form an equilateral triangle, and consequently the three arcs of circles are of equal length. Therefore every two straight lines form an angle of 120° with each other, and thus the solution of the problem is uniquely determined. The above problem was proposed by Todhunter in the Mathematical Tripos Examination of 1865. It is treated by him (Researches in Calculus of Varia- tions, pp. 44 et seq. ). CALCULUS OF VARIATIONS. 251 CHAPTER XVI. THE DETERMINATION OP THE CURVE OF GIVEN LENGTH AND GIVEN END-POINTS, WHOSE CENTER OF GRAVITY LIES THE LOWEST. 216. To solve the problem of this Chapter, let the F-axis be taken vertically with the positive direction upward, and denote by 6" the length of the whole curve. If the coordinates of the center of gravity are x^, y^, then y^ is determined from the equation A A yo^sf yVx'^'+y'^ dt, where S=§ Vx'^+y'^ dt The problem is: So determine x and y as functions of t that the first integral will be a minimum while the second integral re- tains a constant value. (See Art. 16). The property that the center of gravity is to lie as low as possible must also be satisfied for every portion of the curve; for if this were not true, then we could replace a portion 1 2 of the curve by a portion of the same length but with a center of gravity that lies lower, with the result that the center of gravity of the whole curve could be shoved lower down, and consequently the original curve would not have the required minimal property. We have here F--=(^y-\)Vx!^^y'\ 252 CALCULUS OP VAKIATIOJSIS. and therefore dx' v'x'^+y'^ ' 9^' 9/ ~ ( Vx'^+y'^y ' 9/ ~ Vx'^+y'^' We exclude once for all the case where the two given points lie in the same vertical line, because then the integral for 5" does not express for every case the absolute length of the curve ; for example, when a certain portion of the curve overlaps itself. Similarly we exclude the case where the given length ^ is exactly equal to the length between the two points on a straight line ; for, in this case, the curve cannot be varied and at the same time retain the constant length. 217. Since Fi must be positive, a minimum being required, it follows that (_j'—X)>o. Since further, -^ — -, and -^ — - vary in a ax ay continuous manner along the whole curve, and since these quan- tities difEer from the direction-cosines only through the factor y — X, which varies in a continuous manner, it follows that the curve changes everywhere its direction in a continuous manner. The function F is the same as the function F which appeared in Art. 7, except that here we have y — X instead of jk in that prob- lem. Since the differential equation here must be the same as in the problem just mentioned, we must have as the required curve x=a±pt, {y=X+y2l3(ei+e-0, the equation of a catenary. Since _>'—X >-c», it follows that /S is a positive constant. For 5" we have the value A I 4 CALCULUS OF VARIATIONS, 253 218. We have next to investigate whether and how often a catenary may be passed through two points and have the length S/ that is, whether and in how many different ways it is possible to determine the constants a, /8, X in terms of ^ and the coordinates of the given points. If we denote the coordinates of these points by Oo, do, Ui, ^1, then is S = l\^ie^^-e-i.)-(e'.-e-'^)]. It follows that We have assumed that /i^/o, and consequently we have to take the upper or lower sign according as a-^—a^o or ay^—a^^Co. It is clear that we may always take a^~a^o, since we may interchange the point a^i b^ with the point a^, do, and vice versa. We shall accordingly take the upper sign. If we write 2 '^ 2 then /A is a positive quantity and we have «i — ^o = +2/aA S=Nei'-e~''\ (e'' + e\ b,-bo _ i-g-^- ^_ l-^" S 1 + e-^' l+e"' d /b, — bo\ 4 'A S J dv\ S ) {e'+e-'} 254 CALCULUS OF VARIATIONS. Since this derivative is continuously positive, the expression ^ <-. varies in a continuous manner from —1 to +1, while v increases from — 00 to + co . Hence for every real value of v there is one and only one real value of ^"~ " which is situated between —1 and -f 1, and vice versa to every value of '"~ ° situated between — 1 and + 1 there is one and only one real value of v. Since we ex- cluded the case where vS" was equal to the length along a straight line between the two given points, it follows that S is always greater than ^i— ^ and consequently '"7 " is in reality a proper fraction. Hence v is uniquely determined through ^~ " . 219. We have further ^S* _ (g**— g~^) {ev+e-") Ui — ^0 2/i 2 or 2/x g]— gp gi— gp ^M^^y ^"-^"^ S'l' A-^oV vs'-{6,-6oy The right-hand side is a given positive quantity which we may denote by M. It is seen that d ( 2ti \_ ^ [(/x-l)e^ + (/. + l)g-^] By its definition n is always greater than o. If /x is situated between 1 and oo, the right-hand side of the equation is always negative. Since further the differential quotient of the expression (/i— l)e'' + (/^ + l)e~'' is never less than o while /* varies from o to 1, it is seen that this expression increases continuously when fi varies from ^)/ + ^A <2^={'^ — ^)Q + ^q- It follows at once from the preceding Article that Fr{x,y,p, q)-\Ft\x,y,p, q) cannot have values with diflEerent signs for any values of p, q. The converse, however, is not true. (See Art. 160). The condition that (s cannot change its sign in so far as every arbitary direction p, q is concerned has a further significance. For erect lines along the curve 1 perpendicular to the plane of this curve. On these perpendiculars take lengths equal in value to the second integral, where in each case the integration is taken from to the foot of the perpendicular. Then to the curve 1 there corresponds a curve in space 1', where the points in space are marked by indices corresponding to the points in the plane. Thus to every curve through the point and lying in this plane there corresponds a curve in space. We say that a curv^e in space satisfies the difEerential equation of the problem if its pro- jection satisfies the differential equation G^"" — x6^'^'=o, although X need not have the same value for all the curves. 223. Now suppose that we can envelop the curve 1' in space in the following manner : The point is to lie on the boundary, and the point 1' within the space enveloped ; further, it is to be possible to draw from to every point within this enveloped space at least one curve which satisfies the differential equation ; and, when such a curve has been drawn from to any point P within the enveloped space, it must be possible to draw a curve between and a point neighboring to P which also satisfies the differential equation. This curve must lie everywhere as near as we wish to the first curve, and the associated x's can differ from one another only by arbitrarily small quantities. CALCULUS OP VARIATIONS. 259 If the end-point describes a continuous curve in the enveloped space, then we may draw a series of curves, corresponding to the successive positions of the end-point, which satisfy the differential equation. 224. We shall show in the next Chapter that there must ex- ist an enveloped space as described above, if the curve 1 is to offer a maximum or a minimum. There are exceptional cases which are to be treated separately. We may at first assume the existence of such a space in order to make the essential points as clear as possible. We saw above that the function ^(;»;, ;>',/,$', /,5>) along the whole curve for arbi- trary values of p, q could not have different signs. From this we infer that in general S(^x,y,p, q,p,q) will not have values with different signs for other curves which satisf}^ the differential equa- tion. The deviation in the directions of these curves from the position of the original curve, of course, lies within certain limits, and the corresponding x's vary sufl&ciently little from the \ of the original curve. This will certainly be true if the integral 1 c is everywhere different from zero along the first curve. Excepting the case where the above integral becomes zero, we have as a further necessary condition that it must be possible to envelop the portion of curve 1 by a portion of surface, on the boundary of which the point lies, so that within this portion of surface the function ^ (x, y, P, q-,P, ^) does not have values with different signs along any of the curves that pass through 0, and lie within the portion of surface in question, it being assumed that they all satisfy the differential equation, and that the difference in value of X is sufl&ciently small for all the curves. (See Art. 156). 225. The same considerations are also true for a point 6 which lies before along the same curve 1, so that then 1 lies wholly within the corresponding portion of surface, and our orig- inal enveloped space, including the point 1', may be so formed as to lie wholly within the space enveloped by this second surface. 260 CALCULUS OP VARIATIONS. Suppose that the integration of the above integrals begins now with the point instead of with the point as before. Keep- ing our former notation, let the point 0' correspond in space to and join 0' and 1' by a regular curve which lies wholly within the enveloped space. This curve is quite arbitrary and is subjected to the condition that if 2 is the projection of any point 2' upon the ;t:;)/-plane, the sum is equal to the length of the perpendicular projecting the point 2', where we use the notation f to represent an integral that is taken over a definite curve that satisfies the differential equation and / one that is taken over an arbitrary curve, and where the indices represent the limits and the direction of the integration. A curve that satisfies the differential equation may be drawn from to every point 2' of the curve in the enveloped space and this curve also with the exception of the point lies within the enveloped portion of space. These curves are to have the property, which after the assumptions is always possible, that beginning with 1' the following curves are always variations of the preceding. 226. We regard the coordinates of the points of the projec- tion of the arbitrary curve as functions of the length of arc counted from the point 1, and we consider the sum /m + Zji- It follows from the fixed relation regarding the point in space that, wherever the point 2 may lie upon the curve 021, we always have _ r (1) I 71(1) _ rji) -'02 ~r-'2i — ■-'001 ' There is consequently no variation in the integral /'^'. Let the length of the^ portion 1 2 increase by cr. The change thereby produced in I^ + f-n is equal to <^(^2.J^'2,A'?2.A.?2) °'+(°''^''^'^'^) ' where /j, gj ^^e the direction-cosines of 2 at 2, pi, Qx those of 1 2 CALCUI/US OF VARIATIONS. 261 at 2. Again let the length of arc 1 2 decrease by a. The change thereby experienced in /52 + /21 is equal to — '2) A' ?2^A' ^2) represents the differential quotient of the sum /52 + /21, this sum being considered as a function of the length of arc 12 (see Art. 161). If the point 2 coincides with 1, then is hi-^-hx^^Tin, and if 2 coincides with 0, we have -^ 62 + -^ 21 = -^ 60 + Ai • Hence it follows : V) If S along 021 is not positive and not everywhere zero, that ■^ 01 > -' 60 + -^ 01 ) 2) 1/ ^ along 021 is not negative and not everywhere zero, that ■'di*C-'oo + -'oi' 227. It will be shown in the next Chapter that we may as- sume the strips of surface enveloping 1 so narrow that the point may be joined with any other point within this enveloped space by one curve, and only one, w^hich satisfies the diflEerential equation. The curve must of course lie wholly within the envel- oped space. This assumed, it follows that the integral /m is ident- ical with /ooi, and further, that the integral l^ is identical with the portion of the integral /qoi, which is taken over the portion of curve 0. We therefore have in case 1) /oi>7oi; in case 2) /oi<-4i- The maximal and minimal property of the curve that satisfies the differential equation is accordingly proved except in the case where along the whole curve the function (T is zero. 262 CALCULUS OF VARIATIONS. 228. We shall show that for the case where in the constructed realm the integral 1 is not zero, the function 'o = «/'(4 + To. a + a', )8 + /8', X + x'). These equations represent for sufficiently small values of Tq, a', ^\ X', which satisfy the last two equations, all curves in space which satisfy the differential equations and whose projection upon the ;c_j'-plane in its initial direction deviates very little from the initial direction of the projection of the original curve. We may express To, a' and /8' as power-series in X' and the trig- onometrical tangent of the angle which the two initial directions form with each other. If this tangent is denoted by k, we have, as in Art. 148, 4) [f (4)^+V'(4r]>&= [f (4)f' (4)-f(4>/'"(4)]To + [f(4)M4)-f(4)«l'i'(4)]'-' + ['/''(4)<^3'(4)-'^'(4)«l'3'(4)] A'+ [To, a', )8', x']„ where <^3=|^,'/'3=|^. 232. Since the two curves are to go through the same initial point, we have further 266 5) CALCULUS OF VARIATIONS. The determinant of the linear terms on the right-hand side of the equations 4) and 5) is f(4)fX4)-'^'(4)fXaf(4)<^i'(4)-<^X4)'^iX4),'l''(4)<^;(4)-'^'(4)V'X4) In this determinant write 6) dM=^ni)Uf)-4>V)U0. (t=1.2,3) If we multiply the second horizontal row by »/'"(4)) the third by — <^"(4) and add both to the first, the determinant may then be written ^'(4) . <^i(4) . <^2(4) 'P'i^o) , V'i(4) > «l'2(4) or, ^2(4)^i'(4)-^i(4)^;(4). This quantity is not zero, as we shall see later [see the third of of equations 13) in Art. 237]. Hence we may express t^, a, ^' as power-series in ^, X' so that for any pair of values ^, X', which have been taken sufficiently small, there corresponds a curve in space. From the differential equation it follows in a similar manner as was shown in Art. 149, that for one pair of values ^, x' there corresponds only one curve, and that every curve is completely determined through the initial point and the initial direction. We, therefore, conclude, as in Art. 149, that the equations 4) and 5) afford us all the curves which are neighboring the original curve, which have the same end-point with it, and which satisfy the differential equation. CALCULUS OP VARIATIONS. 267 233. We have now to choose the constants in such a way that the new curve in space will go through a point x + ^,y + r), z-\-t, which lies in the neighborhood of any point x, y, z situated on the old curve. If then we give to / a definite value and take sufficiently small values for |, 17, 4, the following equations must be satisfied: 7) 0= f(4)^o+'/'i(4)a'+«/'2(4))8' + 'l'3(4)^' + (^o.a',/8',x'X, ^ dx' ^ But Hence, if we write 8) it follows that 9) ^1 / (v=l,2.3) OX a/r(i) ay -v+®,i A 4 y + H i, 4) /8' + e3( /, t,) \' + (r,a',/8',x'),. If we substitute instead of $ and 17 their power-series in t, a', /8', x' in equation 9), the determinant of the linear terms on the right- hand side of equations 8) and 9) become after a slight transforma- tion 10) D{U,t) = <^'(4) f(4) o o o o o o <^i(4) '/'i(4) ^2(4) ^'2(4) 0i(4,O , ®2(4,0 , ©3(4,/) 268 CALCULUS OF VARIATIONS. We assume that this determinant does not vanish for arbitrary- values of t. This case and the formulae which would follow from it we leave as an exception for future investigation. 234. The first value of t after 4 for which Z?(4, t) vanishes we call the conjugate to 4- We see then that if the upper limit t-^ of the integrals lies be- fore the point that is conjugate to /o. the curve can envelop a portion of space having the property desired. Since in this case, if f, ij, t are chosen sufficiently small, one can always express t, t,,, a', y8', a' as power-series in £, ■»?, C and con- sequently can construct one and only one curve in space which satisfies the differential equation, which passes through the point and the point x-\-i,y-\--r]^ 2-\-l, and which deviates in its position arbitrarily little from the original curve. To these difEerent curves in space there correspond different functions Z>(4, /). If, however, the curves lie sufficiently near the original curve, the functions D which correspond to them will not vanish for any point along them, so that through any point in a sufficiently small neighborhood of any point of these curves a curve starting from can be drawn which satisfies the differential equation. 235. It remains yet to be proved that, if the point conjugate to 4 lies between 4 and t-^, we cannot have a maximum or a mini- mum value of the integral. Since the point can be chosen arbitrarily near and since the point conjugate to t varies in a continuous manner with 4> it is necessary only to show that t-^ cannot lie between and the point conjugate to it. We then will have proved everything except the case where 4 coincides with the point conjugate to 0. This case we must again leave for a special investigation, since the curve may or may not offer a maximum or a minimum, (cf. Art. 132.) A rigorous proof of what has been said requires a close inves- tigation of the function Z?(4. i)- 236. The curve in space which we had through variation of the constants is determined through the initial direction of its CALCULUS OP VARIATIONS. 269 projection at the point and through the differential equation G'°^ —{k+\')G'^^ =0, which it must satisfy. From this the proper- ties of the function -0(4, ^) may also be inferred. We perform the changes which C*" — X C^' suffers when a, /3, X undergo the changes a, )8', x'. The equation G^'°'— X G^^ must van- ish for arbitrary values of a', jS', x'. We have ^G^G'>Xx+i,:y+v)-G^'K^,:y)-x[G%x+^,y+v)-G^'\x,:y)] -x'G%x,:y). In a similar manner as was shown on page 133, formula (3), we have G^%x+i,:y+v)-G^r^,^)=-Fr{y^-x'v)-^^(Fr^^^^^^) and consequently The terms of the first dimension in the development of y i—x' Tj in powers of t, a, 0, x' are which we represent by w. We then have Since this quantity must be zero for arbitrary values of a', j8', x', the coefficients of the individual terms in this expression when de- veloped in power-series must be zero. If we limit ourselves to the linear terms, and use the functional sign for the function itself 270 CALCULUS OF VARIATIONS. when there can be no confusion, we have the following three dif- ferential equations: 12) { pd^Q., dF, de, Q If we multiply the first of these equations by 6^, the second by —01 and add the results, we have iH'-'-^i-'-'M-'-^'' Similarly, if we multiply the second equation by 6^ and the third by —^2' we have upon adding, dt [ F iff ^^2 a dO^X^ g ^(1) Finally, if we multiply the first equation by 62 and the second by —di, we have through addition, dt [^.(^'f-^'f)]-- 237. From these equations it follows that 13) ^ k 0, (4, /) =. J^, G w dtJ\j, ( e, ^ -e, ^^T. , dt]y. .^•("■f-^'f)"^- CALCULUS OP VARIATIONS. The constant C cannot be zero ; for then we would have 271 dt ('°« "')=#.( '"8 "■)• or d^^Ci dj. But, as is easily shown, the determinant Z?(4> ^) oiay be brought to the form 14) Dito,t) UO . ^2(4) , ^3(4) ®i(4,0 , ©2(4,0 , ©3(4, If then ^i(/)= Q 0^ ( ^), it would also follow that 0i( 4, if)= Cie^C 4, ^). and the determinant D{ta,t) would vanish, since two vertical rows differ from each other only by a constant factor; and this is true for arbitrary values of t, which case we have excluded. Hence the constant C cannot be zero. 238. We next prove that the determinant Z?( 4, ^) changes sign, when it vanishes. We have IS) f,D(t,.ty + ^x(4) e^{t,) , ^3(4) eit) . Ut) , ^3(/) ^^ex(4,0 dA^^oJ) ' ^^^3(4,0 ^1(4) U^o) , ^3(4) e^t) , e.'it) , ^3'(0 0,(4, t) , ©2(4, t) , 03(4,0 Owing to 8), we have ^^0.(4,O=G^'''^v(O. 272 CALCULUS OP VARIATIONS. Consequently the first of the determinants vanishes, leaving @i(4, ^) , ®2U,^) , ©3(4,0 0iU) , ^2(4) , ^3(4) We introduce the following notation: ^,Z^(4,0- 16) We can then write D{t,,t)=.^/^{t) ®^{t„t), v=3 ^^i?(4,0=2/^'(0®v(4,0; consequently 17) /3(0 ^^/?(4,^)-/3'(^)/?(4,/)=[/3(O/l'(^)-/l(^)/3'(^)]0l(4,^) -H [/3(0/A -/2( 0/3'( 0] ®2( 4, 0. or 18) dt L /3(/) J [/3( 0//( -/i( t)M i)] Qx( 4, /^) + [fl t)f;{ t) -fl 0/;( 0] e,( 4, 239. The numerator of the right-hand side of the above expression is equal to F^ multiplied by the square of a certain ex- pression, which we shall now determine. CALCULUS OF VARIATIONS. 273 Let us write ^,(4) , ^.(4) , ^3(4) 19) E= e,(t) , e,{t) , e,{t) o^{t) , e^{t) , eat) =0.V)Mt) + em Ait) + em/It). From 16) it follows at once that 19") o^e,{t,)/at)+e,{t,)Mt)^e,{t,)/it), and consequently 20) ei 4) E^ \e^{ t) e,{ t,)-e^{ t) e,( 4)]/,( t) + W{t)eit,)-e^{t)eit,y\/lt) =/3(0//(0-/(0/3'(0. Similarly, we have 21) -er{to)E=/it)/^{t)-Mt)/^{t). Accordingly, the expression 18) may be written: 22^ ^ [ D{t,j) '\_ p [^,(4)e,(4,0-g,(4)e),(4,/)] ^^^ dAf^r [/3(0]^ But owing to the relations 13) 8,(4, /)=[/-, 1^3(0 e^{t)-e,{t) ^3'(0}]^; e,(/o, t)=^F,\eit) e^{t)-e,{t)e^{t)y^^, it follows that e^{ 4) Hto, t)-m) ®2(4, 0=[^i ^3( /) {u 4) ^i'( t)-e,{ 4) e,'( 0} jj^ -[/^i ^3'( 1 02i 4) ^.( 0- ^i( 4) e^i t)} ]^ 274 CALCULUS OF VARIATIONS. 240. Further we have = [f, { e,{ t) e,'( t)-e,i t) e,'( t) y^^ei 4) But owing to the third relation in 13) the expression F-i\ 6-Ji^ f) 6\{ i) — 6^{t)6^{t)\ is independent of t, so that the first term of the right-hand expression is zero, and consequently 23) d^t,) 0i(4, t)-eit,) ©,(/o, t)=F,{t) .F,E. Hence the equation 22) becomes 24^ d[ D{t„ty \_F,{t)E} ^ dtl/it) J- [/,{t)y 241. Suppose that Z>(4, i) is zero of the ki^ order for the value /=/ so that the development of D(^t^, t) begins with t — f to the h{'^ power. If theny^(/) does not vanish for ^=/, the development of begins with the (^— l)*.!' power. But this expression is equal to Fi E'^, and since according to our assumptions F-^ does not become zero or infinity for any point within the interval 4. . . . /j, it is seen that /\i£"^ must begin with an even power. Hence k—\ is an even integer, and consequently k is an odd integer, and therefore D{ta,t) must change signs when it vanishes. CALCULUS OF VARIATIONS. 275 Suppose next thaty^(/) vanishes for /=/, then/3'(/) cannot vanish; for from the equations [see 16)] it would follow, if ^i(/o) and ^2(4) are not simultaneously zero, that But this equation, as also the simultaneous vanishing of fli(^) and &2{t) for the value ^=4, contradicts the equation 13) for, as we have seen, C is difiFerent from o and F^ is neither zero nor infinity. Hence y^( /) and y^'( t) do not vanish simultaneously. If then y^(/) vanishes for /=/', the development of ^(/) in powers of t—f begins with the first power. We may therefore write /,{t)=c{t-i')+ It follows that the development of begins with the term cA{k-l) {t-tfy, except when k—\, in which case the coeflBcient of this term is zero. In this case nothing has been shown, but see the next article. 276 CALCULUS OF VARIATIONS. For i:=l it is evident that Z>(/o. ^) changes sign on vanish- ing. 242. We shall next show that if y^ (/) vanishes, E can be zero only when at the same time/j {t)=o= /^{t). We saw in the pre- ceding article that the quantities B^ (4) and 6^ (4) cannot both be zero. If then tfi (4) >o, it follows, from the relation [formula 21)] that, when E=o andy^(/)=o^ alsoyi(/)=o, and also from o=elu)/lt)+e,{t,)A{t)+elt,)/lt), that/, (0=0. Similarly, when 6i{t(t)>o, it is seen from the equation /lt)f;{t)-A{t)/^{t)=^6M) E, that/(/)=o, if E=o==/i{t), and, consequently, also/(/)=o. But if E does not vanish for t^t', then k=o, and, consequently, also Z>(4, /) does not vanish for /=/. We have thus shown that D(io^ i) does not vanish for /=/. It follows, therefore, that D(^to, /) changes sign on vanishing except when we have simul- taneously In this case it has not been proved whether it changes sign or does not. We must, consequently, consider each separate case for itself (see Art. 255). 243. If we assume that at least one of the quantities /(/), AiO'Ai^) is different from zero, we can give the geometrical sig- nificance of conjugate points : When the constants a, j8, X are increased by a, /8', X', new curves in space are produced. The condition that one of these curves cuts the original curve in the point /' is [see equations 7) and 9)] expressed through the following equations : CALCULUS OF VARIATIONS. 277 'o=f(4)r„ + <^i(4)a' + <^,(4)^' + «^3(4)X'+(r„,a',^',X')2, or, if we eliminate To and t, O=tfi(4)a'+e,(4))8' + 03(4)x' + (a',)8',x')2, ^==<>,(r)a' + fl,(r)^' + (?3(/')x' + (a',)8',x')a. O=0i(4,if)a' + 0,(4,/)/8'+03(4,Ox' + (a',/8',x')2. The elimination of a' and y8' gives 26) o=^D{t,J')x' + {xX or o=Z?(4,/')+(x'). If Z)(/o. tf') is different from zero, we may take for X' a limit as small as we wish such that, for every X' whose absolute value is less than the prescribed limit, we always have \D{t„r)\>\{\')\. Consequently no value of /' can be found which satisfies equation 26). If then / is a definite value of /' for which Z?(/o, /") is differ- ent from zero, there will be no value of /' within a certain interval ^— Tj, . . . . / +T2 which satisfies the equation 26). Hence among all the curves in space for which a', ^', x' have sufficiently small values there will be none which cuts the original curve in the neighbor- hood of ^'. It is quite different, however, if we take for t" an interval t'—T. . . .i'+T which contains /', the point conjugate to to, within which, therefore, Z?(4, t")=o. For then D{to, t") has opposite 278 CALCULUS OF VARIATIONS. signs for t"=t'—T and ^"=/'+t. Hence after an arbitrarily small value T has been fixed, we can always choose x' so small that also has opposite signs for t" — t' —t and t" — ^ -\-t, and consequently there will be within this interval a value of t" for which the equation is satisfied. Hence, if we limit an interval ever so small about the point conjugate to and take arbitrarily small upper limits for a', /8', X', then among the admissible curves there are always such which start from and cut the original curve within this interval. In- deed, if a, /8', A.' are less than a certain quantity, then all the curves in space, for which a', ^', x' have values not greater than this fixed quantity and which go through the point 0, cut the original curve within this interval. This upper limit for a', )8', x' becomes infi- nitely small at the same time with this interval, so that the point conjugate to can be defined as the point which the points of intersection of neighboring curves approach. 244. In a similar manner we may prove that a portion of space as small as we choose may be taken around a point of the curve in space which is not conjugate to 0, and that the points along the curves in space, which are conjugate to 0, do not lie within this limited portion of space, if a', ^', x' are taken suf- ficiently small; but when we limit a portion of space as small as we wish about the point that is conjugate to 0, the points along the curves in space that are conjugate to will with sufficiently small a', /8', x' all lie within this interval. It also follows that, if in Z>(4. t) the quantity 4 varies in a continuous manner, the first value of t, for which D{^t^, i) vanishes, varies in a continuous manner. This follows at once from Z>(4+t,/)=Z>(4,0 + (t,0. where (t, i) becomes infinitely small with t for every value of /. For t—t'—T and t—t'+r, where t' is the point conjugate to the point 4, the function D{to,t) has different signs, however CALCULUS OF VARIATIONS. 279 small T is; and if we take t sufficiently small, it follows that -^(4. ^) + (t, tias different signs for /=/— t and i^f + r and must therefore vanish for some value of / within the interval /— T. . . . / + T. The change in the conjugate point is consequently arbitrarily small for a sufficiently small increment in 4- 245. We come next to the proof of the theorem that a portion of curve which includes 4 cind the point conjugate to it may always be so varied that A/™ may be both positive and negative, while Z'^' rem^ains unchanged. Let us write as in Arts. 180, 181: We have accordingly k A A/<« = £ \G'^*wdt^e^ JG^'^Zi/i^^+Ce, £1)2. Now choose w so that A 27) Kc'^'-^wdt^o. to Then from the condition that A/'^'=i?, it follows that we may ex- press Ci as a power-series in e which begins with a power higher than the first in e. Hence (see Art. 180), it follows that zy=ew+(c)2, and, from Art. 189, that A 280 CALCULUS OP VARIATIONS. or, what is the same thing : where X; is an arbitrarily small quantity over which we have yet a choice. 246. We shall now show that if /j lies beyond the point con- jugate to 4, the absolute value of k may be chosen so small that besides satisfying the condition A J' 4 the quantity w will satisfy also the condition 29) r \F,(^y+(j^,-i^)w^\dt=o 4 without being everywhere zero. If c is chosen sufficiently small, which we are always able to do, it is then seen that the quantity A A/*" has the same sign as k e^j v/di, and consequently the same 4 sign as kj which may be either positive or negative. Since w vanishes for 4 and 4, and since dt tj-. dw~\ !:•( dw^ , ^ d { j~, dw\ ''^'"^r'^\-dt)^'"Ttv^-diy it follows that instead of 29), we may write: 4 4 I- CALCULUS OP VARIATIONS. 281 and instead of this equation and the equation A 4 we may write the two equations: A 30) { 4 to where e^ is a quantity independent of t. 247. Now let e^ (A k), 0^ (A ^), &3 (A k) be three functions of t which satisfy the three differential equations : 31) i ^^(F,^^02U^))-{F,-k)eU^)=o, It follows from the theory of differential equations that for a series of values of t for which F^ is neither zero nor infinite tf, (/, k\ 0^{t, k), 03 (/, k) differ from the three functions ^i (/), 02 (t), 03 (/) by quantities which become infinitely small at the same time with k. Again, let /' be the point conjugate to 4. and write for the stretch from 4 to /", where /" is a point situated before the point f, zu^ei fli( t, k) + Cj 02( t, k) + €3 03{ t, k), and for the stretch from f to A let zy=o. 282 CALCULUS OF VARIATIONS. It is clear that w is not everywhere zero unless 61=0=62=^3. since owing to the differential equations 31 ) which d^{t, k),0-j{t, k),e3{t,k) satisfy, a linear relation for the series of values of t can exist only if for these values 6^'''=o, a case which we excluded (Art. 180). 248. The quantity w satisfies the differential equation dt {^^^)-(^2-'^)^-G^'=o. It must also satisfy the additional conditions that w=o for 4 3-nd for /", and that 4 But we have f t" t" t" Jg ' zvdf^^e^f G''^ eX t,Jc) dt\ e^f G^'' 6^ f,k) df+ e^fG^'' d^{ t, k) dt. to *o 4 '0 If we write t" JQ'^6Xt,k)=&XioJ".k), f (v=i.2.3) then from what was seen above, the functions @,(/o,/', ^) differ from @v( 4. /') by a quantity which becomes infinitely small with k. The conditions which remain to be fulfilled are: / o = e^d^{to, k) + eT,dj{to, k) -\-e^dj,{to, k\ 32) <| o = e,e,{f,k) + e^eit",k) + e,ei^',k\ { o=e,%,{t,, r, k) + e^%lt,, t'\ k) + e,%li„ t" , k). The determinant of these equations differs from D^Iq, /') by a quantity which becomes infinitely small with k (Art. 237). CALCULUS OF VARIATIONS. 283 For f=i — k and f=t'-Vk the quantity /?(/«, /") has different signs, and consequently we may take k so small that the determi- nant of the equation 32) has different signs for t" = t' — k and t" = t' -\-k and consequently vanishes for a value of t" situated be- tween t' —k and f -{-k. We may therefore take /" along the curve before t-^ in such a way that the equations 32) are satisfied by values of e-^, e^ and e^, which are not all zero. If then, returning to equation 28), e is chosen sufficiently small, it follows that A/^"'' has the sign of k and since this is arbitrary, there are among the admissible variations of the curves those for which 7'°' has a negative increm-ent and also those for which the increment of 1°' is positive. The portion of curve 01 cannot therefore extend beyond the point which is conjugate to 0. If we exclude the case where 1 coincides exactly with the point that is conjugate to 0^ it follows that 1 must lie before the point that is conjugate to 0. We may then choose sonear to that 1 lies also before the point that is conjtigate to 0. Along such a portion of curve the ftmction D(^tQ, t) does not vanish and consequently we may envelop such a portion of curve in a portion of space which has the required properties. 249. It only remains, excluding exceptional cases, to show that the function S{x,y,p,q,p,q^ cannot vanish along an entire curve within the portion of space defined above. If we exclude the possibility of the integral 1 j\Fl\x,y,p^, q^)-XFl%x,y,p,,, q^)\ (l-k)dk becoming zero along a portion of the curve in question, then for S to vanish, it is necessary that pq—Pg=o along the whole curve; that is, the direction of the projection of the arbitrary curve in space must coincide at every point with the direction of the pro- jection of the curve that satisfies the differential equation. 284 CALCULUS OF VARIATIONS. If x,y, 2, :r,,jVi, 2 1 are the coordinates of the two curves ex- pressed as functions of their lengths of arc 5=/ and 5' = /, say, then at the point in question we must have _ ^ dx _ dxi dy _ dy-^ ' 'dt~~dV' di~W x=x„y=yi, -^^__^, But since t 2 = jF^%x,y,x',y')di, and 2,= fF^'\x„y,,x,',y,')dt', it follow,s also that dz dzx . ~dt~~di' i. e., the two curves in space have at every point also the same direction. The quantities dF'^^ d^ da' dfi' d\' dk ' dk ' dx' dk^ dy' dk'^ ' dk ^ ^ dk ^ ^ dk' Accordingl}', the three quantities of the first row are proportional to the corresponding quantities of the second row. If we make k^o, the above quantities become ax ay +«,(,., ,)(^) +«,(,, ,)(^)^+e,(,,.)(f)^. CALCULUS OF VARIATIONS. 287 Hence, if we let p denote the factor of proportionality, we have l„=e.(co(^).+eX4.0(f).+«.(4,0(|^), where the third equation is reduced to this form b)^ the applica- tion of the other two. 252. Since the curve which satisfies the differential equation must pass through the point 4. we must in virtue of equations 5) and 6) have the relation and from this it follows that Eliminate p from the first two equations in 33) and write for the diflEerences that appear their values in terms of the ^'s defined by the relations 6). The determinant of the resulting equation, of the last of the equations 33), and of equation 34) is identical vv-ith JD ( 4, A )• [See formula 14), Art. 237]. Hence if D{iQ,t^) does not vanish, these equations have no other solution except \dJi:}o ^dkk \dkh The same conclusions may also be drawn from any small value of k to which the values t, a', y8', x' correspond; there enters here in- stead of the quantity Z?(4, A) the quantity Z?(4,A + T,a-|-a', )8 + ;3', A+X'). 288 CALCULUS OF VARIATIONS. Since this determinant for sufficiently small values of t, a', ^', x' is different from zero, it also follows that da' ^ d^ ^ dx' ^ dk dk dk But the quantities a', ^', x' do not vary with k and are consequently zero, since they are zero for k—O; this means that the curve along which the function /sC^) simultaneously vanish with D{t^, t) (see Art. 242). In reality for /=4-f-2'T, we have D{t„ t)=o, /,{t)=/M=/,{t)^o. Nevertheless, D^tg, /) changes sign when it passes through zero; for the vanishing of Z'(4,/) is effected by making the factor sin " zero. But this factor changes sign, while the second fac- tor retains its sign for /=/o + 2'r. 256. In the problem of finding the curve whose center of gravity lies lowest, we had (Art. 216) 15 =(JK-X) -4= =(^-X)A-|^ =(>^-X)^=^=(>'-X),; x^o-^^t , ;j/=\+|(e' + e-0 ; ^(,x,y,p, q,p, q)={y-K) [1-ipp + qg)]. We saw that jk— X>o, and further x' and y' do not vanish simultaneously at any point Consequently F^ is everywhere differ- ent from o and oo. Since Pp^qq represents the cosine of the angle between the two directions p, q and p, q, its absolute value cannot exceed unity, and in general is less than unity, so that the function S is nowhere negative, as must be the case for a minimum. We have already seen in Art. 219, if the length of arc is sufficient- ly great, that between two arbitrarily given points one curve and only one may be drawn which satisfies the differential equa- tion. It then follows that there can be no conjugate points and consequently the catenary in its whole trace has the desired min- imal property. 292 CALCULUS OF VARIATIONS. 257. That there are no conjugate points is also seen from the consideration of the determinant D{to, /). For we have e,{t) = ~fi,G (1). e.(/o./)=2 f+^:^-;^;°+C) B,(4.o=2 ^"<^^;+^:7-^^^^°-f7'> . ex 4, 0— 2 ^"'^f+^l]r;7^^'+f°> . From these quantities we have D(ta,t^=—- J— Pi 7T multiplied by the determinant e'_e-^ , t(^ef—e-t)-{et^e-'), 2 g/ _|_ e-i — (f?'" + e"^.), ta{e'—e-*) — /((e^- + e"'-), e-''{e' + e"') — e-^(e'« + e~ '«) or D{t„t)^ -2P^[et-t. j^e-^t-K)- ^ (e/-'. - ^('-^.) )- 2| The equation D{to,t)=o, or ^^ — -^{e 2 +e 2 ) — (^ J — ^ 2 ) = o has no real root except t=ta, that is, there exists no point conju- gate to the point /=4. BY THE SAME AUTHOR: I. Lectures on the Theory of MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES. (WEIERSTRASS' THEORY.) II. In Preparation: Lectures on the Theory of MINIMAL SURFACES. III. In Preparation: Lectures on the EXTENDED REALMS OF RATIONALITY. MODULAR SYSTEMS. .jnigalUiN^VERSlTYUBRARi OCT ;J 1991 MATNEMATiCSUBRARV